Collision Experiments with Laser Excited Atoms in Crossed Beams

COLLISION EXPERIMENTS WITH LASER EXCITED ATOMS IN CROSSED BEAMS I . V . HERTEL and W . STOLL* Fachbereich Physik der Llniversitat Kaiserslautern Kaiserslautern, West Germany

I. Introduction

................................................. .... ....

B. The Language of Multipole Moments . . . . . . . . . . . . . . . . . . 111. Excitation of Atoms by Laser Optical Pumping . . . . . . . . . . . . . . . . . . . . . A. Theoretical Considerations . . . . . . . B. Experimental Aspects of Optical Pumping . . . . . . . . . .

IV. Theory of Measurements in Scattering E

A. The Percival-Seaton Hypothesis : Electron Spin and/or Nuclear Spin Uncoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Explicit Expressions for Scattering Multipole Moments in Terms of Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . V. Collision Experiments . . . . . . . . . . . . . . . . . . . A. General Aspects . . . . , . . . . . . . . . . . . . . . . . . . B. Inelastic Electron-Scattering Processes from C. Total Scattering Cross Sections for Low-Energy Electron Scattering from Sodium 3’P,,, Using Recoil Techniques . . . . . . . . . . . . . . . . . . . . . . . . D. Elastic Atom-Excited Atom Scattering at Therm E. Fine-Structure-Changing Transitions in Heavy-Par F. Electronic to Vibrational Energy Transfer . . . . . . VI. Atomic Scattering Processes in the Presence of Strong A. Free-Free Transitions and Similar Phenomena . . . . . . . . . . . . B. Coherent Superposition of Ground and Laser-Ex ..................... mately Resonant Atom Excitation . . . . .

................

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 117 117 119 129 129 142 157 157 162 174 175 179 191 193 200 203 211 212 216 223 224

I. Introduction The past two decades have seen much progress in atomic collision physics, which is now included in standard textbooks of enormous size (e.g., Massey et al., 1969-1974). Both experimental techniques and theoretical methods

* Present address: Messer Griesheim GmbH, Diisseldorf, West Germany. 113

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I . 1/: H u t e l and W Stoll

have been more and more refined and have been applied to a large variety of interacting systems. This has led to an improved understanding of the basic interaction mechanisms underlying atomic scattering processes and has yielded suitable methods of theoretically describing or even predicting them. However, one is still unable to give satisfyingdescriptions of even seemingly simple collision problems. Of course, the Schrodinger equation in the nonrelativistic limit is exactly valid for all atomic collision phenomena so far encountered. It renders, however, little physical insight into specific processes of interest, and what may be called an understanding of a particular process often depends on personal taste. For instance, the elastic scattering of ground-state atoms by other such atoms is adequately described in the range of thermal energies by a knowledge of the interatomic potentials as a function of the internuclear distance only, and even the inversion of the problem is possible, since the underlying differential equation is solvable to a high degree of accuracy (Buck, 1974). In contrast, problems that seem to belong to the simplest situations in atomic collision physics, such as electron scattering by atomic hydrogen and helium, are far from being solved, especially when inelastic processes and ionization are taken into account. They cannot be described in such simple ways as the previous case since the governing equations are an infinite set of coupled integrodifferential equations resisting any attempt to be solved exactly, although even there considerable progress has been achieved. Completely intractable at present, in a more than statistical manner, seems to be the situation where complex heavy-particle processes such as electronic to vibrational energy transfer or even reactive scattering are under consideration. Not only are the computational methods in question, but even the equations to be solved. Thus, the situation is still unsatisfactory, especially if one considers binary atomic collisions as elementary processes for any interaction within any kind of bulk matter. The progress that has been made in atomic collision physics has been due, on the one hand, to the large amount of processes studied, and on the other hand, to the rigorous investigation of more and more refined details of atomic cross sections. It is a long way from predicting a rate constant in which the various averaging procedures over different energies, scattering angles, etc., may statistically cancel computational errors, to (for instance) the understanding of a complex differential scattering amplitude for an inelastic process including, say, spin analysis (Bederson. 1969. 1970), which poses many more critical requirements on any theoretical approach. Steps toward this end have been, to mention just a few. the investigation of resonances in electron scattering (Schulz, 1973),methods of phase shift analysis (Andrick, 1973),experiments with polarized electrons (Kessler, 1976),or in heavy particle scattering the observation of rainbow and supernumerary

COLLISIONS WITH LASER EXCITED ATOMS

115

rainbow oscillations (Pauly and Toennies, 1965). Among the most fruitful methods have been experiments with crossed beams investigating differential cross sections. If one surveys the literature it becomes clear that the bulk of the experimental material is concerned with atomic species in the electronic ground state. Metastable atoms have occasionally been subject to investigations but no crossed-beam experiments with short-living atoms optically accessible from the ground state have been performed until recently. Obvious experimental difficulties have prevented the preparation of excited species in an atomic beam. Such atoms typically live for sec only and at thermal energies travel around cm during that time. However, with the advent of tunable narrow-band lasers, especially cw dye lasers, the situation has changed and it has become clear that it should be possible to excite atoms optically within the scattering region of an otherwise conventional crossed-beam experiment (Fig. 1). In this way a B beam

FIG.1. Schematic arrangement for the investigation of scattering processes from laserexcited atoms (A + h ~ , =~ A*) , in a crossed-beam experiment. After the collision A* B + A' + B'. either A' or B' may be detected.

+

steady-state upper-state population could be reached that may be comparable to the ground-state population. It, for the moment, we assume atom A to be simply a two-level system, then the ratio of the excited-state number density to the total number density would be given by

where A is the exciting wavelength and u,, the spectral radiation density.' At present we just note that with laser intensities of less than 1 W/cm2 and a bandwidth comparable to the natural linewidth ( z 10 MHz) one may obtain nearly equal population of ground and excited state. It is obvious that these possibilities make possible a large variety of novel studies in atomic collision physics that offer more than just an extension of the usual experiments with ground-state atoms and metastables. That is to

' The latter definition will have to be modified somewhat on closer inspection for highintensity fields.

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say, one may make specific use of the inherent properties of the exciting laser light, namely its monochromaticity, well-defined polarization, and high intensity. When an atomic beam is excited, it is free of internal collisions and, for right angle intersection with the laser, free of Doppler broadening. Then the laser properties allow us very specifically to select the state into which the atom is excited. Specific fine- and hyperfine-structure states may be chosen as well as a particular combination of substates. Thus, one may now design experiments to study in detail differential cross sections for finestructure or even hyperfine-structure transitions, or for transitions among substates with different angular momentum projection quantum numbers (so-called phase-changing transitions). In this context it should be remembered that nearly all atoms easily accessible for collision experiments hitherto used have been in an s state or at least in a spherical isotropic mixture of substates. In contrast, the novel techniques allow us to prepare states with an angular momentum different from zero and to vary systematically the alignment and orientation of the resulting nonspherical interaction potentials. Thus, a much more sensitive test of any theory is given than in the usual experiments where one averages over unobserved projection substates. The new area of study is thus a logical step on the way to exploring more and more details of collisional interaction as indicated above. It is obvious that full use of these possibilities can only be made in crossed-beam experiments. Therefore, the present article will be restricted to this type of experiment exclusively. In spite of the attractive features of the new field, few scattering experiments have actually been performed, largely because of the fact that only a few atoms are accessible to this type of preparation, due to the availability of suitable lasers. At present one is essentially restricted to sodium, using the most convenient cw Rhodamine 6G laser to excite the Na(3’P) resonance state. However, rapid progress in laser technology may very soon change this situation. Nevertheless, the possibilities, difficulties, and limitations of the experimental methods are now understood, initial experimental results have emerged, and a theory of measurement has been developed and probed in a variety of interesting cases. Thus, it seems adequate to review the situation as it stands. The present work should therefore be viewed as a means of disseminating the experience hitherto gained, and we hope thereby to stimulate new experiments exploiting these techniques. Most of our knowledge stems from electron-scattering experiments that were first reported by Hertel and Stoll (1973) and followed more recently by Bhaskar rt al. (1976). Electron scattering by laser-excited Na(32P3,2)serves as a test case for understanding the method and its results. Only recently have differential heavy-particle scattering experiments been reported by Carter et al. (1975b). followed by

COLLISIONS WITH LASER EXCITED ATOMS

117

Diiren and collaborators (1976). The interpretation of the latter experiments is still under discussion, and the same is true for an investigation of the quenching of excited sodium by molecules, first reported by Hertel et al. (1976). Besides presenting the available experimental and theoretical material, it is the intent of the present review to clarify some open questions concerning the optical pumping process as well as the theory of measurement, as far as it is specifically related to scattering experiments involving laser-excited atoms. The language of multipole moments is used throughout the paper. We hope. however, to present the necessary mathematical apparatus, together with some perceptual aids that may give easier access into this field, especially to experimental physicists. On the other hand, we think it is worthwhile to give a comprehensive survey of the necessary formulas and to illustrate their use by suitable experiments, thereby omitting some minor errors that have occurred in previous publications. We start with some general remarks on the language of multipole moments and its application to scattering processes by laser-excited atoms, and then give a new interpretation for the meaning of scattering multipole moments. This is followed by a discussion of the optical pumping process for the target preparation, which is the crucial part of the experiment. We then investigate the applicability of rate equations, the time development of the state multipole moments during pumping, and possible excitation mechanisms and other relevant experimental aspects of optical pumping. We then discuss in detail the theory of measurement, and we shall concentrate not only on inelastic electron scattering but also on scattering experiments with spin analysis and on elastic and fine-structure-changingheavy-particle collision problems. Next we report on scattering experiments and illustrate the application of the new techniques to these and other problems. In the final section we make some remarks on scattering processes in strong laser fields. In particular, we discuss some recent experiments on free-free transitions and the effects that may be observed due to a coherent superposition of ground and excited states of the laser-excited atom. It does turn out that these, fortunately, may be neglected as far as the subject of the present review is concerned.

11. Basic Theory A. GENERAL ASPECTS Scattering processes by laser-excited atoms deserve a special theoretical treatment, although in principle the calculation of the relevant transition matrix elements (scattering amplitudes) is subject to standard atomic colli-

sion theory. In fact, the scattering amplitudes fnjm. 61,Jk, k) that describe a ~ a state collisional transition from a state with quantum number’ i i J to with njm are just the complex conjugate to those describing the inverse process from an originally laser-excited atom (the in- and outgoing collision wave vectors G and k are also inverted). However, the usual procedure of averaging over all initial probability p ( m ) and summing over all final states

is not done in such a simple way for scattering by laser-excited atoms. Such atoms are prepared in a well-defined combination of substates I.jm). One can usually find a coordinate frame in which the laser-excited atoms are easily described as an incohcrent superposition of states with different projection quantum number In, each having a probability w(m). That is to say, the excited-state density matrix is diagonal, o,(m. m‘) = d,,,w(m). in a coordinate frame that henceforth is named the photon frame (ph). Its zph axis is parallel to the electric vector E for linear polarization (n light) and parallel to the incident light direction for circular polarization (o light) (see Section 111). However, the collision process is usually described best in a different frame, the collision frame (col), for which the zCoIaxis may be parallel to the initial or final interparticle axis and the x,,, axis lies in the scattering plane. Thus the amplitudes given in the col frame have to be transformed into the photon frame, or the diagonal atomic density matrix has to be rotated into the collision frame. The scattering intensity will be determined by a coherent superposition of different I .jm) states and

Consequently, such experiments provide much more detailed information on scattering processes than do conventional scattering experiments, especially since one has the freedom to choose a variety of relative orientations of the ph frame with respect to the col frame. Equation (3) may be further evaluated in a conventional manner, as was done by Hertel and Stoll (1974a). However, the latter description does not bring out clearly all the possibilities provided by optical pumping. One uses with advantage the language of alignment and orientation parameters as introduced into atomic collision physics by Fano and Macek (1973), which has been applied to the problem under investigation by Macek and Hertel (1974). It presents a

* Here and in the following we shall identify laser-excited states with unbarred quantum numbers, while all other atomic states will be designated by barred symbols.

COLLISIONS WITH

LASER EXCITED ATOMS

119

transparent picture of the experimentally observed phenomena for a variety of accessible parameters. This description in terms of irreducible tensor operators or state multipole moments will be illustrated by experimental data in the following way: (1) Geometrical and dynamical factors are clearly disentangled. (2) The observables reflect the symmetry of the experimental setup, which facilitates the discussion of the possibilities of the experiment and emphasizes the special predictions of different theories. (3) The observables are measured by a suitable choice of experimental parameters and are easily computed from commonly used scattering amplitudes. The language of multipole moments is a standard subject of textbooks (see, e.g., Rose, 1967; Edmonds, 1964; Brink and Satchler, 1971) and is usually applied in optical pumping experiments (see the review by Happer, 1972). Nevertheless, atomic collision physicists seem to be reluctant to use it. Thus some explanatory remarks seem appropriate, since this language is the most convenient description for any type of experiment involving changes in the coordinate frames of target preparation, collisional system, and/or detection device. In particular, spin-polarized collision studies and experiments with polarized molecules might adequately be analyzed in terms of the theory discussed here for atoms prepared by laser optical pumping. All questions arising from coherent or incoherent superposition of scattering amplitudes to and from an electronic level can be elegantly treated in this language.

B. THELANGUAGE OF MULTIPOLE MOMENTS 1. Dejnition of Multipole Moments and Description of’ the Scattering Process

Following Macek and Hertel (1974), the scattering process of particles B (e.g., of an electron) by a laser-excited atom A* with quantum numbers njm leaving the atom in a state q%, A*

+ B + A’ + B’

is described in terms of the time inverse process, where the states fij are usually taken to have an isotropic population (other cases are discussed in Section IV). One may view such a process as the preparation of a particular combination of Injm) states, since the asymptotic wave function has the form (in the center of mass system): exp(ikr) @ ( B ’ ) . exp(ikr) * tpilfi + @(B)- 7 Ye,,,

(4)

I . r/: H i w d and W Stoll

120

with Ycol=

C f:;: m

a1,ii $ n j m

(5)

where $ n j m , $81m stand for the atomic and @(B),@(B)for the other particles' wave functions before and after collision. respectively. In the actual scattering process, the atom is prepared by optical pumping in a state Ypumwhich is usually different from Ycol.Then the scattering intensity I is given by the projection of one state onto the other: I x ~(Ypu,,,~Yc,,l)~2. As an example, the scattering intensity would be ( I (.jmlYcol) I )' = ((Vco,1 jm)(jmI Ycol))if just one state I jm)were excited optically. The brackets ( ) indicate an average over all unobserved states iijtfi and also over unobserved quantum numbers of B and B. such as the electron spin. Obviously 1 jm)(jmI represents the statistical (density) operator of the atom excited in this special way. More generally, one has to sum over all projection quantum numbers according to their statistical weight. In a general form, the differential scattering cross section is given by I ( 9col) = 1 0 Tr w = 10

1

Pmm, am*m

m'm

(6)

where I . is the differential cross section for an isotropic initial iij state population given in Eq. (2). The density operators p and u are those for the atom excited by the time inverse collision from an isotropic fij state distribution [Eqs. (4) and (S)], and for the atom excited by optical pumping, respectively. Since we may choose a particular photon frame where om,, = S,,,,, w(m), Eq. (6) reduces to a single summation over m. The collisional density matrix is given in the collision frame by pmm,= Cq,, , with

where p(B) is the probability of finding a specific particle in state B in the initial collision system and p(B'), p(?), p(tii) are the detection probabilities for particular quantum numbers B', 7, and ti& respectively. Macek and Hertel have evaluated Eq. (6) by making use of the possibilities of expanding the density matrices into state muhipole moments P k q and (Tkq [see Fano, 1953, and Brink and Satchler, 1971, Eqs. (6.64) and 6.65)]: P j m . jm'

=

C

pkq(.i)(k

kq

and inversely P k q W

=

C

mm'

Pjm.j m O

Ij

-~im')(-)~-'-'"

- mjm' I h)(-)k - j - m

(81

(91

COLLISIONS WITH LASER EXCITED ATOMS

or in operator form rtl=

1 I ,jm')(jm I ( j - mjm' Ikq)( -

)k-j-m

mm'

121

(10)

where in our terminology Pkq = (rf'> Both pkqand T : ~are ~ irreducible tensor operators of rank k. Equations (8) and (9) may also be written for an atom excited by optical pumping. Then by inserting these expressions into Eq. (6) one obtains by orthogonality of the Clebsch-Gordon coefficients the scattering intensity z PkqO&-q.Now, since we have chosen a photon frame (ph) where omm, = d,, w(m), we have only one nonzero component of the optically excited multipole moments of the atom:

Zq

ckq= d,, M ' ( k )= dqo

1 w ( m ) ( j - mjm IkO)( -

)"-j-"'

m

(11)

Finally, our scattered intensity becomes I = I,

cw(k)(q)

(12) describe the atom after the time inverse The state multipole moments collision process with respect to the photon frame and may be calculated if Yco, is known.

(rtl)

2. Real Multipole Moments and Frame Transjorination Before actually comparing the experiment one has to transform the (rbkl) from the col frame [where they may be easily computed from Eqs. (6)-(911 to the ph frame to which Eq. (12) refers. This is done by rotation through the appropriate Euler angles. First, however, it is useful to introduce real quantities rather than the usual complex ones. Experimental observables are always real and the number of independent parameters may often be reduced by obvious symmetry relations (see below). Letting p denote k l , the real tensor operators T are connected to the usual complex ones by G kl P

- 1/$(i)(p-1)/2[(-pTI

7?t!= 7-p,

7?#!

= 0,

+

o
pT[!Iq], q=o

(13)

With these relations the real spherical harmonics become

Y t i ( 0 ,a) = n- "28,,(COS

0)cos q@ Y f l ( 0 , a) = n- 1/2:pk,(cos0)sin qa Y t ! @,a)= ( 2 4 - 1/29,,(cos 0).

}

q >0

(14)

I . c! Hertel and W Stoll

122

Here the SP, are the normalized Legendre polynomials of Bethe and Salpeter (1957),which relate to the usual complex spherical harmonics by

Y$)(@,a) = (27r)-1’2( -)qpkq(cos 0) exp(i@),

q 20

with the orthogonality relation

Y r j Y$s*dQ = 6,. 6,,y 6 k k * The real operators exhibit reflection symmetry and hermiticity explicitly. One sees directly that the Yg are real and that they transform as Yiy + pY$ under reflections in the zx plane. For later use we also give the real rotation matrices: 9’‘;.

c y p , = (-)-q[(-p

i2$, 4 p ,

cos(qa

+ q’y) d;k,!(/3)

+ p’ coS(qx - 4’7) dtlq48)1, = (-)-‘Q‘(-Psin(qa + q’y) d$!,(p)

+ sin(qa - q’y) dt!,,(/3)],

9?$,,@,‘= (-p+q’J2 d$( @o+,o+ = dgJ(/3),

cos(qa + q’y), p’ sin(qa q’y),

+

for p = p‘, k Z q, q’ > 0 for p # p‘. k 2 q, q’ > 0 for p = p’, q or q’ = 0 for p # p’, q or q’ = 0

q = q’.= 0

(15)

As a next step we note that the description of the experimental parameters ( 7 : l ) in terms of projection operators is not unique and other representations with more direct physical meaning may be preferable. To obtain such a substitution (Fano and Macek, 1973;Macek and Hertel, 1974)we recall that the Wigner-Eckart theorem allows us to replace any matrix element of an irreducible tensor operator of rank k by the corresponding one of another such operator. The same is true for averages, and thus3

with the reduced matrix element ( j 1 7[k1 11 j ) = (2k + 1)”’. Equation (16)may be inserted into Eq. (12);it allows a great flexibility in the choice of III4kp’ and lends itself to various interpretations of the experimental observables. It may, e.g., be constructed from angular momentum operators j , ,j , ,j z We use the notation of Edmonds (1964) for the redudmatrix elements.

COLLISIONS WITH LASER EXCITED ATOMS

123

using special polynomials: sometimes called the multipole-moment owrators. With this choice of the ratio of reduced matrix elements becomes (Macek and Hertel, 1974)

Using Eqs. ( 1 6 k ( 18) we obtain for the scattering intensity from laser-excited atoms 1 = IokrW(k)V(k)<7$’$(ph)) &=O

where 3”(k)= u(”)(F) and k,,,= 2F, when the atom is described in a hyperfine coupling scheme. We shall discuss in Section IV how to modify this when hyperfine interaction or even spin-orbit interactions do not play an essential role. Finally, transformation to the collision frame is done by

(Q$(ph)) =

2

(TJ(col))Cbk,‘(O,a)

q=o. p = f 1

(20)

where 0 and @ are the spherical coordinates of the photon frame, i.e., the E vector for d i g h t excitation and the light axis for 0 light.5 By variation of these angles, in principle all scattering multipole moments can be measured experimentally. From Eq. (17) (footnote 4) we find that by reflection on the x-z plane, the tensors F:j transform as p j: -,p(-)&T[&l because each pair of j , and Vi in qP Eq. (17) changes sign under reflection. Since the collision system is invariant to reflections in the scattering plane, we have

(q! (col)) = 0,

for k even

(T$!(col)) = 0, for k odd. (21) For reference we give the real spherical harmonics up to rank 2 and the multipole operators derived from them in Table I. These operators are constructed in a general way by polarizing (FalkoB and Uhlenbeck. 1950: Brink and Satchler. 1971) the real (Fano. 1960) solid harmonics,

$*C:L

+ l))1’aY$(C3.

= $(4n/(2k 3

Tl(j,j, ....j) = i t . C._.= il.

@)

( j i ,j i 2 ...j , ) . V i , Vi,... Vii,C$

(17)

1

where i denotes real rectangular coordinates. For consistency. we use here real solid harmonics,which are related to spherical harmonics by Eq. (1 7).

’

124

I . K Hertel and W Stoll TABLE I REALS O L I D HARMONICS A N D MULTIPOLE OPERATORS UP TO RANK 2

The expectation values (qd) = Tr of (T[b-’) and (T,!) relate to the more commonly used orientation and alignment parameters (in the nonstandard normalization by Fano and Macek, 1973):

pqy

(F;!) = j ( j + 1)01-

(qkk,1)

In the following we prefer to use the parameters since they transform under rotation by application of real rotation matrices Eq. (15), in pole moments contrast to the 0 and A parameters. As seen, up to 2kmaa contribute to the scattering intensity, in principle. From Eqs. (9) or (17) we have k,,, = 2j. Thus, if the hyperfine quantum number determines the process k,,, = 2F, and in scattering from Na(3,P3,,, F = 3) one could in principle observe multipole moments of rank 6 (64-pole moment). If the electronic angular momentum J = $ were relevant one could observe rank-3 moments (octopole moments) and if L = 1 determines the scattering process, quadrupole moments would be the highest moments participating in Eq. (19). Of course, the corresponding W(k) have to be nonzero, as discussed in Section 111.

3. Some Examples of Multipole Moments The observables ( F ! ) )describe the atom after the time inverse process. To get a better feeling for the physical significance of the multipole moments we discuss some examples. a. Spin Polarization. The simplest case may be an atom in a ’Sl,, state with the nuclear spin decoupled, just having an electron spin, polarized with a polarization P = (nt - nJ)/(nt + nl), in the usual definition. Here

125

COLLISIONS WITH LASER EXCITED ATOMS

j = s = i,the electron spin. From definition we have p(& = (1 + P)/2 and p( -9, -$) = (1 - P)/2. In the multipole description k,,, = 2s = 1 and the ( 7 $ l ) describe the state. From Eq. (17) we have

4)

7-p=s,,

Fly = s,,

F1” = sy

Since we take p to be diagonal, only (s,) # 0 and we find

( F p ) = (s,)

= (fils,lfi)p(fi)

+ (+ - 41sz14 - $ ) p ( - +

- $) = i

P

Thus, this orientation parameter simply is one-half the spin polarization. From Eq. (21) we see that such an atom cannot be produced in a collision process from an unpolarized atom. Generally the (Ti,)are expectation values of the atomic angular momentum vector. b. Quadrupole Moment. As a second example, we discuss the physical interpretation of the alignment of a collisionally excited p state atom, as it is measured in the time inverse scattering by a laser-excited p state. This scattering quadrupole moment describes the atom by the three numbers (T’,;) (q = 0, 1, 2). Let us first give an upper and lower bound for the zero component. The Wiper-Eckart theorem gives

For the reduced matrix element (L 11 TtZ1 1 L ) we insert the value given by Fano and Racah (1968) p. 79) and evaluate the 3 - j symbol to obtain ((LA4 F,2’ILA4)) = (3M2 - L(L 1)). The averaging ( ) has two limiting cases defined by pure M 2 = L2 and M 2 = 0 states, respectively. Thus the upper and lower bounds are given by

I

+

-L(L

+ 1) < (F:’(L)) < (2L - 1)L

(231

and in particular for a p state, -2

,< ( T p ( L ) ) < 1

This sometimes gives a useful consistency check on the experimental observations. We now try to find a connection to the electronic charge distribution in the atom after collisional excitation, for which one usually has a better physical feeling than for expectation values of irreducible tensor operators. we recall that the elecFrom electrodynamics (e.g., Jackson, 1962, pp. 98ff.) trostatic potential @(r) of a charge distribution p(r) may be expanded into spherical harmonics by

126

I . 1.: Herre1 and PK Stoll

with the multipole moments6 given by

Q?!

=

1d3rfp(r)ybk!(0,@)

(25 1

I

For our problem p(r) = e( Ycol(r)12) is the electric charge distribution in the atom after collision and the potential @(r) gives an image of it. For instance, an equipotential surface could be taken as a simpliJied picture of the atom being a surface-charged rigid body. From the previous terminology we may write Eq. (25) as

Qt:

= e((ycoiIfYbk!I y c 0 1 ) ) = e ( n I f [ n ) ( ( m ’ IV!”I

Im))

Again by use of the Wigner-Eckart theorem we may replace operators Fk),and similarly as in Eq. (16) we obtain

rk)by our

Evaluation of the reduced matrix elements gives

= C(k)U‘”(L)(q2(r* col))

(26) with u(”(L)as given by Eq. (18). One sees again from the 3 - j symbol that only even multipole moments contribute to the potential [the triangular relation A(LLk) # 0 must hold]. We may write the electrostatic potential (charge distribution) of the atom after the collision process iiE -P nL as 6

@(r) =

1

k=O.

even

2 k

C(k)dk’(L) (q](col))C$(O, 0 ) a=O

(27)

This relation gives us a valuable understanding of the scattering intensity for scattering by a laser-excited nL state into a final HLstate as it is described by Eqs. (19) and (20). Apart from numerical factors W(k)and C(k),Eq. (27) is identical to Eqs. (19) and (20) in the even multipole parts. As we shall see later, only even multipoles W(k)are nonzero for linearly polarized excitation. This allows the following interpretation of the measurements at a fixed collision angle Qco, : The scattering intensity for collisions (nL+ EL)with an atom excited by linearly polarized light measured as a function of the polarization angles Since we assume reflection symmetry in the x-z plane, again only p =

+ 1 terms contribute.

COLLISIONS WITH LASER EXCITED ATOMS

127

FIG.2. Electric charge distribution (potential) of the atom after the time-inverse process

iiL -+ nL.It may be measured as scattering intensity from the laser-excited atom nL + iit as a function of the polarization angle of the excited linearly polarized light.

0 and 0 (of the electric vector) gives a direct image of the atomic potential

(charge distribution) that would be the result of a time inverse process (HE -,nL) starting with an isotropic distribution of iiLM sublevels.

If L = 1, k,,, = 2, the charge distribution will usually look as illustrated in Fig. 2. This charge distribution is mirror symmetric to the x,,, - z,,, plane ((q?)= 0). One can always find another frame, which we call the atomic frame (at), in which also (T’:!(at)) = 0.7In the atomic frame we also have reflection symmetry with respect to the z,, - y,, (= y o , ) plane. The angle a between z,, and zCoIis given by tan a =

2(Ty.!(col)) (7-p(col)) ITl,’l(col))

fi(

which follows from simple geometric arguments. It should be noted that the ratios of the three principal axes are determined by the two parameters (Ftl(at)) and (T”)(at)), which are related to the (TJ(co1)) via Eq. (20) with 0 = a, 0 = 0. If one also has (p,2!(at)) = 0, the atomic charge distribution has rotational symmetry about the z,, axis. If (Fzl(at)) < 0 it is cigar-shaped, while for (T[,“(at)) > 0 it looks disklike. Transformation to the atomic frame may also lead to some upper and ’I

In a coordinate representation the quadrupole moment has only diagonal terms Qxx,

Q,, in the atomic frame.

QYy,

128

I . K Hertel and W Stoll

(V~(CO~))

lower bounds for combinations of by exploiting the fact that Eq. (23) must also be valid for (n2](at)).This is done most conveniently in the coordinate representation, but we only give here the following relations without proof. For a p -+ s transition in simple collisions (as with electrons) one finds' that all multipole moments (p;]),(T[:!) that are measured or computed must be within an elliptical area given by [(W'(COl))

H2

+ 41'

+

[<7%(C0l))l2

T 6 l Furthermore, cigar- and disk-shaped regions are separated by

(291

(This is indicated in Fig. 3.) In anticipation of the experimental results given

FIG.3. Possible values for the multipole moments (p;') and (T',21) for an atomic charge distribution after the (inverse) )TS + np excitation, as probed by scattering from a laser-excited atom in an np + AS transition. Regions for disk- and cigarlike atomic charge distribution are indicated.

in Section V, we note that in electron 3p -+ ns scattering by sodium, all multipole moments fall into the cigarlike region. For collision problems involving higher multipole moments no easy interpretation may be given. The k = 3 moment of a 2P3,2state may be seen as a magnetic octupole moment and could possibly be of importance for lowenergy heavy-particle collisions (see Section IV), when spin-orbit interaction is involved.

* Here we have made explicit use of Eq. (81). derived later.

COLLISIONS WITH LASER EXCITED ATOMS

129

111. Excitation of Atoms by Laser Optical Pumping A. THEORETICAL CONSIDERATIONS

1 . Rate Equations and Laser Optical Pumping Processes

It is obvious from the preceding discussion that the target state preparation by optical pumping is the crucial part of any scattering experiment with laser-excited atoms. In order to make full use of the possibilities discussed in Section I1 one needs to know not only the number density of excited atoms but also the excited-state density matrix, or equivalently the multipole moment distribution. Optical pumping has been reviewed by various authors (see, e.g., Happer, 1972, Balling, 1975)including particular aspects of pumping with lasers (Cohen-Tannoudji, 1975) and many related aspects in the field of laser spectroscopy are presented in a monograph edited by Walther (1976a). Some special remarks concerning the experiments under discussion seem appropriate here. First, we discuss whether the pumping process may be described adequately by rate equations. We recall the inherent properties of laser light: (1) Monochromaticity allows us (in atomic beam experiments) to excite specific hyperfine-structure levels F. (2) Polarization (including the high degree of parallelism) implies certain selection rules for the excitation of particular substates I FM). (3) High power enforces a rapid pumping processs. a. A Special Example. To have something tangible in mind we discuss the pumping of Na(32P3,,), which is most commonly used. Figure 4 shows the term scheme of the two atomic sodium states involved, displaying the fine and hyperfine structure, the nuclear spin of Na23 being I = $. The magnetic field is taken to be low enough to prevent the splitting of the sublevels M. In principle, a dye laser may be tuned to excite any of the upper hyperfine states F from any of the two ground states (F= 2 or 1) with the restriction A F = 0, & 1. However, the latter selection rule allows nearly all upper states to decay into both of the ground states. On the other hand, the excitation takes place only from one ground state since we pump with a single laser mode. Thus in general, under equilibrium conditions all atoms are in the ground state and no excited atoms are found.

130

I . K Hertel and W Stoll

I

“’I

32sx

F =1

FIG.4. Hyperfine energy levels of 23Nafor the 32S1i2, 3*P1,, ,and 3’P,,, states, indicating the hyperfine sublevels I F M ) Not to scale.

Only the upper levels belonging to F = 3,’ J = 4 can decay just into the So, if we tune the laser to the 32S1/2(F= 2) -+ 32P312(F= 3) transition we can achieve a finite upper-state population even under stationary conditions. Two cases have to be distinguished: circular polarized ( 0 )light and linear polarized (n)light. In order to interpret the experiment we must know the relative population of the different upper-state magnetic sublevels. In the case of circular polarized light, we only have induced transitions with AM = + 1 (or - l), and so obviously optical pumping leads to a population of only the F = 2, M = 2 level of the ground state and of the F = 3, M = 3 excited state with the relative probability w(M = 3) = 1. For linear polarized light, induced transitions take place with AM = 0 (full lines in Fig. 5), while spontaneous emissions with AM = + 1 are also possible (dotted lines). From Fig. 5 we can see that under equilibrium conditions the

F = 2 ground states.

The F = 0 level partially overlaps the F = 1 level and can also decay into either ground state.

COLLISIONS WITH LASER EXCITED ATOMS

131

FIG.5. Schematic of pumping an F = 3 level from an F = 2 level with linearly polarized light. Solid lines indicate induced transitions, broken lines correspond to spontaneous decay. From Hertel and Stoll (1974a). Copyright by the Institute of Physics.

decay rate from F = 3, M = +2 to F = 2, M = 1 must be equal to the rate for spontaneous decay F = 3, M = 1 to F = 2, M = 2, etc. The (time-dependent) population of the excited-state sublevels would have to be computed by rate equations, at least for low pumping intensities. b. General Formulation. It is the high laser power that gives rise to some doubt about the applicability of ordinary rate equations to describe the pumping process. Of course, perturbation theory, being the basis for the very concept of transition probabilities, is not strictly applicable. In contrast, the optically pumped atom has to be described as a (partially) coherent superposition of ground and excited states. A variety of consequences such as Rabi oscillation or the Bloch-Siegert shift could possibly obstruct the situation. In principle, nonperturbative methods have to be used to treat the excitation process. A number of elaborate theoretical approaches to the multilevel atom in strong radiation fields have emerged recently (e.g., Einwohner et al., 1976; Wong et al., 1976; Lau, 1976b): based essentially on the rotating-wave approximation with inclusion of correction terms. Since we are basically interested only in the number density of atoms in the excited state and in its sublevels, these methods cannot be the subject of the present review. We shall see that one may obtain rate equations even for high laser powers. By high we mean such laser intensities 9(W/cm') and bandwidths that the induced transition time tindx 1/Bu, is shorter than the spontaneous lifetime or at least comparable to it. Here B is the induced transition coefficient as usual, and u, = Y/c 6v:";, , the spectral radiation density. We = o2- o1is the atomic make sure, however, that tind< l/w12, where o12 transition frequency under discussion. This is the case for typical singlemode cw dye lasers of some 10 to 100 mW in a bandwidth of 50 MHz, having a beam diameter of around 1 mm. We apply the discussion given by Cohen-Tannoudji (1975)to this situation. The density matrix for the pumped atom may be written as

&:as2

132

I . 1/: Hertel and W Stoll

where (I, is the submatrix for the ground state and aethat for the excited state. The so-called optical coherence between ground and excited state is contained in the off-diagonal matrices aeg.Possible effects of these coherence terms on the collision process will be discussed in Section VI,where we shall see that they are usually not observable in typical experiments. At present, for scattering processes from laser-excited atoms, we are interested in the evaluation of cre (and (I,) and their time development. The atom interaction with the field D * E(t) gives a purely nondiagonal interaction hamiltonian and acts on uegand neeonly:

where D,, is the dipole operator and E(t) = E, exp(- iw, t ) describes the exciting laser field. One has to solve the Liouville equation, and the radiative decay is taken into account phenomenologically by adding

-roe,

-+r(Iegrci, = s ( a e )

(33) where r is the natural linewidth, equal to the rate of spontaneous decay r = A = l/t; *(ae) takes into account different probabilities for the sublevel decay. Cohen-Tannoudji has shown that one may eliminate oegfrom the equation of motion by formally integrating the nondiagonal terms. In the interaction representation (indicated by ), that is to say by separation of trivial time dependencies, he obtains cie =

bee=

-

cie =

-ree + x

x

.I

dt’ii,(t’)

10

Pge(tPeg(t’)

ex~[-+r(t- t ’ ) I ~ * ( t ) ~ ( t ’ ) }

.I

.(o

dt’6,(tf)

@ge(tPeg(t’)

+ herm. conj.

e x ~ [ - + r ( t - t‘)]~*(t)E(t’)J (34)

and a similar equation for 6,. For not too high intensities and broad-band excitation, contributions to the integrals in Eq. (34) arise only from t’ z t , as Cohen-Tannoudji has shown. Then, if iie(t’) and a#) are slowly varying, they may be taken out of the integrals and ordinary rate equations are obtained. The rate constants for induced transitions are given by the remaining integration over dipole moment-and electric field-correlation functions in the brackets { 3.

COLLISIONS WITH LASER EXCITED ATOMS

133

As is well known from solutions of the optical Bloch equations," for high-intensity fields 6 , and 6, may be rapidly varying functions of time. However, even when the Rabi frequency ORis high compared to l/r, the rapid Rabi oscillations are damped out exponentially in a time scale comparable to the spontaneous lifetime. Thus, for t B t = l/r,i.e., for times long after the beginning of the pumping process, the above argument again holds: Because of the exponential factor in the integral Eq. (34) we again have contributions only from t z t'. Taking i5e and 6, slowly varying we obtain rate equations as previously, which should describe the pumping adequately for not too short times in the order of T. This approximation is sufficient for scattering by laser-excited atoms, since the total pumping time is typically l o p 6 sec, while the spontaneous lifetime is z lo-* sec. Having established the applicability of rate equations we proceed to write them down explicitly. Since no external field is applied, we may choose a coordinate system (the ph frame) in such a way that the matrices ieand 6, become diagonal. For linearly polarized (n)light we choose the zpFaxis of the photon frame parallel to the electric field vector, while for circularly polarized (a)light zph parallels the light axis. In these frames each ground level substate IFA?) couples with only one excited state I FM) by induced transitions. As illustrated in Fig. 5 for our special example, other states are connected only by spontaneous decay (dashed lines). In the photon frame, the off-diagonal matrix elements in ie and i,, which describe the so-called Zeeman coherence, are zero since no phase memory is contained in spontaneous decay according to Eq. (34). In the following, we denote the diagonal terms of ee and 6, by oMand g M , respectively. Then, the rate equations read

b M = -AaM - B tfm

=

+

~ M u ~ o M B ~ M u , o ~

C AMM'OM, +B ~ M u , , o M - B m M u , a ~ M,

(35)

with

-F
-F
M - M = O , k1 f o r n o r a * light

Here BaM u, is the result of the remaining integrals in Eq. (34). For broadband excitation with not too high intensities, u, is the spectral radiation density. The induced spontaneous transition coefficients BmMand AmM (between sublevels I FM) and I I?@)) are equal to those obtained from perturbation theory. A = r = l / is~ the total spontaneous decay rate, which is the same for all substates of the electronically excited upper level. l o A concise treatment of two-level atoms in strong radiation fields is given in the monograph by Allen and Eberly (1975).

134

I . V. Hertel and W Stoll

2. Evaluation of the Transition Probabilities

Numerical data for A -MM and BRM are often useful but are not readily found without some missing factors. We therefore give some relations here. a The Spontaneous Decay Rate. For one particular transition integrated over all emitted photon directions into 471, the spontaneous decay rate is given by A F H F M = (EFM I DY)I nFM)'a (36) with a = 64n4/(31'h). Here 1 is the wavelength of the light and Df) is the dipole operator with q = 1, - 1, or 0 for o+,0 - , or 71 light emission, respectively. Using the Wigner-Eckart theorem, we have ArmFM

=[42F

+ l)](FMlq I FM)'(HFIID(')llnF)

(37)

Obviously M + q = M. In cases of practical interest we wish to excite a finite upper-state population and we must choose an upper hyperfine level F, which may decay into one ground-state level F only. Then we obtain the total decay rate by summing Eq. (37) just over A and q: A =r= l /= ~ a(2F

+ l)-'(iiFIID(''llnF)

(38) In the following we wish to describe all transition probabilities in terms of the one atomic constant 7,characteristic to the transition. From Eqs. (37) and (38) we may write ApH. F M = T - 'C2 FRFM (39) where" C b W F M = (FMlq I FM)' For later use we express the dipole matrix element by Eqs. (36) and (39) in terms of T : I D, I = I (HFM 1 Df' I n F M ) 1 = (a?)- " ' C F M F M (40) and we communicate some special values for the sodium 'P,,, + 2S1,2 transitions with I = $ in Table 11. To find numerical values for the induced transition probability we recall from quantum mechanics that

where u,. is the spectral radiation density of exciting light with the polariza"

If F can decay to more than one final F (but only to one final J) one must substitute

COLLISIONS WITH LASER EXCITED ATOMS

135

TABLE I1

RELATIVE PROBABILITIES FOR AM = O ( x ) TRANSITIONS IN 2p3,2

+

2s,,2"

~~

3 2

0.7746 0

0.7303

0.2886

0.5774 0.5774

' I = $ case for two different upper levels F to one lower level P = 2. Values are for lCFMFMI. tion q. It should be noted that Eq. (41)differs from those commonly used by a factor of three. This arises from integrating only over the small angle of laser divergence, while usually one integrates over 4 ~ . Comparison of Eqs. (39)-(41) leads to BFMFM BFHFMIAFRFM

= (K/t)C$MFM = 27Z~X-'h-~= 3A3/8xh = K

(42)

where K is the usual Einstein relation multiplied by a factor of 3 [see also Eq. (1)l. For excitation with a single-mode laser, having typically a Lorentzian ~ express u,. at the maximum frequency profile with a FWHM of 6 ~ we2 may in terms of the total laser intensity 9: 2

u =-I'

K

.a/c

6vF*

(431

as long as 4 6v;$ and dv:;", 9 v R , where Sv;; is the natural linewidth of the excited state and 2nvR = ORis the Rabi frequency (see below). With these expressions the excitation probability for a two-level atom (e.g., the F = 3, M = 3 level pumped circularly from the F = 2, A = 2 state) becomes

which is identical to Eq. (1) for CgAFM= 1 as in the special example F = 3, M = 3 * F = 2 , M = 2. 3. Saturation

Obviously, for very high intensities or for a very narrow laser linewidth, Eqs. (33) and (44) are no longer applicable and one has to evaluate the

136

I . K Hertel and W Stoll

integral j d(t)a(t') exp[-$r(t - t')](E(t)E*(t') in Eq. (34) in detail: Rabi oscillations occur, and the levels split by the dynamical Stark effect, which leads to power broadening of the resonance line and saturation of the transition. In the intermediate but practically important intensity range the integrals will be difficult to evaluate. We concentrateon the case where the laser linewidth is small compared to the saturated linewidth. Instead of evaluating the above integral directly, we use a plausibility consideration, taking advantage of known solutions for the optical Bloch equations in the two-level case. They may be compared to Eq. (44).Since we always couple any one groundstate sublevel with only one excited-statesublevel by induced transitions (see Sections III,A, l a and b), we deal essentially with a set of two-level systems linked by spontaneous decay only (Fig. 5). The two-level solutions thus should provide a reasonable estimate. The rotating-wave approximation for nearly resonant excitation allows us to determine (T, (see, e.g., the book of Fain and Khanin, 1969; or Allen and Eberly, 1975). If one identifies the longitudinal and transverse relaxation times of the free atom with Tl = Tz/2 = 7 = 1/A = l/r the stationary solution of the optical Bloch equations [see Eq. (3.22~)of Allen and Eberly] reads in our notation: ge=-

1 2n; t/r -n , + ng 2 1 + (2 Ao/r)' + (2Qi7/r) "e

(45)

with the Rabi frequency Q, = I D,,lb/h and the real field amplitude

G = (8x.9/~)~/'(cgs). We insert Eqs. (36) and (40):

.#"'(CgS) QR = ((r7)- " ' C F , ~ Fh-M ' ( 8 n 9 ; / ~ ) - = [313/(2xrhc)]"ZCpa~~

Taking account of 2x Sv;;: = r, we obtain for resonant excitation (Am = 0) from Eq. (45)

This is a nice result since we may compare Eq. (44) directly with Eq. (46); they differ only by normalizing the intensity to 2 dv:;", /n in the weak broadband excitation and to 2 h v ; ; i / a in the strong field or narrow hv& case. We conclude that the rate equation approach may be safely used even for &las << Bysat 1,2 = QR/(xJ2) or for SV?~' Sv;$. We just have to replace the I/' ordinary spectral radiation density Eq. (43) by

+

2 .a/c

u,. = -a

svy;

In the intermediate range some suitable interpolation has to be used.

COLLISIONS WITH LASER EXCITED ATOMS

137

Before continuing, we extract the saturated line profile from Eq. (45), which is

x

+

rT)2 + (I) svs;; 1-'

For the sodium 32P3/2case we report numbers for reference:

For the F = 3 F = 2 and the F = 2 ++ F = 2 transitions excited with linearly polarized light of 2 W/cm2, the power-broadened linewidths are given in Table 111. It shows that a partial overlap of the F = 2 and F = 3 levels (see Fig. 4 ) due to a power broadening of the F = 2 level may become important in these experiments. It will lead to a partial decay from F = 2 into the F = 1 ground state and the atoms are lost. Thus, too high excitation intensities decrease the upper-state populations. In practice the effect is less predominant than it appears from Table 111 since the F = 2, M = 0 level does not overlap at all and the F = 3, M = 0 level contains the maximum population (see next section). t ,

TABLE 111 SATURATED LINEWIDTHS AT 1 W/cm* FOR AM = 0 =2 TRANSITION IN THE SODIUM 32P3,2 + 3'S,/, , CASE'

3 2

96 0

Values are for 6~;": (MHz)

90 36

12 12

x . f - ' l 2 (W/cm2).

4 . Evaluation of Rate Equation

We may now insert A P R F M and B E , F M into Eq. (35). After multiplication of both sides by T, we obtain the (2F + 1 + 2F + 1 ) pumping equations:12 &M = - c M

- B(FM - qlq I FM)'(oM - L T M - q )

(484

l 2 I f F can decay to more than one final state, ( F M l q 1 FM)' has to be replaced by C,,wt, given in Eq. (39a).However, no stationary population may be obtained in this case.

138 and

I . I/: Hrrtc4 and W Stoll

(a= M - q)

in the normalization Tr

0

=

C M

+

R

8,- =

1

Again q = 0. 5- 1 for pumping with n and of light, respectively. The time is measured in units oft and fl = KU,.= 3A3(8nh)- ' ( 2 . l / c ) ( n6v,/,)- I . As discussed in the preceding section, 6 v , / , = 6v'& for low intensity and = 6~;;: for high power. The rate equations (48) may easily be expanded in terms of multipole moments using the definitions Eqs. (8)-(11) and some algebra. Due to the special choice of the photon frame we obtain only zero components. As an example, for linearly polarized light we obtain

.li' ( k ) = - W ( k ) - fl

C [ YF"(k')@(k'F,k F ) - @-(kf)@(k'F,k F ) ] k'

@ ' ( k ) = (-?(2F

+ fl with

(491

+ 1) [ W ( k ' ) @ ( k F ,k'F) - @(kf)@(kF,k'F)]

k'

@(k'F', kf'F")

I

kO)( F - M F"M kO)(FM10 F M ) 2

with the normalization

(2F + l)"'W'(O)

+ ( 2 F + l)1/2YP(0) = 1

This normalization differs somewhat from the one given by Macek and Hertel (1974). While there W ( 0 )= (2F + 1)- 'I2, in the present normalization it directly reflects the excited-state fraction n , / ( n , + ns): W'(0) = (2F

+ 1)- ' I 2 -!!!--< (2F +21)-'l2 ne

+ n,

139

COLLISIONS WITH LASER EXCITED ATOMS

In order to obtain the multipole moments13 needed for analyzing the experiment one can either solve Eq. (49) directly or solve Eq. (48) and then construct X ‘(k) out of o M We . proceed along these lines. 5 . Stationary Condition

We now discuss stationary conditions c ? = ~ a‘M = 0. Equations (48a) + (48b) simply give oy =

2 aMr(FM- qlM’ - M + q1 FM’)’

M’

We see that for stationary conditions the relative population among the upper substates is independent of the pumping power and of the spontaneous lifetime. Also, Eq. (48a) may be reformulated in the stationary case by summing over M and using the normalization Eq. (48c): 1=2

M

+ p-’

M

a M ( F - hf - qlqlFM)-2

(51)

From all atoms in the F and F state (n, and n , is their respective number density) a fraction is in the excited state, given by n , / ( n , n,) = OM. a. Circularly Polarized Light. When pumping with circularly polarized light (a+or a - ) and F = F + 1 then only one sublevel is populated under stationary conditions:

+

and

oM= S M ,+ F o F

C M

= QF

EM

(52)

for left and right circularly polarized light, respectively. One easily verifies that all 2F + 1 Eqs. (50) hold for q = + 1 and - 1, respectively. Inserting Eq. (52) into (51) gives n,/(n,

+ n,) = C a M = (2 + /?-‘)-I M

(53)

which is the two-level formula, Eqs. (44) and (46), with C$,,, = 1. By definition Eq. ( l l ) , the optically excited multipole moments of the stationary pumped atom are

I ’ The excited-state multipole moments may be expressed by orientation and alignment parameters defined similarly to Eq. (22):

I ‘(l)/X.(O)= d”(F)F(F + 1)(2F + l)L’20ph

I ‘(2)/Yf ‘(0) = d2’(F)F(F + 1)(2F + 1)”2Agh

(494

140

I . 1/: H e m 1 and W Stoll

where the + and - signs refer again to left and right circularly polarized light. We note the important symmetry relation PY"(k)= ( - ) " W ' - ( k ) and keep in mind that from Eq. (54) all multipole moments 0 < k

(55)

< 2F are excited by circular polarization. in Eq. (50) b. Linearly Polarized Light. Here we note that the sum contains three terms. They correspond (see Fig. 5 ) to the three possible spontaneous decay processes populating the I FM)- I FAT) subset from I FA4 - 1). I FM), and I F M + 1). One verifies easily that Eq. (50) holds if for all M we have

EM,

oM-*/aM = (FM

- 111 IFM)'/(FMl - 1 I F h f - 1)'

As quoted by Macek and Hertel (1974) one can bring oM into closed form for F = F + 1 by CM =

(FMF - M

IF + FO)'n,/(n,

+ n,)

(56)

This expression is found simply by direct application of explicit formulas for Clebsch-Gordon coefficients. Equation (56) is very suggestive: The optical pumping process couples the ground state (FMI together with the excited state ( F M ) to accept as many photons as possible ( F F) having a projection quantum number q = 0 (K light) with respect to the Zph axis. The multipole moments for stationary linearly polarized pumping are found by inserting Eq. (56) into Eq. (1 1) (Macek and Hertel, 1974). After some Racah algebra ( F = F - l),

+

W ( k )=

~

ne

ne

+n g

'1

(-)2F+k+1(2k + 1)'I2(4F- 1) 2F-1 F

k F-1

2F-1 F

The 3 - j symbol is zero for k odd, and thus by linearly polarized light only even multipole moments are excited with 0 < k,,,, < 2F. This is a consequence of reflection symmetry of linearly polarized light with respect to any plane containing the electric vector. In numbers, we obtain for sodium 32P3,2, F =3 W(2)/W(O)= -0.9623.

We also find a closed expression for the excited-state fraction of atoms by inserting Eq. (56) into Eq. (50):

COLLISIONS WITH LASER EXCITED ATOMS

141

This is the multilevel equivalent to Eqs. (44) and (46), where we just have to replace C i R F MIt. may be used to obtain an appropriate estimate for the excitation fraction obtainable with a certain laser power. 6 . Time Development of Multipole Moments

We continue the discussion of the rate equations (48) and (49) by an investigation of the time-dependent behavior of the multipole moments in the course of the optical pumping, since it is not a priori certain that stationary conditions are always reached in the collision region, especially when pulsed lasers are used. We first discuss the initial moments of the process. For a short time all T (,, z0 and tTm are independent of R. Then from Eq. (48a) the number density fractions oM build up as t

oM z - P(FM - qlq T

I FM)2

(59)

as long as t -4 T/P = T / ( K U V ) , and of course we must have T G t to be able to use rate equations. A comparison for initial moment pumping and the steady state is given in the histograms in Fig. 6 for linearly polarized light. The initial multipole moments are given by Y#”(k)/$-(O) = (2k + 1)1’2(-)1-q

and explicity for sodium 32P3,2, F = 3, F = 2, we have @(2)/4“(0)=

M = -3

-2

-I

0

1

2

3

FIG.6. Relative sublevel population for pumping of an F = 3 from an F = 2 level with linearly polarized light. (a) At the initial moment of pumping (b) for stationary pumping conditions.

I . K Hrrtel and W Stoll

142

-0.6928, which may give a lower limit to the multipole moments that are expected. Strictly Eq. (59) is only applicable to low pumping powers u, << K - ' used in conventional pumping experiments. There only multipole moments up to rank 2 (i.e., orientation and alignment) are usually produced according to Eq. (59). This arises from the fact that only one photon angular momentum (Iph = 1)is coherently absorbed, which allows us to construct only up to 2lPh multipole moments. The pumping Eq. (48) has been solved numerically by Hertel and Stoll (1976) for different pumping intensities. The reader is referred to this paper for details, where he has to correct all " spectral radiation densities," dividing them by 3. One finds that the multipole moments W(1)and W(2)evolve more rapidly than the higher ones. Surprisingly,it takes up to 30 spontaneous Lifetimes to reach the steady state. It should be pointed out, however, that W(O), i.e., the total upper-state population reaches stationarity after a few t. An increase in laser power does not speed up the pumping process, whose time constant is determined by spontaneous decay as long as tind = T / K U , << T. When describing actual experiments one should bear in mind these findings. For low intensities, such as found in the laser beam wings, it may sec before reaching the steady state. Even in the hightake up to intensity limit one should allow at least 20 p e c for W(2) and 100 psec for the higher moments to build up. Circular pumping is especially slow.

B. EXPERIMENTAL ASPECTS OF OPTICAL PUMPING

I. Numerical Example for the Excited-State Number Density In order to get a quantitative estimate of the excited-statenumber density we discuss the example of sodium 3'P3/2, F = 3 excitation from 32S,,2, F = 2. Not all atoms are initially in the F = 2 states; 8 are in the F = 1 hyperfine level. Thus the fraction of total upper-state population is nJn, = 8 c M .With fi = 2.7 x 10- 13/u, we compute nJno from Eqs. (53) and (28). Figure 7 displays nJno for r~ and II light e~citation.'~ Circularly polarized light produces higher densities for excited species than Linearly polarized light. As discussed, u, is given by Eq. (43) for broad-band excitation and by Eq. (43a) for high powers. As realistic examples, we take 6vF2 x 30 and

C

l4 Note the change by a factor of 3 with respect to the original work, since now the correct relation Eq. (42) is used.

COLLISIONS WITH LASER EXCITED ATOMS

143

'SPECTRAL' RADIATION DENSITY

FIG.7. Fraction of the total number n , of atoms in the excited state 3'P,,,, F = 3 for sodium pumped with II or CT laser light under stationary conditions. From Hertel and Stoll (1976).

60 MHz, 6~;;': x 10 MHz and the saturated absorption linewidth dv;:: rz 10091/2(W/cmZ).The dependences of u, on 9 are shown in Fig. 8, where the region between low intensities (- 0.01 and 0.05 W/cm2) and high power (- 0.5 and 2.5 W/cm2) have been interpolated freely.

-

N

-~1 10')

10-2

10-1

100

I ( W / ~ ~ ~

LASER INTENSITY

FIG.8. Radiation density u, per frequency as a function of laser intensity .Ffor two different The dashed lines are freely interpolated between the broad-band lowlaser Linewidths intensity excitation and the high-intensity saturated case.

I . 1/: Hertel and W Stoll

144

The laser beam always has a nearly Gaussian spatial distribution ,f(R) given by its total power P and beam width W: 3 ( R ) = ( P / n W 2 )exp( - R 2 / W 2 ) For some typical intensity distributions we estimate u,,via Fig. 8 and from u, we find n,/no. The resulting spatial distribution of excited-state atom density is given in Fig. 9. One clearly recognizes that due to saturation the

03

02

01

0

>

DISTANCE FROM LASER BEAM AXIS

FIG.9. Spatial profile for excited-state number density (n,)for a typical dye laser. Note the sharp cutoff at around 1.2 mm.

excited-state density profile is much more sharply confined than one might expect from the laser intensity profile. This fact may be very advantageous when performing scattering experiments. As we see, the laser may define a collision volume much more sharply than often possible otherwise in a collision experiment. As we see in Fig. 9 the effect of narrowing the laser bandwidth is of minor importance below 30 MHz. Spatial widening gives a more extended excitation region, which is still relatively well limited.

2. Measurements of Optically Excited Multipole Moments From the discussion in Section III,A,S it is obvious that in an actual experiment one is not sure of finding the stationary multipole moment distribution given in Eqs. (54) and (27). Especially in regions of weaker intensity, long times up to psec may be needed to reach equilibrium. On the other hand, in regions of high power density AvC\ may be typically of the order of the hyperfine splitting. The overlap of the hyperfine levels thus produced changes the characteristics of the pumping mechanism (see Section III,A,3). A similar effect may be caused by radiation trapping in

145

COLLISIONS WITH LASER EXCITED ATOMS

the sodium beam. The reemitted spontaneous radiation has a different polarization and thus disturbs the excitation by n or CJ light. The result, common to all the disturbing effects mentioned, is a poorer degree of anisotropy of the optically pumped state. Thus, the question arises of how to measure the multipole moments, which must be known for a proper interpretation of collision experiments. Fortunately, one often has to know only the multipole moments up to rank 2, i.e., the orientation and alignment optically produced (see Section IV). These (and only these) may be determined by observing the fluorescence light of the atom. This has been outlined in detail by Fano and Macek (1973) in the context of impact excitation and polarization of the emitted light. Fano and Macek give the fluorescence intensity of an aligned and oriented atom as

I x 1 - +h(''A;fec+ $!I(~'A;':cos 2p +

sin 28

(60)

with

while p = 0 or n/2 for z-light detection parallel (III) and rectangular (I,) to a given detector and analyzer direction, and p = +n/4 for left and right circularly polarized light, respectively. The orientation 0;''and the alignment A:" measured in the light detector frame are derived from those in the photon frame by ,:el

=

= Cb"(0, 'Y)Ao"h,

cp(0,0)ogh Adel 2+

1

- 3c:2:(o,WGh

(611

with the real solid harmonics given in Table I. Equation (61) differs somewhat from those of Fano and Macek since at present we refer to optical atom excitation. 0 is the polar angle of the light detector and Y the polarization analyzer angle with respect to the photon frame. Equations (60) and (61) may be used to determine the multipole moments Yl.(l)/#'(O) and I '(2)/ Y4 '(O), which are connected to Ogh and Agh by Eq. (49a). The linear polarization of the emitted light allows us to determine Agh and thus 17 '(2)/ Y4 *(O). Circular polarization leads to a determination of Ogh and % '(l)/W '(0) when Agh is known. For n-light excitation Ogh = 0. Particularly simple is the observation of unpolarized light in the plane of the polarization vector of the exciting laser

146

I. K Hertel and W Stoll

c k! z 5

Z w

E2\8=y; I

FLUORESCENCE

Na-BEAM

V

z

W

u In W a

from behind

N? -BEAM

0

2

4e goo 2700 900

270'

goo

POLARIZATION ANGLE

FIG.10. Fluorescence intensity from a sodium 32P,,2,F = 3 atom, excited with linearly polarized laser light, as a function of the polarization angle 0 of the exciting light relative to the fluorescence detector. Maximum (I,,,,.J and minimum (Imin) positions of the E vector of the light are indicated.

light as a function of its polarization direction 0 with respect to the detector. This geometry together with an experimental signal is shown in Fig. 10. Equation (60) simply gives I K (1 - fh(')AP), Eq. (61) gives A?' = Aghf(3 cos' 0 - l),and Eq. (49a) gives Agh

= W(2)/W(0)(2F

+ 1)- "'[F(F + l)u(')(F)]-

When inserting explicit values for d2)(F)and h") for F = 3, F = 2 one obtains' I K 1 0.1732(3 C O S ~0 - l)W(2)/W(O) (62)

+

The ratio of maximum to minimum fluorescence found in the experimental determination (Fig. 10) is Imax/Imin x 1.47, which is a poor example. For stationary pumping [ W(2)/W(O) = -0.9621 ImuJZmin = 1.75, and for the ini= 1.47 would be tial moment of pumping [W(2)/W(O) = -0.6931 Imux/Zmin expected. The experimental fluorescence anisotropy indicates that in the pumping region we either have no stationary conditions, and/or strong radiation trapping, and/or a partial overlap of the hyperfine levels due to saturation. In fact, we observe a decreasing anisotropy for increasing atom beam intensities, which indicates radiation trapping. If the F = 2 level partially overlaps with the F = 3 level or is excited otherwise, then Eq. (62) gives only a first-order approximation.

COLLISIONS WITH LASER EXCITED ATOMS

147

It should be pointed out that these types of determination of W ( 2 ) and W(1) give values averaged over an area seen by the light detector. This

spatial averaging has to be done in the same way as in the actual scattering experiment for scattered particles by the particle detector. That is to say, the light detector has to see the same area as the particle detector does later. Under these conditions one may safely use the spatially averaged values W ( k ) for evaluation of the collision experiment [see, e.g., Eq. (12)], where the measured scattering intensity would be given by Ispal. 10

C W(k)spal. av.(t[ok'>*

In Section V we describe an alternative way to measure W(2) directly in the collision experiment. This is possible in particularly favorable cases. Some collision experiments may require knowledge of the higher-order multipole moments induced in the optical-pumping processes. These cannot be measured simply by single fluorescence photon detection. In order to detect higher multipole moments many successive photons have to be observed coherently. That is to say, the detection process has to be in itself a pumping process or a time-dependent observation of the atomic response on a defined change in the excitation conditions. We wish to indicate a few possibilities that, however, to our knowledge have not yet been exploited for the determination of higher multipole moments: (1) One could use the exciting laser by rotating its polarization angle once through 360" in a time short compared to the time for reaching stationarity (see Section III,A,5), but long compared to 7 . The absorbed power, or the fluorescence, changes as a function of @(t), O(t), and would depend on W(k)C$'(@,a).

(2) One could use a second laser for pumping excitation of a higher state out of the first laser-excited state. (3) Alternatively, one could apply a magnetic field. If this is made nonparallel to the photon frame, Zeeman coherence results and the optical multipole moments become nondiagonal and time dependent, each characterized by oscillation frequencies up to 2kw,, where wLis the Larmor frequency in the field. The time dependence has to be observed in times short compared to the stationary pumping times. The time origin could be defined by switching the polarization vector from parallel to H into rectangular position.

One also may observe Hanle signals. Such complex signals have been observed by Ducloy et al. (1973) for a gas laser and have been interpreted as determined by a hexadecapole moment in the 2P4neon levels, CohenTannoudji (1975) has discussed such Hanle signals in simple cases, to mention just two references from a field that is related to the present topic and has been reviewed recently by Decomps et al. (1976).

I . K Herrel and W Stoll

148

No experiments on higher multipole moments directly connected to crossed-beam collision processes have been performed and thus we end the discussion here. 3. Methods for Preparation and Detection of Atoms in the Ground State by Optical Pumping

a. It is interesting to discuss within the previous context an experimental method described by Schieder and Walther (1974), which in principle is capable of determining the multipole moments of the ground state after laser pumping. The situation for ground states is somewhat more convenient because Zeeman coherence does prevail for a long time, since no spontaneous decay destroys it. The experiment of Schieder and Walther is shown in Fig. 11. Two laser beams were produced by splitting the output beam of a A

Flucxexence Signal

No- Atomic Beam

cj-

h e r Beam 2

Loser Beam 1

q v 004 Gauss

I

I

I

I

I

I

I

-

FIG.I I . Scheme of the experimental setup (the direction of observation is perpendicular to the laser and atomic beam) and fluorescence signal at the second laser beam as a function of the magnetic field. Both laser beams were circularly polarized. From Schieder and Walther (1974).

dye laser. The first intense beam induced the ground-state coherence by optical pumping, whereas the second beam probed the time development of the ground-state coherence after the flight between the two interaction regions. The fluorescent intensity, observed perpendicular to the atomic and laser beams at the second interaction region, was measured as a function of the magnetic field H,.In principle the set up is similar to a Rabi-type apparatus used in atomic beam resonance work. As Schieder and Walther point out, there might be a quite interesting application in collision studies.

COLLISIONS WITH LASER EXCITED ATOMS

149

For this purpose a second atomic beam could be crossed with the first one in the region between the two laser beams. Observing the fluorescent light induced by a second laser beam, phase and hyperfine level changing collisions could be detected. b. An elaborate method for preparing a beam of ground-state sodium atoms in a particular hyperfine level with a simultaneous velocity selection has been demonstrated by Shouda and Stroud (1973). No application to scattering experiments has been reported so far. But the scheme indicates interesting possibilities. In most cases the nuclear spin is decoupled during collision and the dynamics are essentially determined by elastic-scattering amplitudes (see the similar analysis on fine-structure transitions given in Section IV). Valuable information, e.g., on spin exchange amplitudes could be obtained in such experiments, especially when a pure electron spin state is prepared in an experiment using 0' (or 0 - ) light. Then the 1 F = 2, A = 2) state is populated exclusively and one has a pure electron spin state 1 S = i,M , = +&).Such experiments would have their merits in comparison with the usual magnetic selection methods where a loO'?;, polarization is never obtained.

c. As a general tool for detecting scattered atoms, cw dye lasers may have a number of advantages in collision experiments with ground- or excited-state atoms. In particular, when detecting the particles by fluorescence one can simultaneously detect the atoms and measure their velocity. Such a device can in principle be built in a more compact form than the conventional combination of mechanical selector and surface ionization detector. The feasibility of such a Doppler-shifted fluorescence (DSF) detector has been demonstrated by Hertel et al. (1975b), using a cw dye laser to detect a sodium beam. The atom beam to be detected is intersected by the laser detector beam within a length 1 = d/sin 0 (see Fig. 12). A small fraction of the laser beam crosses the sodium beam under 90" to give an unshifted reference frequency. The atomic fluorescence is observed at right angles to both the atom beam and the reference beam. Assuming the atoms to have only one resonance frequency vreS, fluorescence is observed from atoms flying with a velocity v when the exciting laser light has the frequency v, Doppler-shifted by AV = v,, - v = vv cos O/C (634 i.e., v = A AV/COS 0 (63b) Av is determined with the aid of the reference beam.

I . K Hertel and W Stoll

150

-%-

Top view schematic

(b)

EL

hv

?-

ID4

FIG.12. Schematic diagram of the DSF detector. The sodium beam (Na) is intersected, (a)

by a detector laser beam hv (det) under a small angle Q measuring the Doppler-shifted excitation frequency, and (b) rectangularly by a reference laser beam h ~ ,(ref) , ~ providing a signal at the unshifted resonance frequency.The fluorescencesignal is observed by a photomultiplier PM via a lens L.From Hertel et 01. (1975b). Copyright by the Institute of Physics.

Again, the F = 2 + F = 3 transition is used. Confusion, due to several HFS lines, may then be avoided by optical pumping with high laser intensity, allowing for a sufficiently large number of pumping cycles. Each atom emits several photons when passing through the detection region. During one natural lifetime, each excited atom emits one photon. The total number of photons N emitted by an average atom of velocity u per passage through the detection region of length x is thus given by

This leads to a detection efficiency a l/u, while it is a u with a mechanical selector. Here the passage time is tx = x/u and may be typically 2502 for thermal sodium atoms. A realistic estimate for the overall detection efficiency including the finite collection-and quantum-efficiency of the photomultiplier system may be 35% for each atom entering the detection area. The other property of the DSF detector of interest is its velocity resolu-

COLLISIONS WITH LASER EXCITED ATOMS

151

tion. It is determined by the uncertainty 6(Av) in measuring the Doppler shift Av. From Eq. (63) we obtain for the velocity resolution

6~ = (A/COS 0)~ ( A v )

(65)

Thus the absolute value 6 u for the velocity resolution is independent of u, in contrast to a mechanical selector, which has a constant relative velocity resolution. For sodium we expect a velocity resolution 6u of f 60 msec-' for 0 x 0 and ~ ( A vx) 100 MHz due to saturation. A mechanical selector can easily do better at thermal velocities. However, at suprathermal velocities the DSF detector becomes superior. The results for detecting a velocity selected sodium beam (FWHM x 7%) are shown in Fig. 13 (signals c and b). As the reference point (v = 0, Av = 0)

i\JI I

/ i:, .

*

. .

J

2500 I

2000 1

I1

F = 2-F.2

1500 1

1000 1

500 I

0

AV [MHz]-

Vmox. XI

FIG. 13. Fluorescence intensity from a mechanically selected sodium beam with most probable velocity u,,,~,. The reference beam signal (a) resolves the HFS splitting of the F = 2 + F transitions. Signals (b) and (c) come from the Doppler-shifted F = 1 + F and F = 2 + F excitations, respectively. From Hertel et al. (1975b). Copyright by the Institute of Physics.

here, the F = 2 + F = 3 transition of reference peak (a) is used. Its relatively weak intensity allows the three different HFS transitions to be distinguished. One may state a fair agreement for the expected and measured profile of -9% FWHM and for the position of the maximum determined by the mechanical selector and measured with the DSF detector. Related experiments investigating atomic Na and Naz velocity distributions in supersonic beams have been carried out by Bergmann et al. (1975, 1976) and more recently for fast sodium atoms by Hammer et al. (1976).

152

I . 1/: Hertrl and W Stoll

4 . Atom Beam Deflection by Laser Optical Pumping When an atom beam having a velocity v is excited by photons intersecting it at right angles, the photon momentum h / l is transferred to the excited atoms. In high-intensity fields most excitation processes will be followed by an induced emission transferring the opposite momentum, and no net momentum transfer to the atom results. However, if spontaneous emission follows (which averaged over 4n carries no momentum away) the atom will be deflected from its original direction with an average transverse velocity uI given by mu, = h/l,where m is the atomic mass. If this happens N times the atom beam deflection angle ( 9 4 1) will be 9 = (vJu)N = 90 N (66) where 9, = h/(mAv) is the elementary deflection. If the atom beams passes through an excitation region of length b,, the number of deflections following spontaneous decay will be N = t e x / T , where t,, is the time the atom spends in the excited state. The probability of finding the atom in the excited state is aM= n , / ( n , n,) (see Section II1,A) and t,, = ( b , / v ) a M .We have for the fraction aM < 4 and $aM = n,/n,, where n,/no is given in Fig. 7. The total deflection angle will be

c

c+

1

1

or 3u2 = k A h, bM ,where k A is an atomic constant. This beam deflection has been discussed and investigated by a number of authors (e.g., Ashkin, 1970a,b; Stroke, 1972a,b; Schieder et al., 1972; Picque and Vialle, 1972) mainly since it offers a possibility for isotope separation. Duren et al. (1975) use a well-collimated and velocity-selected sodium beam (FWHM 20/,), which is excited over an effective length of h,. = 2.5 mm. The atom detector is positioned 630 mm away from the excitation region. The experimental results are shown in Fig. 14 for different sodium beam velocities v N p. As described by Eq. (67) for low velocities, the deflection is highest and one clearly recognizes a splitting of the beam into ground state P = 1 atoms, which cannot be excited (and deflected),and into F = 2 atoms, which have been deflected in the pumping region. Duren and collaborators carefully discuss the broadening of the deflected beam by finite atomic and laser beam profiles. The deconvolution is indicated in Fig. 14. The ratios of the F = 1 peak (undeflected) and the F = 2 peak (deflected) should be 5 : 8 = 0.625. Duren et al. find 0.5 to 0.6, which is probably due to neglect of the beam broadening by spontaneous decay. To illustrate the order of magnitude, we cite from Duren et al. (1975) lo9 degrees cm2/sec2 (681 near its maximum, i. This corresponds to

Svi, = 1.05 x for b,z2.5 mm and

c

bM

COLLISIONS WITH LASER EXCITED ATOMS

153

10

5

. o

h

E:

?

P

k

V,

’0

= 1068 [mlsec]

V,.

= 1237 [mlsecj

cd

v

5

0

0

1.0

2.0

3,O

4.0

5

DETECTOR POSITION (am) FIG.14. Experimental distribution of sodium beam intensity as a function of deflection without laser, (.)with laser. (---) 3’S,,, , F = 1 atoms. (---) 3’S,,,, F = 2 atoms. angle. (0) partially excited to the 3’P,,, , F = 3 state. From Duren et al. (1975).

N % 40 to 75 deflections. In principle the measured deflection allows the determination of the excited-state fraction C o M in the deflected beam, while passing through the excitation region. Bhaskar et al. (1976) make use of this possibility in their measurement of the total cross section for the electron scattering of excited sodium (see Section V,C). As they point out, the broadening of the beam due to to first spontaneous decay in each direction is proportional to (9/3 approximation * and will be given by A Qspont = 9( 1/31~)’’~

(69)

l 6 For quantitative investigations it has to be remembered that spontaneous decay is not necessarily isotropic and Eq. (69) has to be modified.

154

1 . 1/. Hertrl and W. Stoll

,290

.335 DETECTOR POSITION (INCHES)

FIG. 15. Fluorescence beam broadening. Laser off. - - - laser on. . Laser is incident at right angles to direction of motion of detector. From Bhaskar et a/. (1976).

This is illustrated in Fig. 15, taken from Bhaskar et al. (1976). Their detector is moved rectangular to the atom and laser beam.” We conclude by saying that in actual scattering experiments the influence of beam deflection by radiation has to be discussed carefully. It may be used with advantage for the determination of total cross sections as done by Bhaskar et al., but it may also be a disturbing effect for small angles in heavy-particle scattering experiments with high angular resolution.

5. Other Excitation Schemes a. Sodium. The pumping scheme described in Section 111,AJ to excite sodium into the 2P,,2, F = 3 state is restricted to one hyperfine level. Among all the hyperfine levels of the p state this one allows the investigation of the largest number of multipole moments, since k,, = 2F. It lends the highest flexibility to the experiment and in principle all dynamical variables ” The

experimental data are uN,, =

lo5 cm/sec, 9, = 3 x

rad, L = 800 mm (32 in.).

COLLISIONS WITH LASER EXCITED ATOMS

155

may be investigated by this type of target preparation. However, it may be of interest to excite other hyperfine levels or even the 'Pli2 state for comparative experiments. By looking at Fig. 4 it may seem that simultaneous excitation by two laser frequencies from the F = 1 and F = 2 states should allow us to pump all upper hyperfine levels F. This is, however, not possible, as Gerritsen and Nienhuis (1975a) have pointed out: Some ground-level substates form a trap. For example, when pumping with linearly polarized light the induced transition probabilities are x ( F M 10 I FM)',i.e., zero for F = F, M = 0. On the other hand, 1 FM = 0) states are populated by spontaneous decay. Thus after some time all atoms will be found in one of these ground levels even when pumping with two frequencies. For circularly polarized light, the I F = 2, M = 2) state is the trap when either the F = 2 or F = 1 excited hyperfine level is pumped, even from both F = 1 and F = 2 . Gerritsen and Nienhuis suggest using at least three optical pumping frequencies in order to obtain up to 50", of the atoms in the 2P,,zor 'P3 state. One may obtain several closely spaced frequencies in different ways, e.g.. as Hertel and Stamatovic (1975) and Hertel et al. (1977~) have demonstrated, one can make use of spatial hole burning in the active medium to operate a cw dye laser on two or more stable modes simultaneously. Gerritsen and Nienhuis (1975a) have proposed a multidirectional D o p pler pumping. A single-frequency laser beam is to be split in order to intersect the sodium beam under two different angles in the scattering center and thus the Doppler shift may be exploited. Each laser beam may be reflected back after passing through the atom beam. Thus, four frequencies may be matched. However as Gerritsen and Nienhuis (1975b) point out elsewhere, one has to be careful when superposing a beam with its reflection. Standing waves or similar interference patterns may be produced. Carter rt al. (1975a) have described and used a different scheme to excite the 3'P1/2 or 3'P,/, state. They use only one laser beam and reflect it back and forth through an angle to fit transitions from both the F = 1 and P = 2 ground states. At u = 1400 m/sec, we obtain a laser-atom-beam intersection angle of 68" (not 23"as cited by Carter et al.). The trap in the M = 0 states is removed by applying a small magnetic field (5G) roughly perpendicular to the photon frame. As described in Section III,B,2 this induces a Zeeman coherence between different magnetic substates, which subsequently may be pumped due to transitions among other magnetic states. The disadvantages common to all these pumping schemes compared to

156

I.

I/: Hertel

and W Stoll

the F = 3 scheme discussed in Section III,A,l is their complexity. It prevents the knowledge of all components of the multipole moments and the intelligible study of polarization effects will become a formidable task. One thereby renounces the powerful tool of investigating the details of scattering dynamics that otherwise could lead to a much more thorough understanding of particular processes. b. Other Atoms and Other Transitions. As mentioned before, excitedstate differential scattering experiments have been reported so far only for the Na(32P)state excited by Rhodamine 6G cw dye lasers. The current improvement of dye lasers, especially with regard to shorter and longer wavelengths, may make other transitions and other atoms accessible soon (see, e.g., Basting et a/., 1976). Some of the alkaline earth elements, some metal atoms, and some other alkalis should be within range of cw dye lasers now in use. Ga-As diodes could in principle be used, operating in a tunable single-mode cw version (Picque, 1974; Picque et al., 1975). Then C S ( ~ ~ P ) could be excited, which may be of great interest for its large spin-orbit interaction. Frequency doubling of dye lasers could widen the scale of possible applications. Very high powers are needed and one probably would have to use a pulsed laser. Quite generally, the frequency range covered by pulsed lasers is much wider. In fact, total scattering cross sections have been measured for fine-structure-changing transitions in a potassium beam excited to the 42P1,2state by a pulsed laser system, as reported by Anderson et al. (1976). The tunable light was produced by an optical parametric oscillator pumped with a frequency-doubled Nd : YAG 'laser. The experiment is described in Section V,E. An alternative to frequency doubling is the direct two-photon excitation of atoms. Two-photon spectroscopy seems to be very attractive of late (see, e.g.. Cagnac, 1975). It has not yet been exploited in connection with scattering experiments in atom beams. As an example, one may excite atomic cesium in the 92D3,2level by absorption of ruby laser light (Ward and Smith, 1975).Theoretically at least, the application of a chirped laser" has been discussed by Garrison er al. (1976). They discuss the possibility of reaching full inversion for the cesium 92D3,2and 62S1;2 levels. The dynamical Stark shift, changing during the pulse duration, necessitates the modulation of the frequency with speeds of around 10'' MHz/sec. To obtain such a chirping speed one could, for instance, wobble one of the laser mirrors, e.g., by ultrasonic resonance, or change the index of refraction inside the laser (Gerlach, 1973; Hutcheson and Hughes, 1974; Taylor et a/., 1971).

'' That is. a pulsed laser with modulated frequency.

COLLISIONS WITH LASER EXCITED ATOMS

157

Most of these techniques are still speculative as far as scattering experiments are concerned. But ambitious experimentalists are called upon to apply them in a prosperous future.

IV. Theory of Measurements in Scattering Experiments by Laser-Excited Atoms In Section I1 we have given general formulas interpreting, in principle, scattering processes of any type by any polarized target. The scattering intensity has been given [Eqs. (18)-(2011 in terms of multipole moments %*(k) describing the target (in our case the laser-excited atoms) and those describing the collision process. The collision multipole moments would be prepared in the inverse scattering process starting with originally unpolarized atoms. After having outlined in the previous section how to prepare the W ( k ) , some more details of scattering multipole moments have to be discussed now. Also, methods will be described that allow us to make use of the technique to extract specific knowledge on the dynamics of the scattering mechanism. A. THEPERCIVAL-SEATON HYPOTHESIS: ELECTRONSPIN AND/OR

NUCLEAR SPIN

UNCOUPLING

In their theory of the polarization of impact radiation, Percival and Seaton (1957) supposed that the hyperfine interaction plays no essential role during the collision. Then the state parameters do not depend in any significant way upon the nuclear spin, but only upon the electronic angular momentum. To prevent any misinterpretation we note that this statement does not imply that hyperfine-structure transitions do not occur in a collision. On the contrary, they usually do take place and the respective cross sections may readily be computed once the electronic transition amplitudes . f J M J are known. The product states l J M J I M , ) [IM, refer to the nuclear spin and J M J to the electronic angular momentum] have to be coupled in the usual way to form hyperfine states I (IJ)FM,). The squared amplitude of the latter is proportional to the differential cross section for exciting that state. In physical terms, this nuclear spin uncoupling means that during the time rco, of collisional interaction, the electronic and the nuclear angular momenta are completely decoupled and M, does not change. Only before and after the collision does the nuclear spin I precess around J under the influence of the atomic I * J interaction. This Larmor precession takes a time tHFs2 h/AEHFSdetermined by the hyperfine splitting AEHFS.As long as

I . 1/: Hertel and W Stoll

158

tHFs % tcol the assumption of spin uncoupling is certainly justified (we may take the hyperhe interaction during collision to be of the same order of sec, the magnitude as the atomic A E H F s ) . Since tHFs is of the order of assumption is probably good for all collision processes of practical interest. Similarly, one often may assume that the spin-orbit interaction AEFSis weak and consequently that the electron spin is uncoupled from orbital angular momentum. The spin projection quantum number M , is conserved during collision. In this case only the orbital angular momentum excitation amplitudes f L M have to be known to fully describe the scattering process, being determined by Coulomb forces only. Again, the assumption is valid if tFs

h/AEFs

% tcoI.

The spin-uncoupling hypothesis should be a good approximation for most atomic collisions with light atoms (tFS z lo-’’ sec) at not too low energies. For electron collisions where f c o , z 10- l 5 sec, it should be possible to apply it, although no experimental proof has yet been given. One should remember that tcol is not a uniquely defined quantity and depends on the magnitude of interaction energy one is willing to regard as negligible. As Macek (1976) has pointed out, the influence of weak long-range forces cannot a priori be neglected when polarization effects are investigated, even though the averaged differential cross sections may well be obtained without them. Furthermore, when dealing with heavier atoms or very low energies, the Percival-Seaton approximation may break down. Departures of a purely statistical population have been observed in the 6’P3/2: excitation of cesium by electron impact in the electron volt range (Hertel and Ross, 1968); additional cases have been discussed by Fano (1970). Also, for the description of photodetechment from negative alkali ions near the atomic first ’P threshold (Lineberger, 1975) the spin-orbit interaction will have to be taken into account. Finally, we note that tFS + tco,may be written as

,,’

where a is a typical interaction range and v the relative collision velocity. This is just the Massay criterion for nonadiabatic fine-structure transitions, which may not always be valid at thermal energies. We shall discuss this point in more detail later. The purpose of the following treatment is to factor out the unimportant nuclear spin and/or electron spin parts from Eqs. (12) and (19) for the scattering intensity. The procedure has been described by Macek and Hertel (1974). For the present review it seems appropriate for clarity to go into somewhat more detail.

COLLISIONS WITH LASER EXCITED ATOMS

159

We have to evaluate the averaged tensor matrix elements ((i' I # F ) I i)), where i, i' stands for an appropriate representation of the laser-excited atom. We note that by definition (10) $'I is constructed by angular momentum states [here Ijm)= I FM) since we have well-defined initial hyperfine quantum numbers] in the coupling scheme (FF)k, or more precisely in the scheme [ ( I J ) F ( I J ) F ;k]. On the other hand, one may define state multipole operators constructed by the nuclear coordinates in the (II)kn,,coupling scheme and by the electron angular momentum in the (JJ)kel scheme. The product may be of rank k. Thus, may be operators [ d k e l 1 ( J ) @ ~[~n"'(I)]Jlk' decomposed into the product representation as one would d o in coupling three angular momentum states, by application of the respective recoupling coefficients:

1( ( I J ) F ( I J ) F ;k I (JJ)ke,(II)knu;k )

rF1(F)=

kcl knu

X [ T [ k c " ( J ) @ T[knY'(f)]jlk'

where

[See, e.g., Brink and Satchler, 1971. Eqs. (3.23) and (5.4).] For averaging we may write I i ) in any representation, in particular in the l J M J IM,) representation. Noting that the averaged quantities ( T ~ I ' ) ~ and ~ ( ~ k ~ lare ) ,irreducible ,~ tensor operators too (the state multipole moments) we may write

((JM;IM,[T""'(J) @ Ttk""'(~)]jlk'1 J M J I M , ) ) = [(T'kpl'(J))@ (T[knu'( I))]!'

(72)

since J and I are uncoupled during the collision. Then in Eq. (71) the averages ( ) may be carried out on either side correspondingly. Again, as discussed in Section I1 the state multipole moments (T!~I'(J)) and ( T ~ ~ ~ ' (may I ) ) be replaced by virtue of the Wigner-Eckart theorem by multipole moments constructed from angular momenta (T',kell(J))and ( T',nul(I)). Using Eq. (16) we obtain (T!I(F)) =

1[(JI)F(JI)F;k I (II)knu(JJ)ke,;k ] kdnu

I

x (kelMkn,,M' k q ) ( T!$")( Tbul)

(73)

160

I . V Hertel and W Sroll

where d k ) ( j )is given in Eq. (18). The conservation of nuclear spin during the collision implies that the nuclear multipole moment is the same before and after collision. If no nuclear spin analysis is performed after the collision and thus M ,is distributed isotropically, this implies in our time-inverse view that (mI(1)) = 0 unless k,, = 0. Then the summation Eq. (73) reduces to one term: (r:’(F)) = [ ( I J ) F ( I J ) F ;k I ( J J ) k , , ( l l P ; k ] ~ ( ~ “ ) ( J ) i ~ ( O ’ ( 1 )

I

x (k,,@O kq)(T[,ke”(J)><71b01(l))

since then k,, = k, (PbO’) = 1, and u c O ’ ( l )= (21 + 1)-

l’’.

Finally with

we obtain

‘(k)(qw) (74) which has to be inserted into Eq. (19) or (12). The multipole moments (ql(J))are now constructed from electronic variables only. Similarly, if the spin-orbit interaction plays no essential role and the spin is uncoupled, we also factor the electron spin to obtain = y

(T:!)

and (T:!) = Y ‘ ( k ) ( q l ( L ) )where , now ( ~ ~ ( Lis )constructed ) from and averaged over orbital angular momentum variables of the electron only. It represents the multipole moment after the time inverse scattering process, i.e., the collisional excitation of a mixture of InWM,) excited states out of 1 ELM). The physical meaning of multipole moments constructed from orbital angular momentum has been discussed in Section I1 in terms of the electrostatic potential of the collisionally excited atom. Equation (75) obviously is valid only if no spin analysis is performed. The factorization carried out above usually implies a reduction in the rank of multipole moments participating in the collision. (1) Obviously, if the nuclear spin cannot be decoupled we have 0
< k,,,=

2F

For sodium 3*P3,*, F = 3 we have k,,, = 6. (2) From Eq. (74) we find in the nuclear spin uncoupled but electron spin coupled case, 0
In our case k,,,=

3.

< k,,,

= Min(2F, 23)

COLLISIONS WITH LASER EXCITED ATOMS

161

(3) From Eq. (75) we find for the totally uncoupled case 0

< k < k,,,

=

Min(2F, 25, 2L)

and specialized k,,, = 2. At this point we recognize the great advantage of this analysis: Without any numerical calculation of the scattering dynamics one may quantitatively investigate the importance of fine- and hyperfine-structure interaction just by observing which is the highest multipole moment participating in the measured scattering intensity as a function of the light polarization angles Eq. (20). This may be of particular interest to heavy-particle collisions at thermal energy. Scattering by sodium 32P3,2, e.g., would lead to an observable octopole moment when the spin is uncoupled, and only quadrupole moments would be observable in the spin-uncoupled case. In conventional experiments this can only be determined by numerical comparison of measured and computed cross sections, such as those given by Reid (1975a,b). Of course, the laser-excited atom-scattering experiment has to be performed in such a way that all multipole moments are actually observable. Linearly polarized light, for instance, prepares the atom in even multipole moments W ( k ) only. Thus, in this case the scattering intensity Eq. (12) allows the determination of even-scattering multipole moments only. In the above case, the critical octupole moment (rank 3) would only be observable when circularly polarized light is used to excite the atom. Even then, the atom has to be irradiated not in the scattering plane (0 = 0), but preferentially at right angles to it. Then a plane and a direction are defined that are needed to observe odd multipole moments having axial symmetry. This may be seen explicitly by Eqs. (19) and (20). For @ = 0, only C$ (0, 0) are ) = 0 for odd k [Eq. (21)]. nonvanishing, while on the other hand ( It should be pointed out that the possibility of preparing and measuring multipole moments of rank k > 2 is a specific feature of the laser optical pumping process for the target preparation. N o such multipoles may be observed when just one photon is involved in the preparation or observation of the atomic-state distribution, as already mentioned in Section 111. Coincidence experiments between scattered particles and the photon emitted after a collisional excitation of the atom have recently been developed into a reliable tool for the measurement of collisional multipole moments (the current state of this art has recently been reviewed by Kleinpoppen et al., 1975). Although these experiments at first sight look like the inverse experiment to the one discussed in the present review they differ distinctively from it by allowing only the determination of collisional orientation and alignment, i.e., tensors up to rank 2. In practice this may become important in the above-mentioned heavy-particle collision problem.

ck)

162

I.

I/: Hertel

and u! Stoll

B. EXPLICIT EXPRESSIONS FOR SCATTERING MULTIPOLE MOMENTS IN TERMS OF SCATTERING AMPLITUDES 1. General Remarks

It is one of the benefits of the multipole language that no explicit use has to be made of quantum-mechanical scattering amplitudes. Indeed in the classical calculations that represent atomic states as an ensemble of orbits (Burgess and Percival, 1968) one could average these irreducible tensors constructed from classical angular momentum components over an ensemble of orbits. However in most practical cases one would like to compare with quantum-theoretical calculations giving results as scattering amplitudes fi. for the time-inverse process, i.e., for the scattering by a state f into the laser-excited state i. It is the intent of the following sections to give explicit expressions for the averaging procedures ( ). We start by writing the first multipole moments up to rank 2 explicitly, as derived from Eq. (17):

0-3= 1 (Ti- ) = (j,) ( T i ) = (3jf

+ 1)OYI - j2) = j ( j + 1)AZ' =j ( j

( T ? + )= Js&j, + j,jJ =jO' + ~ ) A J?) Z (T:+)=$(j:-j;)=j(j+ 1)AYifi

(76)

For reference the relations to the orientation and alignment parameters in the normalization by Fano and Macek (1973) are also given in Eq. (76). Next we note that the averages ( ) may be expressed in terms of the density matrix :

(qk)) = Tr p

=

c (i' 1 qkjli)pii, if'

(77)

and similarly for (z$). The use of the latter [Eq. (1211 may sometimes be more convenient. The density matrix is given by

cr

Pii'

= cqii

l/ci

(78)

with qii,= p(f)firftr and C = q i i . Here p(f) is the probability of finding a particular set of quantum numbersfin the final state (which is the initial state in the time inverse process) (see footnote 2 on p. 118); it contains the statistical weight as well as the detection efficiency for a particular setJ: Equation (77) turns out to be particularly simple when no final-state analysis is performed such as hyperfine- or fine-structure selection or spin analysis. Only with no such selection is Eq. (74) or (75) applicable. We now focus on some cases of practical importance.

163

COLLISIONS WITH LASER EXCITED ATOMS

2. Inelastic and Superelastic Electron Collisions without Spin Analysis a. Multipole Moments. For electron collisions with low-energy electrons (above a few tenths of an electron volt) nuclear spin and electron spin uncoupling is probably a very good assumption. When averaging over all final states J a n d over the electron spin orientation before and after collision, Eq. (75) may be applied directly. The scattering process is usually described by scattering amplitudes with defined total electron spin .CP= S S, including the atomic (S) and scattered (S,) electron spin. Y and .X,yare conserved and in the collision. For evaluating Eq. (78) we have to choosef= i = YLM and to put p ( 9 ) cc 2 9 + 1. Thus,

+ .sPm

i

+

Here f g m denote the ( 2 9 1)-multipletscattering amplitudes for the excitation of the InLM) state out of the I H ~ state. ) In the case of electron = 1) scatscattering by alkali atoms, one has singlet (9’= 0) and triplet (9 tering amplitudes. From conservation of symmetry one may derive (Hertel, M-63 Y f M - m , and find the additional constraint 1977a) f& = (7) M-M q M - M , In . particular, q-l-l = q l l . q- = q l 0 , and of q - M M , = (-) course qol = qT0. Thus we find from Eqs. (74) and (76) for scattering by a laser-excited p state’’ ( T L ) = 2Jzc Im(q0,)

with C = (qoo+ 2qll)-’. In the case of a p + s transition, we have q- 1 1 = - q I 1 and ( T i + ) = -2fiCql,. Then, using the relation 1/C = qoo + 2q1 we find for a p + s transition (ToZ+>+ & T i + )

=

-2

(81)

Obviously this means that the alignment tensor contains in this case only two independent parameters. They correspond to an amplitude ratio and a phase difference. Equation (80) also holds for the spin-uncoupled heavy-particle case. When the other collision partner undergoes changes in angular momentum, it may become necessary to replace 4-11 by Re 4-11 and 401 by (401 - 40-1)/2.

I64

I . V. Ht~r’teland W. Sroll

b. Relations to the 1 and x Parameters. From Eq. (79) we note that q,, x Q u , which is the cross section for the excitation of a particular magnetic sublevel. In electron-photon coincidence work (e.g., Eminyan et ul., 1974; Kleinpoppen et al., 1975)the so-called 1 and x parameters are often used to describe the excitation of a ‘Pstate: 1 = Qo/Q with Q = Qo + 2Q1 and the phase difference x defined by.fo.fT = I .fo I I .fl IeiX.For comparison we define similar quantities for our ’P case. While this is not a problem when using the same definition for the crosssection ratio

A = qoo/(qoo + 2qii) = Qo/Q (82) a difficulty arises from defining a phase x for our case, where two sets of scattering amplitudes, singlet and triplet (or direct and exchange amplitudes), determine the cross section. Hermann et al. (1977a) have chosen the definition

(83) cos x = Re ~ 0 1 / ( ~ o o ~ 1 1 ) ” 2 which, for negligible exchange, yields the same definition as in the ‘Pcase: Then qol +.fb.f: = Ifi IeiX= q&’q:~’ei~. From these definitions we obtain the relations for our scattering multipole moments by

lfol

1 = f(1 - ( T i + ) ) cos x = (T:+)/2J3[1(1 - A)]’” Further evaluation of Eq. (84) leads to 2

cos

x

=

l‘yi’ -

1 1 - [((Ti)

(84)

+ Z1) /3T ] 2

by comparing this expression with Eq. (29) we see that our definitions are reasonable since the phase parameter Icos x I d 1. It should be noted that the definition of the phase parameter cos x is not unique. We may define a second parameter sin cp = ~m(qol)/(~ooqll)1~2 (85) which, for negligible exchange (or singlet scattering only), is limited to the same definition as x. It is interesting to see that sin cp relates to the orientational multipole moments: sin cp = ( T i - )/2[1( 1 - 1)]1/2

In the limit of negligible exchange Icp I = the multipole moments

+

1x I

(86) and we find a relation among

3(T:-)’ (Ti+)’ 4(2 - ( T i ) - ( T i ) ’ ) (87) for only one set of independent scattering amplitudes (negligibleexchange).

165

COLLISIONS WITH LASER EXCITED ATOMS

Since all multipole moments may be determined independently, this relation is remarkable, because it offers the possibility of testing the influence of exchange t o the extent that Eq. (87) does not otherwise hold. This is somewhat surprising, since spin preparation is only done before collision due to the laser excitation. The situation is discussed in detail elsewhere (Hertel, 1977a). 3. Electron Scattering with Spin Analysis

The above-mentioned experiments already yield scattering amplitudes and phases for an excitation of particular projection quantum numbers, averaged over .Y [Eq. (79)]. In contrast, spin analysis should lead to the “perfect scattering experiment” (Bederson, 1969, 1970), disentangling the latter average also. As discussed by Bederson and Miller (1976) and indicated by a recent experiment of Bhasker et al. (1976). it seems feasible to perform an electron-scattering experiment with laser-excited atoms and subsequent analysis of the atomic electron spin. Without going into great detail, we indicate some aspects of the analysis of such a complicated experiment in terms of the language used in the present review: In these experiments the nuclear spin is completely uncoupled by application of an external magnetic field. Thus, the scattering intensity given in Eq. (12) has to be given in the I J M ) representation. (rgl(ph)) and YY ’ ( k ) have to be constructed by the electronic angular momentum J . We assume here that the optically excited multipole moments ‘/t’(k)again have only zero components. This may be achieved by having the magnetic field parallel to the photon frame (see Section 111). Then

I x

kr

k=O

‘/t

’,(k)($](J, ph))

and k,,, = 25. Again, when linearly polarized light is used, the sum, Eq. (88), extends only over even multipole moments. In general, we may recouple as in Eq. (71): (T!’(J))

with

=

1[(a)J(u)J; k 1 (LL)k,(SS)ks;k]([Tkr.

Tks]f)

(89)

kiks

where LM, and SMs are the atomic orbital and spin quantum numbers. The scattering amplitudes are given as previously in the (SSs).Y coupling scheme, S, Ms,being the quantum numbers of the scattered electron, Y’. ti‘, those of the total electron spin. The latter quantum numbers are conserved

166

I . I/: Hrrtel arid W Stoll

during collision (Percival-Seaton hypothesis). The probability of finding a particular M s in the I Y. ti,) state is given by the corresponding ClebscliGordon coefficients. This probability has to be multiplied by the probability of detecting a particular atomic electron spin M, after collision p(M,). We and initial ( M S porientations ) of the scattered have to sum over all final (M,,) electron. For simplicity we confine ourselves to the case where the final atomic state I E M , ) is the ground state in an s-configuration. Then the scattering amplitude is simplyf';, = ,fGl,al,and we have qM1.MsM;.M$

=

d M S ) Ms

1

' , II M.~sMiss

!/ !/

.fGl..f&':

,I

1S ~ , S , M , ) Y'. /?'.,/)(Y. biJ/I S M , S , R , )

(90)

1 ($$(L); Y.Y>(Tt"(s); Jf:y)

(91)

x (SM,S,M,[ .Y'.//!,)(Y.,/d:f x (SM[gSsM,I

+

+

We nqte that . N , = M s + M , , = M, Ms,= M $ Ms,and obviously M$ = M s . If we carry out the averages over orbital and spin quantum numbers in Eq. (89) separately, only zero components of the spin state multipole moments contribute, since M , = M i : ([Trk1.] @ TIksqkq)

=

with (Tfsl(S); .(f.v) =

.v .vg

c ~ ( M s )c as

.HvMs

Ms,Msr

x (SR;isSsRs,I Y&:,)(SMsSsMs,

1 Y .44,)

x ( S M s S s M s ~ ~ . ~ . U y ) ( S M s SJY".U.) sM,~~ x

( M , I rbkS1(S) IM , )

Here, as previously, (TFij) and (tfsl) are the orbital and spin state multipole moments after the time-inverted collision. These multipole moments depend on 9 and Y'. In the general case we now have ks < 2s. If, for example, the atomic electron spin S = i,ks = 0 and k s = 1 moments participate in the scattering process and the scattering intensity has a more complicated structure than the one previously given in Eqs. (75) and (79) together with Eq. (19). Of course, these simple formulas may be regained sx (29' 1) 6,,, and from Eq. (91) by putting p ( M s ) = const. Then is zero unless ks = 0.

( ~ t ~ ] +)

167

COLLISIONS WITH LASER EXCITED ATOMS

The expression for ( T $ " ) in Eq. (91) may be somewhat simplified. After some Racah algebra we obtain

x

c p ( M s ) ( S - MsSMsI k,O)( - )""-

Ms

Ms-S

(92)

This result allows a different interpretation of the spin state multipole moment. By definition [Eq. (9)], the sum over M sis just the state multipole moment of the atomic electron spin measured after the (not time-inverted) collision (i',"sJ(S)). Thus, the scattering intensity finally is given by 2J

Alternatively, the state multipole moments may be replaced by multipole moments with the aid of Eq. (16). Again, Eq. (93) allows great flexibility in choosing the collision frame and the spin detection frame with respect t o the photon frame (ph), into which the multipole moments are easily transferred by rotation through the appropriate Euler angles [Eq. (20)]. We d o not go any further into the analysis of spin selecting, since no actual experiment has been performed yet. 4 . Fine-Structure-Changing Transitions

As another example of how t o apply the theory of Macek and Hertel (1974), we wish t o discuss fine-structure-changing transitions. They are of particular interest in heavy-particle collision problems and have a great experimental tradition in fluorescence cell experiments, where total cross sections for the so-called sensitized fluorescence may be extracted (see, e.g., the review by Krause, 1975). Scattering experiments with laser-excited atoms will bring a completely new degree of detail into this field, especially when use is made of the possibibties opened up by optical pumping discussed in the present review. Recent theoretical calculations of scattering amplitudes have been described, e.g.. by Reid (1973, 1975a) and Bottcher (1976).

168

I. K Herrel and W Stoll

Fine structure transitions are also discussed (Reid, 1975b)in connection with elastic atom-excited atom scattering at thermal energies (Carter et al., 1975b;Duren et a/., 1976).Several authors (see, e.g., Bottcher, 1976, or Reid, 1973)show how to derive scattering amplitudesf,,, Jm for a transition from the final state ISM) to the laser-excited IJM), which we may use in the time-inverse scheme." A priori, we may only decouple the nuclear spin (if hypeene pumping is used to excite the atom) and by inserting Eq. (74)into Eq. (19) we have for the scattering intensity

with

[For the higher moments it may be more convenient to work with the (T!!) related to the (T[gki ) by Eq. (16).]The transformation from collisional frame to photon frame is again done by Eq. (20). As previously, ( are the multipole moments, which would be excited in the inverse-scattering process and k < 25. In the general case this applies whether J is analyzed or not. For not too low energies, to a good approximation (e.g., Bottcher, 1976), M = M' = M = (LIis a good quantum number with respect to the rotating frame. Then we get

qk.))

L"k'(J)(

T',k'(J, col)) = soq

c (- )"-"-"(J (U

- OJO 1 k q ) I f y j z

where the collision frame is now parallel to the incident cms system. Reid (1975a) discusses the importance of spin-orbit interaction and the influence of spin coupling-uncoupling. As stated in Eq. (70), the Massey parameter M has to be small if spin uncoupling is to be expected. Reid (1975a)finds in the case of Li(32P)+ He collisions that spin uncoupling is a good approximation for thermal energies where M G 1. In sodium, higher energies are necessary (Reid, 1975b). In the spin-uncoupled case, Eqs. (71) and (72) have to be applied appropriately. In order to average ( ( L M i < S M ,1 [ ~ [ ~ l@ . ] r[ksr']fl L M L S M s ) ) , we note that the electron spin does not change during collision M, = M, and the scattering amplitudes are independent of it. Thus, only ( T E ~ I ( S )#) 0. On the other hand, the probability of finding a particular M , = M , in any of * O It should be noted that these amplitudes refer to a rotating (body) collision frame. We have adopted a space-fixed frame. However, since Eq. (94)averages over all final A?. a basis that rotates during collision is also an acceptable choice. The col frame is then taken to be the initial internuclear axis between A* and B.

COLLISIONS WITH LASER EXCITED ATOMS

169

where

still depends on the final state R,. In the so-called elastic approximation (which implies more than spin uncoupling) M = M = M‘ = A is a good quantum number and

dk1’(L)(T(qQ(L); A,) =do,

1 (-)”“-“-‘(L-A L A ~ k , ~ 4 ) ~ , f ~ ~ ’ . A

Equation (95) still leads t o multipole moments up to rank k = 25 in spite of the fact that k , = 2 L Only when summing over all final J. due to the orthogonality of 6j symbols, does k = 0 and we recover Eq. (75). We see that fine-structure-changing transitions may not give the clearest test for spin uncoupling, since there k,,, = 23 whether the spin is uncoupled or not. To test spin uncoupling, differential elastic scattering without J analysis is more useful: As an example, in the elastic scattering by ’P,,, state atoms, we would have k,,, = 3 for spin coupling and k,,, = 2 for spin uncoupling. When performing an experiment with circularly polarized excitation incident under an azimuthal angle Q, # 0 with respect to the cms scattering plane, one would observe a c0s3 0 or cos’ 0 dependence on the light incidence direction, respectively [see Eq. (20)]. 5. Born Approximation

We d o not wish to enter into any discussion of how to solve Schrodinger’s equation or how to compute scattering amplitudes. However, we want to point out some special consequences from simple scattering theories such as the Born or Glauber approximation, which have an axis of symmetry.

170

I.

I/: Hrrtel

and W Stoll

The Born theory treats the excitation as an impulsive transfer of momentum AK by the incident electron. Symmetry about the AK direction implies that only q = 0 multipoles are nonzero in a frame, the momentum transfer frame, with the Z axis along AK. Furthermore, owing to reflection symmetry only k even terms are nonzero. (The momentum transfer vector AK has a polar symmetry, while odd multipole moments correspond to an axial symmetry.) Thus, studies of the region of validity of the Born approximation constitute one application of the techniques discussed here. One may either investigate the importance of odd multipole moments or carefully check the symmetry of the scattering intensity when the photon frame is rotated through the AK direction. In addition, we give explicit values for the (Ti),thereby correcting some minor errors in the paper by Macek and Hertel (1974). With respect to the momentum transfer frame (AK), transitions in Born’s approximation obey the selection rule Am = 0. Then

<~ N W = 1 1 .C I’(jm I w I j m ) / c I ~ C m

12

m

(96)

wheref’: is the Born excitation amplitude for a state I njm) out of 16H-1). The transformation to the collision frame (zc,,,parallel to the outgoing electron) is done by Eq. (20):

+

for p = 1, k even, and is 0 for p = - 1, k odd, where OAKis the angle between the momentum transfer vector and z,,, . We specialize to the nuclear and electron spin-uncoupled case. Then from Eq. (80) r

M

M

If we specialize further for sodium 32P3/2(L= l), we find

-I2

The situation is particularly simple, when a p s transition is studied. Since in the final s state L = 0, M = 0 we have I j’: = 0, and thus simply

(Ft’(AK)) = -2, ( P - ] )= 1 for a p c1s transition in the Born approximation.

(100)

COLLISIONS WITH LASER EXCITED ATOMS

171

For this simple but important case we write the transformation to the collision frame: (T’~l(col))= 1 - 3 cos2 O A K (T[:~(co~)) = -J3

sin2 o,,

= -J3
sin’

(101)

OAK

N o explicit dependence on the Born scattering amplitudes is contained in Eq. (101). However, it should be noted that the multipole moments in our normalization refer to relative rather than absolute cross sections. The magnitude of the cross section is contained in the normalization constant lo of Eq. (12). 6. Determination of Scattering Multipole Moments

As we have seen previously, the scattering intensity is given by

p=

f

where V - ( k ) is given by Eq. (74), (75)or (18), corresponding to the coupling scheme appropriate. 0, 4 give the Euler angles of the photon frame with respect to the collision frame. Since they may be varied, in principle all collision multipole moments may be determined experimentally. The question arises as to which are the most adequate experimental procedures. a. Circularly Polarized Light. Since no W(k) vanish, all multipole moments may be observed. However, as discussed in Section III,A it is necessary to measure with the light axis out of the scattering plane to obtain odd multipole moments. From Eqs. (102) and (54)we see that the scattering intensities I + for right and I - for left circular polarization differ, unless @ = 0. Thus I + - I - is given by summing Eq. (102) over odd k only, while I + I - is obtained by summing Eq. ( 102) over even terms only. An experimentally particularly favorable angle is 0 = 4 2 (see Fig. 16a), where the scattering plane is kept perpendicular to the incident light direction, while its inclination is varied. Then +

I+ -I - =I, I+

+ 1- =

c W+(k)V(k)c (7lq!(col))c~kq0. n/2) c W+(k)v-(k)c ( ~ ~ ( C O l ) ) C ~ k !4( 20 ,) k

k odd

q odd

k

10

k even

q even

where Cp) is given in Eq. (17) and Table I.

(1031

I . K Hertel and W Stoll

172

t

b.

a. FIG. 16. (a) Collision and photon coordinate frames for circularly polarized light when the light is incident perpendicular to the scattering plane. (b) Collision (col), electric vector (ph), and incident light (ph) coordinate frames for linearly polarized light excitation. From Macek and Hertel (1974).Copyright by the Institute of Physics.

If we specialize again to the zP3,2case, totally uncoupled, then 1 + - 1- = W (1)V(1)( Ti - (col)) sin 0

I+

+I-

+

= W(O)V(O) +W(2)V(2) x [(a2J(co1))(3 cos2 0 - 1) -&F,”!(col))

and in particular for a p w s transition,

sin2 01 (104)

+ 1- = W(O)V(O)+ +W(2)~-(2)[2cos2 @(F,”l(col)) + 2 sin2 01. The ratio ( I + - l - ) / ( l ++ I-) should be measurable easily. It will have its I+

maximum for 0 = 190”and disappear for 0 = 0. Any departure from the angular dependence given in Eq. (104)is a violation of the spin-uncoupling assumption. The anisotropy is large for low energies and disappears completely in the Born limit. This is a very sensitive test for the range of applicability of any approximation (such as Born’s) with an axis of polar symmetry. b. Linear Polarization. The arguments of the spherical harmonics in Eq. (102) are the polar coordinates of the electric vector E with respect to the collision plane. By varying the direction of this vector one may determine I ( 0 , +) at a sufficient number of points to invert Eq. (102) and determine all of the even multipoles. The plane of polarization of the incident

COLLISIONS WITH LASER EXCITED ATOMS

173

linear polarized laser light may be rotated easily in the experiment, thus varying the direction of E. In general, both angles 0 and 0 vary, but the relative orientation of E is best described by a third angle $ measuring the rotation of the plane of polarization from a standard position. Figure 16b exhibits the axes of the collision frame, the x p h and Zphaxes of the photon frame, and the polar coordinates 01, of the E vector in the collision frame. The vector E parallels the Zphaxis of the old photon frame, while the incident light direction parallels the Zph’axes. The corresponding x , h and Xpht axes lie in the Zph - 2 and z p h # - Z planes. The arc AB is a portion of the circle representing the locus of E vector directions obtained by rotating the linear polarizer about the light direction. Let $ be the angle between the E vector and the Xph,axis. We finally obtain the scattering mtensity (depending on collision angle QEol and energy Ein), k

l($coI+

Ein;

0 1 , @a,

$) = 1 0

C W(k)%”(k) C Cqtpt(7d2,$) q’= o . p ’ = f 1 k

x

C Qq+,q’p,(@ir q=o

o)<~!(col)) (105)

@,i,

From Eq. (17) C$&(7r/2,$) = 0 for k + 4 odd. Since W(k)= 0 for k odd, q has to be even, q = 2Q. Then p’ = - 1 (sin Q)2$, C5,(71/2,$) a (COS Q)2$, p‘ = 1 for Q = 0, 1, . . ., k/2. We see that Eq. (105) represents a Fourier expansion of the electron intensity in a series of sines and cosines of 2$. The maximum value of q is 2[F], where [ F ] is the integer part of F ; thus a total of 2[F] Fourier coefficients can be extracted from measurements with fixed light direction but variable polarization angle. Not all coefficients are independent, since reflection symmetry implies that the intensity with light axis at must equal that at -mi. This requirement takes a simple form when @+ = 0. Then only p even terms of cos q$ terms are nonzero. The number of different nonzero Fourier coefficients again provide a test of the hypothesis of Percival and Seaton (1957). If nuclear spin plays no essential role during the collision, then qmax= 2[J] rather than 2[F], when F > J . Similarly, if the electronic spin plays no role then qmax= 2[L] for J>L Thus in the alkali ’P case we have for 2Pl,2no dependence on ,)I while for 2P3,2, @a = 0, and for a fixed angle of laser incidence

I

= C1

+ C,<’Z$’!(ph’)) + C,
cos 2$

(107)

174

I . K Hertel and W Stoll

We conclude the discussion by giving numerical values for Eq. (105) in the above case: The ratio of the scattering intensities, the electric vector being parallel to the scattering plane (I) = 0") and rectangular to it (I)= 907, respectively, is a function of the angle of light incidence. From Eq. (105) it is found to be

This ratio may be determined easily by experiment, since II is clearly independent of Odfor symmetry reasons. r(Ofi)contains the information on the collision dynamics, i.e., the scattering multipole moments ( T i + ) .The angle of incidence is measured with respect to the Zca,axis, which is chosen in the direction of the outgoing electron in order to compare with theoretical results computed for the inverse process. The relations of the parameters A, B, and C from Eq. (108) to the multipole moments are given by Hermann et al. (1977a): A=- l + a

1 - 2a'

B=-

B

1 -2a'

C=- Y

1 - 2a

(Ti+& COl)>/(700>= f(O)(a - B ) / W ( T:+(kcol))/
(109)

< T 3 + ( Lc01)>/
(110) Thus, while in the general case the ratio of the optical alignment parameters W(2)/W(O) has to be known, in the case of a p + s transition it may be determined from experiment. This is an alternative to the method of measuring W(2)/kn(0)given in Section III,B,2.

V. Collision Experiments As already mentioned, the number of crossed-beam collision experiments with laser-excited atoms that have actually been performed is still small. Nevertheless, these experiments illustrate a number of typical possibilities

COLLISIONS WITH LASER EXCITED ATOMS

175

and limitations of the new field, and thus may give a guideline for future work. We do not wish to describe the experimental methods used in detail. Scattering techniques are treated comprehensively in the series of monographs by Massey et al. (1969-1974)and a survey of the current state of the art of dye lasers is given in Schafer's (1973) book. Also, well-developed commercial laser systems are available now. We shall just discuss some particular experimental aspects, related to the combination of laser excitation and crossed-beam collision schemes. A. GENERAL ASPECTS I . Atom Beam Requirements

a. Collimation. More stringent than sometimes is the case in conventional experiments are the requirements on atom beam collimation. If a substantial fraction of the beam is to be excited, the laser beam must cross the atom beam at right angles and the Doppler spread in this direction has to be small. Ideally, it has to be small compared to the natural linewidth AvYTi, but for high intensities it usually is enough to have a Doppler width small compared to the average saturated linewidth Av;;. If the mean beam velocity is u, the total divergence angle a has to obey the relation

For a supersonic sodium beam (u x 1300 m/sec) irradiated with 1 W/cmz, AvY\ x 90 x lo6 sec- we must have a c 0.08 x 4". b. Radiation Trapping. If one wishes to make full use of the method, polarization of the atomic target has to be exploited and the optical multipole moments W(k)have to be known and should be as large as possible. The optical pumping process described in Section I11 is disturbed, however, when the spontaneous radiation of the atoms becomes comparable to the pumping radiation, since the spontaneous emission is not polarized to 100% as is the laser radiation. Trapped radiation is responsible for a strong atom fluorescence all along an atom beam outside the excitation region. However, its influence is strongest in the excitation center. When increasing the atom beam density, one observes that the optical anisotropy (discussed in Section III,B,2) disappears as a consequence of multipole moment destruction by trapped radiation. We now give an estimate of the maximum atom number density tolerable. The number of spontaneous photons emitted by a volume element dV per unit time is dn,,, = (n,/no)no dV/'lz, where the fraction of excited atoms

'

176

I . !,I Hertel and CI! Stoll

(n,/no) is given in Fig. 7, no is the total number density of atoms, and T the spontaneous lifetime. The spontaneous radiation density in the beam at a position r' is thus given by n u(r') = ." (.)nno o

hvdV(r)

c

--T

1 4nlr-r'I2

The highest value will be found in the center of the excitation region r = 0. The reemitted radiation has an effective Doppler width Av z Ai)112/A [Eq. (65)], when Av1/2is the FWHM of the atom beam velocity distribution. The spectral spontaneous radiation density is thus

which is determined mainly by the smallest beam diameter D within which d V = 4nr2 dr. For sodium 32P3,2excitation where n,/no zO.3, we may neglect the erg sec/cm3 (Fig. 7). Thus we trapped radiation when, say, u,, z 5 x must have n,D/Av1,2 G 8 x lo5 scm-j (114) in order to avoid significant trapped radiation. For a typical beam D z 2 mm and A V ~ z , ' 3~ x lo4 cm/sec, the atom number density must be below z 1 - 10" atoms/cm3. 2. Scattering Geometry

For measuring the angular dependence of a differential cross section accurately, care has to be taken that the detector always sees the same scattering volume. This may become even more difficult when a third beam (the laser beam) participates in defining the scattering volume. Especially cumbersome is the numerical comparison of scattering rates off the ground state and off the excited state. Typical scattering geometries are illustrated in Fig. 17, where the regions for off ground state and/or off excited state scattering are indicated. In Fig. 17a the region for both is much smaller than for ground state scattering only. The ideal geometry, which will rarely be achievable, is displayed in Fig. 17b, where the laser beam is expanded so that both off ground and excited state atom scatterings occur in the same region. Fortunately the excitation area is relatively well defined, as we have seen in Section III,B,l. One may even use this fact with advantage in collisions involving a sodium and a gas beam. As indicated in Fig. 17c, in such a case the collision volume may be defined much better by the laser excitation than

COLLISIONS WITH LASER EXCITED ATOMS

I

\

177

h V from behind

\

e-detector

e- beam

-

-

target gas beam

h V from behind

N

FIG. 17. Scattering geometry for crossed-beam experiments with laser-excited atoms. Scattering off ground state only may occur in the hatched areas while excited-state and ground-state atoms are found in the cross-hatched areas. (a) e + Na. e + Na*: the electron is detected. The excited-state atoms are found in a smaller scattering volume than ground-state atoms. (b) Same as (a), except ground- and excited-state atom scattering comes from identical volumes. (c) Improved angular resolution in a Na* B experiment. Due to laser excitation sodium is detected after collision from a better-defined area.

+

178

I . !I Hertel and W Stoll

by a poorly collimated gas beam, e.g., effusing from a multichannel array. As a consequence, the angular resolution will be much better when scattering processes are observed from within the cross-hatched area than from outside. As Hertel and Stoll (1974b) and Duren et al. (1976) have shown, the extraction of differential cross sections from the scattering rates with light on (Ion)and with light off (Idr)may pose some serious problems. Roughly, these scattering intensities are given by Iofr = const x

[ d3r n,(r)Q, V

(9co~)a~,(9co~ r) dSc0i

(115)

Here no is the total number density of atoms, n, and ne are ground- and excited-state number densities with light on. The apparatus functions ap,( ,9, , r) and ape( 9 ,, , r) refer to the detection efficiency for a particular scattering angle 9,, for the off ground state scattering process (cross section Q,) and the off excited state process (cross section Q,), respectively. In heavyparticle collisions the integrals, Eqs. (115) and (116), have to be extended to an integration over relative velocity and target gas density profile. Regarding the fact that no = n, + n,, we obtain for the difference

1

I , - I,, = const x d3r n,(r)[ap, Q, - up, QJ If the apparatus function is sharp with respect to Qcol and constant otherwise, A = I o n - Ioff [UPeQe - aPgQJNe (118) while Iorfoc up, Q, No ,where No is the total number of atoms in the collision region for off ground state scattering and Ne the number of excited atoms. Only if the apparatus allows us to distinguish the ground and excited state scattering (as, for example, in inelastic electron collisions by energy analysis) can we have, e.g., ap, = 0, and then the difference signal A becomes proportional to the cross section Qe to be investigated. There, one may even determine the fraction of excited atoms when observing a ground state scattering process : NeINo = - A l I o f f (119) In contrast, elastic scattering processes are particularly difficult to analyze, since the elastic scattering off ground and excited state cannot be distinguished : Belast

= const x

[Qe

- QJNe

(120)

COLLISIONS WITH LASER EXCITED ATOMS

179

As Duren and associates (1976) suggest, there may be situations in which it is possible to analyze these data when the elastic ground state scattering cross section is known. Either, by chance, the ground- and excited-state scattering cross sections are equal for some scattering angles or energies. Then from A = 0 one concludes that Q, = Q, at these points. An alternative possibility is given when markedly different structures occur for ground and excited state scattering. In particular, when an oscillatory behavior is observed in A, while no such structures are known in Qs,one may attribute them to Q,. Even there, great care has to be taken, as we shall see later. Otherwise the evaluation of elastic excited state cross sections is an extremely difficult task, necessitating the exact knowledge of n,(r) and $(ScO,).

B. INELASTIC ELECTRON-SCATTERING PROCESSES FROM 32P~/2 STATE

SODIUM IN THE

Inelastic electron collisions are of particular interest for two reasons. First, they may serve as a test case, having relatively simple collision dynamics. The sodium itself may be treated essentially as a one-electron system and the electron collision theory is in general well developed. Thus the principles of scattering experiments by laser-excited atoms may be probed, and the theoretical concepts developed in Sections II-IV may be critically tested. Then, in the more complicated heavy-particle collisions one may rely on this basis and exploit the possibilities to answer critical questions. Second, a numerical comparison of different scattering theories with experiment, probing such sensitive parameters as the collision multipole moments, is now possible and should stimulate further progress in electron-scattering theory. Experiments were first reported by Hertel and Stoll(l973) and have been improved subsequently (Hertel and Stoll, 1974b, Hertel, 1975,1976; Hertel et al., 1975a; Hermann et al., 1977a). 1. Apparatus and Energy Loss Spectra

The sodium 32P3,2, F = 3 excitation scheme discussed in Section 111 is used in these experiments. A schematic diagram is given in Fig. 18. The sodium beam is intersected at right angles in the scattering region by the laser beam. The fluorescence is monitored and used to stabilize the laser frequency. If an occasional mode hop occurs, the laser is scanned automatically to search for maximum fluorescence again. Meanwhile, the data accumulation is halted.21

''

An alternative closed-loop stabilization using digital electronics is described by Diiren and Tischer (1976).

I. K Hertel and W! Stoll

180 I

h

PI(O1OCELL

FIG. 18. Schematic diagram of the experimental setup. From Hermann et al. (1977a). Copyright by the Institute of Physics.

The electron-scatteringsystem is an otherwise conventional system using hemispherical electrostatic analyzers with 60 meV energy resolution, in both the electron gun and detector. Electron gun and detector may be rotated independently around the scattering region. Thus the scattering angle Qcol may be varied, as well as the angle of incidence of the exciting photon beam. The linearly polarized laser beam is incident in the scattering plane. In addition, the polarization angle I)can be rotated with respect to the collision plane. Alternatively, for measurements with circularly polarized light, the electron gun may be inclined perpendicular to the plane defined by the scattered electron (zcol)and the incident light direction (zph).The experiment is controlled on line by a small computer. Energy loss spectra may be taken, or the dependence of the differential cross section on the polarization angle and direction of the incident light is measured. For a fixed collision angle, polarization, and initial electron kinetic energy E , , one can measure the energy of the outgoing electrons E,,, . The energy loss (or gain)AE = Ein- E,, allows us to determine the excitation process that has occurred in a collision. A typical energy loss spectrum is shown in Fig. 19. Collision off ground-state atoms only is shown in the top left part of Fig. 19, where the light is off Na(3s) + e + Na(n1) + e - AE(n1- ns). The most prominent peak corresponds to the resonance transition 3s --* 3p. In the bottom left and enlarged in the top right, the spectrum with light on is displayed. Now, in addition to the ground-state scattering processes from

181

COLLISIONS WITH LASER EXCITED ATOMS

3r-3~

light

ott

rutted intensity (ub. units)

b

scattered int cnsity

1-(

/J@ %-3d

3pds

--JL-rc

3.4p light on

light an- oft

3s-33d _^h-h

FIG. 19. Energy loss spectra for e + Na, e + Na*, Ei, = 30 eV, 9,,, = 0”.

+

sodium 3p we see the processes Na(3p) + e + Na(n’l’) e - AE(n’l’ - 3p) For comparison, the sodium term scheme is shown in Fig. 20, where all processes seen in Fig. 19 are indicated by arrows. The difference “light on” - “light off” is given in the bottom right of Fig. 19. There one sees what is affected by the light. Clearly, the deexcitation process 3p + 3s is seen on the energy gain side. The most prominent feature of this spectrum is, however, the large cross section for the 3p + 3d and 3p -,4sexcitations. Later we shall deal exclusively with these three processes. The depletion of the ground state is demonstrated by the “negative” peak at the position of the resonance line 3s + 3p. There, Eq. (118) would read A = - ap (2.1 eV) x Q (3s + 3p) x N,.By comparison with I, we obtain for the total fraction of excited atoms in the scattering volume, N , / N o x 6%. This reflects an unfavorable scattering geometry. Fractions up to 15% have been observed and one may expect that in the laser beam center the local excitation goes up to 30%. Since the apparatus function up should not change too drastically between AE = 1to 2 eV, one may also obtain a ratio of the differential cross section Q(3p + 4s)/Q(3s + 3p) x 2.7 and = O”, E,, = 30 eV. These numbers are Q(3p + 3d)/Q(3s+ 3p) x 3.1 for meaningful only up to a factor of x 2, since the cross sections still may depend strongly on the polarization of the light, as we shall see next.

182

I . V Hertel and W Stoll (CV)

- 3 -2 - 1

--0

- -1 --2

FIG.20. Sodium energy level diagram. displaying the transitions 3s .+ nl and 3p + nl seen in Fig. 19.

2. Dependence of the Scattering Intensity on Linear Polarization

The scattering geometry is illustrated in Fig. 21. Linearly polarized light incident in the scattering plane is used to excite the atoms. First we give an experimental verification of the theoretical predictions on the dependence of the scattering intensity on the polarization angle I(/ by rotation of the polarization rotator and on the angle of light incidence Ofiat fixed collision angle Qco, by simultaneously rotating the electron gun and the detector around the atom beam (see Fig. 21). Figure 22 displays typical C1+ C2cos 21+bdependencies of the scattering intensity at various angles of light incidence 0 8The . experimental points are given together with a least

atom beam

out

FIG.21. Scattering geometry displaying the scattering angle 9,,,,the angle of light incidence

0,. in the scattering plane, and the polarization angle IJ of the photon electric vector with respect to the scattering plane. The collision system chosen is indicated by Z,,,X c o , . From Hermann et al. (1977a). Copyright by the Institute of Physics.

COLLISIONS WITH LASER EXCITED ATOMS

OJ--

90"

-.

270-

183

-3-Y

FIG.22. Electron scattering intensity for the 3'P -+ 32Stransition as a function of the polarization angle JI for different angles of light incidence @ . The electron kinetic energy before the collision is 10 eV. Measurements for forward scattering 8,,, = 0 are given by + together with a least squares C , + C, cos 2JI fit. From Hermann er al. (1977a). Copyright by the Institute of Physics.

squares fit (from which C1,Czmay be obtained) and illustrate the validity of Eq. (107),thus giving direct experimental proof of the Percival-Seaton hypothesis concerning the conservation of nuclear spin during the collision. No indication of higher multipoie moments has been found in more than 100 such measurements. Of course, this is not at all surprising in the e + Na collisions discussed here, where nuclear relaxation times are at least five orders of magnitude larger than the collision time. Figure 22 also shows that the anisotropy of the cross section strongly depends on the angle of incidence and may altogether disappear for certain This may be displayed in a comprehensive picture by plotting angles 0,. I ( @ , ) = I,,/II = (C, Cz)/(C, - C2), where I = I($ = 0, 0,) and I,. = I($ = No,a,), the latter being independent of 0,for symmetry reasons. Typical data for QEol = 0 are presented in Fig. 23 (for simplicity, we measure 0,from +90 to -90" rather than introducing an azimuthal angle 4 = 180"). Of course, in the case of QEol = 0 the observed " asymmetry lobe " must be symmetric with respect to the collisional z axis, as indicated in Fig. 23.

+

,

I . K Hertel and W Stoll

184

FIG.23. Intensity ratios r = 1 11/1, = (C, + C,)/(C, - C,) as a function c the angle of incidence as obtained from the type of data illustrated in Fig. 22. E,, = 10 eV, Qfo, = 0; 3*P + 3,s transition. Experimental points (+ )are interpolatedby a least squares fit (solid line). From Hermann et a/. (1977a). Copyright by the Institute of Physics.

We remember the interpretation of the scattering intensity given in Section 11: The direction of the E vector of the linearly polarized light probes the electric potential of the atom after the time-inverse collision. Thus the lobe in Fig. 23 (and the ones following) may be directly interpreted as an image of the atomic charge distribution after the inverse collision. The zcoI axis has to be turned around W, however, since E is perpendicular to the incident photon direction with respect to which OAis measured. The lobe in Fig. 23 thus illustrates the cigar-shaped atom (parallel to the electron beam direction) that would be excited in the inverse collision. The measured ratio for Oh= 0 (0= 4 2 ) is 1, which means that the atom is also rotationally symmetrical around the electron beam. Speaking in quantum-mechanical terms, a pure PO (L = 1, M , = 0) state would be excited, which is imaged in Fig. 23, since for forward direction AM = 0 must be valid and the timeinverse initial state is an s state ( L = 0, M, = 0). This symmetry is incorporated in the theory by Eqs. (79),(go), and Eqs. (108), (109): We have qMMt

= dMMlqMM

and

(FfJ) =

=0

thus y = 0 and C = 0, and Eq. (108) simply reads r(O,) = A + B cos 21// for Qco, = 0. Of course, the situation changes when Qcol # 0 and angular momentum is transferred. Some measurements are shown in Fig. 24 for the 3'P + 32Sdeexcitation, together with least squares fits, for gC, = 10" at various incident kinetic electron energies Ei, . The turning of the " anisotropy lobes reflects the very fact of momentum transfer to the atom. The direction OKof the momentum transfer vector AK is indicated by arrows in Fig. 24. We see that the charge cloud is nearly, but not precisely, symmetric in the scattering plane with respect to the momentum transfer vector. (Again the charge cloud is to be seen after rotation through 90"in our plot.) The cigar shape is 'I

COLLISIONS WITH LASER EXCITED ATOMS

185

FIG.24. Intensity ratios r = 111/11 as in Fig. 23 for the 3’P- 3’s transition 9,, = 10” for various incident electron energies. In Born’s approximation these anisotropy lobes would be symmetric through the momentum transfer vector indicated by eK. From Hermann et al. (1977a). Copyright by the Institute of Physics.

in some but not all cases rotationally symmetric with respect to an axis nearly parallel to AK.The deviation from rotational symmetry is small since Min[r(O,)] 2 1. As Hermann et al. (1977a) demonstrate, the alignment of these lobes is reversed, when the 32P-,4% excitation process is observed. The lobes are measured for a variety of collision angles and electron energies. By means of least squares fits to the experimental points according to Eq. (108), the parameters A, B, and C are obtained together with their statistical errors. Since in the p + s case, (T$%”.l) and [ptl)are linearly related, only two parameters determine the scattering dynamics. Thus, in addition one may determine the optical alignment experimentally by Eq. (110). Under ideal stationary pumping conditions W(2)/W(O) = - 0.96 [see Hertel and Stoll, 1974b; Macek and Hertel, 1974, Eq. (48)]. Hermann et al. (1977a)find experimental values between -0.82 and -0.92. This deviation from the theoretical maximum value may be caused by several experimental imperfections, discussed in Section 111. Therefore, the experimental determination of W(2)/W(O) is an essential feature of the method at present. It gives a value averaged over the collision volume. Fortunately, in our case (k even, k,, = 2) the normalized scattering intensity [Eqs. (102) and (105)l depends linearly on W(2)/W(O) only, and thus the use of an average value of this quantity is adequate. Thus, by using Eq. (109) we may obtain (fi,]) and (p:]).

186

I . 1/: Hertel and W Stoll

Nevertheless, the fitting procedure still implies one major experimental assumption, namely that this value of W(2)/W(O) averaged over the collision volume is independent of the angle of laser incidence Ofi,with respect to the electron collision frame. Hermann ef al. (1977a) have made several independent checks to eliminate possible errors. 3. Results and Discussion on Scattering Multipole Moments for 3p + ns Transitions

Hermann et al. (1977a) report results obtained on the scattering multipole moments (P:]) and (Ftj) ((P;”,]) is linearly dependent on (G2])). The experiments are compared with Born’s approximation (here no dynamical calculation is needed; see Section IV,B,5), with scattering multipoles computed by Eqs. (79) and (80) from amplitudes given by Moores and Norcross’ (1972) close-coupling calculations (henceforth denoted by CC) and Kennedy and McDowell’s (1977) distorted wave polarized orbital method (henceforth denoted DWPO). In Figs. 25 and 26 we show the results in terms of the parameters 3, and cos x. Hermann et al. (1977a) summarize their findings as follows: (1) To gain a first idea of the scattering multipole moments (.I:] and )

(pfl) as well as of the 3, parameter, Born’s approximation describes the

experimental points surprisingly well, considering the low energies involved. Of course, only ratios of cross sections are involved and nothing is said about their absolute value. On the other hand, the experiment shows cos x to be markedly different from k 1 (and Born approximation). (2) For low energies, Born’s approximation predicts values too high for 3, (3, 6 eV), while at higher energies (20 eV) it seems to give values too low. The same tendency may be seen in the e He (1’s 2lP) scattering by inspecting the data of Eminyan et al. (1974). (3) At 3 eV, CC and DWPO predict values for 3, that are in fair agreement with experiment, definitely better than Born. At higher energies, DWPO is, in general, somewhat better than Born but the theories do not differ significantly in the prediction of the 1 parameter at small scattering angles. (4) Surprisingly, cos x seems to deviate more from &-1 (and Born’s approximation) as the energy increases. DWPO seems to follow this tendency, but nevertheless fails to give the right numbers for cos x at 10 and 20 eV. At 3 eV for the 32P+ 32Stransition (= 5.1 eV for 32S+ 32P),CC does not give the right values of cos x, in contrast to DWFQ. Higher partial waves may substantially contribute to the phases (as indicated by the good prediction of Born) and are not taken into account accurately in the calcula-

+

187

COLLISIONS WITH LASER EXCITED ATOMS

-.

FIG.25. Measured and calculated parameter 1 = Qo/Qfor the 3'P -+ 3% (left)and 3'P + 42S (right) transition as a function of collision angle at various incident electron energies. CC; _ _ _ , Born; ---, DWPO; experiment with error bars 1 standard deviation. From Hermann et al. (1977a). Copyright by the Institute of Physics.

+,

+r

.-=

tcosx -lo: -0.5

-0.2-

,CDS

x

.co5 x

IE,,;IO.VI

FIG. 26. Measured and calculated phase parameter cos x. Otherwise as Fig. 25. From Hermann et d.(1977a).Copyright by the Institute of Physics.

188

I . V. Hcrtel and W Stoll

tion. It seems especially difficult for theory to predict phase differences where one of the amplitudes is small in magnitude (Qo < Ql).2z ( 5 ) In general, the phase parameter cos x seems to pose more critical requirements to theory than the prediction of the cross-section ratio 1.The discrepancy between DWPO and experiment may possibly be attributed to the influence of the p channel polarization, which is not accounted for by Kennedy and McDowell (1977). This seems to be significant in the e + Na case with its strong p-s coupling. In contrast, the distorted wave methods of Madison and Shelton (1973) yielded excellent results in the e + He case without introduction of the p-state distortion. One should conclude that the theory of measurement by Macek and Hertel (1974) has proved a useful tool in these experiments, applicable also to other experiments with more complicated dynamics. The electron-scatteringtheory itself still needs some improvement. Perhaps theories such as those described by Joachain and associates (Joachain et al., 1977; Joachain, 1977a,b)may lead to better results. Also, more refined close coupling calculations seem to be under way (Burke and Kingston. private communication).

4 . Experiments with 3p + 3d State Excitation

Recently in our laboratory the 3d excitation by electron scattering from laser-excited sodium 32P3,2has been investigated (yet unpublished). Some typical anisotropy lobes are shown in Fig. 27 for 10 and 20 eV incident electron energy. The anisotropy of the cross sections is distinctly observable but small compared to the p s transitions discussed in the previous section, at least for the energy and angular range investigated. The inverse collision, i.e., the deexcitation of an isotropic 3d state into the 3p state, would leave a charge cloud that is nearly but not completely isotropic. Figure 27 is interpreted as a spheroid with somewhat flattened tops rectangular to the scattering plane [since r(O,J is still > 01. This seems plausible, especially since the number of independent transition amplitudes is significantly larger than for a p + s transition: Any of the three angular momentum projection states in the 3p state may be excited out of any of the five substates of the 3d level. This clearly will average out the dependence on polarization. In the language of multipole moments, ( T [ j Y ) is now an independent parameter. Even in the Born approximation with respect to the momentum transfer

’’

Latest CC calculations for cos x at 10 and 20 eV give an improved agreement with our measurements (Moores,1977)

COLLISIONS WITH LASER EXCITED ATOMS

189

O0

O0

FIG. 27. Measured anisotropy for a 32P+ 32Dtransition in the e + Na* scattering for Ei, = 10 eV at different collision angles 3,,, .

vector, we have two independent amplitudes: .fbo and j ; =j - I - 1. They must be calculated explicitly, in contrast to the previous 3p + 3s case.

5. Circularly Polarized Excitation Recently measurements have been carried out (Hermann et al., 1977b),to investigate the difference in the scattering rate for left (I-) and right ( I + ) circular polarization of the exciting laser light. The scattering plane is rectangular to the plane defined by photon and electron detector, so that @ = 7c/2 and 0may be varied.The experimentalsetup corresponds to Fig. 16a (of course, no effect is found when the laser is incident in the collision plane). Measurements for 10 and 20 eV are shown in Fig. 28. Obviously, a strong

190

I . V. Hrr.tc.1 and W Stoll

FIG.28. Measured asymmetry for left and right circularly polarized excitation for the e + Na(32P,I,) -* e + Na(3zS,,2)process. Ratios of scattering intensities I + (fora’) and I (for 0 - light) are measured as a function of scattering angle 9 ,, and fixed 0 (see Fig. 16c) and for fixed 3,, and varying Q, @ is always 90” and the incident energy is 10 eV( + ) and 20 e V ( 0 ) .

asymmetry for left and right circular excitation is observed. It increases from zero with increasing scattering angle and decreases with growing energy. The dependence on the light polar angle 0 should be given by Eq. (104).A detailed comparison would, however, need the knowledge of W(2)/W(O) and W-(l)/W(Owhich ), is not available at present. The magnitude of the asymmetry is remarkably high. Recall that Born’s or Glauber’s approximation would give no such effect. This again lends support to the previous findings: For 10 or 20 eV, which is five to ten times the excitation energy, Born’s approximation fails completely to describe this phase-sensitive effect. We mention also that in photon electron coincidence work not only linearly polarized light has been observed probing the alignments parameters Aq+ (Eminyan et al., 1973, 1974, 1975)but circular polarization has also been measured by Standage and Kleinpoppen (1976),for the helium 3’P excitation. They find that the absolute value of the polarization vector 1 P I = 1 within the limits of error. This is interpreted as a direct experimental proof of coherent excitation. In our language this means that the pa-

191

COLLISIONS WITH LASER EXCITED ATOMS

rameters cp and x [Eqs. (83)and (8611are identical or that Eq. (87) holds. The experiment of Standage and Kleinpoppen thus proves that one amplitude (pure singlet scattering)describes the helium 3'P excitation well. It should be pointed out that circdar asymmetry is a much more critical test of the Born and Glauber approximations than linearly polarized anisotropy and should be applied for a critical study of heavy-particle collision. The method is most sensitive when (D = 4 2 and 0 z 4 2 . C. TOTAL SCATTERING CROSSSECTIONS FOR LOW-ENERGY ELECTRON SCATTERING FROM SODIUM 32P3/2 USING RECOIL TECHNIQUES Instead of detecting the scattered electron directly, one can investigate the deflection of the target atoms due to the momentum transfer in the collision in a way similar to atom-atom collisions. In spite of the small size of the deflection, which depends on the mass ratio of electrons to atoms, this technique has been developed into a reliable tool in low-energy electron scattering by Bederson and co-workers and offers advantages in the determination of absolute cross sections. Detailed discussions of this recoil method when dealing with ground-state atoms are given by Bederson (1968), Rubin et al. (1969), and Collins et al. (1971). Bederson and Miller (1976) have discussed the recoil technique in connection with laser-excited atoms, and Bhaskar et al. (1976)have applied it to the scattering of 4.4eV electrons M , = 3 state. They report preliminary measureby sodium in the 32P3,2, ments of the absolute total cross section. Briefly, a narrow atomic beam is cross-fired by a beam of low-energy electrons (Fig. 29). The atomic beam is velocity- and spin-state-selected before scattering, and can be spin-analyzed after scattering. The spatial dispersion of the scattered atomic beam is measured by an analyzer-detector assembly that rotates about the scattering region. In the scattering-out mode, that is, by measuring the ratio of atomic beam intensities in the

& PHOTON-RECOILED ATOMIC BEAM

DETECTOR

l L i

FIG. 29. Apparatus of Bhaskar et al. (1976).

Y

192

I . K Hertel and W Stoll

forward direction with and without the electron beam operating, one can obtain absolute total cross sections. The atoms are optically excited in a region of uniform magnetic field (- 700 G) oriented along the electron beam axis. This field serves to partially decouple the nuclear and atomic magnetic moments. A cylindrical lens is used to elongate the laser beam along the atomic beam axis. The atomic beam is polarized and velocity-selected by an offset Stern-Gerlach magnet. In addition to the electron recoil, Bhaskar et al. have made use of the atom beam deflection due to the photon recoil, which we have discussed in detail in Section II1,B. They measure the total excited-statescatteringin the part of the beam that is deflected by the photons. They also measure the deflection to determine the excited-state fraction N e / N o = by [Eq. (6711 in this part of the beam and find N e / N o 2 0.4. With the detector placed on the beam axis, let Zo and I, be the atomic beam current with the electron beam off and on, respectively, and let AI = I . - I, be the atom scattering-out signal, related to the total cross section Q by

where h is the height of the atomic beam in the interaction region, u the average atomic beam speed, and I . the electron beam current (electrons/second). Equation (121) refers to a beam consisting of a single constituent. Then, from Eq. (118), one obtains

where AI,,, AXoff are the scattering-out signals with the laser beam on and off, respectively, normalized to the same total currents I . and I,; and Qeand Q, are total cross sections for electron scattering from excited- and groundcm2is known to an accuracy of state atoms, respectively. Q, = 89 x about k 12 % (see Kasdan et al., 1973)and Bhaskin et al. obtain as a preliminary value QJQ, = 3.21. Thus they give Qe(4.4 eV) = (285

55) x 10- l6

cm2

where the errors are attributed primarily to counting statistics and uncertainty in the determination of N e / N o. Moores et al. (1974) have calculated elastic, superelastic, and several inelastic cross sections for scattering of low-energy electrons by sodium in the 3*P states, using reactance matrix elements in a four-state 3s-3p-3d-3s close-coupling approximation. Interpolation of their 4 and 5 eV cross sections and summing all the cross sections cm2. This sum includes the calculated by them yields Qe = 215.86 x

COLLISIONS WITH LASER EXCITED ATOMS

193

contributions 3p-3s, 3p-3p, 3p,,,-3pI,, , and 3p-3d. In view of the few-state nature of the calculations and possible contributions of ionization and higher excitations, as well as of the experimental uncertainties in this initial determination, agreement is very good. It should be pointed out that this method is also applicable in principle to atom-atom and atom-molecule collisions and may stimulate future work in these fields. As far as electron collisions are concerned, the work will be extended by Bederson and co-workers to differential elastic, inelastic, and superelastic processes, possibly with spin analysis after the collision. One looks forward with great interest to the outcome of these ambitious perfect-scattering experiments.” “

D. ELASTIC ATOM-EXCITED ATOM SCATTERING A T THERMAL ENERGIES 1. General Aspects Atom-atom scattering experiments are usually performed to investigate the interatomic potential. The experimental techniques and theoretical methods for reaching this goal are well developed (see the fundamental review by Pauly and Toennies, 1965), at least as far as ground-state atoms are involved. The determination of potentials relies mainly on the marked structures observed in the differential cross sections as a function of scattering angle, known as rainbow and supernumerary rainbow oscillations. Accurate methods have been developed, even for directly inverting the scattering data to obtain the molecular potentials [see, e.g., Buck’s review (1974)l and a variety of interatomic potentials has been determined to great accuracy. With regard to excited states, the situation is much less satisfying. As far as scattering techniques are concerned, one can rely on the experience with ground-state atoms. However, difficulties arise from the fact that only the difference between ground and excited state scattering can be measured experimentally, as discussed in Section V,A [see Eqs. (117) and (118)]. For an excited atomic state, the situation is further complicated. The scattering process may no longer be described by a single interaction potential but, even in the simplest spin uncoupled case, one has to distinguish an AZnand B2X potential, corresponding to the orientation A of the p electron with respect to the internuclear axis. When spin-orbit interaction plays an essen,AzlI,,, , tial role one has to distinguish three different potentials: A2n3,, and B’C,,, . Several authors have discussed how to compute scattering amplitudes (e.g., Reid and Dalgarno, 1969; Wofsy et al., 1971; Mies, 1973a,b;Reid, 1970, 1973, 1975a,b; Bottcher and Dalgarno, 1974; Bottcher et al., 1975; and Bottcher, 1976). The so-called elastic approximation treats the spin-

194

I.

I/: Hertel

and W Stoll

uncoupled-case, i.e., it neglects spin-orbit interaction. In addition, some kinetic coupling terms are discarded, which may be important for low energies. Good quantum numbers are then the projection quantum numbers for the orbital (A) and total ( w ) angular momentum of the electron with respect to the (rotating) internuclear axis. The scattering amplitudes are given by h w . Jw

(l%ol)

=

c A

( U S W

1

- A J4fA(%,I)(JW

1 wso - A)

(123)

The differential scattering cross sections (summed over all projection quan2P,/2 transitions are given by tum numbers) for 2P,,z-,2P,i2and =tI2fn +fr12. Q(3JIIgcoi) = i l f n -hiz (124) The differentialcross section for the elastic process without J and M analysis in the elastic approximation is Q(iiI3coi)

I

) = Q i 2 ( L i ) = Q3,z (gJ = f I .lzIz + 4 .fhIZ (125) Although this approximation is not necessarily valid, such summation over two terms does not facilitate the analysis of experimental data. Even if the elastic approximation is valid, we may not necessarily apply Eq. (125)to the elastic scattering from laser-excited atoms directly, since a particular combination of I 1' is projected out of the scattered wave function. In Sections IV,A and IV,B,4 we have indicated the implications for the cross section. In summary, one can say that at present the complications inherent in excited-state elastic scattering do not allow us to evaluate the experimental data in a conclusive way. Keeping this in mind, we report on the QE(Boi

experiment^.^

2. Elastic Scattering of' Sodium 3'P3,' ,from Neon

Pritchard and Carter (1975) and Carter et al. (1975b) were the first to report experimental results on differential elastic scattering by laser-excited atoms. They used an excitation mechanism involving a magnetic field as described in Section III,B (Carter et al., 1975a). The scattering apparatus otherwise uses standard techniques (Pritchard and Chu, 1970). Figure 30 shows the scattering signal (multiplied by $Jf sin 9,0,) of Carter et al. for scattering by ground-state atoms (Q,) (light off) and Fig. 31 gives the difference signal xQe(3,.o,) - Q,( Qcol) "light on"-"light off." A number of oscillations may be seen in Fig. 30 that allow us to determine the Ne + Na(32S,,2) potential. Carter et al. have also tried to evaluate the excited state *lIand 'Z potentials from the oscillations seen in the difference signal. Their result is in strong disagreement with potentials calculated in an 23

Latest progress has been reported by Dbren (1977).

195

COLLISIONS WITH LASER EXCITED ATOMS

t ~

l

l

!

l

0

~

E8

l

l

I

10

5

I

J

I

( ld40.u.)

FIG.30. Differential cross section for scattering of ground-state Na from Ne. Solid line is an average calculated cross section. The dotted (dashed) line emphasizes the rapid oscillations in the experimental points O(+ ) taken with a 2 ( 5 ) mrad resolution. E = 5.0 x 10- a.u. From Carter et a/. (1975b).

elaborate multistate computation by Pascale and Vandeplanque (1974), which is surprising. We should, however, remember the numerous difficulties in the evaluation procedure discussed above. Since the measured curves are proportional to the difference of the excited and ground state scattering, Qe - Qg,it is difficult to know precisely to which cross section the oscillations are to be attributed. Although at larger E9,,, the oscillations disappear in Fig. 30 for the “light off” signal, this might be due to the angular resolution. This may be markedly improved for the difference signal (Fig. 31), as we have discussed earlier (see Fig. 17c). This is of particular r M

I

e,

U

W

b

0

c -

u)

FIG.31. Differential cross section Q, - Q,. otherwise as Fig. 30. From Carter et a(. (1975b).

I . V . Hrrtrl and W Stoll importance in an experiment with a gas target chamber. Thus the possibility should not be excluded of attributing the oscillations in the difference signal to scattering from ground state atoms, 3c_ -$(9co,). Reid (1975b) has critically discussed the experiment on theoretical grounds. Carter et al. use the elastic approximation, Eq. (125), for their evaluation. As Reid points out, spin uncoupling is a necessary but not sufficient condition for the elastic approximation. Reid computes closecoupling (cc) cross sections for the case of Na(3p2P)+ He at 40 meV (which is below that used in the Na* + Ne experiment). His results nicely illustrate possible differences between elastic approximation (Q") and more exact computations (QCJ). Reid concludes: (1) If the spin is uncoupled, then Q E gives a good representation of the differential cross section as far as the frequency (but not the amplitude) of oscillations is concerned. (2) If the spin is coupled to an appreciable extent, then QE does not represent the oscillatory behavior of the differential cross section adequately. The differential cross section may contain oscillations that do not reflect the molecular potentials directly and that are absent from QE. (3) Hence QE can be used to interpret experimental results only if the impact energy is sufficiently high that the spin is uncoupled during the collision. An indication that the spin is uncoupled is the validity of the equality = Q3,2. 3. Elastic Scattering of' Sodium 32P3,,2from Mercury

Duren er ul. (1976) have performed a crossed-beam experiment with high angular and velocity resolution for the elastic scattering of sodium 32P3,, by mercury. They used the F = 3 excitation scheme (Section IILA). The atom beam is selected before the scattering region with a mechanical selector. Figure 32 gives their signal from the ground state [Eq. (1 15)] together with the difference signal, Eq. (117), between 4 and 70" for a barycentric energy of 43.5 x 10- l 4 ergs. Two sections are displayed: a small-angle part from 4 to 30" and a large-angle part from 40 to 70" (laboratory angles). The small-angle part is characterized by the rainbow structure for the ground state interaction, which is well understood in terms of the respective interaction potential (Buck and Pauly, 1971). In the difference signal one finds oscillations of large amplitude which are exactly in phase with the rainbow oscillations. From this, Duren er al. conclude that these oscillations are due to the ground-state contribution to the difference signal [Eq. (1 17)], and a possible contribution from the excited state is either monotonic or has relatively small amplitude. For this reason they then deduce from the measured signal only an approximate cross section, using the condition that

197

COLLISIONS WITH LASER EXCITED ATOMS

.

I

I

810 -

H

I

R Frr 0

H

U

0-

1

0

I

1- ‘ I

It

10 20 30’’ ~EFLECTIONANGLE

L

1

60 SLAB[GRAD] 50

I

70

FIG.32. Differential cross sections for the ground-state interaction (0) and the difference signal ( 0 )in elastic collisions of sodium 3%, z . 3’P,,, with Hg. From Duren t’t al. (1976).

in the zeros of the difference signal the cross sections of the excited and the ground state are equal [Eq. (117)]. As discussed above, this evaluation gives only the coarse structure of the cross section for the excited state. In contrast to this, the structure of the excited state cross section in the large-angle part can be given precisely, because there the ground state cross section is monotonic and the structure observed can be unambiguously attributed to the excited state. The contribution is clearly seen in Fig. 32 and for another energy we have the result after subtracting a fraction of the ground state signal in Fig. 33. n

:

K

u

bC0

50 DEFLECTION ANGLE

60 *LAB

[GRAD]

m

FIG.33. Differential cross sections for the sodium 3*P, state. From Duren rt al. (1976)

198

I . b! Hrrtc.1 and W Stoll

Diiren (1976) has computed Na-Hg pseudopotentials including the 3'P state. Only the ground state may be compared with experimental data. Diiren et al. (1976) have not yet attempted to interpret their experimental findings with these potentials. They say, however, that the large-angle structure cannot be interpreted in terms of pure potential scattering and conclude that fine-structure transitions may be responsible. In other words, a failure of the elastic approximation, Eqs. (123)-(125), is found. This is especially plausible for collisions involving such a heavy atom as mercury. 4 . Influence of' the Polarizatiori of' the E.uciting Light

The investigation of polarization effects may, in favorable cases, lead to some knowledge of scattering amplitudesfJM,J a ,their phases, and ratios as discussed for electron collision. As mentioned previously, M = M = o to a good approximation for the rotating collision frame. For the elastic approximation, even A = const. This may, however, be more complicated in reality, and polarization effects, if observable, should help to clarify the situation. In contrast to inelastic electron scattering, no dependence of the elastic heavy-particle scattering intensity on the light polarization has yet been reported. Linearly polarized laser light was used in the above experiments for exciting the sodium. Diiren (private communication) explicitly states that within the limits of statistics he does not find any dependence on the polarization angle for the geometry used in his experiment. Figure 34 illus-

lYlab

I

ycO'

FIG. 34. Schematic three-dimensional Newton diagram for the experiment of Duren er al. (1976). Displayed is the photon frame (ph'), the laser E vector, the collision frame (col = cms). and the laboratory angles and velocities of Na and Hg beams.

199

COLLISIONS WITH LASER EXCITED ATOMS

trates the kinematics of the experiment by Duren et al. (1976). The mercury beam (OA) crosses with the initial sodium beam (OC)and the scattered atoms are detected “off plane” in the direction OD. To describe the polarization effect we have to choose the collision frame parallel to the center of mass relative velocity before collision, Zco,IIAC. The cms direction is defined only as well as the angular and velocity spread (especially of the mercury) is small. Thus, the polarization angle between incident E vector and Zco,is known only to a limited extent, which places some critical requirements on the experiment if one wishes to see polarization effects. Even then, if no polarization effect is observed for one direction of incidence (here OA= 90’ and @ > 0 but small) it could possibly be seen for another direction, as we have illustrated in the electron case (Fig. 22). On the other hand, recall that the dependence of the scattering intensity on the E-vector direction probes the charge distribution after the time-inverse process, i.e., the elastic scattering from a p state with isotropically distributed projection quantum numbers. Total isotropy would imply, according to Eqs. (108) and (109), that p = 7 = 0 and thus (T\’J (col)) = 0 and (TL’! (col)) = (Ti’! (col))/& From Eq. (80) this means that Re qol = 0, q l l - qoo = q - l l .Since the number of independent scattering amplitudes describing this process is larger than for a p + s transition, it cannot be excluded that the interference terms responsible for polarization effectscancel in the calculation of qol and b o o - 411) + 4-11. As discussed in Section V,2, this may occur even for electron collisions, as exemplified for the ’P -,’D transitions with its weak anisotropy. However, when the elastic approximation holds, again only two scattering amplitudes determine the process. In addition, only diagonal density matrix terms in Eq. (78)are nonzero: qol = 4- = 0 and qoo =

l.fz1’

and

411

= Ifn1’

where the collision frame is now parallel to the initial center of mass system. This leads to polarization effects unless If’[ = I f n [ (and thus 400 - 411 = 0 = 4-11). As illustrated in Section V,B,4, polarization effects decrease when the number of scattering amplitudes participating gets larger. Obviously, the elastic approximation fails to explain the observed smallness of the effect, if any. For a quantitative understanding, fine-structure interaction has to be considered. But even if one is willing to accept spin uncoupling as a firstorder approximation, one has to discuss all scattering amplitudes f n + , fn- ,j‘-n+ instead of only just two. The relations,f,+ =fn- andfzn+ = 0 are valid only exactly in the adiabatic limit. For finite internuclear velocities the possibility of transitions

.fc.

C ++ ll+ has to be considered, and also A-degeneracy may be removed for

large angular momenta. Thus one would expect at least a phase difference beand .fn-. Then q - is different from zero. Although this would tween jn+ not affect the usual differential cross section, it would be important to the dependence on polarization. However, in the present case no final conclusion can be given. At least one other direction of light incidence has to be probed, preferentially parallel to Z,,, or to Ycor.The latter choice would have the additional advantage that it could be used with the most sensitivity for experiments with circular polarization. Circular polarization should probe at least (F:! ), i.e., the expectation value of the angular momentum transferred in the collision. The elastic approximation would yield no effect. As discussed in Section IV, must be # O and preferably as large as possible for this testing purpose. A particularly interesting experiment would be the search for (Ty! ), which would be an indication of non-uncoupling of spin. As discussed earlier, (Ti1! ) and (T:"_')would be distinguished by a careful measurement of the intensity ratio (I' - I - ) / ( / + + I - ) as a function of 0 for fixed 3co,.The large-angle oscillations found by Duren rt al. (1976) might be especially sensitive to (Tb3-]),since fine-structure transitions may be responsible. Many ambiguities may be resolved by studying polarization effects, especially when using circularly polarized light.

E. FINE-STRUCTURE-CHANGING TRANSITIONS IN HEAVY PARTICLE COLLISIONS Fine-structure-changing transitions of the type A*(nJ)

+B

-+

A*(rtJ) + B

(126)

at thermal energies have for many years been subject to experimental and theoretical studies mainly concerned with fluorescence cells. In the previous section we discussed some implications of these processes for elastic scattering. Directly, one observes the photon emitted in spontaneousdecay A*(nJ) -+ A + hv with a spectral resolution distinguishing it from the A*(nJ) decay. It is outside our present scope to review the voluminous work in this field (see Krause, 1975) even though many experiments use lasers to excite A*. We wish only to report about an experiment by Anderson rt al. (1976), which is to our knowledge the only crossed-beam experiment in this field. It overcomes the usual averaging over thermal energies in a cell in an unambiguous way. Anderson and associates have investigated the integrated (over all scattering angles) cross section for the process K(42P,/2) He K(4'P3/,) He at relative velocities from 1000 to 3500 m/sec. A schematic diagram of the experiment is given in Fig. 35. The

+

-+

+

ev/Fl

COLLISIONS WITH LASER EXCITED ATOMS

20 1

'P3,,- photon-

Filter

K-Supersonicbeam

He Supersonic -beam

Laser -

FIG.

35. A schematic diagram for the experiment of Anderson et a / . (1976).

alkali velocity distribution had a Mach number of 2.5 with a most probable velocity of 900 m/sec. The intersection angle between alkali and gas beam could be choosen as 45 or 90°, thus changing the relative velocity. Alternatively, the temperature T of the helium supersonic beam nozzle was varied to change the helium velocity according to vHe = (5 kT/mH,)li2,where mHcis the helium mass. The tunable light t o excite the K(42P1,2)is generated by an optical parametric oscillator (OPO) pumped by a doubled Nd : YAG laser. The system generates 75 pulses/sec of about 100 nsec duration. The peak power in the axial mode absorbed by the potassium atoms is about 15 W. The various lenses, mirrors, detectors, etc.. serve directing, tuning, and intensity monitoring purposes. The hyperfine states that are excited in these experiments are not determined, but the OPO is tuned to maximize the fluorescence. A fiber optic light pipe conveys the 2P1,2resonance fluorescence to an external photomultiplier. The intramultiplet mixing signal is obtained by a photomultiplier and interference filter device, detecting the 42P3,2fluorescence only. No provision for measurement of polarization (or angular anisotropy) of the 2P3,2emission is provided in these experiments. Less than one photoelectron is obtained for each laser pulse using an S20 photomultiplier. Interference filters blue shift their transmission for nonnormal incidence so that 2P1,2-2P,,2 intramultiplet mixing is seen with less interference from resonance fluorescence than would be the case for 'P, 2-2P, mixing. The intramultiplet count rate S is averaged for typically 4 x lo4 to 4 x lo5 laser pulses. The output F from the total fluorescence monitor is also recorded and

202

I . I/: Hertel and W Stoll

gives a measure for the excited atom density n,. The 2P,,2photon count rate is S 00 ne n~ ore1 Q(f 3 I Ore,) (127) where Q(i 3 I ureJ is the fine-structure-changing cross section ( J = 3 -+ J = 4) and ureIthe relative velocity, determined from uK and uHe, the alkali and gas most probable velocities, and from the intersection angle ;' by ureI= ( u i + u k - 2UKc'G cos I!)"~. The density nG of the gas beam was determined by a pressure rise AP in the beam trap, x AP/uH,. To eliminate dark currents both alkali and gas beam may be switched off separately. The following formula is used:

where the subscripts 00,OC, and CC denote both beams on, gas beam off, and both beams off. The relative cross sections obtained in this way are normalized by means of thermal-averaged absolute cross sections from cell experiments by Krause (1966). The experimental points in Fig. 36 give the results of Anderson rt al. (1976). The cross section rises more or less monotonically from the energetic threshold (at cT)to a remarkably high value at 3500 m/sec of around 200 A2. Considering the relatively large FS splitting (7 meV) corresponding to ENERGY IN KCAL/MOLE

0.2 0.5

1.0

2.0

3.0

5.0

N

5 200

2

2

z

2

0

100

w

fn

fn

In 0

c

0

0

0

10 30 RELATIVE VELOCITY IN lo4 CMISEC

FIG.36. Cross section as a function of relative velocity for K(4pZP,,,) + He + K(4p2P3,,) He. The cross-section scale is normalized to the 95°C bulb result of Krause and 9 0 ' (0. A) are presented to cover the velocity (1966). Two intersection angles, 45' (0) range. v T is the relative velocity corresponding to 58 cm-' endoergicity for the process. The full curve is a fit to the data using the modified Nikitin theory. From Anderson er a / . (1976).

+

203

COLLISIONS WITH LASER EXCITED ATOMS

tFS z 5 x 1513 sec, spin uncoupling cannot be assumed here, especially not when large-distance curve crossings are to be discussed, which are of importance for fine-structure-changing collisions. Then tco,may be of the order of lo-'* sec and the conditions tFS % t,,, for spin uncoupling (Section IV,A) certainly do not hold. Thus, when interpreting their results, Anderson et al. discuss explicitly transitions C,12+ rIIl2and r11/24 l13/2.A semiclassical curve-crossingmodel of Nikitin (1965) is used and a fit is shown in Fig. 36. It uses an intemuclear distance of R , = 14 A for the crossing of the potentials. No satisfactory agreement is obtained with potential calculations and a number of questions remain open.24

F. ELECTRONIC TO VIBRATIONAL ENERGYTRANSFER I . Energy Transjer Spectra

The continuing interest in electronic to vibrational energy transfer from excited atoms A in collisions with molecules M

A*

+ M(u = 0)-

A

+ M(u')

(129) arises from its importance as a basic mechanism in the understanding of many types of chemical reactions. The bulk of the experimental material is concerned with the determination of total cross sections. Spectroscopic methods have been used by Polanyi and collaborators to draw conclusions on the population of the vibrational levels of the molecules in gas cell experiments with Hg* and polar molecules (Karl and Polanyi, 1963; Karl et al., 1967a,b; Heydtmann ef al., 1971). Related techniques have very recently been used to study the process Na* + CO ( u = 0) + Na CO(u')by Hsu and Lin (1976). An involved evaluation procedure is required to understand the experimental signal. A review of the earlier work has been given by Lijnse (1972, 1973). A number of additional references are cited by Barker and Weston (1976) and a detailed account of the current state of the field especially with respect to laser-excited atoms will be given elsewhere (Hertel, 1977b). A survey on theoretical aspects is given by Nikitin (1974, 1975). We restrict ourselves at present to describing a series of experiments that have recently been carried out in our laboratory (Hertel et al., 1976, 1977a,b). They are, to our knowledge, the first crossed-beam experiments with laser-excited atoms in this field and may illustrate how the new technique can be used to broaden the scope of heavy-particle collision physics. Inelastic scattering and other

+

24 A novel technique to observe differential fine-structure-changingcross sections has recently becn demonstrated by Phillips c't a/. (1977).

204

1. K H e m 1 und W Stoll

processes such as reactions hitherto energetically inaccessible may now be subject to studies in detailed differential scattering experiments. Apart from a purely statistical treatment (Levine and Bernstein, 1972; Wilson and Levine, 1974) theoretical models (Bjerre and Nikitin, 1967; Bauer et al., 1969; and Fisher and Smith, 1970, 1971, 1972) usually assume an intermediate ionic state A + + M- to be responsible for the quenching process Eq. (129). The atomic electronic excitation energy E,, is transferred to this state at curve crossings with the A* + M surface and is converted into vibrational and translational energy at crossings with the A + M electronic ground-state system. Although this model is attractive at first sight, its details exhibit some conceptual difficulties. An improved theoretical understanding should be stimulated by as much and as detailed experimental information as possible. The experiment is illustrated schematically in Fig. 37. Briefly, a supersonic NO- BEAM COLLIMATING SYSTEM

n

/ I I'I m

Na-OVEN SU P ERSON IC

ION FOCUSSING SYSTEM

SO'MAGNET

CAPILLARY

'

\

MECHANICAL VELOCITY

U

IR-HOT WIRE DETECTOR

PARTICLEMULTIPLIER

SINGLE MODE DYE LASER-BEAM

FIG 37. Schematic diagram of the cxperiment to determine the energy transfer i n

Na*

+ molecule collisions.

sodium beam with a mean velocity of around 1350 m/sec (FWHM z 23",,) is crossed at right angles with the molecular beam effusing from a capillary array at 300'K or alternatively at 77°K. The sodium atoms are excited in the scattering region into the 32P3,2,F = 3 state by a single-mode linearpolarized cw dye laser. The scattered sodium atoms are velocity analyzed by a mechanical selector and detected in the scattering plane by a hot-wire detector and particle multiplier. A simple kinematic calculation-using the most probable velocities of the beams-allows us to convert the sodium velocity measured in the LAB frame into cms energy E,,,. The kinematics of the experiment are illustrated in a Newton diagram (Fig. 38). The reaction of Eq. (129) has an energy balance (in the crns) given by El" + E,,

=

ECI,

+

AEllhTOt

(130)

COLLISIONS WITH LASER EXCITED ATOMS

205

with the initial kinetic E , , = ,u/20;?,, zz 0.1 to 0.2 eV, the electronic excitation energy E,, = 2.1 eV for Na 3,P, the crns energy after collision E,,, and the vibrational and/or rotational energy transferred to the molecule AEVibrol. The purpose of the experiment is to determine the latter and the cross section d2rr/dQ,, dE,, for a particular energy transfer and crns solid angle Q,,. Since E i n and E,, are known and E , , may be measured, we can determine AEvibrol.Thus, the final molecular vibrational level u‘ may be determined, if for the moment we neglect rotational energy transfer. Hertel er al. (1977b) have investigated the quenching process Eq. (129) for H, , D, , N, , CO, C2H4, 0, , CO,, and N,O. Two typical examples of measured energy transfer spectra are shown in Fig. 39. Displayed is the difference signal ‘‘ light on - light off.” It has been corrected by EZ:/C:,AR for compensation of selector transmission and scale transformation from a measured CLABscale to the E,, scale. For better visibility of structures, dZo/dE,,, dQ,, has been multiplied by E c m . As seen from the Newton diagram, Fig. 38, at fixed laboratory scattering angle, different energies correspond t o the different crns collision angles, which are also given in Fig. 39. Due to a thermal energy spread in the molecular beam, the angular resolution is very limited and no detailed evaluation of angular dependencies can be given at present. Nevertheless, the energy transfer spectra Fig. 39 show that process (129) is determined by clearly nonresonant mechanisms. Especially the Na* N, quenching populates predominantly the P’ = 3 and 4 vibrational levels of N,, while the c ‘ = 7 and 8 levels (near EiIJwould be approximately in resonance with the electronic excitation energy ( E e , = 2.1 eV) for the Na 3,P levels. Quenching by 0 , is a somewhat different matter. Three ”

“

+

I,,,, 0 d'a dvdlln;h [ARB.UNITS]

1.0

2.0

EM

w .,.i-. '

..::,..:".. ._ ... . .,..

02 9ne=1 0 ' T = 100' [K]

FIG. 39. Differential cross sections multiplied by ECMas a function of E,, (relative kinetic energy in the center of mass system after collision). Ei, indicates the initial relative energy of the sodium atoms and the gas molecules calculated for the most probable velocity of each species. EEXis the excitation energy of the 32Pstate of sodium (2.1 eV) The t i scale displays the energetic positions corresponding to an excitation of the target molecule to the vibrational level u'. The variation of the scattering angle OCMin the center of mass system is shown as a function of ECM.From Hertel et al. (197%).

207

COLLISIONS WITH LASER EXCITED ATOMS

-

electronic final states of 0, may be involved. The X 3 X i ground state, the ulAg state at 1 eV and the b’C, state at 1.6 eV. The structures seen in Fig. 39 may easily be identified with the combined vibrational and electronic excitation of these states. Not all structures can be resolved uniquely. The energy resolution of the experiment is determined by velocity and angular spreads of the incident beams and by the analyzer resolution (relative velocity selection is 7”4 FWHM) and is scaled up by the kinematics. The horizontal error bars give the FWHM of the overall energy resolution as obtained by a detailed Monte Carlo study of the kinematics. For the case of the Na* + N, process, such Monte Carlo calculations show that the different final vibrational levels should be distinguishable quite clearly, if only pure vibrational energy is transferred. This is seen in Fig. 40. The lack of these distinct features indi-

-

I

.J

. *

T C

3

e

Is

I

1000

FIG.40. Monte Carlo calculation (-) of pure vibrational excitation in the Na* + N,(o) process, to estimate the experimental velocity resolution (L. in lab. system). The experimental points (...) d o not show structures and thus indicate the influence of rotational excitation. QLAB = lo”, T = 80°K.

cates that a part of the available energy is transferred into rotational energy of the molecule. The advantages of the experimental method presented by Hertel et ul. are obvious: Differential, rather than the usual integrated cross sections can be measured and represent a more sensitive test of theoretical predictions. The method is direct and gives results without the need to disentangle a complex set of reaction rate equations, as necessary in cell experiments. It is applicable to any gaseous or vaporizable quencher. The experiment may thus be seen as a new type of heavy-particle collision spectroscopy. Finally, the possiblities of atomic-state selection by laser excitation may be exploited.

However, in addition to kinematic angular energy uncertainties, one major limitation of the method should also be mentioned: Since resonant transitions show up at E,,, = E,,,, they cannot be distinguished from totally elastic scattering processes. Since neither the elastic cross sections nor the number density of excited atoms is known quantitatively, it is not possible to evaluate the data near E , , Ei,,. The apparent strong rise in the spectrum for N, + Na* (Figs. 39 and 40) near Ei, has to be attributed to elastic, rotationally inelastic, and fine-structure-changing collisions for the system Na* + N, . These latter processes may have total cross sections as large as 190 A (for 0.14 eV) (see Bottcher, 1975), which have to be compared to the total quenching cross sections of about 22 8, (Lijnse, 1973).

-

2. Polurixtioii E&ts A clear dependence of the Na* + N, quenching cross section on the polarization has been observed in the experiment, when one is sure to have a distinct optical alignment Y l (2)/ Y/ (0).The laser is incident perpendicularly to the scattering plane and only small scattering angles are investigated. The experimental findings are summarized as follows (Hertel et ul., 1977a):

(1) A definite but small anisotropy of the differential quenching cross section is observed, when the electric vector E of the exciting laser light is rotated in the scattering plane. Three typical examples are shown in Fig. 41. (2) The anisotropy as a function of the energy transfer seems to follow the energy transfer spectrum (Fig. 39) as shown in Fig. 42, where the ratio of maximum to minimum scattered intensity Zmax/lmill is plotted together with one standard deviation error as obtained from least squares fits to the measured curves. At the maximum the effect is 18",, for quenching by N2 at a tcmperature T 80°K. The anisotropy vanishes in the wings of the energy transfer spectrum. ( 3 ) The anisotropy seems not to depend significantly on the scattering angle in the small-angle range under investigation. (4) The quenching cross section has its maximum when the E vector is approximately parallel to the cms system. (5) Measurements on a circular asymmetry, i.e., a change of the cross section for (T' and (7- light excitation at AE = 1 eV, :Aa,, = 18", 3,,,, = 14.7' have shown no significant effect ( < 4",,). ( 6 ) At 300°K the linear anisotropy decreases to about one-half the value at 80°K. One typical point is shown in Fig. 42. (7) Similar observations have been made for the quenching of Na* by D, . For 0 2 CO, , and CO, measurements have only been carried out for 300"K, where no anisotropy has been observed.

-

-

COLLISIONS WITH LASER EXCITED ATOMS

209

. I

eLAB:loo

SCATT. RATE

EklN :1.255 eV

[ ARB. U N ITS1

I

E k l N :1.256eV I

,

00

360°

.

v

+

FIG.41. Scattering intensity for Na* N, quenching as a function of the polarization angle with respect to the cms system for three different O,,,. AE,,, vih together with a least squares fit. From Hertel c't ul. (1977a).

No final conclusion may be drawn from the observations since several experimental uncertainties may obscure polarization effects. However, it is obvious that the observed polarization effects are much less pronounced than for the corresponding 3p -+ 3s transition in electron collisions. This can only be the case when qoo and q 1 are of similar orders of magnitude, that is to say, the quenching cross sections Q1 for a 3pn orbital (AM = 1) and Qo for a 3p0 orbital (AM = 0) are of similar magnitude. In fact, the latter is somewhat larger. To interpret this finding one has to take account of the fact that in heavy particle collisions large angular momenta may be transferred even for small scattering angles. Recall also that the projection of the molecular rotation may change during collision. This could explain the decrease of the polariza-

2 10

-11

+

f' i.;

0

0

T

FIG.42. Ratio of maximum to minimum scattering intensity taken from fits (as shown in Fig. 41) as a function of the energy transferred to N , . The laboratory scattering angles given correspond at A&,, v,h = 1 eV to O,,, = 1.31(0). 5' (+). 11. ( x ). and 15' (0) at 8 0 ' K and to 1.8" (a)at 300°K. From Hertel et a/. (1977a).

tion effect at higher temperatures, where higher molecular rotational states are populated and the variety of scattering amplitudes becomes larger. We can also give a possible explanation for the fact that the anisotropy is largest where the quenching cross section has its maximum: In an involved curvecrossing process, the largest cross section indicates the most direct process; thus the initial alignment of the atom influences the cross section most strongly. Where the process itself is less probable, the memory is lost during the collision time and 3pa and 3pn quenching may become equally probable. An enhanced rotational momentum transfer provides for the conservation of angular momentum. A theoretical computation would not necessarily have to be a quantummechanical one. Rather, the multipole moment formulation lends itself easily to a semiclassical treatment, as discussed in Section 111. However, in order to gain a quantitative understanding of the experiments the details of the potential energy surfaces have to be known. Thus the type of studies described here will help to critically analyze theoretical models.

21 1

COLLISIONS WITH LASER EXCITED ATOMS

VI. Atomic Scattering Processes in the Presence of Strong Laser Fields We do not wish to conclude the present review without mentioning some pecularities that may occur in strong radiation fields and could possibly obstruct the straightforward analysis of the experiments. However, we hope to make it clear that these influences can be completely neglected for the experimental condition under discussion, i.e., when the laser field is strong but not too strong. To quantify “strong” we recall some numbers for the problems discussed in this paper: The spontaneous lifetime T 2 lo8 sec, the induced transition time rind = 1/Bu,, or equivalently the inverse Rabi frequency l/RR2 10-8-10-10 sec, the Larmor frequency for the finestructure interaction fFS = 10- sec, the inverse electronic transition frequencies t,, z lo-’’ sec, which are of the same order of magnitude as the inverse laser frequency l/v, and finally the collision time tCol2 lO-’’-lO-’* sec. Thus, for the cases studied the field is strong (z > l/QR)but not so strong as to totally destroy the atom (l/QR9 fFS 2 tCol2 tel) and OR < v (we also recall that typical pumping times are t,, = so that essentially stationary systems are investigated). Energetically these numbers imply that the natural linewidth r < Edipol(the energy of the induced dipole) and certainly Edipol% EFS< Eel. Recall also that the energy resolution AE,,, of present-day scattering experiments is some meV at best. Thus we also have Edipol< AEr,%.To violate these conditions the radiation power would have to be at least four orders of magnitude larger than currently used. Even higher laser intensities would be needed off resonance. So, to a high degree of accuracy, the atomic basis sets for the unperturbed atoms may be used to describe the atom under the influence of field and scattering. The laser field could in principle have a direct influence on a collision process A + B + h v in one of two ways:

’’

(a) By direct interaction with the collision system AB: A

+ B + hv

. +

(AB) + hv -+ (AB)*+ A‘ + B’

(131)

(b) By disturbing the excited atom (dressed atom): A

+ B + hv

.+

A:,,,

+ B + A‘ + B’

(132)

This is more clearly seen by inspecting the total Hamiltonian:

H

=

H,

+ HAF +

HF(AB1

+ HAB

with

Ho

= HA0

+ HBo +

HF,

(133)

Here H A , , HBo, and H,, are the free Hamiltonians for the atom A, particle B, and laser field F, respectively. The interaction terms are H A F for the

212

I . I.: Hrrtel arid W Stoll

atom-field interaction, HF(AB) for the field-collision system interaction (typically a dipole operator for the interparticle coordinates), and HAB for the collisional interaction. When the laser is not in resonance with an atomic transition, HAF may be neglected. Then one usually first solves the scattering problem

+ HA,

(134) and treats the interaction HF(AB)with the field as a perturbation that may induce transitions within the scattering continuum, the so-called free-free transitions. In contrast, for the resonant case HAF is dominant. One possibly may neglect HF(AH)and first solve the atom-field problem HI = H,

+ HAF

(135) The collision HA, may be treated as a small perturbation. In this case, the atom is a time-dependent coherent superposition of ground and excited state, which has been neglected throughout the present work. H2 = H,

A. FREE-FREE TRANSITIONS AND SIMILAR PHENOMENA

First we briefly discuss the free-free transitions, going somewhat outside the scope of the present review.25However, we want to show that HF(AB)is of no importance in our experiment. O n the other hand, several very interesting experiments have been reported recently that should not entirely be excluded in an article dealing with scattering and lasers. The most straightforward case is the free-free transition for electron-atom scattering (or inverse bremsstrahlung). There, in the dipole length approximation, H,(,,, = erE, where r is the electronic coordinate and E the electric field. The electron absorbs or emits a photon in the presence of the atom. Theoretical investigations (e.g., Geltman, 1973, Kriiger and Schulz, 1976) predict cross sections for this process. In Born's approximation it is proportional to the elastic cross section:

where i! is the Compton wavelength of the electron, u the radiation density, E the polarization vector of the field, and AK the electronic momentum transferred in the collision. Thus, the free-free cross section increases as AK and fxl/to4. This has been exploited by Andrick and Langhans (1976) to observe the effect experimentally in e + Ar scattering at around 10 eV. The 2s

Recent progress has been reviewed by Gavrila (1977).

COLLISIONS WITH LASER EXCITED ATOMS

213

Scattered electron energy lev1

FIG.43. Energy loss spectrum of e--Ar scattering, upper line with, lower line without laser beam. Counts per data point are plotted against energy of the scattered electrons. Incident energy is 11.55 eV. The arrows indicate an energy gain (or loss) of 117 meV, corresponding to free-free transitions. From Andrick and Langhans (1976).Copyright by the Institute of Physics.

collision angle was 160" (large AK). They used a C 0 2 laser at 10.6 pm wavelength and an intensity of about 6 x lo4 W crn-'. High-energy resolution and extreme background suppression allows us directly t o see the effect on either side of the elastic peak profile (Fig. 43). Andrick and Langhans (1976) give a preliminary experimental value for y2 = 4 x which is in rough agreement with the theoretical value [Eq. (136)] of y 2 x 1.4 x We immediately see that for the scattering from laser-excited atoms this effect is generally completely negligible. Since y 2 sc l/04, in the visible one would need at least lo7 W/cm2 to observe an influence of less than lop3of the cross section. It is exciting to follow the free-free transition through an electronscattering resonance in the initial or final elastic wave. Andrick and Langhans (1978) have done this for the Ar- resonance in elastic scattering. Figure 44 shows the resonance structure (without laser) in the elastic peak as a function of the incident energy. It has a well-known fine-structure splitting (Weingartshofer et al., 1974). Figure 44 also displays the free-free channel where the resonance is doubled due t o its influence in both the incoming and outgoing channels.26 Otherwise the free-free cross section still remains essentially proportional to the elastic scattering without laser.

*' In Fig. 44 the inelastic free-free channel is observed. Thus the resonance contribution from this outgoing channel is shifted 117 meV to higher energies.

I . K Her-trl and W Stoll

214

.-

x lo3

INCID. ELECTRON

ERGY

-

FIG. 44. Resonances in differential electron scattering from argon at 160". -, ordinary elastic scattering; 0 , resonance in the free-free channel. From Andrick and Langhans (1978).

The situation changes completely when the laser line fits the difference between two resonances. Then the free-free process effects a transition between two quasi-bound negative ionic states and a marked increase in the free-free cross section may be expected. Langendam et al. (1976) have recently reported the first experimental observation of this resonant " freefree absorption by an electron in the field of a neon atom. The Ne levels and Ne- resonances involved are shown in Fig. 45. Langendam et al. detect the process by spontaneous emission of either the 5882 or the 6030 A line of the neutral neon, since the upper 3p level may be populated only due to the 18.95 Ne- resonance. The latter is excited by a freefree transition from the 16.8 eV resonance (Fig. 46). N, laser-pumped Rhodamine 6G and Cumarine 47 dye lasers are used, respectively, to induce the free-free transitions. The laser power is lo4 W in 20 nsec.,' Related processes may occur in heavy-particle collisions. These laserinduced transitions take place between the potential energy curves of the colliding atoms A and B and may be seen to some extent inverse to the "

*' Most recently even multiphoton transitionshave been observed in the free-free e-scattering by Weingartshofer et a/. (1977).

215

COLLISIONS WITH LASER EXCITED ATOMS

I

NeNe

llSo ELECTRONS

7, /-

,=/”’

GROUND LEVEL

FIG.45. The Ne and Ne- levels involved in the investigated transitions (Ne level energies from Schulz, 1973). From Langendam et al. (1976). Copyright by the Institute of Physics.

c2 3

ji I-

z

I 1

I.: I

9

2l

>

3

i,i

0:

I l l

>

0

25

2

t-

5

t Ln

5

I

F1

I

I

0:

I

zI

M 15 W

0

53

5891 I

O! 0 5890 5900 5910 5! LASER WAVELENGTH (8)

20

0

ELECTRON ENERGY (eV:

FIG.46. Intensities of Ne I emission at 5882 and 6030 A, induced by optical transitions from an Ne- level at 16.8 eV (see Fig. 45). (a) For a fixed electron energy of 16.8 0.5 eV, as a function of dye laser wavelength. The largest peak occurs at 5897 A; a smaller one lies at 5891 A, i.e., at 2 meV separation. Dots and crosses refer to separate data runs. (b) For a fixed laser wavelength of 5897 A, as a function of electron energy. The 3P,(3p) excitation function (solid curve), obtained as the total 6030 A emission rate without gated counting, serves as a calibration of the energy scale and an indication of the energy width of the beam. From Langendam et al. (1976). Copyright by the Institute of Physics.

216

I . V; Hertel and W Stoll

photon emission in the "spectra of colliding atoms" (Gallagher, 1975). A typical process is A + B + (n)hw--* A B*

+

where ( n ) b (n = 1, 2 , 3 , . . .) is not in resonance with the excitation energy of B* for infinite separation R , . The energy difference of the AB and AB* interatomic potentials becomes resonant to n h o at the smaller internuclear distances formed during collision. Recent theoretical treatments, e.g., by Kroll and Watson (1976) and by Lau (1976a), underline that high laser intensities are needed for these transitions. A similar mechanism for a laserinduced atom-atom transition is A* + B

+ h v + A + B*

where the difference energy between A* and B* excitation at large internuclear distance R , nearly equals hv. Since in the molecular picture the transition is resonant for a large range of internuclear distances, it should have much higher cross sections than the previous case. Such processes have recently been reported by Harris et al. (1976) and Falcone et al. (1977) in the Sr + Ca system at a laser intensity of 5 x lo5 W/cm2. Laser-induced energy transfer to both the calcium 4p2 ' S and 5d'D states has been observed detecting the fluorescence from the excited calcium states:

+ + Ca(4s2 'S) + hv(4711 A) = Sr(5s2 'S) + Ca(5d'D)

~ r ( 5 p ' ~ O+)ca(4s2 's)+ hv(4977 A) = sr(5s2 's) ca(4p2 's)

Sr(5p'P')

The term scheme is shown in Fig. 47. The experimental fluorescence signal as a function of the wavelength of the transfer laser light shows a maximum at around the R = 03 wavelength A = 4976.8 A, with a FWHM of 14 cmAgain, the theoretically predicted cross section depends linearly on the laser x 9 (W/cm') cm2 = 5 x intensity (McGinn, 1969) and is oc = 9 x 10- " an2.The experimental value of 9 x lo-'* cm2 is in good agreement. We conclude that these effects may be completely discarded in the experiments otherwise discussed in the present work.

'.

B. COHERENT SUPERPOSITION OF GROUND AND LASER-EXCITED STATES IN APPROXIMATELY RESONANT ATOM EXCITATION Neglecting the interactions HF(AB) that have been discussed in the previous section, we may now concentrate on the distortions an atom experiences when it is excited with a strong, nearly resonant laser radiation. The problem is closely related to saturation or power broadening, which has been discussed in Section III,A,2. In fact, the behavior of the atomic ensemble in the

COLLISIONS WITH LASER EXCITED ATOMS Sr 1

217

Co I

FIG.47. Energy level diagram for laser-induced transfer from strontium 5p'P0 to calcium 4p2 ' S . From Falkone et a/. (1977).

laser field [H, = H, + HAF,Eq. (1391 may be fully understood in terms of optical Bloch equations. As we have shown, the latter also yield the power broadening. The full atomic density matrix B contains in general nondiagonal terms connecting ground and excited states. These nondiagonal terms have been neglected throughout the paper and only the diagonal excited state part (re has been used. A proper treatment would have to contain the total a(t)as well as the scattering density matrix p(t). The experiment then essentially measures a time-averaged Tr B * p. Several serious problems in defining the basis sets prevent us at present from doing so. Nevertheless, we are concerned about the neglect. A simple but appropriate view is in terms of wave functions. As well known, in the presence of the laser field, the atom is a coherent, timedependent superposition of ground and excited state:28

I + b,,(t)(2),

YAY= a&) 1)

v = 1, 2

(137) where for simplicity we just treat the atom as a two-level system with a stationary unperturbed ground state 11) and an excited state (2). The circularly pumped sodium atom may be taken to represent such a system to a good approximation: ( 1 ) = I F = 2, M = 2) and 12) = ( F = 3, M = 3). The collisional interaction may then be treated in the usual perturbative approximation. However, a fully time dependent treatment has to be applied. 2 8 The two solutions 'PA,(?) and YA,(t)correspond to different initial conditions: ~ ~ (= 01, ) b,(O) = 0, and a,(O) = 0, b,(O)= 1.

218

I . I.: H w r l and W Stoll

The influence of coherent superposition of states on the collision may not be discarded by arguing that during the collision the resonance condition for the photon does not hold. The scattering process is in any case a small perturbation to the atomic ensemble as such, whose time dependence is dominated by the near-resonant laser field ( HAB) < ( HAF). Thus the timedependent wave functions Y A , ( t(there ) are two orthogonal ones) have to serve as a basis set in a proper quantum mechanical treatment of the scattering problem. We will see, nevertheless, that the phenomena arising from coherent superposition will hardly be observable experimentally unless the fields are very large. Hahn and Hertel (1972) were the first to attack the problem and have discussed the experimental implications for the scattering by laser-excited sodium 32P. Gersten and Mittleman (1976) have used an essentially identical treatment and applied it to derive an elaborate formula for electron scattering by laser-excited hydrogen. No discussion is given by the latter authors on the experimental phenomena to be expected; in particular, no numerical estimates of the importance of coherent-state superposition on scattering by laser-excited atoms. Thus we essentially follow Hahn and Hertel (1972), who used a fully quantum mechanical formulation of atom, field, and scattering process. They neglected spontaneous decay. The field quantization is an elegant but unimportant formalism in this case and identical results are obtained in a semiclassical treatment of the laser field (Hahn, 1972). The rotating wave approximation (see, e.g., Paul, 1963; Brunner er al., 1964; or Paul, 1969) is used to obtain the time-dependent atomic wave functions Y &). It is a good approximation for near-resonant excitation (laser frequency w % w2 - wl, the resonance frequency), possibly with a The coupling of the atomic levels by the field small detuning Aw e w 2 - q. is given by K = D,, X [ A , 1/2h = *OR, where D12 is the atomic dipole matrix element, d',, the (real) field amplitude, and OKthe Rabi frequency (see Section III,A,3). The rotating-wave approximation gives a periodic behavior of the timedependent amplitudes a$), b,(t) with essentially where

I b,(r) l2

K sin2 2 Wr

W 2 = K 2 + (Ac0/2)~< w2

The squared amplitudes I a(t)12, I b(r)l2 give the probability of finding the atom in the ground or excited state, respectively. This is illustrated in Fig. 48. K 2 / W 2gives the maximum, and

n,/n, = * K 2 / W 2

(140)

COLLISIONS WITH LASER EXCITED ATOMS

219

FIG.48. Timedependence of the probabilities l u > ( t ) l ’ , Ib,(t)l’ to find an atom i n the excited and ground state, respectively. Curve 1 represents lu2(t)1’ = Ibl(t)l’, while curve 2 is lu,(t)12 = I b2(t)I2.K Z / W Zgives the maximum probability of finding the atom in the excited state, having been originally in the ground state. From Hahn and Hertel (1972). Copyright by the Institute of Physics.

is the time-averaged fraction of upper state p~pulation.~’For resonant excitation, K = W and n , / n , = i,as it must be. The averaged ground state population is

1 - K2/2 W2)

(141) The rotating wave approximation allows an interpretation of these oscillations as splitting of the (nondegenerate) atomic levels, arising from the modulation as side bands. A schematic is given in Fig. 49. These split levels again exist only in a coherent superposition. They cannot be observed in the total fluorescence. When w is tuned through resonance, the splitting just determines the saturated linewidth. They may, however, be seen when the fluorescence light is frequency analyzed as three distinctive components: the Rayleigh scattering w, the anti-Stokes fluorescence w 2W (which is near w2 - w1 for larger detuning), and the Stokes shifted fluorescence w - 2 W. Experiments have resolved these three components (Shouda et al., 1974; Wu et al., 1975). In addition, Carlston and Szoke (1976a,b) and Carlston et ul. (1977) report on collision induced fluorescence under off-resonant conditions (Aw# 0). The rotating-wave approximation even allows us to estimate the relative population ” of these virtual levels. Hahn and Hertel derive a time-dependent close-coupling system [without exchange, which is included by Gersten and Mittleman (1975)l for the collision problem, investigating transitions from yAl-+yAland v~~ --* yA2. The resulting scattering amplitudes would be explicitly time dependent. The (Ilg/H0)(

+

“

”) It should be noted that Eq. (140). which neglects spontaneous emission, ditTers numerically somewhat from the stationary solutions of the optical Bloch equations [Eq. (95)] even in the limit of r = I/T + 0. The numerical consequences of the present discussion therefore have to be modified slightly when spontaneous radiation decay is included.

220

I . I/: Herrel and W Stoll

FIG.49. Schematic illustration of the atomic “level splitting” for a two-state system. The level splitting 2W is small compared to the level distance m2 - wI. A positive detuning AUJis assumed,where w 2 - (ul 2 w;criis thefield frequency. From Hahn and Hertel(l972). Copyright by the Institute of Physics.

experiment time-integrates the scattered wave function. When this is done, a set of delta functions arises describing different energies klm after collision, when the initial energy was homo. The close-coupling system could be solved stationary. Thus, the coherent superposition of states leads to inelastic transitions with a typical energy loss or gain. All transitions among the four levels indicated in Fig. 49 are possible. In addition, multiphoton transitions could occur for very high fields. The close-coupling equations given by Hahn and Hertel allow in principle the following final energies: tic,,,, = hCOmo

+ h[(mo- m)w - 21W]

(142)

where mo refers to the initial and m to the final number of photons in the laser field, and the level-splitting transfer is given by 1 = 0, L 1. For not too high intensities, mo - m = 0, k 1 and, since W < w, we may distinguish the following process: quasielastic superelastic inelastic

1 +0 klm hr-;o,, - ho I hlm = h~~,,,,,

hl,,, = h ~ ~ ,+, ,ho ~

or

+2Wh (143)

=

where ho is nearly the atomic excitation energy. Hahn and Hertel(l972)have given expressions of the differential scattering cross sections for the various processes in first Born approximation.

22 1

COLLISIONS WITH LASER EXCITED ATOMS

TABLE IV DIFFERENTIAL CROSSSECTIONS FOR PARTICLE SCATTERING I N RADIATION FIELDS, CALCULATED I N THE FIRSTBORN APPROXIMATION^

Transition*

‘lrn

1’ 4 1‘; 1“ -+ 1’’ 2’ 4 2’; 2” -+ 2“

2“ -+ 2’; I ’

4

1”

2’ -+ 2”; 1” -+ 1’ 2’ -+ 1”; 2” 4 1‘

2” -+ 1” 2’-+ 1’

I ’ -+ 2”; 1” -+ 2’ 1‘ -+ 2‘

l”42”

T21

EOm0

-w

+2 w

$(I

I2

- & ) 3 ~ ~ 2 1 ~ 2

hi,,,is the energy after collision and T j are Born amplitudes. The designation of the various transitions is to be understood in terms of Fig. 49.

Their results are given in Table IV, where T,, and T22denote the Born elastic scattering amplitudes for pure ground and excited state scattering, and T,, the inelastic and T,, the superelastic Born amplitudes, respectively. Characteristic are the interference terms in the quasielastic scattering, which reflect the coherent superposition of ground and excited state. The question is whether this interference could actually be observed in the experiment. Recall that W is essentially the Rabi frequency (for resonant excitation). It is typically lo* Hz for the intensities used to excite the atoms. This corresponds to an energy splitting of 2 peV, which is hardly observable with any conventional scattering techniques resolving 2 meV at best.

I.

222

I/:

Hortc.1 and W Stoll

Thus one has to sum over the unresolved energy losses. Then one obtains the following cross sections:

superelastic inelastic

da K2 dQ- 2W2

--

do

dR

~

(144)

[ T 1 2 \ 2 h

ko 2

kolno+1 ko

processes

If we recall from Eqs. (140) and (141) that the average population of the ground state is 1 - K 2 / 2W 2 and of the excited state is K 2 / 2W 2 ,these cross sections are just the ones expected if nothing were known about the level splitting and the consequent interference terms. Thus, neglecting coherence terms used throughout the paper is correct in a Born's approximation. We believe, however, that this outcome is typical for any set of scattering amplitudes that do not vary significantly over an energy range of some peV. It should be possible to obtain such a theorem in a more rigorous way, using a density matrix formulation for the laser excitation as well as for the scattering, including the nondiagonal terms. Hahn and Hertel (1972) and Gersten and Mittleman (1976) have neglected spontaneous decay. while in a new paper of Mittleman (1976) it is included. This is advisable, especially under the circumstances relevant to the present paper, where the spontaneous lifetime 7 is of the same order of magnitude as R R .The difficulty with spontaneous decay is that the atomic wave functions become nonnormalized unless a fully quantized treatment of the field arid spontaneous decay is used. On the other hand, the optically excited density matrix is essentially stationary and one could obtain stationary solutions (including the nondiagonal terms) from the optical Bloch equations. Again, a density matrix formalism should be used for the whole problem. However, it is p!ausible that interference terms as given in Table IV should rather have the tendency to wash out. Equally, instead of a pure splitting by W , a broadening of the lines is to be expected when T = n/R,. In summary, we may safely assume that neglecting coherence between ground and excited states does not affect the experimental analysis given in the present review. Possible observation of the laser-induced splittings in collision processes could be made in connection with hyperfine-structure transitions that have the same energy separations. Also, for very high laser fields, six orders of magnitude above the ones presently used, the splitting may become resolvable with scattering techniques.

COLLISIONS WITH LASER EXCITED ATOMS

223

One may also observe multiphoton transitions. As Hahn and Hertel (1972) show, these may become important when the collision time tco,is comparable to n/QR.For these high powers the free-free transition probabilities discussed in Section V1.A may also be of comparable magnitude.

VII. Conclusions It will not have escaped the attention of the reader that the review presented here was not meant t o give a comprehensive survey on all scattering experiments involving lasers. Whole areas, such as excitation of molecules, high rydberg states, or gas cell experiments, have been left out deliberately in order to keep the size of the article within reasonable limits. It has been the intention of the authors to provide a solid basis for planning, preparing, and performing future scattering experiments with laserexcited atoms in crossed beams. Particular attention has been paid to the possibilities of detailed investigations on scattering dynamics, which are typical of the target preparation by laser optical pumping. It is our firm belief that these possibilities open up a novel field in atomic collision physics, whose first successful performance we have just witnessed. We hope we have illustrated this by the considerations of how to test particular theories and by the experimental examples presented. We also have shown how to pose new critical questions on atomic interaction dynahow to obtain answers without the necessity of mics and-possibly-even going into detailed computations. Typical applications are the study of the range of validity of the Born or Glauber approximation, the importance of spin coupling or uncoupling in a collision, or the critical discussion of angular momentum quantum numbers to be conserved in a collision as predicted, e.g., by the so-called elastic” theory in heavy-particle scattering. Thus the new methods may provide fresh insights into old problems. In addition, we have illustrated how a quantitative evaluation of the measurements has to be performed. The language of multipole moments used throughout the review has proved to be a valuable guide thereby. Even though it looks complex at first sight, it gives us a very direct perceptual view of what happens to an atom during collisions and allows intuitive interpretations of many experimental findings. ”

ACKNOWLEDGMENTS We would. of course. like to stimulate further work in this flourishing field. And it is a particular pleasure to us to thank all experimental and theoretical groups actively engaged in these problems for supporting us with their experimental material. numerical data, and helpful com men ts.

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1. VI Hrrtsl and W Stoll

R. Diiren, especially, has been a patient instructor in the field of elastic heavy-particle collision and C. McDowell and D. Moores have continuously supported us with their theoretical results on scattering amplitudes. One of us (I.V.H.)wishes to take the opportunity to give his special thanks to all those who are or were members of our group in Kaiserslautern during the past years. They have in one way or another contributed to part of the work reported here: S. Azarkadeh, L. Hahn, H. Hermann, H. Hofmann, W. Miiller, W. Reiland, K. Rost. A. Stamatovic, W. Sticht, and last but not least my co-author W. Stoll. Without their endeavors we would not have been able to contribute to this fascinating topic in atomic physics. H. Hermann, H. Hofmann, and W. Reiland were particularly helpful in reading the manuscript and suggesting improvements. Finally, we wish to acknowledge gratefully the continuous financial support from both our university and the Deutsche Forschungsgemeinschaft.

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