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Collisional alignment and orientation of atomic outer shells. III. Spin-resolved excitation
COLLISIONAL ALIGNMENT AND ORIENTATION OF ATOMIC OUTER SHELLS. III. SPIN-RESOLVED EXCITATION
Nils ANDERSEN, Klaus BARTSCHAT, John T. BROAD, Ingolf V. HERTEL Joint Institute for Laboratory
Astrophysics and National Institute for Standards and Technology, Boulder, CO 80309-0440, USA
ELSEWIER
AMSTERDAM
- LAUSANNE
- NEW YORK - OXFORD
- SHANNON ~ TOKYO
PHYSICS
REPORTS
Physics Reports 279 (1997) 251-396
ELSEWEiR
Collisional alignment and orientation of atomic outer shells. III. Spin-resolved excitation Nils Andersen’, Klaus Bartschat2, John T. Broad, Ingolf V. Herte13 Joint Institute for Laboratory Astrophysics
and National Institute for Standards and Technology, Boulder, CO 80309-0440, USA
Received December 1995; editor: J. Eichler Contents
1. Introduction 2. Framework, notation, and experimental approaches 2.1. Unpolarized beams 2.2. Framework and notation for polarized beams 2.3. Experimental approaches with polarized beams 3. Excitation by spin polarized electron impact 3.1. The fully coherent case 3.2. The incoherent case with conservation of atomic reflection symmetry 3.3. The incoherent case without conservation of atomic reflection symmetry: Photon polarization analysis 3.4. The incoherent case without conservation of atomic reflection symmetry: Electron polarization analysis
’ Permanent ’ Permanent 3 Permanent
254 255 255 259 260 266 266
266
305
329
3.5. The incoherent
case without conservation of atomic reflection symmetry: Combining photon and electron polarization analysis 3.6. Related work: P + P elastic scattering 4. Conclusions and recommendations for future work Appendix A. Scattering amplitudes A.1. Scattering amplitudes in different coordinate systems A.2. Symmetry properties A.3. Scattering amplitudes in the nonrelativistic limit Appendix B. Density matrices B.l. Reduced density matrix formalism B.2. Generalized STU parameters B.3. State multipoles Appendix C. The fine-structure effect Appendix D. Reduced state multipoles from generalized Stokes parameters References
address: Niels Bohr Institute, 0rsted Laboratory, DK-2100, Copenhagen, Denmark. address: Department of Physics, Drake University, Des Moines, IO 50311, USA. address: Max-Born-Institut, Postfach 1107, D-12474, Berlin, Germany.
0370-1573/97/$32.00 Copyright SSDZ 0370-1573(96)00004-X
0
1997
Elsevier Science B.V. All rights reserved
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Abstract
The review generalizes the formalism of the first part of this series [N. Andersen, J.W. Gallagher and IV. Hertel, Phys. Rep. 165 (1988) l-1881 on electron-atom collisions to cases where the spin polarization of the electron is investigated at least once, i.e., before and/or after the collision. In addition, the target atom may be spin polarized. The preparation of initially polarized beams and/or the analysis of photon and spin polarization in the final state significantly increase the number of cases where a “perfect scattering experiment” can be performed. The connection between the scattering amplitudes and the generalized Stokes and STU parameters is analyzed. Favorable scattering geometries for perfect experiments and possibilities for consistency checks are pointed out. Recommendations are made about directions of future work. PACS: 34.80.D~; 34.80.Bm Ke,vwords: Alignment;
Orientation;
Collisions; Electrons; Polarization;
Excitation
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1. Introduction Besides the “classical” experiment of scattering physics, namely the measurement of cross sections, a wealth of other experiments are currently performed in modern atomic physics laboratories. While enormous progress is being made in terms of experimental techniques including, for example, the preparation of spin polarized projectile and target beams with narrow energy distributions and the detailed determination of the final state (by photon and electron polarization analysis and particle-particle or particle-photon coincidences), the development of new theoretical methods has also been strongly encouraged. This is due, in part, to the construction of extremely fast electronic computers which allow for increasingly complex programs to be run in shorter periods of time. Through close interaction between experimental and theoretical work, a very detailed test of theoretical models is made possible. As a result, it has become evident that in many cases the agreement between theory and experiment is far from being perfect. Although the individual interactions are well known in atomic collision physics, we are in most cases still far away from a complete solution of the quantum mechanical many-body problem. With regard to the variety of processes studied and the experimental and theoretical methods applied, the question arises whether it is possible to establish some connections in the description of the different phenomena. In identifying such links, more systematism can be achieved in both the general formulation and in the numerical calculation of the individual observables. Therefore, an important goal of the present work is to present a general framework in which various, apparently unconnected processes can be discussed within the same formalism. The usefulness of such an approach has been demonstrated in the first part of the present review series [7] (referred to as Vol. I below), which dealt with orientation and alignment phenomena in atomic excitation by electron and atom impact. These phenomena were put on a firm mathematical basis in the influential review by Fano and Macek [44]. Vol. I only dealt with cases where both the projectile and the target beams were unpolarized before the collision, and no spin polarization analysis was performed afterwards. In these cases, the number of independent parameters that can be determined from the “reduced density matrix” (for an introduction, see Blum [28]) is generally not sufficient to extract the individual scattering amplitudes. Hence, a quantum mechanically complete or “perfect scattering experiment” [19] can only be performed for very simple processes, such as electron impact excitation of the He nlP states (see Section 2.1). Early discussions relating to this problem may be found in the pioneering papers of Rubin et al. [84] and Kleinpoppen [62]. The present review will generalize the formalism of Vol. I on electron-atom collisions to cases were the spin polarization of the electron is investigated at least once, i.e., before and/or after the collision. In addition, the target atom may be spin polarized. It will be shown that the preparation of initially polarized beams and/or the analysis of photon and spin polarization in the final state significantly increase the number of cases where a perfect experiment can, in principle, be performed. In particular, we analyze the connection between the scattering amplitudes and the generalized Stokes [4] and STU [lo] parameters. We discuss the information that is contained simultaneously in both sets of observables, as well as the information that is only contained in one of the two sets. Furthermore, we recommend favorable scattering geometries for perfect experiments and suggest consistency checks using data from apparently unrelated experiments.
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The review is organized as follows: In Section 2, the framework of Vol. I for unpolarized beams is summarized and the basic notation is reviewed. This is followed by a brief overview of experimental approaches that have been used with polarized collision partners. Section 3 then describes the excitation process by spin-polarized electrons. We begin with the non-relativistic case where explicitly spin-dependent forces such as the spin-orbit interaction between the projectile and the target nucleus can be neglected during the collision and, therefore, the atomic reflection symmetry is conserved. This is usually the case for light target systems, and we concentrate on electron scattering from alkali and hydrogen atoms, with the most complete example being the Na 32P excitation process. While the goal of a complete experiment has not yet been achieved even in this case, we demonstrate how a combination of data from several partial experiments and state-of-theart theory allows for conversion of available experimental data into a complete set of state parameters. The situation becomes significantly more complicated in the case of heavy target systems, such as Cs and Hg, where spin-dependent effects other than exchange must be taken into account in the description of the collision process. The number of independent scattering amplitudes is, therefore, increased significantly, and the perfect experiment becomes a highly challenging but not unrealistic goal.
2. Framework, notation, and experimental approaches 2. I. Unpolarized beams This section summarizes the framework and parameters used for the description of atomic outer shell alignment and orientation studies introduced in Vol. I, addressing experiments involving unpolarized particles. The presentation will be self-contained, but the reader is referred to Vol. I for a full exposition. For simplicity, the discussion will be focussed on S++P and J = 0-J = 1 transitions. This is justified, since these are the main cases of interest for spin-resolved experimental studies to this date and generalization of the formalism to higher angular momenta is straightforward. Figure 2.1.1(a) shows schematically the basic problem. An atom A is excited from an S state to a P state by impact of a particle B. This projectile B, which may be an electron, atom or ion, is deflected by an angle 0, as subsequently monitored by a detector in this direction. The collisionally excited P state of atom A may be characterized from a study of the radiation pattern (i.e., photon direction and/or polarization). This can be achieved with the geometry shown in Fig. 2.1.1(b), where the scattered particle is detected in coincidence with the photons emitted in a specific direction, including a possible photon polarization analysis. Two approaches are common here: (i) Coherence analysis, i.e., a measurement of the Stokes vector (Pr, P2,P3) in one or several suitably selected directions in space. (ii) Correlation analysis, in which the angular distribution of the photons in one or two planes containing the collision center is mapped. This is equivalent to a measurement of the first two Stokes parameters P1 and P2. This approach, therefore, gives less information than (i), but it is nevertheless used in cases where photon polarization analysis is difficult, such as for resonance
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(b)
(cl
Fig. 2.1.1. (a) Schematic diagram of a collisionally induced charge cloud of an atom A excited to a P state by impact of a particle B, scattered at an angle 8. This event may be studied in two ways: (b) Photons emitted in the P -+ S decay are polarization analyzed (Stokes parameters) in a selected direction and detected in coincidence with the scattered particle. In the time-reversed scheme (c), the atom A is excited by photons coming in from a selected direction, and the number of particles B leading to de-excitation are detected as function of laser polarization. The two approaches yield essentially equivalent information.
excitation of hydrogen or the rare gases. Since correlation analysis has not been applied in spin-resolved studies, we leave it out of the discussion below. An alternative method, which has been used very successfully, exploits the time-reversed scheme. Figure 2.1.1(c) illustrates how this is done. A photon beam from a selected direction excites the atom A. One then maps the P + S de-excitation as function of photon polarization for B particles scattered at an angle 8. For incident energy E, this is equivalent to the information one obtains from the reverse S + P excitation experiment at an impact energy of E + BE, where AE is the S-P energy difference. We now introduce a complete set of parameters, together with their measurement, that is sufficient to fully characterize the P state, thus defining “the perfect scattering experiment” in the
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f+l
k6
f-l
Fig. 2.1.2. Schematic diagram of scattering amplitudes in the natural frame for S -+P transitions without accounting for electron spin.
by electron impact
sense introduced by Bederson [19]. The fact that the reflection symmetry (with respect to the collision plane) of the total wave function describing the system is conserved greatly simplifies the analysis. We proceed in order of increasing complexity: (i) Thefilly coherent case. Neglecting for a moment electron (and possibly nuclear) spin, we can thus describe S + P excitation by scattering amplitudesfM,, with ML = + l,O, - 1. If we choose the “natural coordinate system” (x’, y”, z”) fixed by kin 11x” and quantization axis zn I/ kin x k,,, perpendicular to the scattering plane, see Fig. 2.1.1, the positive reflection symmetry of the wave function with respect to the scattering plane implies& = 0, such that only the (complex) amplitudes and f_ 1 are nonzero. Hence, this process is determined by three parameters only, the f magnitudes off + 1 and f- 1 and their relative phase (see Fig. 2.1.2). A perfect scattering experiment can then be performed, for example, by determining the absolute differential cross section cr and two-dimensionless numbers which fix the relative sizes off+ 1 and f__1, and the angle 6 = arg(f+ i) - arg(f- i) between them. As discussed in detail in Vol. I, the two independent dimensionless parameters may be chosen as the angular momentum transfer (L ) = (0,0,L,) with L
=
L
I.f+112-If-II” =;w.+I12-11.-112L I.f+J + If-II”
(2.1.1)
where cris the differential cross section, and the alignment angle y of the major symmetry axis of the charge cloud in the scattering plane (see Fig. 2.1.1(a)). The alignmer : angle y, defined modrc, is related to the phase difference 6 through 6=-2yfrr.
(2.1.2)
All this may be expressed compactly through the density matrix in the ML) helicity basis, which is given in the natural frame by
!
1 + LI
0
- qe-2iY
0
0
0
- Ple2iY 0
1 - Ll
. i
(2.1.3)
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The parameters (LI, y) are related to the Stokes vector P = (PI, Pz, P3) for the light emitted in the + 2” direction in a subsequent P + S decay through (2.1.4a)
P1 + iPZ = P,e2iY = and
(2.1.4b)
Px=-Ll. Determination of the parameter function can be written as
set (CJ,LI, y) constitutes
a perfect experiment,
since the wave (2.1.5)
This wave function is normalized as ($ ( t,b) = CT,consistent with the trace of the density matrix (2.1.3). The parameter Pl completely describes the shape of the electron charge cloud, with a length/width ratio in the scattering plane given by l/w = (1 + PJ(1 - Pl). In this simple case, the total degree of polarization of the emitted light P = 1PI is unity, i.e., P2 = LT + Pf = 1 .
(2.1.6)
A standard example of an excitation process fulfilling these criteria is He 1‘S -+ n’P electron impact excitation, discussed in detail in Vol. I. (ii) The incoherent case with conservation of atomic rejection symmetry. The situation becomes more complicated if the atomic state has a nonvanishing spin. Standard examples of this are hydrogen and alkali atoms, such as Li, Na, and K. First of all, since atomic fine-structure precession times are much shorter than the lifetime of the excited state, spin-orbit effects within the target will have ample time to depolarize the charge cloud prior to the optical decay, so that the measured linear polarizations become only $ of what they are in the previous case [See Vol. I, Appendix B]. However, by defining the so-called “reduced” Stokes parameters Cl] by (2.1.7a) PI,2 = 3p1,2 Y & = P3 )
(2.1.7b)
one recovers the Stokes parameters characterizing the nascent charge cloud. We will assume implicitly below that this correction has been applied. (An analogous procedure may be used to take into account possible hyperfine-structure effects.) Secondly, singlet and triplet scattering amplitudes come into play, depending on whether the spin of the incoming electron is parallel or antiparallel to the electron spin of the ‘S atomic state. Since the singlet and triplet scattering amplitudes generally differ, particularly at lower impact energies, the outcome of the collision event will depend on the relative spin orientation. In experiments where the spin directions are not observed, the outcome will accordingly be an incoherent sum of the two possibilities. Eqs. (2.1.3) and (2.1.4) still apply, but the light emitted is no longer completely coherent, and Eq. (2.1.6) is replaced by P2 = L: + P; 5 1 .
(2.1.6a)
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2.59
Consequently, the two parameters LI and Pl are now independent. Thus, for the case of partially coherent P state excitation with conservation of reflection symmetry, three parameters are necessary to completely characterize the P state, e.g. (L,, y, Pl). Still, Stokes parameter determination is necessary in one direction only, conveniently chosen along z”. (iii) The incoherent case without conservation of atomic reJection symmetry. The final step in complication arises in the description of cases such as the heavier rare gases, for which spin-orbit interaction may be important during the collision. In this case, spin-dependent scattering amplitudes ,fO may become nonzero (see Section 3.3.1.3). The corresponding charge cloud then has a nonzero height h along the z” axis (see Fig. 3.3.1 of Vol. I), and the equation for the corresponding density matrix generalizes Eq. (2.1.3) to the following form which separates the components of positive and negative reflection symmetry: 1 + L,+
0
- Pl+ee2” 0
(2.1.3a)
1 - L:
Thus, for the most general case of partially coherent P state excitation without conservation of reflection symmetry, four parameters are necessary to characterize the P state: (L: , y, PI’, h). Furthermore, Stokes parameter determination is necessary in two directions, an additional direction being necessary to reveal the charge cloud component perpendicular to the scattering plane. The two directions may be chosen along z” and y”, but other choices are possible, and sometimes more convenient. The Stokes parameters in the direction y” have traditionally been denoted by (P4, 0,O). In a subsequent section we shall, for the sake of generalization, modify this notation. The following relation holds for the height parameter h = poo:
(1 + P,)(l - P4)
(2.1.8)
h = 4 - (1 - P,)(l - P4) .
Note that two parameters, (Pl’, h), are required in this case to determine the shape of the charge cloud. With the normalization condition I + co + h = 1 for the relative length, width, and height, one has 1 = (1 - h)(l + P,+)/2 and o = (1 - h)( 1 + Pl’)/2. Similarly, the degree of light polarization in the zn direction is denoted by P+, with + P[+2 I 1 .
P +2 = Lf2
(2.1.6b)
The main conclusions of this section are summarized in Table 2.1.1.
2.2. Framework
and notation for polarized
beams
In general, transitions from an initial state 1JiMi; kimi) to a final state 1JfMf; kfmf) are described by the scattering amplitudes .f(Mrmr;
Mimi)
=
(JfMf;
kmfIylJiMi;
kimi)
,
(2.2.1)
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Table 2.1.1 Summary of electron-atom interactions for cases of increasing complexity, and the set of orientation parameters describing the excitation process for unpolarized beams
and alignment
Case
Vol. I, Section 3.1
Vol. I, Section 3.2
Vol. I, Section 3.3
Example (excitation of) Interaction(s) responsible Representation Reflection symmetry
He2lP Coulomb Wave function +
H 2’P, Na 3’P + exchange
Ar(4s4s’), Kr, Xe + spin-orbit
Pm =
Pm”
Dimensionless
+
+,-
parameters
Angular momentum ( - P3) Alignment angle Linear polarization Total degree of polarization Height
LI
LL
L:
Y
Y
Y
Pl
PI
p:
P=l h=O
PI1 h=O
P+ I1 h>O
Independent parameters Necessary observation directions
2 1
3 1
4 2
where Y is the transition operator. Furthermore, Jr(&) is the total electronic angular momentum in the initial (final) state of the target and Mi(Mr) its corresponding z-component, while ki(kr) is the initial (final) momentum of the electron and mi(mr) its spin component. The scattering angle 8 is the angle between ki and kr. For simplicity, we will omit it in the notation. We now discuss the number of independent scattering amplitudes. There are 4 *(251 + 1). (25, + 1) possible combinations of magnetic quantum numbers. Due to parity conservation for all interactions determining the outcome of the collision process, the number of independent scattering amplitudes is cut in half, giving a total of 2. (2Ji + 1)*(2Jr + 1) complex amplitudes for each transition between fine-structure levels. Subtracting a common arbitrary phase, the total number of independent real parameters is thus 4*(2Ji + 1) .(2Jr + 1) - 1. These are usually parametrized as one absolute differential cross section and 4 *(2Ji + 1). (2Jr + 1) - 2 dimensionless numbers, namely, 2’(2Ji + 1)‘(2Jr + 1) -1 relative magnitudes and 2. (2Ji + 1) * (25, + 1) - 1 relative phases. Further details on scattering amplitude properties are given in Appendix A. Whereas the scattering amplitudes are the central elements in a theoretical description, some restrictions usually need to be taken into account in a practical experiment. The most important ones are: (i) there is no “pure” initial state, and (ii) not all quantum numbers are simultaneously determined in the final state. The solution to this problem can be found by using the density matrix formalism; see details in Appendix B. 2.3. Experimental approaches with polarized beams To date, three basic experimental setups have been developed to prepare and/or analyze the spin polarization of the projectile and/or target beams. These are summarized in Table 2.3.1. Note that,
N. Andersen et al. /Physics Table 2.3.1 Experimental
approaches
with polarized
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beams
Type
Geometry figure
Spin preparation before Electron Atom
Vol. I I (e hv)
2.1.1 (b,c) 2.3.1 (a) 2.3.2 (a) 2.3.2 (d) 2.3.3 (a) 2.3.3 (a) 2.3.3 (a) 2.3.3 (a) 2.3.3 (e)
~ J J J
2A (1) 2B ( II1 3% 3S, 3T 3u 4v
J J J J
261
Spin analysis after Electron Atom
Photon
analysis
after
” ”
J J
_ J _ J
._
J
J
J
until now, either electron or photon polarizations have been analyzed in the exit channel, but never both simultaneously. The natural extension of the experimental geometry discussed in Section 2.1 is the use of a spin-polarized electron beam in the standard electron-photon coincidence setup (Vol. I). The schematic diagram is shown in Fig. 2.3.1(a), and the geometry and scheme of the experimental setup, as used by the Miinster group [109,90], are shown in Figs. 2.3.l(b,c). The main difference compared to experiments with unpolarized electrons is the source of spin polarized electrons which replaces the electron gun. Recent technological developments have made the GaAs source (see e.g. [SO]) the standard for this kind of experiments because of its favorable beam characteristics in yielding high current, narrow energy width, and an electron spin polarization of up to 50%. (In most experiments discussed in this review, the actual electron beam polarization was between 30% and 40%). We will refer to this setup as “type 1” below. The second type of experiment has been pioneered by the NIST group [70,71]. The main idea is to perform a time-reversed experiment by scattering spin-polarized electrons from spin-polarized laser-excited atoms. The schematic diagram of the experiment is shown in Fig. 2.3.2(a), and more extended schemes in Figs. 2.3.2(b, c). The additional complication in this case is the laser pumping technique to prepare the polarized target beam, but neither polarization nor coincidence analysis is performed after the collision. Consequently, this type of experiment does not suffer from the low count rate problem associated with standard electron-photon coincidence techniques. This experimental setup has been used extensively for electron scattering from sodium atoms (see Sections 3.2.2 and 3.2.3 below), and recently also for a chromium target (Section 3.2.7 [53]). An important experimental restriction was the fact that spin asymmetries of the form (2.3.1) were only determined with electron (P,) and atom (PA) polarizations perpendicular to the scattering plane (type 2A). In Eq. (2.3.1), ot t(0) and ct ,(0) denote the cross sections for parallel and antiparallel polarization vectors, respectively. As will be shown in Section 3.2.2, a measurement
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Coincidence
:
(4
(b)
ght modulator
L
Coincidence
1
dzL
/
Piles-of-plates analyzer
GaAsP cathode
y” Photomultiplier
Cc)
Piles-of-plates anatyzer
f.enr\
Target nozzle
p1ate
Mott detector
Fig. 2.3.1. (a) Schematic diagram of the experiment. Polarized electrons are scattered inelastically from atoms at an angle 0. The photons emitted during the decay of the atom in a specific direction are polarization analyzed in coincidence with the scattered electrons without further spin analysis (type 1). (b) Geometry of the Miinster experiment (from [90]). Directions refer to the collision coordinate frame. (c) Scheme of the actual experiment (from [90]).
with in-plane polarizations can help in the determination of phase differences between individual singlet and triplet amplitudes. The scheme of such an in-plane experiment, where the laser beam makes an angle CDwith the incident projectile direction, is outlined in Fig. 2.3.2(d), referred to as type 2B. Finally, the Mtinster group has also developed an experimental setup where polarized electrons are scattered from unpolarized atoms and the electron polarization after the collision is determined. This type of experiment allows for the determination of the so-called generalized STU parameters [lo] that fully describe the change of an arbitrary initial electron polarization through
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Reports 279 (1997) 251-396 Optical I
Pock& Linear (b)
Loser
I
Pumping Laser
Cell Polarizer Diode
Fluorescence Mon~l
Fig. 2.3.2. (a) Schematic diagram of the NIST analysis is performed after the collision (type polarized-atom scattering apparatus, showing Mott spin analyzer (from [71]). (d) Schematic polarized atoms with in-plane laser pumping
experiment involving polarized electrons and polarized atoms. No spin 2A). (b) Actual scattering geometry (from [71]). (c) Polarized electron GaAs polarized electron source, scattering chamber, sodium oven, and modification of the NIST experiment involving polarized electrons and (type 2B).
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scattering from unpolarized target atoms (type 3); see Fig. 2.3.3(a). The preparation of the projectile and target beams before the collision is identical to the electron-photon coincidence experiment with polarized electrons discussed above, but the photon detector and the coincidence unit for the detection of the final state are replaced by a Mott detector for the spin polarization analysis of the scattered electrons [26, 541.
(4
180” spectrometer
(C)
Mott
detector
Fig. 2.3.3. (a) Physical meaning of the generalized STU parameters for an initial spin polarization P which is changed to a final spin polarization P’ through the scattering process. The polarization function Sp gives the polarization of an initially unpolarized projectile beam after the scattering (type 3Sp), and the asymmetry function SA determines the left-right asymmetry in the differential cross section for scattering of the spin polarized projectiles (type 3SA). The contraction parameters T,, T,, T, describe the change of an initial polarization component along the three Cartesian axes (type 3T), while the parameters U,, and U,, determine the rotation of a polarization component in the scattering plane (type 3U). (b) Schematic diagram of the Miinster experiment to measure the depolarization of electrons after scattering by unpolarized sodium atoms (type 3T). (c) Scheme of an actual experiment (from [54]). (d) Measurement of the change of the electron polarization vector components caused by elastic scattering (from [26]). (e) Schematic diagram of an experiment to determine the rotation of an electron spin polarization out of the plane spanned by orthogonal initial electron and atomic spin polarizations (type 4V).
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LIGHT 90’
GaAsP
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MODULATOR
DEFLECTOR
CATHOOE WIEN
180° DEFLECTOR
FILTER
Xe ATOMIC BEAM
cd)
Fig.2.3.3.
continued
The schematic diagram of the experimental setup is shown in Fig. 2.3.3(b) while the most essential parts are presented in more detail in Figs. 2.3.3(c, d). As discussed in a previous review [lo], there are, in general, eight parameters that describe the reduced spin density matrix of the scattered projectiles completely. Besides the absolute differential cross section G,,for the scattering of unpolarized electrons, there are seven relative generalized STU parameters whose physical meaning is shown in Fig. 2.3.3(a): The polarization function SP gives the polarization of an initially unpolarized projectile beam after the scattering, while the asymmetry function SA determines the left-right asymmetry in the differential cross section for scattering of spin polarized projectiles. Furthermore, the contraction parameters T,, T,, T, describe the change of initial polarization component along the three Cartesian axes while the parameters U,, and U,, determine the rotation of a polarization component in the scattering plane. Note that the number of independent STU parameters may be reduced in certain cases; the most important examples are (i) elastic scattering, (ii) transitions between target states with zero angular momentum in both the initial and the final
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state, and (iii) situations where relativistic effects during the collision may be neglected (for details, see [lo] and further discussion in Section 3.2 and Appendix C). The concept of the generalized ST U parameters can be extended in a systematic way to account for initially polarized targets and even a polarization analysis of the target spin in the final state. Since such experiments have not been performed to date and are not expected to be performed in the near future (at least not for excitation processes), we only mention here one type of parameter that can provide important additional information. The general idea is shown in Fig. 2.3.3(e): one starts with both projectile and target beams spin-polarized, with polarization vectors PA and PB orthogonal to each other; after the scattering, the polarization component P’ of the projectile beam along the direction defined by PA x PB, i.e., perpendicular to the plane defined by the initial polarization vectors, is determined, and the result is normalized to the product of the initial polarization magnitudes (type 4V). We denote the corresponding parameter by the generic symbol I/, i.e., I/ = P’/(P*. PB) .
(2.3.2)
It should be pointed out that many such independent parameters can be defined, particularly in cases where relativistic effects and, consequently, the relative orientation of the scattering plane and the plane defined by the initial polarization vectors must be taken into account.
3. Excitation by spin polarized electron impact In this section we discuss what information can be gained by application of polarized electrons in addition to the knowledge obtained with unpolarized beams. We keep the order of increasing complexity introduced in Section 2.1. 3.1. The fully coherent case 3.1.1. Excitation
of the He n’P1 and n3P, states
A simple example of a fully coherent excitation is He 1iS + nlP excitation, discussed at length in Section 3.1 of Vol. I. As shown, all information for this case can already be obtained from experiments using unpolarized beams, and (0, LI, y) forms a complete set of parameters; see Eq. (2.1.5). Consequently, in this simple case, application ofpolarized electrons will add nothing new: The polarization of the scattered electron beam can be predicted beforehand, since no change is possible in the scattering process. The same statement is true for excitation of the He 3P states, also studied in Vol. I. Excitation can happen by electron exchange only. Consequently, the initial polarization of the electron beam will be changed by a simple factor; for details see Ch. 4.7 of [59]. 3.2. The incoherent case with conservation 3.2.1.
General description
of atomic rejection
symmetry
of the excited 2P state
The targets of interest for this section, typically light alkali atoms or hydrogen, have an electron spin of s = 3. For electron impact, this doubles the number of scattering amplitudes from two to
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four (recall that f0 = 0), since we now have the possibility of triplet (t) and singlet (s) scattering, bringing us into the situation of Fig. 3.2.1. The amplitudes of interest are j:l
= x+eivl+ , i& -
j'L,=x_e
(3.2.la)
(3.2.lb)
,
(3.2.1~) (3.2.ld)
Neglecting an overall phase, we thus need seven independent parameters for each scattering angle 8 to characterize the amplitudes completely. One is traditionally chosen as the differential cross section CJ”corresponding to unpolarized beams. In addition, six dimensionless parameters may be defined, three to characterize the relative lengths of the four vectors, and three to define their relative phase angles. As a start, we parametrize the density matrices in analogy to the unpolarized beam case, Eq. (2.1.3), as [52]
(3.2.2a)
and 1 1 (1s = gs 2
+L",
0
0 0 _ PFe2if0
_ pce-2i7 0
(3.2.2b)
l-L”,
where (3.2.3a) (3.2.3b) (3.2.3~)
(3.2.3d) p;eW’
= Pi + iPG = - (2rx+cr_/a’)eCi6’,
p;e2iy’ =
Pj + iP; = - (2P+&/os)e-i6‘
(3.2.3e) .
(3.2.3f)
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N. Andersen et al. /Physics
Fig. 3.2.1. Schematic diagram transitions by electron impact.
of triplet (t) and singlet (s) scattering Note that A+ + 6” = A- + 6’.
Reports 279 (1997) 251-396
amplitudes
in the natural
frame for ‘S + ‘P
In the case where unpolarized beams are used, the total density matrix becomes the weighted sum of the two matrices p” and pf, i.e.,
pu =
0”
0
0
0
!
1 2
=
1+ Ll
0
- P1e2iy 0 1 + Lk
pie-W
_
1
- Ll
0
- P[epzir’
_ p:eziY’ 0 0 0
3w’# ; i.
+ w%, ;
I
1
1 + L”,
0
0
0
_ p,“&W
0
-0 L:
I
- Pfe-2iys
(3.2.4)
i
where wi = at/(aS + 3a’) = C+/46”
(3.2.5a)
ws = cq(8 + 3cr’) = 0740, = 1 - 3w’
(3.2.5b)
0 ” = [3wt + wq0 ” = jai + ;as .
(3.2.5~)
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269
The parameters w’ and ws are related to the parameter Y= (T’/G’used by Hertel et al. [52] through r = w’/wS = w’/(l - 3w’) = (1 - w”)/3w” .
(3.2.6)
However, we prefer the use of w’s’ to r for reasons of mathematical symmetry and simplicity. At this point we have thus introduced a total of six parameters, namely CJ,,,w’, L:, L;, y’, y’, leaving one parameter, a relative phase, still to be chosen. Inspection of Fig. 3.2.1 suggests, for example, the angle A+. Note that the fourth angle, A-, is then fixed through the relation A+ - A- = 6’ - 6” = 2(?” - y’) .
(3.2.7)
The first equality sign follows from inspection of Fig. 3.2.1, the second one from Eq. (2.1.2), applied individually to the singlet and triplet components. In the spirit of [52] and Vol. I, we therefore suggest the following complete set: (a,; w’, L:, L;; yt,y”,A+) . We have thus introduced
(3.2.8)
parameters that
allow for a complete description of the scattering process; l are a natural generalization of the parameters used for unpolarized beams; l can be interpreted in simple physical pictures; l and are accessible in “partial” (i.e., noncomplete) experiments as will be illustrated in the next section. The reduced Stokes vector P of the unpolarized beam experiment is given by the weighted sum of the singlet and triplet (unit) Stokes vectors Psvt as l
P = 3w’P’ + w”P”
(3.2.9)
from which the set of parameters (L,, y, 4) for the unpolarized beam experiment may be evaluated by using Eqs. (2.1.4a, b). In particular, LJ_ = 3w’L: + w”L”, .
(3.2.10)
Since, in general, L\ # L”, and yt # y”, this causes the (reduced) degree of polarization P to be smaller than unity, i.e., P I 1, cf. Eq. (2.1.6a). We now want to express the STU and I/ parameters discussed in Section 2.3 both in terms of the amplitudes (3.2.1) and in terms of the complete set of parameters (3.2.8) [3]. This can be done in the case of light (quasi) one-electron atoms for which the so-called fine-structure efSect is now well documented in the literature (see e.g. [SO, 52, 10,48, 1l] and Appendix C). Provided that electron exchange is the only spin-dependent effect of importance for the excitation, and that the finestructure energy splitting is negligible compared to the initial and final energies of the projectile, the STU parameters get vastly simplified. The seven polarization, symmetry, contraction, and rotation parameters for each fine-structure level reduce to the following set of only four independent parameters (the superscripts denote the J-value of the excited target state): (3.2.1 la) (3.2.1 lb)
270
N. Andersen et ai. J Physics Reports 279 (1997) 251-396 T
T,1’2 = 7-7’
E
= T,“2
= T,“/”
= 772
= T,3’2 ,
(3.2.1 lc)
U = U.j/’ = U:x’2= - 2Ux3y/2 = - 2Uy”x’” .
(3.2.1 Id)
The results in terms of scattering amplitude parameters are: Sp = - (1/4C”)[a: - lx2 + p: - p1-j )
(3.2.12a)
SA = (1/40,)[Re(2cr+P+ei(~+-++) - 2cx_p_ei(#--1L-J} - 2(a$ - a?)] ,
(3.2.12b)
T = (1/4~,)[Re{2a+fi+ei(~+-*+) + 2a_fi_ei(@~-ti~)} + 2(& + cx?)] ,
(3.2.12~)
U = (1/40,)[Im{2cc+~+ei(~+-~+) - 2a_P_ei(+~-ti-))]
,
(3.2.12d)
I/ = (1/4~,)[Im{2ct+/3+ei(~+-~+) + 2a_fi-ei(~--ti-))]
,
(3.2.12e)
where Re{X} and Im{X> d enote the real and the imaginary parts of the complex quantity X, or sp= - w'L\+ wSLT,
(3.2.13a)
SA = (1/20,)[a+/?+ cosA+ - a-P_ cos A-] - 2w’L:,
(3.2.13b)
T = (1/20,)[a+P+ cosA+ + cr_j_ cos A-] + 2w’,
(3.2.13~)
U = (1/2~,)[a+j?+
sin
A+ - K/K sinA-
,
(3.2.13d)
I/ = (1/2a,)[a+fl+
sin
A+ + a-/3_ sinA-
.
(3.2.13e)
Note that the o[+ and pi a,&J20,
may be eliminated from Eqs. (3.2.13) by using
= Jw’wS(1 & L\)(l f L”,) .
(3.2.14)
As an example of an in-plane A-parameter, cf. Eq. (2.3.1), we give B = A(4 = 90”, e), where the angle @ is defined in Fig. 2.3.2(d). The result for this type 2B experiment is [3] B
=
_
w’(l-
Pi) - wS(l - Pf) (1-P1)
(3.2.15) .
The amount of information contained in the atomic density matrix (i.e., the Stokes parameters) and the reduced density matrix of the scattered electrons (i.e., the STU parameters) is illustrated in Fig. 3.2.2. From a Stokes parameter analysis, one obtains information about the relative phase between the two f:r and j? 1 amplitudes and the relative phase between the two fsl and f? 1 amplitudes, as well as the relative sizes of all four amplitudes. However, none of the relative phases between any triplet and singlet amplitude can be determined. The STU parameters, on the other hand, determine the relative phase A’ between the two j’:i and fS1 amplitudes and the relative phase A- between the two fl I and f?1 amplitudes, provided that the relative sizes of all four amplitudes are known from a Stokes parameter measurement. In this case, none of the relative phases between ML = + 1 and ML = - 1 amplitudes can be obtained. Note also that the electron spin polarization parameter SP, Eq. (3.2.13a), may be derived from results of a type 2A experiment.
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N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
STU
fl,
Triplet : :
G f’,
0
Singlet fSll
Fig. 3.2.2. (a) This diagram shows which relative amplitude analysis. (b) This diagram shows which relative amplitude parameters.
d’ f’,
.’
-:. ‘...
..
“‘(-J
ff ,
sizes and phases can be evaluated from a Stokes parameter sizes and phases enter into the Eqs. (3.2.13) for the STU
We finish this subsection with a comment on Eq. (3.2.13a) which shows the necessary conditions for a nonvanishing polarization function Sp in S -+ P excitation of alkali-like systems, based on the fine-structure effect alone: (1) electron exchange, i.e., a dynamical dependence of the scattering amplitudes on the individual spin channels (triplet and singlet in 3.2.13a); (2) a nonvanishing orbital angular momentum orientation in at least one of the spin channels; (3) experimental resolution of individual fine-structure levels. (4) negligible dependence of the nonrelativistic scattering amplitudes on the small energy difference between the fine-structure states; although this condition, in a strict sense, is incompatible with (3), it is only required for exact relationships like (3.2.11) to hold between generalized STU parameters for individual multiplet members. These conditions are well fulfilled, for example, for electron impact excitation of sodium, provided the collision energy is large compared to the fine-structure splitting [ 111. 3.2.2. Excitation of the Na 32P state: General considerations 3.2.2.1. The present experimental situation. At this point, it is appropriate to investigate which of the seven independent parameters of Eq. (3.2.8) have actually been measured experimentally, i.e., to what degree the perfect scattering experiment has been achieved to date. We begin with the type 2A experiment performed by the NIST group [71] and described in Section 2.3. Recall that spin polarized electrons with polarization vector perpendicular to the scattering plane were scattered superelastically from excited spin-polarized P state sodium atoms produced by pumping with circularly polarized laser light. By reversing the directions of the two polarizations individually and applying time-reversal invariance to relate the superelastic de-excitation to the inelastic excitation process, this experiment allows for the determination of the angular momentum transfers L\ and L; for the two spin channels individually, as well as for the relative weights w’ and ws of triplet and singlet contributions to the differential cross section (for details, see Hertel et al. [52]). The angular
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Reports 279 (1997) 251-396
momentum transfer LI for unpolarized beams can be recovered from the set (L:, Ly,w')through Eq. (3.2.10). This experiment does not allow for a unique determination of the alignment angles yt and y”. The off-diagonal elements of Eq. (3.2.4) show that pie2ir
=
3wtp/e2iy’
+ wspfe2iy’
.
(3.2.16)
As illustrated in Fig. 3.2.3, this complex equation corresponds to the addition of two vectors P;” and P/, multiplied by weighting factors 3w’ and w’, respectively, to form the resulting vector Pl. Hence, elementary geometry can be applied to obtain two pairs of solutions, (y’,Ys)true(the true solution) and (Y’,f)ghost (the other possibility):
Y’=YT xl2 Y” =
Y+
(3.2.17a)
7
(3.2.17b)
$12>
where the angles x and $ are defined in the figure. Hence, provided experimental data are available for the parameter set (Pl, y,w', L\,L",) at a given collision energy and scattering angle, two sets of possible angles (y’,y”) can be determined, as will be further discussed below. It was pointed out by Hertel et al. [52] that the above ambiguity is mathematically identical to that found in S -+ D excitation processes [2]. Inspection of Eq. (3.2.15) shows that the ambiguity could be resolved directly through an in-plane measurement of the B-parameter: A consistency check between the results of Eqs. (3.2.15) and (3.2.16) will eliminate the ghost solution. Finally, the NIST experiment contains 120information about the missing phase angle A+ relating singlet and triplet amplitudes. Some information about this angle can, however, be obtained by combining the NIST data for (y’,y”) with results from the type 3T experiment (see Figs. 2.3.3(b,c)) performed by the Miinster group [54,55]. As seen from Eq. (3.2.13c), the T-parameter contains the phase angles A’ r, but not in a unique way.
Fig. 3.2.3. Vector diagram
corresponding
to Eq. (3.2.16). For a discussion,
see text.
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273
3.2.2.2. Combining experimental and theoretical data. The question therefore arises whether more can be learned from the existing experimental data, especially when combined with state-of-the-art theoretical approaches. This problem will now be addressed. (if Determination of the alignment angles y’ and y”: As pointed out in the discussion of Eq. (3.2.16) it is possible to determine two sets of angles (yt,ys), provided experimental data are available for the set of parameters (Pl, y, w’, L\, L”,) at a given collision energy and scattering angle. While data for all these parameters have indeed been obtained for electron-sodium (de-)excitation, the energy and scattering angle combinations investigated by the Adelaide [103] and the NIST [71,91] groups, unfortunately, do not overlap at all. Consequently, we demonstrate the feasibility of combining various experimental data sets by replacing the missing experimental data for Pl and y with theoretical results of Bray’s calculation [33] at a total energy of 4.1 eV to “invert” the NIST data at this energy. This approach seems justified in light of the excellent agreement between Bray’s theory and experiment for all available collision energies; see Section 3.2.3. The results of the inversion are shown in Fig. 3.2.4 for both angles y’ and “J’.The error bars on the experimental points were obtained by first changing the theoretical results for the set of input parameters (w’, L\, L”,) (recall that q’t’ = dm for the two spin channels separately) by a very small amount and then looking at the effect on the inverted theoretical results for the two pairs of (y’,7”). This gives the partial derivatives of y’ and y” with respect to these three input values and allows for the calculation of error bars through the formula
6p
+ (ap/aL;)2(6L;)2
= J(ap/aL:)2(scJ
+ (aytvyawt)2(6wt)2
(3.2.18)
It should be noted that this is a somewhat pessimistic estimate of the error bars, since we assume an independent determination of the parameters (w’, Ll, L”,). Consequently, the error bars shown in Fig. 3.2.4 are generahy overestimated. The important point, however, is that both the true and the ghost solutions, after inverting the experimental and the theoretical data, are in very good agreement with each other. Hence, it seems justified to select the “true” experimental data as those that follow the true theoretical solution. It is interesting to note that the two sets of solutions can cross each other and that it is impossible to stay on the “true” experimental curve by assuming, for example, a smooth angle and energy dependence of the phase angles In attempts to map the geometry of the triangle of Fig. 3.2.3 [3], we encountered a problem with the conversion procedure. In regions where the three vectors Pr, P; and Pf are almost parallel or anti-parallel, error bars or minor deviations between theory and experiment produced instances where the sum of two sides was slightly smaller than the third side. We solved this problem by setting x and $ (see Fig. 3.2.3) to 0” or 180”, respectively. For the same reason, an attempt to invert purely experimental data - necessarily at slightly different energies - failed miserably. (ii) Determination of the singlet-triplet angles A+ and A-: As mentioned previously, it is impossible to obtain the phase difference between singlet and triplet amplitudes from any set of type 2 experiments, but part of the information can be gained by combining type 2 results with those from a 3T-type experiment. For the simple case of elastic scattering, this is evident from Eq. (4.40) in Kessler’s standard treatise [59]. Eq. (3.2.13~) is a generalization of the expression for elastic S-S scattering which, in our notation, reads T = 2(w’ + mcosA)
.
(3.2.19)
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274
30
Reports 279 (1997) 251-396
60 Scattering
90 60 -
-“O
c
120
150
f
I
0
60
90
120
150
theory
----
ghost
Scattering
180
Angle (deg)
e
~
30
90
180
Angle (deg)
Fig. 3.2.4. Alignment angles y’ and y” calculated from the NIST [71] data for (w’, L\, Ll) and theoretical results for (Pi, y) from scattering amplitudes of Bray [33] for electron impact excitation of the 3 ‘P state of sodium at an incident electron energy of 4.1 eV; (0) two sets of inverted experimental data as well as true (__ ) and ghost (- - -) theoretical solutions.
This formula was applied to experimental data by McClelland et al. [74] to extract cos A, leaving the sign of A( = yst in their notation) undetermined. We now show that a similar, though more complicated procedure, can be applied to the inelastic case [3]. Type 3T experiments were performed by Hegemann et al. [54,55] at 4.0 eV and 12.1 eV total collision energy, while the type 2A NIST experiments were done at 4.1 and 10.0 eV, respectively. Thus, due to the lack of data from different experiments at the same energy, we again demonstrate the inversion method by using theoretical data [33] for the contraction parameter T at 4.1 eV. The main idea of our inversion procedure is illustrated in Fig. 3.2.5. Eq. (3.2.13~) for the T-parameter gives a nonlinear equation for A+ and A-. On the other hand, these two angles are not independent, since their difference is related to the difference between the alignment angles y’ and y” through Eq. (3.2.7). Consequently, solutions for A+ and A- can be found by searching for crossings between the lines determined by AcosA+ + BcosA-
= C ,
(3.2.20)
N. Andersen et al. / Phvsics Reports 279 (1997) 2S1-396
AcosA+
+ BcosA-
21.5
= C
180
-60
-180
180
-120
-60
0
60
120
180
A+ (deg) Fig. 3.2.5. Determination (see text).
of the singlet-triplet
phase
angles
A+ and A _ from experimentally
observable
parameters
where the constants A, B and C are evaluated from Eq. (3.2.13~) and the lines defined by Eq. (3.2.7) in Fig. 3.2.5. Due to the ambiguity in the sign of the arguments in the labeled (J’ - Yt)t,Ue,ghost cosines, as well as the ambiguity in the proper pair of (y’,ys), one will usuallrfindfour solutions, only one of which is correct. This is illustrated in Fig. 3.2.5 for inversion of the theoretical data at a scattering angle of 40” and a total collision energy of 4.1 eV [3]. Note that the problem can be reduced to searches in the first quadrant, since the “ghosts” in the second and fourth quadrants may be found via intersections with the dashed lines in Fig. 3.2.5 which are mirror images of the difference lines in those quadrants seen in the first quadrant. Since the slopes of the dashed mirror lines are reversed compared to the original difference lines, the actual crossing points in the second and fourth quadrants can easily be reconstructed, while the only remaining crossing, in the third quadrant, is related to the one in the first quadrant through a simultaneous sign change in A+ and A-. Furthermore, crossings can only occur in ranges where Eq. (3.2.20) has real solutions. The typical pattern of graphs corresponding to a numerical search for line crossings is shown in Fig. 3.2.6 for a set of four scattering angles at a collision energy of 4.1 eV. We first tested the algorithm using theoretical data; the agreement between the original and the inverted values for A+ and A- was usually better than 0.001%. The procedure was then applied using experimental data for the set of parameters (w’,L\, L”,) [71] to construct experimental pairs of (y’,7’) with error bars evaluated by Eq. (3.2.18) above. The results for A+ and A- as a function of the scattering angle for a collision energy of 4.1 eV are shown in Fig. 3.2.7. For simplicity, only one “ghost” solution (where A+ > 0) is shown. Again, it can be seen how the theoretical results help to
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N. Andersen et al. ,f Physics Reports 279 (1997) 251-396
Fig. 3.26. Typical structure of curves for numerical determination experimentally observable parameters (see text).
of the singlet-triplet
phase angles A + and d - from
theory
-
---~ ghost .
Scattering
experiment
Angle
(deg)
120 G 9) = &
60 0 -60 -120 -I 80
0
30
60 Scattering
90 Angle
120
150
180
(deg)
Fig. 3.2.7. Singlet-triplet phase angles A’ and A- calculated from the type 2A data [71] for (NJ’:L\,L;), the corresponding alignment angles (y’, v”), and theoretical type 3T results from scattering amplitudes of Bray [33] for electron impact excitation of the 3 ‘P state of sodium at an incident electron energy of 4.1 eV; (a) two sets of inverted experimental data as and one ghost (- - -) solution (see text). well as the krue ( ----)
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271
identify, in most cases unambiguously, the true solution among the various possibilities obtained from an inversion of experimental data alone. 3.2.2.3. Consistency checks of the formalism. The discussion above also demonstrates how the inversion procedure may serve as a consistency check among separate experimental data sets. Consistent experimental data should always allow for inversion within experimental uncertainties and the ambiguities pointed out. Of course, this can only be done if the individual data sets are obtained at the same total collision energy. This important coordination of efforts has, unfortunately, not taken place in most cases until now, thereby preventing desirable consistency tests of results from separate, very sophisticated experimental setups. A type 3s experiment was performed by the Miinster group [79]. Its geometry is similar to a type 2A setup, Fig. 2.3.2(a), except that both the (3p)‘Pli2 and the (3p)“Pj12 fine-structure levels were pumped individually, and the target density was chosen high enough to ensure essentially unpolarized target atoms due to the depolarization effect of radiation trapping. For each finestructure level, the left-right asymmetry in the differential cross section given by CUlcr,(o)-
Oright(e)llCOleft(Q) + gright(e)l G S*(8)pL
(3.2.21)
was measured. There are several interesting points to be made regarding the asymmetry function SA. Because of time-reversal invariance of the interaction, this type 3SA experiment for the de-excitation process is equivalent to determination of the polarization function Sp, a type 3Sp experiment, for excitation. Interestingly, in the approximation of the pure fine-structure effect, one can use Eq. (3.2.13a) to predict the results of this experiment from those of the type 2A experiment [52,25]. The data from the Nickich et al. [79] experiment and the results predicted from the NIST experiment [91] are shown together with theoretical results calculated from Bray’s amplitudes [33] in Fig. 3.2.8. The
-0.4
0
20
40
60
80
Scattering
100
120
Angle
(deg)
140
160
180
Fig. 3.2.8. Spin polarization function &. for electron impact excitation of the (~P)‘P,,~,~,~ states of sodium at an incident electron energy of 10 eV; data of Nickich et al. [79] for Sp (2PI,2) (0) and Sp (‘P& (W); prediction of Sp (‘PIi2) (0) and Sp (‘P3,J (0) from the NIST data [91] for (w’, L:, L”,); The theoretical curves for Sp (2P1!2) (~) and SP (2P~,~) (~ - ) were calculated from scattering amplitudes of Bray [33].
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Table 3.2.1 Alignment and orientation
parameters
Symbol
0 0 0 0
0 + 0 A A
A X
;
rc
for excitation
Reports 279 (1997) 251-396
of the Na 3’P state by spin-polarized
electrons
Source
Fig. 3.2.9
Comment
[lo21
f h
a
d,f c, e, g, h d,f
L,,Y,P,,P
C361 c931 c941 C891 [651 C651 c911 c711 c711 [791 c541 c31 c341 c341 ClW Cl001 [@I c251 C781 WI
a
L&P 0
;h
a
c, g a, e h
L,,LYL,LLW LI,L:,L;,W’
b, c, d, f a, d a, b, c, e, g ash a-h : b-h i,f e
L, SP T ‘it,‘is,Ll+,d-,SP ccc cc0 1occ 1occ DWB2 DWBlOP DWBlLE DWBlLE
consistency between the two independent experimental data sets and their agreement with the theoretical prediction underlines the validity of the present formalism. 3.2.2.4. Conclusions. We conclude this section by pointing out that the somewhat complicated inversion procedure could be avoided and the perfect scattering experiment could be achieved, for example, as follows: measurement of the in-plane asymmetry parameter B, i.e., a type 2B experiment, would resolve the ambiguity in the two (y’, y”) pairs. We also notice from Eqs. (3.2.12, 13) that, for example, the equations for the pairs (SA, U) as well as (T, V) can be recast in a form similar to the real and imaginary parts of Eq. (3.2.16). Consequently, determination of one of the pairs gives a geometrical ambiguity in the (A’,A-) pair, as discussed previously for (‘it,?“). The ambiguity can be removed by measurement of (at least) one more of the remaining STUI/ parameters. If (L,, y, P,) are known from unpolarized beam experiments and (L\, L”,, w’) from a type 2A experiment, any three of the five parameters B, SA, T, U, I/ will suffice to achieve a perfect scattering experiment. 3.2.3. Excitation
of the Na 32P state: Data
After this general discussion we are now prepared for inspection of the theoretical and experimental results for excitation of the Na 32P state at various energies. Fig. 3.2.9 shows the differential cross sections and the various alignment and orientation parameters, including all theoretical and experimental results in the literature available to us at the time of writing.
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Reports 279 (1997) 251-396
Na(3'P) 4.1 eV
-
I
0.8 0 (a)
P
-I
I, 90
I
180 0
--
u I
I
I
90
I
I,
180 0
--
T I
I 90
I
I 180 0
v I
/
I 90
I
I
_, 180
g(deg)
Fig. 3.2.9. Survey of alignment and orientation parameters for excitation of the Na 3’P state by spin-polarized electrons. See Table 3.2.1 for sources and symbols. Throughout this paper, differential cross sections are given in units of &sr.
Extensive sets of theoretical results originate from the “close coupling plus optical potential” (CCO) and the “convergent close coupling” (CCC) methods of Bray and collaborators [33,34], and the “second-order distorted wave Born” (DWB2) approach of Madison et al. [68]. All three methods account, to some extent, for the effect of other discrete and continuum target states on the results for a particular transition. This is achieved in the DWB2 through a numerically exact evaluation of the second Born term, and in the KC0 method through the close coupling of y1discrete target states and an approximate optical potential to represent the coupling to other
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Na(3’P) 8.1 eV
I
0 @)
I
I
90
I
I
180 0
I
I
,
90
I
I
I
I
180 0
I
90
I
I
180 0
90
180
Wed Fig. 3.2.9. continued
states. Finally, the CCC method corresponds to a close-coupling expansion with a large basis including low-lying physical discrete target states and a large set of “pseudo states” that simulate the effect of the higher-lying discrete as well as the continuum states in the interaction region to sufficient accuracy. These results are supplemented by n CC close-coupling calculations from Trail et al. [loo] and by some first-order DWBl calculations. Although the y1CC approach only includes the coupling between IZdiscrete target states, it is expected to be fairly accurate at collision energies below the
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N. Andersen et al. J Physics Reports 279 (1997) 251-396 Na(3’P)
0.0 ’
0 (c)
’
’
’ ’ ’ ’ ’ 90 180 0
’
10 eV
’ ’ ’ ’ ’ 90 180 0
’
1 I I I ’ ’ ’ 90 180 0 90
I
-1 180
O(de.4 Fig. 3.2.9. continued
ionization threshold. On the other hand, the DWBlOP approach of Balashov and GrumGrzhimailo [25] accounts for the effect of other target states via a complex optical model potential, representing exchange with the target electrons, charge cloud polarization and loss of flux into inelastic channels. Such a method is expected to do well at higher collision energies above the ionization threshold. Finally, the DWBlLE calculations by Mathur and Puroit [78, Xl] include a local exchange potential only. We first note that the CCC and CC0 results are very close to each other in most cases, with deviations mainly to be observed in the singlet channel. In these cases, such as for the parameter
282
N. Andersen et al. / Physics Reports 279 (1997) 251-396
Na(3’P) 12.1 eV
-
0.0 L ’ 0 (4
--
P ’
’ ’ ’ 90 180 0
u I
-I
I 90
,
I
T t
‘(
180 0
* 90
,
I 180 0
v !
I
I 90
1
I
d_1 180
~(deg) Fig. 3.2.9. continued
L”, at 10 eV, the CCC approach performs consistently superior to CCO. Including also the DW-results in the discussion, we see a general convergence of the various methods at the higher energies. As expected, the DW-predictions become gradually less reliable with decreasing energy, with significant deviations from both the experimental data and the close-coupling results developing below 10 eV. Fig. 3.2.10 analyses the convergence with basis size n = 2,4,7,10 for the y1CC calculations at low energy. For most parameters, the 1OCC calculation seems to have sufficiently converged at 4.1 eV.
N. Andersen et al. /Physics Na(32P)
I
0.8 0 Cc)
I
I 90
I
I 180 0
I
I
I 90
I
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I
I
I
180 0
I 90
I
I 180 0
I
1
1 90
I
I
_, 180
o(deg) Fig. 3.2.9. continued
Turning now to the experimental results, the situation is generally quite good at several energies for many of the parameters, in particular when including the additional data extracted by the inversion technique outlined in the preceding paragraph. As stated above, however, a better coordination of the experimental efforts would have been beneficial. Additional experimental results obtained, for example, with the in-plane laser pumping scheme described above, could finally achieve “the perfect scattering experiment”. This would be desirable for at least one convenient electron impact energy, such as 10.0 eV, to remove remaining ambiguities.
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Na(3’P)
22.1 eV
f8ded Fig. 3.2.9. continued
When comparing theory and experiment for Na 32P excitation, the most strik :ing general observation is the very good results obtained by the CCC method at all energies. 3.2.4. Excitation of the Li 2, 32P states Corresponding, but considerably less extensive results are available for the Li 2’P state. Fig. 3.2.11 shows results from the CCC, CCO, DWB2 codes already mentioned for Na, supplemented at 20.0 eV by DWBlLE results of Mathur [66]. Only CCC data are given at 4.1 eV, since the CC0 method produces essentially the same graphs. We notice that many features in the graphs are similar to the corresponding results for the Na 32P state at the same collision energy. There is
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_ ...f’. I
I
1
i
Y:
I
,
0 0
0
(Ed
e(ded Fig. 3.2.9. continued
again qualitative agreement between the predictions of the various theories, especially at higher energies. The CCC and CC0 curves only differ noticeably for the singlet scattering channel. Experimental data only exist for the r parameter [22] which have been transformed to w’ results. As for Na, the comparison favors the CCC/CC0 predictions. For the Li 32P state, theoretical data exist which were obtained by the CC0 and the CCC methods [20], as well as first-order [S6] and second-order [75] DW methods. The results at 10 and 20 eV are displayed in Fig. 3.2.12. As in Na, the CC0 results are often very similar to those
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Na(3’P) 54.4 eV
0
o.gol
I
0 (h)
I
I
90
I
I
I
180 0
90
180 0
90
180 0
90
180
o(deg) Fig. 3.2.9. continued
obtained with CCC, except for some significant deviations in the singlet spin channel. Furthermore, there are often large discrepancies between the predictions from the CC and the DW approaches. Note also that the agreement between the two sets of DW results is not very satisfactory. The differences could be due to the importance of second-order effects and/or the use of different atomic wave functions and distortion potentials. Nevertheless, on this limited basis we notice some similarities between the results for the Li 22P and the Li 32P states at the same collision energy, in accordance with the approximate n-independence noticed earlier for the alignment and orientation
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0.L
1
0
-90
l30.5
0.8 0
(4
P
L I
--
I 90
I
I 18010
u
I
--
I
I 90
I
, 18010
T
I
--
1 I 90
I
I 18010
v
I
I
I 90
I
,
-1 180
Q(k)
Fig. 3.2.10. Convergence with basis size in n CC calculations (n = 2,4, 7, 10) for alignment and orientation parameters 2CC; for excitation of the Na 3’P state by spin-polarized electrons. The results are taken from Trail et al. [loo]. ( ~--) (,‘.“,,4cc;(-----)7cc;( p) 1occ.
parameters for the He n’P Rydberg series, cf. Vol. I, Section 3.1.3 and Fig. 3.1.16. Such similarities within a Rydberg series are even more prominent for electron scattering from atomic hydrogen and will be further discussed in Section 3.2.6. Considering the similarities between the data for the Li, 2, 32P and the Na32P states, the fairly satisfactory situation for Na, and the very considerable investment in time and other resources involved in the experimental investigations, it seems unlikely to us that much significant new information can be gleaned from further studies of the Li case at the present time.
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Na(3’P)
I
(b)
I
I
I
8.1 eV
I
1 I
I
I
I
O(detd Fig. 3.2.10 continued
3.2.5. Excitation
of the K 42P state
Information on the K 42P state is even more sparse. Fig. 3.2.13 shows theoretical results from the CCC, CC0 and DWB2 methods. Again, we notice similar trends at higher energies, but also big differences at lower energies, in particular for the singlet channel parameters L”, and y”. No experimental results exist, not even for unpolarized electron and atom beams. In view of the richer structures predicted for this atom, and the intriguing differences between the (angle-integrated) ionization asymmetries observed between Li and Na on one side and K on the other [32], this element might offer a more attractive experimental challenge than Li.
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N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
0.5
0.0 0
(a)
90
18010
90
18010
90
18010
90
180
@(ded
Fig. 3.2.11. Survey of alignment and orientation parameters for excitation of the Li 2 *P state by spin-polarized electrons. -)CCC[20];(-----)CCO[20];(.--...)DWB2[75];(..p-)DWBlLE
Finally, Fig. 3.2.14 shows a comparison of results from the early pioneering experiments in Bederson’s group [47] with various theoretical predictions. The parameter reported, here called W, is the fractional depolarization of the polarized atom beam, observed as the change of spin state of a potassium beam resulting from resonance excitation with subsequent decay back to the ground state. To approximate the effect of the finite experimental resolution, we convoluted the various theoretical results using a 2-D Gaussian with CJ= 5”.
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Li(22P) 1OeV
,
I
-90
-180 180
I,
10
0.5
-.
I
‘..
.,.,,.
.,’ ,:
SP
5‘4
I3
,
I
0.5 -
:t ;:
I I
I ,
1 I
t !
-1 1
0
\' p
:\ ; ..,.:
-I-
Fig. 3.2.11. continued
The CCC and CC0 calculations show, once again, the best overall agreement with the experimental data. However, given the importance of the convolution procedures (demonstrated in Fig. 3.2.14 by CCC results with and without convolution), this statement should be regarded more on a qualitative than on a quantitative basis. 3.2.6, Excitation of the H 2, 3, 42P states Electron impact excitation of hydrogen is a fundamental process, which justifies its position as a benchmark system for state-of-the-art quantum theory. It was found in Vol. I that the situation,
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Li( 2*P) 20eV
0.8 -
: . P
0.7
I 0
Cc)
I
I 90
I
, 180 0
I
I
I 90
U
1 1 180 0
I
I
, 90
,
T
, 180 0
1 I
1 I 90
V
1
_, 1x0
f4deg) Fig. 3.2.11. continued
even for unpolarized electron excitation, was only satisfactory at relatively small scattering angles, and this situation has not improved dramatically in the meantime. Perhaps not surprisingly, considering the requirements for such an experiment, no experimental results for alignment and orientation parameters exist yet for hydrogen excitation by polarized electrons, but some theoretical results are available. Fig. 3.2.15 shows data for excitation of the H 22P state. We notice an overall agreement between the various theories, CCC, CC0 and DWB2. Furthermore, the agreement with the (unpolarized)
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Li(YP) 10eV
()*I
1
0 (a)
I
I
90
I
I
I
180 0
I
I
I
90
II
I
180 0
I
I
I
90
I
I
I,
180 0
1-1
,,I,
90
180
@(de4
Fig. 3.2.12. Survey of alignment and orientation parameters for excitation of the Li 3 ‘P state by spin-polarized electrons: ccc [ZO]; (- - - - -) cc0 [20]; (. . . . . .) DWB2 [75]; (- - -) DWBILE [56]. ( ---)
experimental data is good up to about 90”, but from then on there is a marked disagreement with the experimental results. These latter ones have been obtained by two groups with different methods and are internally consistent. Further experimental results with unpolarized electron beams, in particular for the difficult P3 parameter (providing further information on LI and P) at large scattering angles, are highly desirable for this system, which persists to challenge our understanding of the perhaps most basic inelastic process in the whole field. In view of the
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Li(3’P) 20eV
I
1000
I
I
,
I
u
I
1 I
--
I
1
u*
I
r
I
I
I
I
1 I
I
,
1.0
-0.5
0.0 180
0
: P 0.7
I 0
(b)
,
I 90
I
U I 180 0
I
I
I 90
I
T I 180 0
!
1
1 90
I
V 1 180 0
I
I
I 90
/
I
-1 180
@(deg) Fig. 3.2.12. continued
experience on the e-Na system with the singlet spin channel being the source of theoretical difficulties, experiments with polarized beams may be important to gain information about the individual spin channels. On the other hand, the theoretical predictions are very similar for the individual spin channels and, therefore, for the spin-averaged results. This is not surprising, since the importance of electron exchange is already quite small at 35 eV incident energy. Hence, we recommend a refined experiment with unpolarized beams as a first step.
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K(4’P) 4.1 eV
(4
*(de4
Fig. 3.2.13. SUI-vey ofalignment and orientation parameters for excitation of the K 4 ‘P state by spin-polar :ized electrons: ) ccc [21]; (- - - - -) CC0 [21]; (. 1. .) DWB2 [73]. (--
Fig. 3.2.16 shows data for excitation of the H 3’P state. The theoretical curves (CCC, CCO, and DWB2) look in almost all respects very similar to the results just discussed for the H 22P state, and the conclusions therefore remain the same. The similarities in shape for y1= 2,3 encourage comparison of the alignment and orientation parameters within the Rydberg series. This is done in Fig. 3.2.17, which shows results for it = 2,3,4, as obtained in the CCC (a) and the DWB2 (b) methods. As was the case for the corresponding series
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295
K(4’P) 10 eV
0.5
0.0
IO 0.5
0.0
(h)
O(d-4 Fig. 3.2.13. continued
in lithium, we notice very similar curves for the various n values, suggesting an approximate scaling law, or n-independence, for the parameters. 3.2.7. Excitation of the Cr 4’P state The Cr 47S -+ 47P transition is an example of excitation in a multi-electron system with high spin, S = 3. Thus, in analogy with the case of Na 32P, electron impact excitation may lead to sextet and octet scattering. The number of independent scattering amplitudes is four and a
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K(4*P) 20 eV
0.5
.._,._. ...,. I: P
0
T
U 0.90
’ 0
’
’ ’ 90
I ’ 180 0
(cl
I
I 90
I
I
I
1
180 0
I 90
I
V I 180 0
I
I
I 90
I
I
-1 180
@(%I Fig. 3.2.13. continued
parametrization in a set like Eq. (3.2.8) applies. Only one experiment has been performed for this system [53 3. The NIST group determined the parameters (r, L”,, L”,) (indices “0” for “octet” and “s” for “sextet”) at 6.8 and 13.6 eV. Similarly to Eq. (3.2.6), the r-parameter is related to octet and sextet weights through r = @/cY = w”/ws = 6w”/(l-
8~“) = (1- 6w”)/8w” ,
(3.2.22)
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0.25
0
5
10
15
20
0.40
1 -
7 eV
W
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0.40
-’ 10 eV
W
0.35
0.30
0.25
0.20
(,,_,,,___,_ 0
..I 5 Scattering
10
. . . . . j 15
Angle
20
0.20~,_.._,,,..,__..,_.__, 0
4 5
(deg)
Scattering
10 Angle
15
20
(deg)
Fig. 3.2.14. The Bederson W-parameter for excitation of the K 4 ‘P state by spin-polarized electrons at incident energies ) 17CC (for 3 and 5 eV) and CCC of 3,5, 7. and 10 eV. Experimental data are from [47]. Theoretical curves are: (-~ (for 7 and 10 eV) [21]; (- - - - - ) 2CC [SS]; (- -) 3CC [77]; (-- .) Ochkur approximation [47]; (. ) “distorted wave polarized orbital” calculation [64] for 5 eV and DWB2 [73] for 1OeV. Also shown for 1OeV are CC0 results (- - - - -). The theoretical curves have been convoluted with a two-dimensional Gaussian of half-width 5” to simulate the experimental angular distribution. The effect of this averaging is demonstrated in the figure for 7 eV where the dashed line is the CCC result without convolution.
Table 3.2.2 Alignment and orientation
parameters
for excitation
Symbol
Source
0 q
cw Cl101 Cl111 Cl081
0 a
of the H 2’P state by spin-polarized
electrons
Fig. 3.2.15
Comment
a. c
;‘.
c351 C671
b b b a-c a c
c271 c951 Cl061
a, b a, b b, c
P, fl’, ;‘>P, L,, P ;‘, P, CCC DWB2 cc0 IERM CCPS
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Reports 279 (1997) 251-396
H(2*P) 35 eV
(a)
o(k)
Fig. 3.2.15. Survey of alignment and orientation parameters for excitation of the H 2 ‘P state by spin-polarized electrons. See Table 3.2.2 for sources and symbols.
where we have used that 8w”+6w”=l.
(3.2.23)
The experimental results, together with the spin-averaged angular momentum LI, are shown in Fig. 3.2.18. We notice substantial differences between L”, and L; , as well as a significant angular dependence of SW” at the lower energy. Due to lack of data for (Pl,y) from an experiment with
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Reports 279 (1997) 251-396
H(2’P) 54.4 eV
b4
.
3
_,
o,oI
I
0 (b)
I
I
90
I
I
I
180 0
c
!
._
.-
I
I
B
br !
,-!
I
I
!
!
!
I
I
I,
!
.I
!
!
!
!
!
I
I
I
I
I
\
U /
3
I
90
I
180 0
90
I
180 0
90
J_, 180
~(k4 Fig. 3.2.15. continued
unpolarized electrons, and the spin contraction parameter T, application of the inversion procedure to retrieve the remaining phase angles in analogy to the Na case is infeasible. Recently, Bartschat [ 153 performed two-state nonrelativistic R-matrix (close-coupling) calculations, with just the two atomic states of interest, (3d54s)‘S, (3d54p)‘P, included in the closecoupling expansion. From a theoretical point of view, an interesting feature of the chromium atom concerns the question to what extent the 3d-shell remains “inert” in the scattering process. If this shell is essentially unaffected by the collision, similarities to electron scattering from alkali atoms,
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H(2’P) 100 eV 100
Fe----x u
i, 1. \
‘\ .\
‘..._ ..’ .....
i..‘\,
I
0.8 0 Cc)
.. ..
-0
0.9 -
C I 90
I
P
( 180 0
I
I
I 90
I
U
1 180 0
I
1
I 90
I
T
I 180 0
I
1
I 90
V 1 I
-1 180
d(deg) Fig. 3.2.1.5. continued
where one has singlet and triplet channels for the valence plus the continuum electron, can be expected. In this case, it should be possible to use a simple model potential for the core, including the half-filled 3d-shell. The results of two models, an “all-electron calculation” and a “quasi-two-electron calculation” are also shown in Fig. 3.2.18. In the former model, the 24 electrons of chromium and the continuum electron were treated explicitly, whereas in the latter model, the nucleus and the inner 23 electrons of the Cr target were represented by a core potential. Only the outer target electron and the continuum electron interacted with each other and this inert core.
N. Andersen et al. /Physics
0.8
I 0
(a)
I
I 90
I
P
I 180 0
I
1 1 I 90
U
301
Reports 279 (1997) 251-396
I
I
180 0
1 I 90
I
T
I 180 0
I
I
I 90
,
V
I
-1 180
@(deg)
Fig. 3.2.16 Survey of alignment and orientation parameters for excitation of the H 32P state by spin-polarized electrons: CCC [35]; (. . . . .) DWB2 [67]. (0) Cl 121;‘( ---)
Beginning with the angular momentum transfers, we see that the results are, in most cases, in agreement with the experimental data of the NIST group C-531,although quantitative agreement is only achieved for small scattering angles. This indicates that two-state close-coupling is not sufficient at these rather high collision energies. (The ionization potential of Cr is 6.763 eV.) We note, however, that the positive values of the angular momentum transfers at small scattering angles are in agreement with general propensity rules, as discussed in Vol. I. qualitative
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H(3”P) 54.4 eV 10
L
’
’
’
’
’
1
I
I
T
I
I
I
I
I
I
,
I
I
I
1
I 1
1.0
u’
0 :: : ..:
0.8
I 0
(b)
I
, 90
I
P
I 18010
1
I
1 90
I
U
T I 18010
I
,
, 90
I
V I 18010
I
I
I 90
I
I
-1 180
*(deg)
Fig. 3.2.16. continued
Similarly, qualitative agreement between theory and experiment can be seen for the octet weight w“. As a general trend, we note that 8w” should approach 4 for high energies, since exchange effects become negligible and, therefore, there is less and less difference between scattering in the sextet and the octet channels. For a given energy, this effect will first occur near forward scattering angles. This overall expectation is, indeed, qualitatively supported by experiment and theory at both 6.8 and 13.6 eV total collision energy.
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N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
I
10
I
I
I
I
,
I
H(32P) 100 eV , 1 I 1
I
u
I
rl’
I
I
,
I
I
I
I
1.0
uJ
LKU
YU
0
0
-
r
t
+
+
PY
0.0
I
1n
,,,I
-Yr--J.FYY-
I.”
y
_,,.
. .
: :
I,,,,
...
+
SA
SP III,,
III
-
I I
I I,
I
I
I
B, I
.. . ..
__
1 1 1
,I
:. ,:
0.9
P I I
II,
0.8 0 (cl
0
-
90
180 0
I
I,,, 90
U
1
lI
180 0
,
90
I
I
180 0
I
I I 90
V I I
__1 180
Wed Fig. 3.2.16. continued
The differences between the results of the core potential and the all-electron calculations at large scattering angles, however, confirm the importance of core effects for small classical impact parameters in this collision system. Consequently, model potential approaches are not expected to yield reliable results for this problem over the entire angular range. For the energies investigated
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H(n*P) 54.4 eV 1
10-5 r
0
-1 90
0
-90 10
0.5
0.0 1.00
0.95
0.90 0 (a)
90
18010
90
18010
90
18010
90
180
e(k)
Fig. 3.2.17. Scaling with principal quantum number (n = 2, -; n = 3,. . . ; n = 4,- ~ -) of alignment and orientation parameters for excitation of the H n2P state by spin-polarized electrons at an incident energy of 54.4eV, as calculated in (a) the CCC [35] and (b) the DWB2 [67] approximation.
experimentally, which correspond essentially to one and two times the ionization threshold, the preliminary two-state close-coupling approaches are not sufficiently accurate. This is not surprising, since such energies are well known to be extremely difficult for theoretical investigations, even for such a simple system as atomic hydrogen where the structure part of the problem is known.
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N. Andersen et al. / Physics Reports 279 (1997) 251-396 H (nZP) 54.4 eV I
2.0 180
10-s r
0
,180 -i80
-1 90
0
-90 10
0.5
0.0 IO
0.9
0.8 0
90
18010
90
(b)
18010
90
18010
90
180
@(deg) Fig. 3.2.17. continued
3.3. The incoherent case without conservation Photon polarization analysis
of atomic reJection
We now discuss the more complicated case where parameter. This is the case when explicitly spin-dependent or other relativistic effects, must be taken into account or simply in the description of the target states, for
symmetry:
the density matrix has a finite height forces, such as the spin-orbit interaction either in the F-operator for the collision example, by an intermediate coupling
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Cr
(4’P) 1 3.6 eV
6.8 eV 1.0 I
,
o
-1 .o;
-0.5
.
-I”0
I
30
60
90
120
150
10
’
20
30
40
30
40
!
180 1
-0.5
.
I
-108
’ 0
1.0
30 a
60 ..I
90
120
150
180
120
150
180
m
SW0
0.01 0
30
60 Scattering
90
Angle (deg)
0.0;
10
20 Scattering
Angie (deg)
3
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Reports 279 (1997) 251-396
307
scheme. As stated in Vol. I, the density matrix elements no longer describe the electronic charge cloud but rather the excited state (J = 1) distribution. Since this state radiates like a set of classical oscillators completely analogous to a ‘P, state, we shall maintain our previous notation, simply replacing the term charge cloud density by oscillator density. For simplicity, we also maintain the parameter name L, though, strictly speaking, J,_ would be more appropriate. As shown in Vol. I (see also Section 2.1, Eq. (2.1.8)), this fact increases the number of independent parameters and, therefore, requires the measurement from another direction, such as y”, of an additional Stokes parameter, usually denoted P4, to determine the form of the charge cloud for unpolarized projectile and target beams. This, however, is not sufficient for a complete experiment because of the incoherence introduced by the averaging and summation over unresolved spin states. The extension necessary for a complete experiment for the J = 0’ + J = 1” transition will now be discussed. In this section, we only analyze the photon polarization in the exit channel; the electron spin parameters will be discussed in Section 3.4.
3.3.1. Analysis of the J = 0 + J = I case 3.3.1.1. Scattering amplitudes and state multipoles. There are six independent scattering amplitudes for a J = 0 + J = 1 transition, thereby requiring the determination of one absolute differential cross section, five relative magnitudes, and another five relative phases of the scattering amplitudes.
This large number of independent parameters will necessarily lead to considerable complications. Nevertheless, the natural coordinate system will enable us to disentangle the scattering amplitudes and generalize the parametrization of the density matrix for the case of unpolarized beams in a straightforward way. Applying a similar analysis as in [16] for the collision frame, the nonvanishing amplitudes f”(Mf,mf,mi) in the natural frame (see Fig. 3.3.1) for a Ji = 0 + Jf = 1 transition are [4]: (3.3.la) (3.3.lb) (3.3.lc) (3.3.ld) (3.3.le) (3.3.lf) Fig. 3.2.18. Excitation of the 4 ‘S + 4 ‘P transition in electron-chromium scattering with a total energy of 6.8 eV (left column) and 13.6 eV (right column). The three parameters 8w”, L; and L”, are plotted together with the spin averaged angular momentum L,. The experimental data from [53] (obtained via superelastic scattering) are compared with theoretical results from ail electron ( -) and model potential (- - - - -) two-state close-coupling (R-matrix) calculations [ 153.
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Fig. 3.3.1. Schematic diagram of scattering amplitudes in the natural frame for J = 0 + J = 1 transitions impact. Note that A+ + 6’ = A- + 8.
by electron
where we have omitted Jf = 1 and Ji = Mi = 0. Note that (3.3.la-d) represent no-JEipamplitudes that leave the projectile spin unchanged while (3.3.le, f) describe the cases where the electron spin is flipped.
The analysis becomes most systematic by using the reduced density matrix formalism, in connection with an expansion of the density matrix elements in terms of state multipoles (for details, see Appendix B). The state multipoles (tko) in the natural frame can be expressed in terms of these amplitudes as [4] CtEO)U= (1/2$)(a:
+ P: + aZ_+ p”- + pi + @$)= (1/2J5)(O,,.rri* + Cfrip)3
(3.3.2a)
(tlo)” = (1/2&X2,
+ p’+ - cz - /3’-) )
(3.3.2b)
(t?o)U = (1/‘2fi)(a:
+ fi: + aZ- + p” - 2[pi + a:]) = (1/2&)(b”+nrp
(tz2)” = (tz_2),* = ~(a+a_e’(~+-+-)
(61)” = (tt-1)”
= (Gl)” = (tl-1)”
+ /?+P-eicti+-@))
= 0
)
,
- 2orrip) 3 (303.2~) (3.3.2d) (3.3.2e)
(3.3.3a) (3.3.3b) (3.3.3c)
N. Andersen et al. /Physics
(t);2)Pz =
(tf&)$z
- /?+fi_ei(JI*-@)) ,
= (Pz/2)(a+z_ei(4+-4-)
(tl*LL = (G-1&,
= (GJP,
MO)P,
=
NO)P,
=
(WP,
=
-
=
-
(pX/2~)(cl+poei(9+-“o)
= 0-JP, =
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(3.3.3d) (3.3.3e)
= 0,
G)P,
=
+
(t”z-Z)P,
=
(3.3.4a)
0 ,
~+~oei(~+-40)
+
x_floe-i(4m-*o)
+
fl_goe-i(tiL
-40))
,
(3.3.4b)
= - (P,/2JZ)(a+P,e’(~+-~“)
+ /?+ccoei(ti+-bo)
_ [CC_poe-i(6--W + p_ccoe-iW-&d]), GO)P,
=
O?O>P,
(WP,
=
-
=
(iP,/2~)(~+Poei(h+-~“) _
=
(GO)P,
=
W2)P,
=
(3.3.4c)
WdP,
=
0
(3.3.5a)
Wl%~
[x_poe-i(d-
-ILO) _
/3+aoei($+-40)
p_aoe-iW
-&)I)
,
(3.3.5b)
=
(iP,/2~)(a+Poeit~+-“~)
- P+010ei(ti+-60)
+ z-floe -U$--W _ p_Oloe-i~#--&)) ?
(3.3.5c)
where the subscript “u” indicates that a quantity is independent of the projectile spin polarization while the other subscripts denote the dependence on the individual polarization components. Inspection of Eqs. (3.3.2,3) shows that the six real state multipoles with rank Q = 0 contain all information about the magnitudes of the scattering amplitudes; also, these equations are linearly independent. Consequently, if the absolute differential cross section cU = $(t;lo)U is known, the five relative magnitudes can be determined using onZy one projectile polarization, P,, perpendicular to the scattering plane. Note that the “unpolarized” multipoles can be determined by taking suitable combinations of results obtained with polarized electron beams, i.e., there is no need to repeat the experiment with unpolarized beams. A detailed discussion is given in the next section. It is also interesting to note that the real and imaginary parts of (t!jz)U and (t!j2)p: allow for unambiguous determination of the phase differences 6T = $+ - 4_ and d1 T I,$+ - I,!_ (see Fig. 3.3.1), i.e., transversally polarized incident electrons with polarization vector perpendicular to the scattering plane can be used to measure not only all the magnitudes of the scattering amplitudes but also two of the five independent relative phases. For a determination of other
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relative phases, one needs to measure the four state multipoles (til)p,, (L;~)~,, (~7~)~~and (til)P,. In principle, not all relative phases that enter into Eqs. (3.3.4,5) need to be determined, due to the following relationships: (4+ - $0) - (b- - $0) = b+ -6
(3.3.6a)
(ti+ - 40) - (ti- - Al) = ti- -$+.
(3.3.6b)
On the other hand, determination of relative phases in different ways can serve as a valuable consistency check in these very sophisticated experiments. 3.3.1.2. Generalized Stokes parameters. The emitted radiation can be analyzed systematically in terms of the generalized Stokes parameters introduced in [4]. They are defined in such a way that all four possible combinations of photon polarization analyzer and initial electron polarizations enter on an equal footing. We first introduce the following notation: The quantity Z$PX,Y,,(p) is the light intensity transmitted by a linear polarization analyzer oriented at an angle fi for incident electron polarizations in the x”, y” or z” direction, with the light being observed in the direction denoted by ri. Similar definitions are made for the intensities transmitted by circular polarization analyzers. This gives, for example, G(45”) - _
= 4K,
JL(135”) - .
= tlx,
+ ~~&)yiP,l 3 -
(3.3.7a)
V~2&1
and the total intensity for unpolarized incident electrons can be constructed z,y = ~[Zy,,(45”) + ZY,(45”) + Zy,,(135”) + ZY,(135”)] .
(3.3.7b) as (3.3.8)
We define the “generalized Stokes parameters” Qfy measured with a photon detector in the direction n^ and electron spin polarization along the direction A by taking the three other independent linear combinations of the four intensities. The second subscriptj = 1,2,3 refers to the sign combinations + + - -, + - - + and + - + -, while the first subscript i = 1,2,3 refers to the photon polarizer settings (O’,90”), (45”, 135”) and (a-, o+), as for the standard Stokes vector. For example, we get Q;; s
Z’+p,(OO)+ ZY,(o”) - Zy,,(90”) - ZY_,(90”) Z”+p,(OO) + ZY,(O”) + z:pZ(900) + ZYpJ90°) ’ - Gz(O”) - C.(90°) + %Z(900) Z’,,(OO) + ZY,(O”) + Zy,,(90”) + ZYpx(900) ’
QYZ _ ~Y,,V)
l2 Qf; E
Q
2’;
Z’,,(OO) - ZY,(O”) + Z”,pJ90°) - ZY_,(90”)
Zy+p;(OO) + Z’,(O”) + Z”+p,(90°)+ ZYpj90°) ’
Zy,,(45”) + ZY,(45”) - Zy,,(135”) - 1’,(135”) = Zy,,(45”) + ZY,(45”) + Zy,,(135”) + ZY,(135”) ’
(3.3.9a)
(3.3.9b)
(3.3.9c)
(3.3.10a)
N. Andersen et al. J Physics Reports 279 (1997) 251-396
Q;; E
1:pJ45”)
- P&(45”)
Zy,,(45”) + IY,(45”)
- Zy,pJ135”) + IY,(135”) + Iy,,(135”)
+ IY,(135”)
(3.3.10b)
’
yz _ &(45”)
- P-,(45”)
+ &(135”)
- 1’,(135”)
Q23 = Iy,,(45”)
+ 1’,(45”)
+ ly,,(135”)
+ 1KpJ135”) ’
(3.3.1Oc)
z:p;(G-) + ZYpz(a-) - ZY,p&‘) - z’p;(o+)
QE=It&-)
(3.3.1 la)
+ ZYpz(G-)+ zy+p:(rJ+) + Z“pZ(CT+) ’ + zKpz(o+)
z:p.(o-) - Z~‘,(o-) - zQT+)
Qg; =Z’;,(C)+ ZYpr(fT)+ zQJ+)
+ I!++)
311
(3.3.1 lb)
’
Z?pja-) - zYpjc-) + zY,pja+)- Z“pZ(f7+) ZY,p:jF) + ZYpz(C)+ zy+pJJ+)+ Z”pZ(CT+) .
(3.3.1 lc)
Qc3 =
The standard three-component Stokes vector is thus replaced by a 3 x 3 matrix. Note that the first three components (3.3.9a, 3.3.10a, 3.3.1 la) form the usual Stokes vector (Pi, P,, P3), as measured with an unpolarized beam, and that all denominators in Eqs. (3.3.9-11) are equal to the intensity 1: of Eq. (3.3.8). Next, we define relative state multipoles which are normalized to the monopole term (t$o)U. i.e., essentially the absolute differential cross section. For example, (3.3.12a)
(3.3.12b) Inserting the relative state multipoles’ from Eqs. (3.3.2-5) into Eqs. (3.3.9-l 1) for the generalized Stokes parameters and using equations (3.3.7,8), we find for the light intensities in the (z”, y”, x”) directions after unpolarized electron beam impact: C = 1 + (I/$)
(3.3.13a)
(Go)”
(3.3.13b)
1: = 1 - (l/2$)
(Go)ll
- (,/5/2)
1,” = 1 - (l/24)
(f%)”
+ (&I
(f%)”
(3.3.13c)
.
In Eqs. (3.3.13), we have normalized the intensity in such a way that the monopole term simply produces a factor of unity. This is justified since, in practice, only ratios of light intensities will be measured, i.e., the absolute differential cross section is assumed to be determined in a separate experiment that does not require the measurement of absolute light intensities in an electron-photon coincidence setup. Furthermore, we note that corrections due, for example,
’ F’or J = 1, the (r;(u),’
s and (&),
s are identical
to the ( T,,
Jo)’
s vol.
I,
Appendix A
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Reports 279 (1997) 2.5-396
to hyperfine structure effects in the target, can be implemented by multiplying the above state multipoles by standard perturbation coefficients (see e.g. [107] for the present case of interest). The expression of the various generalized Stokes parameters (3.3.9-11) in terms of the relative state multipoles requires some tedious but straightforward algebra. The final results can be compactly expressed in matrix form. With I referring to the total light intensity emitted in the direction of the photon detector for unpolarized incident electrons, we find for the z” direction [4]:
(UQ:;l=[
;$::;: % j.
(3.3.14c)
In the y” direction one gets: (IQi’j”)= C-$j(60>~+~(~~2)J
I
C-~(GO)P,+~(GZ)P,I
I 7
00
0 0
0 0
(3.3.15a)
C- -jg (rZn0 >” + (IQ;)
C(~;Io)p,-~(r~o)pi-~(r;Z)p,]
=
0 i
0
SFM2 >“I
&i> I1
- &
P,
(3.3.15b)
(ii&,
(3.3.15c)
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Reports 279 (1997) 251-396
313
Finally, we obtain for the Y direction:
c-+o%d” x8 (IQ?)=
-~"l 0
(3.3.16b)
0
(3.3.16~)
The left column in all these matrices represents the standard Stokes parameters Pi, P2 and P3 that one obtains with unpolarized incident electron beams. It is, however, important to recall the definition of the angle fi for the linear polarization analyzers, as given in Fig. 3.3.2. For example, it turns out that Qy; = QfT = QfT = - P4 and Q45 = - P5 while Q{; = P6 in the notation of [90]. Finally, we point out that observation directly along the incident beam axis, which could be difficult from an experimental point of view, is not necessary; a choice of (0 = 45, @ = 45”), for example, can be used to replace Eqs. (3.3.16) and obtain the same information. 3.3.1.3. Density matrix parametrization. We now proceed to the parametrization of the density matrix for the classical oscillator density, thereby generalizing and unifying the approaches used in Section 3.2.1 of this review for spin-polarized electron impact excitation of hydrogen-like targets (see Eq. (3.2.4) for excitation of sodium) and in Vol. I, Section 3.3.1, for unpolarized electron beam excitation of heavy atoms. The latter results have been summarized in Section 2.1, Eqs. (2.1.3a) and (2.1.X) of the present review. We recall from Eq. (3.2.4) that the density matrix for impact excitation of sodium with unpolarized beams was decomposed into an incoherent sum of two parts, namely a singlet and a triplet component. The parameters for the individual spin channels, such as L\ and Lp, could then be disentangled by means of polarized electron and atom beams. In Eq. (2.1.3a), the density matrix for target atoms such as Ne, Ar, Kr, Xe or Hg was decomposed in a similar way for excitation with an unpolarized electron beam, namely into a pair of matrices with one having positive reflection symmetry with respect to the scattering plane and the other one
314
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Fig. 3.3.2. Coordinate frame for definition of generalized Stokes parameters. The linear polarizer settings in the directions 6 = xn, y”, z” are shown for polarizer angles /I = O”, 45”, 90”, and 135”, following the notation of Blum [28]. The incident polarized electron beam is characterized by polarizations + P,, +I’,, or *P,, as indicated.
having negative reflection symmetry, respectively. The importance of excitation of states with negative reflection symmetry and the corresponding spin-flip processes is measured by the height parameter h. The most important physical effect that causes a nonvanishing h is the spin-orbit interaction. It can manifest itself both in the description of the atomic target states (the dominant effect in a light system such as Ne) and in the magnetic interaction of the continuum electron with the target nucleus. (i) Electron beam polarization perpendicular to the scattering plane: We begin the discussion of polarized electron beam excitation by first assuming a spin polarization perpendicular to the scattering plane, i.e., along the z” direction of the natural coordinate system. The natural extension of the above-mentioned decomposition to the case of polarized electron beams is again the introduction of a pair of density matrices, one for spin-up electron impact excitation and one for spin-down excitation where “up” and “down” correspond to the initial spin component orientation with respect to the scattering plane. Hence,
(1 - h) ;
pu = cu
[
l+Lf
0
0
0
0
_ p+1 e2iy 0
1-L:
(
1
=
(1 - hf) ;
wb, I
-P:e-2iY
+LfT
0
0
0
_ pl+f e2iYT
0
+h
0
0
0
1 0
0
0
-pP:Te-2iYf 0 1
-
LfT
+ hT
0
0
N. Andersen et al. /Physics
Reports 279 (1997) 251-396
1-t Ll’i
0
_ Pl+lep2i$
0 _ PI+1e2iG
0
0
0
: 0
+ hi
1 -Ll’l
r
\ i‘j. 315
01
0
(3.3.17)
Here we have defined: 2 LfT
=
Lll
2
%+
-
xP
2:
+
ix2
=
_
pT 3 ’
-fit
-
-/I:
fl’ = _ pl
+/I”
PI + ipi = -
pt+le2i+
p;
+
ipj
=
(3.3.18b)
3’
pt+Te2iyT =
=
(3.3.18a)
_
2-j+c(_ei(4--4+)
ax:+ x4
(3.3.18~)
9
2’+~~~‘~~““, +
(3.3.18d)
,a;)
(3.3.18e)
d = fi: + p’ + fig,
(3.3.18f)
ITT= cl”, + x2 +
0” = +
(d + x2 + 3; + p”++ /I’ + p;, = ;
((JT+ 01) ,
(3.3.18g)
x;/c+)
(3.3.18h)
h1 = /3$/d ,
(3.3.18i)
WT= oT/(2cJ” ) 3
(3.3.18j)
= oL/(2a,)
(3.3.18k)
h’ =
d
= 1 - WT.
From these definitions, (1 - h)Lf
it follows that
= wt(l - h’)~I’r + wl(l -
(1 - h)P:e2’;
= wt(l _ ht)pl+te2irf
Ir = w’h’ + wlh’ = (x; + /3;)/(24
hl)Lfl , + W1(l - h1)P:le2i71
,
(3.3.19a) ,
(3.3.19b) (3.3.19c)
P;t=Jc-@j$/cFp,
(3.3.19d)
P:-=Jc-@p=Jc-p,
(3.3.19e)
2;~‘=4_-~+fTC=-~+T,
(3.3.19f)
21/L=$_-$+_t7T=-6+T,
(3.3.198)
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Consequently, the decomposition of the density matrix (3.3.17) is described in terms of the cross section 0” and seven dimensionless independent parameters: ((r& wr, LI’T, Lf 1,IIt, hJ; yt, $) .
(3.3.20)
This set leaves three relative phases still unknown. Information about these phases can be obtained with an electron polarization in the scattering plane, as will be discussed in (ii) and Section 3.4 below. 3.3.1.4. Density matrix parameters from generalized Stokes parameters. We now analyze the relationship between the parameter set (3.3.20) and the generalized Stokes parameters. For an electron beam polarized along the z” axis, the generalized Stokes vector matrix for observation in the + zn direction is given by (cf. Eqs. (3.3.2-5) and (3.3.14a)):
+ j?+p- cos Sl]
cos 8 - p+p_ cos $1
2[cl+a_ sin 8 + fJ+fl_ sin $1
2[cr+a_ sin dt - fi+/L sin $1
- (B: + B” - 2@)3
-[(LY: -or’)
$(U4 + a2 - 2U;)
-v:
- (P: + p” - ml
- 2[cc+a_ cos 8
(IQ;;)
= ;
-[(a:
-a’)
+ (B: - 891
- 2[a+a-
-P’)l
3[(cl”+ + cr2_- 2ai) - (B”++ BE - 2/%)1 &lx”+ + CZ - 2a;)
or, in terms of the density matrix parameters,
I
(3.3.21)
(IQ;;) = wT(l - hT)PI - w’(1 - /+)Pi ;
$[wT(l - 3h’T)- wl(l - 3V)]
WT(1- h’)PJ - w”(1 - /+)Pi ;
$[wT(l - 3hT) - wl(l - 3P)]
wT(l - h’)Pl - wJ(l - /+)Pj ;
$[wT(l - 3hT) - wJ(l - 3h”)]
)
‘i (3.3.22a)
with the (normalized) light intensity (3.3.22b)
I = rZ,= $(l - h) and (1 - h)Pi = ~‘(1 - hT)P’ + wl(l - hl)P/ ;
i = 1,2,3 .
(3.3.22~)
Assuming that the height of the charge cloud for unpolarized electron impact, h, is known from a standard P4 measurement, one may use the sum and the difference of the elements in the first two columns (i.e., six parameters) of Eq. (3.3.22) to obtain wt(l - hT)Pt/(l - h) and wJ(l - hi)PL/(l - h), where Pt = (PI,Pz, Pi) and P1 = (Pi,Pi,Pi),respectively. Since the degree of polarization P+T,L E (Prvl 1= 1, one can extract cf = wt(l - hT)/(l - h) and ci = wl(l - h’)/(l - h) from the sum of the squares of the individual components, and subsequently Lf t, LIi, 1)Tand y1from the Stokes vectors Pt and Pi.Since ct + c1 = 1 theoretically, any
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deviation from this relationship should be remedied by renormalizing all elements of the generalized Stokes matrix by a common factor. The last column determines [ wt(l - 3hT) wl(l - 3h1)]/(1 - h) which, when combined with CTand cl, allows for determination of wT,hT and hl. Thus, knowing h, the seven dimensionless parameters of (3.3.20) can be determined from QiJ. It is important to note that the independent determination of h cannot be replaced by Eq. (3.3.19~); this equation is not independent from the others and, therefore, can only be used as a consistency check. A description of how to extract the parameter set (3.3.20), including hyperfine-structure depolarization effects, is given in Appendix D. The equations for the nonvanishing elements in the first row of the matrices (QG) and (QF) (see Eqs. (3.3.15a) and (3.3.16a)) are (written as a column for convenience):
=-
- [wT(l - 3hT) + wl(l - 3h1)] - [wT(l - hr)P[ + wi(l - hL)Pf]
3 A
i
-
‘\
[w’(l - 3hT) - wl(l - 3h1)] - [wf(l - ht)Pi - wJ(l - hl)P;]
[wT(l + ht)
,
(3.3.23a)
- wL(l + hl)] + [wT(l - hT)PI - wl(l - hJ)P;]
with the normalized light intensity I = I,y = 2
[(l + h) + (1 - h)P,)]
(3.3.23b)
,
and
IQ;; IQ:'2 i IQ:"3 I
1 / - cwT(l - 3hT) + wl(l - 3h1)] + [wf(l - h’)Pi + wl(l - /+)I’;] \ - [w’(l - 3hT) - wl(l - 3h1)] + [wT(l - hl)P; - wl(l - hl)Pi]
=-;
!
[wT(l + hT) - wl-(1 + hl)] - [wt(l - hT)Pl - wl(l - hl)P;]
I
,
(3.3.24a)
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Reports 279 (1997) 2516396
with the normalized light intensity I = 1; = ; [(l + h) - (1 - h)P,)]
(3.3.24b)
.
Except for the P4 measurement with unpolarized electrons, no further information can therefore be obtained from generalized Stokes parameters observed in other directions (such as X”or y”) with electron beam polarization vector perpendicular to the scattering plane. On the other hand, such additional measurements with photon detectors in the scattering plane can provide valuable consistency checks. (ii) Electron beam polarization in the scattering plane: As mentioned above, three more relative independent phases are needed to determine the scattering amplitudes uniquely. In analogy to the sodium case, we define A+++-$+,
(3.3.25a)
A-~~_-$-,
(3.3.25b)
A0 s cjo - t+bo.
(3.3.25~)
Inspection of Fig. 3.3.1, however, shows that only two of these are independent, A + - A - = $ - 61 = 2(yi - yt)
since (3.3.26)
in analogy to Eq. (3.2.7). Therefore, it remains to fix the phase of the spin-flip amplitudes fJ,l relative to the nonflip amplitudes. As will become clear below, a convenient choice for the remaining phase angle is $1 G C#)+- e. . A complete set of independent (0” ; wT,LfT,Lf”,ht,hi;
(3.3.27) parameters is then given by yt,y1,A+,Ao,6t”),
(3.3.28)
i.e., one absolute cross section, five relative sizes and five relative phases. Information about the remaining three phase angles may be sought for in experiments with in-plane spin polarization, a possibility that we will now explore. Inspection of Eqs. (3.3.14b,c) shows that no additional information may be obtained with such a beam polarization if the light is observed along the z” direction. Instead, Eqs. (3.3.15b,c) and (3.3.16b,c) reveal eight nontrivial components, namely IQ?; and IQ$‘i with &n^= xx, xy, yx and yy. Nonvanishing values for the latter generalized Stokes parameters require both a “twist” and a “tilt” of the classical oscillator density with respect to the collision plane. The related twist and tilt of the atomic charge cloud density have been discussed in [90] and [83] and were characterized by two additional rotation angles, 6 and E, respectively. An example of such a “twisted-tilted” charge cloud is shown in Fig. 3.3.3 [83]. We will here omit the parametrization of the density matrices for in-plane polarizations P, and P,, in terms of twist and tilt angles for spin up and spin down with regard to these two directions. Instead, we give the expressions for the nonvanishing generalized Stokes parameter components in terms of the relative magnitudes of the scattering amplitudes and the phase angles defined in
N. Andersen et al. /Physics
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Reports 279 (I 997) 251-396
*n
c
+pY
X”
Fig. 3.3.3. Example of a tilted and twisted charge cloud due to an in-plane spin polarized incident electron beam (from [83]). The figure is for Hg 6 3P, excitation with impact energy 8 eV, initial spin polarization P, = I and scattering angle 0 = 30
Eqs. (3.3.25). To clarify the algebraic structure of the general expressions for these components,
we
(3.3.29a) (3.3.29b) (3.3.29~) (3.3.29d) (3.3.30a) (3.3.30b) (3.3.3Oc) (3.3.30d)
(3.3.31a)
(3.3.31b)
(3.3.31c)
(3.3.31d)
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Inspection of Eqs. (3.3.31) shows that a measurement of the generalized Stokes parameters in the y” and x” directions with in-plane electron beam polarizations PYand P, will provide the four relative magnitudes and the four relative phases angles defined in Eqs. (3.3.30,31). Provided that the analysis in the zn direction with polarization P, described has been done before, all relative magnitudes are already known, and the four A parameters may serve as consistency checks. For example, the parameters LIT and ,!,lL are derived as LfT
=
(4 - &)I(& + 4) >
L;1 = (AZ - A:)/(&
+ A;) .
(3.3.32a) (3.3.32b)
Additional information, however, may be obtained from the o angles. They are seen to determine the relative phase angles within the amplitude triplets (fj r, j? 1, f,i) and (fj i, fi 1, f,), corresponding to final spin up and down, respectively. However, the phase of the two triplets with respect to each other cannot be obtained from the co angles. We notice the relationships 01
-
co3
=
CY =
w2 - 04 = 8=
)
(3.3.33a)
- 2yl + 7c*
(3.3.33b)
-
2yt
+
n
Provided that yTand yLare already known, these relations allow us to extract two additional angles only, e.g. ~5~~ and (4’ + d - ). This is possible, for example, by photon observation in the y” direction only, as can be seen from Eqs. (3.3.31a,c) after eliminating co3 and o4 with (3.3.33). For a perfect experiment, a determination of any of the three angles d +, d - or do will suffice. This, however, cannot be achieved without spin analysis in the exit channel, since the incoherent summation over the unobserved final electron spins destroys the phase information between the amplitude triplets that belong to final electron spins up and down, respectively. This aspect will be further discussed in Section 3.4. 3.3.2. Excitation of the Hg(6s6p)3Pl state Results from an electron-polarized-photon coincidence experiment with a spin-polarized incident electron beam have been reported by Goeke et al. [46] and by Sohn and Hanne [90] for electron impact excitation of the (6~6p)~Pi in mercury. (We keep the notation of a triplet state that is well established in the literature, although the state is more appropriately described in an “intermediate coupling” scheme.) Sohn and Hanne used a transversally polarized electron beam with polarization vector either perpendicular to the scattering plane (i.e., P,” # 0 in the natural frame) or in the scattering plane (i.e., Py” # 0). The photons were observed either in the z” or in the negative y” direction. The latter sign change, however, does not affect the basic information that can be obtained. It can be incorporated in our theory by deriving Eqs. (3.3.15) with polar angles (0 = 90”, @ = 180”). The individual count rates needed for the determination of these parameters have been obtained by Sohn and Hanne. They combined them to give the nonvanishing Stokes parameters PI, . . . , P4 for unpolarized electrons, and they also defined light polarizations where the analyzer setting was switched for fixed electron polarization. Furthermore, they defined asymmetry parameters by
321
N. Andersen et al. / Physics Reports 279 (1997) 251-396 Hg 6’S,, -+ 63P1 1.0
0.0
-1.0 TO
0.0
-1.0
-1.0
7.0
I.0
0.0
0.0
- I .o
-1.0 0
180’ 0
90
90
180’0
180
90
f4deg)
(a)
Hg 6’So + 63P,
-1.oL 0 (b)
’
’
’ 90
’
’
’ lSd0
’
90
1804I
I
’
90
’
’
-1.0 1x0
~(ded
Fig. 3.3.4. Generalized Stokes parameters for electron impact excitation of Hg(6s6p)3 P, at an incident electron energy of 8 eV: (0) measured (PI-P.+) [46] and (P2, PJ, P,, P6) [90]; (0) unpublished data of [90]; (A) derived from measured polarization and asymmetry data of [46,90]; ( x ) average of measured (A) Qy3 and Q;Z3data; (7) prediction based on properties of the generalized Stokes parameters (see text). The experimental results are compared with the predictions from a five-state Breit-Pauli R-matrix calculation based on [SS].
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Hg 6’5, -+ 63P, 1.0
1.0
0.0
0.0
,
1.0
I
I
I
90
0
4
I
I
b
180 0
I
I
90
Cc)
1 180 0
I
1
I
90
I
_-1.0 180
‘4deg)
Hg 65,, + 63P, 1.0
1.0 ++
4::
Q;; 0.0
0.0
180’ 0
90
Cd)
90
180
e(deg)
Fig. 3.3.4. continued
switching the electron polarization for fixed linear or circular polarization analyzer settings, i.e., asymmetries A”” where AA are defined in the same way as for the Qfj!‘.For example: Ay”(o”) ~ 1 I(O”, +P=) - r(o”, -PA P, I(O”, + PZ) + 1(0°, - PZ) ’
(3.3.34a)
AY’(90”)_ I 1(90°, +P=) - 1(90”? -PA P,1(90°, +P,) + 1(90”, -Pz) ’
(3.3.34b)
AY”(45”)_ i 1(45”, +P,) - 1(45”, -PA P,1(45”, +P,) + 1(45”, -Pz) ’
(3.3.34c)
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323
Hg 6’S,, -+ 63P,
180’0
90
0
90
1x0
‘4deg)
(e)
Fig. 3.3.4. continued
Ay”(135”) _ 11(135”> +P,) - J(135”> -PJ Pz 1(135”, +P,) + 1(135”, -Pz) ’ AY’(rT+)
1 I(o+, +Pz) - I@+ -Pz) Pz I(o+, +P,) + I(o+, -Pz) ’ 3
E
-
Ay”(g_)EII(PzI(a-,
+Pz)-lb-, +P,)
+ I@,
-Pz)
(3.3.34d)
(3.3.34e)
(3.3.34f)
-Pz)
and similarly for P, and Py. These parameters contain similar information as the generalized Stokes parameters, though in a nonsymmetric form. We have used data from [46] and [90] to calculate the generalized Stokes parameters for the case of initial electron spin polarization perpendicular to the scattering plane. Experimental results for an incident electron energy of 8 eV are shown in Fig. 3.3.4. Unfortunately, the full set of generalized Stokes parameters for the z” direction could only be obtained for three scattering angles (lo’, 20” and 307, due to some missing raw data for the circular polarization parameters. Although Q’1; = QZ;; = Qy3 according to Eq. (3.3.22), the experimental data for Qy3 and Qys do not agree within the specified error bars, thus indicating some internal inconsistency between them. We have therefore averaged these results and plotted them as x in the field for QZ;3. Further consistency checks may be performed by using data from the y” direction to evaluate, for example,
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the third column of (QT]F)in yet another way. Inspection of Eqs. (3.3.22) and (3.3.23) shows that 1 + PI
2Q"1"2 + 3Q”1”3+ 4 1+p 4Q;;=0.
(3.3.35)
We have solved for Q;‘3 and plotted the results as V in the third column of Fig. 3.3.4(a). Once again, the trend is correct, but the data are not entirely consistent. Finally, using the available experimental data, we can predict experimental results for generalized Stokes parameters that have not been measured to date, such as those that would be observed in the X” direction, i.e., along the incident beam axis. Inspection of Eqs. (3.3.22-24) yields (3.3.36a)
Qf"z =
1
1_+p;;4 Q;; + ’ +p4Qyz , 1 - PrP4
(3.3.3613)
(3.3.36~) where we have used that Q”1;= PI and Qfi = - P4. The predictions are plotted in Fig. 3.3.4(c). The agreement between the experimental results in Figs. 3.3.4(a-c) and the theoretical predictions based on a five-state Breit-Pauli R-matrix calculation [SS] is satisfactory, bearing in mind the complexity of the collision problem, the level of detail in the comparison, and the remaining inconsistencies within the experimental data set. The agreement between experiment and theory is good for the “in-plane” polarization parameters Qi; and Qg; shown in Fig. 3.3.4(d), except for a theoretical underestimation of Q z; at small scattering angles. The extraction of the set (3.3.20) of spin-dependent density matrix parameters (except for the cross section 0,) from the generalized Stokes parameters is significantly complicated by depolarization effects from hyperfine-structure (hfs) interaction in mercury isotopes with nonvanishing nuclear spin. In order to apply the recipe outlined above, we used the measured generalized Stokes parameters, together with known hyperfine depolarization coefficients Gr = 0.887 and Gz = 0.789 for state multipoles with ranks K = 1, 2 Cl071 to first determine the state multipoles. These can then be used to calculate “reduced generalized Stokes parameters” without hfs-depolarization. Since the system of equations derived from (3.3.14a) and (3.3.14b), relating the generalized Stokes parameters to the state multipoles, is overdetermined for the multipoles (r&,)p,, (r&,)p, and (r~z)p, (see Appendix D), the outcome of this procedure is not unique - except for a hypothetical, entirely consistent experimental data set with insignificant error bars. The results shown in Fig. 3.3.5 [117] were therefore obtained as follows. It turns out that the relative state multipoles (r&)“, (rTo)“, (r&,)U, (rzz),, (rlo)p, and (i;2)p, are determined through a unique set of equations given in Appendix D. Furthermore, three different sets of equations, each omitting one of the three generalized Stokes parameters QyZ, Q”1;and Q:;, were used to obtain the then uniquely determined set (rl;o)p,, (r&)p, and (r&)pz. The average of the latter set obtained with the three possibilities was also calculated. This procedure yields four different sets of state multipoles which were used to calculate the reduced generalized Stokes
N. Andersen et al. /Physics
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Reports 279 (1997) 251-396
Hg 6’50 -+ 63PI at 8 eV
-90
IO 0.5
0.5
o.ot/t 0
h’
--#It, I
I
I
90
I
I
180 0
I
90
I
L
Y,
180 0
I
I
90
I
U-v, I
180 0
,\I 90
t I_,,, 180
e(W Fig. 3.3.5. Spin-dependent coherence parameters for electron impact excitation of Hg(6~6p)~P, at an incident electron energy of 8 eV [117]. The experimental data were calculated from the generalized Stokes parameters presented in Fig. 3.3.4 (see text): ( x) prediction based on data of [46] and [90]; ( + ) prediction based on unpublished data of [90]. The experimental data are compared with the results from a five-state BreittPauli R-matrix calculation based on [SS].
parameters and, finally, the set of spin-dependent alignment and orientation parameters (wT,L; t, L;l, hT,hi; i’T,$). F ig. 3.3.5 shows the results obtained with the average values for (r;lo)p,, (r;& and (rb >P,. Th e vertical lines connect the maximum and minimum results obtained with the three individual sets of equations for (r&,)pz, (Y;~)~, and (T&)~~, thus reflecting inconsistencies within the experimental data set. Purely statistical errors have been omitted; inspection of
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Fig. 3.3.4 indicates that they would be of the order of the symbol size and, therefore, mostly insignificant compared to the systematical errors. The two sets of results plotted in Fig. 3.3.5 were calculated from published data of Goeke et al. [46] and of Sohn and Hanne [90] ( x ), and also from unpublished data of Sohn and Hanne ( + ) for scattering angles of lo”, 20” and 30” [117]. As mentioned above, the important difference between these data sets is the fact that the latter set corresponds to a full determination of the generalized Stokes parameters shown in Figs. 3.3.4(a, b), i.e., all parameters that can be determined from the photon radiation pattern with electron polarization perpendicular to the scattering plane. On the other hand, the first set lacks data for QyZ and QZ;. While data for Qys are desirable for a consistency check between the three elements in the third column of the (QTT)matrix, data for QyZ are mandatory for performing the evaluation procedure outlined below Eqs. (3.3.22). In particular, the normalization P+T = PfJ = 1 ensuring full coherence in the individual spin channels, is crucial for the numerical stability of the inversion. Only a complete set of generalized Stokes parameters can be tested for this normalization and, consequently, be renormalized to fulfill this fundamental condition exactly. If any element in the first two columns of the (QfJ) matrix is missing, the normalization condition can be used without verijcation to calculate this element (except for the sign). Obviously, this is a much less satisfactory procedure and leads to significantly larger systematical errors, as seen in Fig. 3.3.5. This is a striking example of the importance of complete data sets. In the above case, the additional data for Q’;z not only provide more information about the spin-dependent orientation, but also stabilize the inversion procedure for all other parameters through built-in internal consistency checks. Interesting features can be seen in Fig. 3.3.5 after transformation of the generalized Stokes parameters into the spin-dependent coherence parameters defined in Eqs. (3.3.18). The differential cross sections for the two spin directions are very similar, with wTz 0.5 at all angles. The coherence parameters, however, are vastly different for the two spin channels. Note, for example, that LTt > 0 while Lll < 0 at small scattering angles! While the positive value of LTT is in agreement with well-established propensity rules (for details, see Vol. I), the negative value of J!,:~ might seem surprising. Note, however, that both Lf’ and ,!,I1 are nonzero for forward scattering, and that LfT(Oo) = - L,+‘(O”) by symmetry requirements. Also, the alignment angles yf and y1 show no similarities, with the directions of the major axes often being perpendicular to each other (once again, yf '(0')= - yf “(0”) # 0). Furthermore, there is a large difference between the height parameters hT and h1 (which measure the relative importance of spin-flips) in this angular range, with h1 assuming a maximum value of 75% near a scattering of 40”. This means that spin-flips are very likely for spin down electrons, but those spin down electrons, whose spin is not flipped, tend to transfer a negative angular momentum to the atom. This situation is illustrated in Fig. 3.3.6 [117]. These effects can only be revealed by using spin-polarized electrons beams. Due to the large spin-flip probability for spin down electrons and the comparable cross sections for both spin orientations, the spin averaged angular momentum transfer L1 is positive at small scattering angles, in agreement with standard expectations. The theoretical (Breit-Pauli R-matrix) prediction of LI1 < 0 is clearly confirmed at 10” scattering angle while the error bars are too large at 20” and 30” to draw definite conclusions. We recommend a detailed experimental investigation of this effect, particularly of the prediction LI1 z - 1 near 40”.
N. Andersen et al. /Physics
T
Reports 279 (1997) 251-396
327
t
0
63
(al
(bl
Fig. 3.3.6. Illustration of angular momentum transfer by spin up (a) and spin down(b) impact excitation of Hg(6s6p)3PI for small scattering angles at an incident electron
electrons, as predicted energy of 8 eV [117].
for electron
Fig. 3.3.7 shows further theoretical results for the spin-dependent coherence parameters for excitation of the Hg(3P1) state, as calculated in the five-state Breit-Pauli R-matrix approximation at 6.5 and 7.5 eV incident energy and in the semi-relativistic first-order DBWA at 15 and 40 eV. The comments made previously on the R-matrix results for 8 eV apply equally well to those at 6.5 and 7.5 eV. In the DWBA results, we also notice significant differences between the predictions for the “spin up” and the “spin down” parameters. However, with increasing energy, both LIT and L,‘l start out with values near zero values in the forward direction and become positive at small scattering angles. Also, the DWBA results predict that LT1 increases faster with increasing scattering angle than LTt - the opposite situation from the (Breit-Pauli R-matrix) results at the lower energies. Given the importance of semi-classical treatments in many areas of atomic collision physics, we recommend further experimental tests of these predictions. Combined with theoretical efforts, such tests may yield new insight into the validity of propensity rules for spin-resolved collisions [ 1171. 3.3.3. Excitation of the Xe 6s,6s’ states Fig. 3.38 displays the results of first-order DWBA calculations for the generalized Stokes parameters after electron impact excitation of the Xe (5~~6s)~Pi state at an incident electron energy of 25 eV. Surveys of the spin-dependent coherence parameters for excitation of the (5~~6s)~Pi and the (5p56s’)‘P1 are presented in Figs. 3.3.9 and 3.3.10. Compared to excitation of the Hg 63P1 state, the differences in the predicted results for “spin up” and “spin down” excitation are much less pronounced for the xenon target, especially at 40 eV incident energy. Furthermore, the results for both J = 1 states become very similar with increasing energy. This is to be expected because of the very strong singlet admixture in an intermediate coupling representation of both states. Since excitation of this singlet component becomes more and more predominant with increasing energy, all the relative parameters essentially reflect the results for excitation of this component.
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The only electron-photon coincidence experiment for impact excitation of xenon with spinpolarized electrons was reported by Uhrig et al. [104], who measured the spin asymmetry given in Eq. (3.3.34a) with an electron polarization P, perpendicular to the scattering plane and photon observation in this direction. The results for an incident electron energy of 25 eV, normalized to an electron polarization of lOO%, are shown in Fig. 3.3.11 and compared with predictions from a first-order semi-relativistic DWBA calculation based on [29]. In light of the difficulty of the
Hg 61So + 63P1 at 6.5 eV
(4
e(deg)
Fig. 3.3.7. Survey of spin-dependent coherence parameters for electron impact excitation of Hg(6s6p)3PI by spin) five-state Breit-Pauli R-matrix calculation at 6.5 and 7.5 eV based on [SS]; (. . . . .) polarized electrons: (__ semi-relativistic first-order DWBA calculation at 15 and 40 eV based on [31].
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Hg 61S0 -+ 63P1 at 7.5 eV
V__
LfI -1
I
1
I
I
L+’ I I
--v
LT’ I
I
90
I
I
I
, !
,
,
w
,
,
-180 180
h/
0
~(de.4
@I
Fig. 3.3.7. continued
experiment and the smallness of the spin-dependent for further studies of this kind. 3.4. The incoherent case without conservation Electron polarization analysis
effect, we do not recommend the xenon target
of atomic reflection symmetry:
In this section, we only analyze the electron spin parameters in the exit channel while the photons are not detected. The initial atomic states are assumed to be unpolarized. The spin
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N. Andersen et al. /Physics
10 1
%: ' 1 1 1 1 u
Reports 279 (1997) 251-396
Hg 6'504 63P1 at 15 eV I ~ I I I ! I "..I, .. ..
0.01 1
0
=........
,.,“...(( ,,.. ... ,..' \. .i I i I I !
-
.,....‘.
.... : I I I .> I.;:I
,
I,,
I
I
,;. .’
:I":!, ;'., ; i ...,. : j' ::,, >Z '.. \ ,..._..,_ ;.. ...._..." ii :: "
:
90
1 I
1.0
\
\
I
I
I
I
I
II
yt ... ‘.., ..’ ‘.,, ..,,,...’..‘..
I ..,,,‘I,,: ...11,‘.;‘.,.I .. . i,,,,: i;
‘I
::"'>i i'.. -:j i: : '.. ..... : I;: ii :,: .,,i
0.5 -'\
1.0
I
‘. ‘::,..’ II ,,I , I II
II
i,,j
h
II
I I I ‘; I I, :, :: : .”..,, :: :: !,, i ,: \; __i /
1 1 II
i
I
I I L’, ,i_ /i f j;: j ii / ji :i iiii
:: /j .\,,.,; :
: i ii i p+t -_ :,
II
0.0 180
I
I
,
;:
P+l c -_
II
hr
I1 II ht
L f
, I
, I
0
-180
, I
180
A+’
q
;
e ,1 /I
I
1 I
(,.." ., ../ .. . . ...'. ._..
: .... i : ; : : ;:
; / - / I ;/ _: ,’ II ,I I, II , I ..i.-.. ; .;.i __ :..: ::
I,...........:i;
v
0.0
I L I
,..’
i..,.._.,..... ‘i
b -90
j I '.. I
\. ......_-
.‘:...
;
1
0.4
br'
..I., 0
I,
L+'
I ‘..,I ‘..,
0.8
-
. ; 2 ;” - .,,,.,.. ......
‘_
... “...._., _,.. .,” \
L+' 1
Lf
1 I 1 I I p ’ “Y. ’ .,_ (, ’
-
‘/,
“‘.L .,,,
1.0
; ”... ;: ...._
“
: .......,.x.,... i
-1
I I ! Wf
- i.. ..(
I.... 0.1
11
I,
0r
,
0 ;.-. . ._ .\ i ).,: , ,..i , -180 I: I I 180
:I I j ,;: *-I’, :
.....L,
,.,.: . .’ :. , , t; : , L 41.1 .,
k..,_
, :)
0
-180
Ao 'i.,J -
180
(.. ...'. x,, 0 !. ..,, ..." \ ,, ,...."..,, ........ .,;.' :;, .., .._.. 7- ; ..' .'... : ...." '. : :..,, i: -,(I :, ...” : '.., \, : ,.: ,:' _/ I '+---....'I ! I I I I I I Ii, I -180 0.0 l..y ' ' ' ' ' 90 90 180 0 180 0 90 180 0 180 0 90 0.5 -
(cl
."_'...
Q(detd Fig. 3.3.7. continued
observables studied are the generalized STU parameters introduced in Section 2.3. Explicit formulas in terms of reduced density matrix elements are given in Appendix B. The study of these parameters provides important, though seldom complete, information about the spin-dependent dynamics of the collision process. For heavy targets, simple relationships such as Eqs. (3.2.11-13) for e-Na scattering are generally not valid. New information can thus be obtained from the generalized ST U parameters.
N. Andersen et al. /Physics
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Reports 279 (1997) 251-396
Hg 6’S,, -+ 63P1 at 40 eV
1
.:
--
‘:._yT
:
-- 1
‘. y’
;
.‘:..., ; :,.., !: :: / ..” ; i : ___ :, : ; j ; : \ ‘\ ;. : j ; : : / i : : j ; .. . i:>,,; ; -:,:, i:, ::.,,,j ; -‘.,,, -90 T:,,i :> I I I I I ..,I1,.:.:I,:.r:. I: II 1I .‘: I., ;‘:: I :ii I i. 1 1 I I , 1.0 :,: i / ::.I ::: : ; .“> ,: : :.... :, i:
O
:j 0.F) -1
(I.0
:
-i.p: ?
1
1.0
.. ::: : :: :: :: ::: :., :.;: ‘::,:i \ /: -:;,:: -‘;::
::
I
II
,I
II
h
:,:
g
,
:_” ./..,,,.. __,..,_,~ .; ,.._ : ; -.I : ,.. // I
/ :j :’ i / : i: ; :i. i,., i // :::: 1: :;:i ;//, :: i:: :,:/ /:: ___::i:., :: :/ :: ./ : ii:: :/ ,i :: i;
p+, e I , 1 1 1 I
I
t
I
I
,
i
1
I
-
I
0 (d)
90
180 0
90
I
f/ ;II
I
II
: A0
..”j
,.
.
;.’
180 0
90
,...
..
-180 180
. ..
0
;’ !I
P180 -
180
:
0
’
‘-180 180
..I
:
180 0
II
... ; ‘: : : .:
:i‘... : “.., ;“..,, ... .... ,:’, ;..y , ‘.,,,./ ..,I ,.,,,,, / I “..,.,i...” I 1
I
II
Am1
0.5 -
o,.
I
“...: ....
h’
h’
At1
--
3
’ ’ 90
6Ydeg)
Fig. 3.3.7. continued
3.4. I. Excitation of the Cs 62P112,312 states We begin with electron impact excitation of the 62Si,2 -+ 62Plj2,3,2 transition in cesium. While neither electron-photon coincidence studies nor the time-reversed superelastic scattering experiment have been carried out to date, there are some theoretical results available for the generalized STU parameters [99, 131. In the present case of e-Cs (62S1,2 --+62P1,2,3,2) excitation, there are 24 independent amplitudes, 8 for the 62Sr,2 + 62Pr,2 transition and 16 for 62S1,2 + 62P3/2. Given one undetermined overall phase for each fine-structure transition, the complete experiment would
332
N. Andersen et al. f Physics Reports 279 (1997) 251-396
Xe 5’S,, + 63P1 at 25 eV
0
90
180’ 0
180’ 0
90
90
180
@(de&d
(a)
Xe 5lS0 + 63P1 at 25 eV 1.0
0.0
-1.0 0
(b)
90
180’ 0
90
180’ 0
90
180
Wed
Fig. 3.3.8. Generalized Stokes parameters for electron impact excitation of Xe (5~~6s)~P, at an incident electron energy of 25 eV, calculated in a semi-relativistic first-order DWBA calculation based on [29].
N. Andersen et al. /Physics
Reports 279 (1997) 251-396
Xe 5’So A3P1
333
at 25 eV
1.0
0.0
I
-1.0 180’ 0
I
,
90
I 90
180’ 0
180
e(deg)
Xe 51Sa -+ 63P1 at 25 eV 1.0
Q:;
QYIS
0.0
-1.0 1.0-
-1.0 -1.0
0.0
0.0
-1.0 0
(4
90
180’ 0
90
-1.0 180
o(de.4
Fig. 3.3.8. continued
require the determination of one absolute cross section for each transition, together with 7 (15) relative magnitudes as well as 7 (15) relative phases for the J = i (p) fine-structure levels. So there are now 46 (!) parameters instead of 7 in the case of e-Na sodium. In light of the complications described for the latter case in Section 3.2.2, it is not surprising that no attempts towards a complete experiment have been made to date, and we do not expect such attempts in the foreseeable future. Nevertheless, the study of the generalized STU parameters reveals interesting information. An example is shown in Figs. 3.4.l(a,b) where the differential cross section CT,and the seven
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N. Andersen et al. /Physics
Reports 279 (1997) Z-396
Xe 5lS0 -+63P1at 25 eV
(e)
o(ded Fig. 3.3.8. continued
independent parameters T,, TX, T,,, Sp, SA, U,,, and U,, are plotted for electron impact excitation of the h2S1,2 + 62P1,2,312 transitions in cesium at incident electron energies of 1.63 and 2.04 eV. These results have been obtained using scattering amplitudes from semi-relativistic five-state [86,87,13] and eight-state [13] Breit-Pauli R-matrix (close-coupling) calculations, as well as a fully relativistic five-state Dirac R-matrix calculation [lOl, 991. There are several interesting aspects to be seen in these figures. First, the results of the three Breit-Pauli calculations are in fair agreement with each other. As discussed in detail by Bartschat [13], the differences between the five-state results of [86] and [13] are mostly due to the use of different target wavefunctions. Second, as shown in [13], the discrepancies between the results of Scott et al. [S6] and those from other theoretical calculations have been removed by improving these target wave functions. Consequently, the five-state Breit-Pauli calculation of [13] is expected to yield the more reliable results. Third, the good agreement between the results from the five-state and the eight-state Breit-Pauli calculations indicate that the close-coupling expansion has sufficiently converged at these low energies. The most interesting observation, however, is the fact that the results from the five-state Dirac calculation is in quite good agreement with the Breit-Pauli results in some cases (for example, S,) while there are severe discrepancies in others. In particular, we mention the spin polarization function Sp. The Dirac results not only differ significantly from those of the Breit-Pauli approach, but also from the predictions of the fine-structure effect (see Section 3.2.1, Eq. (3.2.11a)). Note that the Breit-Pauli results at 2.04 eV collision energy are still in qualitative agreement with these predictions; one finds, for example, Sp,A(2P1,2) z - 2Sp,A(2P3,2). The deviations from the approximate relations at 1.6 eV for the Breit-Pauli calculations are expected to be primarily due to the
335
N. Andersen et al. / Ph.ysics Reports 279 (1997) 251-396
energy difference between the two fine-structure levels. Such an energy effect has even been demonstrated for the sodium target [ll]. Experiments with a cesium target, like the one performed with sodium by Nickich et al. [79], would therefore be most valuable for a deeper understanding of relativistic effects in the collision dynamics, particularly with regard to the approximate validity of the fine-structure effect for heavy targets. We strongly recommend future experimental investigations in this area.
Xe !?SO -+ 63P1 at 15 eV 10
1 0.1
1,
o.o_l_ 1
0
4
-190
0
-90 10
0.5 -F 0.0 0.E
0.05
0.00 0
(4
90
180 10
90
180 10
90
180 lo
90
180
@(detz)
Fig. 3.3.9. Survey of spin-dependent coherence parameters for electron a semi-relativistic first-order DWBA calculation based on [29].
impact excitation
of Xe (5~~6s)~P~, calculated
in
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N. Andersen et al. /Physics
@I
Reports 279 (1997) 251-396
‘4de.s) Fig. 3.3.9. continued
3.4.2. The fine-structure effect in S + P excitation of Na, K, Rb, Cs, Cu and Au Following up on the above discussion of the fine-structure effect in low-energy electron impact excitation of cesium, we now analyze recent results of the Toronto group [115,116], who performed first-order full-relativistic distorted wave calculations for electron impact excitation of the resonance transitions (ns)?Sljz + (np)2PIj2, aI2 in the alkali atoms Na, K, Rb and Cs, as well as for Cu and Au. The results for the fine-structure resolved spin polarization function Sp are shown in Fig. 3.4.2 for impact energies between 20 and 100 eV. As one might have expected, the symmetry
N. Andersen et al. / Physics Reports 279 (1997) 251-396
337
Xe 5’S0 -+ 63P1 at 40 eV
(c)
f?(deg) Fig. 3.3.9. continued
relationshrp Sp(2P112) = - 2Sp(2P312) predicted in the nonrelativistic limit is very well fulfilled for low-energy small-angle scattering from Na and K. However, even in these light target systems, strong deviations from the predictions of the fine-structure effect are seen at larger scattering angles, and the deviations increase with increasing collision energy. Qualitatively, these results can be understood as follows. With increasing energy, the importance of electron exchange (one of the necessary conditions for spin polarization through the finestructure effect, see Section 3.2.1) decreases, and large scattering angles correspond to small classical impact parameters. Consequently, one might expect Mott scattering, which involves
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interaction with the target nucleus, to become more and more important with increasing nuclear charge, collision energy and scattering angle. Since Mott scattering is independent of the angular momentum coupling of the valence electron, one would also expect similar results for both fine-structure levels. This overall trend is, indeed, seen in the results for all the alkali targets. Also note the absolute size of the Mott scattering effect - starting at about 3% for Na and K but increasing to 30% in Rb at 60 eV and almost 50% in Cs at 50 eV. A more detailed energy scan may even reveal higher absolute values of SP for these systems. Such an analysis is currently in preparation [116]. Xe 51S0 + 6’P1 at 15 eV
0.05 -
(a)
Ydeg)
Fig. 3.3.10. Survey of spin-dependent coherence parameters for electron in a semi-relativistic first-order DWBA calculation based on [29].
impact excitation
of Xe (5~~6s ‘)‘P,, calculated
339
N. Andersen et al. /Physics Reports 279 (1997) 251-396 Xe 5’50 -+ 61P1 at 25 eV
0
@I
90
18010
90
180 10
90
180 10
90
180
o(deg) Fig. 3.3.10. continued
Similar trends can be seen in the results for the Cu and the Au target. The most surprising feature is the apparent dominance of Mott scattering over fine-structure effect at 40 eV incident energy for electron impact excitation of Cu. The results for both fine-structure levels are nearly identical, and the absolute value of the spin-polarization function has a maximum approaching 90%(!) near 65” scattering angle for a target with nuclear charge Z = 29. Note that an experimental test of this prediction would be possible in a standard electron polarization measurement. Since the predicted results are nearly identical for both fine-structure levels, no laser-pumping or other means to
340
N. Andersen et al. / Physics Reports 279 (1997) 251-396
0
(cl
90
18010
90
180 10
90
180 10
90
180
fl(deg) Fig. 3.3.10. continued
separate them are required. Therefore, an experimental test of these results seems both feasible and highly desirable. 3.4.3. P + S excitation: In and Tl in In(n = 5) and Tl(n = 6) have been studied experiThe n2h,2, 3/2 -+ (n + 1)2S1,2 excitations mentally by the Miinster group [24,45]. Fig. 3.4.3 shows the spin asymmetry parameter SA measured at several energies from 4.0 to 9.0 eV for the 52P1j2 + 62S1,2 (left column) and
N. Andersen et al. / Physics Reports 279 (1997) 251-396
341
Fig. 3.3.11. Spin asymmetry ~“(0”) of Eq. (3.3.34a) for electron impact excitation of Xe (5~‘6s)~P, at an incident electron energy of 25 eV. The experimental data of Uhrig et al. [ 1041 are compared with results from a semi-relativistic first-order DWBA calculation based on [29]. The two theoretical curves are for an ideal photon polarizer analyzer (solid) and for the actual experimental analyzer power of 0.84 (dashed).
--) 6*Si,2 (right column) In transitions. We note a strong angular dependent variations for 5*p3,2 the i -+ i transition, whereas the curves for the $ + 3 transition exhibit much less structure. The experimental results at 4.0 and 5.0 eV are compared with seven-state semi-relativistic and four-state nonrelativistic Breit-Pauli R-matrix calculations of Bartschat [12]. It is interesting to note that algebraic recoupling of the nonrelativistic results accounts predominantly for the nonvanishing values of SA, i.e., including the spin-orbit interaction explicitly does not change the theoretical predictions dramatically. Also, the agreement between the semi-relativistic (and even the nonrelativistic) results and the experimental data is very satisfactory for the 52P1,2 -+ 6*S1/2 transition. Based on these findings, the large discrepancy between theory and experiment for the -+ 6*Si,2 transition is surprising. Whereas the theoretical results are in qualitative agree5*p3,2 ment with expectations based on the fine-structure effect, the experimental data clearly violate these predictions. Since the original publication [24], the experimental raw data for both transitions have been carefully re-analyzed and found to be consistent with the published results. Hence, we can only state the surprising result that one fine-structure transition seems to be described adequately in a semi-relativistic model while a much more sophisticated treatment is apparently required for the other transition within the same multipiet. A full-relativistic calculation of this problem is, therefore, highly desirable. Fig. 3.4.4 displays a summary of the generalized STU parameters for electron impact excitation of the (5p)*Pr,2,3,2 -+ (6s)*S1,2 transitions in indium for a total collision energy of 5 eV. As pointed out above, the theoretical R-matrix results are in qualitative agreement with the predictions of the fine-structure effect (see Eqs. (3.2.11)) for all parameters. For the corresponding cases in Tl experimental data are only available for the 6*P1!2 -+ 7*S1,* transition. They are shown in Fig. 3.4.5. Again, a strong variation with scattering angle and impact energy is observed. Semi-relativistic Breit-Pauli R-matrix calculations have been carried out [48,14], for which results at 4.0,5.0, and 6.0 eV are shown in the figure. Good, though not perfect,
N. Andersen et al. /Physics
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agreement is seen - once again, even when relativistic effects are only accounted for by recoupling of nonrelativistic scattering amplitudes. To further investigate the role of the fine-structure effect, Fig. 3.4.6 shows the STU parameters corresponding to excitation from the two fine-structure components of the Tl ground state configuration (6~~6p)~Pl,2, 3,2. In addition, the weighted average value defined in Appendix C is plotted. A clear deviation from zero is seen for all four averaged S and U parameters, showing the role of relativistic effects. Also, the fine-structure energy splitting is probably significant, thus contributing further to the breakdown of the pure fine-structure effect.
Cs 6’ PI/~
1.6 eV
Cs
6’P3,2
(a) Fig. 3.4.1. (a) Differential cross section and the seven generalized STU parameters Sp, SA, T,, T,, T,, U,, and U,, for electron impact excitation of cesium 62SI,Z -+ 62Pr12.3,2 at incident electron energies of 1.63 and 2.04 eV: ( -----) eight-state Breit-Pauli calculation [13]; (- ~ -) five-state Breit-Pauli calculation [13]; (. . . .) five-state Breit-Pauli calculation [86]; (- - .) five-state Dirac calculation [lOl, 991.
N. Andersen et al. /Physics
cs62P,,2
Reports 279 (1997) 251-396
2.04 eV
343
Cs 62 P3,2
fYdeg1
(b)
Fig. 3.4.1. continued
3.4.4. Excitation
of the Hg (6~6p)“~P states
We now move on to transitions of the form (ns2)‘So -+ (n~n’p)‘,~P~,~,~. Again, we use the established notation of singlet and triplet states, but keep in mind the more appropriate “intermediate coupling” scheme. The general description also applies to excitation of states with configuration (np5n’s) which will be discussed for electron scattering from inert gases. We first recall that the number of independent scattering amplitudes is given by 2. (2Ji + 1). (25, + 1). For the J = 0 + J = 0, 1,2 transitions, this leaves us with two (for 3P0), six (for each of the ‘s3P1 states) and ten (for 3P2) amplitudes, respectively. Once again, the level of complexity has increased dramatically, and the complete experiment becomes a much harder challenge compared to the nonrelativistic case. The Hg term diagram with the states of interest for this section is shown in Fig. 3.4.7.
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3.4.4.1. Excitation of the Hg (6~6p)~P~,~ states. We begin with excitation of the metastable (6~6p)~P~,~ states of mercury. While these states are well described in M-coupling (i.e., S = 1 and L = 1 are good quantum numbers), a fully relativistic description requires the use of 6p,,, and 6~312 radial orbitals (see, e.g. [97]). The simplest case is the excitation of the (6~6p)~P, state with only two independent amplitudes. A complete experiment can be performed by measuring the individual STU parameters in the way described by Berger and Kessler [26] for elastic scattering. It has not been performed to date, however, due to the small excitation cross section and the experimental problems associated with the isolation of the signal from the individual multiplet level with J = 0. It is important to note that the generalized STU parameters are not independent of each other for this particular case. It was shown in Eqs. (2.26) of [lo] that the following relations hold for a J = 0 + 0 transition: (3.4.la) (3.4.lb) (3.4.lc)
0.2
'. I "
Na 22.1
SP
-o.20""""' 30 0.1
I..
I "
I "
I
0.2
1 "-
I
1
eV
60
90
.“‘.I
120
150
‘I
180
.0.2~,""'..""""' 30 60 0.1
..1..1..1"1"1"
‘.
1.
I ‘.
90
I.
No 54.4eV
-O” (4
0
30
60
90
120
150
180
-"'O
30
60
90
I I Na 27.2 eV
120
150
180
I ” I No 100 eV
120
150
1 0
ScotteringAngle(deg)
Fig. 3.4.2. Spin polarization function Sp for the resonance transitions (ns)‘SIiz + (np)‘Pliz (solid) and (ns)‘S1,, -+ (np)zP3,2 (dashed) in Na (n = 3), K (n = 4), Rb (n = S), Cs (n = 6), Cu (n = 4) and Au (n = 6), as calculated in a full-relativistic first-order DWBA approach [116].
N. Andersen et al. /Physics 0.2
I..
I '. I
Reports 279 (1997) 251-396
345
'I "
K20eV
1
K 35 eV
-0.1 -
SP -O.*o 0.1
30
60
90
120
..1.'1....'1
150
180
.I..
-O.'o 0.1
30 '.I'
60
90
30
60
K 100
90
120
(b)
150
180
Scattering
-"'O
30
150
1
I"I"I'.I"
K 50 eV
-O.’ 0
120
60
90
120
eV
150
180
Angle (deg) 0.2
"
I
I "
I "
I "
'.
Rb 40eV 0.1 -
0.1
"I
'I.'I.
I
Rb 100
90
120
150
180
Scattering
-"'O
30
Angle (deg)
Fig. 3.4.2. continued
60
90
120
.I'
eV
150
180
346
N. Andersen et al. /Physics 0.2
I.. ;
I. .I..
! ,.,.
Reports 279 (1997) 251-396 -
0.2
I..
I.
I
I
..,
Cs 20eV
Cs 35 eV
0.1 -
SP
-O.Zo"""" 0.6
30
..I.
60
I.' P" 90 120
,..n
* .' 150 180
.,..I..
-0.20 0.4
30 "
I..
60 I,,
90
120
I.
.,
150
160
..,
Cs 100eV 0.2 -
-"'60
30
60
90
120
150
160
-"'40
30
60
90
120
150
180
120
150
180
ScotteringAngle(deg)
Cd)
-0.5 -
-1.0
0
..'.~~"'..'.~~'~ 30 60 90
: Cu 60 eV
Cu 100 eV
0.1 -
-0.1 -
-"'O
(4
1..
8.. 30
I..
60
I
90
.
I
120
.
6
150
I 160
-o~o"""“""""' 30 60
ScotteringAngle(deg)
Fig. 3.4.2. continued
90
120
150
180
N. Andersen et al. /Physics
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Reports 279 (1997) 251-396
0.3
I
I
I
*
Au 50eV
Au 30eV
Scattering
:
Angle (deg)
Fig. 3.4.2. continued
Uy.y=
7lf
’
7li
’
U,y
)
(3.4.ld)
where ni and rrf are the parities of the initial and final states, respectively. This leads to Sp = - SA, T, = - 1, T, = - TX and U,, = - U,, for our case of interest. As mentioned above, the situation is much more complicated for excitation of the (6~6p)~P~ state with ten independent amplitudes and almost no chance for a complete experiment. The study of the generalized STU parameters will, nevertheless, provide some information about the importance of relativistic effects on the outcome of the collision process. This can be seen by investigating the results predicted in the approximation of the pure fine-structure effect. One finds [49, 50, 311: s P=
-
S‘4
3
(3.4.2a)
T, z - 0.4 ,
(3.4.2b)
TX + T, z - 0.6 ,
(3.4.2~)
u 4’1% - UX,
(3.4.2d)
Furthermore, related by
the results for the S- and the U-parameters for the two 3Po,Z fine-structure levels are
SP,‘4(3Po) * - 2SP,A(3P2) >
(3.4.3a)
UqQJ3Po) = - 2U.X,,,J3P2) .
(3.4.3b)
The validity of the approximate relationship (3.4.3a) can be assessed in Fig. 3.4.8 where recent experimental data of the Miinster group [38] for the asymmetry function SA are compared for the two fine-structure levels. The agreement with the prediction of the fine-structure effect improves with increasing collision energy since the energy difference of 0.77 eV is relatively less important at 15 eV than, for example, at 6.5 eV incident energy. In fact, one might expect the approximate relationships (3.4.3) to be better fulfilled by comparing the results for the 3P0 state at 6.5 eV with
348
N. Andersen et al. / Physics Reports 279 (1997) 251-396
I
‘..,
7L
I
0
++++-I -1
6.0
e\,I
1
6.0 eV
0
I
0
3
9.0eV
-1 1
@@I@%+
-1
L---u-0
90
1:SO 0
90
O
-1 180
Fig. 3.4.3. Angular dependence of the asymmetry parameter SA for inelastic scattering for the In 52P1,2 + column) and In 5’P,,, + 62S1,2 (right) transitions at 4.0,5.0,6.0, and 9.0 eV: (0) experimental data from [24]; semi-relativistic calculation including the spin-orbit interaction within the data based on [12]: ( -) between the projectile electron and the target nucleus through the operator (~‘2/2) Ci(l/r”)li.si; (relativistic calculation with the spin-orbit operator (~‘12) xi (l/rJ [dl’(rJ/dri] li ‘Si; (. . . .) nonrelativistic with subsequent recoupling of the scattering amplitudes.
62S1,2 (left theoretical target and - -) semicalculation
those for the 3Pz state at 7.5 eV, since the percentage deviations of the initial and final projectile energies are smaller than they would be in comparing the results at the same total collision energy of either 6.5 or 7.5 eV. Inspection of Fig. 3.4.8 indeed supports the above argument, thus indicating that not only explicitly spin-dependent effects during the collision are responsible for violations of the
N. Andersen et al. J Physics Reports 279 (1997) 251-396
349
Fig. 3.4.4. Generalized STU parameters for electron impact excitation of the transitions (5~)‘Pi,~,~,~ --t (~s)‘S,;~ in indium for a total collision energy of 5 eV. The individual theoretical curves based on [ 121 are: (__ ) semi-relativistic calculation including the spin-orbit interaction within the target and between the projectile electron and the target nulceus through the operator (a’Z/2) xi (l/r?)Zi.si; (- -- -) semi-relativistic calculation with the spin-orbit operator (a2/2) C,(l/ri) [dl’(ri)/dri] li.si; (. . .) nonrelativistic calculation with subsequent recoupling of the scattering amplitudes. The experimental data are from [24].
predictions from the fine-structure effect; as mentioned before, such violations can also be caused by the energy dependence of the Y operator [ll]. Theoretical results for these parameters have been obtained in a five-state Breit-Pauli R-matrix (BPRM) calculation (based on work of the Belfast group [SS]), and in semi-relativistic [31] as well as full-relativistic [97] first-order distorted wave approaches (SRDW and FRDW). The agreement between the experimental data and the theoretical predictions is not very satisfactory, indicating the need for improved theoretical models for this complicated collision process. Note, however, that the theoretical data, by themselves, also support the validity of the approximate relationship (3.4.3a). Fig. 3.4.9 shows the differential cross section crUand the generalized ST U parameters for the Hg 63P0 state (left) and 63P2 state (right) at 8, 15 and 40 eV. Experimental points are from the Miinster
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N. Andersen et al. /Physics
-1
I
1
I
1
I
Reports 279 (1997) 2.5-396
I
1
1
I
I
I
1
I
I
1
-1 1
11.0eV
5.0eV
d(detd Fig. 3.4.5. Angular dependence of the asymmetry parameter SA for inelastic scattering for the Tl 62Pr,2 + 72S,,z ) semi-relativistic transition at 4.0, 5.0, 6.0, 9.0, 11.0, and 14.0 eV (from [45]). The theoretical curves are: (calculation including the spin-orbit interaction within the target and between the projectile electron and the target nucleus through the operator (c~‘Z/2) xi (l/r?)li.si (from [48]); (-.-.) semi-relativistic calculation including the spin-orbit interaction within the target and between the projectile electron and the target nucleus through the operator (a2Z,rr/2) Ci(l/r!)li.si (based on [14]);(- - -) semi-relativistic calculation with the spin-orbit operator (g2/2) Ci(l/ri) x [dI/(ri)/dri] li.si (based on [14]); (. . . . .) nonrelativistic calculation with subsequent recoupling of the scattering amplitudes (based on [ 141). The experimental data are from [45].
group [38,23,69], and the theoretical curves are again the Breit-Pauli R-matrix calculations from [SS], and semi-relativistic [31] and fully relativistic distorted wave calculations [97]. In Figs. 3.4.9(b,c) we note a good overall agreement between the FRDW and SRDW predictions. At 8 eV, there is also fair agreement between the SRDW and BPRM theories, except for the
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
-1.0
I
0
I
I
I
I
I
I
1800 &ded
90
I
I
I
90
I
I
351
l-1.0 180
Fig. 3.4.6. Angular dependence of the generalized STU parameters for electron impact excitation of the transitions T16*P riZ + 7’SIjZ (~ ~ -) and Tl 62P3,2 --* 72S1,2 (. .) at a total collision energy of 5 eV. Also plotted are the sum of the cross sections and the weighted averages ( -----) of the STff parameters, as defined in Appendix C (from [48]). These averages vanish in the approximation of the pure fine-structure effect.
differential cross sections and the T_ parameters for the 3P0 state. In addition to the SA parameter discussed above, experimental results for Sp and T, are available at 40 eV for the 3P2 state. As for the SA parameter at this energy, the agreement with the FRDW calculation is very good. 3.4.4.2.
Excitation
six amplitudes
As discussed above, this case requires in general For the ‘PI state, which we will consider first, an
of the Hg (6~6p)‘*~P, states.
for a complete description.
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Reports 279 (1997) 251-396
v
Hg I
6'S.
eV
I
Fig. 3.4.7. Energy level scheme for the HgI 6lS ground state and the excited 6is3P levels and optical transitions studied in alignment and orientation experiments with polarized electron impact excitation.
6%
Fig. 3.4.8. Angular dependence of the spin asymmetry parameter SAfor excitation of the Hg 63P, (left column) and 63P, (right column) state at collision energies of 6.5, 7.5, 8.0 and 15.0 eV. Experimental points (0) are from [38], theoretical curves from (__ ) [SSJ, (- - - - -) [31] and (. . . .) [97].
IV. Andersen et al. /Physics
approximate SP
=
Reports 279 (1997) 251-396
353
symmetry pointed out in [17] causes the following equations to be nearly valid: SA
(3.4.4a)
2
T z z 1.0 )
(3.4.4b)
Tx =
(3.4.4c)
u YX
Ty ,
z
u,,
(3.4.4d)
.
Thus, effectively, one absolute differential cross section and six dimensionless parameters need to be determined. While the three independent STU parameters may be derived from electron spin
a eV
Hn 63P,
0.0 ‘\ __ ‘\
-1.0
1 I
I
I
c,
-a L
- ,’
-.__
I
‘\
I II tp_,’ 1 I I
,
U
\
I
Hg 63P2
II
1 , II 1 1
,I
TX
I II 1
rI 1 I ,I UYT --__-
1.
-1.0 1.0
. 0.0
-1.0 1.0
-1.0 180’0
90
180’0 (‘Ydeg)
90
180’0
90
180
(a) Fig. 3.4.9. Differential cross section and STU parameters for the Hg 63P,, state (left side) and 63P2 state (right side) at 8, 15 and 40 eV. Experimental points are from [23] (A) and [38] (Cl) for SA, [72] (Cl) for Sp and T,, and [63] (0) for T,, T,, U,, and U,,; theoretical curves are from [SS] ( -), [31] (- - - - -) and [97] (. . . . .).
354
N. Andersen et al. /Physics
Reports 279 (1997) 251-396 15 eV
Hg 63P0
A i: ::
::
,..“.,
,J_\ ,
0.0
I’ /
-1.0
/
0
(b)
,
5.. \.,
:i
‘<,
.;
L.-
i
I
90
,
:
\
:
‘,
-
I.:
180 0
I
,
,
90
:
I
:.,
,: i
,
\;’ ,
I
180 0
I
I
90
I
I
o,.
“.‘N-_L’
’
180 0
‘...
:’ r-;‘j.
___*.%.T)
+i i ; ,’ e: ; I _______..‘-~+ ‘%,__ r: ‘i 1; _I ;
\ : I
‘..
:I
i
!,._/: : ~‘~....,,.,. “,, ;
; ., Z’.
,;
*‘,\
:
Hg fj3P2
’
’
90
’
’
-1.0
180
f-%ded Fig. 3.4.9. continued
analysis, the remaining three relative parameters could be deduced from polarized photonscattered particle coincidence experiments. No such experiments have been performed at the time of writing, due to the somewhat inconvenient (VUV) photon wavelength, cf. Fig. 3.4.7. However, this approach should be no more difficult for Hg than for resonance excitation of the heavy rare gases which have been extensively studied in recent years, so future progress in this direction is a possibility. The pioneering spin polarization measurements by Eitel and Kessler [42] showed a striking similarity between the results for elastic and inelastic scattering in the structures of the differential cross sections as well as for the spin polarization of the scattered electrons. This is particularly evident when the incident electron energy becomes considerably larger than the excitation energy. Fig. 3.4.10 shows a key graph from the paper, illustrating this fact at 50 and 80 eV impact energy. The results were interpreted in terms of a two-step model, first proposed by Mohr and Nicoll[76]
N. Andersen et al. /Physics
Hg 6”P2
40 eV
Hg 63P0
355
Reports 279 (1997) 251-396
0.02
1.0
0.01
0.0
0.00
1.0
-1.0 I.0
0.0
0.0
-1.0 I.0
-1.0 -i.n
0.0
0.0
-1.0 1.0
-1.0 -LO
0.0
0.0
-1.0 0 (cl
90
180’ 0
90
180’ 0
90
180’ 0
90
-1.0 180
@(deg) Fig. 3.4.9. continued
for large-angle inelastic scattering. In this model, the excitation takes place in the first step, giving rise to only small-angle scattering; in the second step, the electron, now having a slightly smaller energy, is elastically scattered at a large angle by the atom. The two steps may also take place in the opposite order. For the 3P1 state, no simplifications such as Eqs. (3.4.4) can be made, and all seven generalized ST U parameters are independent. Here, we restrict ourselves to an overview of the parameters, for which theoretical results exist in the same BPRM, SRDW and FRWD approximations just discussed for the (6~6p)~P0,2 cases. Experimental results for the SA parameter have been obtained [23], as well as Sp, T,,, T, and U,,, data for ‘PI excitation at 40 eV. Fig. 3.4.11 shows the differential cross section 0” and the STU parameters at 8, 15, and 40 eV. We first notice that, theoretically, the approximations (3.4.4) are excellently fulfilled for the (6s6p)‘Pr state. Secondly, except for ‘PI at
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N. Andersen et al. /Physics
Reports 279 (1997) 251-396
30
90 8 (de@
Fig. 3.4.10. Differential cross section (~ ) and spin polarization (0) for electrons scattered elastically (upper row) and inelastically (6lPr excitation) from Hg atoms at 50 eV (left column) and 180 eV (right column). Note the similarities between the elastic and inelastic results for fixed energy (adapted from [42]).
40eV, considerable
discrepancies exist between theory and the (sparse) experimental results, indicating that more work is needed on both the experimental and theoretical side. Fig. 3.4.12 shows a measurement [54] of the spin parameter T,for the Hg 63P0,1,2 states at 15 eV. Due to problems with energy resolution of the close-lying 3P0, 1,2 levels, the results are a weighted sum of the Tz's for the three levels, i.e., CC) = CL(O) + rJ,(l)
+ @‘G(2) .
(3.4.5)
Here rj = w(J)/CJ w(J) wh ere w(J) is the product of the excitation cross section, G,(J), and the fraction of electrons exciting the fine-structure level 3PJ. Because of the ‘PI admixture in the intermediate coupling description of the 3P1 state, however, this is the predominantly excited fine-structure level within the multiplet. Theoretical predictions for T,(J) and a,(J) are again available from SRDW [30] and FRDW [97]. These can be used to calculate the parameter (Tz) of Eq. (3.4.5) by combining theoretical cross sections with the excitation fractions given in [SS]. Agreement between theory and experiment for this parameter is fairly good, though systematic deviations are noticed, particularly at smaller scattering angles. These results will be put into a broader context in Section 3.5.2.
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351
3.4.5. Excitation of Ca, Sr, Ba and Cd Recently, full-relativistic first-order distorted wave calculations have been reported by the Toronto group for other quasi two-electron systems such as Ca, Sr, Ba [98] and Cd [96]. Example results for an incident projectile energy of 40 eV are shown in Figs. 3.4.13-17. As one would expect, relativistic effects become more prominent within the alkali-earth sequence Ca (Fig. 3.4.13), Sr (Fig. 3.4.14) and Ba (Fig. 3.4.15) with increasing nuclear charge of the target. Furthermore, these
(4
Fig. and and (. ”
Wed
3.4.11. Differential cross section and STU parameters for the Hg 6 ‘PI (left side) and 6 3P1 (right side) states at 8, 15 40 eV. Experimental points are from [23] (A) and [38] (0) for SA, [72] (Cl) for Sp and T,, [63] (0) for T, and 1U,.Y,
[82] (0) for the differential cross section; theoretical curves are from [85] ( -)), . .I.
[31] (- - - - -) and c971
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N. Andersen et al. /Physics
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15 eV
Hg 6’P,
i'
__
1,’
I
-1.0 0 @I
I
I 90
I
1 180 0
I
I
I 90
I
Hg CI~P,
I
--
I
I
180 0
I 90
I
\
I 180 0
\A’-------
’
’
’ 90
’
’
-1.0 180
@Cdeg) Fig. 3.4.11. continued
effects are more visible in the excitation of the triplet states than they are in singlet-singlet transitions. In a semi-relativistic picture, this is explained as an “intermediate coupling effect”, i.e., an interference between the excitation of the singlet and the triplet parts of the target wave function. Since optically allowed singlet excitation will dominate at high energies, such an interference effect will be relatively large if there is a large triplet and a small singlet coefficient in the target wave function (as in the “triplet” states), and it will almost disappear for the opposite case (as in the “singlet” states). The results for the cadmium target are somewhat similar to the predictions for mercury, as one might expect from the fact that Cd is one row above mercury in the periodic system. Hence, the predictions of the fine-structure effect are very well reflected in the results for excitation of the (5~5p)~P~,~ states (Fig. 3.4.16). Once again, the “intermediate coupling interference effect”
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
0
(cl
90
180’ 0
Hg 63P,
40 eV
Hg 6’P,
90
180’ 0
359
90
f?deg) Fig. 3.4.11. continued
discussed above is more significant for the (5~5p)~P~ than for the (5s5p)‘P1 state, as can be seen in Fig. 3.4.17. 3.4.6. Excitation of heavy rare gases: Xe 6s,6s’ states Figs. 3.4.18 and 3.4.19 show results for the differential cross section and the generalized STU parameters for impact excitation of the 63P0,2 and the 61y3P1 states in xenon at incident electron energies of 15, 25 and 40 eV. The theoretical curves were obtained in semi-relativistic [31] and full-relativistic [ 1141 (40 eV only) first-order DWBA calculations; for the asymmetry function SA, they are compared with experimental data of the Miinster group [40]. In general, the two theoretical predictions look very similar, in particular for the 63P ,,2 states for which the results agree very well with the symmetry relationships predicted from the fine-structure effect. The agreement with the experimental data, however, is not very satisfactory, indicating that first-order
360
N. Andersen et ai. j Physics Reports 279 (1997) 251-396
Fig. 3.4.12. The spin parameter T, = (T,) for electron impact excitation of the Hg 63P states at 15 eV. Experimental [97] and(- - - - -) [31]. points (a) are from [54], theoretical curves from ( -)
Ca 43P1
Ca 4’P,
1.U
T, % 1
-1.0
’ 0
’
’ 90
UTV
’
’ 180 0
I
I 90
TV
L I 180 0
I
I
I 90
180 0
’ 90
’
-1.0 180
Wed
Fig. 3.4.13. Differential cross section and STU parameters for the Ca 4lP, (left side) and 43P1 (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [98].
361
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
Sr 53P, 1.0
0.0
-1.0 I.0
““i j-+-i++-
0.0
T,ol
-1.0 I.0
0.95
0.90
1
1.00
I ’
, -
c n
c
0.0
TZ ,
I
,
1
s
G
,
I
I
# I
I
, I
I
I I
Uw
I
I
I
I
!
I
I
I
I
I
I
,
I ,
U=‘I
T, A
0.95 -
I ,
n
/\
-1.0 3.0
A_
0.0
T,
’
0.90 0
’
’ 90
’
I ’ lsd o
4 I 90
,
I MO’ 0
1 4 I 90
I
I 180’0
I
I
I
90
I
-1.0 180
@(ded
Fig. 3.4.14. Differential cross section and STU parameters for the Sr 5’P1 (left side) and 53P, (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [98].
theories are not sufficiently accurate for these collision energies. Although the experimental data for the 63P0,2 states vary rapidly with the scattering angle, a tendency towards fulfillment of Eq. (3.4.3a) can be recognized for scattering angles below 90”. However, this tendency is not as clear in these data as it is in the two theoretical predictions. 3.5. The incoherent case without conservation Combining photon and electron polarization
of atomic rejection analysis
symmetry:
In this section we discuss the relationship between the generalized Stokes parameters of Section 3.3 and the generalized STU parameters of Section 3.4. In particular, we analyze how data from one of these sets of results may be used to complement, predict, and check results obtained for the other set. We concentrate on the J = 0 + J = 1 transition where experimental and theoretical data are available.
362
N. Andersen et al. /Physics Ba 6’P1
40 eV
Ba 63P,
A
0.95 -
0.90
Reports 279 (1997) 251-396
’
I
’ ’ 90
0
I 180 0
I
I
I
I
90
I
I 180 0
I
I 90
1
I
’
180 0
’
’
’
’
-1.0 180
90
6Yde.d
Fig. 3.4.15. Differential cross section and STU parameters for the Ba 61P1 (left side) and 6lP, (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [98].
3.5. I. Generalized STU parameters for J = 0 -+ J = 1 excitation In this case, the generalized STU parameters can be expressed in terms of scattering amplitude parameters as follows [4]: SA = (l/2(7”) L-(X”+ + a? + c(t) - (B: + P” + pa1 = <&l)P,, SP
=
(l/20”)
I@:
+
2
-
&3
-
w:
+
P”
-
(3.5.la)
ml = scwcJP, + q/5
T, = (l/20,) [(cl”++ CX?- cl;) + (/3”++ /3’- - ,8;)] = f[l + 2&;,),1
,
,
(3.5.lb) (3.5.lc)
T,, = (l/o”) Re{a+j?+e’(~+-@+) + cI_p-ei(‘#-*~)
- ~O,&ei(~O-“o)),
(3.5.ld)
TX = (l/a,) Re(cc+fi+e’(@+-ti+)
+ cq,floei(@~-@O)} ,
(3.5.le)
U,, = - (l/au) Im{a+lj+e’(@+-$L+)
+ ~-fl-e’(~--+L-) + x_p_eiC~--IL-)
+ aoPoei(*o-bo))
,
u,, = - (l/a”) Im{a+fl+e’(@+-JI+) + a_P_e’(@--@-) - ~OjjOei(@Q-~o))
(3.5.lf) (3.5.lg)
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N. Andersen et al. 1 Physics Reports 279 (1997) 251.. 396
Cd VP0
Cd 53P2
180’0
90
180’0
90
180’0
90
180
Q(ded
Fig. 3.4.16. Differential cross section and STU parameters for the Cd 5 3P0 (left side) and 5 3P2 (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [96].
and in terms of the density matrix parameters as S* = NJ - M’J, sp = d(
(3.5.2a)
1 - 2hT) - w’(1 - 2h”) )
(3.5.2b)
T, = d( 1 - 2hT) + wl(l - 2h9 = 1 - 2h , Ty = (l/d
{a+P+ cosd+
+ a-p-COSK
TX = (l/o”) {r+b+ cos A+ + z-p-
(3.5.2~) - r&)cosLlO)
)
(3.5.2d)
cos A- + aojo cos A’} ,
(3.5.2e)
Cry, = - (l/o”) (x+P+ sin A+ + r-/L
sin A- - cco~o sin A’} ,
(3.5.2f)
U,, = - (l/a,)
sin A- + cxofio sin A’} .
(3.5.2g)
(c(+b+ sin A+ + x-p_
N. Andersen et al. / Physics Reports 279 (1997) 251-396
364
Cd
5’P,
Cd
!i3P,
Fig. 3.4.17. Differential cross section and STU parameters for the Cd 5 ‘PI (left side) and 5 3P1 (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [96].
For brevity, we kept the products of the amplitude magnitudes in the TU parameters. They can also be expressed in terms of density matrix parameters by using aJ*
/CT”=
wTwl(l - hT)(l - U)(l + LfT’)(l * Lfl’)
cto/?o/a” = &&am
.
)
(3.5.3a) (3.5.3b)
Note that the three parameters SA, SP and T, can be predicted directly from coherence parameters extracted in electron-photon coincidence experiments (see Section 3.3.1) and vice versa; the parameter T, = 1 - 2h with an unpolarized electron beam, and SA,P with an electron beam polarized perpendicular to the scattering plane. Thus the three relative sizes hT, h” and wT(= 1 - WI) can be evaluated. The two missing relative sizes LTT and Lf ” may be obtained by circular polarization measurements only. The parameters Ty, TX, U,, and U,, contain bilinear combinations of amplitudes that do not appear in the state multipoles of equations (3.3.2-5). It is therefore not possible to express these
N. Andersen et al. /Physics Reports 279 (1997) 251-396
365
parameters in terms of the light polarizations measured in the electron-photon coincidence setup. Measurements of the STU parameters, therefore, provide complementary data to the information that can be obtained from the coincidence arrangements (see Section 3.5.2). In particular, linear combinations of the form Ty * TX and U,, + U,, allow for determination of the two complex numbers (r+fl+e”‘+ + a_ p_ein- and j&,aOeiAO, thereby providing the crucial phase difference A0 not obtainable with the electron-polarizedPphoton coincidence technique without simultaneous electron spin analysis in the exit channel. Hence, the combination ofthe two techniques described in Sections 3.3 and 3.4 constitutes the ultimate goal of a perfect scattering experiment,for important J = 0 + f = 1 excitation problem in its most general form.
(4
the
f-VW
Fig. 3.4.18. Differential cross section and STU parameters for the Xe 63P0 (left side) and 63P, (right side) states at 15,25 and 40 eV. The theoretical curves were obtained in semi-relativistic (___-- ) [313 and full-relativistic ( .) [ 1141 first-order DWBA calculations, the experimental data are from 1403.
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N. Andersen et al. /Physics
Xe 6”P0
(b)
Reports 279 (1997) 251-396
25 eV
Xe f.i3P2
8(ded Fig. 3.4.18. continued
The considerations above, and the fundamental difference between the information extracted from generalized Stokes and ST U parameters, may be succinctly summarized in the way shown in Fig. 35.1. Here, (a) illustrates that generalized Stokes parameter analysis in the zn direction with electron spin polarization P, perpendicular to the scattering plane determines all relative amplitude magnitudes, and the phase relationship between the f+ 1 and f_ 1 amplitudes - provided P4 is also known. Additional analysis in the y” (or x”) diretion with in-plane spin polarization PYor P, yields the four relative phases within the two triples (ff 1, f! 1, fof) and (j-2 r, fl 1, fd), respectively. None of the relative phases (d +, A’, A -) between f 1 1 and fi 1, etc. enter. On the other hand, Fig. 3.5.1 (b) illustrates that the generalized STU parameters SA, SP and T, depend only on the relative sizes of the amplitudes, while TX, T,,, U,,,, and U,, depend also on the relative phases (A+, A’, A-) within the three amplitude pairs (fl r, f! 1), (fl r, f! r), and (fo, fi). N o m ’ formation on the relative phase angles between these pairs can be extracted from generalized STU parameter analysis.
N. Andersen et al. J Physics Reports 279 (1997) 251-396
Xe 63P0
40 eV
367
Xe 63P2
f4ded Fig. 3.4.18. continued
3.5.2. Example:
The S,, S, and T, parameters for Hg(6s6p)3P,
For the Hg(6~6p)~P 0, I,2 states, fine-structure averaged T, results [SS] were shown for an incident energy of 15 eV in Fig. 3.4.12. Electron-photon coincidence data also exist for this energy [92,90]. Eq. (3.5.1~) thus allows for a consistency check between the two experimental approaches. Hyperfine structure effects modify Eq. (2.1.8) for the height parameter h, and thus the expression for T,(J = 1) to [4] 1 3G, - Pr(4 - G,) + P4(8 + G2 + P1[4 - G,]) T,(l) = 3G 2 PA1 - PI) + (3 + PI) The hyperfine perturbation
coefficient is taken as G2 = 0.789 [107].
(3.5.4)
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Reports 279 (1997) 251-396
The asymmetry and polarization functions, SA and Sp, can also be expressed directly in terms of the experimentally measured generalized Stokes parameters. With the depolarization correction for the hyperfme structure effects included, the corresponding equations become very lengthy. However, Eqs. (3.5.2a, b) can be used instead, with the parameters wt, wl, h? and h1 determined as described in Section 3.3.1.4 below Eq. (3.3.22). As can be seen from Fig. 3.52, the agreement between the experimental data and the theoretical predictions from the Breit-Pauli R-matrix calculation for T,(l) is satisfactory. Regarding SA and Sp, we first note that there are some discrepancies between our conversion using Eqs. (3.5.2a-c) and the indirect measurements reported by Goeke et al. [46], particularly at small scattering angles.
I
-1.0 0 (a)
,
I 90
1
I 180 0
I
1
I 90
I
, 180 0
0
I
I 90
I
0 180 0
1
1
’ 90
’
’
-1.9 180
fl(deg)
Fig. 3.4.19. Differential cross section and STU parameters for the Xe 6lP, (left side) and 63PI (right side) states at 1525 [31] and full-relativistic (. . . . .) [114] and 40 eV. The theoretical curves were obtained in semi-relativistic ( ----) first-order DWBA calculations, the experimental data are from [40].
369
N. Andersen et al. /Physics Reports 279 (1997) 251-396 Xe 63P1
25 eV
Xe 6’P,
1.0
0.0
-1.0 -I.0
0.0
-1.0 I.0
0.0
-1.0 I.0
0.0
0
(b)
90
1sd 0
90
180’0
90
180’0
90
-1.0 1X0
o(h) Fig. 3.4.19. continued
These discrepancies are due to additional experimental correction factors that were included in the analysis presented by Goeke et al., such as detector efficiencies and angular resolutions that go beyond the general analysis presented above. However, the good agreement between the direct measurement for SA of Borgmann et al. [23] and the indirect extraction of the same parameter by Goeke et al. demonstrates the value of consistency checks. Also, the agreement between the data sets extracted by the Miinster group and the Breit-Pauli R-matrix results is satisfactory. Fig. 3.5.3 [6] displays again the spin depolarization parameter T, for Hg(6s2)lS0 -+ (6~6p)~P excitation at 8 eV impact energy, both the fine-structure averaged result TL [SS] and T,(l) as extracted from the electron-photon coincidence data of [92,90]. Note that the large-angle error bars are significantly shorter for the electron-photon coincidence data, and the photon channel improves the energy resolution tremendously. Using T,(O) = - 1 for this case (see Eq. (3.4.lb)), experimental values for TI = (Tz) and T,(l), and known relative excitation fractions [55], it is in
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N. Andersen et al. /Physics
Xe 61P1
(cl
Reports 279 (1997) 251-396
40 eV
Xe 63P1
Q(ded Fig. 3.4.19. continued
principle possible to extract even T,(2) from the experimental data. We have performed this extraction using theoretical cross sections to estimate the rj factors in Eq. (3.4.5). While the results are in agreement with theoretical predictions for T,(2), the experimental error bars are presently too large for a detailed test of the theories. We have also applied the conversion formula (3.5.2~) to recent coincidence data for resonance line excitation in the heavy rare gases [37] to predict the electron spin depolarization in these cases, for which no direct experimental determination presently exists. In the energy range 30-80 eV covered by their data, the predicted value of T, is unity within experimental uncertainty, in agreement with theoretical predictions [29, 1141. At lower collision energies, however, considerable variation in T, with scattering angle is predicted theoretically, and here the coincidence technique may be an attractive alternative approach to a direct determination of T,.
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N. Andersen et al. / Physics Reports 279 (1997) 251-396
STU
Stokes
QYh)
---
SA,SP,TZ .
pxSPY
Tx.y
1 "YCXY
(b)
(a)
Fig. 3.5.1. (a) This diagram shows which relative amplitude sizes and phases can be evaluated from a generalized Stokes parameter analysis in the z” direction with electron polarization P, and from in-plane measurements with polarizations P, and P,,, respectively. (b) This diagram shows which relative amplitude sizes and phases enter into Eqs. (3.5.2) for the generalized STU parameters. Note that SA, Sp and T, only depend on the relative sizes of the amplitudes and can, therefore. be predicted from a generalized Stokes parameter analysis.
Hg 63P, at 8 eV 1.0
0.0
-1.0 0
90
180’ 0
90
180’ 0
90
-1.0 180
@(W
function Sp and contraction parameter T for electron impact Fig. 3.5.2. Asymmetry function S,, spin polarization excitation of Hg(6s6p)3P, at an incident electron energy of 8 eV: (0) direct measurements by Borgmann et al. [23] and Goeke et al. [46]; (0) parameters derived from the electron-photon coincidence parameters by [46]; ( x) and ( + ) parameters derived using Eqs. (3.5.1). The theory curve is from a five-state Breit-Pauli R-matrix calculation based on
P51.
3.5.3. Sublevel-resolved generalized STU parameters A further refinement of experimental technique is the combination of photon polarization analysis with simultaneous determination of STU parameters. Whereas such a direct experiment seems infeasible in the near future, it is possible to use a similar approach as in the previous sections
372
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
-1.011111111111l,1
0
,I
30
60
90
0 (degl
Fig. 3.5.3. The spin depolarization parameters T, for Hg(6sZ)‘S0 + (6~6p)~P excitation at 8 eV impact energy. The experimental points for T, = (T,) are from [SS] (0) and for T, (1) converted from [92] (A) and [90] (0). Theoretical curves are from [30] (- - -) and [97] (-). The upper curves are for T, (1) while the lower curves are for (T,).
to predict some of the results from presently available data. In fact, such results have been reported in [46], with a parameterization for the collision coordinate frame, which we will now extend using the natural coordinate frame. From Eqs. (3.51) it is seen, with obvious notation, that the following equations hold for the states with positive reflection symmetry (MJ = f 1) with respect to the scattering plane W+1 = (1/2D”) Cc& + /%I = (a ‘/2%) ,
(3.5.5a)
s,”
(3.5.5b)
=S,”
=(1/G)
[IX!‘,-@J,
T,‘l = 1 )
(3.5.5c)
TX*1 = Ty* = (l/a * ) 2a+j3+ cos A * )
(3.5.5d)
U&l = U& = -(l/o+)
(3.5.5e)
2cr+p+ sin A * ,
while we get for the state with negative reflection symmetry (MJ = 0) w” = (l/20,) [a; + p;] = (o”/20,) = h ,
(3.5.6a)
s: = - s,” = (l/CO) [ai - /IO’])
(3.5.6b)
7,; = - 1)
(3.5.6~)
T,O = - T$ = (l/Do) 2aoPo cos A0 )
(3.5.6d)
U& = - U$ = - (l/fro) 2ctopo sin do .
(3.5.6e)
From the experimental data of [46] alone, it is not possible to extract the parameters for the three individual magnetic sublevels. However, one can extract the parameters for M = 0 and the average parameters for excitation of sublevels with M = _+ 1 that correspond to positive reflection
373
N. Andersen et ai./Physics Reports 279 (1997) 251.-396
symmetry. We define the latter parameters by W+=~+‘+~-‘=~-~“=l-hh,
(3.5.7a)
(S,)+
= (S,)+
= w+lsAf’ + w-lsi’
(7’,)+
= (T,)+
= w+lT,f’
(U,,)’
= (U,,)’
= w+q?
+ w-‘T,’
(3.5.7b)
,
(3.5.7c)
,
+ w-w;yl
(3.5.7d)
.
Breit--Pauli R-matrix results for the various sublevel-resolved parameters are shown in Fig. 3.5.4 and, where possible, compared with the experimental data of Goeke et al. [46]. The agreement
Hg 63P,
8 lV
1.0
0.5
0.0 1.0
0.0
-1.0 1.0
0.0
-1.0 l.Yi
0.0
-1.0 0
90
l&O
90
180'0
90
180'0
90
180
@(deg)
Fig. 3.5.4. Sublevel-resolved generalized STU parameters for electron impact excitation of Hg(6s6p)3P, at an incident electron energy of 8 eV: (x) and ( + ) parameters derived using Eqs. (3.5.6,7). The theory curve is from a five-state Breit-Pauli R-matrix calculation based on [SS].
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N. Andersen et al. / Physics Reports 279 (1997) 251-396
between theory and experiment is not as good as one might have expected, and also not as good as in the original paper of Goeke et al. We believe that the same reason applies as given above in the discussion of Fig. 3.52, namely that knowledge of more experimental details would slightly modify our conversion formulas, apparently improving the agreement between the experimental data and the R-matrix results. Nevertheless, the overall potential of detailed data analysis is clearly demonstrated. 3.6. Related work: P + P elastic scattering Up to this point, we have only discussed excitation (and the time-reversed de-excitation) of atomic targets by electron impact. In these cases, the combination of data for generalized Stokes and generalized STU parameters allows for a detailed investigation of spin-dependent alignment and orientation effects. One of the important mechanisms for obtaining a spin polarization of an initially unpolarized electron beam through scattering was the fine-structure effect, i.e., the energy-resolved excitation of individual members of a multiplet. It is important to note that the necessary conditions for the effect, outlined in Section 3.2.1, can also apply to elastic scattering from targets with nonvanishing spin and orbital angular momenta, such as (np)‘P,,, and (~P*)~P~ ground states in the third and fourth column of the periodic system. Therefore, one may expect the fine-structure effect to be an important spin polarization mechanism in such collisions as well, although standard methods to determine the angular momentum orientation via a circular light polarization measurement cannot be applied. The first experimental attempt to demonstrate the importance of the fine-structure effect in elastic scattering was undertaken by Kaussen et al. [61], who compared the spin polarization of an initially unpolarized electron beam after scattering from Hg (6s2)‘So, T1(6p)*Pi,2, Pb (~P*)~P~ and Bi (6~~)~S3,2. However, no clear indication of the fine-structure effect was found in the experimental data. Furthermore, Haberland and Fritsche [Sl] reproduced most of the experimental data in an ab initio calculation based on a nonlocal exchange approximation. According to Haberland and Fritsche, their model only allowed for spin polarization effects through Mott scattering. Consequently, they concluded that this was, indeed, the dominating mechanism. A subsequent model calculation by Bartschat [9] for elastic scattering from B (2p)*Pi12 and C (~P*)~P,,, however, revealed that the fine-structure effect can be an important polarization mechanism. The results of this model calculation are shown in Fig. 3.6.1. As expected, the explicit inclusion of the spinorbit interaction does not change the theoretical results in any substantial way; the nonvanishing results for the Sherman function S (= Sp = S,) for elastic scattering are thus clearly due to orbital angular momentum orientation combined with electron exchange. Surprisingly, the generic form of the curves for the Sherman function look very similar to well-known results for Mott scattering from heavy targets - an increase with scattering angle, followed by a rapid sign change near 90”, and a correlation between large absolute values of S and minima in the differential cross section. Unfortunately, the small fine-structure splitting in boron and carbon does not allow for a direct experimental check of these theoretical predictions, since the required energy resolution can presently not be achieved with a count rate sufficient for standard polarization experiments. In order to reduce the Mott scattering effect to some extent, the Miinster group repeated the Kaussen
N. Andersen et al. / Physics Reports 279 (1997) 251-396
0
90
375
180
e(de.4 Fig. 3.6.1. Differential cross section cr and spin asymmetry function S, ( = S,) for elastic electron scattering from boron -) and carbon (- - - - -) atoms at 1.36 eV, obtained in a nonrelativistic R-matrix calculation with re-coupling of the scattering amplitudes. The difference between the dotted and dashed curves for carbon shows the effect of explicitly taking the spin-orbit interaction into account (from [9]).
(~
et al. experiment for elastic scattering and excitation from the In (~P)‘P~,~,~,~ states [24]. The experimental results for incident energies of 1.0, 2.0 and 3.0 eV are shown in Fig. 3.6.2 and compared with Breit-Pauli R-matrix calculations [12]. In the calculation, the spinorbit interaction was included in two different ways, once by using the operator (r2Z/2) xi l/r? li.si (this is the form implemented in the original Breit-Pauli R-matrix codes) and once by using (a22/2) xi (l/ri) x [d t’(ri)/dri] li . si, where x z & is the fine-structure constant and V(r) is the scattering potential. Note that the latter form is expected to be more accurate if only one-electron terms of the Breit-Pauli Hamiltonian are taken into account. In a third model, explicit relativistic effects were omitted and the problem was solved in M-coupling, with subsequent recoupling of the scattering amplitudes to account for fine-structure resolved transitions. To allow for a direct comparison between experimental and theoretical results for elastic scattering, the results for elastic scattering from both the (5~)~Pr,~ state and the (~P)‘P~,~ were taken into account, weighted with the corresponding cross sections and the ratio of 84/l 6 obtained from the multiplicity and the Boltzmann factors representing an initial thermal occupation of the (5P)2PW state [24]. It is seen in Fig. 3.6.2 that inclusion of the spin-orbit interaction between the projectile and the target nucleus, together with the use of nondegenerate fine-structure states, does not change the results dramatically. Also, the agreement between theory and experiment is not improved any further in the Breit-Pauli calculations; in fact, for 1.0 and 2.0 eV incident energy, the results obtained with the spin-orbit operator (0?2/2)Ci (l/r;) Zi‘si predict a deep minimum in the asymmetry function for elastic scattering at small angles that is not found in the experimental data.
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In 52P1/2 +52p1/2,3/2 l/r3
nr
l/r dV/dr
SA
0
-1
1-
0
-1 0
90
180’ 0
90
180’ 0
90
180
0Cde.s)
Fig. 3.6.2. Asymmetry function SA for elastic and inelastic electron scattering from indium atoms at incident electron weighted average for energies of 1.0, 2.0 and 3.0 eV (from [12]): (- - .) elastic scattering from (SP)‘P~,~ only; (-) elastic scattering from (5p)‘PO1,2 and (SP)~P$, (see text); (- - -) inelastic transition (Sp)‘P& + (5p)‘P&. Left column: nonrelativistic calculation with subsequent recoupling of the scattering amplitudes. Center column: Semi-relativistic calculation including the spin-orbit interaction within the target and between the projectile electron and the target nucleus through the operator (a2 Z/2) xi (l/r;) li’si. Right column: Semi-relativistic calculation with the spin-orbit operator (a2/2) xi (lh) [dV (ri)/dri] li’si. The experimental data for elastic scattering (A) and excitation (V) are from Bartsch et al. [24].
On the other hand, the purely algebraic recoupling procedure yields impressive agreement between the experimental and the theoretical data. Especially for elastic scattering at 1.0 eV incident energy, the nonrelativistic calculation is in much better agreement with the experimental results than any of the two semi-relativistic calculations. This indicates that angular momentum orientation is the dominating mechanism that causes the left-right asymmetry, in contrast to the
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conventional Mott scattering effect. Experimental studies with even lighter targets such as Ga (4p)‘PIj2. 3I2 with a fine-structure splitting of approximately 100 meV might be within reach of current experimental technology and could further clarify the role of the fine-structure effect in elastic scattering. Fig. 3.6.3 displays the asymmetry function SA for elastic electron scattering from thallium atoms. Recent experimental data of the Muenster group [39,45] are compared with one-state, three-state and seven-state BreittPauli R-matrix calculations [14] where, once again, relativistic effects were accounted for either by recoupling alone or explicitly by two different versions of the spin orbit operator (see below). The surprise in Fig. 3.6.3 is the fact that the close-coupling expansion has apparently not converged, even at the lowest collision energy of 1.0 eV. Accidentally, the most primitive one-state calculation (labeled LCC), yields very good agreement with the experimental data, provided the spinorbit interaction contains the operator Zerr/r3, where the effective charge Zen is obtained by scaling the strength of the interaction in such a way that the fine-structure splittings in the target states of interest are well reproduced. On the other hand, the three-state (KC) and seven-state (7CC) expansions with the same representation of the spin-orbit interaction deviate significantly from the experimental data. What one might intuitively expect to be the “best” approximation, namely 7CC, yields the largest discrepancy from experiment. For this case, the full-relativistic “generalized density functional” (GDF) calculation of Fritsche et al. 1431 is in good agreement with the experimental data. On the other hand. it also corresponds to a one-state calculation, although with a very sophisticated and fully optimized potential for the elastic channel alone. The situation is further confused by the fact that predictions of the nonrelativistic seven-state calculation agree with the experimental data. Whereas this might indicate another example of the fine-structure effect, it could also be interpreted as a coincidence in light of the dramatic changes seen in the theoretical results when the form of the spinorbit operator and the number of states in the close-coupling expansion are changed. A full-relativistic close-coupling calculation would be desirable to shed more light on this problem, which is most likely related to the many nearthreshold resonances. When one compares in detail the different theoretical approximations discussed above, it is found that some of these structures move from negative ion bound states into the scattering continuum and vice versa, depending on the number of states in the close-coupling expansion and the form of the spinorbit operator. Such changes. of course. strongly affect the results [ 141. As our final example, results for the Sherman function in elastic electron scattering from Pb (6p’)‘P0 atoms are shown in Fig. 3.6.4. Once again, the GDF calculation of Haberland and Fritschc [Sl] reproduces most of the features in the experimental data of the Munster group [61,45] for collision energies above 6 eV, whereas a five-state BreittPauli R-matrix calculation [S] achieves qualitative but not always quantitative agreement with experiment for energies below 6 eV. No study about the potential importance of the fine-structure effect versus Mott scattering has been performed for this collision system to date, but we expect that (i) both effects will play an important role at low collision energies and (ii) it will be very difficult, if not impossible, to disentangle them, theoretically and experimentally. Finally, Fig. 3.6.5 shows some results for excitation of the (6~‘) levels of lead, demonstrating again the need for more theoretical work on this collision system.
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TI 6*P,,* +
TI @P,,z
3cc
l.OeV
SA
3.0eV
W
Fig. 3.6.3. Asymmetry function SA for elastic electron scattering from thallium atoms in the 1CC (left column), 3CC (center column) and 7CC (right column) close-coupling approximations (from [14]): ( -) semi-relativistic calculation including the spinorbit interaction within the target and between the projectile electron and the target nucleus through the operator (a2 Z,rr/2) xi (l/r:) li. si; (- - .) semi-relativistic calculation including the spin-orbit interaction through the operator (a2/2)Ci (l/ri) Cdl/ (ri)/dri] Z,.si; (. .) nonrelativistic calculation with subsequent recoupling of the scattering amplitudes; (- - - - -) full-relativistic “generalized density functional” calculation by Fritsche et al. [43]. The experimental are from Geesmann et al. [45].
N. Andersen et al. /Physics
Pb
-1 0
60
0
60
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379
63Po-63Pc
0
60
60
120
8 ldegl Fig. 3.6.4. Asymmetry function SA ( = S,) for elastic electron scattering from lead atoms. The experimental data of Kaussen et al. [61] (A) and Geesmann et al. [45] (0) are compared with the results from a five-state BreittPauli R-matrix calculation (-----) [8] and a full relativistic “generalized density functional” calculation by Haberland and Fritsche (- - - - -1 [Sl].
f4de.4 Fig. 3.65. Asymmetry function SA for electron impact excitation of lead atoms at an incident electron energy of 6.0 eV. The experimental data of Geesmann et al. [45] are compared with the results from a five-state Breit-Pauli R-matrix calculation [S].
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4. Conclusions and recommendations
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for future work
We have presented a systematic generalization of the analytical framework of Vol. I to include spin-dependent alignment and orientation effects for outer-shell excitation in electron-atom collisions. It has been demonstrated that the “perfect scattering experiment” is within reach with presently available technology for some of the most important processes, particularly *S + *P excitation in e-Na and I!!&,+ 3P, excitation in e-Hg collisions. Our results are summarized in Table 4.1. Valuable information can also be extracted in cases where full information is not within reach. For a general review on the status of perfect experiments, including also elastic scattering and excitation of D states, see [S]. Our recommendations about directions in which future work is encouraged or discouraged will now be summarized. Starting with excitation of the Na 3*P state (see Sections 3.2.2 and 3.2.3), the situation is generally good at several energies for many of the parameters, in particular when including the additional data extracted by the inversion technique outlined in Section 3.2.2. However, a better coordination of the experimental efforts among different groups would be beneficial. Additional experimental results obtained, for example, with the in-plane laser pumping scheme, could finally achieve “the perfect scattering experiment”. This may be desirable for at least one convenient electron impact energy, such as 10.0 eV, to remove remaining ambiguities. Considering the similarities between the data for the Li 2, 3*P (Section 3.2.4) and the Na 32P states, the fairly satisfactory situation for Na, and the very considerable investment in time and other resources involved in the experimental investigations, it seems unlikely to us that much significant new information can be gleaned from further studies of the Li case at the present time. For excitation of the K 4*P state (Section 3.2.5), no experimental results for coherence parameters exist, not even for unpolarized electron and atom beams. In view of the richer structures predicted for this atom, and the intriguing differences between the (angle-integrated) ionization asymmetries observed between Li and Na on one side and K on the other [32], this element offers a more attractive experimental challenge than Li. For excitation of the H 2*P state (Section 3.2.6) at incident electron energies above 35 eV, the theoretical predictions are very similar for the individual spin channels and, therefore, for the spin-averaged results. Hence, we recommend a refined experiment with unpolarized beams as a first step. In light of possible scaling laws, the similarities in shape for the y1= 2,3 states encourage comparison of the alignment and orientation parameters within Rydberg series. Excitation of the Hg 63P1 state (Section 3.3.2) is the primary example of a collision system involving a many-electron heavy target for which a complete experiment is within reach with presently available technology. We strongly recommend a detailed study at one or two selected impact energies, such as 8 eV, where the full set of independent generalized Stokes and generalized STU parameters should be determined. In order to provide reliable benchmark data for comparison with theory, the numerous opportunities for consistency checks must be fully exploited. Given the importance of semi-classical treatments in many areas of atomic collision physics, we also recommend further experimental studies of the validity of propensity rules for spin-resolved collisions; see Fig. 3.3.6. Regarding excitation of the Xe 6s,6s’ states (Section 3.3.3), the difficulty of the experiment and the smallness of the spin-dependent effects suggest low priority of noble gas targets for further electron-photon coincidence studies. On the other hand, the disagreement between experimental
N. Andersen et al. /Physics
Table 4.1 Summary of electron-atom interactions for cases of increasing parameters describing the excitation process for spin polarized 3. I
Sections
Na 3’P + exchange
Hg 6”P, + spin orbit
P:n,,7Ph” +
,L
L; ^,
L;. L; .,t ;“
b, P=l h=O II’ = I
Ipj, P;
L:’ .Ll-* “9. .,* ;,+r . p’.I
Section
Example (excitation of) Interaction(s) responsible Representation Reflection symmetry
He 2’P Coulomb Wave function +
p’=
2 2 0 1
l.P‘=
6 5
)
3.3~ 5
Km,
+.-
1
h’ = 0. h’ = 0 31\+ + \1” = 1 A*(A
lndependent parameters Parameters from Stokes analysis Parameters from STU analysis Necessary observation directions .‘These numbers are the minimum independent electron spin polarizations
complexity, and the set of orientation and alignment beams. Compare to table 2.1.1 for unpolarized beams Section 3.2
CXC
Dirnrmionlrss purumeter,s Angular momentum ( -PJ Alignment an& Linear polarization Total degree of polarization Height Weights Marc angles
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p+t = 1. p+, = 1 h’ 2 0, h’ 2 11
11.’+ \VI = I A+ (A ). A”, 3‘. 10 9
I
I
I”
2;’
required for Stokes parameter analysis. They may increase directions can be prepared. See text for details.
if not all three
data and theoretical predictions for the generalized STU parameters (Section 3.4.6) indicates that first-order theories are not sufficiently accurate for these collision energies. More theoretical work is required to resolve the discrepancies. Studies of the Cs 62S1,2tt62P,,Z,3,z transitions (Section 3.4.1), similar to the experiment performed by Nickich et al. [79] for sodium, would be most valuable for a deeper understanding of relativistic effects in the collision dynamics, particularly with regard to the approximate validity of the tine-structure effect for heavy targets. We strongly recommend further experimental investigations of generalized STU parameters in this area. Similar statements hold for other quasi one-electron systems (Section 3.4.2). In particular, we recommend an experimental test of the theoretical prediction for the spin polarization function at 40 eV incident energy for electron impact excitation of the Cu 42P1,2,3,2 states. A check of this prediction, indicating the dominance of Mott scattering over the fine-structure effect. would be possible in a standard electron polarization setup. Since the theoretical results are nearly identical for both fine-structure levels, no laser-pumping or other means to separate them are required. For the In 52P3,2 + 62S1!2 transition (Section 3.4.3) the large discrepancy between theory and experiment is surprising. Whereas the theoretical results are in qualitative agreement with expectations based on the fine-structure effect, the experimental data clearly violate these predictions. Apparently, one fine-structure transition is adequately described in a semi-relativistic model while a much more sophisticated treatment is required for the other transition within the same multiplet.
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A full-relativistic calculation of this problem is desirable. The same observation is made for transitions in Tl (Section 3.4.3) and Pb (Section 3.6), for which semi-relativistic treatments are inadequate. As can be seen from the above comments, excitation of the resonance transitions in light quasi one-electron targets is sufficiently well understood. Remaining discrepancies between experimental and theoretical results point to the need for further reduction of experimental uncertainties. However, the situation is much less clear for heavy quasi-one-electron atoms and rare gases, and more complex targets in general. Experimental benchmark data for such collision systems would be very valuable for comparison with and further development of theory. Even with unpolarized beams, experiments on targets such as carbon, nitrogen, or oxygen, which are crucial in the interpretation and modeling of observations in astronomy, planetary atmospheric studies, plasma physics and laser physics, seem worthwhile to us. Regarding further theoretical work, the success of the “convergent close-coupling” approach for the light targets and the rapid development of computer resources suggest the development of a full-relativistic CCC method for complex targets - a certainly ambitious, but not unrealistic goal for theorists in this field. A new field that may be emerging from combining the techniques presented here with those developed for (e,2e) studies is the analysis of simultaneous ionization and excitation, with e + He 1‘S + 2e + He: (22P) as the generic process. Coincidence measurements between the He+ 22P + 12S 304 A photon and one or even two outgoing electrons provide a new test of ionization theories since, in contrast to standard (e,2e) cross sections, relative phases between ionization amplitudes become important. Finally, we want to reiterate an earlier statement [3]: “In light of the substantial amount of human power and equipment invested in these studies over the past years, we feel obliged to comment on how these impressive resources may be used most effectively in the future. To support the efforts, we strongly encourage theoreticians to report complete sets of parameters from which everything else can be derived. Also, close coordination between experimental groups concerning a common choice of collision systems, energies and scattering angles, as well as the systematic application of state-of-the-art theory for data reduction, seem mandatory to us.”
Acknowledgements
We dedicate this final volume of our review series to Professor Ugo Fano on the occasion of his Enrico Fermi Award. His contribution to this field has been seminal for our approach and we are grateful for his continuous interest in our project. We thank Igor Bray, Friedrich Hanne, Mike Kelley, Steve Lorentz, Don Madison, Jabez McClelland, Bob McEachran, David Norcross, Al Stauffer, Peter Teubner, Wayne Trail and Martin Uhrig for readily communicating their data in numerical form, and Ben Bederson and Joachim Kessler for their long-term support, input and encouragement. We also thank the referee for carefully reading the manuscript and making valuable suggestions for improvement. This work was funded by a grant from the National Institute of Standards and Technology, Office of Standard Reference Data. We are much indebted to Jean Gallagher for continuous support. Financial assistance was also provided, in part, by the Danish Natural Science
N. Andersen et al. / Physics Reports 279 (1997) 251. 396
Research Council (NA), the National Science Foundation (KB).
383
(KB), and a JILA Visiting Fellowship
Appendix A. Scattering amplitudes In the following appendixes we summarize some mathematical details that were used in the data analysis for this review. Most of the equations will not be derived from basic principles; interested readers should consult the standard literature for more information. Instead, the formulas presented here should be sufficient to retrace our steps and reproduce the results presented in the main text. The scattering amplitudes .f(~,~I,~oWI;~)
= (J,M,;krmr
l.Fl JoMo; kJ%)
(A.1)
describe the scattering of electrons (or, more general, spin-; particles) with initial linear momentum k0 and spin component m. (with regard to a given quantization axis) from a target with total by the projectile angular momentum Jo and component MO. The final state is characterized momentum kI, the spin component m, , and the target quantum numbers Jr and MI, respectively. The scattering angle 0 is the angle between k. and kl, and ?F is the transition operator, usually presented as the .F matrix. A possible normalization factor in Eq. (A.l) is omitted for simplicity in
the general formulation A. 1. Scattering
amplitudes
presented here. in di!erent
coordinate .frames
In practical applications, it is necessary to define the scattering amplitudes with respect to a quantization axis for the angular momentum components. A standard choice for numerical calculations is the so-called “collision system” where the incident beam axis is the quantization (z’) axis while the y’ axis is perpendicular to the scattering plane. On the other hand, the algebra often becomes simpler and many observables can be interpreted more easily in the “natural system” where the quantization axis coincides with the normal vector to the scattering plane and the xn axis is defined by the incident beam direction. The transformation of the scattering amplitudes from one system to another can be achieved in a straightforward way by transforming the initial and final states through standard rotation matrices and using the fact that the action of the F-operator must be independent of the particular coordinate system. This transformation is performed as follows. Consider states [j,m’), defined in a coordinate system (X,, Yr,Zr ). Suppose a second coordinate system (X2, YZ,Z2) is obtained from (X 1. Y, , 2,) through a rotation by a set of three Euler angles (SI,j?, 11)as defined by Edmonds [41]. The states I,j, nz’)r can then be expressed in terms of states Ij,vn), defined in the system (X2, Y1, 2,) as
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N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
and d( b)Am,are rotation matrix elements. Note that the angular momentum j is invariant against such rotations. As an explicit example, consider amplitudes for an S’ + P” transition without consideration of the electron spin. Omitting the arguments kO, kI and 6’for simplicity, the corresponding amplitudes are f(M) =f(L,
= l,M&
= M; Lo = O,ML” = 0) E (l,M1,F[O,O)
(A.4)
The natural coordinate system evolves from the collision system through rotation by the Euler angles c1= - rc/Z, fi = - rc/Z and y = 0, and the collision frame from the natural frame through a = 0, ,Q= 7c/2, and ‘/ = n/2. Furthermore, only the P states have to be transformed since the spherically symmetric S states are invariant against such a rotation. Consequently, the amplitude (1,1 (YIO, 0), in the natural system (subscript “n”) is related to amplitudes in the collision system (subscript “c”) as
(Lll~lO,O>, = -(i/2)
(l,ll~lO,O),
+ (i/2) (1, -
- (l/a)
llYlO,O),
.
(l,OlF\O,O),
(A.5)
It will be shown below that Eq. (A.5) can be further simplified by using parity conservation and the planar symmetry of the collision process, A.2. Symmetry properties A very important point for the discussion of scattering amplitudes is the fact that certain symmetry properties of the projectile-target interaction lead to conservation laws through the F-operator. These, in turn, will cause interdependences between various scattering amplitudes or simply require some amplitudes to vanish. Consider, for instance, the conservation of the total parity. This is an extremely good approximation in atomic collision physics. Consequently, computer programs in this field will generally only include projectile-target interactions that conserve the total parity, with the most important example being the Coulomb interaction. For our case of interest, electron-atom scattering in a plane, the process must be invariant against reflection in this plane. This reflection operation can be constructed as the parity operation, followed by a 180” rotation around the normal axis n^ of the scattering plane. If we consider a general matrix element of the form (j, ml 15 1j, mo), reflection invariance implies that (j,mII~ljj,mO>
= (jImI
I~+~NOkJ~
64.6)
where 39 = ~;(180”)~~
(A.7)
is the operator for the reflection, constructed as a product of the parity operator 9’ and the rotation operator Bi, (1SO’). The operation of the parity operator yields the same (eigen)state multiplied by the parity II of the state, and rotations are handled again through the Euler angles. Fortunately, rotations by 180” around the z” axis in the natural frame or the y’ axis in the collision frame are straightforward.
N. Andersen et al. /Physics
Applying the general transformation
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385
(A.2) one finds
.9?l(.jom,) = 2?(0,0,1~)~~~j~m,) = I7,( -l)“oIj,m,)
(A.8a)
for the natural frame and BjjOvnO) = p(O, - rt,O)O~ljjOm,) for the collision frame. Consequently,
= LI,( - l)jn-*filjO -mo)
Eq. (A.6) becomes
~.i~~~l~~lI.j~~~~=~~~~~-~~~~~~~~j~~~I.~I~~~~~ for the natural
(A.8b)
(A.9a)
frame and
(,iI m, ~~TJjomo) = I7,I?,( - l)j~-m~+j~-“‘~(j, - m,)FJj,
- mo)
(A.9b)
for the collision frame, where & and n, are the parities of the initial and final states, respectively. The above symmetry relationships can be generalized to our amplitudes of interest. The results are .f(M,mi,M~m();8)
= n,n,(
-1)“o-M,+mo-m,f(M~m,,Mom,;8)
(A.lOa)
for the natural frame and .f(W~,,MZl%;Q) x
.f(
-Ml
-
= n,n,
ml,
-MO
( -1)
J,-M,+J,,~M”+(l:2)-m,,+(1/2)~m,
(A.lOb)
-Q;@
for the collision frame. Instead of providing phase relationships ( + ) between amplitudes with a given set of magnetic quantum numbers and those with the sign-reversed quantum numbers in the collision frame, see Eq. (A.lOb), Eq. (A.lOa) shows that many amplitudes simply vanish in the natural frame (namely those where the exponent is an even or odd integer, depending on the product of the parities). This fact is one of the many advantages that can be used when formulating the general theory in this frame. Numerical calculations, on the other hand, are much simpler in the collision frame. If desired, however, the resulting amplitudes can easily be transformed into the natural frame. Finally, the corresponding form of Eq. (A.lOb) for S’ -+P” transitions can be used to eliminate the amplitude (1, - 1 I .T IO,0), from eq. (A.5). This leads to the well-known results (1.1 iZ/O,O),
= - i(l,l(Y(O,O),
(l,-lIr~IO,O),= (l.o~qo,o),
-(1/Jz)(1,0~~(0,0),,
-i(1,1~~~0,0),+(1/~)(1,0~~~0,0),, G 0
(A. 1la) (A.llb)
(A. 1lc)
for this case (see also Eq. (F.38) of Vol. I). A.3. Scattering In electron
amplitudes
in the nonrelativistic
limit
scattering from light targets, it is often assumed that the total spin S and the total L of the combined target + projectile system are conserved during the
angular momentum
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collision. This approximation is very good, for example, for many of the alkali-like targets discussed in this review. It is also the basis for the “fine-structure effect” which will be further discussed in Appendix C. In this nonrelativistic approximation, the scattering amplitudes depend in a purely algebraic way on the spin quantum numbers, and transitions between fine-structure levels are described by standard recoupling techniques. Specifically, the scattering amplitudes (A.l) can then be expressed as
xfSWt>,
ML,; ‘3 >
(A.12)
where SO, S1, MS,, MS,, Lo, L1, ML,,, and ML, are the spins as well as the angular momenta and the corresponding z components of the initial and final target states, and (A.13) are nonrelativistic scattering amplitudes that describe transitions between orbital angular momentum states. Eq. (A.12) expresses the conservation of the total spin S and its component MS = MS, + ml = MS, + m,
(A.14)
through the Clebsch-Gordan coefficients (jl, ml; j,, m2 1j,, m3). Note that this approximation will reduce the number of independent scattering amplitudes significantly, since the scattering dynamics are assumed to be identical for each member of a fine-structure multiplet.
Appendix B. Density matrices A thorough introduction to the theory of density matrices with emphasis on their applications in atomic physics can be found in the book by Blum [28]. Extended applications to electron collisions have been given in Vol. I and the review by Bartschat [lo]. The main advantage of the densitymatrix formalism is its ability to deal with pure and mixed states in the same consistent manner. The preparation of the initial state and the details regarding the observation of the final state can be treated in a systematic way. In particular, averages over quantum numbers of unpolarized beams in the initial state and incoherent sums over nonobserved quantum numbers in the final state can be accounted for by the “reduced density matrix”. Furthermore, expansion of the density matrix in terms of “irreducible tensor operators” and the corresponding “state multipoles” (see Section B.3 below) allows for the use of advanced angular momentum techniques. The “complete” density operator after the collision process is given by
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where Pin is the density operator before the collision. The corresponding by (P,“JZXfO =
387
matrix elements are given
c Pmbm,PM;M, m;,m,MGMo xf(M; m’r, W mb; 0) f* (MI ml, MO mo; Q) ,
03.2)
where the pm6,,,,pM;M,describe the preparation of the initial state and the star denotes the complex conjugate quantity. (We assume that the projectile and target beams are prepared independently, thereby allowing for this factorization.) B. I. Reduced density matrix formalism
The “reduced” density matrices account for the fact that, in practical experiments, not all quantum mechanically possible quantum numbers are determined simultaneously. For example, if only the scattered projectiles are observed, the corresponding elements of the reduced density matrix are obtained by summing over the target quantum numbers as follows:
(PouJm;m,:H =
M;=M,
(H.3)
The differential cross section is then given by
s =c
1 (PouJm;m,:o ?
(B.4)
m; =rn,
where C is a constant that depends on the normalization of the continuum waves as well as a possible normalization factor in Eq. (A.l) in a numerical calculation. These reduced densitymatrix elements contain information about the projectile spin. The information can be extracted through a measurement of the generalized STU parameters which will be discussed in the next section. On the other hand, if only the atoms are observed (for example, by analyzing the light emitted in optical transitions), the elements
determine the “integrated Stokes parameters” [18], i.e., the polarization of the emitted light independent of the electron scattering angle. These parameters contain information about the angular momentum distribution in the excited target ensemble. Finally, for electron-photon coincidence experiments without projectile spin analysis in the final state, the elements
u3.6) contain information about the projectiles and the target. This information can be extracted by measuring the angle-differential generalized Stokes parameters discussed in detail in Section 3.3.
simultaneously
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The density-matrix formalism outlined above is very useful for obtaining a qualitative description of the geometrical and sometimes also of the dynamical symmetries of the collision process. An example are the generalized STU parameters which will now be discussed in some detail. B.2. Generalized STU parameters The generalized STU parameters describe the scattering of a spin polarized electron beam from unpolarized targets. They can be expressed conveniently in terms of the quantities (m; mb;m, mo) = 2J ‘+ 1 C fWbmL&mbV*W~ 0
MI
m&f0m0).
(B-7)
MO
The derivation of the general formulas for these parameters proceeds as follows. First, the unpolarized target beam in the initial state is represented by matrix elements pM; M0= 6, Moin Eq. (B.2), while a spin-polarized projectile beam is represented by elements of the polarization matrix [59] Pproj
1 l+P, 2 ( Px-iP,
=
-
P, - iPy l-P,
03.8)
>
Then the reduced density matrix elements defined in Eq. (B.3) are calculated, followed by the differential cross section given in Eq. (B.4). Finally, the spin polarization components Pi,y,z after the scattering are obtained through PX.Y.2
=
tm,,,,
Pdltr
fhd
03.9)
,
where &.d is the reduced density matrix for the scattered projectiles with the matrix elements given in Eq. (B.3), %,,,, are the three Pauli matrices, and tr denotes the trace operation. For the natural system, one finds the general result, see Fig. 2.3.3(a): p = 6% + T,P,)I
+ (T,P,
+ U,,P,)j 1 + SAP,
+ (T,P,
- U,, P,)k
(B.lO)
The equivalent equation for the collision system [lo] is obtained by making the index substitutions x+-z, yttx and ztty in Eq. (B.lO). The final formulas can be simplified substantially by using the hermiticity of the density matrix and assuming parity conservation in the projectile-target interaction. The hermiticity condition yields [lo] (B.11) (mi &;mI mo> =
m&; ml
of Eqs. (A.lO) to the parameters mo), ,
(m; m&;ml mo)c = ( -l)m~-mb+m~-m~(- m; - mb; - m, - mo)c,
defined in (B.12a) (B.12b)
where the subscripts again denote the parameters in the natural and in the collision frame, respectively. Once more, we see that reflection invariance with respect to the scattering plane
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389
yields & phase relationships between parameters in the collision system, whereas half of the possible combinations of the magnetic quantum numbers will not give any contribution in the natural frame. Omitting a possible normalization factor, the differential cross section for unpolarized projectile beams is given by
in both the natural frame and the collision frame. For the remaining seven generalized STU parameters one finds: Sp = (1/20”)[(~~;+3),
+ (3 - 4;s - i), - ( - 3f; -+f),
- ( - 4 - 4; - 4 - i)“]
T, =
(B.14a)
- ( - f - 3; - 4 _ i),]
,
(B.14b)
(l/a,)Re{(a$;
+ (-~$;~-~),},
(B.14~)
-3
-3)”
T, = (l/o”) Re ((33; - t - f), - ( - $$;f - $),), TZ = (l/20,) [(+&++),
- (4 - f;+ - +), - ( - if; -++),
+ ( - f - 3; - + _ f),] ) Uyx=
-(l/o,)Im{(~+;--3-+),-(
U,, = - (l/a,)Im
(B.14d)
(B.14e) -f+;i-*),},
{($$; - 3 - f), + ( - ++;+ - i),}
(B.14f) (B.14g)
in the natural frame and (B.15a) (B.15b) (B.15~) (B.lSd) (B.15e) (B.15f) (B.15gl
390
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
B.3. State multipoles
The general density-matrix theory can be elegantly formulated by decomposing the density operator in terms of irreducible components whose matrix elements become the so-called “state multipoles”. In such a formulation, full advantage can be taken of the sophisticated techniques developed in angular momentum algebra. For a thorough introduction to this method, we refer again to the book by Blum [28]. The density operator for an ensemble of particles in quantum states labeled as IJ, M) where J and M are the total angular momentum and its magnetic component, respectively, can be written as P= c
&MIJM’)
(11.15)
(JMI >
M’M
where &M
(JM'bI
=
(B.16)
JM)
are the matrix elements. Since it is sufficient for our cases of interest, we have assumed that J is well defined, but coherences (M’ # M) between different magnetic sublevels are possible. Alternatively, one may write P =
c
KQ
T(J)KQ
(B.17)
3
where the irreducible tensor operators are defined as
T(J)
KQ
=
1
(
-l)J-M(J,M’;J,
- MjK,Q)jJM’)(JMj
(B.18)
M’M
and the “state multipoles” or “statistical tensors” are given by (T(J)Z;Q)
= C ( -l)J-M(J,M’;
J, - MIK,Q)(JM’Ip(
JM)
.
(B.19)
M’M
Note that the selection rules for the Clebsch-Gordan
coefficients imply
OIK12J,
(B.20a)
M’-M=Q.
(B.20b)
Eq. (B.19) can be inverted through the orthogonality to give (J’M’lp(JM)
= I(
-l)J-M(J,
condition of the Clebsch-Gordan
M’; J, - M(K, Q)(T(J)&Q)
.
coefficients (B.21)
KQ
The transformation properties of the tensor operators and the state multipoles defined in two different coordinate systems are given by expressions similar to Eq. (A.2) as
T(J) KQ
=
~T(JhA~,P> 4
?>,“Q 7
(B.22a)
N. Andersen et al. j Physics Reports 279 (1997) 251-396
391
i.e., the rank K of the tensor operator is invariant, and U(J&p)
(B.22b)
=~UV)LJ~@JY)y”;;~ 4
The irreducible tensor operators fulfill the orthogonality
condition (8.23)
With (B.24) being proportional to the unit operator 1, it follows that all tensor operators have vanishing trace, except for the monopole T(J),,. The hermiticity condition for the density matrix implies (T(&_$*
=
(
.
-l)"
(B.25)
Hence, the state multipoles (T (J)kO) are real numbers. Furthermore, the transformation property (B.22b) of the state multipoles imposes restrictions on non-vanishing state multipoles to describe systems with given symmetry properties. In detail, one finds the following: (a) For spherically symmetric systems: (T(J):,Q)
=
(B.26)
(T(~)2(,)rot
for all sets of Euler angles. This implies that only the monopole term (T(J):,) from zero. (b) For axially symmetric systems: (T(diQ)
=
can be different
(B.27)
(~(~)~,)rot
for all Euler angles 4 that describe a rotation around the z axis. Since this angle enters through a factor exp { -iQ$} into the general transformation formula (B.22b), it follows that only state multipoles with Q = 0 can be different from zero in such a situation. (c) For systems with properties invariant under reflection in the xy plane: (r(f):,Q)
=
( -ljK
(T(&Q)
.
(B.28)
In this case, state multipoles with odd rank K must vanish. This gives the well-known result for the description of electron-photon coincidence experiments with unpolarized beams in the natural frame (see, for example, Eqs. (3.3.2)). (d) For systems with properties invariant under reflection in the xz plane:
=
( -l)K
(B.29)
In this case, state multipoles with even rank K are real numbers, while those with odd rank are purely imaginary. This is the equivalent result to (c) for unpolarized beams in the collision frame.
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N. Andersen et al. /Physics Reports 279 (1997) 251-396
Appendix C. The fine-structure effect
The “fine-structure effect”, in its pure form, is derived by assuming the validity of Eq. (A.12). It was discussed in detail by Hanne [SO] for the special cases of S” -+ P” transitions in alkali-like systems and ‘Se + 3P0 transitions in quasi-two-electron targets. Some of the dynamical requirements for the realization of the fine-structure effect were discussed in Section 3.2.1 and will not be repeated here. Instead, we give a generalization of Hanne’s treatment that was derived by Goerss et al. [48]. Using Eq. (A.12), one can obtain an interesting result for the sum over fine-structure levels of the parameters defined in Eq. (B.7). In detail, we evaluate
=
C(L,,M(t,;S1,M$,IJ1,M,)C(Lo,M~,;So,M$oIJ,,M,) c f , ,allM s
J,,J,,S,S
x C(S1,M~l;~,m;lS’,M$)C(So,M$,;~,mbIS’,M$) xC(L1,ML,;S1,Ms,IJ1,M1)C(~o,MLo;So,Ms,IJ,,~o)
xC(S1,Ms,;f,m,IS,Ms)C(So,Ms,;~,molS,Ms) (C.1)
XP’ UC.,>Jc,,)fS UK, 9ML,)* .
In this case, the sums over J1 and Ml as well as over Jo and MO can be performed by using the well-known relations for the Clebsch-Gordan coefficients [41]. The result W&J%,)
WW’%)
W&J’%)
W%,J’%)
can then be used in the remaining Clebsch-Gordan rule
(C.2)
coefficients in Eq. (C.l) to yield the selection
M~-Ms=m~-m,=m~-mo
(C-3)
Hence, the sums (C.l) will be zero if rn; - ml # rnb - m,. From Eqs. (B. 15) in the collision system, it follows that the mean values <&) = c
(250 + l)a(J,,Jo)Sp(J1,JO)/~tot
>
(C.4a)
(S‘4)
(2.40 +
3
(C.4b)
J,,Jo
=
1
l)a(J1,Jo)Sp(J1,Jo)I(T,,,
(C.4c)
(CAd) J,,Jo
with ctot = c
J,,J,
(250 + I)~(J,YJO)
(C.5)
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
393
will vanish in the approximation of the pure fine-structure effect. With the index substitutions outlined below Eq. (B.lO), these relationships will also hold for the equivalent parameters in the natural frame. Examples have been given in Eqs. (3.2.11) and (3.4.3).
Appendix D. Reduced state multipoles from generalized Stokes parameters
This appendix lists the equations that allow for the extraction of the reduced state multipoles (3.3.12) from measured generalized Stokes parameters, including the possibility of depolarization effects in the emitted radiation due to hyperfine structure in the target. In the latter case, state multipoles of rank K >O in Eqs. (3.3.14-16) must be multiplied by perturbation coefficients GK(G, E 1). As mentioned in the discussion of Fig. 3.3.5, the set of inversion equations is unique for the following state multipoles: (r;o>” = - $P3(1
(D. 1.a)
+ P,)lD, 3
(D.1.b)
(%>u
= - $
(P, - 2P, - Pi PJ /& 7
(r’&
= - $P,
(1 + P4)/D2 ,
(D.1.c)
(i!%
= ,:‘5P,(l
+ P,)l&
,
(D.1.d)
+ P,)lD, ,
(D.1.e)
(CC&, = - &!Z;Z(I
(D.1.f)
(iL>p; = & QZ2Z2 (1 + P4)l& , where we have defined the denominators DK = GK(3 + PI + P4 - PI P4) .
(D.2)
Furthermore, when omitting any one of the three generalized Stokes parameters Qj;, Qf; and QP;, the following three sets of equations can be used to derive the state multipoles (r”,,),z, ( r~o)r, and (r;2)pz in a unique way: (1) Omitting Qf;: (&& (&Jp:=
= 3 KQ'1; + QWU 3$
+ Pd + 2Q:W +
EdllD,
Q:;U + RJ&,
+%)P, = $[2QT% (1+ P,)+ 3 Q:;(l + P4)]/D2,
,
(D.3.a) (D.3.b) (D.3.c)
where QZ = CQ;; + Q;Z3+ Q”;33/3
(D.4)
is the experimental average value for the elements in the third column of the Stokes matrix (Q;;).
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Reports 279 (1997) 2.5-396
(2) Omitting Q$: (&)p,
= 3C2 QP; (1+ J'd- (Q;Z2 - QfZ3)U+ RJ1/(2&),
0%)~; = 34 (GA,
QI;(l+ pa)/&,
= - fiQqzZ(l + PA/D,.
(D.5.a) (D.5.b) (D.5.c)
(3) Omitting Qy3: %dP,
= - (QP;- 3QYN
%A,
= - ~PQpz2U
+&I + 2Q",;U +f%)/&, + f'd+ Q"1;(1 + p,)l/& ,
(Gdp, = - fi Q"1;(1+ p4)lD2.
(D.6.a) (D.6.b) (D.6.c)
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Nils ANDERSEN, Klaus BARTSCHAT, John T. BROAD, Ingolf V. HERTEL Joint Institute for Laboratory
Astrophysics and National Institute for Standards and Technology, Boulder, CO 80309-0440, USA
ELSEWIER
AMSTERDAM
- LAUSANNE
- NEW YORK - OXFORD
- SHANNON ~ TOKYO
PHYSICS
REPORTS
Physics Reports 279 (1997) 251-396
ELSEWEiR
Collisional alignment and orientation of atomic outer shells. III. Spin-resolved excitation Nils Andersen’, Klaus Bartschat2, John T. Broad, Ingolf V. Herte13 Joint Institute for Laboratory Astrophysics
and National Institute for Standards and Technology, Boulder, CO 80309-0440, USA
Received December 1995; editor: J. Eichler Contents
1. Introduction 2. Framework, notation, and experimental approaches 2.1. Unpolarized beams 2.2. Framework and notation for polarized beams 2.3. Experimental approaches with polarized beams 3. Excitation by spin polarized electron impact 3.1. The fully coherent case 3.2. The incoherent case with conservation of atomic reflection symmetry 3.3. The incoherent case without conservation of atomic reflection symmetry: Photon polarization analysis 3.4. The incoherent case without conservation of atomic reflection symmetry: Electron polarization analysis
’ Permanent ’ Permanent 3 Permanent
254 255 255 259 260 266 266
266
305
329
3.5. The incoherent
case without conservation of atomic reflection symmetry: Combining photon and electron polarization analysis 3.6. Related work: P + P elastic scattering 4. Conclusions and recommendations for future work Appendix A. Scattering amplitudes A.1. Scattering amplitudes in different coordinate systems A.2. Symmetry properties A.3. Scattering amplitudes in the nonrelativistic limit Appendix B. Density matrices B.l. Reduced density matrix formalism B.2. Generalized STU parameters B.3. State multipoles Appendix C. The fine-structure effect Appendix D. Reduced state multipoles from generalized Stokes parameters References
address: Niels Bohr Institute, 0rsted Laboratory, DK-2100, Copenhagen, Denmark. address: Department of Physics, Drake University, Des Moines, IO 50311, USA. address: Max-Born-Institut, Postfach 1107, D-12474, Berlin, Germany.
0370-1573/97/$32.00 Copyright SSDZ 0370-1573(96)00004-X
0
1997
Elsevier Science B.V. All rights reserved
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N. Andersen et al. / Physics Reports 279 (1997) 251-396
253
Abstract
The review generalizes the formalism of the first part of this series [N. Andersen, J.W. Gallagher and IV. Hertel, Phys. Rep. 165 (1988) l-1881 on electron-atom collisions to cases where the spin polarization of the electron is investigated at least once, i.e., before and/or after the collision. In addition, the target atom may be spin polarized. The preparation of initially polarized beams and/or the analysis of photon and spin polarization in the final state significantly increase the number of cases where a “perfect scattering experiment” can be performed. The connection between the scattering amplitudes and the generalized Stokes and STU parameters is analyzed. Favorable scattering geometries for perfect experiments and possibilities for consistency checks are pointed out. Recommendations are made about directions of future work. PACS: 34.80.D~; 34.80.Bm Ke,vwords: Alignment;
Orientation;
Collisions; Electrons; Polarization;
Excitation
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1. Introduction Besides the “classical” experiment of scattering physics, namely the measurement of cross sections, a wealth of other experiments are currently performed in modern atomic physics laboratories. While enormous progress is being made in terms of experimental techniques including, for example, the preparation of spin polarized projectile and target beams with narrow energy distributions and the detailed determination of the final state (by photon and electron polarization analysis and particle-particle or particle-photon coincidences), the development of new theoretical methods has also been strongly encouraged. This is due, in part, to the construction of extremely fast electronic computers which allow for increasingly complex programs to be run in shorter periods of time. Through close interaction between experimental and theoretical work, a very detailed test of theoretical models is made possible. As a result, it has become evident that in many cases the agreement between theory and experiment is far from being perfect. Although the individual interactions are well known in atomic collision physics, we are in most cases still far away from a complete solution of the quantum mechanical many-body problem. With regard to the variety of processes studied and the experimental and theoretical methods applied, the question arises whether it is possible to establish some connections in the description of the different phenomena. In identifying such links, more systematism can be achieved in both the general formulation and in the numerical calculation of the individual observables. Therefore, an important goal of the present work is to present a general framework in which various, apparently unconnected processes can be discussed within the same formalism. The usefulness of such an approach has been demonstrated in the first part of the present review series [7] (referred to as Vol. I below), which dealt with orientation and alignment phenomena in atomic excitation by electron and atom impact. These phenomena were put on a firm mathematical basis in the influential review by Fano and Macek [44]. Vol. I only dealt with cases where both the projectile and the target beams were unpolarized before the collision, and no spin polarization analysis was performed afterwards. In these cases, the number of independent parameters that can be determined from the “reduced density matrix” (for an introduction, see Blum [28]) is generally not sufficient to extract the individual scattering amplitudes. Hence, a quantum mechanically complete or “perfect scattering experiment” [19] can only be performed for very simple processes, such as electron impact excitation of the He nlP states (see Section 2.1). Early discussions relating to this problem may be found in the pioneering papers of Rubin et al. [84] and Kleinpoppen [62]. The present review will generalize the formalism of Vol. I on electron-atom collisions to cases were the spin polarization of the electron is investigated at least once, i.e., before and/or after the collision. In addition, the target atom may be spin polarized. It will be shown that the preparation of initially polarized beams and/or the analysis of photon and spin polarization in the final state significantly increase the number of cases where a perfect experiment can, in principle, be performed. In particular, we analyze the connection between the scattering amplitudes and the generalized Stokes [4] and STU [lo] parameters. We discuss the information that is contained simultaneously in both sets of observables, as well as the information that is only contained in one of the two sets. Furthermore, we recommend favorable scattering geometries for perfect experiments and suggest consistency checks using data from apparently unrelated experiments.
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The review is organized as follows: In Section 2, the framework of Vol. I for unpolarized beams is summarized and the basic notation is reviewed. This is followed by a brief overview of experimental approaches that have been used with polarized collision partners. Section 3 then describes the excitation process by spin-polarized electrons. We begin with the non-relativistic case where explicitly spin-dependent forces such as the spin-orbit interaction between the projectile and the target nucleus can be neglected during the collision and, therefore, the atomic reflection symmetry is conserved. This is usually the case for light target systems, and we concentrate on electron scattering from alkali and hydrogen atoms, with the most complete example being the Na 32P excitation process. While the goal of a complete experiment has not yet been achieved even in this case, we demonstrate how a combination of data from several partial experiments and state-of-theart theory allows for conversion of available experimental data into a complete set of state parameters. The situation becomes significantly more complicated in the case of heavy target systems, such as Cs and Hg, where spin-dependent effects other than exchange must be taken into account in the description of the collision process. The number of independent scattering amplitudes is, therefore, increased significantly, and the perfect experiment becomes a highly challenging but not unrealistic goal.
2. Framework, notation, and experimental approaches 2. I. Unpolarized beams This section summarizes the framework and parameters used for the description of atomic outer shell alignment and orientation studies introduced in Vol. I, addressing experiments involving unpolarized particles. The presentation will be self-contained, but the reader is referred to Vol. I for a full exposition. For simplicity, the discussion will be focussed on S++P and J = 0-J = 1 transitions. This is justified, since these are the main cases of interest for spin-resolved experimental studies to this date and generalization of the formalism to higher angular momenta is straightforward. Figure 2.1.1(a) shows schematically the basic problem. An atom A is excited from an S state to a P state by impact of a particle B. This projectile B, which may be an electron, atom or ion, is deflected by an angle 0, as subsequently monitored by a detector in this direction. The collisionally excited P state of atom A may be characterized from a study of the radiation pattern (i.e., photon direction and/or polarization). This can be achieved with the geometry shown in Fig. 2.1.1(b), where the scattered particle is detected in coincidence with the photons emitted in a specific direction, including a possible photon polarization analysis. Two approaches are common here: (i) Coherence analysis, i.e., a measurement of the Stokes vector (Pr, P2,P3) in one or several suitably selected directions in space. (ii) Correlation analysis, in which the angular distribution of the photons in one or two planes containing the collision center is mapped. This is equivalent to a measurement of the first two Stokes parameters P1 and P2. This approach, therefore, gives less information than (i), but it is nevertheless used in cases where photon polarization analysis is difficult, such as for resonance
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(b)
(cl
Fig. 2.1.1. (a) Schematic diagram of a collisionally induced charge cloud of an atom A excited to a P state by impact of a particle B, scattered at an angle 8. This event may be studied in two ways: (b) Photons emitted in the P -+ S decay are polarization analyzed (Stokes parameters) in a selected direction and detected in coincidence with the scattered particle. In the time-reversed scheme (c), the atom A is excited by photons coming in from a selected direction, and the number of particles B leading to de-excitation are detected as function of laser polarization. The two approaches yield essentially equivalent information.
excitation of hydrogen or the rare gases. Since correlation analysis has not been applied in spin-resolved studies, we leave it out of the discussion below. An alternative method, which has been used very successfully, exploits the time-reversed scheme. Figure 2.1.1(c) illustrates how this is done. A photon beam from a selected direction excites the atom A. One then maps the P + S de-excitation as function of photon polarization for B particles scattered at an angle 8. For incident energy E, this is equivalent to the information one obtains from the reverse S + P excitation experiment at an impact energy of E + BE, where AE is the S-P energy difference. We now introduce a complete set of parameters, together with their measurement, that is sufficient to fully characterize the P state, thus defining “the perfect scattering experiment” in the
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251
f+l
k6
f-l
Fig. 2.1.2. Schematic diagram of scattering amplitudes in the natural frame for S -+P transitions without accounting for electron spin.
by electron impact
sense introduced by Bederson [19]. The fact that the reflection symmetry (with respect to the collision plane) of the total wave function describing the system is conserved greatly simplifies the analysis. We proceed in order of increasing complexity: (i) Thefilly coherent case. Neglecting for a moment electron (and possibly nuclear) spin, we can thus describe S + P excitation by scattering amplitudesfM,, with ML = + l,O, - 1. If we choose the “natural coordinate system” (x’, y”, z”) fixed by kin 11x” and quantization axis zn I/ kin x k,,, perpendicular to the scattering plane, see Fig. 2.1.1, the positive reflection symmetry of the wave function with respect to the scattering plane implies& = 0, such that only the (complex) amplitudes and f_ 1 are nonzero. Hence, this process is determined by three parameters only, the f magnitudes off + 1 and f- 1 and their relative phase (see Fig. 2.1.2). A perfect scattering experiment can then be performed, for example, by determining the absolute differential cross section cr and two-dimensionless numbers which fix the relative sizes off+ 1 and f__1, and the angle 6 = arg(f+ i) - arg(f- i) between them. As discussed in detail in Vol. I, the two independent dimensionless parameters may be chosen as the angular momentum transfer (L ) = (0,0,L,) with L
=
L
I.f+112-If-II” =;w.+I12-11.-112L I.f+J + If-II”
(2.1.1)
where cris the differential cross section, and the alignment angle y of the major symmetry axis of the charge cloud in the scattering plane (see Fig. 2.1.1(a)). The alignmer : angle y, defined modrc, is related to the phase difference 6 through 6=-2yfrr.
(2.1.2)
All this may be expressed compactly through the density matrix in the ML) helicity basis, which is given in the natural frame by
!
1 + LI
0
- qe-2iY
0
0
0
- Ple2iY 0
1 - Ll
. i
(2.1.3)
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The parameters (LI, y) are related to the Stokes vector P = (PI, Pz, P3) for the light emitted in the + 2” direction in a subsequent P + S decay through (2.1.4a)
P1 + iPZ = P,e2iY = and
(2.1.4b)
Px=-Ll. Determination of the parameter function can be written as
set (CJ,LI, y) constitutes
a perfect experiment,
since the wave (2.1.5)
This wave function is normalized as ($ ( t,b) = CT,consistent with the trace of the density matrix (2.1.3). The parameter Pl completely describes the shape of the electron charge cloud, with a length/width ratio in the scattering plane given by l/w = (1 + PJ(1 - Pl). In this simple case, the total degree of polarization of the emitted light P = 1PI is unity, i.e., P2 = LT + Pf = 1 .
(2.1.6)
A standard example of an excitation process fulfilling these criteria is He 1‘S -+ n’P electron impact excitation, discussed in detail in Vol. I. (ii) The incoherent case with conservation of atomic rejection symmetry. The situation becomes more complicated if the atomic state has a nonvanishing spin. Standard examples of this are hydrogen and alkali atoms, such as Li, Na, and K. First of all, since atomic fine-structure precession times are much shorter than the lifetime of the excited state, spin-orbit effects within the target will have ample time to depolarize the charge cloud prior to the optical decay, so that the measured linear polarizations become only $ of what they are in the previous case [See Vol. I, Appendix B]. However, by defining the so-called “reduced” Stokes parameters Cl] by (2.1.7a) PI,2 = 3p1,2 Y & = P3 )
(2.1.7b)
one recovers the Stokes parameters characterizing the nascent charge cloud. We will assume implicitly below that this correction has been applied. (An analogous procedure may be used to take into account possible hyperfine-structure effects.) Secondly, singlet and triplet scattering amplitudes come into play, depending on whether the spin of the incoming electron is parallel or antiparallel to the electron spin of the ‘S atomic state. Since the singlet and triplet scattering amplitudes generally differ, particularly at lower impact energies, the outcome of the collision event will depend on the relative spin orientation. In experiments where the spin directions are not observed, the outcome will accordingly be an incoherent sum of the two possibilities. Eqs. (2.1.3) and (2.1.4) still apply, but the light emitted is no longer completely coherent, and Eq. (2.1.6) is replaced by P2 = L: + P; 5 1 .
(2.1.6a)
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2.59
Consequently, the two parameters LI and Pl are now independent. Thus, for the case of partially coherent P state excitation with conservation of reflection symmetry, three parameters are necessary to completely characterize the P state, e.g. (L,, y, Pl). Still, Stokes parameter determination is necessary in one direction only, conveniently chosen along z”. (iii) The incoherent case without conservation of atomic reJection symmetry. The final step in complication arises in the description of cases such as the heavier rare gases, for which spin-orbit interaction may be important during the collision. In this case, spin-dependent scattering amplitudes ,fO may become nonzero (see Section 3.3.1.3). The corresponding charge cloud then has a nonzero height h along the z” axis (see Fig. 3.3.1 of Vol. I), and the equation for the corresponding density matrix generalizes Eq. (2.1.3) to the following form which separates the components of positive and negative reflection symmetry: 1 + L,+
0
- Pl+ee2” 0
(2.1.3a)
1 - L:
Thus, for the most general case of partially coherent P state excitation without conservation of reflection symmetry, four parameters are necessary to characterize the P state: (L: , y, PI’, h). Furthermore, Stokes parameter determination is necessary in two directions, an additional direction being necessary to reveal the charge cloud component perpendicular to the scattering plane. The two directions may be chosen along z” and y”, but other choices are possible, and sometimes more convenient. The Stokes parameters in the direction y” have traditionally been denoted by (P4, 0,O). In a subsequent section we shall, for the sake of generalization, modify this notation. The following relation holds for the height parameter h = poo:
(1 + P,)(l - P4)
(2.1.8)
h = 4 - (1 - P,)(l - P4) .
Note that two parameters, (Pl’, h), are required in this case to determine the shape of the charge cloud. With the normalization condition I + co + h = 1 for the relative length, width, and height, one has 1 = (1 - h)(l + P,+)/2 and o = (1 - h)( 1 + Pl’)/2. Similarly, the degree of light polarization in the zn direction is denoted by P+, with + P[+2 I 1 .
P +2 = Lf2
(2.1.6b)
The main conclusions of this section are summarized in Table 2.1.1.
2.2. Framework
and notation for polarized
beams
In general, transitions from an initial state 1JiMi; kimi) to a final state 1JfMf; kfmf) are described by the scattering amplitudes .f(Mrmr;
Mimi)
=
(JfMf;
kmfIylJiMi;
kimi)
,
(2.2.1)
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Table 2.1.1 Summary of electron-atom interactions for cases of increasing complexity, and the set of orientation parameters describing the excitation process for unpolarized beams
and alignment
Case
Vol. I, Section 3.1
Vol. I, Section 3.2
Vol. I, Section 3.3
Example (excitation of) Interaction(s) responsible Representation Reflection symmetry
He2lP Coulomb Wave function +
H 2’P, Na 3’P + exchange
Ar(4s4s’), Kr, Xe + spin-orbit
Pm =
Pm”
Dimensionless
+
+,-
parameters
Angular momentum ( - P3) Alignment angle Linear polarization Total degree of polarization Height
LI
LL
L:
Y
Y
Y
Pl
PI
p:
P=l h=O
PI1 h=O
P+ I1 h>O
Independent parameters Necessary observation directions
2 1
3 1
4 2
where Y is the transition operator. Furthermore, Jr(&) is the total electronic angular momentum in the initial (final) state of the target and Mi(Mr) its corresponding z-component, while ki(kr) is the initial (final) momentum of the electron and mi(mr) its spin component. The scattering angle 8 is the angle between ki and kr. For simplicity, we will omit it in the notation. We now discuss the number of independent scattering amplitudes. There are 4 *(251 + 1). (25, + 1) possible combinations of magnetic quantum numbers. Due to parity conservation for all interactions determining the outcome of the collision process, the number of independent scattering amplitudes is cut in half, giving a total of 2. (2Ji + 1)*(2Jr + 1) complex amplitudes for each transition between fine-structure levels. Subtracting a common arbitrary phase, the total number of independent real parameters is thus 4*(2Ji + 1) .(2Jr + 1) - 1. These are usually parametrized as one absolute differential cross section and 4 *(2Ji + 1). (2Jr + 1) - 2 dimensionless numbers, namely, 2’(2Ji + 1)‘(2Jr + 1) -1 relative magnitudes and 2. (2Ji + 1) * (25, + 1) - 1 relative phases. Further details on scattering amplitude properties are given in Appendix A. Whereas the scattering amplitudes are the central elements in a theoretical description, some restrictions usually need to be taken into account in a practical experiment. The most important ones are: (i) there is no “pure” initial state, and (ii) not all quantum numbers are simultaneously determined in the final state. The solution to this problem can be found by using the density matrix formalism; see details in Appendix B. 2.3. Experimental approaches with polarized beams To date, three basic experimental setups have been developed to prepare and/or analyze the spin polarization of the projectile and/or target beams. These are summarized in Table 2.3.1. Note that,
N. Andersen et al. /Physics Table 2.3.1 Experimental
approaches
with polarized
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beams
Type
Geometry figure
Spin preparation before Electron Atom
Vol. I I (e hv)
2.1.1 (b,c) 2.3.1 (a) 2.3.2 (a) 2.3.2 (d) 2.3.3 (a) 2.3.3 (a) 2.3.3 (a) 2.3.3 (a) 2.3.3 (e)
~ J J J
2A (1) 2B ( II1 3% 3S, 3T 3u 4v
J J J J
261
Spin analysis after Electron Atom
Photon
analysis
after
” ”
J J
_ J _ J
._
J
J
J
until now, either electron or photon polarizations have been analyzed in the exit channel, but never both simultaneously. The natural extension of the experimental geometry discussed in Section 2.1 is the use of a spin-polarized electron beam in the standard electron-photon coincidence setup (Vol. I). The schematic diagram is shown in Fig. 2.3.1(a), and the geometry and scheme of the experimental setup, as used by the Miinster group [109,90], are shown in Figs. 2.3.l(b,c). The main difference compared to experiments with unpolarized electrons is the source of spin polarized electrons which replaces the electron gun. Recent technological developments have made the GaAs source (see e.g. [SO]) the standard for this kind of experiments because of its favorable beam characteristics in yielding high current, narrow energy width, and an electron spin polarization of up to 50%. (In most experiments discussed in this review, the actual electron beam polarization was between 30% and 40%). We will refer to this setup as “type 1” below. The second type of experiment has been pioneered by the NIST group [70,71]. The main idea is to perform a time-reversed experiment by scattering spin-polarized electrons from spin-polarized laser-excited atoms. The schematic diagram of the experiment is shown in Fig. 2.3.2(a), and more extended schemes in Figs. 2.3.2(b, c). The additional complication in this case is the laser pumping technique to prepare the polarized target beam, but neither polarization nor coincidence analysis is performed after the collision. Consequently, this type of experiment does not suffer from the low count rate problem associated with standard electron-photon coincidence techniques. This experimental setup has been used extensively for electron scattering from sodium atoms (see Sections 3.2.2 and 3.2.3 below), and recently also for a chromium target (Section 3.2.7 [53]). An important experimental restriction was the fact that spin asymmetries of the form (2.3.1) were only determined with electron (P,) and atom (PA) polarizations perpendicular to the scattering plane (type 2A). In Eq. (2.3.1), ot t(0) and ct ,(0) denote the cross sections for parallel and antiparallel polarization vectors, respectively. As will be shown in Section 3.2.2, a measurement
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Coincidence
:
(4
(b)
ght modulator
L
Coincidence
1
dzL
/
Piles-of-plates analyzer
GaAsP cathode
y” Photomultiplier
Cc)
Piles-of-plates anatyzer
f.enr\
Target nozzle
p1ate
Mott detector
Fig. 2.3.1. (a) Schematic diagram of the experiment. Polarized electrons are scattered inelastically from atoms at an angle 0. The photons emitted during the decay of the atom in a specific direction are polarization analyzed in coincidence with the scattered electrons without further spin analysis (type 1). (b) Geometry of the Miinster experiment (from [90]). Directions refer to the collision coordinate frame. (c) Scheme of the actual experiment (from [90]).
with in-plane polarizations can help in the determination of phase differences between individual singlet and triplet amplitudes. The scheme of such an in-plane experiment, where the laser beam makes an angle CDwith the incident projectile direction, is outlined in Fig. 2.3.2(d), referred to as type 2B. Finally, the Mtinster group has also developed an experimental setup where polarized electrons are scattered from unpolarized atoms and the electron polarization after the collision is determined. This type of experiment allows for the determination of the so-called generalized STU parameters [lo] that fully describe the change of an arbitrary initial electron polarization through
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Reports 279 (1997) 251-396 Optical I
Pock& Linear (b)
Loser
I
Pumping Laser
Cell Polarizer Diode
Fluorescence Mon~l
Fig. 2.3.2. (a) Schematic diagram of the NIST analysis is performed after the collision (type polarized-atom scattering apparatus, showing Mott spin analyzer (from [71]). (d) Schematic polarized atoms with in-plane laser pumping
experiment involving polarized electrons and polarized atoms. No spin 2A). (b) Actual scattering geometry (from [71]). (c) Polarized electron GaAs polarized electron source, scattering chamber, sodium oven, and modification of the NIST experiment involving polarized electrons and (type 2B).
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scattering from unpolarized target atoms (type 3); see Fig. 2.3.3(a). The preparation of the projectile and target beams before the collision is identical to the electron-photon coincidence experiment with polarized electrons discussed above, but the photon detector and the coincidence unit for the detection of the final state are replaced by a Mott detector for the spin polarization analysis of the scattered electrons [26, 541.
(4
180” spectrometer
(C)
Mott
detector
Fig. 2.3.3. (a) Physical meaning of the generalized STU parameters for an initial spin polarization P which is changed to a final spin polarization P’ through the scattering process. The polarization function Sp gives the polarization of an initially unpolarized projectile beam after the scattering (type 3Sp), and the asymmetry function SA determines the left-right asymmetry in the differential cross section for scattering of the spin polarized projectiles (type 3SA). The contraction parameters T,, T,, T, describe the change of an initial polarization component along the three Cartesian axes (type 3T), while the parameters U,, and U,, determine the rotation of a polarization component in the scattering plane (type 3U). (b) Schematic diagram of the Miinster experiment to measure the depolarization of electrons after scattering by unpolarized sodium atoms (type 3T). (c) Scheme of an actual experiment (from [54]). (d) Measurement of the change of the electron polarization vector components caused by elastic scattering (from [26]). (e) Schematic diagram of an experiment to determine the rotation of an electron spin polarization out of the plane spanned by orthogonal initial electron and atomic spin polarizations (type 4V).
N. Andersen et al. /Physics
LIGHT 90’
GaAsP
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MODULATOR
DEFLECTOR
CATHOOE WIEN
180° DEFLECTOR
FILTER
Xe ATOMIC BEAM
cd)
Fig.2.3.3.
continued
The schematic diagram of the experimental setup is shown in Fig. 2.3.3(b) while the most essential parts are presented in more detail in Figs. 2.3.3(c, d). As discussed in a previous review [lo], there are, in general, eight parameters that describe the reduced spin density matrix of the scattered projectiles completely. Besides the absolute differential cross section G,,for the scattering of unpolarized electrons, there are seven relative generalized STU parameters whose physical meaning is shown in Fig. 2.3.3(a): The polarization function SP gives the polarization of an initially unpolarized projectile beam after the scattering, while the asymmetry function SA determines the left-right asymmetry in the differential cross section for scattering of spin polarized projectiles. Furthermore, the contraction parameters T,, T,, T, describe the change of initial polarization component along the three Cartesian axes while the parameters U,, and U,, determine the rotation of a polarization component in the scattering plane. Note that the number of independent STU parameters may be reduced in certain cases; the most important examples are (i) elastic scattering, (ii) transitions between target states with zero angular momentum in both the initial and the final
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state, and (iii) situations where relativistic effects during the collision may be neglected (for details, see [lo] and further discussion in Section 3.2 and Appendix C). The concept of the generalized ST U parameters can be extended in a systematic way to account for initially polarized targets and even a polarization analysis of the target spin in the final state. Since such experiments have not been performed to date and are not expected to be performed in the near future (at least not for excitation processes), we only mention here one type of parameter that can provide important additional information. The general idea is shown in Fig. 2.3.3(e): one starts with both projectile and target beams spin-polarized, with polarization vectors PA and PB orthogonal to each other; after the scattering, the polarization component P’ of the projectile beam along the direction defined by PA x PB, i.e., perpendicular to the plane defined by the initial polarization vectors, is determined, and the result is normalized to the product of the initial polarization magnitudes (type 4V). We denote the corresponding parameter by the generic symbol I/, i.e., I/ = P’/(P*. PB) .
(2.3.2)
It should be pointed out that many such independent parameters can be defined, particularly in cases where relativistic effects and, consequently, the relative orientation of the scattering plane and the plane defined by the initial polarization vectors must be taken into account.
3. Excitation by spin polarized electron impact In this section we discuss what information can be gained by application of polarized electrons in addition to the knowledge obtained with unpolarized beams. We keep the order of increasing complexity introduced in Section 2.1. 3.1. The fully coherent case 3.1.1. Excitation
of the He n’P1 and n3P, states
A simple example of a fully coherent excitation is He 1iS + nlP excitation, discussed at length in Section 3.1 of Vol. I. As shown, all information for this case can already be obtained from experiments using unpolarized beams, and (0, LI, y) forms a complete set of parameters; see Eq. (2.1.5). Consequently, in this simple case, application ofpolarized electrons will add nothing new: The polarization of the scattered electron beam can be predicted beforehand, since no change is possible in the scattering process. The same statement is true for excitation of the He 3P states, also studied in Vol. I. Excitation can happen by electron exchange only. Consequently, the initial polarization of the electron beam will be changed by a simple factor; for details see Ch. 4.7 of [59]. 3.2. The incoherent case with conservation 3.2.1.
General description
of atomic rejection
symmetry
of the excited 2P state
The targets of interest for this section, typically light alkali atoms or hydrogen, have an electron spin of s = 3. For electron impact, this doubles the number of scattering amplitudes from two to
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four (recall that f0 = 0), since we now have the possibility of triplet (t) and singlet (s) scattering, bringing us into the situation of Fig. 3.2.1. The amplitudes of interest are j:l
= x+eivl+ , i& -
j'L,=x_e
(3.2.la)
(3.2.lb)
,
(3.2.1~) (3.2.ld)
Neglecting an overall phase, we thus need seven independent parameters for each scattering angle 8 to characterize the amplitudes completely. One is traditionally chosen as the differential cross section CJ”corresponding to unpolarized beams. In addition, six dimensionless parameters may be defined, three to characterize the relative lengths of the four vectors, and three to define their relative phase angles. As a start, we parametrize the density matrices in analogy to the unpolarized beam case, Eq. (2.1.3), as [52]
(3.2.2a)
and 1 1 (1s = gs 2
+L",
0
0 0 _ PFe2if0
_ pce-2i7 0
(3.2.2b)
l-L”,
where (3.2.3a) (3.2.3b) (3.2.3~)
(3.2.3d) p;eW’
= Pi + iPG = - (2rx+cr_/a’)eCi6’,
p;e2iy’ =
Pj + iP; = - (2P+&/os)e-i6‘
(3.2.3e) .
(3.2.3f)
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Fig. 3.2.1. Schematic diagram transitions by electron impact.
of triplet (t) and singlet (s) scattering Note that A+ + 6” = A- + 6’.
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amplitudes
in the natural
frame for ‘S + ‘P
In the case where unpolarized beams are used, the total density matrix becomes the weighted sum of the two matrices p” and pf, i.e.,
pu =
0”
0
0
0
!
1 2
=
1+ Ll
0
- P1e2iy 0 1 + Lk
pie-W
_
1
- Ll
0
- P[epzir’
_ p:eziY’ 0 0 0
3w’# ; i.
+ w%, ;
I
1
1 + L”,
0
0
0
_ p,“&W
0
-0 L:
I
- Pfe-2iys
(3.2.4)
i
where wi = at/(aS + 3a’) = C+/46”
(3.2.5a)
ws = cq(8 + 3cr’) = 0740, = 1 - 3w’
(3.2.5b)
0 ” = [3wt + wq0 ” = jai + ;as .
(3.2.5~)
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The parameters w’ and ws are related to the parameter Y= (T’/G’used by Hertel et al. [52] through r = w’/wS = w’/(l - 3w’) = (1 - w”)/3w” .
(3.2.6)
However, we prefer the use of w’s’ to r for reasons of mathematical symmetry and simplicity. At this point we have thus introduced a total of six parameters, namely CJ,,,w’, L:, L;, y’, y’, leaving one parameter, a relative phase, still to be chosen. Inspection of Fig. 3.2.1 suggests, for example, the angle A+. Note that the fourth angle, A-, is then fixed through the relation A+ - A- = 6’ - 6” = 2(?” - y’) .
(3.2.7)
The first equality sign follows from inspection of Fig. 3.2.1, the second one from Eq. (2.1.2), applied individually to the singlet and triplet components. In the spirit of [52] and Vol. I, we therefore suggest the following complete set: (a,; w’, L:, L;; yt,y”,A+) . We have thus introduced
(3.2.8)
parameters that
allow for a complete description of the scattering process; l are a natural generalization of the parameters used for unpolarized beams; l can be interpreted in simple physical pictures; l and are accessible in “partial” (i.e., noncomplete) experiments as will be illustrated in the next section. The reduced Stokes vector P of the unpolarized beam experiment is given by the weighted sum of the singlet and triplet (unit) Stokes vectors Psvt as l
P = 3w’P’ + w”P”
(3.2.9)
from which the set of parameters (L,, y, 4) for the unpolarized beam experiment may be evaluated by using Eqs. (2.1.4a, b). In particular, LJ_ = 3w’L: + w”L”, .
(3.2.10)
Since, in general, L\ # L”, and yt # y”, this causes the (reduced) degree of polarization P to be smaller than unity, i.e., P I 1, cf. Eq. (2.1.6a). We now want to express the STU and I/ parameters discussed in Section 2.3 both in terms of the amplitudes (3.2.1) and in terms of the complete set of parameters (3.2.8) [3]. This can be done in the case of light (quasi) one-electron atoms for which the so-called fine-structure efSect is now well documented in the literature (see e.g. [SO, 52, 10,48, 1l] and Appendix C). Provided that electron exchange is the only spin-dependent effect of importance for the excitation, and that the finestructure energy splitting is negligible compared to the initial and final energies of the projectile, the STU parameters get vastly simplified. The seven polarization, symmetry, contraction, and rotation parameters for each fine-structure level reduce to the following set of only four independent parameters (the superscripts denote the J-value of the excited target state): (3.2.1 la) (3.2.1 lb)
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N. Andersen et ai. J Physics Reports 279 (1997) 251-396 T
T,1’2 = 7-7’
E
= T,“2
= T,“/”
= 772
= T,3’2 ,
(3.2.1 lc)
U = U.j/’ = U:x’2= - 2Ux3y/2 = - 2Uy”x’” .
(3.2.1 Id)
The results in terms of scattering amplitude parameters are: Sp = - (1/4C”)[a: - lx2 + p: - p1-j )
(3.2.12a)
SA = (1/40,)[Re(2cr+P+ei(~+-++) - 2cx_p_ei(#--1L-J} - 2(a$ - a?)] ,
(3.2.12b)
T = (1/4~,)[Re{2a+fi+ei(~+-*+) + 2a_fi_ei(@~-ti~)} + 2(& + cx?)] ,
(3.2.12~)
U = (1/40,)[Im{2cc+~+ei(~+-~+) - 2a_P_ei(+~-ti-))]
,
(3.2.12d)
I/ = (1/4~,)[Im{2ct+/3+ei(~+-~+) + 2a_fi-ei(~--ti-))]
,
(3.2.12e)
where Re{X} and Im{X> d enote the real and the imaginary parts of the complex quantity X, or sp= - w'L\+ wSLT,
(3.2.13a)
SA = (1/20,)[a+/?+ cosA+ - a-P_ cos A-] - 2w’L:,
(3.2.13b)
T = (1/20,)[a+P+ cosA+ + cr_j_ cos A-] + 2w’,
(3.2.13~)
U = (1/2~,)[a+j?+
sin
A+ - K/K sinA-
,
(3.2.13d)
I/ = (1/2a,)[a+fl+
sin
A+ + a-/3_ sinA-
.
(3.2.13e)
Note that the o[+ and pi a,&J20,
may be eliminated from Eqs. (3.2.13) by using
= Jw’wS(1 & L\)(l f L”,) .
(3.2.14)
As an example of an in-plane A-parameter, cf. Eq. (2.3.1), we give B = A(4 = 90”, e), where the angle @ is defined in Fig. 2.3.2(d). The result for this type 2B experiment is [3] B
=
_
w’(l-
Pi) - wS(l - Pf) (1-P1)
(3.2.15) .
The amount of information contained in the atomic density matrix (i.e., the Stokes parameters) and the reduced density matrix of the scattered electrons (i.e., the STU parameters) is illustrated in Fig. 3.2.2. From a Stokes parameter analysis, one obtains information about the relative phase between the two f:r and j? 1 amplitudes and the relative phase between the two fsl and f? 1 amplitudes, as well as the relative sizes of all four amplitudes. However, none of the relative phases between any triplet and singlet amplitude can be determined. The STU parameters, on the other hand, determine the relative phase A’ between the two j’:i and fS1 amplitudes and the relative phase A- between the two fl I and f?1 amplitudes, provided that the relative sizes of all four amplitudes are known from a Stokes parameter measurement. In this case, none of the relative phases between ML = + 1 and ML = - 1 amplitudes can be obtained. Note also that the electron spin polarization parameter SP, Eq. (3.2.13a), may be derived from results of a type 2A experiment.
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N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
STU
fl,
Triplet : :
G f’,
0
Singlet fSll
Fig. 3.2.2. (a) This diagram shows which relative amplitude analysis. (b) This diagram shows which relative amplitude parameters.
d’ f’,
.’
-:. ‘...
..
“‘(-J
ff ,
sizes and phases can be evaluated from a Stokes parameter sizes and phases enter into the Eqs. (3.2.13) for the STU
We finish this subsection with a comment on Eq. (3.2.13a) which shows the necessary conditions for a nonvanishing polarization function Sp in S -+ P excitation of alkali-like systems, based on the fine-structure effect alone: (1) electron exchange, i.e., a dynamical dependence of the scattering amplitudes on the individual spin channels (triplet and singlet in 3.2.13a); (2) a nonvanishing orbital angular momentum orientation in at least one of the spin channels; (3) experimental resolution of individual fine-structure levels. (4) negligible dependence of the nonrelativistic scattering amplitudes on the small energy difference between the fine-structure states; although this condition, in a strict sense, is incompatible with (3), it is only required for exact relationships like (3.2.11) to hold between generalized STU parameters for individual multiplet members. These conditions are well fulfilled, for example, for electron impact excitation of sodium, provided the collision energy is large compared to the fine-structure splitting [ 111. 3.2.2. Excitation of the Na 32P state: General considerations 3.2.2.1. The present experimental situation. At this point, it is appropriate to investigate which of the seven independent parameters of Eq. (3.2.8) have actually been measured experimentally, i.e., to what degree the perfect scattering experiment has been achieved to date. We begin with the type 2A experiment performed by the NIST group [71] and described in Section 2.3. Recall that spin polarized electrons with polarization vector perpendicular to the scattering plane were scattered superelastically from excited spin-polarized P state sodium atoms produced by pumping with circularly polarized laser light. By reversing the directions of the two polarizations individually and applying time-reversal invariance to relate the superelastic de-excitation to the inelastic excitation process, this experiment allows for the determination of the angular momentum transfers L\ and L; for the two spin channels individually, as well as for the relative weights w’ and ws of triplet and singlet contributions to the differential cross section (for details, see Hertel et al. [52]). The angular
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momentum transfer LI for unpolarized beams can be recovered from the set (L:, Ly,w')through Eq. (3.2.10). This experiment does not allow for a unique determination of the alignment angles yt and y”. The off-diagonal elements of Eq. (3.2.4) show that pie2ir
=
3wtp/e2iy’
+ wspfe2iy’
.
(3.2.16)
As illustrated in Fig. 3.2.3, this complex equation corresponds to the addition of two vectors P;” and P/, multiplied by weighting factors 3w’ and w’, respectively, to form the resulting vector Pl. Hence, elementary geometry can be applied to obtain two pairs of solutions, (y’,Ys)true(the true solution) and (Y’,f)ghost (the other possibility):
Y’=YT xl2 Y” =
Y+
(3.2.17a)
7
(3.2.17b)
$12>
where the angles x and $ are defined in the figure. Hence, provided experimental data are available for the parameter set (Pl, y,w', L\,L",) at a given collision energy and scattering angle, two sets of possible angles (y’,y”) can be determined, as will be further discussed below. It was pointed out by Hertel et al. [52] that the above ambiguity is mathematically identical to that found in S -+ D excitation processes [2]. Inspection of Eq. (3.2.15) shows that the ambiguity could be resolved directly through an in-plane measurement of the B-parameter: A consistency check between the results of Eqs. (3.2.15) and (3.2.16) will eliminate the ghost solution. Finally, the NIST experiment contains 120information about the missing phase angle A+ relating singlet and triplet amplitudes. Some information about this angle can, however, be obtained by combining the NIST data for (y’,y”) with results from the type 3T experiment (see Figs. 2.3.3(b,c)) performed by the Miinster group [54,55]. As seen from Eq. (3.2.13c), the T-parameter contains the phase angles A’ r, but not in a unique way.
Fig. 3.2.3. Vector diagram
corresponding
to Eq. (3.2.16). For a discussion,
see text.
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273
3.2.2.2. Combining experimental and theoretical data. The question therefore arises whether more can be learned from the existing experimental data, especially when combined with state-of-the-art theoretical approaches. This problem will now be addressed. (if Determination of the alignment angles y’ and y”: As pointed out in the discussion of Eq. (3.2.16) it is possible to determine two sets of angles (yt,ys), provided experimental data are available for the set of parameters (Pl, y, w’, L\, L”,) at a given collision energy and scattering angle. While data for all these parameters have indeed been obtained for electron-sodium (de-)excitation, the energy and scattering angle combinations investigated by the Adelaide [103] and the NIST [71,91] groups, unfortunately, do not overlap at all. Consequently, we demonstrate the feasibility of combining various experimental data sets by replacing the missing experimental data for Pl and y with theoretical results of Bray’s calculation [33] at a total energy of 4.1 eV to “invert” the NIST data at this energy. This approach seems justified in light of the excellent agreement between Bray’s theory and experiment for all available collision energies; see Section 3.2.3. The results of the inversion are shown in Fig. 3.2.4 for both angles y’ and “J’.The error bars on the experimental points were obtained by first changing the theoretical results for the set of input parameters (w’, L\, L”,) (recall that q’t’ = dm for the two spin channels separately) by a very small amount and then looking at the effect on the inverted theoretical results for the two pairs of (y’,7”). This gives the partial derivatives of y’ and y” with respect to these three input values and allows for the calculation of error bars through the formula
6p
+ (ap/aL;)2(6L;)2
= J(ap/aL:)2(scJ
+ (aytvyawt)2(6wt)2
(3.2.18)
It should be noted that this is a somewhat pessimistic estimate of the error bars, since we assume an independent determination of the parameters (w’, Ll, L”,). Consequently, the error bars shown in Fig. 3.2.4 are generahy overestimated. The important point, however, is that both the true and the ghost solutions, after inverting the experimental and the theoretical data, are in very good agreement with each other. Hence, it seems justified to select the “true” experimental data as those that follow the true theoretical solution. It is interesting to note that the two sets of solutions can cross each other and that it is impossible to stay on the “true” experimental curve by assuming, for example, a smooth angle and energy dependence of the phase angles In attempts to map the geometry of the triangle of Fig. 3.2.3 [3], we encountered a problem with the conversion procedure. In regions where the three vectors Pr, P; and Pf are almost parallel or anti-parallel, error bars or minor deviations between theory and experiment produced instances where the sum of two sides was slightly smaller than the third side. We solved this problem by setting x and $ (see Fig. 3.2.3) to 0” or 180”, respectively. For the same reason, an attempt to invert purely experimental data - necessarily at slightly different energies - failed miserably. (ii) Determination of the singlet-triplet angles A+ and A-: As mentioned previously, it is impossible to obtain the phase difference between singlet and triplet amplitudes from any set of type 2 experiments, but part of the information can be gained by combining type 2 results with those from a 3T-type experiment. For the simple case of elastic scattering, this is evident from Eq. (4.40) in Kessler’s standard treatise [59]. Eq. (3.2.13~) is a generalization of the expression for elastic S-S scattering which, in our notation, reads T = 2(w’ + mcosA)
.
(3.2.19)
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30
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60 Scattering
90 60 -
-“O
c
120
150
f
I
0
60
90
120
150
theory
----
ghost
Scattering
180
Angle (deg)
e
~
30
90
180
Angle (deg)
Fig. 3.2.4. Alignment angles y’ and y” calculated from the NIST [71] data for (w’, L\, Ll) and theoretical results for (Pi, y) from scattering amplitudes of Bray [33] for electron impact excitation of the 3 ‘P state of sodium at an incident electron energy of 4.1 eV; (0) two sets of inverted experimental data as well as true (__ ) and ghost (- - -) theoretical solutions.
This formula was applied to experimental data by McClelland et al. [74] to extract cos A, leaving the sign of A( = yst in their notation) undetermined. We now show that a similar, though more complicated procedure, can be applied to the inelastic case [3]. Type 3T experiments were performed by Hegemann et al. [54,55] at 4.0 eV and 12.1 eV total collision energy, while the type 2A NIST experiments were done at 4.1 and 10.0 eV, respectively. Thus, due to the lack of data from different experiments at the same energy, we again demonstrate the inversion method by using theoretical data [33] for the contraction parameter T at 4.1 eV. The main idea of our inversion procedure is illustrated in Fig. 3.2.5. Eq. (3.2.13~) for the T-parameter gives a nonlinear equation for A+ and A-. On the other hand, these two angles are not independent, since their difference is related to the difference between the alignment angles y’ and y” through Eq. (3.2.7). Consequently, solutions for A+ and A- can be found by searching for crossings between the lines determined by AcosA+ + BcosA-
= C ,
(3.2.20)
N. Andersen et al. / Phvsics Reports 279 (1997) 2S1-396
AcosA+
+ BcosA-
21.5
= C
180
-60
-180
180
-120
-60
0
60
120
180
A+ (deg) Fig. 3.2.5. Determination (see text).
of the singlet-triplet
phase
angles
A+ and A _ from experimentally
observable
parameters
where the constants A, B and C are evaluated from Eq. (3.2.13~) and the lines defined by Eq. (3.2.7) in Fig. 3.2.5. Due to the ambiguity in the sign of the arguments in the labeled (J’ - Yt)t,Ue,ghost cosines, as well as the ambiguity in the proper pair of (y’,ys), one will usuallrfindfour solutions, only one of which is correct. This is illustrated in Fig. 3.2.5 for inversion of the theoretical data at a scattering angle of 40” and a total collision energy of 4.1 eV [3]. Note that the problem can be reduced to searches in the first quadrant, since the “ghosts” in the second and fourth quadrants may be found via intersections with the dashed lines in Fig. 3.2.5 which are mirror images of the difference lines in those quadrants seen in the first quadrant. Since the slopes of the dashed mirror lines are reversed compared to the original difference lines, the actual crossing points in the second and fourth quadrants can easily be reconstructed, while the only remaining crossing, in the third quadrant, is related to the one in the first quadrant through a simultaneous sign change in A+ and A-. Furthermore, crossings can only occur in ranges where Eq. (3.2.20) has real solutions. The typical pattern of graphs corresponding to a numerical search for line crossings is shown in Fig. 3.2.6 for a set of four scattering angles at a collision energy of 4.1 eV. We first tested the algorithm using theoretical data; the agreement between the original and the inverted values for A+ and A- was usually better than 0.001%. The procedure was then applied using experimental data for the set of parameters (w’,L\, L”,) [71] to construct experimental pairs of (y’,7’) with error bars evaluated by Eq. (3.2.18) above. The results for A+ and A- as a function of the scattering angle for a collision energy of 4.1 eV are shown in Fig. 3.2.7. For simplicity, only one “ghost” solution (where A+ > 0) is shown. Again, it can be seen how the theoretical results help to
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N. Andersen et al. ,f Physics Reports 279 (1997) 251-396
Fig. 3.26. Typical structure of curves for numerical determination experimentally observable parameters (see text).
of the singlet-triplet
phase angles A + and d - from
theory
-
---~ ghost .
Scattering
experiment
Angle
(deg)
120 G 9) = &
60 0 -60 -120 -I 80
0
30
60 Scattering
90 Angle
120
150
180
(deg)
Fig. 3.2.7. Singlet-triplet phase angles A’ and A- calculated from the type 2A data [71] for (NJ’:L\,L;), the corresponding alignment angles (y’, v”), and theoretical type 3T results from scattering amplitudes of Bray [33] for electron impact excitation of the 3 ‘P state of sodium at an incident electron energy of 4.1 eV; (a) two sets of inverted experimental data as and one ghost (- - -) solution (see text). well as the krue ( ----)
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271
identify, in most cases unambiguously, the true solution among the various possibilities obtained from an inversion of experimental data alone. 3.2.2.3. Consistency checks of the formalism. The discussion above also demonstrates how the inversion procedure may serve as a consistency check among separate experimental data sets. Consistent experimental data should always allow for inversion within experimental uncertainties and the ambiguities pointed out. Of course, this can only be done if the individual data sets are obtained at the same total collision energy. This important coordination of efforts has, unfortunately, not taken place in most cases until now, thereby preventing desirable consistency tests of results from separate, very sophisticated experimental setups. A type 3s experiment was performed by the Miinster group [79]. Its geometry is similar to a type 2A setup, Fig. 2.3.2(a), except that both the (3p)‘Pli2 and the (3p)“Pj12 fine-structure levels were pumped individually, and the target density was chosen high enough to ensure essentially unpolarized target atoms due to the depolarization effect of radiation trapping. For each finestructure level, the left-right asymmetry in the differential cross section given by CUlcr,(o)-
Oright(e)llCOleft(Q) + gright(e)l G S*(8)pL
(3.2.21)
was measured. There are several interesting points to be made regarding the asymmetry function SA. Because of time-reversal invariance of the interaction, this type 3SA experiment for the de-excitation process is equivalent to determination of the polarization function Sp, a type 3Sp experiment, for excitation. Interestingly, in the approximation of the pure fine-structure effect, one can use Eq. (3.2.13a) to predict the results of this experiment from those of the type 2A experiment [52,25]. The data from the Nickich et al. [79] experiment and the results predicted from the NIST experiment [91] are shown together with theoretical results calculated from Bray’s amplitudes [33] in Fig. 3.2.8. The
-0.4
0
20
40
60
80
Scattering
100
120
Angle
(deg)
140
160
180
Fig. 3.2.8. Spin polarization function &. for electron impact excitation of the (~P)‘P,,~,~,~ states of sodium at an incident electron energy of 10 eV; data of Nickich et al. [79] for Sp (2PI,2) (0) and Sp (‘P& (W); prediction of Sp (‘PIi2) (0) and Sp (‘P3,J (0) from the NIST data [91] for (w’, L:, L”,); The theoretical curves for Sp (2P1!2) (~) and SP (2P~,~) (~ - ) were calculated from scattering amplitudes of Bray [33].
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Table 3.2.1 Alignment and orientation
parameters
Symbol
0 0 0 0
0 + 0 A A
A X
;
rc
for excitation
Reports 279 (1997) 251-396
of the Na 3’P state by spin-polarized
electrons
Source
Fig. 3.2.9
Comment
[lo21
f h
a
d,f c, e, g, h d,f
L,,Y,P,,P
C361 c931 c941 C891 [651 C651 c911 c711 c711 [791 c541 c31 c341 c341 ClW Cl001 [@I c251 C781 WI
a
L&P 0
;h
a
c, g a, e h
L,,LYL,LLW LI,L:,L;,W’
b, c, d, f a, d a, b, c, e, g ash a-h : b-h i,f e
L, SP T ‘it,‘is,Ll+,d-,SP ccc cc0 1occ 1occ DWB2 DWBlOP DWBlLE DWBlLE
consistency between the two independent experimental data sets and their agreement with the theoretical prediction underlines the validity of the present formalism. 3.2.2.4. Conclusions. We conclude this section by pointing out that the somewhat complicated inversion procedure could be avoided and the perfect scattering experiment could be achieved, for example, as follows: measurement of the in-plane asymmetry parameter B, i.e., a type 2B experiment, would resolve the ambiguity in the two (y’, y”) pairs. We also notice from Eqs. (3.2.12, 13) that, for example, the equations for the pairs (SA, U) as well as (T, V) can be recast in a form similar to the real and imaginary parts of Eq. (3.2.16). Consequently, determination of one of the pairs gives a geometrical ambiguity in the (A’,A-) pair, as discussed previously for (‘it,?“). The ambiguity can be removed by measurement of (at least) one more of the remaining STUI/ parameters. If (L,, y, P,) are known from unpolarized beam experiments and (L\, L”,, w’) from a type 2A experiment, any three of the five parameters B, SA, T, U, I/ will suffice to achieve a perfect scattering experiment. 3.2.3. Excitation
of the Na 32P state: Data
After this general discussion we are now prepared for inspection of the theoretical and experimental results for excitation of the Na 32P state at various energies. Fig. 3.2.9 shows the differential cross sections and the various alignment and orientation parameters, including all theoretical and experimental results in the literature available to us at the time of writing.
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Na(3'P) 4.1 eV
-
I
0.8 0 (a)
P
-I
I, 90
I
180 0
--
u I
I
I
90
I
I,
180 0
--
T I
I 90
I
I 180 0
v I
/
I 90
I
I
_, 180
g(deg)
Fig. 3.2.9. Survey of alignment and orientation parameters for excitation of the Na 3’P state by spin-polarized electrons. See Table 3.2.1 for sources and symbols. Throughout this paper, differential cross sections are given in units of &sr.
Extensive sets of theoretical results originate from the “close coupling plus optical potential” (CCO) and the “convergent close coupling” (CCC) methods of Bray and collaborators [33,34], and the “second-order distorted wave Born” (DWB2) approach of Madison et al. [68]. All three methods account, to some extent, for the effect of other discrete and continuum target states on the results for a particular transition. This is achieved in the DWB2 through a numerically exact evaluation of the second Born term, and in the KC0 method through the close coupling of y1discrete target states and an approximate optical potential to represent the coupling to other
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Na(3’P) 8.1 eV
I
0 @)
I
I
90
I
I
180 0
I
I
,
90
I
I
I
I
180 0
I
90
I
I
180 0
90
180
Wed Fig. 3.2.9. continued
states. Finally, the CCC method corresponds to a close-coupling expansion with a large basis including low-lying physical discrete target states and a large set of “pseudo states” that simulate the effect of the higher-lying discrete as well as the continuum states in the interaction region to sufficient accuracy. These results are supplemented by n CC close-coupling calculations from Trail et al. [loo] and by some first-order DWBl calculations. Although the y1CC approach only includes the coupling between IZdiscrete target states, it is expected to be fairly accurate at collision energies below the
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N. Andersen et al. J Physics Reports 279 (1997) 251-396 Na(3’P)
0.0 ’
0 (c)
’
’
’ ’ ’ ’ ’ 90 180 0
’
10 eV
’ ’ ’ ’ ’ 90 180 0
’
1 I I I ’ ’ ’ 90 180 0 90
I
-1 180
O(de.4 Fig. 3.2.9. continued
ionization threshold. On the other hand, the DWBlOP approach of Balashov and GrumGrzhimailo [25] accounts for the effect of other target states via a complex optical model potential, representing exchange with the target electrons, charge cloud polarization and loss of flux into inelastic channels. Such a method is expected to do well at higher collision energies above the ionization threshold. Finally, the DWBlLE calculations by Mathur and Puroit [78, Xl] include a local exchange potential only. We first note that the CCC and CC0 results are very close to each other in most cases, with deviations mainly to be observed in the singlet channel. In these cases, such as for the parameter
282
N. Andersen et al. / Physics Reports 279 (1997) 251-396
Na(3’P) 12.1 eV
-
0.0 L ’ 0 (4
--
P ’
’ ’ ’ 90 180 0
u I
-I
I 90
,
I
T t
‘(
180 0
* 90
,
I 180 0
v !
I
I 90
1
I
d_1 180
~(deg) Fig. 3.2.9. continued
L”, at 10 eV, the CCC approach performs consistently superior to CCO. Including also the DW-results in the discussion, we see a general convergence of the various methods at the higher energies. As expected, the DW-predictions become gradually less reliable with decreasing energy, with significant deviations from both the experimental data and the close-coupling results developing below 10 eV. Fig. 3.2.10 analyses the convergence with basis size n = 2,4,7,10 for the y1CC calculations at low energy. For most parameters, the 1OCC calculation seems to have sufficiently converged at 4.1 eV.
N. Andersen et al. /Physics Na(32P)
I
0.8 0 Cc)
I
I 90
I
I 180 0
I
I
I 90
I
283
Reports 279 (1997) 251-396 20 eV
I
I
I
180 0
I 90
I
I 180 0
I
1
1 90
I
I
_, 180
o(deg) Fig. 3.2.9. continued
Turning now to the experimental results, the situation is generally quite good at several energies for many of the parameters, in particular when including the additional data extracted by the inversion technique outlined in the preceding paragraph. As stated above, however, a better coordination of the experimental efforts would have been beneficial. Additional experimental results obtained, for example, with the in-plane laser pumping scheme described above, could finally achieve “the perfect scattering experiment”. This would be desirable for at least one convenient electron impact energy, such as 10.0 eV, to remove remaining ambiguities.
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Na(3’P)
22.1 eV
f8ded Fig. 3.2.9. continued
When comparing theory and experiment for Na 32P excitation, the most strik :ing general observation is the very good results obtained by the CCC method at all energies. 3.2.4. Excitation of the Li 2, 32P states Corresponding, but considerably less extensive results are available for the Li 2’P state. Fig. 3.2.11 shows results from the CCC, CCO, DWB2 codes already mentioned for Na, supplemented at 20.0 eV by DWBlLE results of Mathur [66]. Only CCC data are given at 4.1 eV, since the CC0 method produces essentially the same graphs. We notice that many features in the graphs are similar to the corresponding results for the Na 32P state at the same collision energy. There is
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_ ...f’. I
I
1
i
Y:
I
,
0 0
0
(Ed
e(ded Fig. 3.2.9. continued
again qualitative agreement between the predictions of the various theories, especially at higher energies. The CCC and CC0 curves only differ noticeably for the singlet scattering channel. Experimental data only exist for the r parameter [22] which have been transformed to w’ results. As for Na, the comparison favors the CCC/CC0 predictions. For the Li 32P state, theoretical data exist which were obtained by the CC0 and the CCC methods [20], as well as first-order [S6] and second-order [75] DW methods. The results at 10 and 20 eV are displayed in Fig. 3.2.12. As in Na, the CC0 results are often very similar to those
286
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Na(3’P) 54.4 eV
0
o.gol
I
0 (h)
I
I
90
I
I
I
180 0
90
180 0
90
180 0
90
180
o(deg) Fig. 3.2.9. continued
obtained with CCC, except for some significant deviations in the singlet spin channel. Furthermore, there are often large discrepancies between the predictions from the CC and the DW approaches. Note also that the agreement between the two sets of DW results is not very satisfactory. The differences could be due to the importance of second-order effects and/or the use of different atomic wave functions and distortion potentials. Nevertheless, on this limited basis we notice some similarities between the results for the Li 22P and the Li 32P states at the same collision energy, in accordance with the approximate n-independence noticed earlier for the alignment and orientation
N. Andersen et al. /Physics
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0.L
1
0
-90
l30.5
0.8 0
(4
P
L I
--
I 90
I
I 18010
u
I
--
I
I 90
I
, 18010
T
I
--
1 I 90
I
I 18010
v
I
I
I 90
I
,
-1 180
Q(k)
Fig. 3.2.10. Convergence with basis size in n CC calculations (n = 2,4, 7, 10) for alignment and orientation parameters 2CC; for excitation of the Na 3’P state by spin-polarized electrons. The results are taken from Trail et al. [loo]. ( ~--) (,‘.“,,4cc;(-----)7cc;( p) 1occ.
parameters for the He n’P Rydberg series, cf. Vol. I, Section 3.1.3 and Fig. 3.1.16. Such similarities within a Rydberg series are even more prominent for electron scattering from atomic hydrogen and will be further discussed in Section 3.2.6. Considering the similarities between the data for the Li, 2, 32P and the Na32P states, the fairly satisfactory situation for Na, and the very considerable investment in time and other resources involved in the experimental investigations, it seems unlikely to us that much significant new information can be gleaned from further studies of the Li case at the present time.
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Na(3’P)
I
(b)
I
I
I
8.1 eV
I
1 I
I
I
I
O(detd Fig. 3.2.10 continued
3.2.5. Excitation
of the K 42P state
Information on the K 42P state is even more sparse. Fig. 3.2.13 shows theoretical results from the CCC, CC0 and DWB2 methods. Again, we notice similar trends at higher energies, but also big differences at lower energies, in particular for the singlet channel parameters L”, and y”. No experimental results exist, not even for unpolarized electron and atom beams. In view of the richer structures predicted for this atom, and the intriguing differences between the (angle-integrated) ionization asymmetries observed between Li and Na on one side and K on the other [32], this element might offer a more attractive experimental challenge than Li.
289
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
0.5
0.0 0
(a)
90
18010
90
18010
90
18010
90
180
@(ded
Fig. 3.2.11. Survey of alignment and orientation parameters for excitation of the Li 2 *P state by spin-polarized electrons. -)CCC[20];(-----)CCO[20];(.--...)DWB2[75];(..p-)DWBlLE
Finally, Fig. 3.2.14 shows a comparison of results from the early pioneering experiments in Bederson’s group [47] with various theoretical predictions. The parameter reported, here called W, is the fractional depolarization of the polarized atom beam, observed as the change of spin state of a potassium beam resulting from resonance excitation with subsequent decay back to the ground state. To approximate the effect of the finite experimental resolution, we convoluted the various theoretical results using a 2-D Gaussian with CJ= 5”.
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Li(22P) 1OeV
,
I
-90
-180 180
I,
10
0.5
-.
I
‘..
.,.,,.
.,’ ,:
SP
5‘4
I3
,
I
0.5 -
:t ;:
I I
I ,
1 I
t !
-1 1
0
\' p
:\ ; ..,.:
-I-
Fig. 3.2.11. continued
The CCC and CC0 calculations show, once again, the best overall agreement with the experimental data. However, given the importance of the convolution procedures (demonstrated in Fig. 3.2.14 by CCC results with and without convolution), this statement should be regarded more on a qualitative than on a quantitative basis. 3.2.6, Excitation of the H 2, 3, 42P states Electron impact excitation of hydrogen is a fundamental process, which justifies its position as a benchmark system for state-of-the-art quantum theory. It was found in Vol. I that the situation,
N. Andersen et al. /Physics
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Reports 279 (1997) 251-396
Li( 2*P) 20eV
0.8 -
: . P
0.7
I 0
Cc)
I
I 90
I
, 180 0
I
I
I 90
U
1 1 180 0
I
I
, 90
,
T
, 180 0
1 I
1 I 90
V
1
_, 1x0
f4deg) Fig. 3.2.11. continued
even for unpolarized electron excitation, was only satisfactory at relatively small scattering angles, and this situation has not improved dramatically in the meantime. Perhaps not surprisingly, considering the requirements for such an experiment, no experimental results for alignment and orientation parameters exist yet for hydrogen excitation by polarized electrons, but some theoretical results are available. Fig. 3.2.15 shows data for excitation of the H 22P state. We notice an overall agreement between the various theories, CCC, CC0 and DWB2. Furthermore, the agreement with the (unpolarized)
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Li(YP) 10eV
()*I
1
0 (a)
I
I
90
I
I
I
180 0
I
I
I
90
II
I
180 0
I
I
I
90
I
I
I,
180 0
1-1
,,I,
90
180
@(de4
Fig. 3.2.12. Survey of alignment and orientation parameters for excitation of the Li 3 ‘P state by spin-polarized electrons: ccc [ZO]; (- - - - -) cc0 [20]; (. . . . . .) DWB2 [75]; (- - -) DWBILE [56]. ( ---)
experimental data is good up to about 90”, but from then on there is a marked disagreement with the experimental results. These latter ones have been obtained by two groups with different methods and are internally consistent. Further experimental results with unpolarized electron beams, in particular for the difficult P3 parameter (providing further information on LI and P) at large scattering angles, are highly desirable for this system, which persists to challenge our understanding of the perhaps most basic inelastic process in the whole field. In view of the
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Reports 279 (1997) 251-396
Li(3’P) 20eV
I
1000
I
I
,
I
u
I
1 I
--
I
1
u*
I
r
I
I
I
I
1 I
I
,
1.0
-0.5
0.0 180
0
: P 0.7
I 0
(b)
,
I 90
I
U I 180 0
I
I
I 90
I
T I 180 0
!
1
1 90
I
V 1 180 0
I
I
I 90
/
I
-1 180
@(deg) Fig. 3.2.12. continued
experience on the e-Na system with the singlet spin channel being the source of theoretical difficulties, experiments with polarized beams may be important to gain information about the individual spin channels. On the other hand, the theoretical predictions are very similar for the individual spin channels and, therefore, for the spin-averaged results. This is not surprising, since the importance of electron exchange is already quite small at 35 eV incident energy. Hence, we recommend a refined experiment with unpolarized beams as a first step.
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K(4’P) 4.1 eV
(4
*(de4
Fig. 3.2.13. SUI-vey ofalignment and orientation parameters for excitation of the K 4 ‘P state by spin-polar :ized electrons: ) ccc [21]; (- - - - -) CC0 [21]; (. 1. .) DWB2 [73]. (--
Fig. 3.2.16 shows data for excitation of the H 3’P state. The theoretical curves (CCC, CCO, and DWB2) look in almost all respects very similar to the results just discussed for the H 22P state, and the conclusions therefore remain the same. The similarities in shape for y1= 2,3 encourage comparison of the alignment and orientation parameters within the Rydberg series. This is done in Fig. 3.2.17, which shows results for it = 2,3,4, as obtained in the CCC (a) and the DWB2 (b) methods. As was the case for the corresponding series
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295
K(4’P) 10 eV
0.5
0.0
IO 0.5
0.0
(h)
O(d-4 Fig. 3.2.13. continued
in lithium, we notice very similar curves for the various n values, suggesting an approximate scaling law, or n-independence, for the parameters. 3.2.7. Excitation of the Cr 4’P state The Cr 47S -+ 47P transition is an example of excitation in a multi-electron system with high spin, S = 3. Thus, in analogy with the case of Na 32P, electron impact excitation may lead to sextet and octet scattering. The number of independent scattering amplitudes is four and a
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K(4*P) 20 eV
0.5
.._,._. ...,. I: P
0
T
U 0.90
’ 0
’
’ ’ 90
I ’ 180 0
(cl
I
I 90
I
I
I
1
180 0
I 90
I
V I 180 0
I
I
I 90
I
I
-1 180
@(%I Fig. 3.2.13. continued
parametrization in a set like Eq. (3.2.8) applies. Only one experiment has been performed for this system [53 3. The NIST group determined the parameters (r, L”,, L”,) (indices “0” for “octet” and “s” for “sextet”) at 6.8 and 13.6 eV. Similarly to Eq. (3.2.6), the r-parameter is related to octet and sextet weights through r = @/cY = w”/ws = 6w”/(l-
8~“) = (1- 6w”)/8w” ,
(3.2.22)
N. Andersen et al. /Physics
0.25
0
5
10
15
20
0.40
1 -
7 eV
W
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0.40
-’ 10 eV
W
0.35
0.30
0.25
0.20
(,,_,,,___,_ 0
..I 5 Scattering
10
. . . . . j 15
Angle
20
0.20~,_.._,,,..,__..,_.__, 0
4 5
(deg)
Scattering
10 Angle
15
20
(deg)
Fig. 3.2.14. The Bederson W-parameter for excitation of the K 4 ‘P state by spin-polarized electrons at incident energies ) 17CC (for 3 and 5 eV) and CCC of 3,5, 7. and 10 eV. Experimental data are from [47]. Theoretical curves are: (-~ (for 7 and 10 eV) [21]; (- - - - - ) 2CC [SS]; (- -) 3CC [77]; (-- .) Ochkur approximation [47]; (. ) “distorted wave polarized orbital” calculation [64] for 5 eV and DWB2 [73] for 1OeV. Also shown for 1OeV are CC0 results (- - - - -). The theoretical curves have been convoluted with a two-dimensional Gaussian of half-width 5” to simulate the experimental angular distribution. The effect of this averaging is demonstrated in the figure for 7 eV where the dashed line is the CCC result without convolution.
Table 3.2.2 Alignment and orientation
parameters
for excitation
Symbol
Source
0 q
cw Cl101 Cl111 Cl081
0 a
of the H 2’P state by spin-polarized
electrons
Fig. 3.2.15
Comment
a. c
;‘.
c351 C671
b b b a-c a c
c271 c951 Cl061
a, b a, b b, c
P, fl’, ;‘>P, L,, P ;‘, P, CCC DWB2 cc0 IERM CCPS
298
N. Andersen et al. /Physics
Reports 279 (1997) 251-396
H(2*P) 35 eV
(a)
o(k)
Fig. 3.2.15. Survey of alignment and orientation parameters for excitation of the H 2 ‘P state by spin-polarized electrons. See Table 3.2.2 for sources and symbols.
where we have used that 8w”+6w”=l.
(3.2.23)
The experimental results, together with the spin-averaged angular momentum LI, are shown in Fig. 3.2.18. We notice substantial differences between L”, and L; , as well as a significant angular dependence of SW” at the lower energy. Due to lack of data for (Pl,y) from an experiment with
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Reports 279 (1997) 251-396
H(2’P) 54.4 eV
b4
.
3
_,
o,oI
I
0 (b)
I
I
90
I
I
I
180 0
c
!
._
.-
I
I
B
br !
,-!
I
I
!
!
!
I
I
I,
!
.I
!
!
!
!
!
I
I
I
I
I
\
U /
3
I
90
I
180 0
90
I
180 0
90
J_, 180
~(k4 Fig. 3.2.15. continued
unpolarized electrons, and the spin contraction parameter T, application of the inversion procedure to retrieve the remaining phase angles in analogy to the Na case is infeasible. Recently, Bartschat [ 153 performed two-state nonrelativistic R-matrix (close-coupling) calculations, with just the two atomic states of interest, (3d54s)‘S, (3d54p)‘P, included in the closecoupling expansion. From a theoretical point of view, an interesting feature of the chromium atom concerns the question to what extent the 3d-shell remains “inert” in the scattering process. If this shell is essentially unaffected by the collision, similarities to electron scattering from alkali atoms,
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H(2’P) 100 eV 100
Fe----x u
i, 1. \
‘\ .\
‘..._ ..’ .....
i..‘\,
I
0.8 0 Cc)
.. ..
-0
0.9 -
C I 90
I
P
( 180 0
I
I
I 90
I
U
1 180 0
I
1
I 90
I
T
I 180 0
I
1
I 90
V 1 I
-1 180
d(deg) Fig. 3.2.1.5. continued
where one has singlet and triplet channels for the valence plus the continuum electron, can be expected. In this case, it should be possible to use a simple model potential for the core, including the half-filled 3d-shell. The results of two models, an “all-electron calculation” and a “quasi-two-electron calculation” are also shown in Fig. 3.2.18. In the former model, the 24 electrons of chromium and the continuum electron were treated explicitly, whereas in the latter model, the nucleus and the inner 23 electrons of the Cr target were represented by a core potential. Only the outer target electron and the continuum electron interacted with each other and this inert core.
N. Andersen et al. /Physics
0.8
I 0
(a)
I
I 90
I
P
I 180 0
I
1 1 I 90
U
301
Reports 279 (1997) 251-396
I
I
180 0
1 I 90
I
T
I 180 0
I
I
I 90
,
V
I
-1 180
@(deg)
Fig. 3.2.16 Survey of alignment and orientation parameters for excitation of the H 32P state by spin-polarized electrons: CCC [35]; (. . . . .) DWB2 [67]. (0) Cl 121;‘( ---)
Beginning with the angular momentum transfers, we see that the results are, in most cases, in agreement with the experimental data of the NIST group C-531,although quantitative agreement is only achieved for small scattering angles. This indicates that two-state close-coupling is not sufficient at these rather high collision energies. (The ionization potential of Cr is 6.763 eV.) We note, however, that the positive values of the angular momentum transfers at small scattering angles are in agreement with general propensity rules, as discussed in Vol. I. qualitative
302
N. Andersen et al. / Physics Reports 279 (1997) 251-396
H(3”P) 54.4 eV 10
L
’
’
’
’
’
1
I
I
T
I
I
I
I
I
I
,
I
I
I
1
I 1
1.0
u’
0 :: : ..:
0.8
I 0
(b)
I
, 90
I
P
I 18010
1
I
1 90
I
U
T I 18010
I
,
, 90
I
V I 18010
I
I
I 90
I
I
-1 180
*(deg)
Fig. 3.2.16. continued
Similarly, qualitative agreement between theory and experiment can be seen for the octet weight w“. As a general trend, we note that 8w” should approach 4 for high energies, since exchange effects become negligible and, therefore, there is less and less difference between scattering in the sextet and the octet channels. For a given energy, this effect will first occur near forward scattering angles. This overall expectation is, indeed, qualitatively supported by experiment and theory at both 6.8 and 13.6 eV total collision energy.
303
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
I
10
I
I
I
I
,
I
H(32P) 100 eV , 1 I 1
I
u
I
rl’
I
I
,
I
I
I
I
1.0
uJ
LKU
YU
0
0
-
r
t
+
+
PY
0.0
I
1n
,,,I
-Yr--J.FYY-
I.”
y
_,,.
. .
: :
I,,,,
...
+
SA
SP III,,
III
-
I I
I I,
I
I
I
B, I
.. . ..
__
1 1 1
,I
:. ,:
0.9
P I I
II,
0.8 0 (cl
0
-
90
180 0
I
I,,, 90
U
1
lI
180 0
,
90
I
I
180 0
I
I I 90
V I I
__1 180
Wed Fig. 3.2.16. continued
The differences between the results of the core potential and the all-electron calculations at large scattering angles, however, confirm the importance of core effects for small classical impact parameters in this collision system. Consequently, model potential approaches are not expected to yield reliable results for this problem over the entire angular range. For the energies investigated
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H(n*P) 54.4 eV 1
10-5 r
0
-1 90
0
-90 10
0.5
0.0 1.00
0.95
0.90 0 (a)
90
18010
90
18010
90
18010
90
180
e(k)
Fig. 3.2.17. Scaling with principal quantum number (n = 2, -; n = 3,. . . ; n = 4,- ~ -) of alignment and orientation parameters for excitation of the H n2P state by spin-polarized electrons at an incident energy of 54.4eV, as calculated in (a) the CCC [35] and (b) the DWB2 [67] approximation.
experimentally, which correspond essentially to one and two times the ionization threshold, the preliminary two-state close-coupling approaches are not sufficiently accurate. This is not surprising, since such energies are well known to be extremely difficult for theoretical investigations, even for such a simple system as atomic hydrogen where the structure part of the problem is known.
305
N. Andersen et al. / Physics Reports 279 (1997) 251-396 H (nZP) 54.4 eV I
2.0 180
10-s r
0
,180 -i80
-1 90
0
-90 10
0.5
0.0 IO
0.9
0.8 0
90
18010
90
(b)
18010
90
18010
90
180
@(deg) Fig. 3.2.17. continued
3.3. The incoherent case without conservation Photon polarization analysis
of atomic reJection
We now discuss the more complicated case where parameter. This is the case when explicitly spin-dependent or other relativistic effects, must be taken into account or simply in the description of the target states, for
symmetry:
the density matrix has a finite height forces, such as the spin-orbit interaction either in the F-operator for the collision example, by an intermediate coupling
306
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Cr
(4’P) 1 3.6 eV
6.8 eV 1.0 I
,
o
-1 .o;
-0.5
.
-I”0
I
30
60
90
120
150
10
’
20
30
40
30
40
!
180 1
-0.5
.
I
-108
’ 0
1.0
30 a
60 ..I
90
120
150
180
120
150
180
m
SW0
0.01 0
30
60 Scattering
90
Angle (deg)
0.0;
10
20 Scattering
Angie (deg)
3
N. Andersen et al. /Physics
Reports 279 (1997) 251-396
307
scheme. As stated in Vol. I, the density matrix elements no longer describe the electronic charge cloud but rather the excited state (J = 1) distribution. Since this state radiates like a set of classical oscillators completely analogous to a ‘P, state, we shall maintain our previous notation, simply replacing the term charge cloud density by oscillator density. For simplicity, we also maintain the parameter name L, though, strictly speaking, J,_ would be more appropriate. As shown in Vol. I (see also Section 2.1, Eq. (2.1.8)), this fact increases the number of independent parameters and, therefore, requires the measurement from another direction, such as y”, of an additional Stokes parameter, usually denoted P4, to determine the form of the charge cloud for unpolarized projectile and target beams. This, however, is not sufficient for a complete experiment because of the incoherence introduced by the averaging and summation over unresolved spin states. The extension necessary for a complete experiment for the J = 0’ + J = 1” transition will now be discussed. In this section, we only analyze the photon polarization in the exit channel; the electron spin parameters will be discussed in Section 3.4.
3.3.1. Analysis of the J = 0 + J = I case 3.3.1.1. Scattering amplitudes and state multipoles. There are six independent scattering amplitudes for a J = 0 + J = 1 transition, thereby requiring the determination of one absolute differential cross section, five relative magnitudes, and another five relative phases of the scattering amplitudes.
This large number of independent parameters will necessarily lead to considerable complications. Nevertheless, the natural coordinate system will enable us to disentangle the scattering amplitudes and generalize the parametrization of the density matrix for the case of unpolarized beams in a straightforward way. Applying a similar analysis as in [16] for the collision frame, the nonvanishing amplitudes f”(Mf,mf,mi) in the natural frame (see Fig. 3.3.1) for a Ji = 0 + Jf = 1 transition are [4]: (3.3.la) (3.3.lb) (3.3.lc) (3.3.ld) (3.3.le) (3.3.lf) Fig. 3.2.18. Excitation of the 4 ‘S + 4 ‘P transition in electron-chromium scattering with a total energy of 6.8 eV (left column) and 13.6 eV (right column). The three parameters 8w”, L; and L”, are plotted together with the spin averaged angular momentum L,. The experimental data from [53] (obtained via superelastic scattering) are compared with theoretical results from ail electron ( -) and model potential (- - - - -) two-state close-coupling (R-matrix) calculations [ 153.
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N. Andersen et al. /Physics
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Fig. 3.3.1. Schematic diagram of scattering amplitudes in the natural frame for J = 0 + J = 1 transitions impact. Note that A+ + 6’ = A- + 8.
by electron
where we have omitted Jf = 1 and Ji = Mi = 0. Note that (3.3.la-d) represent no-JEipamplitudes that leave the projectile spin unchanged while (3.3.le, f) describe the cases where the electron spin is flipped.
The analysis becomes most systematic by using the reduced density matrix formalism, in connection with an expansion of the density matrix elements in terms of state multipoles (for details, see Appendix B). The state multipoles (tko) in the natural frame can be expressed in terms of these amplitudes as [4] CtEO)U= (1/2$)(a:
+ P: + aZ_+ p”- + pi + @$)= (1/2J5)(O,,.rri* + Cfrip)3
(3.3.2a)
(tlo)” = (1/2&X2,
+ p’+ - cz - /3’-) )
(3.3.2b)
(t?o)U = (1/‘2fi)(a:
+ fi: + aZ- + p” - 2[pi + a:]) = (1/2&)(b”+nrp
(tz2)” = (tz_2),* = ~(a+a_e’(~+-+-)
(61)” = (tt-1)”
= (Gl)” = (tl-1)”
+ /?+P-eicti+-@))
= 0
)
,
- 2orrip) 3 (303.2~) (3.3.2d) (3.3.2e)
(3.3.3a) (3.3.3b) (3.3.3c)
N. Andersen et al. /Physics
(t);2)Pz =
(tf&)$z
- /?+fi_ei(JI*-@)) ,
= (Pz/2)(a+z_ei(4+-4-)
(tl*LL = (G-1&,
= (GJP,
MO)P,
=
NO)P,
=
(WP,
=
-
=
-
(pX/2~)(cl+poei(9+-“o)
= 0-JP, =
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Reports 279 (1997) 251-396
(3.3.3d) (3.3.3e)
= 0,
G)P,
=
+
(t”z-Z)P,
=
(3.3.4a)
0 ,
~+~oei(~+-40)
+
x_floe-i(4m-*o)
+
fl_goe-i(tiL
-40))
,
(3.3.4b)
= - (P,/2JZ)(a+P,e’(~+-~“)
+ /?+ccoei(ti+-bo)
_ [CC_poe-i(6--W + p_ccoe-iW-&d]), GO)P,
=
O?O>P,
(WP,
=
-
=
(iP,/2~)(~+Poei(h+-~“) _
=
(GO)P,
=
W2)P,
=
(3.3.4c)
WdP,
=
0
(3.3.5a)
Wl%~
[x_poe-i(d-
-ILO) _
/3+aoei($+-40)
p_aoe-iW
-&)I)
,
(3.3.5b)
=
(iP,/2~)(a+Poeit~+-“~)
- P+010ei(ti+-60)
+ z-floe -U$--W _ p_Oloe-i~#--&)) ?
(3.3.5c)
where the subscript “u” indicates that a quantity is independent of the projectile spin polarization while the other subscripts denote the dependence on the individual polarization components. Inspection of Eqs. (3.3.2,3) shows that the six real state multipoles with rank Q = 0 contain all information about the magnitudes of the scattering amplitudes; also, these equations are linearly independent. Consequently, if the absolute differential cross section cU = $(t;lo)U is known, the five relative magnitudes can be determined using onZy one projectile polarization, P,, perpendicular to the scattering plane. Note that the “unpolarized” multipoles can be determined by taking suitable combinations of results obtained with polarized electron beams, i.e., there is no need to repeat the experiment with unpolarized beams. A detailed discussion is given in the next section. It is also interesting to note that the real and imaginary parts of (t!jz)U and (t!j2)p: allow for unambiguous determination of the phase differences 6T = $+ - 4_ and d1 T I,$+ - I,!_ (see Fig. 3.3.1), i.e., transversally polarized incident electrons with polarization vector perpendicular to the scattering plane can be used to measure not only all the magnitudes of the scattering amplitudes but also two of the five independent relative phases. For a determination of other
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relative phases, one needs to measure the four state multipoles (til)p,, (L;~)~,, (~7~)~~and (til)P,. In principle, not all relative phases that enter into Eqs. (3.3.4,5) need to be determined, due to the following relationships: (4+ - $0) - (b- - $0) = b+ -6
(3.3.6a)
(ti+ - 40) - (ti- - Al) = ti- -$+.
(3.3.6b)
On the other hand, determination of relative phases in different ways can serve as a valuable consistency check in these very sophisticated experiments. 3.3.1.2. Generalized Stokes parameters. The emitted radiation can be analyzed systematically in terms of the generalized Stokes parameters introduced in [4]. They are defined in such a way that all four possible combinations of photon polarization analyzer and initial electron polarizations enter on an equal footing. We first introduce the following notation: The quantity Z$PX,Y,,(p) is the light intensity transmitted by a linear polarization analyzer oriented at an angle fi for incident electron polarizations in the x”, y” or z” direction, with the light being observed in the direction denoted by ri. Similar definitions are made for the intensities transmitted by circular polarization analyzers. This gives, for example, G(45”) - _
= 4K,
JL(135”) - .
= tlx,
+ ~~&)yiP,l 3 -
(3.3.7a)
V~2&1
and the total intensity for unpolarized incident electrons can be constructed z,y = ~[Zy,,(45”) + ZY,(45”) + Zy,,(135”) + ZY,(135”)] .
(3.3.7b) as (3.3.8)
We define the “generalized Stokes parameters” Qfy measured with a photon detector in the direction n^ and electron spin polarization along the direction A by taking the three other independent linear combinations of the four intensities. The second subscriptj = 1,2,3 refers to the sign combinations + + - -, + - - + and + - + -, while the first subscript i = 1,2,3 refers to the photon polarizer settings (O’,90”), (45”, 135”) and (a-, o+), as for the standard Stokes vector. For example, we get Q;; s
Z’+p,(OO)+ ZY,(o”) - Zy,,(90”) - ZY_,(90”) Z”+p,(OO) + ZY,(O”) + z:pZ(900) + ZYpJ90°) ’ - Gz(O”) - C.(90°) + %Z(900) Z’,,(OO) + ZY,(O”) + Zy,,(90”) + ZYpx(900) ’
QYZ _ ~Y,,V)
l2 Qf; E
Q
2’;
Z’,,(OO) - ZY,(O”) + Z”,pJ90°) - ZY_,(90”)
Zy+p;(OO) + Z’,(O”) + Z”+p,(90°)+ ZYpj90°) ’
Zy,,(45”) + ZY,(45”) - Zy,,(135”) - 1’,(135”) = Zy,,(45”) + ZY,(45”) + Zy,,(135”) + ZY,(135”) ’
(3.3.9a)
(3.3.9b)
(3.3.9c)
(3.3.10a)
N. Andersen et al. J Physics Reports 279 (1997) 251-396
Q;; E
1:pJ45”)
- P&(45”)
Zy,,(45”) + IY,(45”)
- Zy,pJ135”) + IY,(135”) + Iy,,(135”)
+ IY,(135”)
(3.3.10b)
’
yz _ &(45”)
- P-,(45”)
+ &(135”)
- 1’,(135”)
Q23 = Iy,,(45”)
+ 1’,(45”)
+ ly,,(135”)
+ 1KpJ135”) ’
(3.3.1Oc)
z:p;(G-) + ZYpz(a-) - ZY,p&‘) - z’p;(o+)
QE=It&-)
(3.3.1 la)
+ ZYpz(G-)+ zy+p:(rJ+) + Z“pZ(CT+) ’ + zKpz(o+)
z:p.(o-) - Z~‘,(o-) - zQT+)
Qg; =Z’;,(C)+ ZYpr(fT)+ zQJ+)
+ I!++)
311
(3.3.1 lb)
’
Z?pja-) - zYpjc-) + zY,pja+)- Z“pZ(f7+) ZY,p:jF) + ZYpz(C)+ zy+pJJ+)+ Z”pZ(CT+) .
(3.3.1 lc)
Qc3 =
The standard three-component Stokes vector is thus replaced by a 3 x 3 matrix. Note that the first three components (3.3.9a, 3.3.10a, 3.3.1 la) form the usual Stokes vector (Pi, P,, P3), as measured with an unpolarized beam, and that all denominators in Eqs. (3.3.9-11) are equal to the intensity 1: of Eq. (3.3.8). Next, we define relative state multipoles which are normalized to the monopole term (t$o)U. i.e., essentially the absolute differential cross section. For example, (3.3.12a)
(3.3.12b) Inserting the relative state multipoles’ from Eqs. (3.3.2-5) into Eqs. (3.3.9-l 1) for the generalized Stokes parameters and using equations (3.3.7,8), we find for the light intensities in the (z”, y”, x”) directions after unpolarized electron beam impact: C = 1 + (I/$)
(3.3.13a)
(Go)”
(3.3.13b)
1: = 1 - (l/2$)
(Go)ll
- (,/5/2)
1,” = 1 - (l/24)
(f%)”
+ (&I
(f%)”
(3.3.13c)
.
In Eqs. (3.3.13), we have normalized the intensity in such a way that the monopole term simply produces a factor of unity. This is justified since, in practice, only ratios of light intensities will be measured, i.e., the absolute differential cross section is assumed to be determined in a separate experiment that does not require the measurement of absolute light intensities in an electron-photon coincidence setup. Furthermore, we note that corrections due, for example,
’ F’or J = 1, the (r;(u),’
s and (&),
s are identical
to the ( T,,
Jo)’
s vol.
I,
Appendix A
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to hyperfine structure effects in the target, can be implemented by multiplying the above state multipoles by standard perturbation coefficients (see e.g. [107] for the present case of interest). The expression of the various generalized Stokes parameters (3.3.9-11) in terms of the relative state multipoles requires some tedious but straightforward algebra. The final results can be compactly expressed in matrix form. With I referring to the total light intensity emitted in the direction of the photon detector for unpolarized incident electrons, we find for the z” direction [4]:
(UQ:;l=[
;$::;: % j.
(3.3.14c)
In the y” direction one gets: (IQi’j”)= C-$j(60>~+~(~~2)J
I
C-~(GO)P,+~(GZ)P,I
I 7
00
0 0
0 0
(3.3.15a)
C- -jg (rZn0 >” + (IQ;)
C(~;Io)p,-~(r~o)pi-~(r;Z)p,]
=
0 i
0
SFM2 >“I
&i> I1
- &
P,
(3.3.15b)
(ii&,
(3.3.15c)
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313
Finally, we obtain for the Y direction:
c-+o%d” x8 (IQ?)=
-~
(3.3.16b)
0
(3.3.16~)
The left column in all these matrices represents the standard Stokes parameters Pi, P2 and P3 that one obtains with unpolarized incident electron beams. It is, however, important to recall the definition of the angle fi for the linear polarization analyzers, as given in Fig. 3.3.2. For example, it turns out that Qy; = QfT = QfT = - P4 and Q45 = - P5 while Q{; = P6 in the notation of [90]. Finally, we point out that observation directly along the incident beam axis, which could be difficult from an experimental point of view, is not necessary; a choice of (0 = 45, @ = 45”), for example, can be used to replace Eqs. (3.3.16) and obtain the same information. 3.3.1.3. Density matrix parametrization. We now proceed to the parametrization of the density matrix for the classical oscillator density, thereby generalizing and unifying the approaches used in Section 3.2.1 of this review for spin-polarized electron impact excitation of hydrogen-like targets (see Eq. (3.2.4) for excitation of sodium) and in Vol. I, Section 3.3.1, for unpolarized electron beam excitation of heavy atoms. The latter results have been summarized in Section 2.1, Eqs. (2.1.3a) and (2.1.X) of the present review. We recall from Eq. (3.2.4) that the density matrix for impact excitation of sodium with unpolarized beams was decomposed into an incoherent sum of two parts, namely a singlet and a triplet component. The parameters for the individual spin channels, such as L\ and Lp, could then be disentangled by means of polarized electron and atom beams. In Eq. (2.1.3a), the density matrix for target atoms such as Ne, Ar, Kr, Xe or Hg was decomposed in a similar way for excitation with an unpolarized electron beam, namely into a pair of matrices with one having positive reflection symmetry with respect to the scattering plane and the other one
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Fig. 3.3.2. Coordinate frame for definition of generalized Stokes parameters. The linear polarizer settings in the directions 6 = xn, y”, z” are shown for polarizer angles /I = O”, 45”, 90”, and 135”, following the notation of Blum [28]. The incident polarized electron beam is characterized by polarizations + P,, +I’,, or *P,, as indicated.
having negative reflection symmetry, respectively. The importance of excitation of states with negative reflection symmetry and the corresponding spin-flip processes is measured by the height parameter h. The most important physical effect that causes a nonvanishing h is the spin-orbit interaction. It can manifest itself both in the description of the atomic target states (the dominant effect in a light system such as Ne) and in the magnetic interaction of the continuum electron with the target nucleus. (i) Electron beam polarization perpendicular to the scattering plane: We begin the discussion of polarized electron beam excitation by first assuming a spin polarization perpendicular to the scattering plane, i.e., along the z” direction of the natural coordinate system. The natural extension of the above-mentioned decomposition to the case of polarized electron beams is again the introduction of a pair of density matrices, one for spin-up electron impact excitation and one for spin-down excitation where “up” and “down” correspond to the initial spin component orientation with respect to the scattering plane. Hence,
(1 - h) ;
pu = cu
[
l+Lf
0
0
0
0
_ p+1 e2iy 0
1-L:
(
1
=
(1 - hf) ;
wb, I
-P:e-2iY
+LfT
0
0
0
_ pl+f e2iYT
0
+h
0
0
0
1 0
0
0
-pP:Te-2iYf 0 1
-
LfT
+ hT
0
0
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Reports 279 (1997) 251-396
1-t Ll’i
0
_ Pl+lep2i$
0 _ PI+1e2iG
0
0
0
: 0
+ hi
1 -Ll’l
r
\ i‘j. 315
01
0
(3.3.17)
Here we have defined: 2 LfT
=
Lll
2
%+
-
xP
2:
+
ix2
=
_
pT 3 ’
-fit
-
-/I:
fl’ = _ pl
+/I”
PI + ipi = -
pt+le2i+
p;
+
ipj
=
(3.3.18b)
3’
pt+Te2iyT =
=
(3.3.18a)
_
2-j+c(_ei(4--4+)
ax:+ x4
(3.3.18~)
9
2’+~~~‘~~““, +
(3.3.18d)
,a;)
(3.3.18e)
d = fi: + p’ + fig,
(3.3.18f)
ITT= cl”, + x2 +
0” = +
(d + x2 + 3; + p”++ /I’ + p;, = ;
((JT+ 01) ,
(3.3.18g)
x;/c+)
(3.3.18h)
h1 = /3$/d ,
(3.3.18i)
WT= oT/(2cJ” ) 3
(3.3.18j)
= oL/(2a,)
(3.3.18k)
h’ =
d
= 1 - WT.
From these definitions, (1 - h)Lf
it follows that
= wt(l - h’)~I’r + wl(l -
(1 - h)P:e2’;
= wt(l _ ht)pl+te2irf
Ir = w’h’ + wlh’ = (x; + /3;)/(24
hl)Lfl , + W1(l - h1)P:le2i71
,
(3.3.19a) ,
(3.3.19b) (3.3.19c)
P;t=Jc-@j$/cFp,
(3.3.19d)
P:-=Jc-@p=Jc-p,
(3.3.19e)
2;~‘=4_-~+fTC=-~+T,
(3.3.19f)
21/L=$_-$+_t7T=-6+T,
(3.3.198)
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Consequently, the decomposition of the density matrix (3.3.17) is described in terms of the cross section 0” and seven dimensionless independent parameters: ((r& wr, LI’T, Lf 1,IIt, hJ; yt, $) .
(3.3.20)
This set leaves three relative phases still unknown. Information about these phases can be obtained with an electron polarization in the scattering plane, as will be discussed in (ii) and Section 3.4 below. 3.3.1.4. Density matrix parameters from generalized Stokes parameters. We now analyze the relationship between the parameter set (3.3.20) and the generalized Stokes parameters. For an electron beam polarized along the z” axis, the generalized Stokes vector matrix for observation in the + zn direction is given by (cf. Eqs. (3.3.2-5) and (3.3.14a)):
+ j?+p- cos Sl]
cos 8 - p+p_ cos $1
2[cl+a_ sin 8 + fJ+fl_ sin $1
2[cr+a_ sin dt - fi+/L sin $1
- (B: + B” - 2@)3
-[(LY: -or’)
$(U4 + a2 - 2U;)
-v:
- (P: + p” - ml
- 2[cc+a_ cos 8
(IQ;;)
= ;
-[(a:
-a’)
+ (B: - 891
- 2[a+a-
-P’)l
3[(cl”+ + cr2_- 2ai) - (B”++ BE - 2/%)1 &lx”+ + CZ - 2a;)
or, in terms of the density matrix parameters,
I
(3.3.21)
(IQ;;) = wT(l - hT)PI - w’(1 - /+)Pi ;
$[wT(l - 3h’T)- wl(l - 3V)]
WT(1- h’)PJ - w”(1 - /+)Pi ;
$[wT(l - 3hT) - wl(l - 3P)]
wT(l - h’)Pl - wJ(l - /+)Pj ;
$[wT(l - 3hT) - wJ(l - 3h”)]
)
‘i (3.3.22a)
with the (normalized) light intensity (3.3.22b)
I = rZ,= $(l - h) and (1 - h)Pi = ~‘(1 - hT)P’ + wl(l - hl)P/ ;
i = 1,2,3 .
(3.3.22~)
Assuming that the height of the charge cloud for unpolarized electron impact, h, is known from a standard P4 measurement, one may use the sum and the difference of the elements in the first two columns (i.e., six parameters) of Eq. (3.3.22) to obtain wt(l - hT)Pt/(l - h) and wJ(l - hi)PL/(l - h), where Pt = (PI,Pz, Pi) and P1 = (Pi,Pi,Pi),respectively. Since the degree of polarization P+T,L E (Prvl 1= 1, one can extract cf = wt(l - hT)/(l - h) and ci = wl(l - h’)/(l - h) from the sum of the squares of the individual components, and subsequently Lf t, LIi, 1)Tand y1from the Stokes vectors Pt and Pi.Since ct + c1 = 1 theoretically, any
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deviation from this relationship should be remedied by renormalizing all elements of the generalized Stokes matrix by a common factor. The last column determines [ wt(l - 3hT) wl(l - 3h1)]/(1 - h) which, when combined with CTand cl, allows for determination of wT,hT and hl. Thus, knowing h, the seven dimensionless parameters of (3.3.20) can be determined from QiJ. It is important to note that the independent determination of h cannot be replaced by Eq. (3.3.19~); this equation is not independent from the others and, therefore, can only be used as a consistency check. A description of how to extract the parameter set (3.3.20), including hyperfine-structure depolarization effects, is given in Appendix D. The equations for the nonvanishing elements in the first row of the matrices (QG) and (QF) (see Eqs. (3.3.15a) and (3.3.16a)) are (written as a column for convenience):
=-
- [wT(l - 3hT) + wl(l - 3h1)] - [wT(l - hr)P[ + wi(l - hL)Pf]
3 A
i
-
‘\
[w’(l - 3hT) - wl(l - 3h1)] - [wf(l - ht)Pi - wJ(l - hl)P;]
[wT(l + ht)
,
(3.3.23a)
- wL(l + hl)] + [wT(l - hT)PI - wl(l - hJ)P;]
with the normalized light intensity I = I,y = 2
[(l + h) + (1 - h)P,)]
(3.3.23b)
,
and
IQ;; IQ:'2 i IQ:"3 I
1 / - cwT(l - 3hT) + wl(l - 3h1)] + [wf(l - h’)Pi + wl(l - /+)I’;] \ - [w’(l - 3hT) - wl(l - 3h1)] + [wT(l - hl)P; - wl(l - hl)Pi]
=-;
!
[wT(l + hT) - wl-(1 + hl)] - [wt(l - hT)Pl - wl(l - hl)P;]
I
,
(3.3.24a)
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with the normalized light intensity I = 1; = ; [(l + h) - (1 - h)P,)]
(3.3.24b)
.
Except for the P4 measurement with unpolarized electrons, no further information can therefore be obtained from generalized Stokes parameters observed in other directions (such as X”or y”) with electron beam polarization vector perpendicular to the scattering plane. On the other hand, such additional measurements with photon detectors in the scattering plane can provide valuable consistency checks. (ii) Electron beam polarization in the scattering plane: As mentioned above, three more relative independent phases are needed to determine the scattering amplitudes uniquely. In analogy to the sodium case, we define A+++-$+,
(3.3.25a)
A-~~_-$-,
(3.3.25b)
A0 s cjo - t+bo.
(3.3.25~)
Inspection of Fig. 3.3.1, however, shows that only two of these are independent, A + - A - = $ - 61 = 2(yi - yt)
since (3.3.26)
in analogy to Eq. (3.2.7). Therefore, it remains to fix the phase of the spin-flip amplitudes fJ,l relative to the nonflip amplitudes. As will become clear below, a convenient choice for the remaining phase angle is $1 G C#)+- e. . A complete set of independent (0” ; wT,LfT,Lf”,ht,hi;
(3.3.27) parameters is then given by yt,y1,A+,Ao,6t”),
(3.3.28)
i.e., one absolute cross section, five relative sizes and five relative phases. Information about the remaining three phase angles may be sought for in experiments with in-plane spin polarization, a possibility that we will now explore. Inspection of Eqs. (3.3.14b,c) shows that no additional information may be obtained with such a beam polarization if the light is observed along the z” direction. Instead, Eqs. (3.3.15b,c) and (3.3.16b,c) reveal eight nontrivial components, namely IQ?; and IQ$‘i with &n^= xx, xy, yx and yy. Nonvanishing values for the latter generalized Stokes parameters require both a “twist” and a “tilt” of the classical oscillator density with respect to the collision plane. The related twist and tilt of the atomic charge cloud density have been discussed in [90] and [83] and were characterized by two additional rotation angles, 6 and E, respectively. An example of such a “twisted-tilted” charge cloud is shown in Fig. 3.3.3 [83]. We will here omit the parametrization of the density matrices for in-plane polarizations P, and P,, in terms of twist and tilt angles for spin up and spin down with regard to these two directions. Instead, we give the expressions for the nonvanishing generalized Stokes parameter components in terms of the relative magnitudes of the scattering amplitudes and the phase angles defined in
N. Andersen et al. /Physics
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Reports 279 (I 997) 251-396
*n
c
+pY
X”
Fig. 3.3.3. Example of a tilted and twisted charge cloud due to an in-plane spin polarized incident electron beam (from [83]). The figure is for Hg 6 3P, excitation with impact energy 8 eV, initial spin polarization P, = I and scattering angle 0 = 30
Eqs. (3.3.25). To clarify the algebraic structure of the general expressions for these components,
we
(3.3.29a) (3.3.29b) (3.3.29~) (3.3.29d) (3.3.30a) (3.3.30b) (3.3.3Oc) (3.3.30d)
(3.3.31a)
(3.3.31b)
(3.3.31c)
(3.3.31d)
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Inspection of Eqs. (3.3.31) shows that a measurement of the generalized Stokes parameters in the y” and x” directions with in-plane electron beam polarizations PYand P, will provide the four relative magnitudes and the four relative phases angles defined in Eqs. (3.3.30,31). Provided that the analysis in the zn direction with polarization P, described has been done before, all relative magnitudes are already known, and the four A parameters may serve as consistency checks. For example, the parameters LIT and ,!,lL are derived as LfT
=
(4 - &)I(& + 4) >
L;1 = (AZ - A:)/(&
+ A;) .
(3.3.32a) (3.3.32b)
Additional information, however, may be obtained from the o angles. They are seen to determine the relative phase angles within the amplitude triplets (fj r, j? 1, f,i) and (fj i, fi 1, f,), corresponding to final spin up and down, respectively. However, the phase of the two triplets with respect to each other cannot be obtained from the co angles. We notice the relationships 01
-
co3
=
CY =
w2 - 04 = 8=
)
(3.3.33a)
- 2yl + 7c*
(3.3.33b)
-
2yt
+
n
Provided that yTand yLare already known, these relations allow us to extract two additional angles only, e.g. ~5~~ and (4’ + d - ). This is possible, for example, by photon observation in the y” direction only, as can be seen from Eqs. (3.3.31a,c) after eliminating co3 and o4 with (3.3.33). For a perfect experiment, a determination of any of the three angles d +, d - or do will suffice. This, however, cannot be achieved without spin analysis in the exit channel, since the incoherent summation over the unobserved final electron spins destroys the phase information between the amplitude triplets that belong to final electron spins up and down, respectively. This aspect will be further discussed in Section 3.4. 3.3.2. Excitation of the Hg(6s6p)3Pl state Results from an electron-polarized-photon coincidence experiment with a spin-polarized incident electron beam have been reported by Goeke et al. [46] and by Sohn and Hanne [90] for electron impact excitation of the (6~6p)~Pi in mercury. (We keep the notation of a triplet state that is well established in the literature, although the state is more appropriately described in an “intermediate coupling” scheme.) Sohn and Hanne used a transversally polarized electron beam with polarization vector either perpendicular to the scattering plane (i.e., P,” # 0 in the natural frame) or in the scattering plane (i.e., Py” # 0). The photons were observed either in the z” or in the negative y” direction. The latter sign change, however, does not affect the basic information that can be obtained. It can be incorporated in our theory by deriving Eqs. (3.3.15) with polar angles (0 = 90”, @ = 180”). The individual count rates needed for the determination of these parameters have been obtained by Sohn and Hanne. They combined them to give the nonvanishing Stokes parameters PI, . . . , P4 for unpolarized electrons, and they also defined light polarizations where the analyzer setting was switched for fixed electron polarization. Furthermore, they defined asymmetry parameters by
321
N. Andersen et al. / Physics Reports 279 (1997) 251-396 Hg 6’S,, -+ 63P1 1.0
0.0
-1.0 TO
0.0
-1.0
-1.0
7.0
I.0
0.0
0.0
- I .o
-1.0 0
180’ 0
90
90
180’0
180
90
f4deg)
(a)
Hg 6’So + 63P,
-1.oL 0 (b)
’
’
’ 90
’
’
’ lSd0
’
90
1804I
I
’
90
’
’
-1.0 1x0
~(ded
Fig. 3.3.4. Generalized Stokes parameters for electron impact excitation of Hg(6s6p)3 P, at an incident electron energy of 8 eV: (0) measured (PI-P.+) [46] and (P2, PJ, P,, P6) [90]; (0) unpublished data of [90]; (A) derived from measured polarization and asymmetry data of [46,90]; ( x ) average of measured (A) Qy3 and Q;Z3data; (7) prediction based on properties of the generalized Stokes parameters (see text). The experimental results are compared with the predictions from a five-state Breit-Pauli R-matrix calculation based on [SS].
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Hg 6’5, -+ 63P, 1.0
1.0
0.0
0.0
,
1.0
I
I
I
90
0
4
I
I
b
180 0
I
I
90
Cc)
1 180 0
I
1
I
90
I
_-1.0 180
‘4deg)
Hg 65,, + 63P, 1.0
1.0 ++
4::
Q;; 0.0
0.0
180’ 0
90
Cd)
90
180
e(deg)
Fig. 3.3.4. continued
switching the electron polarization for fixed linear or circular polarization analyzer settings, i.e., asymmetries A”” where AA are defined in the same way as for the Qfj!‘.For example: Ay”(o”) ~ 1 I(O”, +P=) - r(o”, -PA P, I(O”, + PZ) + 1(0°, - PZ) ’
(3.3.34a)
AY’(90”)_ I 1(90°, +P=) - 1(90”? -PA P,1(90°, +P,) + 1(90”, -Pz) ’
(3.3.34b)
AY”(45”)_ i 1(45”, +P,) - 1(45”, -PA P,1(45”, +P,) + 1(45”, -Pz) ’
(3.3.34c)
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323
Hg 6’S,, -+ 63P,
180’0
90
0
90
1x0
‘4deg)
(e)
Fig. 3.3.4. continued
Ay”(135”) _ 11(135”> +P,) - J(135”> -PJ Pz 1(135”, +P,) + 1(135”, -Pz) ’ AY’(rT+)
1 I(o+, +Pz) - I@+ -Pz) Pz I(o+, +P,) + I(o+, -Pz) ’ 3
E
-
Ay”(g_)EII(PzI(a-,
+Pz)-lb-, +P,)
+ I@,
-Pz)
(3.3.34d)
(3.3.34e)
(3.3.34f)
-Pz)
and similarly for P, and Py. These parameters contain similar information as the generalized Stokes parameters, though in a nonsymmetric form. We have used data from [46] and [90] to calculate the generalized Stokes parameters for the case of initial electron spin polarization perpendicular to the scattering plane. Experimental results for an incident electron energy of 8 eV are shown in Fig. 3.3.4. Unfortunately, the full set of generalized Stokes parameters for the z” direction could only be obtained for three scattering angles (lo’, 20” and 307, due to some missing raw data for the circular polarization parameters. Although Q’1; = QZ;; = Qy3 according to Eq. (3.3.22), the experimental data for Qy3 and Qys do not agree within the specified error bars, thus indicating some internal inconsistency between them. We have therefore averaged these results and plotted them as x in the field for QZ;3. Further consistency checks may be performed by using data from the y” direction to evaluate, for example,
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the third column of (QT]F)in yet another way. Inspection of Eqs. (3.3.22) and (3.3.23) shows that 1 + PI
2Q"1"2 + 3Q”1”3+ 4 1+p 4Q;;=0.
(3.3.35)
We have solved for Q;‘3 and plotted the results as V in the third column of Fig. 3.3.4(a). Once again, the trend is correct, but the data are not entirely consistent. Finally, using the available experimental data, we can predict experimental results for generalized Stokes parameters that have not been measured to date, such as those that would be observed in the X” direction, i.e., along the incident beam axis. Inspection of Eqs. (3.3.22-24) yields (3.3.36a)
Qf"z =
1
1_+p;;4 Q;; + ’ +p4Qyz , 1 - PrP4
(3.3.3613)
(3.3.36~) where we have used that Q”1;= PI and Qfi = - P4. The predictions are plotted in Fig. 3.3.4(c). The agreement between the experimental results in Figs. 3.3.4(a-c) and the theoretical predictions based on a five-state Breit-Pauli R-matrix calculation [SS] is satisfactory, bearing in mind the complexity of the collision problem, the level of detail in the comparison, and the remaining inconsistencies within the experimental data set. The agreement between experiment and theory is good for the “in-plane” polarization parameters Qi; and Qg; shown in Fig. 3.3.4(d), except for a theoretical underestimation of Q z; at small scattering angles. The extraction of the set (3.3.20) of spin-dependent density matrix parameters (except for the cross section 0,) from the generalized Stokes parameters is significantly complicated by depolarization effects from hyperfine-structure (hfs) interaction in mercury isotopes with nonvanishing nuclear spin. In order to apply the recipe outlined above, we used the measured generalized Stokes parameters, together with known hyperfine depolarization coefficients Gr = 0.887 and Gz = 0.789 for state multipoles with ranks K = 1, 2 Cl071 to first determine the state multipoles. These can then be used to calculate “reduced generalized Stokes parameters” without hfs-depolarization. Since the system of equations derived from (3.3.14a) and (3.3.14b), relating the generalized Stokes parameters to the state multipoles, is overdetermined for the multipoles (r&,)p,, (r&,)p, and (r~z)p, (see Appendix D), the outcome of this procedure is not unique - except for a hypothetical, entirely consistent experimental data set with insignificant error bars. The results shown in Fig. 3.3.5 [117] were therefore obtained as follows. It turns out that the relative state multipoles (r&)“, (rTo)“, (r&,)U, (rzz),, (rlo)p, and (i;2)p, are determined through a unique set of equations given in Appendix D. Furthermore, three different sets of equations, each omitting one of the three generalized Stokes parameters QyZ, Q”1;and Q:;, were used to obtain the then uniquely determined set (rl;o)p,, (r&)p, and (r&)pz. The average of the latter set obtained with the three possibilities was also calculated. This procedure yields four different sets of state multipoles which were used to calculate the reduced generalized Stokes
N. Andersen et al. /Physics
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Hg 6’50 -+ 63PI at 8 eV
-90
IO 0.5
0.5
o.ot/t 0
h’
--#It, I
I
I
90
I
I
180 0
I
90
I
L
Y,
180 0
I
I
90
I
U-v, I
180 0
,\I 90
t I_,,, 180
e(W Fig. 3.3.5. Spin-dependent coherence parameters for electron impact excitation of Hg(6~6p)~P, at an incident electron energy of 8 eV [117]. The experimental data were calculated from the generalized Stokes parameters presented in Fig. 3.3.4 (see text): ( x) prediction based on data of [46] and [90]; ( + ) prediction based on unpublished data of [90]. The experimental data are compared with the results from a five-state BreittPauli R-matrix calculation based on [SS].
parameters and, finally, the set of spin-dependent alignment and orientation parameters (wT,L; t, L;l, hT,hi; i’T,$). F ig. 3.3.5 shows the results obtained with the average values for (r;lo)p,, (r;& and (rb >P,. Th e vertical lines connect the maximum and minimum results obtained with the three individual sets of equations for (r&,)pz, (Y;~)~, and (T&)~~, thus reflecting inconsistencies within the experimental data set. Purely statistical errors have been omitted; inspection of
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Fig. 3.3.4 indicates that they would be of the order of the symbol size and, therefore, mostly insignificant compared to the systematical errors. The two sets of results plotted in Fig. 3.3.5 were calculated from published data of Goeke et al. [46] and of Sohn and Hanne [90] ( x ), and also from unpublished data of Sohn and Hanne ( + ) for scattering angles of lo”, 20” and 30” [117]. As mentioned above, the important difference between these data sets is the fact that the latter set corresponds to a full determination of the generalized Stokes parameters shown in Figs. 3.3.4(a, b), i.e., all parameters that can be determined from the photon radiation pattern with electron polarization perpendicular to the scattering plane. On the other hand, the first set lacks data for QyZ and QZ;. While data for Qys are desirable for a consistency check between the three elements in the third column of the (QTT)matrix, data for QyZ are mandatory for performing the evaluation procedure outlined below Eqs. (3.3.22). In particular, the normalization P+T = PfJ = 1 ensuring full coherence in the individual spin channels, is crucial for the numerical stability of the inversion. Only a complete set of generalized Stokes parameters can be tested for this normalization and, consequently, be renormalized to fulfill this fundamental condition exactly. If any element in the first two columns of the (QfJ) matrix is missing, the normalization condition can be used without verijcation to calculate this element (except for the sign). Obviously, this is a much less satisfactory procedure and leads to significantly larger systematical errors, as seen in Fig. 3.3.5. This is a striking example of the importance of complete data sets. In the above case, the additional data for Q’;z not only provide more information about the spin-dependent orientation, but also stabilize the inversion procedure for all other parameters through built-in internal consistency checks. Interesting features can be seen in Fig. 3.3.5 after transformation of the generalized Stokes parameters into the spin-dependent coherence parameters defined in Eqs. (3.3.18). The differential cross sections for the two spin directions are very similar, with wTz 0.5 at all angles. The coherence parameters, however, are vastly different for the two spin channels. Note, for example, that LTt > 0 while Lll < 0 at small scattering angles! While the positive value of LTT is in agreement with well-established propensity rules (for details, see Vol. I), the negative value of J!,:~ might seem surprising. Note, however, that both Lf’ and ,!,I1 are nonzero for forward scattering, and that LfT(Oo) = - L,+‘(O”) by symmetry requirements. Also, the alignment angles yf and y1 show no similarities, with the directions of the major axes often being perpendicular to each other (once again, yf '(0')= - yf “(0”) # 0). Furthermore, there is a large difference between the height parameters hT and h1 (which measure the relative importance of spin-flips) in this angular range, with h1 assuming a maximum value of 75% near a scattering of 40”. This means that spin-flips are very likely for spin down electrons, but those spin down electrons, whose spin is not flipped, tend to transfer a negative angular momentum to the atom. This situation is illustrated in Fig. 3.3.6 [117]. These effects can only be revealed by using spin-polarized electrons beams. Due to the large spin-flip probability for spin down electrons and the comparable cross sections for both spin orientations, the spin averaged angular momentum transfer L1 is positive at small scattering angles, in agreement with standard expectations. The theoretical (Breit-Pauli R-matrix) prediction of LI1 < 0 is clearly confirmed at 10” scattering angle while the error bars are too large at 20” and 30” to draw definite conclusions. We recommend a detailed experimental investigation of this effect, particularly of the prediction LI1 z - 1 near 40”.
N. Andersen et al. /Physics
T
Reports 279 (1997) 251-396
327
t
0
63
(al
(bl
Fig. 3.3.6. Illustration of angular momentum transfer by spin up (a) and spin down(b) impact excitation of Hg(6s6p)3PI for small scattering angles at an incident electron
electrons, as predicted energy of 8 eV [117].
for electron
Fig. 3.3.7 shows further theoretical results for the spin-dependent coherence parameters for excitation of the Hg(3P1) state, as calculated in the five-state Breit-Pauli R-matrix approximation at 6.5 and 7.5 eV incident energy and in the semi-relativistic first-order DBWA at 15 and 40 eV. The comments made previously on the R-matrix results for 8 eV apply equally well to those at 6.5 and 7.5 eV. In the DWBA results, we also notice significant differences between the predictions for the “spin up” and the “spin down” parameters. However, with increasing energy, both LIT and L,‘l start out with values near zero values in the forward direction and become positive at small scattering angles. Also, the DWBA results predict that LT1 increases faster with increasing scattering angle than LTt - the opposite situation from the (Breit-Pauli R-matrix) results at the lower energies. Given the importance of semi-classical treatments in many areas of atomic collision physics, we recommend further experimental tests of these predictions. Combined with theoretical efforts, such tests may yield new insight into the validity of propensity rules for spin-resolved collisions [ 1171. 3.3.3. Excitation of the Xe 6s,6s’ states Fig. 3.38 displays the results of first-order DWBA calculations for the generalized Stokes parameters after electron impact excitation of the Xe (5~~6s)~Pi state at an incident electron energy of 25 eV. Surveys of the spin-dependent coherence parameters for excitation of the (5~~6s)~Pi and the (5p56s’)‘P1 are presented in Figs. 3.3.9 and 3.3.10. Compared to excitation of the Hg 63P1 state, the differences in the predicted results for “spin up” and “spin down” excitation are much less pronounced for the xenon target, especially at 40 eV incident energy. Furthermore, the results for both J = 1 states become very similar with increasing energy. This is to be expected because of the very strong singlet admixture in an intermediate coupling representation of both states. Since excitation of this singlet component becomes more and more predominant with increasing energy, all the relative parameters essentially reflect the results for excitation of this component.
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The only electron-photon coincidence experiment for impact excitation of xenon with spinpolarized electrons was reported by Uhrig et al. [104], who measured the spin asymmetry given in Eq. (3.3.34a) with an electron polarization P, perpendicular to the scattering plane and photon observation in this direction. The results for an incident electron energy of 25 eV, normalized to an electron polarization of lOO%, are shown in Fig. 3.3.11 and compared with predictions from a first-order semi-relativistic DWBA calculation based on [29]. In light of the difficulty of the
Hg 61So + 63P1 at 6.5 eV
(4
e(deg)
Fig. 3.3.7. Survey of spin-dependent coherence parameters for electron impact excitation of Hg(6s6p)3PI by spin) five-state Breit-Pauli R-matrix calculation at 6.5 and 7.5 eV based on [SS]; (. . . . .) polarized electrons: (__ semi-relativistic first-order DWBA calculation at 15 and 40 eV based on [31].
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Hg 61S0 -+ 63P1 at 7.5 eV
V__
LfI -1
I
1
I
I
L+’ I I
--v
LT’ I
I
90
I
I
I
, !
,
,
w
,
,
-180 180
h/
0
~(de.4
@I
Fig. 3.3.7. continued
experiment and the smallness of the spin-dependent for further studies of this kind. 3.4. The incoherent case without conservation Electron polarization analysis
effect, we do not recommend the xenon target
of atomic reflection symmetry:
In this section, we only analyze the electron spin parameters in the exit channel while the photons are not detected. The initial atomic states are assumed to be unpolarized. The spin
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N. Andersen et al. /Physics
10 1
%: ' 1 1 1 1 u
Reports 279 (1997) 251-396
Hg 6'504 63P1 at 15 eV I ~ I I I ! I "..I, .. ..
0.01 1
0
=........
,.,“...(( ,,.. ... ,..' \. .i I i I I !
-
.,....‘.
.... : I I I .> I.;:I
,
I,,
I
I
,;. .’
:I":!, ;'., ; i ...,. : j' ::,, >Z '.. \ ,..._..,_ ;.. ...._..." ii :: "
:
90
1 I
1.0
\
\
I
I
I
I
I
II
yt ... ‘.., ..’ ‘.,, ..,,,...’..‘..
I ..,,,‘I,,: ...11,‘.;‘.,.I .. . i,,,,: i;
‘I
::"'>i i'.. -:j i: : '.. ..... : I;: ii :,: .,,i
0.5 -'\
1.0
I
‘. ‘::,..’ II ,,I , I II
II
i,,j
h
II
I I I ‘; I I, :, :: : .”..,, :: :: !,, i ,: \; __i /
1 1 II
i
I
I I L’, ,i_ /i f j;: j ii / ji :i iiii
:: /j .\,,.,; :
: i ii i p+t -_ :,
II
0.0 180
I
I
,
;:
P+l c -_
II
hr
I1 II ht
L f
, I
, I
0
-180
, I
180
A+’
q
;
e ,1 /I
I
1 I
(,.." ., ../ .. . . ...'. ._..
: .... i : ; : : ;:
; / - / I ;/ _: ,’ II ,I I, II , I ..i.-.. ; .;.i __ :..: ::
I,...........:i;
v
0.0
I L I
,..’
i..,.._.,..... ‘i
b -90
j I '.. I
\. ......_-
.‘:...
;
1
0.4
br'
..I., 0
I,
L+'
I ‘..,I ‘..,
0.8
-
. ; 2 ;” - .,,,.,.. ......
‘_
... “...._., _,.. .,” \
L+' 1
Lf
1 I 1 I I p ’ “Y. ’ .,_ (, ’
-
‘/,
“‘.L .,,,
1.0
; ”... ;: ...._
“
: .......,.x.,... i
-1
I I ! Wf
- i.. ..(
I.... 0.1
11
I,
0r
,
0 ;.-. . ._ .\ i ).,: , ,..i , -180 I: I I 180
:I I j ,;: *-I’, :
.....L,
,.,.: . .’ :. , , t; : , L 41.1 .,
k..,_
, :)
0
-180
Ao 'i.,J -
180
(.. ...'. x,, 0 !. ..,, ..." \ ,, ,...."..,, ........ .,;.' :;, .., .._.. 7- ; ..' .'... : ...." '. : :..,, i: -,(I :, ...” : '.., \, : ,.: ,:' _/ I '+---....'I ! I I I I I I Ii, I -180 0.0 l..y ' ' ' ' ' 90 90 180 0 180 0 90 180 0 180 0 90 0.5 -
(cl
."_'...
Q(detd Fig. 3.3.7. continued
observables studied are the generalized STU parameters introduced in Section 2.3. Explicit formulas in terms of reduced density matrix elements are given in Appendix B. The study of these parameters provides important, though seldom complete, information about the spin-dependent dynamics of the collision process. For heavy targets, simple relationships such as Eqs. (3.2.11-13) for e-Na scattering are generally not valid. New information can thus be obtained from the generalized ST U parameters.
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Hg 6’S,, -+ 63P1 at 40 eV
1
.:
--
‘:._yT
:
-- 1
‘. y’
;
.‘:..., ; :,.., !: :: / ..” ; i : ___ :, : ; j ; : \ ‘\ ;. : j ; : : / i : : j ; .. . i:>,,; ; -:,:, i:, ::.,,,j ; -‘.,,, -90 T:,,i :> I I I I I ..,I1,.:.:I,:.r:. I: II 1I .‘: I., ;‘:: I :ii I i. 1 1 I I , 1.0 :,: i / ::.I ::: : ; .“> ,: : :.... :, i:
O
:j 0.F) -1
(I.0
:
-i.p: ?
1
1.0
.. ::: : :: :: :: ::: :., :.;: ‘::,:i \ /: -:;,:: -‘;::
::
I
II
,I
II
h
:,:
g
,
:_” ./..,,,.. __,..,_,~ .; ,.._ : ; -.I : ,.. // I
/ :j :’ i / : i: ; :i. i,., i // :::: 1: :;:i ;//, :: i:: :,:/ /:: ___::i:., :: :/ :: ./ : ii:: :/ ,i :: i;
p+, e I , 1 1 1 I
I
t
I
I
,
i
1
I
-
I
0 (d)
90
180 0
90
I
f/ ;II
I
II
: A0
..”j
,.
.
;.’
180 0
90
,...
..
-180 180
. ..
0
;’ !I
P180 -
180
:
0
’
‘-180 180
..I
:
180 0
II
... ; ‘: : : .:
:i‘... : “.., ;“..,, ... .... ,:’, ;..y , ‘.,,,./ ..,I ,.,,,,, / I “..,.,i...” I 1
I
II
Am1
0.5 -
o,.
I
“...: ....
h’
h’
At1
--
3
’ ’ 90
6Ydeg)
Fig. 3.3.7. continued
3.4. I. Excitation of the Cs 62P112,312 states We begin with electron impact excitation of the 62Si,2 -+ 62Plj2,3,2 transition in cesium. While neither electron-photon coincidence studies nor the time-reversed superelastic scattering experiment have been carried out to date, there are some theoretical results available for the generalized STU parameters [99, 131. In the present case of e-Cs (62S1,2 --+62P1,2,3,2) excitation, there are 24 independent amplitudes, 8 for the 62Sr,2 + 62Pr,2 transition and 16 for 62S1,2 + 62P3/2. Given one undetermined overall phase for each fine-structure transition, the complete experiment would
332
N. Andersen et al. f Physics Reports 279 (1997) 251-396
Xe 5’S,, + 63P1 at 25 eV
0
90
180’ 0
180’ 0
90
90
180
@(de&d
(a)
Xe 5lS0 + 63P1 at 25 eV 1.0
0.0
-1.0 0
(b)
90
180’ 0
90
180’ 0
90
180
Wed
Fig. 3.3.8. Generalized Stokes parameters for electron impact excitation of Xe (5~~6s)~P, at an incident electron energy of 25 eV, calculated in a semi-relativistic first-order DWBA calculation based on [29].
N. Andersen et al. /Physics
Reports 279 (1997) 251-396
Xe 5’So A3P1
333
at 25 eV
1.0
0.0
I
-1.0 180’ 0
I
,
90
I 90
180’ 0
180
e(deg)
Xe 51Sa -+ 63P1 at 25 eV 1.0
Q:;
QYIS
0.0
-1.0 1.0-
-1.0 -1.0
0.0
0.0
-1.0 0
(4
90
180’ 0
90
-1.0 180
o(de.4
Fig. 3.3.8. continued
require the determination of one absolute cross section for each transition, together with 7 (15) relative magnitudes as well as 7 (15) relative phases for the J = i (p) fine-structure levels. So there are now 46 (!) parameters instead of 7 in the case of e-Na sodium. In light of the complications described for the latter case in Section 3.2.2, it is not surprising that no attempts towards a complete experiment have been made to date, and we do not expect such attempts in the foreseeable future. Nevertheless, the study of the generalized STU parameters reveals interesting information. An example is shown in Figs. 3.4.l(a,b) where the differential cross section CT,and the seven
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Xe 5lS0 -+63P1at 25 eV
(e)
o(ded Fig. 3.3.8. continued
independent parameters T,, TX, T,,, Sp, SA, U,,, and U,, are plotted for electron impact excitation of the h2S1,2 + 62P1,2,312 transitions in cesium at incident electron energies of 1.63 and 2.04 eV. These results have been obtained using scattering amplitudes from semi-relativistic five-state [86,87,13] and eight-state [13] Breit-Pauli R-matrix (close-coupling) calculations, as well as a fully relativistic five-state Dirac R-matrix calculation [lOl, 991. There are several interesting aspects to be seen in these figures. First, the results of the three Breit-Pauli calculations are in fair agreement with each other. As discussed in detail by Bartschat [13], the differences between the five-state results of [86] and [13] are mostly due to the use of different target wavefunctions. Second, as shown in [13], the discrepancies between the results of Scott et al. [S6] and those from other theoretical calculations have been removed by improving these target wave functions. Consequently, the five-state Breit-Pauli calculation of [13] is expected to yield the more reliable results. Third, the good agreement between the results from the five-state and the eight-state Breit-Pauli calculations indicate that the close-coupling expansion has sufficiently converged at these low energies. The most interesting observation, however, is the fact that the results from the five-state Dirac calculation is in quite good agreement with the Breit-Pauli results in some cases (for example, S,) while there are severe discrepancies in others. In particular, we mention the spin polarization function Sp. The Dirac results not only differ significantly from those of the Breit-Pauli approach, but also from the predictions of the fine-structure effect (see Section 3.2.1, Eq. (3.2.11a)). Note that the Breit-Pauli results at 2.04 eV collision energy are still in qualitative agreement with these predictions; one finds, for example, Sp,A(2P1,2) z - 2Sp,A(2P3,2). The deviations from the approximate relations at 1.6 eV for the Breit-Pauli calculations are expected to be primarily due to the
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N. Andersen et al. / Ph.ysics Reports 279 (1997) 251-396
energy difference between the two fine-structure levels. Such an energy effect has even been demonstrated for the sodium target [ll]. Experiments with a cesium target, like the one performed with sodium by Nickich et al. [79], would therefore be most valuable for a deeper understanding of relativistic effects in the collision dynamics, particularly with regard to the approximate validity of the fine-structure effect for heavy targets. We strongly recommend future experimental investigations in this area.
Xe !?SO -+ 63P1 at 15 eV 10
1 0.1
1,
o.o_l_ 1
0
4
-190
0
-90 10
0.5 -F 0.0 0.E
0.05
0.00 0
(4
90
180 10
90
180 10
90
180 lo
90
180
@(detz)
Fig. 3.3.9. Survey of spin-dependent coherence parameters for electron a semi-relativistic first-order DWBA calculation based on [29].
impact excitation
of Xe (5~~6s)~P~, calculated
in
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N. Andersen et al. /Physics
@I
Reports 279 (1997) 251-396
‘4de.s) Fig. 3.3.9. continued
3.4.2. The fine-structure effect in S + P excitation of Na, K, Rb, Cs, Cu and Au Following up on the above discussion of the fine-structure effect in low-energy electron impact excitation of cesium, we now analyze recent results of the Toronto group [115,116], who performed first-order full-relativistic distorted wave calculations for electron impact excitation of the resonance transitions (ns)?Sljz + (np)2PIj2, aI2 in the alkali atoms Na, K, Rb and Cs, as well as for Cu and Au. The results for the fine-structure resolved spin polarization function Sp are shown in Fig. 3.4.2 for impact energies between 20 and 100 eV. As one might have expected, the symmetry
N. Andersen et al. / Physics Reports 279 (1997) 251-396
337
Xe 5’S0 -+ 63P1 at 40 eV
(c)
f?(deg) Fig. 3.3.9. continued
relationshrp Sp(2P112) = - 2Sp(2P312) predicted in the nonrelativistic limit is very well fulfilled for low-energy small-angle scattering from Na and K. However, even in these light target systems, strong deviations from the predictions of the fine-structure effect are seen at larger scattering angles, and the deviations increase with increasing collision energy. Qualitatively, these results can be understood as follows. With increasing energy, the importance of electron exchange (one of the necessary conditions for spin polarization through the finestructure effect, see Section 3.2.1) decreases, and large scattering angles correspond to small classical impact parameters. Consequently, one might expect Mott scattering, which involves
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Reports 279 (1997) 251-396
interaction with the target nucleus, to become more and more important with increasing nuclear charge, collision energy and scattering angle. Since Mott scattering is independent of the angular momentum coupling of the valence electron, one would also expect similar results for both fine-structure levels. This overall trend is, indeed, seen in the results for all the alkali targets. Also note the absolute size of the Mott scattering effect - starting at about 3% for Na and K but increasing to 30% in Rb at 60 eV and almost 50% in Cs at 50 eV. A more detailed energy scan may even reveal higher absolute values of SP for these systems. Such an analysis is currently in preparation [116]. Xe 51S0 + 6’P1 at 15 eV
0.05 -
(a)
Ydeg)
Fig. 3.3.10. Survey of spin-dependent coherence parameters for electron in a semi-relativistic first-order DWBA calculation based on [29].
impact excitation
of Xe (5~~6s ‘)‘P,, calculated
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N. Andersen et al. /Physics Reports 279 (1997) 251-396 Xe 5’50 -+ 61P1 at 25 eV
0
@I
90
18010
90
180 10
90
180 10
90
180
o(deg) Fig. 3.3.10. continued
Similar trends can be seen in the results for the Cu and the Au target. The most surprising feature is the apparent dominance of Mott scattering over fine-structure effect at 40 eV incident energy for electron impact excitation of Cu. The results for both fine-structure levels are nearly identical, and the absolute value of the spin-polarization function has a maximum approaching 90%(!) near 65” scattering angle for a target with nuclear charge Z = 29. Note that an experimental test of this prediction would be possible in a standard electron polarization measurement. Since the predicted results are nearly identical for both fine-structure levels, no laser-pumping or other means to
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0
(cl
90
18010
90
180 10
90
180 10
90
180
fl(deg) Fig. 3.3.10. continued
separate them are required. Therefore, an experimental test of these results seems both feasible and highly desirable. 3.4.3. P + S excitation: In and Tl in In(n = 5) and Tl(n = 6) have been studied experiThe n2h,2, 3/2 -+ (n + 1)2S1,2 excitations mentally by the Miinster group [24,45]. Fig. 3.4.3 shows the spin asymmetry parameter SA measured at several energies from 4.0 to 9.0 eV for the 52P1j2 + 62S1,2 (left column) and
N. Andersen et al. / Physics Reports 279 (1997) 251-396
341
Fig. 3.3.11. Spin asymmetry ~“(0”) of Eq. (3.3.34a) for electron impact excitation of Xe (5~‘6s)~P, at an incident electron energy of 25 eV. The experimental data of Uhrig et al. [ 1041 are compared with results from a semi-relativistic first-order DWBA calculation based on [29]. The two theoretical curves are for an ideal photon polarizer analyzer (solid) and for the actual experimental analyzer power of 0.84 (dashed).
--) 6*Si,2 (right column) In transitions. We note a strong angular dependent variations for 5*p3,2 the i -+ i transition, whereas the curves for the $ + 3 transition exhibit much less structure. The experimental results at 4.0 and 5.0 eV are compared with seven-state semi-relativistic and four-state nonrelativistic Breit-Pauli R-matrix calculations of Bartschat [12]. It is interesting to note that algebraic recoupling of the nonrelativistic results accounts predominantly for the nonvanishing values of SA, i.e., including the spin-orbit interaction explicitly does not change the theoretical predictions dramatically. Also, the agreement between the semi-relativistic (and even the nonrelativistic) results and the experimental data is very satisfactory for the 52P1,2 -+ 6*S1/2 transition. Based on these findings, the large discrepancy between theory and experiment for the -+ 6*Si,2 transition is surprising. Whereas the theoretical results are in qualitative agree5*p3,2 ment with expectations based on the fine-structure effect, the experimental data clearly violate these predictions. Since the original publication [24], the experimental raw data for both transitions have been carefully re-analyzed and found to be consistent with the published results. Hence, we can only state the surprising result that one fine-structure transition seems to be described adequately in a semi-relativistic model while a much more sophisticated treatment is apparently required for the other transition within the same multipiet. A full-relativistic calculation of this problem is, therefore, highly desirable. Fig. 3.4.4 displays a summary of the generalized STU parameters for electron impact excitation of the (5p)*Pr,2,3,2 -+ (6s)*S1,2 transitions in indium for a total collision energy of 5 eV. As pointed out above, the theoretical R-matrix results are in qualitative agreement with the predictions of the fine-structure effect (see Eqs. (3.2.11)) for all parameters. For the corresponding cases in Tl experimental data are only available for the 6*P1!2 -+ 7*S1,* transition. They are shown in Fig. 3.4.5. Again, a strong variation with scattering angle and impact energy is observed. Semi-relativistic Breit-Pauli R-matrix calculations have been carried out [48,14], for which results at 4.0,5.0, and 6.0 eV are shown in the figure. Good, though not perfect,
N. Andersen et al. /Physics
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agreement is seen - once again, even when relativistic effects are only accounted for by recoupling of nonrelativistic scattering amplitudes. To further investigate the role of the fine-structure effect, Fig. 3.4.6 shows the STU parameters corresponding to excitation from the two fine-structure components of the Tl ground state configuration (6~~6p)~Pl,2, 3,2. In addition, the weighted average value defined in Appendix C is plotted. A clear deviation from zero is seen for all four averaged S and U parameters, showing the role of relativistic effects. Also, the fine-structure energy splitting is probably significant, thus contributing further to the breakdown of the pure fine-structure effect.
Cs 6’ PI/~
1.6 eV
Cs
6’P3,2
(a) Fig. 3.4.1. (a) Differential cross section and the seven generalized STU parameters Sp, SA, T,, T,, T,, U,, and U,, for electron impact excitation of cesium 62SI,Z -+ 62Pr12.3,2 at incident electron energies of 1.63 and 2.04 eV: ( -----) eight-state Breit-Pauli calculation [13]; (- ~ -) five-state Breit-Pauli calculation [13]; (. . . .) five-state Breit-Pauli calculation [86]; (- - .) five-state Dirac calculation [lOl, 991.
N. Andersen et al. /Physics
cs62P,,2
Reports 279 (1997) 251-396
2.04 eV
343
Cs 62 P3,2
fYdeg1
(b)
Fig. 3.4.1. continued
3.4.4. Excitation
of the Hg (6~6p)“~P states
We now move on to transitions of the form (ns2)‘So -+ (n~n’p)‘,~P~,~,~. Again, we use the established notation of singlet and triplet states, but keep in mind the more appropriate “intermediate coupling” scheme. The general description also applies to excitation of states with configuration (np5n’s) which will be discussed for electron scattering from inert gases. We first recall that the number of independent scattering amplitudes is given by 2. (2Ji + 1). (25, + 1). For the J = 0 + J = 0, 1,2 transitions, this leaves us with two (for 3P0), six (for each of the ‘s3P1 states) and ten (for 3P2) amplitudes, respectively. Once again, the level of complexity has increased dramatically, and the complete experiment becomes a much harder challenge compared to the nonrelativistic case. The Hg term diagram with the states of interest for this section is shown in Fig. 3.4.7.
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3.4.4.1. Excitation of the Hg (6~6p)~P~,~ states. We begin with excitation of the metastable (6~6p)~P~,~ states of mercury. While these states are well described in M-coupling (i.e., S = 1 and L = 1 are good quantum numbers), a fully relativistic description requires the use of 6p,,, and 6~312 radial orbitals (see, e.g. [97]). The simplest case is the excitation of the (6~6p)~P, state with only two independent amplitudes. A complete experiment can be performed by measuring the individual STU parameters in the way described by Berger and Kessler [26] for elastic scattering. It has not been performed to date, however, due to the small excitation cross section and the experimental problems associated with the isolation of the signal from the individual multiplet level with J = 0. It is important to note that the generalized STU parameters are not independent of each other for this particular case. It was shown in Eqs. (2.26) of [lo] that the following relations hold for a J = 0 + 0 transition: (3.4.la) (3.4.lb) (3.4.lc)
0.2
'. I "
Na 22.1
SP
-o.20""""' 30 0.1
I..
I "
I "
I
0.2
1 "-
I
1
eV
60
90
.“‘.I
120
150
‘I
180
.0.2~,""'..""""' 30 60 0.1
..1..1..1"1"1"
‘.
1.
I ‘.
90
I.
No 54.4eV
-O” (4
0
30
60
90
120
150
180
-"'O
30
60
90
I I Na 27.2 eV
120
150
180
I ” I No 100 eV
120
150
1 0
ScotteringAngle(deg)
Fig. 3.4.2. Spin polarization function Sp for the resonance transitions (ns)‘SIiz + (np)‘Pliz (solid) and (ns)‘S1,, -+ (np)zP3,2 (dashed) in Na (n = 3), K (n = 4), Rb (n = S), Cs (n = 6), Cu (n = 4) and Au (n = 6), as calculated in a full-relativistic first-order DWBA approach [116].
N. Andersen et al. /Physics 0.2
I..
I '. I
Reports 279 (1997) 251-396
345
'I "
K20eV
1
K 35 eV
-0.1 -
SP -O.*o 0.1
30
60
90
120
..1.'1....'1
150
180
.I..
-O.'o 0.1
30 '.I'
60
90
30
60
K 100
90
120
(b)
150
180
Scattering
-"'O
30
150
1
I"I"I'.I"
K 50 eV
-O.’ 0
120
60
90
120
eV
150
180
Angle (deg) 0.2
"
I
I "
I "
I "
'.
Rb 40eV 0.1 -
0.1
"I
'I.'I.
I
Rb 100
90
120
150
180
Scattering
-"'O
30
Angle (deg)
Fig. 3.4.2. continued
60
90
120
.I'
eV
150
180
346
N. Andersen et al. /Physics 0.2
I.. ;
I. .I..
! ,.,.
Reports 279 (1997) 251-396 -
0.2
I..
I.
I
I
..,
Cs 20eV
Cs 35 eV
0.1 -
SP
-O.Zo"""" 0.6
30
..I.
60
I.' P" 90 120
,..n
* .' 150 180
.,..I..
-0.20 0.4
30 "
I..
60 I,,
90
120
I.
.,
150
160
..,
Cs 100eV 0.2 -
-"'60
30
60
90
120
150
160
-"'40
30
60
90
120
150
180
120
150
180
ScotteringAngle(deg)
Cd)
-0.5 -
-1.0
0
..'.~~"'..'.~~'~ 30 60 90
: Cu 60 eV
Cu 100 eV
0.1 -
-0.1 -
-"'O
(4
1..
8.. 30
I..
60
I
90
.
I
120
.
6
150
I 160
-o~o"""“""""' 30 60
ScotteringAngle(deg)
Fig. 3.4.2. continued
90
120
150
180
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0.3
I
I
I
*
Au 50eV
Au 30eV
Scattering
:
Angle (deg)
Fig. 3.4.2. continued
Uy.y=
7lf
’
7li
’
U,y
)
(3.4.ld)
where ni and rrf are the parities of the initial and final states, respectively. This leads to Sp = - SA, T, = - 1, T, = - TX and U,, = - U,, for our case of interest. As mentioned above, the situation is much more complicated for excitation of the (6~6p)~P~ state with ten independent amplitudes and almost no chance for a complete experiment. The study of the generalized STU parameters will, nevertheless, provide some information about the importance of relativistic effects on the outcome of the collision process. This can be seen by investigating the results predicted in the approximation of the pure fine-structure effect. One finds [49, 50, 311: s P=
-
S‘4
3
(3.4.2a)
T, z - 0.4 ,
(3.4.2b)
TX + T, z - 0.6 ,
(3.4.2~)
u 4’1% - UX,
(3.4.2d)
Furthermore, related by
the results for the S- and the U-parameters for the two 3Po,Z fine-structure levels are
SP,‘4(3Po) * - 2SP,A(3P2) >
(3.4.3a)
UqQJ3Po) = - 2U.X,,,J3P2) .
(3.4.3b)
The validity of the approximate relationship (3.4.3a) can be assessed in Fig. 3.4.8 where recent experimental data of the Miinster group [38] for the asymmetry function SA are compared for the two fine-structure levels. The agreement with the prediction of the fine-structure effect improves with increasing collision energy since the energy difference of 0.77 eV is relatively less important at 15 eV than, for example, at 6.5 eV incident energy. In fact, one might expect the approximate relationships (3.4.3) to be better fulfilled by comparing the results for the 3P0 state at 6.5 eV with
348
N. Andersen et al. / Physics Reports 279 (1997) 251-396
I
‘..,
7L
I
0
++++-I -1
6.0
e\,I
1
6.0 eV
0
I
0
3
9.0eV
-1 1
@@I@%+
-1
L---u-0
90
1:SO 0
90
O
-1 180
Fig. 3.4.3. Angular dependence of the asymmetry parameter SA for inelastic scattering for the In 52P1,2 + column) and In 5’P,,, + 62S1,2 (right) transitions at 4.0,5.0,6.0, and 9.0 eV: (0) experimental data from [24]; semi-relativistic calculation including the spin-orbit interaction within the data based on [12]: ( -) between the projectile electron and the target nucleus through the operator (~‘2/2) Ci(l/r”)li.si; (relativistic calculation with the spin-orbit operator (~‘12) xi (l/rJ [dl’(rJ/dri] li ‘Si; (. . . .) nonrelativistic with subsequent recoupling of the scattering amplitudes.
62S1,2 (left theoretical target and - -) semicalculation
those for the 3Pz state at 7.5 eV, since the percentage deviations of the initial and final projectile energies are smaller than they would be in comparing the results at the same total collision energy of either 6.5 or 7.5 eV. Inspection of Fig. 3.4.8 indeed supports the above argument, thus indicating that not only explicitly spin-dependent effects during the collision are responsible for violations of the
N. Andersen et al. J Physics Reports 279 (1997) 251-396
349
Fig. 3.4.4. Generalized STU parameters for electron impact excitation of the transitions (5~)‘Pi,~,~,~ --t (~s)‘S,;~ in indium for a total collision energy of 5 eV. The individual theoretical curves based on [ 121 are: (__ ) semi-relativistic calculation including the spin-orbit interaction within the target and between the projectile electron and the target nulceus through the operator (a’Z/2) xi (l/r?)Zi.si; (- -- -) semi-relativistic calculation with the spin-orbit operator (a2/2) C,(l/ri) [dl’(ri)/dri] li.si; (. . .) nonrelativistic calculation with subsequent recoupling of the scattering amplitudes. The experimental data are from [24].
predictions from the fine-structure effect; as mentioned before, such violations can also be caused by the energy dependence of the Y operator [ll]. Theoretical results for these parameters have been obtained in a five-state Breit-Pauli R-matrix (BPRM) calculation (based on work of the Belfast group [SS]), and in semi-relativistic [31] as well as full-relativistic [97] first-order distorted wave approaches (SRDW and FRDW). The agreement between the experimental data and the theoretical predictions is not very satisfactory, indicating the need for improved theoretical models for this complicated collision process. Note, however, that the theoretical data, by themselves, also support the validity of the approximate relationship (3.4.3a). Fig. 3.4.9 shows the differential cross section crUand the generalized ST U parameters for the Hg 63P0 state (left) and 63P2 state (right) at 8, 15 and 40 eV. Experimental points are from the Miinster
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-1
I
1
I
1
I
Reports 279 (1997) 2.5-396
I
1
1
I
I
I
1
I
I
1
-1 1
11.0eV
5.0eV
d(detd Fig. 3.4.5. Angular dependence of the asymmetry parameter SA for inelastic scattering for the Tl 62Pr,2 + 72S,,z ) semi-relativistic transition at 4.0, 5.0, 6.0, 9.0, 11.0, and 14.0 eV (from [45]). The theoretical curves are: (calculation including the spin-orbit interaction within the target and between the projectile electron and the target nucleus through the operator (c~‘Z/2) xi (l/r?)li.si (from [48]); (-.-.) semi-relativistic calculation including the spin-orbit interaction within the target and between the projectile electron and the target nucleus through the operator (a2Z,rr/2) Ci(l/r!)li.si (based on [14]);(- - -) semi-relativistic calculation with the spin-orbit operator (g2/2) Ci(l/ri) x [dI/(ri)/dri] li.si (based on [14]); (. . . . .) nonrelativistic calculation with subsequent recoupling of the scattering amplitudes (based on [ 141). The experimental data are from [45].
group [38,23,69], and the theoretical curves are again the Breit-Pauli R-matrix calculations from [SS], and semi-relativistic [31] and fully relativistic distorted wave calculations [97]. In Figs. 3.4.9(b,c) we note a good overall agreement between the FRDW and SRDW predictions. At 8 eV, there is also fair agreement between the SRDW and BPRM theories, except for the
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
-1.0
I
0
I
I
I
I
I
I
1800 &ded
90
I
I
I
90
I
I
351
l-1.0 180
Fig. 3.4.6. Angular dependence of the generalized STU parameters for electron impact excitation of the transitions T16*P riZ + 7’SIjZ (~ ~ -) and Tl 62P3,2 --* 72S1,2 (. .) at a total collision energy of 5 eV. Also plotted are the sum of the cross sections and the weighted averages ( -----) of the STff parameters, as defined in Appendix C (from [48]). These averages vanish in the approximation of the pure fine-structure effect.
differential cross sections and the T_ parameters for the 3P0 state. In addition to the SA parameter discussed above, experimental results for Sp and T, are available at 40 eV for the 3P2 state. As for the SA parameter at this energy, the agreement with the FRDW calculation is very good. 3.4.4.2.
Excitation
six amplitudes
As discussed above, this case requires in general For the ‘PI state, which we will consider first, an
of the Hg (6~6p)‘*~P, states.
for a complete description.
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Reports 279 (1997) 251-396
v
Hg I
6'S.
eV
I
Fig. 3.4.7. Energy level scheme for the HgI 6lS ground state and the excited 6is3P levels and optical transitions studied in alignment and orientation experiments with polarized electron impact excitation.
6%
Fig. 3.4.8. Angular dependence of the spin asymmetry parameter SAfor excitation of the Hg 63P, (left column) and 63P, (right column) state at collision energies of 6.5, 7.5, 8.0 and 15.0 eV. Experimental points (0) are from [38], theoretical curves from (__ ) [SSJ, (- - - - -) [31] and (. . . .) [97].
IV. Andersen et al. /Physics
approximate SP
=
Reports 279 (1997) 251-396
353
symmetry pointed out in [17] causes the following equations to be nearly valid: SA
(3.4.4a)
2
T z z 1.0 )
(3.4.4b)
Tx =
(3.4.4c)
u YX
Ty ,
z
u,,
(3.4.4d)
.
Thus, effectively, one absolute differential cross section and six dimensionless parameters need to be determined. While the three independent STU parameters may be derived from electron spin
a eV
Hn 63P,
0.0 ‘\ __ ‘\
-1.0
1 I
I
I
c,
-a L
- ,’
-.__
I
‘\
I II tp_,’ 1 I I
,
U
\
I
Hg 63P2
II
1 , II 1 1
,I
TX
I II 1
rI 1 I ,I UYT --__-
1.
-1.0 1.0
. 0.0
-1.0 1.0
-1.0 180’0
90
180’0 (‘Ydeg)
90
180’0
90
180
(a) Fig. 3.4.9. Differential cross section and STU parameters for the Hg 63P,, state (left side) and 63P2 state (right side) at 8, 15 and 40 eV. Experimental points are from [23] (A) and [38] (Cl) for SA, [72] (Cl) for Sp and T,, and [63] (0) for T,, T,, U,, and U,,; theoretical curves are from [SS] ( -), [31] (- - - - -) and [97] (. . . . .).
354
N. Andersen et al. /Physics
Reports 279 (1997) 251-396 15 eV
Hg 63P0
A i: ::
::
,..“.,
,J_\ ,
0.0
I’ /
-1.0
/
0
(b)
,
5.. \.,
:i
‘<,
.;
L.-
i
I
90
,
:
\
:
‘,
-
I.:
180 0
I
,
,
90
:
I
:.,
,: i
,
\;’ ,
I
180 0
I
I
90
I
I
o,.
“.‘N-_L’
’
180 0
‘...
:’ r-;‘j.
___*.%.T)
+i i ; ,’ e: ; I _______..‘-~+ ‘%,__ r: ‘i 1; _I ;
\ : I
‘..
:I
i
!,._/: : ~‘~....,,.,. “,, ;
; ., Z’.
,;
*‘,\
:
Hg fj3P2
’
’
90
’
’
-1.0
180
f-%ded Fig. 3.4.9. continued
analysis, the remaining three relative parameters could be deduced from polarized photonscattered particle coincidence experiments. No such experiments have been performed at the time of writing, due to the somewhat inconvenient (VUV) photon wavelength, cf. Fig. 3.4.7. However, this approach should be no more difficult for Hg than for resonance excitation of the heavy rare gases which have been extensively studied in recent years, so future progress in this direction is a possibility. The pioneering spin polarization measurements by Eitel and Kessler [42] showed a striking similarity between the results for elastic and inelastic scattering in the structures of the differential cross sections as well as for the spin polarization of the scattered electrons. This is particularly evident when the incident electron energy becomes considerably larger than the excitation energy. Fig. 3.4.10 shows a key graph from the paper, illustrating this fact at 50 and 80 eV impact energy. The results were interpreted in terms of a two-step model, first proposed by Mohr and Nicoll[76]
N. Andersen et al. /Physics
Hg 6”P2
40 eV
Hg 63P0
355
Reports 279 (1997) 251-396
0.02
1.0
0.01
0.0
0.00
1.0
-1.0 I.0
0.0
0.0
-1.0 I.0
-1.0 -i.n
0.0
0.0
-1.0 1.0
-1.0 -LO
0.0
0.0
-1.0 0 (cl
90
180’ 0
90
180’ 0
90
180’ 0
90
-1.0 180
@(deg) Fig. 3.4.9. continued
for large-angle inelastic scattering. In this model, the excitation takes place in the first step, giving rise to only small-angle scattering; in the second step, the electron, now having a slightly smaller energy, is elastically scattered at a large angle by the atom. The two steps may also take place in the opposite order. For the 3P1 state, no simplifications such as Eqs. (3.4.4) can be made, and all seven generalized ST U parameters are independent. Here, we restrict ourselves to an overview of the parameters, for which theoretical results exist in the same BPRM, SRDW and FRWD approximations just discussed for the (6~6p)~P0,2 cases. Experimental results for the SA parameter have been obtained [23], as well as Sp, T,,, T, and U,,, data for ‘PI excitation at 40 eV. Fig. 3.4.11 shows the differential cross section 0” and the STU parameters at 8, 15, and 40 eV. We first notice that, theoretically, the approximations (3.4.4) are excellently fulfilled for the (6s6p)‘Pr state. Secondly, except for ‘PI at
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N. Andersen et al. /Physics
Reports 279 (1997) 251-396
30
90 8 (de@
Fig. 3.4.10. Differential cross section (~ ) and spin polarization (0) for electrons scattered elastically (upper row) and inelastically (6lPr excitation) from Hg atoms at 50 eV (left column) and 180 eV (right column). Note the similarities between the elastic and inelastic results for fixed energy (adapted from [42]).
40eV, considerable
discrepancies exist between theory and the (sparse) experimental results, indicating that more work is needed on both the experimental and theoretical side. Fig. 3.4.12 shows a measurement [54] of the spin parameter T,for the Hg 63P0,1,2 states at 15 eV. Due to problems with energy resolution of the close-lying 3P0, 1,2 levels, the results are a weighted sum of the Tz's for the three levels, i.e., CC) = CL(O) + rJ,(l)
+ @‘G(2) .
(3.4.5)
Here rj = w(J)/CJ w(J) wh ere w(J) is the product of the excitation cross section, G,(J), and the fraction of electrons exciting the fine-structure level 3PJ. Because of the ‘PI admixture in the intermediate coupling description of the 3P1 state, however, this is the predominantly excited fine-structure level within the multiplet. Theoretical predictions for T,(J) and a,(J) are again available from SRDW [30] and FRDW [97]. These can be used to calculate the parameter (Tz) of Eq. (3.4.5) by combining theoretical cross sections with the excitation fractions given in [SS]. Agreement between theory and experiment for this parameter is fairly good, though systematic deviations are noticed, particularly at smaller scattering angles. These results will be put into a broader context in Section 3.5.2.
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351
3.4.5. Excitation of Ca, Sr, Ba and Cd Recently, full-relativistic first-order distorted wave calculations have been reported by the Toronto group for other quasi two-electron systems such as Ca, Sr, Ba [98] and Cd [96]. Example results for an incident projectile energy of 40 eV are shown in Figs. 3.4.13-17. As one would expect, relativistic effects become more prominent within the alkali-earth sequence Ca (Fig. 3.4.13), Sr (Fig. 3.4.14) and Ba (Fig. 3.4.15) with increasing nuclear charge of the target. Furthermore, these
(4
Fig. and and (. ”
Wed
3.4.11. Differential cross section and STU parameters for the Hg 6 ‘PI (left side) and 6 3P1 (right side) states at 8, 15 40 eV. Experimental points are from [23] (A) and [38] (0) for SA, [72] (Cl) for Sp and T,, [63] (0) for T, and 1U,.Y,
[82] (0) for the differential cross section; theoretical curves are from [85] ( -)), . .I.
[31] (- - - - -) and c971
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15 eV
Hg 6’P,
i'
__
1,’
I
-1.0 0 @I
I
I 90
I
1 180 0
I
I
I 90
I
Hg CI~P,
I
--
I
I
180 0
I 90
I
\
I 180 0
\A’-------
’
’
’ 90
’
’
-1.0 180
@Cdeg) Fig. 3.4.11. continued
effects are more visible in the excitation of the triplet states than they are in singlet-singlet transitions. In a semi-relativistic picture, this is explained as an “intermediate coupling effect”, i.e., an interference between the excitation of the singlet and the triplet parts of the target wave function. Since optically allowed singlet excitation will dominate at high energies, such an interference effect will be relatively large if there is a large triplet and a small singlet coefficient in the target wave function (as in the “triplet” states), and it will almost disappear for the opposite case (as in the “singlet” states). The results for the cadmium target are somewhat similar to the predictions for mercury, as one might expect from the fact that Cd is one row above mercury in the periodic system. Hence, the predictions of the fine-structure effect are very well reflected in the results for excitation of the (5~5p)~P~,~ states (Fig. 3.4.16). Once again, the “intermediate coupling interference effect”
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
0
(cl
90
180’ 0
Hg 63P,
40 eV
Hg 6’P,
90
180’ 0
359
90
f?deg) Fig. 3.4.11. continued
discussed above is more significant for the (5~5p)~P~ than for the (5s5p)‘P1 state, as can be seen in Fig. 3.4.17. 3.4.6. Excitation of heavy rare gases: Xe 6s,6s’ states Figs. 3.4.18 and 3.4.19 show results for the differential cross section and the generalized STU parameters for impact excitation of the 63P0,2 and the 61y3P1 states in xenon at incident electron energies of 15, 25 and 40 eV. The theoretical curves were obtained in semi-relativistic [31] and full-relativistic [ 1141 (40 eV only) first-order DWBA calculations; for the asymmetry function SA, they are compared with experimental data of the Miinster group [40]. In general, the two theoretical predictions look very similar, in particular for the 63P ,,2 states for which the results agree very well with the symmetry relationships predicted from the fine-structure effect. The agreement with the experimental data, however, is not very satisfactory, indicating that first-order
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N. Andersen et ai. j Physics Reports 279 (1997) 251-396
Fig. 3.4.12. The spin parameter T, = (T,) for electron impact excitation of the Hg 63P states at 15 eV. Experimental [97] and(- - - - -) [31]. points (a) are from [54], theoretical curves from ( -)
Ca 43P1
Ca 4’P,
1.U
T, % 1
-1.0
’ 0
’
’ 90
UTV
’
’ 180 0
I
I 90
TV
L I 180 0
I
I
I 90
180 0
’ 90
’
-1.0 180
Wed
Fig. 3.4.13. Differential cross section and STU parameters for the Ca 4lP, (left side) and 43P1 (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [98].
361
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
Sr 53P, 1.0
0.0
-1.0 I.0
““i j-+-i++-
0.0
T,ol
-1.0 I.0
0.95
0.90
1
1.00
I ’
, -
c n
c
0.0
TZ ,
I
,
1
s
G
,
I
I
# I
I
, I
I
I I
Uw
I
I
I
I
!
I
I
I
I
I
I
,
I ,
U=‘I
T, A
0.95 -
I ,
n
/\
-1.0 3.0
A_
0.0
T,
’
0.90 0
’
’ 90
’
I ’ lsd o
4 I 90
,
I MO’ 0
1 4 I 90
I
I 180’0
I
I
I
90
I
-1.0 180
@(ded
Fig. 3.4.14. Differential cross section and STU parameters for the Sr 5’P1 (left side) and 53P, (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [98].
theories are not sufficiently accurate for these collision energies. Although the experimental data for the 63P0,2 states vary rapidly with the scattering angle, a tendency towards fulfillment of Eq. (3.4.3a) can be recognized for scattering angles below 90”. However, this tendency is not as clear in these data as it is in the two theoretical predictions. 3.5. The incoherent case without conservation Combining photon and electron polarization
of atomic rejection analysis
symmetry:
In this section we discuss the relationship between the generalized Stokes parameters of Section 3.3 and the generalized STU parameters of Section 3.4. In particular, we analyze how data from one of these sets of results may be used to complement, predict, and check results obtained for the other set. We concentrate on the J = 0 + J = 1 transition where experimental and theoretical data are available.
362
N. Andersen et al. /Physics Ba 6’P1
40 eV
Ba 63P,
A
0.95 -
0.90
Reports 279 (1997) 251-396
’
I
’ ’ 90
0
I 180 0
I
I
I
I
90
I
I 180 0
I
I 90
1
I
’
180 0
’
’
’
’
-1.0 180
90
6Yde.d
Fig. 3.4.15. Differential cross section and STU parameters for the Ba 61P1 (left side) and 6lP, (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [98].
3.5. I. Generalized STU parameters for J = 0 -+ J = 1 excitation In this case, the generalized STU parameters can be expressed in terms of scattering amplitude parameters as follows [4]: SA = (l/2(7”) L-(X”+ + a? + c(t) - (B: + P” + pa1 = <&l)P,, SP
=
(l/20”)
I@:
+
2
-
&3
-
w:
+
P”
-
(3.5.la)
ml = scwcJP, + q/5
T, = (l/20,) [(cl”++ CX?- cl;) + (/3”++ /3’- - ,8;)] = f[l + 2&;,),1
,
,
(3.5.lb) (3.5.lc)
T,, = (l/o”) Re{a+j?+e’(~+-@+) + cI_p-ei(‘#-*~)
- ~O,&ei(~O-“o)),
(3.5.ld)
TX = (l/a,) Re(cc+fi+e’(@+-ti+)
+ cq,floei(@~-@O)} ,
(3.5.le)
U,, = - (l/au) Im{a+lj+e’(@+-$L+)
+ ~-fl-e’(~--+L-) + x_p_eiC~--IL-)
+ aoPoei(*o-bo))
,
u,, = - (l/a”) Im{a+fl+e’(@+-JI+) + a_P_e’(@--@-) - ~OjjOei(@Q-~o))
(3.5.lf) (3.5.lg)
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N. Andersen et al. 1 Physics Reports 279 (1997) 251.. 396
Cd VP0
Cd 53P2
180’0
90
180’0
90
180’0
90
180
Q(ded
Fig. 3.4.16. Differential cross section and STU parameters for the Cd 5 3P0 (left side) and 5 3P2 (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [96].
and in terms of the density matrix parameters as S* = NJ - M’J, sp = d(
(3.5.2a)
1 - 2hT) - w’(1 - 2h”) )
(3.5.2b)
T, = d( 1 - 2hT) + wl(l - 2h9 = 1 - 2h , Ty = (l/d
{a+P+ cosd+
+ a-p-COSK
TX = (l/o”) {r+b+ cos A+ + z-p-
(3.5.2~) - r&)cosLlO)
)
(3.5.2d)
cos A- + aojo cos A’} ,
(3.5.2e)
Cry, = - (l/o”) (x+P+ sin A+ + r-/L
sin A- - cco~o sin A’} ,
(3.5.2f)
U,, = - (l/a,)
sin A- + cxofio sin A’} .
(3.5.2g)
(c(+b+ sin A+ + x-p_
N. Andersen et al. / Physics Reports 279 (1997) 251-396
364
Cd
5’P,
Cd
!i3P,
Fig. 3.4.17. Differential cross section and STU parameters for the Cd 5 ‘PI (left side) and 5 3P1 (right side) states at 40 eV, calculated in a full-relativistic first-order DWBA calculation [96].
For brevity, we kept the products of the amplitude magnitudes in the TU parameters. They can also be expressed in terms of density matrix parameters by using aJ*
/CT”=
wTwl(l - hT)(l - U)(l + LfT’)(l * Lfl’)
cto/?o/a” = &&am
.
)
(3.5.3a) (3.5.3b)
Note that the three parameters SA, SP and T, can be predicted directly from coherence parameters extracted in electron-photon coincidence experiments (see Section 3.3.1) and vice versa; the parameter T, = 1 - 2h with an unpolarized electron beam, and SA,P with an electron beam polarized perpendicular to the scattering plane. Thus the three relative sizes hT, h” and wT(= 1 - WI) can be evaluated. The two missing relative sizes LTT and Lf ” may be obtained by circular polarization measurements only. The parameters Ty, TX, U,, and U,, contain bilinear combinations of amplitudes that do not appear in the state multipoles of equations (3.3.2-5). It is therefore not possible to express these
N. Andersen et al. /Physics Reports 279 (1997) 251-396
365
parameters in terms of the light polarizations measured in the electron-photon coincidence setup. Measurements of the STU parameters, therefore, provide complementary data to the information that can be obtained from the coincidence arrangements (see Section 3.5.2). In particular, linear combinations of the form Ty * TX and U,, + U,, allow for determination of the two complex numbers (r+fl+e”‘+ + a_ p_ein- and j&,aOeiAO, thereby providing the crucial phase difference A0 not obtainable with the electron-polarizedPphoton coincidence technique without simultaneous electron spin analysis in the exit channel. Hence, the combination ofthe two techniques described in Sections 3.3 and 3.4 constitutes the ultimate goal of a perfect scattering experiment,for important J = 0 + f = 1 excitation problem in its most general form.
(4
the
f-VW
Fig. 3.4.18. Differential cross section and STU parameters for the Xe 63P0 (left side) and 63P, (right side) states at 15,25 and 40 eV. The theoretical curves were obtained in semi-relativistic (___-- ) [313 and full-relativistic ( .) [ 1141 first-order DWBA calculations, the experimental data are from 1403.
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Xe 6”P0
(b)
Reports 279 (1997) 251-396
25 eV
Xe f.i3P2
8(ded Fig. 3.4.18. continued
The considerations above, and the fundamental difference between the information extracted from generalized Stokes and ST U parameters, may be succinctly summarized in the way shown in Fig. 35.1. Here, (a) illustrates that generalized Stokes parameter analysis in the zn direction with electron spin polarization P, perpendicular to the scattering plane determines all relative amplitude magnitudes, and the phase relationship between the f+ 1 and f_ 1 amplitudes - provided P4 is also known. Additional analysis in the y” (or x”) diretion with in-plane spin polarization PYor P, yields the four relative phases within the two triples (ff 1, f! 1, fof) and (j-2 r, fl 1, fd), respectively. None of the relative phases (d +, A’, A -) between f 1 1 and fi 1, etc. enter. On the other hand, Fig. 3.5.1 (b) illustrates that the generalized STU parameters SA, SP and T, depend only on the relative sizes of the amplitudes, while TX, T,,, U,,,, and U,, depend also on the relative phases (A+, A’, A-) within the three amplitude pairs (fl r, f! 1), (fl r, f! r), and (fo, fi). N o m ’ formation on the relative phase angles between these pairs can be extracted from generalized STU parameter analysis.
N. Andersen et al. J Physics Reports 279 (1997) 251-396
Xe 63P0
40 eV
367
Xe 63P2
f4ded Fig. 3.4.18. continued
3.5.2. Example:
The S,, S, and T, parameters for Hg(6s6p)3P,
For the Hg(6~6p)~P 0, I,2 states, fine-structure averaged T, results [SS] were shown for an incident energy of 15 eV in Fig. 3.4.12. Electron-photon coincidence data also exist for this energy [92,90]. Eq. (3.5.1~) thus allows for a consistency check between the two experimental approaches. Hyperfine structure effects modify Eq. (2.1.8) for the height parameter h, and thus the expression for T,(J = 1) to [4] 1 3G, - Pr(4 - G,) + P4(8 + G2 + P1[4 - G,]) T,(l) = 3G 2 PA1 - PI) + (3 + PI) The hyperfine perturbation
coefficient is taken as G2 = 0.789 [107].
(3.5.4)
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The asymmetry and polarization functions, SA and Sp, can also be expressed directly in terms of the experimentally measured generalized Stokes parameters. With the depolarization correction for the hyperfme structure effects included, the corresponding equations become very lengthy. However, Eqs. (3.5.2a, b) can be used instead, with the parameters wt, wl, h? and h1 determined as described in Section 3.3.1.4 below Eq. (3.3.22). As can be seen from Fig. 3.52, the agreement between the experimental data and the theoretical predictions from the Breit-Pauli R-matrix calculation for T,(l) is satisfactory. Regarding SA and Sp, we first note that there are some discrepancies between our conversion using Eqs. (3.5.2a-c) and the indirect measurements reported by Goeke et al. [46], particularly at small scattering angles.
I
-1.0 0 (a)
,
I 90
1
I 180 0
I
1
I 90
I
, 180 0
0
I
I 90
I
0 180 0
1
1
’ 90
’
’
-1.9 180
fl(deg)
Fig. 3.4.19. Differential cross section and STU parameters for the Xe 6lP, (left side) and 63PI (right side) states at 1525 [31] and full-relativistic (. . . . .) [114] and 40 eV. The theoretical curves were obtained in semi-relativistic ( ----) first-order DWBA calculations, the experimental data are from [40].
369
N. Andersen et al. /Physics Reports 279 (1997) 251-396 Xe 63P1
25 eV
Xe 6’P,
1.0
0.0
-1.0 -I.0
0.0
-1.0 I.0
0.0
-1.0 I.0
0.0
0
(b)
90
1sd 0
90
180’0
90
180’0
90
-1.0 1X0
o(h) Fig. 3.4.19. continued
These discrepancies are due to additional experimental correction factors that were included in the analysis presented by Goeke et al., such as detector efficiencies and angular resolutions that go beyond the general analysis presented above. However, the good agreement between the direct measurement for SA of Borgmann et al. [23] and the indirect extraction of the same parameter by Goeke et al. demonstrates the value of consistency checks. Also, the agreement between the data sets extracted by the Miinster group and the Breit-Pauli R-matrix results is satisfactory. Fig. 3.5.3 [6] displays again the spin depolarization parameter T, for Hg(6s2)lS0 -+ (6~6p)~P excitation at 8 eV impact energy, both the fine-structure averaged result TL [SS] and T,(l) as extracted from the electron-photon coincidence data of [92,90]. Note that the large-angle error bars are significantly shorter for the electron-photon coincidence data, and the photon channel improves the energy resolution tremendously. Using T,(O) = - 1 for this case (see Eq. (3.4.lb)), experimental values for TI = (Tz) and T,(l), and known relative excitation fractions [55], it is in
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N. Andersen et al. /Physics
Xe 61P1
(cl
Reports 279 (1997) 251-396
40 eV
Xe 63P1
Q(ded Fig. 3.4.19. continued
principle possible to extract even T,(2) from the experimental data. We have performed this extraction using theoretical cross sections to estimate the rj factors in Eq. (3.4.5). While the results are in agreement with theoretical predictions for T,(2), the experimental error bars are presently too large for a detailed test of the theories. We have also applied the conversion formula (3.5.2~) to recent coincidence data for resonance line excitation in the heavy rare gases [37] to predict the electron spin depolarization in these cases, for which no direct experimental determination presently exists. In the energy range 30-80 eV covered by their data, the predicted value of T, is unity within experimental uncertainty, in agreement with theoretical predictions [29, 1141. At lower collision energies, however, considerable variation in T, with scattering angle is predicted theoretically, and here the coincidence technique may be an attractive alternative approach to a direct determination of T,.
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N. Andersen et al. / Physics Reports 279 (1997) 251-396
STU
Stokes
QYh)
---
SA,SP,TZ .
pxSPY
Tx.y
1 "YCXY
(b)
(a)
Fig. 3.5.1. (a) This diagram shows which relative amplitude sizes and phases can be evaluated from a generalized Stokes parameter analysis in the z” direction with electron polarization P, and from in-plane measurements with polarizations P, and P,,, respectively. (b) This diagram shows which relative amplitude sizes and phases enter into Eqs. (3.5.2) for the generalized STU parameters. Note that SA, Sp and T, only depend on the relative sizes of the amplitudes and can, therefore. be predicted from a generalized Stokes parameter analysis.
Hg 63P, at 8 eV 1.0
0.0
-1.0 0
90
180’ 0
90
180’ 0
90
-1.0 180
@(W
function Sp and contraction parameter T for electron impact Fig. 3.5.2. Asymmetry function S,, spin polarization excitation of Hg(6s6p)3P, at an incident electron energy of 8 eV: (0) direct measurements by Borgmann et al. [23] and Goeke et al. [46]; (0) parameters derived from the electron-photon coincidence parameters by [46]; ( x) and ( + ) parameters derived using Eqs. (3.5.1). The theory curve is from a five-state Breit-Pauli R-matrix calculation based on
P51.
3.5.3. Sublevel-resolved generalized STU parameters A further refinement of experimental technique is the combination of photon polarization analysis with simultaneous determination of STU parameters. Whereas such a direct experiment seems infeasible in the near future, it is possible to use a similar approach as in the previous sections
372
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
-1.011111111111l,1
0
,I
30
60
90
0 (degl
Fig. 3.5.3. The spin depolarization parameters T, for Hg(6sZ)‘S0 + (6~6p)~P excitation at 8 eV impact energy. The experimental points for T, = (T,) are from [SS] (0) and for T, (1) converted from [92] (A) and [90] (0). Theoretical curves are from [30] (- - -) and [97] (-). The upper curves are for T, (1) while the lower curves are for (T,).
to predict some of the results from presently available data. In fact, such results have been reported in [46], with a parameterization for the collision coordinate frame, which we will now extend using the natural coordinate frame. From Eqs. (3.51) it is seen, with obvious notation, that the following equations hold for the states with positive reflection symmetry (MJ = f 1) with respect to the scattering plane W+1 = (1/2D”) Cc& + /%I = (a ‘/2%) ,
(3.5.5a)
s,”
(3.5.5b)
=S,”
=(1/G)
[IX!‘,-@J,
T,‘l = 1 )
(3.5.5c)
TX*1 = Ty* = (l/a * ) 2a+j3+ cos A * )
(3.5.5d)
U&l = U& = -(l/o+)
(3.5.5e)
2cr+p+ sin A * ,
while we get for the state with negative reflection symmetry (MJ = 0) w” = (l/20,) [a; + p;] = (o”/20,) = h ,
(3.5.6a)
s: = - s,” = (l/CO) [ai - /IO’])
(3.5.6b)
7,; = - 1)
(3.5.6~)
T,O = - T$ = (l/Do) 2aoPo cos A0 )
(3.5.6d)
U& = - U$ = - (l/fro) 2ctopo sin do .
(3.5.6e)
From the experimental data of [46] alone, it is not possible to extract the parameters for the three individual magnetic sublevels. However, one can extract the parameters for M = 0 and the average parameters for excitation of sublevels with M = _+ 1 that correspond to positive reflection
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N. Andersen et ai./Physics Reports 279 (1997) 251.-396
symmetry. We define the latter parameters by W+=~+‘+~-‘=~-~“=l-hh,
(3.5.7a)
(S,)+
= (S,)+
= w+lsAf’ + w-lsi’
(7’,)+
= (T,)+
= w+lT,f’
(U,,)’
= (U,,)’
= w+q?
+ w-‘T,’
(3.5.7b)
,
(3.5.7c)
,
+ w-w;yl
(3.5.7d)
.
Breit--Pauli R-matrix results for the various sublevel-resolved parameters are shown in Fig. 3.5.4 and, where possible, compared with the experimental data of Goeke et al. [46]. The agreement
Hg 63P,
8 lV
1.0
0.5
0.0 1.0
0.0
-1.0 1.0
0.0
-1.0 l.Yi
0.0
-1.0 0
90
l&O
90
180'0
90
180'0
90
180
@(deg)
Fig. 3.5.4. Sublevel-resolved generalized STU parameters for electron impact excitation of Hg(6s6p)3P, at an incident electron energy of 8 eV: (x) and ( + ) parameters derived using Eqs. (3.5.6,7). The theory curve is from a five-state Breit-Pauli R-matrix calculation based on [SS].
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N. Andersen et al. / Physics Reports 279 (1997) 251-396
between theory and experiment is not as good as one might have expected, and also not as good as in the original paper of Goeke et al. We believe that the same reason applies as given above in the discussion of Fig. 3.52, namely that knowledge of more experimental details would slightly modify our conversion formulas, apparently improving the agreement between the experimental data and the R-matrix results. Nevertheless, the overall potential of detailed data analysis is clearly demonstrated. 3.6. Related work: P + P elastic scattering Up to this point, we have only discussed excitation (and the time-reversed de-excitation) of atomic targets by electron impact. In these cases, the combination of data for generalized Stokes and generalized STU parameters allows for a detailed investigation of spin-dependent alignment and orientation effects. One of the important mechanisms for obtaining a spin polarization of an initially unpolarized electron beam through scattering was the fine-structure effect, i.e., the energy-resolved excitation of individual members of a multiplet. It is important to note that the necessary conditions for the effect, outlined in Section 3.2.1, can also apply to elastic scattering from targets with nonvanishing spin and orbital angular momenta, such as (np)‘P,,, and (~P*)~P~ ground states in the third and fourth column of the periodic system. Therefore, one may expect the fine-structure effect to be an important spin polarization mechanism in such collisions as well, although standard methods to determine the angular momentum orientation via a circular light polarization measurement cannot be applied. The first experimental attempt to demonstrate the importance of the fine-structure effect in elastic scattering was undertaken by Kaussen et al. [61], who compared the spin polarization of an initially unpolarized electron beam after scattering from Hg (6s2)‘So, T1(6p)*Pi,2, Pb (~P*)~P~ and Bi (6~~)~S3,2. However, no clear indication of the fine-structure effect was found in the experimental data. Furthermore, Haberland and Fritsche [Sl] reproduced most of the experimental data in an ab initio calculation based on a nonlocal exchange approximation. According to Haberland and Fritsche, their model only allowed for spin polarization effects through Mott scattering. Consequently, they concluded that this was, indeed, the dominating mechanism. A subsequent model calculation by Bartschat [9] for elastic scattering from B (2p)*Pi12 and C (~P*)~P,,, however, revealed that the fine-structure effect can be an important polarization mechanism. The results of this model calculation are shown in Fig. 3.6.1. As expected, the explicit inclusion of the spinorbit interaction does not change the theoretical results in any substantial way; the nonvanishing results for the Sherman function S (= Sp = S,) for elastic scattering are thus clearly due to orbital angular momentum orientation combined with electron exchange. Surprisingly, the generic form of the curves for the Sherman function look very similar to well-known results for Mott scattering from heavy targets - an increase with scattering angle, followed by a rapid sign change near 90”, and a correlation between large absolute values of S and minima in the differential cross section. Unfortunately, the small fine-structure splitting in boron and carbon does not allow for a direct experimental check of these theoretical predictions, since the required energy resolution can presently not be achieved with a count rate sufficient for standard polarization experiments. In order to reduce the Mott scattering effect to some extent, the Miinster group repeated the Kaussen
N. Andersen et al. / Physics Reports 279 (1997) 251-396
0
90
375
180
e(de.4 Fig. 3.6.1. Differential cross section cr and spin asymmetry function S, ( = S,) for elastic electron scattering from boron -) and carbon (- - - - -) atoms at 1.36 eV, obtained in a nonrelativistic R-matrix calculation with re-coupling of the scattering amplitudes. The difference between the dotted and dashed curves for carbon shows the effect of explicitly taking the spin-orbit interaction into account (from [9]).
(~
et al. experiment for elastic scattering and excitation from the In (~P)‘P~,~,~,~ states [24]. The experimental results for incident energies of 1.0, 2.0 and 3.0 eV are shown in Fig. 3.6.2 and compared with Breit-Pauli R-matrix calculations [12]. In the calculation, the spinorbit interaction was included in two different ways, once by using the operator (r2Z/2) xi l/r? li.si (this is the form implemented in the original Breit-Pauli R-matrix codes) and once by using (a22/2) xi (l/ri) x [d t’(ri)/dri] li . si, where x z & is the fine-structure constant and V(r) is the scattering potential. Note that the latter form is expected to be more accurate if only one-electron terms of the Breit-Pauli Hamiltonian are taken into account. In a third model, explicit relativistic effects were omitted and the problem was solved in M-coupling, with subsequent recoupling of the scattering amplitudes to account for fine-structure resolved transitions. To allow for a direct comparison between experimental and theoretical results for elastic scattering, the results for elastic scattering from both the (5~)~Pr,~ state and the (~P)‘P~,~ were taken into account, weighted with the corresponding cross sections and the ratio of 84/l 6 obtained from the multiplicity and the Boltzmann factors representing an initial thermal occupation of the (5P)2PW state [24]. It is seen in Fig. 3.6.2 that inclusion of the spin-orbit interaction between the projectile and the target nucleus, together with the use of nondegenerate fine-structure states, does not change the results dramatically. Also, the agreement between theory and experiment is not improved any further in the Breit-Pauli calculations; in fact, for 1.0 and 2.0 eV incident energy, the results obtained with the spin-orbit operator (0?2/2)Ci (l/r;) Zi‘si predict a deep minimum in the asymmetry function for elastic scattering at small angles that is not found in the experimental data.
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In 52P1/2 +52p1/2,3/2 l/r3
nr
l/r dV/dr
SA
0
-1
1-
0
-1 0
90
180’ 0
90
180’ 0
90
180
0Cde.s)
Fig. 3.6.2. Asymmetry function SA for elastic and inelastic electron scattering from indium atoms at incident electron weighted average for energies of 1.0, 2.0 and 3.0 eV (from [12]): (- - .) elastic scattering from (SP)‘P~,~ only; (-) elastic scattering from (5p)‘PO1,2 and (SP)~P$, (see text); (- - -) inelastic transition (Sp)‘P& + (5p)‘P&. Left column: nonrelativistic calculation with subsequent recoupling of the scattering amplitudes. Center column: Semi-relativistic calculation including the spin-orbit interaction within the target and between the projectile electron and the target nucleus through the operator (a2 Z/2) xi (l/r;) li’si. Right column: Semi-relativistic calculation with the spin-orbit operator (a2/2) xi (lh) [dV (ri)/dri] li’si. The experimental data for elastic scattering (A) and excitation (V) are from Bartsch et al. [24].
On the other hand, the purely algebraic recoupling procedure yields impressive agreement between the experimental and the theoretical data. Especially for elastic scattering at 1.0 eV incident energy, the nonrelativistic calculation is in much better agreement with the experimental results than any of the two semi-relativistic calculations. This indicates that angular momentum orientation is the dominating mechanism that causes the left-right asymmetry, in contrast to the
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conventional Mott scattering effect. Experimental studies with even lighter targets such as Ga (4p)‘PIj2. 3I2 with a fine-structure splitting of approximately 100 meV might be within reach of current experimental technology and could further clarify the role of the fine-structure effect in elastic scattering. Fig. 3.6.3 displays the asymmetry function SA for elastic electron scattering from thallium atoms. Recent experimental data of the Muenster group [39,45] are compared with one-state, three-state and seven-state BreittPauli R-matrix calculations [14] where, once again, relativistic effects were accounted for either by recoupling alone or explicitly by two different versions of the spin orbit operator (see below). The surprise in Fig. 3.6.3 is the fact that the close-coupling expansion has apparently not converged, even at the lowest collision energy of 1.0 eV. Accidentally, the most primitive one-state calculation (labeled LCC), yields very good agreement with the experimental data, provided the spinorbit interaction contains the operator Zerr/r3, where the effective charge Zen is obtained by scaling the strength of the interaction in such a way that the fine-structure splittings in the target states of interest are well reproduced. On the other hand, the three-state (KC) and seven-state (7CC) expansions with the same representation of the spin-orbit interaction deviate significantly from the experimental data. What one might intuitively expect to be the “best” approximation, namely 7CC, yields the largest discrepancy from experiment. For this case, the full-relativistic “generalized density functional” (GDF) calculation of Fritsche et al. 1431 is in good agreement with the experimental data. On the other hand. it also corresponds to a one-state calculation, although with a very sophisticated and fully optimized potential for the elastic channel alone. The situation is further confused by the fact that predictions of the nonrelativistic seven-state calculation agree with the experimental data. Whereas this might indicate another example of the fine-structure effect, it could also be interpreted as a coincidence in light of the dramatic changes seen in the theoretical results when the form of the spinorbit operator and the number of states in the close-coupling expansion are changed. A full-relativistic close-coupling calculation would be desirable to shed more light on this problem, which is most likely related to the many nearthreshold resonances. When one compares in detail the different theoretical approximations discussed above, it is found that some of these structures move from negative ion bound states into the scattering continuum and vice versa, depending on the number of states in the close-coupling expansion and the form of the spinorbit operator. Such changes. of course. strongly affect the results [ 141. As our final example, results for the Sherman function in elastic electron scattering from Pb (6p’)‘P0 atoms are shown in Fig. 3.6.4. Once again, the GDF calculation of Haberland and Fritschc [Sl] reproduces most of the features in the experimental data of the Munster group [61,45] for collision energies above 6 eV, whereas a five-state BreittPauli R-matrix calculation [S] achieves qualitative but not always quantitative agreement with experiment for energies below 6 eV. No study about the potential importance of the fine-structure effect versus Mott scattering has been performed for this collision system to date, but we expect that (i) both effects will play an important role at low collision energies and (ii) it will be very difficult, if not impossible, to disentangle them, theoretically and experimentally. Finally, Fig. 3.6.5 shows some results for excitation of the (6~‘) levels of lead, demonstrating again the need for more theoretical work on this collision system.
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TI 6*P,,* +
TI @P,,z
3cc
l.OeV
SA
3.0eV
W
Fig. 3.6.3. Asymmetry function SA for elastic electron scattering from thallium atoms in the 1CC (left column), 3CC (center column) and 7CC (right column) close-coupling approximations (from [14]): ( -) semi-relativistic calculation including the spinorbit interaction within the target and between the projectile electron and the target nucleus through the operator (a2 Z,rr/2) xi (l/r:) li. si; (- - .) semi-relativistic calculation including the spin-orbit interaction through the operator (a2/2)Ci (l/ri) Cdl/ (ri)/dri] Z,.si; (. .) nonrelativistic calculation with subsequent recoupling of the scattering amplitudes; (- - - - -) full-relativistic “generalized density functional” calculation by Fritsche et al. [43]. The experimental are from Geesmann et al. [45].
N. Andersen et al. /Physics
Pb
-1 0
60
0
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379
63Po-63Pc
0
60
60
120
8 ldegl Fig. 3.6.4. Asymmetry function SA ( = S,) for elastic electron scattering from lead atoms. The experimental data of Kaussen et al. [61] (A) and Geesmann et al. [45] (0) are compared with the results from a five-state BreittPauli R-matrix calculation (-----) [8] and a full relativistic “generalized density functional” calculation by Haberland and Fritsche (- - - - -1 [Sl].
f4de.4 Fig. 3.65. Asymmetry function SA for electron impact excitation of lead atoms at an incident electron energy of 6.0 eV. The experimental data of Geesmann et al. [45] are compared with the results from a five-state Breit-Pauli R-matrix calculation [S].
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4. Conclusions and recommendations
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for future work
We have presented a systematic generalization of the analytical framework of Vol. I to include spin-dependent alignment and orientation effects for outer-shell excitation in electron-atom collisions. It has been demonstrated that the “perfect scattering experiment” is within reach with presently available technology for some of the most important processes, particularly *S + *P excitation in e-Na and I!!&,+ 3P, excitation in e-Hg collisions. Our results are summarized in Table 4.1. Valuable information can also be extracted in cases where full information is not within reach. For a general review on the status of perfect experiments, including also elastic scattering and excitation of D states, see [S]. Our recommendations about directions in which future work is encouraged or discouraged will now be summarized. Starting with excitation of the Na 3*P state (see Sections 3.2.2 and 3.2.3), the situation is generally good at several energies for many of the parameters, in particular when including the additional data extracted by the inversion technique outlined in Section 3.2.2. However, a better coordination of the experimental efforts among different groups would be beneficial. Additional experimental results obtained, for example, with the in-plane laser pumping scheme, could finally achieve “the perfect scattering experiment”. This may be desirable for at least one convenient electron impact energy, such as 10.0 eV, to remove remaining ambiguities. Considering the similarities between the data for the Li 2, 3*P (Section 3.2.4) and the Na 32P states, the fairly satisfactory situation for Na, and the very considerable investment in time and other resources involved in the experimental investigations, it seems unlikely to us that much significant new information can be gleaned from further studies of the Li case at the present time. For excitation of the K 4*P state (Section 3.2.5), no experimental results for coherence parameters exist, not even for unpolarized electron and atom beams. In view of the richer structures predicted for this atom, and the intriguing differences between the (angle-integrated) ionization asymmetries observed between Li and Na on one side and K on the other [32], this element offers a more attractive experimental challenge than Li. For excitation of the H 2*P state (Section 3.2.6) at incident electron energies above 35 eV, the theoretical predictions are very similar for the individual spin channels and, therefore, for the spin-averaged results. Hence, we recommend a refined experiment with unpolarized beams as a first step. In light of possible scaling laws, the similarities in shape for the y1= 2,3 states encourage comparison of the alignment and orientation parameters within Rydberg series. Excitation of the Hg 63P1 state (Section 3.3.2) is the primary example of a collision system involving a many-electron heavy target for which a complete experiment is within reach with presently available technology. We strongly recommend a detailed study at one or two selected impact energies, such as 8 eV, where the full set of independent generalized Stokes and generalized STU parameters should be determined. In order to provide reliable benchmark data for comparison with theory, the numerous opportunities for consistency checks must be fully exploited. Given the importance of semi-classical treatments in many areas of atomic collision physics, we also recommend further experimental studies of the validity of propensity rules for spin-resolved collisions; see Fig. 3.3.6. Regarding excitation of the Xe 6s,6s’ states (Section 3.3.3), the difficulty of the experiment and the smallness of the spin-dependent effects suggest low priority of noble gas targets for further electron-photon coincidence studies. On the other hand, the disagreement between experimental
N. Andersen et al. /Physics
Table 4.1 Summary of electron-atom interactions for cases of increasing parameters describing the excitation process for spin polarized 3. I
Sections
Na 3’P + exchange
Hg 6”P, + spin orbit
P:n,,7Ph” +
,L
L; ^,
L;. L; .,t ;“
b, P=l h=O II’ = I
Ipj, P;
L:’ .Ll-* “9. .,* ;,+r . p’.I
Section
Example (excitation of) Interaction(s) responsible Representation Reflection symmetry
He 2’P Coulomb Wave function +
p’=
2 2 0 1
l.P‘=
6 5
)
3.3~ 5
Km,
+.-
1
h’ = 0. h’ = 0 31\+ + \1” = 1 A*(A
lndependent parameters Parameters from Stokes analysis Parameters from STU analysis Necessary observation directions .‘These numbers are the minimum independent electron spin polarizations
complexity, and the set of orientation and alignment beams. Compare to table 2.1.1 for unpolarized beams Section 3.2
CXC
Dirnrmionlrss purumeter,s Angular momentum ( -PJ Alignment an& Linear polarization Total degree of polarization Height Weights Marc angles
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p+t = 1. p+, = 1 h’ 2 0, h’ 2 11
11.’+ \VI = I A+ (A ). A”, 3‘. 10 9
I
I
I”
2;’
required for Stokes parameter analysis. They may increase directions can be prepared. See text for details.
if not all three
data and theoretical predictions for the generalized STU parameters (Section 3.4.6) indicates that first-order theories are not sufficiently accurate for these collision energies. More theoretical work is required to resolve the discrepancies. Studies of the Cs 62S1,2tt62P,,Z,3,z transitions (Section 3.4.1), similar to the experiment performed by Nickich et al. [79] for sodium, would be most valuable for a deeper understanding of relativistic effects in the collision dynamics, particularly with regard to the approximate validity of the tine-structure effect for heavy targets. We strongly recommend further experimental investigations of generalized STU parameters in this area. Similar statements hold for other quasi one-electron systems (Section 3.4.2). In particular, we recommend an experimental test of the theoretical prediction for the spin polarization function at 40 eV incident energy for electron impact excitation of the Cu 42P1,2,3,2 states. A check of this prediction, indicating the dominance of Mott scattering over the fine-structure effect. would be possible in a standard electron polarization setup. Since the theoretical results are nearly identical for both fine-structure levels, no laser-pumping or other means to separate them are required. For the In 52P3,2 + 62S1!2 transition (Section 3.4.3) the large discrepancy between theory and experiment is surprising. Whereas the theoretical results are in qualitative agreement with expectations based on the fine-structure effect, the experimental data clearly violate these predictions. Apparently, one fine-structure transition is adequately described in a semi-relativistic model while a much more sophisticated treatment is required for the other transition within the same multiplet.
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A full-relativistic calculation of this problem is desirable. The same observation is made for transitions in Tl (Section 3.4.3) and Pb (Section 3.6), for which semi-relativistic treatments are inadequate. As can be seen from the above comments, excitation of the resonance transitions in light quasi one-electron targets is sufficiently well understood. Remaining discrepancies between experimental and theoretical results point to the need for further reduction of experimental uncertainties. However, the situation is much less clear for heavy quasi-one-electron atoms and rare gases, and more complex targets in general. Experimental benchmark data for such collision systems would be very valuable for comparison with and further development of theory. Even with unpolarized beams, experiments on targets such as carbon, nitrogen, or oxygen, which are crucial in the interpretation and modeling of observations in astronomy, planetary atmospheric studies, plasma physics and laser physics, seem worthwhile to us. Regarding further theoretical work, the success of the “convergent close-coupling” approach for the light targets and the rapid development of computer resources suggest the development of a full-relativistic CCC method for complex targets - a certainly ambitious, but not unrealistic goal for theorists in this field. A new field that may be emerging from combining the techniques presented here with those developed for (e,2e) studies is the analysis of simultaneous ionization and excitation, with e + He 1‘S + 2e + He: (22P) as the generic process. Coincidence measurements between the He+ 22P + 12S 304 A photon and one or even two outgoing electrons provide a new test of ionization theories since, in contrast to standard (e,2e) cross sections, relative phases between ionization amplitudes become important. Finally, we want to reiterate an earlier statement [3]: “In light of the substantial amount of human power and equipment invested in these studies over the past years, we feel obliged to comment on how these impressive resources may be used most effectively in the future. To support the efforts, we strongly encourage theoreticians to report complete sets of parameters from which everything else can be derived. Also, close coordination between experimental groups concerning a common choice of collision systems, energies and scattering angles, as well as the systematic application of state-of-the-art theory for data reduction, seem mandatory to us.”
Acknowledgements
We dedicate this final volume of our review series to Professor Ugo Fano on the occasion of his Enrico Fermi Award. His contribution to this field has been seminal for our approach and we are grateful for his continuous interest in our project. We thank Igor Bray, Friedrich Hanne, Mike Kelley, Steve Lorentz, Don Madison, Jabez McClelland, Bob McEachran, David Norcross, Al Stauffer, Peter Teubner, Wayne Trail and Martin Uhrig for readily communicating their data in numerical form, and Ben Bederson and Joachim Kessler for their long-term support, input and encouragement. We also thank the referee for carefully reading the manuscript and making valuable suggestions for improvement. This work was funded by a grant from the National Institute of Standards and Technology, Office of Standard Reference Data. We are much indebted to Jean Gallagher for continuous support. Financial assistance was also provided, in part, by the Danish Natural Science
N. Andersen et al. / Physics Reports 279 (1997) 251. 396
Research Council (NA), the National Science Foundation (KB).
383
(KB), and a JILA Visiting Fellowship
Appendix A. Scattering amplitudes In the following appendixes we summarize some mathematical details that were used in the data analysis for this review. Most of the equations will not be derived from basic principles; interested readers should consult the standard literature for more information. Instead, the formulas presented here should be sufficient to retrace our steps and reproduce the results presented in the main text. The scattering amplitudes .f(~,~I,~oWI;~)
= (J,M,;krmr
l.Fl JoMo; kJ%)
(A.1)
describe the scattering of electrons (or, more general, spin-; particles) with initial linear momentum k0 and spin component m. (with regard to a given quantization axis) from a target with total by the projectile angular momentum Jo and component MO. The final state is characterized momentum kI, the spin component m, , and the target quantum numbers Jr and MI, respectively. The scattering angle 0 is the angle between k. and kl, and ?F is the transition operator, usually presented as the .F matrix. A possible normalization factor in Eq. (A.l) is omitted for simplicity in
the general formulation A. 1. Scattering
amplitudes
presented here. in di!erent
coordinate .frames
In practical applications, it is necessary to define the scattering amplitudes with respect to a quantization axis for the angular momentum components. A standard choice for numerical calculations is the so-called “collision system” where the incident beam axis is the quantization (z’) axis while the y’ axis is perpendicular to the scattering plane. On the other hand, the algebra often becomes simpler and many observables can be interpreted more easily in the “natural system” where the quantization axis coincides with the normal vector to the scattering plane and the xn axis is defined by the incident beam direction. The transformation of the scattering amplitudes from one system to another can be achieved in a straightforward way by transforming the initial and final states through standard rotation matrices and using the fact that the action of the F-operator must be independent of the particular coordinate system. This transformation is performed as follows. Consider states [j,m’), defined in a coordinate system (X,, Yr,Zr ). Suppose a second coordinate system (X2, YZ,Z2) is obtained from (X 1. Y, , 2,) through a rotation by a set of three Euler angles (SI,j?, 11)as defined by Edmonds [41]. The states I,j, nz’)r can then be expressed in terms of states Ij,vn), defined in the system (X2, Y1, 2,) as
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N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
and d( b)Am,are rotation matrix elements. Note that the angular momentum j is invariant against such rotations. As an explicit example, consider amplitudes for an S’ + P” transition without consideration of the electron spin. Omitting the arguments kO, kI and 6’for simplicity, the corresponding amplitudes are f(M) =f(L,
= l,M&
= M; Lo = O,ML” = 0) E (l,M1,F[O,O)
(A.4)
The natural coordinate system evolves from the collision system through rotation by the Euler angles c1= - rc/Z, fi = - rc/Z and y = 0, and the collision frame from the natural frame through a = 0, ,Q= 7c/2, and ‘/ = n/2. Furthermore, only the P states have to be transformed since the spherically symmetric S states are invariant against such a rotation. Consequently, the amplitude (1,1 (YIO, 0), in the natural system (subscript “n”) is related to amplitudes in the collision system (subscript “c”) as
(Lll~lO,O>, = -(i/2)
(l,ll~lO,O),
+ (i/2) (1, -
- (l/a)
llYlO,O),
.
(l,OlF\O,O),
(A.5)
It will be shown below that Eq. (A.5) can be further simplified by using parity conservation and the planar symmetry of the collision process, A.2. Symmetry properties A very important point for the discussion of scattering amplitudes is the fact that certain symmetry properties of the projectile-target interaction lead to conservation laws through the F-operator. These, in turn, will cause interdependences between various scattering amplitudes or simply require some amplitudes to vanish. Consider, for instance, the conservation of the total parity. This is an extremely good approximation in atomic collision physics. Consequently, computer programs in this field will generally only include projectile-target interactions that conserve the total parity, with the most important example being the Coulomb interaction. For our case of interest, electron-atom scattering in a plane, the process must be invariant against reflection in this plane. This reflection operation can be constructed as the parity operation, followed by a 180” rotation around the normal axis n^ of the scattering plane. If we consider a general matrix element of the form (j, ml 15 1j, mo), reflection invariance implies that (j,mII~ljj,mO>
= (jImI
I~+~NOkJ~
64.6)
where 39 = ~;(180”)~~
(A.7)
is the operator for the reflection, constructed as a product of the parity operator 9’ and the rotation operator Bi, (1SO’). The operation of the parity operator yields the same (eigen)state multiplied by the parity II of the state, and rotations are handled again through the Euler angles. Fortunately, rotations by 180” around the z” axis in the natural frame or the y’ axis in the collision frame are straightforward.
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Applying the general transformation
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(A.2) one finds
.9?l(.jom,) = 2?(0,0,1~)~~~j~m,) = I7,( -l)“oIj,m,)
(A.8a)
for the natural frame and BjjOvnO) = p(O, - rt,O)O~ljjOm,) for the collision frame. Consequently,
= LI,( - l)jn-*filjO -mo)
Eq. (A.6) becomes
~.i~~~l~~lI.j~~~~=~~~~~-~~~~~~~~j~~~I.~I~~~~~ for the natural
(A.8b)
(A.9a)
frame and
(,iI m, ~~TJjomo) = I7,I?,( - l)j~-m~+j~-“‘~(j, - m,)FJj,
- mo)
(A.9b)
for the collision frame, where & and n, are the parities of the initial and final states, respectively. The above symmetry relationships can be generalized to our amplitudes of interest. The results are .f(M,mi,M~m();8)
= n,n,(
-1)“o-M,+mo-m,f(M~m,,Mom,;8)
(A.lOa)
for the natural frame and .f(W~,,MZl%;Q) x
.f(
-Ml
-
= n,n,
ml,
-MO
( -1)
J,-M,+J,,~M”+(l:2)-m,,+(1/2)~m,
(A.lOb)
-Q;@
for the collision frame. Instead of providing phase relationships ( + ) between amplitudes with a given set of magnetic quantum numbers and those with the sign-reversed quantum numbers in the collision frame, see Eq. (A.lOb), Eq. (A.lOa) shows that many amplitudes simply vanish in the natural frame (namely those where the exponent is an even or odd integer, depending on the product of the parities). This fact is one of the many advantages that can be used when formulating the general theory in this frame. Numerical calculations, on the other hand, are much simpler in the collision frame. If desired, however, the resulting amplitudes can easily be transformed into the natural frame. Finally, the corresponding form of Eq. (A.lOb) for S’ -+P” transitions can be used to eliminate the amplitude (1, - 1 I .T IO,0), from eq. (A.5). This leads to the well-known results (1.1 iZ/O,O),
= - i(l,l(Y(O,O),
(l,-lIr~IO,O),= (l.o~qo,o),
-(1/Jz)(1,0~~(0,0),,
-i(1,1~~~0,0),+(1/~)(1,0~~~0,0),, G 0
(A. 1la) (A.llb)
(A. 1lc)
for this case (see also Eq. (F.38) of Vol. I). A.3. Scattering In electron
amplitudes
in the nonrelativistic
limit
scattering from light targets, it is often assumed that the total spin S and the total L of the combined target + projectile system are conserved during the
angular momentum
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collision. This approximation is very good, for example, for many of the alkali-like targets discussed in this review. It is also the basis for the “fine-structure effect” which will be further discussed in Appendix C. In this nonrelativistic approximation, the scattering amplitudes depend in a purely algebraic way on the spin quantum numbers, and transitions between fine-structure levels are described by standard recoupling techniques. Specifically, the scattering amplitudes (A.l) can then be expressed as
xfSWt>,
ML,; ‘3 >
(A.12)
where SO, S1, MS,, MS,, Lo, L1, ML,,, and ML, are the spins as well as the angular momenta and the corresponding z components of the initial and final target states, and (A.13) are nonrelativistic scattering amplitudes that describe transitions between orbital angular momentum states. Eq. (A.12) expresses the conservation of the total spin S and its component MS = MS, + ml = MS, + m,
(A.14)
through the Clebsch-Gordan coefficients (jl, ml; j,, m2 1j,, m3). Note that this approximation will reduce the number of independent scattering amplitudes significantly, since the scattering dynamics are assumed to be identical for each member of a fine-structure multiplet.
Appendix B. Density matrices A thorough introduction to the theory of density matrices with emphasis on their applications in atomic physics can be found in the book by Blum [28]. Extended applications to electron collisions have been given in Vol. I and the review by Bartschat [lo]. The main advantage of the densitymatrix formalism is its ability to deal with pure and mixed states in the same consistent manner. The preparation of the initial state and the details regarding the observation of the final state can be treated in a systematic way. In particular, averages over quantum numbers of unpolarized beams in the initial state and incoherent sums over nonobserved quantum numbers in the final state can be accounted for by the “reduced density matrix”. Furthermore, expansion of the density matrix in terms of “irreducible tensor operators” and the corresponding “state multipoles” (see Section B.3 below) allows for the use of advanced angular momentum techniques. The “complete” density operator after the collision process is given by
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where Pin is the density operator before the collision. The corresponding by (P,“JZXfO =
387
matrix elements are given
c Pmbm,PM;M, m;,m,MGMo xf(M; m’r, W mb; 0) f* (MI ml, MO mo; Q) ,
03.2)
where the pm6,,,,pM;M,describe the preparation of the initial state and the star denotes the complex conjugate quantity. (We assume that the projectile and target beams are prepared independently, thereby allowing for this factorization.) B. I. Reduced density matrix formalism
The “reduced” density matrices account for the fact that, in practical experiments, not all quantum mechanically possible quantum numbers are determined simultaneously. For example, if only the scattered projectiles are observed, the corresponding elements of the reduced density matrix are obtained by summing over the target quantum numbers as follows:
(PouJm;m,:H =
M;=M,
(H.3)
The differential cross section is then given by
s =c
1 (PouJm;m,:o ?
(B.4)
m; =rn,
where C is a constant that depends on the normalization of the continuum waves as well as a possible normalization factor in Eq. (A.l) in a numerical calculation. These reduced densitymatrix elements contain information about the projectile spin. The information can be extracted through a measurement of the generalized STU parameters which will be discussed in the next section. On the other hand, if only the atoms are observed (for example, by analyzing the light emitted in optical transitions), the elements
determine the “integrated Stokes parameters” [18], i.e., the polarization of the emitted light independent of the electron scattering angle. These parameters contain information about the angular momentum distribution in the excited target ensemble. Finally, for electron-photon coincidence experiments without projectile spin analysis in the final state, the elements
u3.6) contain information about the projectiles and the target. This information can be extracted by measuring the angle-differential generalized Stokes parameters discussed in detail in Section 3.3.
simultaneously
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The density-matrix formalism outlined above is very useful for obtaining a qualitative description of the geometrical and sometimes also of the dynamical symmetries of the collision process. An example are the generalized STU parameters which will now be discussed in some detail. B.2. Generalized STU parameters The generalized STU parameters describe the scattering of a spin polarized electron beam from unpolarized targets. They can be expressed conveniently in terms of the quantities (m; mb;m, mo) = 2J ‘+ 1 C fWbmL&mbV*W~ 0
MI
m&f0m0).
(B-7)
MO
The derivation of the general formulas for these parameters proceeds as follows. First, the unpolarized target beam in the initial state is represented by matrix elements pM; M0= 6, Moin Eq. (B.2), while a spin-polarized projectile beam is represented by elements of the polarization matrix [59] Pproj
1 l+P, 2 ( Px-iP,
=
-
P, - iPy l-P,
03.8)
>
Then the reduced density matrix elements defined in Eq. (B.3) are calculated, followed by the differential cross section given in Eq. (B.4). Finally, the spin polarization components Pi,y,z after the scattering are obtained through PX.Y.2
=
tm,,,,
Pdltr
fhd
03.9)
,
where &.d is the reduced density matrix for the scattered projectiles with the matrix elements given in Eq. (B.3), %,,,, are the three Pauli matrices, and tr denotes the trace operation. For the natural system, one finds the general result, see Fig. 2.3.3(a): p = 6% + T,P,)I
+ (T,P,
+ U,,P,)j 1 + SAP,
+ (T,P,
- U,, P,)k
(B.lO)
The equivalent equation for the collision system [lo] is obtained by making the index substitutions x+-z, yttx and ztty in Eq. (B.lO). The final formulas can be simplified substantially by using the hermiticity of the density matrix and assuming parity conservation in the projectile-target interaction. The hermiticity condition yields [lo] (B.11) (mi &;mI mo> =
m&; ml
of Eqs. (A.lO) to the parameters mo), ,
(m; m&;ml mo)c = ( -l)m~-mb+m~-m~(- m; - mb; - m, - mo)c,
defined in (B.12a) (B.12b)
where the subscripts again denote the parameters in the natural and in the collision frame, respectively. Once more, we see that reflection invariance with respect to the scattering plane
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389
yields & phase relationships between parameters in the collision system, whereas half of the possible combinations of the magnetic quantum numbers will not give any contribution in the natural frame. Omitting a possible normalization factor, the differential cross section for unpolarized projectile beams is given by
in both the natural frame and the collision frame. For the remaining seven generalized STU parameters one finds: Sp = (1/20”)[(~~;+3),
+ (3 - 4;s - i), - ( - 3f; -+f),
- ( - 4 - 4; - 4 - i)“]
T, =
(B.14a)
- ( - f - 3; - 4 _ i),]
,
(B.14b)
(l/a,)Re{(a$;
+ (-~$;~-~),},
(B.14~)
-3
-3)”
T, = (l/o”) Re ((33; - t - f), - ( - $$;f - $),), TZ = (l/20,) [(+&++),
- (4 - f;+ - +), - ( - if; -++),
+ ( - f - 3; - + _ f),] ) Uyx=
-(l/o,)Im{(~+;--3-+),-(
U,, = - (l/a,)Im
(B.14d)
(B.14e) -f+;i-*),},
{($$; - 3 - f), + ( - ++;+ - i),}
(B.14f) (B.14g)
in the natural frame and (B.15a) (B.15b) (B.15~) (B.lSd) (B.15e) (B.15f) (B.15gl
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B.3. State multipoles
The general density-matrix theory can be elegantly formulated by decomposing the density operator in terms of irreducible components whose matrix elements become the so-called “state multipoles”. In such a formulation, full advantage can be taken of the sophisticated techniques developed in angular momentum algebra. For a thorough introduction to this method, we refer again to the book by Blum [28]. The density operator for an ensemble of particles in quantum states labeled as IJ, M) where J and M are the total angular momentum and its magnetic component, respectively, can be written as P= c
&MIJM’)
(11.15)
(JMI >
M’M
where &M
(JM'bI
=
(B.16)
JM)
are the matrix elements. Since it is sufficient for our cases of interest, we have assumed that J is well defined, but coherences (M’ # M) between different magnetic sublevels are possible. Alternatively, one may write P =
c
KQ
T(J)KQ
(B.17)
3
where the irreducible tensor operators are defined as
T(J)
KQ
=
1
(
-l)J-M(J,M’;J,
- MjK,Q)jJM’)(JMj
(B.18)
M’M
and the “state multipoles” or “statistical tensors” are given by (T(J)Z;Q)
= C ( -l)J-M(J,M’;
J, - MIK,Q)(JM’Ip(
JM)
.
(B.19)
M’M
Note that the selection rules for the Clebsch-Gordan
coefficients imply
OIK12J,
(B.20a)
M’-M=Q.
(B.20b)
Eq. (B.19) can be inverted through the orthogonality to give (J’M’lp(JM)
= I(
-l)J-M(J,
condition of the Clebsch-Gordan
M’; J, - M(K, Q)(T(J)&Q)
.
coefficients (B.21)
KQ
The transformation properties of the tensor operators and the state multipoles defined in two different coordinate systems are given by expressions similar to Eq. (A.2) as
T(J) KQ
=
~T(JhA~,P> 4
?>,“Q 7
(B.22a)
N. Andersen et al. j Physics Reports 279 (1997) 251-396
391
i.e., the rank K of the tensor operator is invariant, and U(J&p)
(B.22b)
=~UV)LJ~@JY)y”;;~ 4
The irreducible tensor operators fulfill the orthogonality
condition (8.23)
With (B.24) being proportional to the unit operator 1, it follows that all tensor operators have vanishing trace, except for the monopole T(J),,. The hermiticity condition for the density matrix implies (T(&_$*
=
(
.
-l)"
(B.25)
Hence, the state multipoles (T (J)kO) are real numbers. Furthermore, the transformation property (B.22b) of the state multipoles imposes restrictions on non-vanishing state multipoles to describe systems with given symmetry properties. In detail, one finds the following: (a) For spherically symmetric systems: (T(J):,Q)
=
(B.26)
(T(~)2(,)rot
for all sets of Euler angles. This implies that only the monopole term (T(J):,) from zero. (b) For axially symmetric systems: (T(diQ)
=
can be different
(B.27)
(~(~)~,)rot
for all Euler angles 4 that describe a rotation around the z axis. Since this angle enters through a factor exp { -iQ$} into the general transformation formula (B.22b), it follows that only state multipoles with Q = 0 can be different from zero in such a situation. (c) For systems with properties invariant under reflection in the xy plane: (r(f):,Q)
=
( -ljK
(T(&Q)
.
(B.28)
In this case, state multipoles with odd rank K must vanish. This gives the well-known result for the description of electron-photon coincidence experiments with unpolarized beams in the natural frame (see, for example, Eqs. (3.3.2)). (d) For systems with properties invariant under reflection in the xz plane:
=
( -l)K
(B.29)
In this case, state multipoles with even rank K are real numbers, while those with odd rank are purely imaginary. This is the equivalent result to (c) for unpolarized beams in the collision frame.
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N. Andersen et al. /Physics Reports 279 (1997) 251-396
Appendix C. The fine-structure effect
The “fine-structure effect”, in its pure form, is derived by assuming the validity of Eq. (A.12). It was discussed in detail by Hanne [SO] for the special cases of S” -+ P” transitions in alkali-like systems and ‘Se + 3P0 transitions in quasi-two-electron targets. Some of the dynamical requirements for the realization of the fine-structure effect were discussed in Section 3.2.1 and will not be repeated here. Instead, we give a generalization of Hanne’s treatment that was derived by Goerss et al. [48]. Using Eq. (A.12), one can obtain an interesting result for the sum over fine-structure levels of the parameters defined in Eq. (B.7). In detail, we evaluate
=
C(L,,M(t,;S1,M$,IJ1,M,)C(Lo,M~,;So,M$oIJ,,M,) c f , ,allM s
J,,J,,S,S
x C(S1,M~l;~,m;lS’,M$)C(So,M$,;~,mbIS’,M$) xC(L1,ML,;S1,Ms,IJ1,M1)C(~o,MLo;So,Ms,IJ,,~o)
xC(S1,Ms,;f,m,IS,Ms)C(So,Ms,;~,molS,Ms) (C.1)
XP’ UC.,>Jc,,)fS UK, 9ML,)* .
In this case, the sums over J1 and Ml as well as over Jo and MO can be performed by using the well-known relations for the Clebsch-Gordan coefficients [41]. The result W&J%,)
WW’%)
W&J’%)
W%,J’%)
can then be used in the remaining Clebsch-Gordan rule
(C.2)
coefficients in Eq. (C.l) to yield the selection
M~-Ms=m~-m,=m~-mo
(C-3)
Hence, the sums (C.l) will be zero if rn; - ml # rnb - m,. From Eqs. (B. 15) in the collision system, it follows that the mean values <&) = c
(250 + l)a(J,,Jo)Sp(J1,JO)/~tot
>
(C.4a)
(S‘4)
(2.40 +
3
(C.4b)
J,,Jo
=
1
l)a(J1,Jo)Sp(J1,Jo)I(T,,,
(C.4c)
(CAd) J,,Jo
with ctot = c
J,,J,
(250 + I)~(J,YJO)
(C.5)
N. Andersen et al. 1 Physics Reports 279 (1997) 251-396
393
will vanish in the approximation of the pure fine-structure effect. With the index substitutions outlined below Eq. (B.lO), these relationships will also hold for the equivalent parameters in the natural frame. Examples have been given in Eqs. (3.2.11) and (3.4.3).
Appendix D. Reduced state multipoles from generalized Stokes parameters
This appendix lists the equations that allow for the extraction of the reduced state multipoles (3.3.12) from measured generalized Stokes parameters, including the possibility of depolarization effects in the emitted radiation due to hyperfine structure in the target. In the latter case, state multipoles of rank K >O in Eqs. (3.3.14-16) must be multiplied by perturbation coefficients GK(G, E 1). As mentioned in the discussion of Fig. 3.3.5, the set of inversion equations is unique for the following state multipoles: (r;o>” = - $P3(1
(D. 1.a)
+ P,)lD, 3
(D.1.b)
(%>u
= - $
(P, - 2P, - Pi PJ /& 7
(r’&
= - $P,
(1 + P4)/D2 ,
(D.1.c)
(i!%
= ,:‘5P,(l
+ P,)l&
,
(D.1.d)
+ P,)lD, ,
(D.1.e)
(CC&, = - &!Z;Z(I
(D.1.f)
(iL>p; = & QZ2Z2 (1 + P4)l& , where we have defined the denominators DK = GK(3 + PI + P4 - PI P4) .
(D.2)
Furthermore, when omitting any one of the three generalized Stokes parameters Qj;, Qf; and QP;, the following three sets of equations can be used to derive the state multipoles (r”,,),z, ( r~o)r, and (r;2)pz in a unique way: (1) Omitting Qf;: (&& (&Jp:=
= 3 KQ'1; + QWU 3$
+ Pd + 2Q:W +
EdllD,
Q:;U + RJ&,
+%)P, = $[2QT% (1+ P,)+ 3 Q:;(l + P4)]/D2,
,
(D.3.a) (D.3.b) (D.3.c)
where QZ = CQ;; + Q;Z3+ Q”;33/3
(D.4)
is the experimental average value for the elements in the third column of the Stokes matrix (Q;;).
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Reports 279 (1997) 2.5-396
(2) Omitting Q$: (&)p,
= 3C2 QP; (1+ J'd- (Q;Z2 - QfZ3)U+ RJ1/(2&),
0%)~; = 34 (GA,
QI;(l+ pa)/&,
= - fiQqzZ(l + PA/D,.
(D.5.a) (D.5.b) (D.5.c)
(3) Omitting Qy3: %dP,
= - (QP;- 3QYN
%A,
= - ~PQpz2U
+&I + 2Q",;U +f%)/&, + f'd+ Q"1;(1 + p,)l/& ,
(Gdp, = - fi Q"1;(1+ p4)lD2.
(D.6.a) (D.6.b) (D.6.c)
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