Electron impact formation of metastable atoms

ELECTRON IMPACT FORMATION OF METASTABLE ATOMS

I.I. FABRIKANT

Institute of Physics, Academy of Sciences of the Latvian SSR, 229021 Riga, Salaspils, USSR O.B. SHPENIK, A.V. SNEGURSKY and A.N. ZAVILOPULO

Institute for Nuclear Research of the Ukrainian SSR, Academy of Sciences, Uzhgorod Department, Uzhgorod, USSR

NORTH-HOLLAND

- AMSTERDAM

L I. Fabrikant et al., Electron impaa formation of metastable atoms

3

1. Introduction

Among the elementary processes occurring duringlow-energy electron-atom collisions those of elastic scattering, excitation of discrete atomic levels, and single and multiple ionization are the most probable. A special place among them is taken by the formation of metastable particles. The latter include electronically excited atoms, molecules and ions for which the radiative electric dipole transition into the ground state is forbidden by at least one of the selection rules, so that their lifetimes in the excited state are long (>10 -5 s). There exist quite a few elementary processes leading to the formation of metastable particles, in the first place those occurring in electron-atom collisions. Metastable particles dan be efficiently produced also in atomic collisions, especially during charge-exchange processes and in the neutralization of ions at the surface of solids. Within the great variety of all these elementary processes the most efficient and, therefore, significant are binary electron-atom collisions, which have been studied fairly well by now. Metastable particles, being long-lived excited systems, absorb some of the kinetic energy of the bombarding particles (electrons, ions, fast neutral atoms), converting it into their internal excitation energy, and, thus, play an important role for the energy balance in different kinds of plasma both of natural and laboratory origin. Metastable atoms and molecules are found to be exceptionally important in gas discharges. All this makes it highly desirable that reliable and authentic data concerning the cross sections of excitation of forbidden levels, their production and decay rates, mobility and spatial distribution are available. Investigation of the production and destruction of metastable particles is a matter of fundamental interest as well, since it allows one to check different theoretical models describing electron excitation of atoms. Besides, metastable levels can be efficiently populated by cascade transitions from higher levels. Hence, a detailed investigation of the cross section for metastable-particle production as a function of energy enables one, in many cases, to extract certain information about the excitation of higher levels. Up to now most of the work on the excitation of metastable states has concerned the excitation of helium and other gases, hydrogen and some diatomic molecules. Among the metals, only the mercury atom has been investigated in some detail. For all these objects, sufficient data on the relative dependences of the excitation cross sections have been accumulated, and data on the absolute values of effective excitation cross sections are available as well. At the same time, there exist a number of experiments using electron beams with a high energy resolution which were carded out to study the effect of resonance processes in the excitation of metastable states of atoms. In recent years, investigations of angular and velocity distributions of metastable particles produced in collisions with slow electrons have also been reported. These data allow one to investigate both kinematic and dynamic singularities in the inelastic scattering process. The present review concerns only the formation of metastable atoms in binary electron-atom collisions. We have aimed to present a survey of the main directions of research of the processes of excitation of lower metastable levels at their present stage of development. Here we have deliberately left out of consideration a number of long-lived highly excited Rydberg states and excitation of the metastable autoionization states of atoms, which are beyond the scope of our review and can be discussed elsewhere. The review also covers the experimental procedures and the results obtained, as compared with the theoretical calculations, which were not sufficiently elucidated in the books published long ago by Hasted (1961) [1], McDaniel (1964) [2] and the reviews by Moiseiwitsch and Smith [3], and Bransden and McDowell [4, 5]. The second section of this review describes the experimental procedures adopted for excitation of

4

1.1. Fabrikant et al., Electron impact formation of metastable atoms

metastable states of atoms, including photoabsorption, electron energy loss spectroscopy, and the trapped electron method. Special attention is paid to the technique of intersecting monoenergetic electron beams and gas dynamical atomic beams, which enables one to study angular and velocity distributions of metastable particles. The third Section deals with the theoretical aspects of electron-atom collisions resulting in excitation of metastable levels. The perturbation and close-coupling theories, the R-matrix approach, the contribution of relativistic effects and the methods of many-body theory are considered. The threshold singularities in excitation cross sections and resonances are discussed as well. The fourth section summarizes the data on differential and integral cross sections of metastable-state formation for the hydrogen atom, the helium atom, other inert gas atoms, and some metal atoms. The fifth section deals with resonance effects, that is, the effect of formation and decay of short-lived negative ion states in the process of excitation of metastable atoms (H, He, Ne, At, Kr, Xe, Hg) during electron-atom collisions. The sixth section is concerned with the investigations of the differential (with respect to the drift angle of the target particle) cross section of the production of metastable particles. Also the data concerning the velocity dependence of metastable atoms formed by electron impact are presented.

2. Experimental methods Many experimental methods have hitherto been developed to determine cross sections for the excitation of metastable levels in electron-atom collisions. They are based on the well-known Lambert-Beer relation, which, in the case of a thin target (o"nl ~ 1) may be written as (2.1)

Is(E ) = Io(E)o"nl ,

where I s is the total current of inelasticaUy scattered electrons which have lost an energy equal to that of the metastable level excitation; I 0 is the current of incident electrons with energy E; n is the number density of the target atoms; l is the length of the interaction region; and o" is the integral (total) excitation cross section. Suppose that the experimental setup detects inelastically scattered electrons with an energy loss equal to that of the metastable level excitation and a scattering angle 0 relative to the direction of the primary beam within a solid angle do. Then the scattering cross section is defined by do" i~(E) = Io(E ) ~ nl dto .

(2.2)

It is evident that in this case the total cross sections of the level excitation may be obtained by integration of the differential cross section over all angles, o" =

ff

d--d~d o ,

o -- {0, ~0}.

(2.3)

Relations (2.1) and (2.2) hold true, however, only under the assumption of a constant electron density in different cross sections of the beam (i.e., the beam is supposed to be properly collimated)

I.I. Fabrikant et al., Electron impact formation of metastable atoms

5

and a constant atom number density in the whole collision volume. Therefore, to find the excitation cross section o- of the metastable level for the given energy E of the incident electrons, one must let the monoenergetic electron beam pass through a gas layer with constant number density and measure the total number of electrons which have lost a portion of their energy corresponding to the excitation of the metastable level. Next, on the basis of relation (2.1), one should determine the scattering cross section tr and, for each value of the electron energy, measure the angular distribution of inelastically scattered electrons involved in the metastable level excitation; then, using relations (2.2) and (2.3), one may find the excitation cross section tr. The experimentalist may also determine the excitation cross section of a metastable level following an alternative way, namely one which involves measuring the number of the metastable particles produced in the course of collisions. However, in this case, the population of metastable levels may take place either by direct transitions of the atom from the ground state into the excited state (see fig. 1) or by cascade transitions from higher levels [6]. In this case, the experiment may yield, rather than the excitation cross section o', the cross section for metastable state formation, trm, which can be presented in the following way: o"m = or + ~

(2.4)

o'i(Aim/Ai) , i

where o-is the total cross section determined from (2.1); o-i is the excitation cross section of the ith level; Aim and A i are the probabilities of spontaneous transitions from level i to level m and from level i to all the lower levels, respectively. Metastable particles can be detected: (1) by the selective absorption of light which transfers the atom from the metastable state to a higher state, or by recording the radiation accompanied by spontaneous decay of this higher excited state into the optically resolved state; (2) by the number of secondary electrons emitted from a metal or semiconductor surface when it is hit by a metastable particle; (3) by the number of positive ions or free electrons produced in a Penning ionization. Furthermore, an external electric or magnetic field may transfer the metastable particles into one of the neighboring excited states, from which a radiative transition into the ground state or another excited state may occur. The optical radiation that emerges in this case, is recorded in the usual way. All this accounts for the great variety of experimental methods for studying the processes of production of metastable particles in electron-atom collisions. Figure 2 illustrates the possible experimental methods employed in the studies of metastable-particle formation. Each of them, featured by its own specific sensitivity,

i

k

Fig. 1. Schematic diagram of atomic energy levels.

6

I.I. Fabrikant et al., Electron impact formation of raetastable atoms

~1 SECONDARYELECTRON ~[ E~LTSSION J ELECTRONENERGYLOSS --[ sPECTROSCOPY _

~h

Aln,t) Sm%

e,A÷AM, e

_I -I

I A

H I

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PENNING IONIZATION

1

,:e:'.

A* B(n,t}

~,,.e

PHOTOABSORPTION

I~%

-1

PENNINGIONIZATION I ELECTRONSPECTROSCOPY

EXCITATION TRANSFER

I

~o~.~o~

Fig. 2. Schematicoverviewof experimentalmethodsfor the studyof electronimpactformationprocessesof metastableatoms. provides certain information about the mechanism of electron-atom interaction and possesses some advantages and drawbacks. The hitherto developed specific methods and setups designed for determining the cross sections of metastable-particle formation may be divided into two classes: gas cell methods and intersecting beam methods. Moreover, as seen from fig. 2, they also can be classified as direct and indirect methods depending upon the detection method. Direct methods involve measuring the number of inelastically scattered particles or of metastable particles, while indirect methods provide only information concerning the probability of metastable level excitation, e.g. about the absorption of optical radiation by metastable particles or about the number of photons resulting from secondary collisions between metastable particles and the buffer gas atoms. Consider the most typical devices used for the investigation of metastable-particle formation processes.

1.1. Fabrikant et al., Electron impact formation of metastable atoms

7

2.1. The trapped-electron method This method was developed by Schulz [7] and employed, at first, for studies of the production cross sections of metastable helium atom states close to the excitation threshold. All the inelastically scattered electrons are trapped by the potential well, which originates from the positive potential of the cylinder M, which partially penetrates through the grid G into the collision chamber (fig. 3). The electron beam, formed in the longitudinal magnetic field, enters into the collision chamber filled with the gas under investigation. Electrons, which have lost almost all their energy for the excitation of atoms, are trapped by the potential well with depth W. Due to collisions with atoms, electrons diffuse towards the cylinder M through the grid G, while unscattered or elastically scattered (by small angles) electrons are held in the beam by the longitudinal magnetic field. Therefore, the magnitude of the current flowing to the cylinder is proportional to the total cross section of metastable level excitation. With a quasi-monochromatic electron beam or with the use of a trochoidal electron monochromator, one can investigate the excitation due to monoenergetic electrons. The drawbacks of this method include, firstly, the small energy range in which the excitation of a level (0.5-3.0eV) may be investigated, because of the contribution of the higher states to the total excitation current and, secondly, the fact that the energy distribution of the electrons is broadened due to inhomogeneity of the potential well.

2.2. Electron spectroscopy measurements of the differential excitation cross sections Electron spectrometers with sufficient angular and energy resolution of the scattered electrons are more adaptable to experimental applications. With their aid it is possible to measure the energy

6 E LEC TRON COLLECTOR

I

w"

l'--

REAM C U~J~ENT .COLLISION

MONOC HRO~ATOE

M!

I

w.

r

od

_

_

W

Fig. 3. Schematic diagram of a trapped-electron experiment and potential distribution on the electrodes.

8

1,1. Fabrikant et al., Electron impact formation of metastable atoms

dependences of differential cross sections at fixed energy loss and scattering angle, as well as the angular distributions of elastically and inelastically scattered electrons. In addition, these spectrometers are sensitive to resonances in the excitation cross section. As regards metastable states, this method is rather convenient, since, as a rule, metastable levels are sufficiently "isolated" in energy and, therefore, it is unnecessary to impose strict requirements upon the monoenergetic properties of the exciting electrons and the resolving power of the analyzers. Trajmar and Williams designed a typical device for such experiments [8]. A schematical drawing of the electron spectrometer is given in fig. 4. A hemispherical electrostatic energy analyzer and a system of electrostatic lenses are used to create a monoenergetic electron beam. The beam of neutral atoms makes an angle of 90° with the electron beam. The former is generated by an effusion source with a ratio of the channel length to its diameter of 10:1. The electron energy spread of the primary beam was from 0.05 to 0.08 eV. Electrons scattered by an angle 0 relative to the direction of the primary electron beam are analyzed with respect to their energies with the aid of a similar spherical capacitor and detected by an electron multiplier. The angular resolution of the electron spectrometer was equal to ---2°. The measurements can be carried out within the angular interval from 0° to 140° and in the energy region from 10 to 140 eV. The experiment gives the possibility to obtain the energy dependence of the differential excitation cross section at different scattering angles or to measure the angular distribution of inelastically scattered electrons (differential excitation cross sections of the level) for each energy of the incident electrons. Integration of the differential cross sections at different energies yields the total (integral) excitation cross section of the level. The advantage of this method lies in the fact that a single ACCELERATOR

ACCBLERATOR

BEAH

I II--II-IC!

DDDDDI

I!

I

0

I IF--II-IF-I OlqOOEII II I L~SB8 ~ DEC~T~mATOR BLmCTR08T£~ZC IK)NOCHROUa~OR

C0LLIN~OR ~

I IF]I

II

I

llO~ImllflL~OR Fig. 4. Typical electron spectrometer for measurement of energy loss spectra and investigationof the scattering cross section [8].

1.L Fabrikant et al., Electron impact formation of metastable atoms

9

experiment makes it possible to study excitation of both optically forbidden (metastable) and optically allowed states. However, in order to obtain the integral cross sections, one has to extrapolate the differential cross sections to large scattering angles (140°-180°), which results in an additional error. Moreover, difficulties arise in detecting the scattered electrons at small scattering angles (0-10 °) owing to a considerable background caused by the primary electron beam. Finally, noteworthy are the complications involved in measurements of the differential cross sections versus energy since the transmission of the electron lenses and of the electron analyzer depends on energy and this fact proves particularly pronounced near the level excitation threshold. When carrying out measurements by this method one has to take special care of cleaning the spectrometer surfaces. In order to avoid contamination of electrodes and insulators, it is desirable to heat the spectrometers to 100-200°C. To compensate the external magnetic field the whole setup is usually placed inside the field of two or three mutually perpendicular pairs of Helmholtz coils and, furthermore, the vacuum chamber is screened by permalloy. Trajmar et al. used three methods to determine the absolute values of excitation cross sections. The first method stems from the known relation between the generalized oscillator.strength fo,'(K) and the scattering differential cross section ( d % n / d w ) ( K ) , E,, - E o k o

f°,'(/C) = ----T--- k,"

K2 do'o,, ~

(/()'

(2.5)

where hk o and h k , are the electron momenta before and after the collision, and K is the value of the momentum transfer. The differential cross section was taken in the first Born approximation. It is known that for K--->0 the generalized oscillator strength tends to the optical oscillator strength f0,'(0), which is available for resonance transitions of most atoms. When extrapolating to zero momentum transfer at low and intermediate electron energies one meets with difficulties which become even more complicated due to considerable errors in the measurements of the electrons scattered at small angles. At high collision energies such a normalization method possesses a high accuracy. According to the second normalization method suggested by Trajmar [8], the total scattering cross section obtained by integrating the differential cross sections is compared with the data of other experiments. As was already mentioned, the main errors inherent in this method arise from the extrapolation of differential cross sections to zero angles and angles close to 180°, where limited technical facilities make it extremely difficult to measure cross sections. The third normalization method is based on comparing the differential cross section of electron scattering by the atoms under investigation with known theoretical calculations and data on elastic differential cross sections of electron scattering by helium atoms. However, this method requires both accurate knowledge of the geometrical dimensions of the atomic beams and the availability of reliable data on the ratio of the number density of helium atoms to that of the atoms under investigation in the collision region. 2.3. The method o f metastable atoms

The physical effect underlying this method is secondary electron emission from the surface of a metal or semiconductor exposed to a flux of metastable particles. The method is valid if the following

1.1. Fabrikant et al., Electron impact formation of metastable atoms

10

condition is satisfied: ET > e ¢ ,

(2.6)

where E r is the total energy of an incident particle and etp is the work function of the detector material. Since the kinetic energy of metastable particles is close to thermal (for particles in the gas volume in nozzle and effusion sources this is just the case), condition (2.6) can be written as e > e~,

(2.7)

where e is the excitation energy of the metastable level. For most metals and semiconductor compounds used as detectors for metastable particles, the work function e¢ ranges from 3 to 5 eV [9], and hence, only few metastable atoms and molecules can be detected by means of this method. However, it should be noted that thermal heating of the detector [10] makes it possible also to detect atoms featuring a low excitation energy of the metastable level (e.g. mercury). In earlier times, plates made of pure metals (Ta, Mo, Au, Pt) were used for detectors of metastable particles, but in recent years also various kinds of secondary electron multiplier and channeltrons [11, 12] with different channel shapes have been widely applied. Quite recently [13], applications of microchannel plates have been reported where the spatial distribution of excitation products can also be analyzed. Employment of electron multipliers for detectors of metastable particles allows one to greatly enhance the sensitivity of this method. However, determination of the qualitative properties (absolute fluxes of metastable particles and effective cross sections of the processes in which they participate) remains the main difficulty inasmuch as the values of the coefficients of secondary electron emission of the detector materials are strongly affected by the experimental parameters. The basic problem in determining the absolute number of metastable particles involves the necessity of ensuring the cleanness of the detecting surface, which depends upon the thickness of the gas layer adsorbed at the surface. In order to get the required cleanness of the detector surface there exist several methods, which involve, however, serious difficulties. For instance, according to the method reported by Dunning [14], the cleanness of the detector surface was achieved and maintained by continuous spray coating with cadmium vapor on the rotating detector. There also exist methods of cleaning the detecting surface by electron bombardment or thermal treatment. The scheme of the device [15] intended for measuring the cross sections for production of metastable states is given in fig. 5. It consists of an electron gun, a collision chamber and an electron collector E. Electrons are emitted by a tungsten wire F, then get accelerated until the required energy is achieved and are registered by the collector E. The collision chamber containing the gas under a pressure of 10 -2 to 10 -4 t o r r is surrounded by two concentric grids G 1 and G 2. The metallic cylinder M is mounted on an insulator in such a way that it is possible to measure a small current supplied to the cylinder. Special attention was given to obtaining a monoenergetic electron beam. For this purpose the retarding potential difference method was used together with an axial magnetic field which served to suppress the defocusing of the electron beam. Noteworthy is that a trochoidal electron monochromator can also be used efficiently in such experiments for monochromatization of the electron beam. The produced metastable particles drift from the center of the collision chamber to the cylinder M, thus causing secondary electron emission. Then the electrons get accelerated to the grid G 2 located beneath the

L I. Fabrikant et al., Electron impact formation of metastable atoms

i1

-VA-'] M

?IIIP I Fig. 5. Schematic diagram of an experimental set-up for measurement of the integral cross sections of formation of metastable states (from Schulz and Fox [15]).

positive potential. By measuring the dependence of the secondary electron current upon the energy of the incident electrons one obtains the metastable production cross section versus energy for the state considered. To determine the absolute value of the cross section one should know the secondary electron emission ratio. In order to improve the sensitivity of this technique some authors (e.g. refs. [16, 17]) suggested the method of intersecting electron and atomic beams, in which a channeltron placed at a small angle (14 °) relative to the atomic beam served as the metastable particle detector. The displacement of the metastable particle detector relative to the direction of motion of the atoms is due to the recoil effect to which the atom under investigation is exposed as a result of momentum transfer from an incident electron (this phenomenon will be described in detail later). The use of a channel electron multiplier permits a great improvement of the monoenergetic properties of the exciting electrons; in the latest experiments an energy spread of 0.02 eV (FWI-IM) was achieved.

2.4. The laser fluorescence method This method is a modern version of the absorption technique, which was employed as early as in the thirties for studies of the excitation of metastable levels in inert gases. The method is based on the phenomenon that an atom in a metastable state absorbs radiation with a definite wavelength. For all this, it is possible either to measure the radiation attenuation corresponding to an atomic transition from the metastable state to a higher state or to detect the radiation emitted by atoms in optically allowed levels populated by optical pumping from the metastable level. At present, the efficiency of this method is connected with the availability of sufficiently powerful lasers tunable in a wide spectral range. The schematic drawing of this method and the experimental setup reported in ref. [18] are given in fig. 6. The cross section for the metastable ls 3 state of the neon atom was measured by Phillips et al. [18] in the following way. Neon atoms are transferred from the ground state into the excited lS 3 state by electron collision; then radiation with "wavelength A = 616.4nm from a tunable laser transfers metastable atoms into the higher 2p2 state, from which the emission (2pz-ls2) with wavelength h = 659.9 nm was recorded by a photoelectronic device. As shown in ref. [18], the intensity variation of the emission due to the 2p2-1S 2 transition which results from tuning the laser wavelength away from the l s 3 - 2 p 2 transition is proportional to the excitation cross section of the metastable level. This procedure makes it

12

1.1. Fabrikant et al., Electron impact formation of metastable atoms N e ENERGY LEVEI~

6~99x~ ..i, ° (6~64~)

I I

'So

Fig. 6. Schematic diagram of the laser-induced fluorescence method and the energy level diagram of the lowest neon levels [18].

possible to determine the absolute value of cross sections for metastable state production, including the contribution due to cascade transitions, and to resolve the resonance structure in the excitation cross sections. Nevertheless, this method is fraught with complications caused by the use of bulky optical machinery and by the low efficiency for the detection of the emerging emission. A modification of the method described above was suggested by Stebbings et al. [19] to measure the production cross sections for metastable H(2 2S1/2) atoms. In this case the metastable hydrogen atoms were detected by recording the emission of the Lyman-alpha line which emerged during the quenching of H(2 2S,/2) atoms within the field of an electrostatic capacitor. This method makes it possible to study the energy and angular distributions of metastable atoms with high accuracy. However, the method possesses some drawbacks, including a large background signal due to the optical decay of allowed energy levels of hydrogen atoms (especially, the resonance signal) and certain limitations to its applicability since one needs optical devices operating in a specific spectral range. In addition, this method is confined to a small number of objects because of the considerable energy splitting between the resonance and metastable states, which requires application of electric fields actually inaccessible to experiment today. 2.5. Use of atom-atom collisions as an instrument for the detection of metastable atoms

The energy transfer processes in atom-atom and atom-molecule collisions exhibit high values of effective cross sections, comparable with the gas-kinetic ones. The effect of these processes becomes especially pronounced in mixtures [20, 180,229]. Collisions of an excited A* atom with a B atom, the latter being in the ground state, may result in the following reactions: A* + B ~ A 0 + B* +- AE----~A 0 + B o + hv 1 +- A E , +

A*+B~,m-Ao+B+*+e_+AE--->A0+Bo + e + h v 2.

(2.8) (2.9)

With a binary mixture of A and B atoms excited by an electron beam under single-collision conditions of the atom-beam interaction, the processes (2.8) and (2.9) may take place if the A and B atoms have similar energy levels. Suppose that state I of the A* atom possesses a large cross section for

1~I. Fabrikant et al., Electron impact formation of metastable atoms

13

electron excitation and state K of the B atom is connected by a radiative transition l/Kiwith a lower state i. In general, there exists an energy gap AE between the levels I and K. Then, the energy transfer process in inelastic collisions according to (2.8) and (2.9) may lead to additional population of the group of levels of the B atoms. The probability for this process increases with decreasing AE. Thus, provided that component B be present as a small impurity and the cross section of electron excitation of the level be small, processes (2.8) and (2.9) proceed predominantly in one direction (that of the upper arrow). This enables one to study the excitation energy transfer under excitation by an electron beam with fixed energy measuring the emission of the line Pi~iof the B atoms. On the other hand, in the case of single-scattering electron collisions with an A atom, the number density of A atoms in the excited state I depends upon the electron energy and is proportional to the excitation function of the level I. 2.6. The metastable spectroscopy method An original technique for studies of metastable states was suggested by Zavilopulo et al. [21]. It stems from the method of (perpendicularly) intersecting monoenergetic electron and gas-dynamical molecular beams and involves detection of the spatial distribution of the metastable particles produced. A specific feature of such a beam interaction pattern is the fact that the target cannot be considered as being at rest. The momentum transferred from bombarding electrons to neutral atoms results in a recoil of the latter, leading to a shift of the initial trajectory by an angle X (this angle was called the drift angle or angle of observation in ref. [21], see fig. 7) and affects the angular distribution of metastable particles as compared to that of the initial atomic beam. The angular range depends on such factors as the geometry of the beams, the velocity of the colliding particles, the excitation potentials, and the mass of the target atoms. The drift angle is an important parameter for describing the process of excitation. The conceptual description of this experimental technique is given in fig. 7. An intense beam of inert gas atoms was created by the gas-dynamical sources of the molecular beam (nozzle 1 and slits 2, 3 in fig. 7) and intersected, under fight angles, with the monoenergetic electron beam ejected from a 127° cylindric electrostatic monochromator (4). The metastable atoms produced in the beam interaction were detected in the beam intersection plane with the aid of a channel electron multiplier (6) mounted on a pivoted platform, with its rotation center located at the beam intersection. The high intensity of the atomic beam (~1015 cm -2 S-l), its negligible energy straggling (<0.1 eV), the small spatial divergence (not more than 1°) as well as the good energy resolution of the exciting electrons (AE = 0.08 eV, current ~10 -8 A) enable one to study thoroughly both the angular distribu-

,J Fig. 7. Schematic diagram of the metastable spectroscopy method [21]. 1, nozzle; 2, skimmer; 3, collimation slit; 4, 127° electron monochromator; 5, Faraday cup; 6, channeltron.

I.I. Fabrikant et al., Electron impact formation of metastable atoms

14

tions of metastable particles and the excitation cross sections as functions of the energy at different observation angles. The angular resolution fl of the metastable-particle detector (see fig. 7) amounted to 0.1 °, while the rotation angle g of the detector with respect to the neutral beam direction could be varied from -10 ° to +30 ° and was set with accuracy of +_0.2°. The zero angle of the detector readout was determined with the aid of a laser or mercury lamp, whose emission passed through all forming slits of the gas-dynamical source of the molecular beam and hit the detector. The metastable-pa~ticle recording setup consisted of the channel electron multiplier (6), preamplifier, amplifier-discriminator, shaper and multichannel pulse analyzer designed for legitimate signal accumulation. The measuring procedure starts with the determination of the range of angles )t' for each kind of atom. Then, the excitation function of the metastable state of the atom under consideration is measured for selected fixed angles X- It should be noted that the excitation functions obtained in such a way are, in fact, the energy dependences of the metastable production cross sections for fixed observation angle, and further integration over all angles g permits one to obtain the total (integral) cross section of the process trm. Allowing for the fact that for all inert gasses the drift angles of the metastable atoms lie within a rather narrow region limited by the drift angles at which the differential cross section is different from zero, there is no need to extrapolate the differential cross sections as in the case of scattered electrons. The experiment [21] showed that in the plane perpendicular to that of the beam intersection, metastabie atoms were observed within a narrow range of angles ( - 2 °) and thus almost all of them were inside the angular span of the detector. This allows one to calculate the total (integral) cross section of metastable particle production trm(E ) as Xmax

trm(E) = f

dtr.,(E) dx dx ,

(2.10)

groin

where dtrm(E )/dx is the differential (with respect to the drift angle) cross section of metastable particle production. As will be shown in section 6, in the vicinity of the threshold there is a one-to,one correspondence between dtrm/d X and the differential cross section of excitation of the level dtr/dx, although, with increasing electron energy, a certain contribution to dtrm/d X arises due to cascade transitions. Therefore, measurements of the differential cross sections with respect to the drift angle make it possible to evaluate, in principle, the differential cross sections dtr/dto (and vice versa), although this is a pretty laborious task. It is natural that, in order to obtain the absolute values of the excitation cross section by using the method of metastable spectroscopy, it is necessary, just as with other methods, to allow for the efficiency with which metastable atoms are detected by the channel electron multiplier, which changes with atoms and states. In this connection, the experimental results should be referred to the total production of helium atoms in the triplet and singlet states and, for other inert gases, to that of atoms in the n 3P2 and n 3P0 states, since the separation of these states involves many difficulties. The basic parameters of the setup [21] and the experimental conditions are given in table 1. A considerable amount of data, e.g. the differential cross section for excitation of a metastable level in the vicinity of the threshold, may be obtained by supplementing the procedure described above with measurements of the time-of-flight (TOF) spectra of metastable particles at different drift angles [22]. Recording of the TOF spectra is achieved with short-pulse modulation of the electron beam, which

L I. Fabrikant et al., Electron impact formation of metastable atoms

15

Table 1 Neutral atomic beam characteristics [21] Atom

Mass (au)

Intensity (1015 cm -2 s -1)

Density (101° cm -3)

Energy (10 -2 eV)

Energy spread (10 -2 eV)

Angular divergence (deg)

He Ne Ar Kr Xe

4 20 40 84 131

46 4.9 1.6 1.0 0.3

27.0 6.20 2.77 2.60 1.38

12.0 6.5 6.9 6.4 6.2

6.0 0.70 0.80 0.60 0.53

1.0 1.0 1.0 1.0 1.0

gives rise to the production of packets of metastable atoms. The velocity distribution of metastable atoms can be deduced from the time of flight, corresponding to a fixed distance between the beam intersection point and the detector, measured at different delays between the electron current pulses and the gate pulse which triggers the recording channel of the setup.

3. Theoretical description of electron impact excitation of metastable levels In principle, to calculate the cross sections for electron impact excitation of metastable levels one may use all conventional methods of electron-atom collision theory, which have been properly considered in monographs and reviews (see, for instance, refs. [4, 23-25]). However, excitation of metastable states has some specific features, which we will dwell upon without entering into details of the calculation procedure. First of all, it should be noted that, as we have already mentioned, electron-atom collisions may result both in direct population of metastable levels and in cascade processes occurring through the radiative decay of higher states. Usually cascades, especially close to the excitation threshold, give a small contribution to the cross section. This contribution can be evaluated by estimating, e.g. in the framework of the Born approximation, the excitation cross section of a higher state and the relative probability of its radiative decay into the metastable state. Some relevant examples of such a contribution will be considered in section 4. In the present section we will touch upon the problems referring to the direct excitation theory. 3.1. Perturbation methods

So far as metastable levels are optically forbidden, their electron-induced excitation cannot take place due to the long-range dipole interaction but only due to the relatively weak exchange or quadrupole interaction. Exchange interaction is important if the transition is accompanied by changes of the atomic state multiplicity, whilst quadrupole interaction is considerable provided the multiplicity and the state parity remain constant (in the latter case the exchange interaction may be no less significant in the vicinity of the threshold). As a result of the relative weakness of exchange and quadrupole interactions, their contribution to the excitation process can theoretically be considered within the framework of perturbation theory. However, its simplest modifications (Born and Born-Oppenheimer approximations) are usually not appropriate for this case, since the electron-atom interaction in the initial and final states may be rather strong and the approximation of plane waves may be unsuitable as well. Nevertheless, these approxima-

16

1.I. Fabrikant et al., Electron impact formation of metastable atoms

tions, in virtue of their relative simplicity, were widely used in studies of the excitation cross sections of metastable states, especially at high energies of the incident electrons when these approximations are more valid. In the event of transitions with changing multiplicity the Ochkur approximation [26] and the first-order exchange approximation [3] give better results than the Born-Oppenheimer approximation. In the case of quadrupole excitation (the most characteristic example being the excitation of metastable 1D levels in Ca, Sr, and Ba atoms), it is more expedient to employ the unitary Born approximation [27]. The data on a number of transitions, calculated in this approximation, were reported by Vainshtein et al. [28]. Application of the second Born approximation is of academic interest since for practical purposes it is more expedient to use more accurate methods. The second Born approximation was employed in studies of excitation of metastable states in hydrogen and helium atoms [29-31]. Burkova and Ochkur [32] obtained a simplified formula of the second Born approximation, which happened to be useful in studies of metastable helium excitation at intermediate energies. The distorted-wave method, which takes the electron-atom interaction in the initial and final states into account, is more consistent for the investigation of metastable state excitation. The simplest and most common version of this method takes only the static field of the atom into account (Hartree-Fock distorted-wave method). However, atomic polarization induced by an incident electron may be rather considerable and then taking only the static field into account may be insufficient, especially in the final (excited) state. The difficulties of constructing the polarization (or optical) potential, which is a nonlocal and energy-dependent operator, have led to the development of quite a few approximate methods which include the atomic polarization induced by an incident electron. One of the simplest is the Temkin method of polarized orbitals [33], which, together with the distorted-wave approximation (adiabaticexchange distorted-wave method), was used in studies of excitation of metastable states in Cu and Si [34]. Another method of constructing the optical potential will be outlined in section 3.2.

3.2. The close-coupling method and its modifications The main idea of this method is well known (see, for instance, refs. [3, 23-25]), and we are not going to dwell upon it here. The discussion presented in section 3.1 shows that in applications to the problem of excitation of metastable states, which concern us here, it is most expedient to employ such a version of the method that would account for the coupling between the initial and the final state and for their coupling with all the states that give the main contributions to the polarization of the former and the latter. However, realization of this calculational procedure may lead to considerable difficulties, technical (if one must allow for a great number of discrete levels) and fundamental (if one must allow for the continuous spectrum). For this purpose, two modifications of the close-coupling method have been created: the method of pseudostates and that of pseudopotentials. It is noteworthy that hitherto both methods have been applied almost exclusively to considerations of excitation in the simplest systems, i.e. in H and He atoms. The pseudostate method was first suggested by Damburg and Karule [35]. They introduced the 2p-function of the hydrogen atom, which provided the full atomic polarizability. Since then, the method was repeatedly extended and modified in order to obtain better allowance for the correlation effects and extend it to the energy region above the ionization threshold. In the latter case, it becomes necessary to introduce pseudostates whose energies lie within the continuous spectrum, and consequently, nonphysical resonances emerge above the ionization threshold [36]. Burke et al. [37] proposed a procedure

1.1. Fabrikant et al., Electron impact formation of metastable atoms

17

which eliminated these resonances with the aid of the rational approximation of the scattering amplitude. One of the latest applications of this method has been made by Callaway [38] in the study of scattering on the hydrogen atom (including excitation of a metastable state); these results will be discussed in section 4. The pseudopotential method was rather completely described by Bransden and McDowell [4]. The explicit expression for pseudopotentials is obtained by exclusion of the coupling between the main channels (in our case these are the channels of scattering from the ground and metastable states) and the additional ones--by the use of the Green function. To simplify the expression for the pseudopotential further approximations are used, e.g. the closure procedure. We will not discuss here other modifications of the close-coupling method (method of pseudopotentials in momentum space, quasiclassical close-coupling method, exchange in the local approximation, etc.), which were also described by Bransden and McDowell [4] and have no specific features in their applications to the problem of metastable state excitation. The studies involving applications of these methods to the problem of excitation of the metastable states of H and He will be mentioned in section 4. The equations derived in the framework of the close-coupling approach can be properly solved by means of the variational technique. Regarding this subject the reader is referred to the reviews by Callaway [39] and Nesbet [40]. 3.3. The R-matrix method The R-matrix method was introduced into the theory of electron-atom collisions by Burke et al. [41]. It employs the close-coupling expansion in the external region of the configuration space and adopts the calculational procedures intended for the bound states in the internal region. The R-matrix method is especially efficient in the region of small energies, where one can retain just a few pole terms in the R-matrix expansion. This fact becomes especially important for the investigation of the resonance structure of the metastable state excitation cross section near threshold. So far, theoretical investigations of the resonance structure in the excitation of metastable states of inert gas atoms have been carried out exclusively by the R-matrix method [42-45]. The most complete results for the helium atom have also been obtained by this method [46, 47]. 3.4. Taking relativistic effects into account With increasing atomic number, relativistic effects become considerable, especially those of the spin-orbit interaction in the atom. Taking them into account when considering metastable states is particularly important if the initial and final states exhibit a fine structure. In this case, several approaches are feasible. Provided the effects of the spin-orbit interaction are small (e.g. for light atoms), the collision dynamics may be treated in the framework of LS-coupling, and then, according to Saraph [48], one may transform the K- or T-matrix to the ]j-coupling representation with the aid of recoupling coefficients. To allow for fine-structure effects one may introduce the electron energy shift i n the final state corresponding to the fine-structure splitting. Such a calculation was carried out by Le Dourneuf and Nesbet [49] for transitions between the fine-structure sublevels in oxygen atoms. The effects of spin-orbit interaction can be taken into account more consistently by introducing atomic wave functions at intermediate coupling (see, for instance, the calculation of excitation in the Hg atom by McConnell and Moiseiwitsch [50]). All relativistic effects in not very heavy atoms can be

18

I.I. Fabrikant et al., Electron impact formation of metastable atoms

accounted for in the framework of the Breit-Pauli Hamiltonian. This method was first applied to electron-atom collisions by Jones [51], while Scott and Burke [52] advanced the R-matrix formulation of the method to solve this problem. The excitation cross sections of metastable levels in Hg [53,321,322], TI [54] and Pb [55] were calculated just in this way. This method was also employed in studies of the fine-structure effects involved in elastic scattering of electrons by Ar [56], but results concerning the metastable-state excitation of Ar have not yet been obtained.

3.5. Methods of many-body theory: the Bethe-Goldstone equations The accuracy of the close-coupling method and of the R-matrix method is limited because one cannot take into account an infinite number of states (or pseudostates) of the target atom. This difficulty can be surmounted by solving the Bethe-Goldstone two-particle equation for the system consisting of the incident electron plus that of the atom embedded in the Fermi sea of other atomic electrons. Such an approach was suggested by Mittleman [57] and Nesbet [58]. Later Nesbet [25] developed a variational procedure for solving the Bethe-Goldstone equation for electron-atom scattering' and generalized this technique to solve the hierarchy of the Bethe-Goldstone equations of the third, fourth and in general, nth order. In the two-particle approximation the contribution of various atomic subshells to the polarization caused by the bombarding electron is of an additive nature: this is an inherent property of the polarized-orbital method and the close-coupling approach where pseudostates are used. However, even in this approximation, the Bethe-Goldstone equation provides a more accurate description of the short-range correlations. It should be noted that, when being actually implemented in practice, the method of solving the Bethe-Goldstone equation involves the same difficulties as the dose-coupling method, which are caused by the limited basis of trial functions. Therefore, the results obtained in particular for excitation of metastable states of He [59] and C, N, O [60-62] have roughly the same accuracy as those of the R-matrix calculations and of the pseudostate method.

3.6. Methods of many-body theory: Green function and diagram technique Solution of the Bethe-Goldstone equations corresponds to the accurate treatment of the two-particle (three-particle etc.) interaction in the many-body theory. As we have seen, excitation of metastable states often occurs as a result of a relatively weak electron-atom interaction. In this case, one can use the perturbation technique which, in the framework of the many-body approach, can be formulated in terms of Feynman diagrams. To formulate the method, it is convenient to use the field-theoretic Green's functions. Such an approach to inelastic scattering was suggested by Csanak et al. [63]. Derivation of a dosed set of equations for the multi-particle Green functions and the self-energy operator inevitably involves certain approximations. The simplest way is to introduce the first-order approximation for the double-point vertex function [63]. As was shown by Csanak et al., this is equivalent to the random phase approximation. From the standpoint of the methods discussed above, the first-order approximation of the method of many-body theory is a version of the distorted-wave method. It was employed to calculate the excitation of metastable states in He [64], Ar [65], Kr [66], Ne [67] and Mg [68]. Amusia et al. [69] also investigated the effect of the second-order diagrams upon the differential cross sections of excitation of the 2 3S-state in He and showed that the second-order effects considerably improve agreement with experiment, especially for small scattering angles.

L I. Fabrikant et al., Electron impact formation of metastable atoms

19

3. 7. Resonances and threshold structure Irregular behavior of the excitation cross sections as a function of energy usually results from two effectshresonances and threshold effects. As a rule, resonances give rise to a more pronounced structure. Their traditional theoretical description provided by common collision methods amounts to the calculation of cross sections performed in relatively small energy steps in order to obtain the resonance curve. In should be stressed that not all of the methods mentioned above are capable of predicting the resonance structure, but only those whose trial electron-atom wave function allows for the formation of negative ions. Such methods include the close-coupling method and its modifications, the R-matrix method, the Bethe-Goldstone method, and the versions of many-body theory in which a certain sequence of diagrams is summed. Resonances are usually observed either under the excitation threshold of some excited state of the target atom or above it [70]. In the former case, they are either the core-excited Feshbach resonances or those provided by a virtual state of the excited atom-electron system. The structure above the threshold is usually attributed to the shape resonances. One should note that, in the presence of two close excitation thresholds, it is sometimes difficult to classify the resonance occurring between them. For a long time, in particular, there was an opinion that the 2p-resonance observed near the excitation threshold of the 2 3S-state of the He atom is a shape resonance due to the presence of the 2 3S-threshold. However, calculations [71-73] have shown that the 2p-resonance is of the Feshbach-type supported by the 2 1S-threshold. In order to ascertain such facts the methods of collision theory are insufficient. Therefore, when calculating the resonances, the methods most widely employed are similar to those applied to calculations of bound states, in particular, the stabilization method [74, 307] and that of the complex rotation of coordinates [75]. Recently intense experimental and theoretical research has been devoted to the Feshbach resonances associated with the motion of two highly excited electrons in the field of a grandparent ion with charge + 1. This motion is highly correlated and the independent-particle model fails to provide its description. Fano [76] described this motion starting from the model of the Wannier potential ridge, while Herrick et al. [77, 78] did the same on the basis of the 0(4) symmetry. As was demonstrated by Read [79], the position of the resonances caused by these states is properly described by the generalized Rydberg formula. The application of this formula to a concrete system will be discussed in section 5. In contrast to resonances, threshold singularities do not give rise to a sharp structure in the vicinity of the threshold but appear as irregularities (cusps) in the cross section versus energy curve. These irregularities may be observed both in the excitation threshold of the state under investigation and the thresholds of higher states. The multichannel effective range theory [81] predicts that, if only short-range forces act between the escaping electron with angular momentum I and the atom, then the partial excitation cross section obeys the following law: o't = aK2t+1[1 + brloK + O(K2)],

(3.1)

where a, b are some constant coefficients, and K is the wave number of the escaping electron. Thus, the cross section as a function of energy has a branch point. Analysis of the behavior of the cross section near the excitation threshold of a higher atomic state based on formula (3.1) aswell as on analyticity and unitarity leads to the conclusion that there should

20

1.1. Fabrikant et al., Electron impact formation of metastable atoms

be a cusp in this threshold, i.e. the Wigner [82]-Baz' [83] singularity. These singularities were experimentally observed by Huetz et al. [84] in the excitation cross section of the 2 3S-state of the He atom near the 2 IS-, 2 3 p . and 2 ~P-thresholds. Now consider the effects of the long-range forces upon resonances and threshold singularities. If the excitation is caused by a dipole interaction, then, according to Damburg [85] and Gailitis [86], the correction term in the brackets in formula (3.1) is of the order of K 2 In K. Therefore, in an excitation of the metastable state due to a quadrupole, that is, even more short-range interaction, the correction term becomes negligible and the main effect upon the threshold behavior comes from the interaction in the final state. Most important is the case of excitation of the metastable state of a hydrogen atom when the degeneracy of the 2s- and 2p-levels gives rise to an effective electron-dipole interaction in the final state and a finite value of the cross section at the threshold [87]. In experiments, however, the finiteness of the excitation cross section of the 2s-state is not observed directly at the threshold; therefore, in comparing with the data one must average the theoretical results over the energy distribution of the electrons in the beam. Pioneering studies of this kind were carried out by Chamberlain et al. [88]. Quadrupole interaction in the final state does not result in such notable changes, but it can also give rise to an interesting effect. Let, for instance, the metastable P-state be excited. Then, the electron angular momentum may assume the values l = L - 1 and l = L + 1 if ~r = (-1) L, or l = L if ~-= (-1) L-~, where L and rr are the total angular momentum and total parity of the atom-electron system. Consider the first case. According to (3.1) o-L_1 ~ K 2L-1 ,

(3.2)

orL+1 oc K 2L+3 .

(3.3).

However, Fabrikant [89] showed that due to the presence of a quadrupole interaction between the atom and the escaping electron we have instead of (3.3) O-L+ 1 o~ K 2L+1 ,

and the ratio o"L+l/crL_l depends only upon L and the quadrupole momentum of the atom. By analogy, if the D-state is excited, we have OrL_ 2 oc K 2 L - 3 ,

orL oc K 2 L - 1 ,

OrL_ 1 oc K 2 L - 1 ,

O-L+ 1 oc K 2 L + l ,

OrL+ 2 oc g 2 L - 1 ,

f o r ~r = ( - 1 ) L ,

and for 1r =

(-1) L-1 ,

i.e. the Wigner law [82] crl oc K 2t+1 fails for waves with angular momenta exceeding the minimal one at given L, It. Usually all the threshold laws considered above are of pretty narrow applicability because of the presence of resonance structures close to the threshold and the large electron-atom interaction radius. Quite a few efforts were taken to develop a wider theory of the threshold behavior (e.g. refs. [90, 91]), but all of them involved a considerable increase in the number of parameters entering the formulae of the threshold behavior, which must be extracted either from ab initio calculations or from experimental data.

I.I. Fabrikant et al., Electron impact formation of metastable atoms

21

Long-range polarization often affects, in the most complicated way, the energy dependence of the excitation cross section close to the threshold, since atoms in excited states usually exhibit a high polarization. The presence of a virtual or resonance level close to the threshold may also give rise to a considerable effect.

4. Experimental and theoretical results

In this section we shall consider only the latest studies which were carried out with good energy and angular resolution and under "pure" experimental conditions. The reader who is interested in the earlier research or in more detailed information on some specific problems is referred to the corresponding reviews, monographs and papers, refs. [1-6]. In our consideration of the basic points concerning excitation of metastable levels we skip completely such issues as the fine structure of cross sections and excitation functions, autoionization, Rydberg states and so forth. We did not think it necessary to overstock our review, already rich in factual data, with extra material and left the above-mentioned aspects beyond the scope of our consideration. As regards such important effects as the formation of states of negative atomic ions and their contribution to the excitation cross sections of metastable levels, we shall dwell upon this subject in section 5. 4.1. The hydrogen atom

The lifetime of the 2s-state of the hydrogen atom is 0.142 s [92]. Population of this state due to collisions with electrons comes from direct excitation of the 2s-state and from the excitation of np-states (n >- 3) followed by cascade transitions to the 2s-level. Hummer and Seaton [93] (see also ref. [94]) have shown that the contribution due to the cascade transitions roughly amounts to 0.23 tr3p and that the cross section of metastable atomic hydrogen formation is equal to o'~ = o-2s+ 0.23 o'3v.

(4.1)

Regardless of the fact that from the theoretical point of view the hydrogen atom is the simplest system possible, there still exist some disparities between the theoretical and experimental values of the cross section O'2s[95] though in general the agreement is rather good. The recent theoretical calculations [38, 96] of the excitation cross section of the 2s-level include the ls-, 2s-, 2p- and 3d-states of the target and seven pseudostates, three of which lie in the discrete part of the spectrum close to the ionization threshold, while the remaining four states are in the continuum. To eliminate the emerging pseudoresonances a polynomial approximation for the transition amplitude was employed [37]. According to Williams [98], the experimental data [97] should be multiplied by the factor 1.1. New results obtained by Williams [99] for o-2sare compared with calculations [38, 96] in fig. 8. In the range of energies E > 12 eV the results reported in ref. [96] are properly approximated by the following formula:

l a,

O'2s ~" " ~

i=1

xi-l'

(4.2)

22

I.I. Fabrikant et al., Electron impact formation of metastable atoms

0.40 0,30 t'~t o

~D

010

0.75

I

0179

I

I

0.83 k2lRydl

Fig. 8. H(2s) excitation cross section [38]. Solid curve, calculation of Callaway [38]. Circles, experimental data of Williams [99].

where x = 4K2; the coefficients a i were determined by Callaway [96] by a fit to the calculated cross section at 12 eV < E < 54 eV, the high-energy Born asymptotic behavior being either taken into account (constrained method) or not (unconstrained method). These coefficients are given in table 2. In addition to the data on the total excitation cross sections of the 2s-level there are also experimental results obtained by Williams [100] on the differential cross sections at E = 54.4 eV, which agree extremely well with the calculations of Edmundus et al. [101]; these calculations were carried out in the ten-states (n = 1, 2, 3, 4) coupling approximation and allowed for the exchange in the local approximation [102]. A calculation performed in the three-states (ls, 2s, 2p) coupling approximation taking exchange into account [103] also yielded good results at this energy. Edmundus et al. [101] calculated the differential cross sections at E = 35 eV as well. Callaway [104] presented the differential cross sections at E = 15 eV obtained in the same approximation as in refs. [37, 96]. As regards the absolute values of the total excitation cross sections of the 2s-level in atomic H, there exist a number of experiments in which these values were determined by different methods. For instance, KaupiUa et al. [97] extracted the energy dependence of the total cross section of formation of the 2s-level upon the threshold ( E = 10.20 eV-1000 eV) by measuring the ratio of the excitation cross sections of the 2s-level to that of the optically allowed 2p-level. The maximum of the cross section was achieved at E = 11.6 --- 0.2 eV and amounted to (0.168 +--0.20)~'a02. Hills et al. [105], by normalizing the measured cross section dependence to that calculated in the Born approximation at 500 eV, obtained a value of 0.11~'a 2 for the cross section at its maximum as a function of energy. However, as was shown Table 2 Coefficients a i of eq. (4.2), data of Callaway [96]

1 2 3 4 5 6

Constrained

Unconstrained

0.88789 -2.07216 -0.56183 11.74003 - 17.63091 8.07849

0.96999 -3.08712 4.12630 1.57342 -7.20537 4.00358

I.I. Fabrikant et al., Electron impactformation of metastable atoms

23

by Damburg and Propin [106], the normalization to the results of the Born approximation must be handled with care. A similar magnitude of the cross section was obtained by Stebbings et al. [19] by measuring the ratio of the excitation cross section of the 2s-level to the Lyman-alpha line. Later Lichten [107] pointed out an error in ref. [19], which involved incomplete registration of the angular distribution of the emitted photons. The refined value of the cross section is 0.167ra 2, which is in good agreement with the data of Lichten and Schulz [94]. Much theoretical consideration has been devoted to the excitation of H (2s) in the region of intermediate incident electron energies (50 eV ~
Theoretical approach

Energy (eV) for which the differential cross section has been calculated

[30]

Second Born approximation

54.4

[108]

ls-2s-2p close-coupling + polarization potential, built from pseudostates

54, 100, 200

[109, 111]

ls-2s-2p momentum space dose-coupling + polarization potential in diagonal approximation

54.4, 100, 200

[110]

The same as refs. [109, 111] + 3s-3p-3d

54.4

[112]

The same as refs. [109, 111] with nonadiabatic polarization potential

54.4

[113]

Is-2s-2p close-coupling + second Born approximation with closure

54.4, 100,200,300,400,500,680

[116]

Comparison of models from ref. [108] and refs. [109-112]

54.4, 200

[117, 118]

Glauber exchange approximation

100,200,300

[119]

Modified Glauber approximation

Total cross sections from threshold up to 200

[120]

Distorted-wave polarized orbital method

13.88, 16.46, 19.0, 50.0, 54.4,100

[121]

Two-potential eikonal approximation

54,100

[122]

Unitarized eikonal-Born series

100, 200, 300, 400

[123]

Faddeev-Watson approximation

54.4, 100

[124]

Second-order eikonal approximation

54.4, 100,200

[125]

Asymptotic Green function approximation

54.4

[126]

Asymptotic Green function approximation

Total cross sections from threshold up to 24 Ry

[127]

Close-coupling method including n = 2 and n = 3 levels

13.6, 19.6, 30.6, 54.4

[1281

Momentum space close-coupling

54.4

[1291

Pseudostate + second distorted-wave Born approximation

(00 < 0 < 30*)

24

1.1. Fabrikant et al., Electron impact formation of metastable atoms I

I

I

I

I

.!.., 1J 1D

16

0

4

i

8

I

12

~(Ryd)

i

16

J

20

24

Fig. 9. ls-2s cross sections for electron scattering from H [126]. - - . - --, asymptotic Green function approximation, basis set ls-2s-2pm; - variable-charge Coulomb-projected Born approximation (Schaub-Schaver and Stauffer); . . . . , calculation of McDowell et al. (see ref. [126])i . . . . , close-coupling calculation, Burke et al. (see ref. [126]); O, experimental data of Hills et al. [105]; O, experimental data of KaupiUa et al. [97].

is taken into account by the close-coupling method while the remaining part of the discrete spectrum and the continuum are considered with the aid of optical (polarization) potentials. In ref. [108] the close-coupling equations are solved in coordinate space while the polarization potentials are constructed in terms of the wave functions of the target pseudostates. In refs. [109-112] the close-coupling equations are solved in momentum space and the polarization potentials are calculated approximately with the aid of Green's functions. Considerable simplification when forming the polarization potentials may be achieved through the closure procedure [113]. Ermolayev and Waiters [30] showed that this procedure did not introduce a considerable error into calculations of the excitation differential cross section of the 2s-level. Recently Van Wyngaarden and Waiters [114] have analyzed the sources of disagreement between the results on the excitation cross sections obtained in the framework of the pseudostates approach in three studies [96, 115,116] for 2s- and 2p-states at E = 54.4 eV. It was shown that in Callaway's calculations [96] there were two considerable inaccuracies, i.e., the d-pseudostates for the total angular momenta L = 4, 5 and the coupling with the states with l > 2 were neglected. However, the corrections due to both effects are of approximately equal magnitude and of opposite sign, thus, in the end the result obtained by Van Wyngaarden and Waiters [115] is almost the same as that reported by Callaway [96]. 4.2. The helium atom

In our discussion of excitation of the metastable states of helium, as in the case of hydrogen, main emphasis will be laid upon the near-threshold energy region. As regards the intermediate energy region we will just touch upon the works not included in the review of ref. [5]. The metastable states of the helium atom are 2 3S (lifetime ~-- 104 s) and 21S (~-- 2 x 10 -2 s). These

I.L Fabrikant et al., Electron impact formation of metastable atoms

25

levels become populated as a result of direct excitation by electrons and cascade transitions from higher levels. The latter mechanism is especially important when the intermediate state is a triplet. Indeed, in the case of a singlet level, the most probable process of radiation decay occurs either via a direct transition to the ground state or via the 2 mp-level but not via the 2 ~S-level. Thus, o-m= o-(1 ~S---~2 ~S) + ~ o-(1 15.---->n 3L),

(4.3)

n,L

where o'(1 1S---~n 3L) is the electron excitation cross section of n 3L-levels. As was shown in refs. [130, 131], the most significant contribution to the sum over triplet levels is due to tr(1 1S~ 2 3S) and tr(1 IS----~23p), while other terms in the sum may be neglected, as was shown by Zapesochny [132].

4.2. I. Experimental results Though the number of experimental studies on the differential excitation cross sections of metastable levels in He is pretty impressive, only few of them provide detailed information especially concerning the absolute values of these cross sections. A review of the basic relevant works may be found in the reviews of Trajmar et al. [133] and Bransden and McDowell [5]. Here we will dwell upon the most important efforts, as we see them, to measure the absolute values of the cross section for metastable particle production. The experiments carried out by Trajmar [134] and Hall et al. [135] are seemingly the first endeavors of this kind. These experiments involved the electron spectroscopy method and resulted in the differential cross sections of the 2 3S- and 2 1S-levels at electron energies of 29.2, 39.2, 48 eV [135] and 29.6, 40.1 eV [134] in the angular range 0 = 10-125 ° (3-138 ° [134]). The absolute values of the cross sections were determined by normalization to the known cross sections of elastic scattering [135] and excitation of the 2 IP-level [134]. Table 4 presents the absolute values of the cross sections for the 2 3Sand 2 1S-levels taken from refs. [134, 135]. The accuracy of these values is between 14 and 20% according to Trajmar [134], and between 10 and 26% according to Hall et al. [135] (depending upon the angle 0). One cannot help noting excellent agreement of the results in both works, especially in the range of energies around 40eV (39.2eV [135] and 40.1 eV [134]). Hall et al. [135] did not observe the deep minimum in the 2 ~S-level cross section in the 50° range, but, in general, the behavior of the differential cross section is in an excellent agreement as regards the general shape of the curves and the absolute values. In a similar way Opal and Beaty [136] determined the differential cross sections of 2 3S-level excitation for an electron energy of E = 82 eV, while Yagishita et al. [137] did it for E = 50-500 eV. Yagishita et al. [137] found a definitely pronounced prevalence of forward scattering, while around 90° at 50 eV they observed the deep minimum first reported by Crooks et al. [138] in their experiments on excitation of the 2 3S-level in the range of scattering angles 0 from 25° to 150°. Pichou et al. [139] employed the electron spectroscopy method for measuring the excitation differential cross section of both metastable levels in the angular range of 0 = 10°-120 ° and at electron energies exceeding the threshold value by 1.6 eV, with an energy resolution of about AE = 55 meV. The absolute values of the cross sections measured in ref. [139] (see fig. 10) were found by normalizing the data to the known cross sections for elastic scattering (Andrick and Bitsch [140]). These authors also determined energy dependences of the absolute differential cross sections of both levels at energies exceeding the threshold by 3.6 eV. Figures 11 and 12 present these dependences for scattering angles O = 30°, 60°, 90° and 120°.

1.I. Fabrikant et al., Electron impact formation of metastable atoms

26

Table 4 Differential cross sections for excitation of the 2 3S- and 2 ~S-states of helium in 10-19 cln2/sr E = 30eV

E=40eV

2 3S

2 iS

2 3S

2 1S

0

Experiment c)

Theorya)

Experiment °)

Theoryb)

Experiment ~)

Theorya)

Experiment ~)

Theoryb)

(deg)

[134]

(in [134])

[1341

(in [134])

[134]

(in [134])

[134]

(in [134])

0 5 10 20 30 40 50 60 70 80 90 100 110 120 130 140 160 180

(7.57) 6.25 4.98 3.02 1.72 1.12 1.08 1.32 1.18 1.58 1.05 1.11 0.82 0.61 0.83 (1.9) (4.4) (6.1)

0.646 0.674 0.753 1.08 1.62 2.30 2.99 3.55 3.94 4.14 4.17 4.11 3.97 3.80 3.61 3.47 3.24 3.19

(42.7) 36.4 25.6 8.14 2.09 0.204 0.048 0.16 0.19 0.36 0.53 0.95 1.48 2.09 2.44 (3.9) (8.6) (12.5)

20.83 20.60 19.96 17.63 14.39 11.08 8.28 6.13 4.65 3.47 2.69 2.12 1.71 1.40 1.19 1.03 0.85 0.80

(9.39) 7.15 6.17 3.43 2.10 0.94 0.42 0.30 0.22 0.18 0.12 0.12 0.27 0.62 1.12 (2.1) (3.1) (3.8)

0.225 0.251 0.330 0.697 1.33 2.06 2.65 2.96 3.07 2.91 2.69 2.44 2.19 1.97 1.79 1.65 1.47 1.41

(72.9) 47.7 32.3 26.7 5.94 7.32 1.03 1.23 0.14 0.22 0.017 0.024 0.087 0.108 0.26 0.31 0.46 0.58 0.75 0.82 1.06 1.23 1.38 1.36 1.74 1.55 2.04 (2.7) (2,7) (2,7)

[135]

4.42 3.24 2.09 1.61 1.70 1.64 1.53 1.70 1.72 1.24 0.86 0.71

[1351

22.6 8.37 1.97 0.26 0.19 0.26 0.38 0.61 0.95 1.24 1.56 1.97

[135]

6.0 4.6 2.5 1.13 0.60 0.372 0.301 0.248 0.205 0.196 0.287 0.420

[1351

29.96 29.12 27.55 21.75 14.95 9.45 5.72 3.52 2.25 1.47 0.99 0.69 0.50 0.38 0.30 0.25 0.20 0,18

a) Ochkur-Rudge approximation. b) Born-Ochkur-Rudge approximation. c) Data in ref. [134] were measured at E = 29.6 and 40.1 eV; data in ref. [135] were measured at E = 29.2 and 39.2 eV.

8-

I

I

1

i

I

!

I

I

7 I v

I

I

I

6i 'n'~-

60* 0

I

0 20

I

I

60

I

I

I

I

I

100 140180 O (deg)

Fig. 10. Differential excitation cross sections for the 2 3S (O) and 2 ~S (©) states of helium at 1.6 eV above threshold [139].

20

21

22

23

E{eV) Fig. 11.2 3S excitation functions for helium at 0 = 30°, 60°, 90° and 120° [139].

I.I. Fabrikant et al., Electron impact formation o[ metastable atoms

27

3 2 ~---.1 E C

i

,20.18

.

9~

•o 0 1.2

I

0B

O.4 0 4 2 0

~

.:\\

sS

20.28

30 °

21

22

23

24

Fig. 12. 21S excitation functions for helium at 0 = 30 °, 60 °, 90 ° and 120" [139].

0

60

120 180 0

0(deg}

60 120 180

Fig. 13. Differential cross sections for excitation of the 2 3S-state of helium near threshold [22]. - - , experiment of Zajonc et al. [22]; . . . . , experiment of Pichou et al. [139]; - - - - . , experiment of Andrick et al. [143].

It is interesting to compare the data of ref. [139] with the experimental results of Phillips and Wong [141] obtained with higher (40 meV) energy resolution. The energy dependences of the differential cross sections at 0 = 55° and 90° exhibit good agreement, although their absolute values coincide only to an accuracy of a factor 2 [141]. At the same time their higher energy resolution gave Phillips and Wong the possibility to suggest that there might be a second shape 2p-resonance at E = 20.8 eV. In addition, in the differential cross section of the 2 3S-level they found a definitely pronounced angular dependence of the structure in the cross section, particularly of the discovered 2p-resonance (see section 5). The excitation differential cross sections for the 2 3S-level were also studied by Zajonc et al. [22, 142], who employed the TOF technique to investigate the energy range 19.9 to 20.4eV. They measured relative dependences of the differential cross section in the range of electron scattering angles 8 from 0 to 180° (fig. 13); in ref. [22] these were compared with the experimental results of Pichou et al. [139] and Andrick et al. [143] and with the calculations made by Burke et al. [144] in the framework of the close-coupling method. Trajmar [134], Hall et al. [135], and Pichou et al. [139] determined the total excitation cross sections of the 2 3S- and 2 1S-levels from the data on differential cross sections, integrating the latter over all possible angles of electron scattering. These data are summarized in table 5 together with the results of Borst [145] obtained by the direct detection of metastable atoms in both states. Table 5 also presents the data reported by other authors on partial and total excitation cross sections of the metastable levels; these data were obtained by different techniques. The data are compared with some calculated values of ~rm in fig. 14. Lately, a number of papers have appeared on measurements of the total excitation cross sections of the 2 3S- and 2 1S-levels in helium for the energy range close to the threshold. We should mention here,

I.I. Fabrikant et al., Electron impact formation of metastable atoms

28

Table 5 Integral cross sections for excitation of metastable states of the helium atom, in 10-1° cm 2 2 38

2 3'18

2 18

E

(eV)

Experiment

20

62±6

Theory

Experiment

Theory

1181.6b) 43.46 ¢)

29.1 -+7.0 [134] 24.1 -+ 7.0 [135]

42.56 d) 58.80 ° 28.84 f)

48.5-+ 11.51134] 41.2-+ 12.01135]

496.2 b) 27.88 c)

21.1 +--4.0 [134] 21.0 -+ 6.0 [135]

39.20 d) 44.24 ~) 26.60 o

33.3 -+ 8.5 [134] 33.0 ± 18.0 [135]

Experiment

[1481

25 30

62-+20 19.4 -+4.5 [134] 17.1 -+ 5.0 [135]

40

12.2 -+4.5 [134] 14.0 ± 12.0 [135]

50

80+!200. _ . [135]

19.6-+5.01135]

[145]")

27.6 +1700 [135]

') Data on o- m . b) Born-Ochkur approximation (see ref. [134]). °) Ochkur-Rudge approximation (see ref. [134]). d) Born approximation (data of Truhlar, see ref. [134]). e) Born-Ochkur-Rudge approximation (see ref. [134]). f) Born approximation (data of Van den Bos, see ref. [134]).

in the first place, studies due to Brunt et al. [147], Johnston and Burrow [148] and Buckman et al. [80], which were carried out with high (AE = 20 meV) energy resolution. For instance, Johnston and Burrow [148] found that the maximum of the 2 3S-level cross section at E = 20.35 eV amounted to (6.2 - 0.6) x 10 -18 c m 2, thus being in good agreement with the data of other authors (see table 5). Insofar as these studies mainly aimed at investigation of the resonance structure in the excitation cross sections they will be analyzed in more detail in section 5.

50

•

I

I

I

xO

40 I

.

e--~o30 "7"C~

2(] I /

~oo

.,

10 ! O|

20

V i

30

,,

40

,

50

BO

EleV) Fig. 14. Integral cross sections of helium for metastable excitation. - - - - - - , Born approximation (Kim and Inokuti, see ref. [135]); Born-Ochkur-Rudge approximation (Tmhlar et al., see ref. [135]); 2 'S: O experiment of Trajmar [134]; O, experiment of Hall et al. [135]; ~, data of Rice et al. (see ref. [135]). 2 as: --, Ochkur approximation (see ref. [135]); x, data of Trajmar [134]; A, data of Hall et aL [135]; ~7, data of Crooks and Rudd (see ref. [135]). I , data of Zavilopulo et al. [168] (2 SS + 2 ~S).

29

I.I. Fabrikant et al., Electron impact formation of metastable atoms

7 6

5 ~'-o 4

1D

2 1 0

I

I

20

I

21

I

E(eV)

I

I

22

I

23

Fig. 15. Excitation function for the 2 3S-state of neutral helium [46]. --, eleven-state R-matrix calculation, Freitas et al. [46]; - - - - - - , five-state R-matrix calculation, Fon et al. [152]; +, measurement of the 1 'S-2 3S peak (Johnston and Burrow [148]); the size of the symbol indicates the experimental uncertainty in position and magnitude.

4.2.2. Theoretical results Theoretical calculations of the excitation cross sections of the 2 1S-, 2 3p. and 2 3S-levels were done mainly by the R-matrix method and by solving the continuum Bethe-Goldstone equations with the use of the variational technique. Employing the R-matrix method in the close-coupling approximation accounting for the 1 'S-, 2 3S-, 2 is'-, 2 3p_; and 2 'P-states, Burke and coworkers calculated differential cross sections in the entire range of scattering angles for energies of 26.5, 29.6, 81.63, 100, 120, 150, and 200 eV [149, 150] as well as differential cross sections at the scattering angles 0 = 30°, 60°, 90 °, and 120° for energies close to the threshold [151]. Also given were detailed data on the total cross sections Table 6 Overall data on theoretical calculationsof the metastable helium 2 3S and 2 1S states at intermediate energies Energies (eV) for which the cross sections were calculated Reference

Theoretical approach

differential

[32]

Asymptotic second Born approximation (2 'S, 2 3S, 2 3p)

200

[154]

1 lS-2 3S-2 'S-2 3P-2 ,p close-couplingapproximation

29.6, 40.1, 60, 80, 100

[155]

Distorted wave approximation (2 3S)

[156]

1 lS-2 1S-2 1p without exchange 1 'S-2 'S-2 'P-2 3S-2 3p with localized exchange 1 'S-2 'S-2 1P-k 1P-4 'S without exchange 1 'S-2 'S-2 ~P-3 'S-3 'P without exchange 1 1S-2 'S-2 'P-2 3S-2 3P-k XP-4'S with localized exchange

[157[

[158]

total

30-200 60-200

S- and P-waves from the experimental differential cross section data D, F . . . . distorted-wave polarized orbital approximation

29-100

Variable charge Coulomb-Born approximation

100, 200

30-50

[69]

Many-body diagram method (2 3S)

30.8, 79,111,192

20-240

[16o1

Integral equation method for T-matrix

50, 81.63,100, 150,200, 300,400, 500

30

L I. Fabrikant et al., Electron impact formation of metastable atoms

calculated in the same approximation [152]. Calculations in the approximation taking five states into account overestimated the values of the excitation cross sections and failed to describe the rich resonance structure, produced by n = 3 thresholds, which was found experimentally [80, 141,147]. The calculations by Oberoi and Nesbet [59], who took into account virtual 3s-orbitals when solving the Bethe-Goldstone equations, partly described the effects under consideration, whereas a complete description of all these effects was achieved by Nesbet [153] in calculations taking the whole set of n = 3 orbitals into account. In recent work by Freitas et al. [46, 47], the R-matrix method enabled the authors to take into account eleven states of the helium atom, all states with n = 3 included; this study gives the first precise description of the excitation cross sections in the vicinity of n = 3 thresholds. Recent measurements [148] of the height of the first peak in the cross section 0"(2 3S).are in a good agreement with the calculations presented in refs. [46, 59]. The latest experimental [148] and theoretical [46] values of the peak position are equal to 20.35 and 20.32 eV, respectively. The best experimental [148] and theoretical [46] results are compared in fig. 15. In conclusion, in table 6 we list the main theoretical works on excitation of metastable states of helium at intermediate energies which were published after the reviews of refs. [4, 5]. 4.3. The neon, argon, krypton, and xenon atoms In neon, argon, krypton, and xenon atoms the metastable levels are the n 3P2- and n 3p0-1evels, where the principal quantum number n acquires the values n = 3, 4, 5 and 6, respectively. In all these elements, the electric dipole transition from the metastable states to the ground 1S0-state is forbidden because of the total angular momentum J. Nevertheless, transitions from the n 3p0-1evelto the n 3Paand n 3p2-1evels are allowed but their probabilities are small indeed [161]. Two-photon emission in the case of the electric dipole transitions from both metastable levels to the ground state is forbidden by the parity selection rules, while the probabilities of magnetic quadrupole transitions are so small [161,162] that we neglect them here, All this determines large lifetimes of atoms in the n 3P2- and n 3P0- states. For instance, according to ref. [161], the lower bound of the integral lifetimes of both levels amounts to 0.8 s, 1.3 s, and 1.0 s for neon, argon, and krypton, respectively. In ref. [162] the following values of the lifetimes of the naP 2and n 3p0-1evelsare given: Ne, 24.4 s and 430 s; Ar, 55.9 s and 44.9 s, Kr, 85.1 s and 0.488 s; Xe, 149.5 s and 0.078 s. Regardless of considerable discrepancies in the data given in refs. [161] and [162], one can confidently use the classification of Mushlitz [163], who referred these levels to the class of "very long-lived levels", apparently inferior only to the 2 as-state of the He atom and the 2D5/2,3/2state of the N atom [163]. The basic parameters of the states are given in table 7. Similarly to the case of the helium atom, production of inert gas atoms in n 3P2 and n 3po-states by electron impact may take place not only as a result of a direct electron transition from the ground state but also due to cascade transitions from higher levels [336]. When considering these levels, one should keep in mind that their classification (as well as that of metastable levels) according to the LS-coupling scheme is not quite correct, because this scheme is essentially violated even in neon, the lightest of these atoms. More adequate is the j - l coupling scheme, according to which the total angular momentum Jc of the core (n - 1)p 5 couples, first, with the orbital angular momentum l of the excited electron to form the angular momentum K, which then couples with the spin of the excited electron and produces the total angular momentum J. The terms are denoted, according to Racah, as nl[K] ° or nl'[K] °, where the prime implies that Jc = 1/2 and its absence means that J¢ = 3/2, In this notation, the excitation cross section of metastable levels (for instance, of the Kr atom) is given by the following sum

I.L Fabrikant et al., Electron impactformation of metastable atoms

31

Table 7 Metastable states of noble gas atoms /'/-coupling notation

LS-eoupling configuration

Energy (eV)

Lifetime (s)

Neon

3s[3/2]o° 3s'[1/2]~

2pS(2P~/:)3s3P0

2pS(2P~/z)3s 3P2

16.619 16.716

24.4 430

Argon

4s[3/2]~ 4s'[l/2]°0

3p~(2P~/~)4s 3P2 3pS(2P~/2)4s 3P0

11.548 11.723

55.9 44.9

Krypton

5s[3/2]~ 5s'[1/2]°o

4pS(2P~,2)Ss 3P2 4pS(2p~n)5s 3P0

9.915 10.563

85.1 0.488

6s[3/2]~ 6s'[1/2]°0

5p'(2p~:2)6s 3P2 5pS(2p°)6s 3P0

8.315 9.447

149.5 0.078

Atom

Xenon

of partial excitation cross sections of separate levels (within the energy region where all these levels are energetically accessible): 0.m = O'(5S[3/212) + 0.(5S'[1/2]0) + O'(5p[1/211) + O'(5p[5/213) + 0.400"(5p[5/212) +0.37o-(5p[3/212 ) + o-(4d[1/2]o ) + 0.34o-(5p'[3/211 ) + 0.50o-(4d[3/212 ) +0.64o-(5p'[1/211) + 0.59o'(4d[7/2]3 ) + . . . .

(4.4)

This equation shows that quite a few levels with the same principal quantum number n contribute to the electronic excitation cross section of metastable levels. Undoubtedly, this fact complicates analysis and interpretation of experimental data and requires a great number of levels to be taken into account in theoretical calculations. All this is likely to account for the relative scarcity of theoretical and experimental studies of this problem, especially as regards heavy atoms (krypton, xenon). In this section we will refrain from listing all relevant papers but will dwell upon only the latest ones. A detailed analysis of earlier works is given, for instance, by Bransden and McDowell [5].

4.3.1. Neon One of the latest works [164] has reported on measurements of the excitation differential cross sections of certain levels in the neon atom in the range of electron scattering angles from 10° to 140° at 30 and 50 eV. The cross sections were extracted from electron energy loss spectra and normalized to the known values of the excitation cross section of the 3 1p:level. The differential cross sections thus found were about 10 -19 cm2/sr (table 8). Extrapolation of the differential cross sections obtained in ref. [164] to the range of electron scattering angles around 0° and around 180° made it possible to obtain absolute values of the integral excitation cross section of each separate level at the electron energies mentioned above. In ref. [164] the total cross sections were evaluated by integrating the differential cross sections over the electron scattering angles, and in fig. 16 they are compared with the measured values of the total cross section obtained by Teubner et al. [165] and Phillips et al. [335, 18, 166]. In refs. [18, 166,335], the laser fluorescence method was used to measure both the excitation cross section 0. of the level and the cross section o-m of metastable production, the latter being given by the sum of the direct cross sections and the partial cross sections of cascade transitions from higher levels j, o):

I.I. Fabrikant et al., Electron impact [ormation of metastable atoms

32

Table 8 Differential cross sections for excitation of metastable states of neon, in 10 -19 cm2/sr. Upper data: experimental values [164], lower data: calculated values [67] E (eV)

20

O (deg)

3P2

25 3Po

0

10

.

1.72 1.787

0.358

1,50

1.868

60

70

-

80

-

1.507

90

100

110

120

130

140

160

0.010

-

1.99

0.466

-

-

0.610

0.378

-

0.686

0.137

0.518

0.104

0.465

0.094

0.287

2.05

-

-

-

0.540

-

-

-

0.848

0.340

1.437

0.288

1.224

0.244

1.72

0.398

1.84

0.399

-

-

1.05

0.303

0.389

-

-

2.170

0.434

2.025

0.403

1.551

0.311

1.84

0.416

2.01

0.319

-

-

1.00

0.229

0.389

-

-

2.383

0.476

1.994

0.398

1.367

0.271

1.45

0.280

1.79

0.395

-

0.837

0.175

0.375

-

-

2.117

0.428

1.596

0.319

0.991

0.199

1.33

0.294

1.38

0.319

-

0.538

0.113

0.341

-

-

1.699

0.339

1.145

0.229

0.683

0.137

1.14

0.235

1.30

0.304

-

-

0.401

0.079

-

-

1.263

0.253

0.784

0.156

0.473

0.106

0.815

0.162

0.84

0.174

-

-

0.143

0.034

0.260

-

-

0.947

0.189

0.524

0.105

0.316

0.064

0.704

0.138

0.42

0.097

-

-

0.111

0.029

0.224

-

-

0.767

0.154

0.364

0.074

0.203

0.126

0.452

0.112

0.65

0.093

-

0.196

-

-

0.711

0.143

0.291

0.322

0.114

0.17

0.036

-

0.174

-

-

0.748

0.150

0.294

0.227

0,071

0.21

0.051

-

-

-

0.840

0.168

0.361

0.123

0.039

0.058

0.137

0.027

0.134

0.020

0.059

0.120

0.024

0.116

0.051

0.072

0.150

0.090

0.094

0.019

0.097

0.225

0.045

0.784

0.157

-

-

0.314

0.060

0.17

0.040

-

0.708

0.141

-

-

0.958

0.192

0.482

0.308

0.080

0.17

0.044

-

0.079

0.032

0.128

-

-

1.081

0.216

0.636

0.127

0.330

0.066

.

. 0.792

. 0.158

0.442

0.088

.

. 0.185

0.540

0.108

-

. 0.577

.

. 0.524

.

. 0.115

.

.

.

.

.

-

. 0.098

. -

.

0.490

. 0.095

. -

.

0.479

15

22

180

. 1.193

0.105

170

Exp. error ( % )

0.052

0.663

-

0.638 150

0.012

0.297

0.871

0.061

1.484

0.980

0.066

2.06

1.123

.

0.327

0.291

0.302

1.302

.

3Po

-

1.708

.

3P2

1.66

1.871

.

3Po

-

1.952

50

aP2

0.372

1.943

3Po

.

-

-

40

3P2

.

-

20

50

.

0.350

-

40

3Po

1.750

15

30

30

3P2

0.238 .

1.280

0.256

. 0.927

. 1.336

. 0.267

. 1.019

. 0.204

0.610

0.122

. 1.355

. 0.271

. 1.050

. 0.210

0.638

0.127

19

35

33

40

I.I. Fabrikant et al., Electron impact formation of metastable atoms w

i

i

,

i

i

i

~

,

33

,

4oI

%

\

0,)o

~ o 2~ L~

0-e

\

40

80

120 E(eV)

160

200

Fig. 16. Metastable excitation integral cross sections for neon. --, experiment of Phillips et al. [18, 166] (~r=); D, experimental data of Zavilopulo et al. [168] (~rm); O, experiment of Register et al. [164]; O, experimental data of Teubner et al. [165] (trm); - - ' --, direct cross section measurements of Phillips et al. [18, 166].

O'm= O'+ ~ %..

(4.5)

J

Apart from the studies mentioned above, the absolute total cross sections of production of Ne atoms in the 3 3P2- and 3 3p0-states were measured by Korotkov et al. [167] through relaxation of metastable atoms on the surface of pure metal and by Zavilopulo et al. [168] by integrating the drift angle differential cross sections. Korotkov et al. [167] obtained the following values of the cross sections at the maximum as a function of energy (E = 22 eV): for the 3 3p2-1evel5.3 × 10 -18 cm2, for the 3 3p0-1evel 1.2 × 10 -18 cm2 As is seen here, the total cross section for both metastable levels exceeds the values of Zavilopulo et al. [168] by almost one order of magnitude and lies above the results of refs. [165, 18, 166] by a factor 1.5-2.0. In turn, the data of ref. [168] are almost two times smaller than those of ref. [164] and 4.5-5 times smaller than those reported in refs. [18, 165, 166]. Table 9 summarizes the data on integral cross sections for neon atoms (3 3P2,0) produced by electron impact, which are quoted from the studies mentioned, as well as from a study using the photoabsorption method [169]. The discussion of the accuracy of the values of the cross sections obtained deserves special attention. In ref. [164] the accuracy of determining the integral cross sections was governed mainly by that of the measurements of the differential cross sections (table 8), while the integration procedure introduced a small error (not exceeding 5%); however, application of indirect methods [165, 168] involves, in addition to the experimental error, also the inaccuracy of cross section calibration with respect to the calculation [186] or experiment [165]. On the whole, the total error in the reported integral cross sections was, for instance, as large as 47% (at E = 25 eV) or 83% (E = 50eV) in ref. [164], 25% (E = 20 eV) in the experiment by Phillips et al. [18, 166] and 33% (E = 20 eV) in ref. [168]. Of special interest is the research performed by Teubner and coworkers [165] concerning estimates of the cascade contributions to the cross section of metastable production. Close to the threshold, where the contribution due to cascades is minimal, the experimental data were found to agree with the results of the R-matrix calculation carried out by Taylor et al. [43]. The effect of cascades increases at 30 eV, while at E---40 eV the cascade mechanism of populating metastable states becomes dominant.

34

1.I. Fabrikant et al., Electron impact formation of metastable atoms

Table 9 Integral cross sections for excitation of the metastable 3 3P2- and 3 3p0-1evelsof the neon atom in J=0

1 0 -19 c m 2

E (eV)

1=2 Experiment

Theory

Experiment

Theory

Experiment

Theory

20

16 -+4 [166] 34"~ [169]

16.83[43] 15.25[67]

2.8 +-0.7 [181 13.) [169]

3.37 [43] 3.05 [67]

18.8 -+4.7 [18, 166] 47.) [169]

20.20 [431 18.30 [67]

25

10.3 37a) 33-) 10

[1641 [167] [335] [335]

30

10.1 14') 31.) 7.3

[164] [167] [335] [335]

50

4.25 [164] 26-) [335] 2.3 [166]

15.46[67]

3.7 [334] 6.23 [67]

J=2,0

2.40 10-) 5.8 ') 1.8

[164] [167] [335] [335]

2.31 6.5 a) 6.0 a) 1.4

[164] [167] [335] [335]

1.21 5.5 .) 0.61

[164] [166] [18]

12.7 47a) 39.) 11.8

[1641 [167] [335] [335]

3.09 [67]

12.4 20.5') 37.) 8.7

[164] [167] [335] [335]

0.70 [334] 1.26 [67]

5.46 32"1 2.9

[164] [335] [18, 166]

18.55 [67]

4,40 [334] 7.49 [67]

") Data on %.

Experiments with high energy resolution intended to determine the cross sections in the range from the threshold to 22 eV (Buckman et al. [170]) and from 42 to 52 eV (Dassen etal. [171]) are of great interest as regards resonance studies. These experiments will be discussed in section 5. Among the theoretical calculations close to the excitation threshold, those by Taylor et al. [42, 43] and Noro et al. [44] are known. Both of them were carried out with the neglect of fine-structure effects; the former allowed for coupling of the 2p 6 (1S)-, 2p53s (3p, 1p). and 2p53p (X'3S, ~'3D)-states of the target atom, while in the latter the 2p54s (1.3p). and 2p53d (rap, 1.3D, ~,3F).state s were also added. Taylor's calculations [43] are in good agreement with the experimental data obtained by Phillips et al. [18,166], thus proving the applicability of the LS-coupling scheme to this case (in particular, the cross section is proportional to the statistical weight of the final state). As far as intermediate energies are concerned, one should mention here, in addition to the calculations mentioned by Bransden and McDoweU [4, 5]; computations carried out by Machado etal. [67] within the framework of the first-order many-body perturbation theory (see table 8).

4.3.2. Argon Employing the TOF technique, Borst [145] measured the total production cross section for argon atoms in the metastable 4 3P2,0-states. His results are in good agreement with the relative measurements by Lloyd etal. [146]; however, as was mentioned in ref. [5], their comparison with the results obtained by Kuprianov [172] leads to a considerable discrepancy as regards the energy dependence. Later, Schearer-Izumi [173] showed that it was not correct to compare the data in refs. [145] and [172], because in ref. [172] the production of highly excited Rydberg states whose energies lie in the region of the ionization potential of the neutral atom, had been measured rather than that of the metastable 4 3P2,o-states. The contribution of metastable levels was revealed by Tam and Brion [174] in their electron energy loss spectra recorded in the range of electron scattering angles from 5° to 90°. These data proved helpful in determining the relative differential cross sections for excitation of the 4 3p2,0-states at 30 eV and the ratio of the excitation cross sections of the 4 3P2- and 4 3Pl-levels at the same energy. Forward electron

1.1. Fabrikant et al., Electron impactformation of metastable atoms

35

Table 10 Differential cross sections for excitation of metastable states of argon, in 10 -19 cnl2/sr. Upper data: experimental values of Chutjian and Cartwright [178], lower data: calculation of Padial et al. [65]

e (eV)

16

0 (deg)

J=2

20 J=0

J=2

30 J=0

J=2

50

100

J=0

J=2

J=0

J=2

J=0

4.15 9.772

1.12 1.954

1.02 0.915 1.52 0.1183 0.02365

2.65

(0)

1.70 0.505 6.40 2.30 0.1906 0.03812 5.063 1.013

10

0.951 0.275 0.6420 0.1284

3.94 1.14 6.137 1.227

5.60 10.96

1.32 2.191

1.45 0.725 0 . 7 7 0 0.7195 0.1439

0.890

20

0.860 1.765

0.215 0.3530

2.40 0.446 8.472 1.694

5.80 12.58

1.30 2.515

1.38 1.184

0.238

30

1.03 3.029

0.213 2.45 0.6059 1 0 . 3 5

0.510 2.070

4.75 11.94

1.04 2.388

1.07 0.345 0.7940 0.1588

40

1.41 3.952

0.259 2.58 0.7904 1 0 . 7 7

0.470 2.153

3.45 9.176

0.670 1.835

0 . 7 2 0 0 . 1 9 7 0.0275 0.0365 0.4924 0.09847

50

2.30 4.333

0.360 0.8667

2.14 0.370 9 . 8 6 7 1.973

1.59 6.058

0.302 1.212

0 . 3 6 8 0 . 1 0 0 0.0143 0.0139 0.4118 0.08236

60

2.85 4.257

0.495 0.8514

1.95 0.420 8 . 2 7 2 1.655

0.360 3.736

0.0940 0 . 1 5 3 0.0620 0.0119 0.00951 0.7472 0.3294 0.06588

70

3.20 3.942

0.580 0.7883

1.79 0.415 6 . 5 1 6 1.303

0.101 2.389

0.0440 0 . 1 1 9 0.0440 0.0140 0.0100 0.4778 0.2990 0.05978

80

3.20 3.600

0.580 0.7200

1.61 0,347 4 . 9 3 3 0.9865

0.189 0.0380 0 . 1 3 1 0.0320 0.0189 0.0124 1 . 7 6 9 0.3538 0.3759 0.07518

90

3.08 3.381

0.530 0.6761

1.51 0.307 3 . 7 7 5 0.7550

0.355 0.0385 0 . 1 5 0 0.0290 0.0217 0.0141 1 . 5 4 0 0.3080 0.4767 0.09535

I00

2.95 3.352

0.510 0.6705

1.70 0.348 3 . 2 5 0 0.6500

0.400 0.0460 0 . 1 6 9 0.0295 0.0202 0.0140 1 . 4 4 6 0.2892 0.4824 0.09647

110

2.91 3.501

0.522 0.7001

2.03 0.405 3 . 4 5 0 0.6901

0.405 0.0685 0 . 1 8 0 0.0330 0.0161 0.0130 1 . 4 1 8 0.2836 0.3870 0.07741

120

3.12 3.732

0.578 0.7464

2.50 0.498 4 . 2 6 9 0.8538

0.415 0.0845 0 . 1 6 0 0.0340 0.0110 0.0112 1 . 5 3 4 0.3069 0.2730 0.05461

130

3.80 3.911

0.680 0.7821

3.10 0.618 5 . 4 0 6 1.081

0.605 0 . 1 4 9 0 . 1 0 2 0.0290 0.00910 0.00975 1.872 0.3744 0.2256 0.04512

140

5.60 3.925

0.850 0.7850

3.90 0.790 6 . 4 8 9 1.298

1.31 2.472

0 . 5 6 5 0.0785 0.0292 0.0135 0.00980 0.4854 0.2881 0.05760

(150)

7.30 3.751

1.04 0.7503

4.90 0.960 7 . 2 5 1 1.450

2.15 3.117

1.02 0.109 0.0375 0.0209 0.0120 0.6233 0.4198 0.08397

(160)

8.80 3.469

1.24 0.6938

6.00 1.13 7 . 6 2 6 1.525

3.12 3.783

1.49 0.173 0.0515 0.0305 0.0167 0.7567 0.5452 0.1090

(170)

10.2 3.216

1.39 0.6431

7.35 1.32 7 . 7 3 1 1.546

4.12 4.253

1.92 0.263 0.0670 0.0435 0.0229 0.8507 0.6313 0.1263

(180)

11.3 3.115

1.53 0.6229

8.90 1.52 7 . 7 4 0 1.548

4.90 4.419

2.40 0.339 0.0840 0.0540 0.0289 0.8839 0.6638 0.1328

Exp. error(%) 26

31

26

31

26

45

31

0.520 0 . 3 1 5 0.2368

45

0.102

45

0.150

45

36

1.1. Fabrikant et al., Electron impact formation of metastable atoms

'T2...

%

IdI

.

.

" . . ' . " ~..__ .__

'

-----.

,o

20

40

60

80

100

E(eV) Fig. 17. Integral cross sections for the excitation of the metastable levels of the argon atom. , calculation for all "metastable" levels (see ref. [178]); , calculation of Padial et al. by the FOMBT method [65], 4 3p2.o; ©, experiment of Chutjian and Cartwright for the 4 3P2.o-levels [178]; - - . - - , the same for 16 levels [178]; - - - - - - , experimental data of Borst [145] (o,~); . . . . , experiment of Lloyd et al. [146], normalized on data of Borst [145] (Crm); @, experiment of Zavilopulo et al. [168] (~rm).

scattering was found to be markedly dominant; this observation differs from the results on excitation in helium obtained by Kupperman et al. [133,175]. It was interpreted in ref. [174] as a result of the j-j coupling in the argon atom. One should note that such a manifestation of the 3 3P2- and 3 3p0-states was not observed in the loss spectra of the neon atom measured by Tam and Brion [174]. Similar measurements of differential cross sections were carded out by Lewis et al. [176] at 30, 40, 50, 60, 80, 100, and 120 eV, where they were compared with the calculations of Sawada et al. [177] and the experiment of Tam and Brion [174]. In both cases, fairly good agreement was observed. In a recently published study of Chutjian and Cartwright [178], differential and total excitation cross sections of the metastable levels were measured. They were tabulated and, according to these data, a unified Table 11 Integral cross sections for excitation of the metastable 4 3P2- and 4 3p0-1eveis of the argon atom, in 10-19 cm 2 E

J=2

J=0

~)

Expe~t

~eo~]

Expe~ent

16

41.9 [1781 340.) [336] (at E = 18 eV)

45.4

6.09 [1781 90a) [336] (at E = 18 eV)

9.08

15.73

370-) [145] 38.3 [178]

94.4

9.29

21.1 [178] 142"~ [167]

55.7

1.08

4.73 [178]

20

31.9 [178] 78.7 143') [167] (at E = 21.5 eV)

6.44 [178] 31 .) [167] (at E = 22 eV)

30

16.6 [1781 116-) [167]

4.54 [178] 26") [167]

50

3.61 [178]

80

46.4 5.40 9.4

') D a t a o n ~rm.

1.12 [178]

J=2,0 ~eo~5]

E~e~ent

~eo~l

48.0 [178] 430.) [336]

54.4

(at e=18eV)

0.19

6.48

1.I. Fabrikant et al., Electron impactformation of metastable atoms

37

table (table 10) has been composed which enables one to estimate the differential cross section versus excitation energies. The total excitation cross sections of the 4 aP2,0-1evelswere obtained in ref. [178] with the use of the electron spectroscopy method. Figure 17 presents the total excitation cross sections of these levels as compared to data of other authors [145, 146, 168, 179]. It is interesting to follow the behavior of the cross section for excitation of 16 levels of Ar considered by Chutjian and Cartwright [178] as metastable because they cannot decay to the ground state according to the selection rule for the total angular momentum J. As is seen from fig. 17, this curve fits best of all to the data obtained by Borst [145], which have to be multiplied by a factor of 2, and to those due to Lloyd et al. [146] obtained by the TOF method. The agreement of the results with the data obtained by Korotkov et al. [167] can also be considered as quite satisfactory. Only the cross section found in ref. [168] is one order of magnitude smaller compared with other works, which is due to the inaccuracy of the normalization procedure using the calculations of ref. [65] (see ref. [168]). The values of the cross sections are presented in the summary table 11. This table also contains experimental results .obtained by Korotkov et al. [167] by the secondary electron emission technique and by Mityureva and Smirnov [336] by the photoabsorption method. The results of a detailed study of the cross sections within the energy range from the threshold up to 16 eV (Buckman et al. [187]) and from 24 to 33 eV (Dassen et al. [171]) will be discussed in section 5. Padial et al. [65] calculated differential and total excitation cross sections for four levels (involving the metastable ones) of the argon atom from the same energies at which the measurements of Chutjian and Cartwright [178] were performed. The calculation was made within the first-order many-body theory, equivalent to the distorted-wave method, taking the spin-orbit interaction in the atom into account. The theoretical and experimental results are in good agreement (see table 10). A theoretical calculation in the near-threshold energy region was carried out by Ojha et al. [45] in the framework of the nonrelativistic R-matrix method with the 3p6(1S)-, 3p54s(l'3P)- and 3p54p (I,3S, ~,3p, 1,3D).state s of the target taken into account, at electron energies from 10.8 to 13.4 eV. 4.3.3. Krypton and xenon As far as the heavier atoms of inert gases are concerned, the number of theoretical and experimental studies of excitation of the metastable 5 3P2,o (Kr) and 6 3p2,0 (Xe) states is rather small. This is probably explained by the complex structure of the electron shell of these atoms and, hence, the difficulty of interpreting the results obtained. Among the few hitherto available works we may distinguish the measurements of the differential excitation cross sections carried out by Phillips [181] and Swanson et al. [182] for Kr and those by Swanson et al. [183] for Xe. For instance, Phillips [181] determined the absolute cross sections for the production of the krypton atom in the 5 3P2- and 5 3P0-states in the near-threshold energy region at electron scattering angles of 30°, 55°, and 90°. The ratio of the absolute values of the excitation cross sections of these levels was found to remain constant over the entire energy range, equal to 5.6:1. Figure 18 presents some differential cross sections versus energy [181] at the angles mentioned above. Similar curves but in relative units were obtained by Swanson et al. [182] for 45°. In table 12 the excitation differential cross sections obtained for the krypton atom by Trajmar et al. [184] within the range of electron scattering angles from 10° to 135° at 15, 20, 30, 50, and 100 eV are given. In ref. [184] these data are compared with the results reported by Lewis et al. [176] and Delage and Carrette [185]. Trajmar et al. [184] noted good agreement in the behavior of the differential cross sections. In ref. [184] absolute total cross sections for the excitation of the krypton 5 3P2,o-levelswere

I.I. Fabrikant et al., Electron impact formation of metastable atoms

38

I

I

i

:

-

i

i

I

i

p.,:

-

i

90*

I

-

i

~ "

0

i

1

i

I

3

i

1

i

J

t

t

t

t

.'. ~O

.I"X.-

0.1

E Goo

O'

l

I ,'41~"I

I

I

l

I

I

I

I

I

I

I

I

2

t

-

/

55*

•

/ t-i ",

o

0.4

~

'o I

~1.

I

I/"~" - -

I

I

I

.." :. /

I

I

~.~

~,f'-.~.- - v ~ " i

i

i

A

L

I

F j' 0t,

' 10

'J'¢: 11

,

," . . . . 12 13

/ it,

Fig. 18. Differential excitation cross sections for the 4p55s ~P2,o-states of Kr versus incident electron energy at 30°, 55° and 90* [181],

Table 12 Differential cross sections for excitation of metastable states of krypton, data of Trajmar et ai. [184] (experiment), in 10-19 cm2/sr. Experimental error +-35% "X~(eV)

15

0 (deg)

J=2

J=O

J=2

J=O

J=2

J=O

J=2

J=0

1.9 3.1 3.6 4.0 4.4 4.7 4.9 3.5 2.7 2.3 2.7 3.9 5.0 5.5

1.34 0.56 0.35 0.47 0.73 0.99 0.88 0.59 0:48 0.46 0.51 0.70 0.90 1.00

12.0 13.4 11.2 5.3 2.9 2,2 2.8 2.9 2.4 2.4 2.6 3.2 4.3 5.9

3.4 2.3 1.4 1.0 0.72 0.76 0.79 0,68 0,52 0.51 0.58 0.76 1.1 1.7

3.5 4.9 4.1 3.1 1.9 0,96 0.43 0.43 0.50 0.54 0.99 1.10 1.00 (1.0)

6.6 1.3 0.68 0.53 0.37 0.23 0.12 0.09 0.10 0.10 0.19 0.24 0.24 (0.25)

8.2 3.0 1.2 0.40 0.24 0.25 0.14 0.17 0.18 0.16 0.18 0.19 0.21 0.20

3.3 0.90 0.33 0.15 0.08 0.050 0.030 0.044 0.050 0.040 0.036 0.032 0.026 0.024

10 20 30 40 50 60 70 80 90 100 110 120 130 135

~,

20

30

50

LI. Fabrikant et al., Electron impactformation of metastable atoms Table 13 Integral cross sections for excitation of metastable states of the krypton atom in

e (eV) 15

5%

9.1__.3.51184]

[167] (E = 14.5 eV) [333] (E = 16 eV)

(experimental data)

5 %,o

5%

5o__.19 [184] 210 390.)

10 -]9 c m 2

39

59-+23 [184]

[167] (E = 13.5 eV) [333] (E = 16 eV)

65 100")

490.)

[333] (E = 16 eV)

20

544- 20

[184]

13 - 4.9 [184]

30

164-6.1

[184]

5.6__.2.1118,1]

21.6_+8.21184]

6.3 _+2.4 [184]

2.0 _+0.8 [184]

8.3 _+3.2 [184]

50

6 7 - 25

[184]

") Data on trm. Oi

I

I

i

~\~

1(~9

I

I

40

20

~

i

I

5s'1~/2~ I

I

60 80 E(eV)

I

100

Fig. 19. Integral cross sections for the excitation of metastable states of krypton: 5 3P2: ~ . , FOMBT data of Meneses etal. [66]; O, experiment of Korotkov etal. [167] (~m); A, experiment of Trajmar etal. [184]. 5 3Po: - - - - FOMBT calculation [66]; O, data of Korotkov et al. [167] (c%); A, experiment of Trajmar etal. [184].

1J

I

*

5~,r.2~(÷Io) 10221

m 20

~

, 60 O(deg)

I

i 100

f

I

140

10

20

I

I

~

6O

I

I00

i

140

Oldeg)

Fig. 20. Differential cross sections for the excitation of metastable states of Kr at 30 eV (left) and 50 eV (fight) [184]. Meneses et al. [66]; G, O, experiment of Trajmar etal. [184].

, FOMBT calculation of

1.1. Fabrikant et al., Electron impactformation of metastable atoms

40

obtained by integrating the differential cross sections, that is, in a similar manner as for neon [164] and argon [178]. Figure 19 presents these cross sections, which are compared with those given by the calculation by Meneses et al. [66] and with the experimental data obtained by Korotkov et al. [167]. Table 13 comprises an analysis of the data for total cross sections for excitation of the metastable levels of the krypton atom which were obtained by various authors with the use of different techniques. The results of a detailed study of the cross sections close to threshold (Buckman et al. [187], Jureta et al. [188]) and at 22-32 eV for Kr (Dassen et al. [171]) will be discussed in section 5. It is worth noting that the absence of data on absolute cross sections for Xe atom excitation in Buckman et al. [187] was eliminated by Blagoev et al. [189], who obtained absolute cross sections using the results of independent measurements of the de-excitation rates. Meneses et al. [66] have recently carried out calculations of the differential cross sections for excitation of Kr levels (involving the metastable ones). This calculation made it possible to compare the results of the first-order many-body theory with the experiment of Trajmar et al. [184] (see fig. 20). Finally note that recently two experimental papers [338,339] were published concerning excitation cross sections for the n 3P2 state of Ar and Xe at energies from the threshold up to 20 and 30 eV, respectively. The maximum values of the cross sections are 3.2 x 10 -17 c m 2 and 4.5 x 10 -17 c m 2 for Ar and Xe, respectively.

4.4. Atoms of the second group of the periodic table Here, we will dwell upon excitation of metastable states of Be, Mg, Ca, Sr, Ba, Zn, Cd, and Hg atoms. In all these atoms except barium the n o 3P2- and n o 3P0-1evelsare metastable (n o is the principal quantum number of the ground state). Electric dipole transitions from these states to the ground state are forbidden by the selection rules for the total angular momentum J. In the LS-coupling approximation the decay of the n o 3p:state is also forbidden; therefore for the light atoms (Be and Mg) this level may also be considered as metastable. However, already in Ca, the lifetime of this state becomes equal to 1.5 x 10-4s [92]; therefore we will not classify the n o 3p:level for zinc, cadmium, mercury and strontium as metastable. Besides the direct excitation, excitation of higher triplet states followed by cascade transitions to the n o 3p2,o-levels also contributes to the production of the n o 3P2,0-states. The 3 1D2- and 4 1D2-1evelsshould be taken into account in Ca and Sr atoms since electric dipole transitions from these states to the ground state are forbidden by the parity and total angular momentum selection rules and they decay into the 4 3P1,2- and 5 3p1,2-states, respectively. Similarly to the case of the He atom, the higher singlet states do not contribute significantly to trm, since the relative probability of radiative decay of these states into the 3 ID 2- (4 ID2-) states is small. The 5 3Da,2,3- and 5 1D2-states are metastable in the Ba atom, and all higher triplet levels contribute to their production. The decay rate of the higher singlet levels is small, similarly to the case of Ca and Sr atoms. Taking into account these factors, we suppose that tr~ = ~ o', 3L ,

Be, Mg, Zn, Cd, Hg,

(4.6)

n,L

trm= ~] tr 3L + tr(,_l)1~,

Ca, Sr, Ba,

n,L

where the sum is taken over all possible values of n and L.

(4.7)

I.I. Fabrikant et al., Electron impactformation of metastable atoms

41

In the case of Zn, Cd, Hg and Sr atoms, only the 3p2,o-States are taken into account in the sum for n = no, while for n > n o each term of the sum must be taken with a factor allowing for the relative

possibility of decay to the n o 3p2,o- (but not the n o 3p1-) state. Thus one has to know the excitation cross sections of all triplet and singiet (n o - 1) 1D-states in order to theoretically estimate the metastable production of the atoms belonging to the second group. These cross sections were calculated in the region of small energies by the close-coupling method [190-193]. Moreover, there are calculations by Savchenko [194, 195] on the excitation of the n o 3P2,1,0-1evelsin the Born-Ochkur approximation allowing for spin-orbit interaction in the atom. However, these calculations give too small values (compared with experiment and with results of the close-coupling method), which results from the inapplicability of the Born-Ochkur approximation in the region under study. Also available are the data of Vainshtein et al. [28] on excitation of the (n0-1) 1D-states obtained in the Born approximation with unitarization using semi-empirical atomic wave functions. According to these data, the results of the numerical calculations are described by the formula A ( u ~

or(,0_1) 1o -

u+~o

1/2

\ ~-j-/

[Tra~]

(4.8)

where u = ( E / A E ) - 1. The parameters A, ~o and AE are given in table 14. When determining the cross sections for excitation of the metastable state n o 3pj, investigations of the optical excitation functions of the radiative n o 3 p l - n 0 1S0 transitions (see, e.g., refs. [196-199]) may be helpful. Although the n o 3pl-state may be considered as a metastable state only in Be, Mg and Ca, nevertheless the availability of data on the excitation cross section of this state enables one to extract information about the excitation cross sections of the "true" metastable n o 3P2- and no 3P0-states, since in the LS-coupling approximation [200] applied to the nonresonant energy region we have (4.9)

0"3[00 : Og3P1 : O'3P2 = 1 : 3 : 5.

One should note, however, that relation (4.9) is considerably violated in the case of heavy atoms (Cd, Ba, Hg). Moreover, in the course of measurements of the opticial excitation functions a large portion of excited atoms leaves the observation region of the photon detector without emission owing to the relatively long lifetimes of the n o 3pl-states. Therefore, one needs a special formula to calculate the cross section from the parameters of the setup and the lifetime of the n o 3pl-state [196-199]. Now we will present estimates of the contribution due to the most significant lower levels, which will be based upon analysis of different theoretical and experimental results. 4.4.1. Beryllium

We know only one theoretical calculation [190] of the cross section of the 2 3p-level excitation, which was carried out in the close-coupling approximation including the 2 IS-, 2 3p. and 2 lp-levels. Its results are given in table 15. Table 14 A, ~o and AE values from eq. (4.8), data of Vainshtein et al. [28] Atom

AE (Ry)

A

~o

Calcium Strontium Barium

0.1991 0.1836 0.1038

63.9 112.5 176.7

0.821 0.867 0.926

I.I. Fabrikant et al., Electron impact formation of metastable atoms

42

Table 15 Calculated 2 3p excitation cross sections for the beryllium atom [190], in 10-16 cm 2 E (Ry) tr

0.22 11.1

0.25 11.8

0.30 11.2

0.35 10.5

0.45

0.50

0.60

8.45

7.48

5.98

As in the case of Mg (see below) the contribution of cascade transitions from higher levels is unlikely to be large.

4.4.2. Magnesium The optical excitation function for the 3 ~S---)33p~-transition was measured by Aleksakhin et al. [196]. In addition, the line emission excitation cross section was estimated and calculated at its maximum, taking into account the escape of excited atoms from the observation volume of the photon detector. The contribution of the cascade transitions to the population of the 3 3pl-level was evaluated as well. The maximum value of the latter does not exceed 8% of the maximum value of the excitation cross section; the implication involved is that the contribution of cascade transitions is small. A similar result follows from the data on the excitation of magnesium atoms by monoenergetic electrons [201]. According to (4.9) O'3p = 30r3P1 .

Theoretical calculations [190-193] of the excitation of the 3 3P-level of the magnesium atom were performed in the close-coupling approximation including the 3 1S-, 3 3p. and 3 1p-states. Their results are very sensitive to the choice of the theoretical model. In his calculations Robb [193] employed the R-matrix method and pointed out the necessity of taking into account the exchange of the incident electron with those in the atomic core. Figure 21 gives a comparative summary of the different results. As regards differential cross sections, there are experimental results of Williams and Trajmar [202] obtained at E = 10, 20 and 40eV with the electron spectroscopy technique, as well as results of Avdonina and Amusia [68] calculated in the first-order approximation of the many-body theory. Comparison of the theoretical and experimental results [68] shows their satisfactory agreement. The values of the total cross sections for the same energies are given in table 16.

4.4.3. Calcium Suppose the 4 apl-level with a lifetime of 1.5 × 10 -4 s to be metastable. In this case formula (4.7) holds. The excitation cross sections of the 4 3pl-level were measured by investigating the optical excitation function [197] within the energy range from the threshold up to 15 eV. Measurements of the cross sections for all triplet 4 3p-sublevels performed by detection of metastable atoms (E ~ 20 eV) are known as well [203]. Another study [190] reported calculations of the excitation cross section of the 4 3P-level Table 16 3 3p excitation cross sections for the magnesium atom, in *ra20 E (eV) Experiment [202] Calculation [68]

10 3.1 3.1

20 0.8 0.5

40 0.03 0.05

I.I. Fabrikant et al., Electron impactformation of metastable atoms

2~

.

I-

I

30

,

,

p

,

43

,

,

,

,

I

I

,,.¢-Q

1E

VSY

R 2C ,

1;

1C //

4 0

I

5

!

E(eV|

10

Fig. 21. 3 JP excitation cross sections for the magnesium atom. R, calculation of Robb [193]; F, calculation of Fabrikant [190, 192]; x, experiment [202] and theory [68].

I

I

I

0 1.5 2

3

t, 5 E(eV) 10

I

20

Fig. 22. Integral excitation cross sections for the calcium atom. P, 4 ap-level, close-coupling calculation (Fabfikant [190]); (P), the same calculation averaged over the electron energy resolution in the experiment [203]; R, 4 3P-level, experimental data [203]; D, 3 ~D-level, calculation of Fabrikant [190]; VSY, 3 1D-level, calculation from eq. (4.8).

in the close-coupling approximation including the 4 tS-, 4 3p. and 4 tP-states, and of the excitation cross sections of the 3 ID-level where the 4 tS-3 1DW-4 ZP-coupling was taken into account (the superscript w in 3 ~D means that in the given approximation all exchange terms associated with the 3 ~D-level have been neglected). Kazakov and Christoforov [204] measured the energy dependences of the differential excitation cross sections of the 4 3p. and 3 3D-levels at O = 90° in the energy range from the threshold up to 6 eV. Similarly as for Mg the contribution of cascade transitions to the 4 ap-level is small [197]; therefore, 0"m is mainly determined by the excitation of the 4 3p. and 3 ~D-levels. Figure 22 presents the data on the excitation cross section for these levels. The experimental data [197] are likely to be overestimated, as was the case with Mg, which was due to the difficulty of accounting for the escape of excited atoms from the observation volume of the photon detector. At high energies the main contribution to 0"m is given by the cross section 0"3'D since o~e decreases more rapidly. One may use formula (4.8) to describe the high-energy behavior of the cross section 0"3lo. Comparison with the experimental data on Ba shows that this formula results in values that are 2-3 times overestimated, up to 50 eV. At high energies the accuracy of the formula is seemingly about 50%. Such a slow approach of the cross section to the Born asymptotic value is probably explained by a great effect of the 4 tP-3 XD-coupling on the excitation of the 3 1D-level. 4.4.4. Strontium The lifetime of the 5 3pl-state of the Sr atom is 0.83 x 10 -5 s [92]; thus, we may suppose that 0.m = 0"53Po+ 0"53P2 + 0"41D2 •

(4.10)

In the LS-coupling approximation we have 0"53P2 + 0"53P0 = 3~0"3P --'~20"3P1 '

(4.11)

44

1.I. Fabrikant et al., Electron impact formation of metastable atoms

The error given by formula (4.11) amounts to several percent since, according to ref. [205], the coefficient accounting for the admixture of the 5 1Pl-state to the 5 3pl-state is as small as 0.037. The data for excitation of strontium metastable states are very scarce. We have only the calculations of ref. [190] of the excitation cross sections of the 5 3p. and 4 1D-levels, carried out in the 5 1S-5 3p_ 5 1p_ and 5 1S-4 1DW-5 IP-coupling approximations, and the calculations by Peterkop and Liepin'sh [249] in the Born-Ochkur approximation, whose results are given in table 17. Excitation of the 5 3p2,1,0 and other lower excited states of the Sr atom was observed in ref. [206], where the scattered-electron loss spectra were recorded for a scattering angle of 90 ° in the energy range from 6 to 15 eV. As is seen from table 17, excitation of the 4 ~D-level gives the main contribution to 0-m at an energy as large as 5 eV. Therefore, at high energies the cross section is governed by formula (4.8). According to the estimate given in ref. [198] and formula (4.11), the excitation cross section 0-53~,2.0 is equal to 124 x 10-16 cm 2 at E = 2.8eV (0.206Ry), i.e., it is one order of magnitude larger than the theoretical prediction (refer to the discussion of the discrepancy between experimental data and theory in the case of Mg and Ca). 4.4.5. Barium As was already mentioned, the 5 3D 1,2,3 - and 5 1D2-1evels are metastable in Ba. The radiative decay of the 6 3p2,1,0-1evels proceeds mainly to the 5 3D1,2,3-1evels. Therefore O"m "~" 0"5 3D1,2,3 "3i- 0"6 3P0,1,2 +

(4.12)

0-5 1D 2 "

The contribution of cascade transitions from higher triplet levels is rather small [199].

Table 17 Calculated 5 3p and 4 1D excitation cross sections for the strontium atom, in 1 0 - 1 6 c m 2. R e f . [190]: close-coupling approximation; ref. [249]: Born-Ochkur approximation; ref. [28]: calculation from eq. (4.8) 5 3p

4 ID

E (eV)

[1901

[2491

[190]

[281

[2491

2.04

15.2

-

-

-

2.79

-

17

32.6

-

3.0

8.57

55.4

-

37.8

5.62

3.40

7.27

-

11

51.0

-

4.08

6.22

-

8.2

44.7

-

4.76

1.29

-

7.0

39.6

-

0.59

-

6.8

35.4 29.7

7.79

5.0 5.44 7.0

-

15.55

-

-

5.83

10

-

2.00

22.2

6.59

20

-

0.25

11.8

4.13

30

-

0.075

50

0.016

100

-

-

-

200

-

-

400 600

-

-

8.34

7.98

2.97

4.85 2.45

1.89 0.99

-

1.23

0.51

-

0.62

0.26

-

0.41

0.17

I.I. Fabrikant et al., Electron impact formation of metastable atoms i

r

f

i

I

I

10

20

45

i

63-

oo .;2 1C

"'-

5

o

o

4 I 2 ,

I

b~Jp 2

OI

I

3 4 5

+

EleV)

f

100

Fig. 23. Inte~al excitation cross sections for the barium atom. O, 6 3P-level, 6 :S-6 SP-61P-coupling approximation (Fabfikam [190]); • . . . . 51D-ievd, 61S-5 ~DW-61p-coupling approximation (Fabfikant [190, 208]); ,5 ~D-levd, calculation from eq. (4.8); +, experimental data of Je~en et aL

[207].

According to experimental data [207], at energies as small as 20 eV the excitation cross section of the triplet levels is one order of magnitude smaller than o"51D2. However, experimental data are not available at lower energies. An estimate of °'63~,1[199] gives the value of 8.2 x 10 -16 cm2 at E = 5.8 eV (0.43Ry); then in the LS-coupling approximation one has o.63p=24.6x 10-16cm2. Trajmar and Williams [8] presented the results of their measurements of the differential cross sections for the 5 :D-level at E = 5 eV. As far as the range of intermediate energies is concerned, there exist experimental data on the differential cross sections due to Jensen et al. [207]. Results on the total cross sections are given in fig. 23. 4.4.6. Zinc and cadmium The production of metastable atoms of zinc and cadmium was investigated in a number of studies [209-216]. For instance, Barrat and Duclos [209] used the method of photoabsorption of the 4810 A, line, corresponding to the 4 3p2-5 3Sl-transition, to study metastable zinc atom production. They obtained the following value for the cross section for excitation of the 4 3p/-level: O.4 3P2 "~- (18

- 5)

x 10 -16 c m 2 .

Employing the same method, Barrat et al. [210] measured the excitation cross section in cadmium for the 5 3p2-state. They investigated absorption of the 5085 .A, line (5 3 p 2 - 6 3Si-transition). The results in ref. [210] concerning the zinc atom confirm those obtained earlier in ref. [209], while for the Cd atom the following value of the cross section was reported: o"53v2= (31 --+5)

x 10 -16 cm 2 .

In both cases, the cross section was derived from the following expression for the collision probability Z: Z = 4No.~,

(4.13)

46

I.I. Fabrikant et al., Electron impact formation of metastable atoms

where N is the density of atoms, T is the absolute temperature, R is the universal gas constant and M is the molecular mass. Expression (4.13) was extracted from the experimentally measured time dependence of NX/-T. Therefore, the values of 0.43~,2 (Zn) and 0.53p, (Cd) given in refs. [209,210] were, in fact, averaged over a Maxwell distribution. As regards the Zn atom, there also exists the study of ref. [211], where the following values for the cross sections were obtained: 0.43P2 =

1.5

x 10 -16 cm2

0"4 3P0 = 0 . 5 X 10 -16 c m 2

It is seen that these results are in strong disagreement with the data of refs. [209,210], even if one takes into account the error of 60-70% stipulated by the authors of ref. [211]. Employing a high energy resolution technique, Shpenik et al. [212] studied the optical excitation functions of the naPl-n 1S0-transitions in Zn and Cd as well as those of a large number of spectral lines giving the cascade contribution to the metastable 3P2,0- and resonant 3P1-states. The analysis of the results obtained showed that the cascade contribution may be neglected in the near-threshold region of energies since their magnitudes do not exceed several percent. Figure 24 shows the Born approximation calculations of 0.%.~ versus electron energy reported by Savchenko [195] for zinc and cadmium. In table 18 they are compared with the data obtained in the experiments of refs. [211,213]. According to the data in ref. [195], the ratio of the excitation cross sections for the 4 3Po- and 4 3p2-1evels in zinc is 1:5, while in the experiment of ref. [211] this ratio was found to be 1:3. At the same time, for the Cd atom this ratio is reportedly equal to 1:1.5 [213]; however, the ratio of the statistical weights calculated in the LS-coupling approximation amounts to 1:5, while the ratio of the calculated cross sections equals 1:4, according to the data given in ref. [195]. Williams and Bozinis [214] have measured differential cross sections for excitation of the 4p 3p. multiplet of Zn at E = 40 eV using the electron spectroscopy method. The value of the integral cross section for this energy is 0.086 x 10 -16 c m 2.

4.4.7. Mercury The Hg(6 3P2)- and Hg(6 3p0)-states have lifetimes of 5.67 and 5.56 s, respectively [217]; therefore, they can be regarded as metastable. Their excitation energies are 4.6670 and 5.460 eV [217]. Transition I

1.0

I

i

I

I

1.£

I

3 p2

Cd

!

0

~---R E

~a° o-~ O2 I

0

5

i0

~

I

15

E(eV} 20 E

0

I

I

I

I

4

5

6

7

I

EleV) 9

Fig. 24. Integral cross sections for metastable Zn and Cd level excitation (Born approximation [195]). (a) J = 2; (b) J = 0.

1.1. Fabrikant et al., Electron impact formation of metastable atoms

47

Table 18 Effective excitation cross sections for metastable levels of zinc and cadmium (maximum values as a function of energy), in 10 -16 cm2 J=2

J=0

Atom

Experiment

Theory

Zinc

1.5 -+0.3 [2111 18 -+ 5 [209,2101

2.08 [195]

0.5 -+ 0.2 [2111

0.44 [1951

4.80 [195]

1.2 -+0.8 [213]

1.13 [195]

Cadmium

1.8 -+ 1.2 [213]

31-+5

- Experiment

Theory

[21Ol

from these levels to the ground state 6 1S0 may occur with emission of the 2269.8/~ and 2655.6/~ lines forbidden by the J selection rule. Garstang [218] calculated the probability for the magnetic quadrupole transition 6 3p2-6 1S0, which turned out to be 42 times smaller than that of the electric dipole transition. As regards the probability for the magnetic quadrupole transition 6 3P0-6 1S0, it is reportedly even smaller than the electric quadrupole transition [217]. Borst [219] presented the data concerning the higher metastable level 6' 3D3, whose excitation energy amounts to 8.79 eV. In earlier studies (e.g. Borst [220]) the 6' 3D3-state was not considered because of the error in the spectroscopic data. The excitation cross sections for metastable states are mainly due to the exchange interaction [195]. Similar to the case of the atoms discussed above, the population of metastable states results not only from direct excitation but also from cascade transitions. Thus, from the structural conformity of the integral excitation cross section of the n 3S1-, n 3Di_and 6 3pl-levels and the excitation function of the 63p-6 1S0-transition , Jongerius [221] concludedt'hat in the energy range from 8 t o 15eV cascade transitions contribute significantly (>50%) to population of the 6 3pl-level. One might believe that the contribution of cascade transitions to the population of the 63p2- and 6 3p0-1evelshas approximately the same value. However, the suggestion is not well founded since the absolute value of the cross section of the resonant 6 3p1-6 1So-transition was not measured in ref. [221] because of the self-absorption of the resonance line. Figures 25 and 26 present the excitation functions of the 6 3P2- and 6 3p0-1evels of the Hg atom obtained by the use of different procedures, such as secondary electron emission [220], superelastic electron scattering [222], and an optical method [217] based on measurements of the intensity of the 2655.6 and 2269.8 :k lines. Figure 25 also presents the calculation by McConnell and Moiseiwitsch [50]. It is seen that the data obtained by Krause et al. [217] for the 6 3p2-1evel agree well with those due to Borst [220], the threshold region apparently being an exception, while the results reported by Korotkov and Prilezhayeva [222] differ significantly from those of refs. [217, 220] in the energy range E > 5.8 eV. A similar disagreement between the results of Korotkov and Prilezhayeva [222] and Krause et al. [217] is also seen in the energy dependence of the 6 3p0-excitation (fig. 26). Near the peak of the cross section (at E = 5.8 eV for 3P2 and at E = 5.2 eV for 3p0), the ratio of the J = 2 to J = 0 cross sections 0-2/0-o is 5.2 [217]. According to Korotkov and Prilezhayeva [222] this ratio is equal to 6.7 (0-3pz= 2.8 x 10 -16 cm 2, X -16 rap0 = 0.42 10 cm2). The values of the ratio o-2[o-o at E = 5.8 eV (maximum of O'3p2) were 9.11 and 22, respectively. Figure 27 presents the excitation function for the 6' 3D3-1evel taken from Borst's paper [219]; the behavior of this function is in good agreement with that described in the later study by Koch et al. [2231.

48

I.I. Fabrikant et al., Electron impact formation of metastable atoms

I

I

I

I

I

1.0

,- IJ0

°iI

. ..,..,~

4

o°°

"

5

"I~--" I-

o.o

•

• f I~ ,

"" ----... .

"'.

•

,,

6

7

8

-....

.'......

•

E(eV)

0

ooL * °

Fig: 25. 6 3p2-1evel excitation cross sections for the mercury atom (normalized at maximum values) [217]. - - - - - , calculation of McConnell and Moiseiwitsch [50]; . . . . , experimental data of Kranse et al. [217]; . . . . , measurements of Korotkov and Prilezhayeva [222] (trm); , experiment of Borst [220].

I

5

4

6

E{eV)

7

8

Fig. 26. 6 ~'P0-1evel excitation cross sections for the mercury atom (normalized at peak values) [217]. -----., calculation of McConnell and Moiseiwitsch [50]; ..., experiment of Krause et al. [217]; experimental data of Korotkov and Pfilezhayeva [222] (tr,).

Within the framework of the Born-Ochkur approach, Savchenko [195] calculated the excitation cross sections of the 6 3P2- and 6 3p0-1evels(see figs. 25 and 26) and arrived at the following values. For the 6 3p2-6 1So transition the maximum of the cross section amounted to 7.4 x 1 0 -17 c m 2 at E = 6.1 eV, whereas Borst [220] reported the value (3.2 +- 1.2) x 10 -16 c m 2 and the calculation of McConnell and Moiseiwitsch [50] resulted in 3.08 x 10 -16 cm2. For the 6 3p0-1evel Savchenko's computations yielded 0.23 x 10 -16 cm2 for the excitation cross section, against the value of 0.88 x 1 0 -16 c m 2 obtained by McConnell and Moiseiwitsch [50]. The mean data concerning calculated and experimentally measured values of the excitation cross sections for these two metastable levels are compared in table 19. Noteworthy is that, according to the data of ref. [195] in mercury the effect of the spin-orbit coupling is rather important and taking it into account leads to the following ratio of the excitation cross sections for these levels: o'3P0: tr3p2= 1:3,

(4.14)

whereas the LS-coupling approximation predicts the ratio of the statistical weights of both levels to be equal to 1 : 5. Now consider more recent experimental work and theoretical calculations. Webb et al. [224] and 16

I

I

t

I

I

4

0

8

g

I

~o

E(ev)

Fig. 27. Cross section for the formation of the 6' 3D~-state of the mercury atom, data of Borst [219] (experiment).

I.I. Fabrikant et al., Electron impact formation of metastable atoms

49

Table 19 Effective cross sections for excitation of metastable states of mercury (maximum values as a function of energy), in 10 -16 cm 2 Level

Experiment

Theory

6 3P2

3.2 -+ 1.2 [220]

0.74 [195] 2.48 [2161 3.08 [50]

6 3Po

2.8

[222]

0.41

[2221

0.23 [1951 0.24 [216]

0.88 [501

Hanne et al. [225] investigated the excitation of the 6 3P2- and 6 3p0-1evels of Hg by the electron-laser stepwise excitation method developed earlier by Phillips et al. [18, 166] for Ne (see section 2). After the electron excitation to the 6 3P2- (6 3Po-) level, the atom was transferred to the intermediate 7 3Sl-level by the laser, and the fluorescence due to the transition to the 6 3Po- (or 6 3p2-) level was detected [224, 225]. Webb et al. [224] have obtained relative excitation cross sections for the Zeeman sublevels of the 6 3p2-1evel in the electron energy range from 5.50 to 6.75 eV and the ratios of these cross sections to the total excitation cross section at threshold (5.46 eV). Their results are in good agreement with the relativistic R-matrix calculations of Scott et al. [321]. Some disagreement at threshold was explained [224] by the influence of the (6s6p 2) 2Ds/2-resonance at 5.5 eV. The improved calculations of Bartschat and Burke [322], with accurate inclusion of this resonance in the R-matrix expansion, have removed this disagreement. The R-matrix calculations of refs. [321,322] are most accurate for Hg because they include all important effects: close coupling, exchange, correlation and spin-orbit interaction. Hanne et al. [225] have obtained relative excitation cross sections for the 6 3P2- and 6 3p0-1evels at energies from threshold up to 8 eV. A small maximum in the excitation cross section for the 6 3p0-1evel at 4.7 eV, which is due to (6s6p 2) 4p3/2 resonance, was observed. A broad peak at E = 5.2 eV due to ,

1.0

,

,

,

,

06

C12 =

~°%t,°,°°°°,~.°,° ..:

0

,

,

,

I

l

,

,

°" I °°°°,

l

1.0

0.6 ~

,°%°°°

°'° °°°.°

° °°°-.°°°°° °°°

0.2 0

5

6

7 ElgV)

Fig. 28. Measured fluorescence emission intensity ( e ) normalized to calculations of Scott et al. [321] ( excitation [225].

) for Hg(6 3p2) and Hg(6 3Po)

50

I.I. Fabrikant et al., Electron impact formation of metastable atoms

(6s6p 2) 2D3/2 resonance was also found. According to Hanne et al. [225] the maximum in the excitation cross section of the 6 3p0-1evel (5.2 eV) was masked in the results of Koch et al. [223] and Newman et al. [319] by the increase of the cross section at the 6 3p2-threshold (5.46 eV). Besides, the short lifetime of the 2D3/2 resonance does not allow one to observe it in other scattering channels. The results of Hanne et al. [225] are in good agreement with the earlier measurements of Krause et al. [217] and the R-matrix calculations of ref. [321] (see fig. 28). Reasonable agreement with the theoretical data of ref. [322] is observed also for the ratios of the cross sections for the Zeeman sublevels with M = 2, 1, 0. However, the experimental data are larger than the theoretical values by a factor of 1.2 to 2. Thorough studies of the excitation of the lower excited states including the metastable ones were performed by Shpenik et al. [226] and Kazakov et al. [227]. These and a number of other relevant investigations carried out with high energy resolution (e.g. ref. [228]) intended to resolve the fine structure of the excitation cross sections of levels in the Hg atom. Their results will ]~e discussed in section 5. 4.5. Some other atoms

In this section we do not aim at reviewing in detail the studies on the excitation of metastable states in some complex atoms, but just touch upon those most important for applications. 4.5.1. Atoms with an unfilled p-subshell

Experimental data on metastable excitation of these atoms are extremely rare since the production of C, N, O, and Si atomic beams entails considerable experimental difficulties. There is only one experiment performed by Stone and Zipf [230], who determined the excitation cross section of the 3s 5S°-state in the oxygen atom by measuring the optical excitation function. Inasmuch as the electric dipole transition from this state to the ground state occurs due to violation of LS-coupling, its lifetime is relatively long (-1.9 x 10-4s) and this state may be regarded as metastable. However, electron impact-induced excitation of the levels with much longer lifetimes, belonging to the ground configuration of the atoms under consideration, has not been reported yet. Spence and Burrow [231] obtained atomic nitrogen in discharge and using the electron spectroscopy method measured the excitation cross sections of some levels. However, they did not manage to observe excitation of metastable states. Equally variable are the experimental results of Wells et al. on the translational spectroscopy of metastable fragments produced under dissociation of atmospheric molecular components [232,233]. Carbon, nitrogen, oxygen. Excitation of the C, N and O atoms was considered theoretically by Henry and coworkers [234,235] with allowance for close coupling of the terms belonging to the ground configuration and for a part of the correlation effects (3P~-ID~-~S~ for C, 4S°-2D°-2P° for N, 3P~-~DC-IS~ for O). This method does not enable one to take into account the long-range dipole polarization but it seems to be expedient at sufficiently high energies. Therefore, the formula for the high-energy behavior of the excitation cross sections, given in ref. [235], seems to be useful: tr = A / E 3

[10 -16

cm/l,

where E is the energy and the values of the parameter A are given in table 20.

(4.15)

I.I. Fabrikant et al.,Electron impact formation of metastable atoms

51

Table 20 Values of the parameter A of eq. (4.15),data from reL [235] Carbon

A

Nitrogen

Oxygen

3p_1D

3p_tS

4S_2D

4S_2P

3p_tD

3p_1S

5950

732

12800

4340

7500

900

Q I

1.0

\

I

I

I

I

I

I

I

b

\

1.0

\ \ \ ,%

LO

O~

\ ,%

I' O'(=pJs)

2

3

4

5

O.2

6

ll I

7

E(eV)

"~ . . . . . . . .

I

//

2

4

6

8

10

E(eV)

Q32"C

I

I

1

/~

T

f

!

I

I

~(JP'JD)

-- "" ..

©

O~ 0:!~ Q04 0 2

/

7 ,

OOP ,-,"~-i'-~--~

4

6

.

s)

,

8

1'0

E(eV) Fig. 29. Calculated cross sections for metastable-state excitation for C, N, and O. (a) e-C excitation: , data of Thomas and Nesbet [60]; , data of Henry et al. [235]; (b) e-N excitation: - - - , data of Thomas and Nesbet [61]; - - - - - , data of Ormonde et al. [236]; (c) e-O excitation: ~ . , data of Thomas and Nesbet [62]; - - - - - - , data of Vo Ky Lan et al. [337].

I.I. Fabrikant et al., Electron impact formation of metastable atoms

52

Inclusion of the terms of excited configurations into the close-coupling expansion makes it possible to take part of the long-range polarization into account. When calculating excitations in nitrogen Ormonde et al. [236] used the multi-configuration close-coupling theory to include, in addition to terms of the ground configuration 2s22p3, the terms , p c and 2p~ belonging to the configuration 2s22p23s, as well as the term 4p of the configuration 2s2p 4. In oxygen they took into account the term 3D belonging to the configuration ls22s22p3(4S°)3p and the term 3 D ° from the configuration ls22s22p3(2D°)3s. Other possible ways to account for the long-range polarization lead to the method of polarized orbitals and solution of the continuum Bethe-Goldstone equations. Pindzola et al. [34] calculated the excitation of the 1D-state of C and Si atoms by the distorted-wave method allowing for polarization in the adiabatic exchange approximation. Thomas and Nesbet [60-62] presented results of the variational solution of the continuum Bethe-Goldstone equations for oxygen (from threshold to 11 eV), N (from threshold to 10 eV) and C (from threshold to 7 eV). Their calculations seem to be more accurate since they allow for the main effects of short-range correlations and long-range polarization. The various experimental results for C, N and O are compared in fig. 29. !

I

i

I

I

I

30

40

0.8

--3 0

l_O O~

0

•

/

\\

0

0.2

l

10

I EII~dl

20

50

E(~I

Fig. 30. 3P--3s SS° excitation cross sections for the O atom [238]; , dose-coupling calculation of Rountree [238]; I , distortedwave data of Sawada and Ganas [237]; O, experimental results of Stone and Zipf [230], scaled by 1/5.6.

Fig. 31.3P-~D excitation cross sections for the Si atom [34]. - adiabatic-exchange distorted-wave method; . . . . , Hartree-Fock distorted-wave method; . . . . , single-configuration close-coupling method [240].

Table 21 Calculated cross sections for excitation of metastable states of silicon, dose-coupling data of John and Wilfiams [240], in units of Ira~ E (eV)

3P-1D

3p-lS

1.23 2.18 3.40 4.9O 6.67 8.71

0.306 0.565 0.741 0.841 0.868 0.833

0.032 O.06O 0.073 0.079

I.I. Fabrikant et al., Electron impact formation of metastable atoms

53

Figure 30 presents the data on the excitation cross section of the comparatively long-lived 3s 5S°-state of the oxygen atom. As was mentioned above, this is the only example in the group of atoms under consideration where experimental results can be compared with theoretical predictions [230]. Sawada and Ganas [237] carried out their calculations in the distorted-wave approximation, while Rountree [238] employed the close-coupling method including the states 3p, 3s 5S° and 3s 3S° Rountree [238] supposed that disagreement between experimental results and theory was mainly caused by the contribution of cascades and the undetermined lifetime of the 3s 5S°-states; knowledge of the latter is needed for processing experimental data. We have already pointed out that a similar problem arises when one determines the excitation cross sections of 3pl-levels in atoms of the second group (see section 4.3). Using a quasistatistical approximation Andreev and Bodrov [239] obtained approximate analytical formulae for the excitation cross sections of metastable levels. The results for the N and O atoms are in rather good agreement with the numerical calculations. Silicon. John and Williams [240] calculated the excitation of the silicon atom using a single-

configuration version of the close-coupling approach (table 21). Their results show considerable disagreement with the data of Pindzola et al. [34] (see fig. 31). in conclusion we will briefly discuss the situation with atoms of the third group. The ground o configuration of these atoms does not give metastable terms (with the exception of the sublevel 2P3/2 of the ground term, see later); however, the term 4pe of the configuration n0sn0p 2 is metastable in B and A1. In Ga, In and Tl these terms decay rather quickly due to the violation of the LS-coupling and such transitions are observed in investigations of the optical excitation functions [241,242]. Figure 32 presents the excitation cross section of the 2s2p 2 4Pe-level of boron calculated by Ryabikh [243] in the Born-Ochkur approximation. We have no information pertaining to excitation of the 3s3p 2 4p%level of Al.

1D

1

05

0

10

20 E(eV)

Fig. 32. (1) 2s22p 2Pl:z-2s2p2 4Pt/2 and (2) 2s22p 2pa/2-2s2p2 4P~:2 excitation cross sections for the B atom [243].

54

I.I. Fabrikant et al., Electron impact formation of metastable atoms

!

i

I

I

i

I

.

I

I

I

I

15

w-~

I0

f

102

,~"

5

I

5

10 20

!

50 100 2130 T(IO2KI

Fig. 33. e-O collision strength for the 3P2-3P1 transition [49]. ~ - computation from eq. (4.16); - - . - - , K Ls matrix method, data of Tambe and Henry [244b]; , jj-coupling scattering equation method, data of Tambe and Henry [244a].

,

L I

2

3

E{eV)

Ill

4

,

5

Fig. 34. Total cross section for the (6p)2P1,2-(6p)2Ps/2 transition in the TI atom [54] with J= resonance definition. (J is the total momentuna, ~r the electron-atom total parity: + even, - odd).

4.5.2. Atoms with an unfilled p-subshell; excitation of fine-structure components of the ground term In the atoms with an unfilled p-subsheU, the ground term exhibits a fine structure with metastable excited sublevels. The calculations listed above neglect the fine-structure effects. There are two possibilities to take this structure into account: either by transforming the scattering matrix from LS to j-j coupling or by solving the equations of the close,coupling theory within the framework of the j-] coupling. LeDourneuf and Nesbet [49] tried the first possibility in order to calculate the collision strengths for 3p2-3p0-transitions in oxygen, which are important for understanding the mechanism of electron cooling in the Earth's atmosphere. In their calculations they made use of the matrix K Ls computed by Thomas and Nesbet [62]. The collision strengths obtained were modified to allow for the fine-structure splitting. It turned out that at low temperature T they resemble the results due to Tambe and Henry [2214a], who solved the equations derived in the framework of the j-j coupling. This shows that in this case the effect of spin-orbit interaction may be adequately treated by means of perturbation theory. The results of calculations concerning the 3P2-aPl-transition are given in fig. 33, which also supplies the data calculated by Tambe and Henry [244b] with the aid of a transformed K cs matrix that had originally been constructed by the method of pseudostates. The calculations of ref. [49] are more accurate since they take short-range correlation effects into account more completely. The approximate formulae for the collision strengths due to 3p2-3p1,0-transitions are as follows:

O(3P2-3P,)~ 9.76 x 10-6(T -

228) + 3.46 x 10-'I(T - 228) 2 ,

(4.16)

/2(3pE-3po) ~ 3.29 × 10-6(T - 326)- 2.90 × 10-n(T - 326)2 .

(4.17)

The relativistic R-matrix approach (Burke et al. [41]) makes it possible to take the effect of spin-orbit interaction in heavy atoms consistently into account. This approach enabled Bartschat and Scott [54] to calculate the excitation cross sections of the 6p 2P3/2 sublevel of the thallium atom with

1.I. Fabrikant et al., Electron impact formation of metastable atoms

55

account of the coupling between the 6p 3p1/2,3/2- , 7s 2S1/2.3/2", 7p 2p1/2,3/2_ and 6d 2D3/2.5/2-states. The results are given in fig. 34. Experimental data for the 6p 3p3/2-1evel of the TI atom were obtained by Vu~kovi6 et al. [245], who carried out electron spectroscopy measurements of the cross section in the angular range from 5° to 120°.

4.5.3. Copper, lead, bismuth, manganese Excitation of the metastable 3d94s2 2D3/2,5/2-levels of copper was experimentally studied by Trainor et al. [8,246,247] by electron spectroscopy; the corresponding theoretical calculations were carried out by Leonard [248], who utilized the classical model, by Trainor and coworkers, who employed the impact parameter method (their results were reported in ref. [8]), and by Winter (also reported in ref. [8]) and Peterkop and Liepin'sh [249] in the Born approximation. More sophisticated calculations were performed by the close-coupling method including the (3dl°4s)2S-, (3d94s2)2D- and (3dl°4p)2p-states. Some preliminary results were due to Smith and Wade [250, 251]. In particular, they calculated the excitation cross sections at E = 60 eV, which are in fair agreement with the data obtained by Williams and Trajmar [246]. Recently Msezane and Henry [252,253] presented new calculations. They also investigated the effect of inclusion of the 3d~°4d-state into the close-coupling expansion at small energies. Figure 35 provides the data on the total excitation cross sections of the 3d94s2 2D-state in copper. The effects of spin-orbit interaction in this case are small; thus the following relation holds with good accuracy: O'2D5/2 " O'2D3/2 = 3 : 2 .

The same figure gives the total cross section O'2D ~ O'2D3/2 "~- O'2D5/2 •

Trajmar et al. [247] obtained the absolute values of the cross sections by extrapolating the data on the differential excitation cross sections of the resonance transition in Cu to O< 15° and determining the integral cross sections. Msezane and Henry [252], having compared the cross sections for 0 < 15° with the extrapolated values obtained by Trajmar et al., found the correction factor for the experimental values. The cross sections renormalized in this way are given in fig. 35. To make the picture more complete, we also present the data on the excitation cross sections for higher energies, as calculated by Peterkop and Liepin'sh [249] (table 22). The data on the differential excitation cross sections of Cu at three energy values are given in fig. 36. The outer electrons in the atoms of Pb and Bi have configurations 6s26p2 and 6sZ6p3, respectively. Therefore, they resemble, to a certain extent, the C and N atoms. However, the Pb and Bi atoms are rather heavy and the magnitude of the fine-structure splitting of the terms is comparable in this case with that of the electrostatic splitting of the terms of the ground configuration. Therefore, the effects of Table 22 Cross sections for excitation of the 2D-state of the copper atom (in 10-17 cm2)7 calculated in the BornOchkur approximation [249]

E (eV)

100

200

400

600

o"

0.57

0.23

0.12

0.18

56

1.I. Fabrikant et al., Electron impact formation of metastable atoms

101 I

I

I

I

I

I

,go

I

I

I

20

,~0

60

EteV)

Fig. 35.4s-3d94s 2 excitation cross sections for the copper atom [252]. Curve A: Born approximation (Winter, quoted in ref. [247]); curve B: Born approximation [249]; curve C: impact parameter method (Winter and Hazi, quoted in ref. [252]); curves D, E: four-state and three-state close-coupling approximations; (3, experimental data of Trajmar et al. [247]; 0, renormalized experimental data of ref. [247].

I

Id

0

E

I

o

0

30

60 90 g(deg)

120

Fig. 36. Differential scattering cross sections for s-d excitation of copper [253]. Solid curves: close-coupling approximation; A: E = 6eV; B: 20eV; C: 100 eV. 0, experimental data of Trajmar et al. [247] at 6 eV; ©, the same at 20 eV.

spin-orbit interaction need to be taken consistently into account to calculate the excitation cross sections in these atoms. Such a c~tlculation performed for Pb in the framework of the R-matrix method involving the Breit-Pauli Hamiltonian was reported by Bartschat [55]. All the five terms (3P2,t,0, tD2, XS0) of the ground configuration were taken into account in the expansion of the total wave functions. Figure 37 shows the total excitation cross sections of all metastable levels for energies ranging from threshold up to 7 eV. Trajmar and Williams [8,254] used the electron spectroscopy procedure to measure the excitation differential cross sections for four metastable levels, 3p~, 3p2' ~D2' and tS 0 at E = 40 eV. Normalization of the absolute values using the value of the oscillator strength of the resonance transition and extrapolation of the differential cross sections to small and large angles made it possible to obtain the integral cross sections [254] of three levels: o'3et= 0.15, O"3P2= 1.5, o'~D2= 0.050 (in units of 10 -16 cmZ). The differential excitation cross section of the tS0-1evel acquires very small values for 0 < 80°, and the authors did not succeed in estimating the corresponding total cross section. As regards the Bi atom, we only know of the measurements carried out by Williams et al. [8,255], concerning the excitation differential cross sections of the metastable levels 2D3/2, 2D5/2, and 2pu2 of the ground configuration at E = 40 eV. The values of the total cross sections, obtained in the same way as in ref. [254], were O"2D3/2 = 0.29, O'2D5/2-----0.46, O'2P1/2----0.16 (in units of 10-16 cm2). Concluding this section, we will consider the manganese atom as an example of an atom with an unfilled d-subshell. The ground term of the Mn atom is 6S5/2, belonging to the configuration 3d54s2. The first excited configuration 3d6(SD)4s gives rise to five metastable 6D1-terms: J = 9/2, 7/2, 5/2, 3/2, 1/2. Williams et al. [255] measured the total differential excitation cross section of these levels for E = 20 eV. The total integral excitation cross section proved to be equal to 52 x 10 -16 Cl~ 2. The next 8 o excited configuration 3d54s(7S)4p provides, among others, the term P9/2, which yields a cascade contribution to the population of the metastable l e v e l s 6D7/2,9/2. However, Williams et al. presented only the total excitation cross section of the aP~-levels, where J = 5/2, 7/2, 9/2.

I.I. Fabrikant et al., Electron impactformation of metastable atoms 1

I

i

i

i

i

I

I

[

hI 1

,

57 I

I

I

10

0.10

5

Q05

0 I,I, f, 1

LO

3

i

5 i

i

I

7

I,

[,

3

, I~

5

l

1.5 ~,Oe ~i.-ie

r o - . - r4

10

o

1.0 0.5

,),

), 3

, ;

7

0

E(eV}

I,I 1

,

I, 3

f

I

5

Fig. 37. Calculated excitation cross sections for different transitions in the lead atom [55].

5. Resonances in the excitation of metastable states

It has already been shown in section 3.7 that close to the threshold an excitation of atomic levels by electron impact can take place through the formation of short-lived compound states of the negative ion. The formation of these states leads to structures in the cross sections of elastic and inelastic scattering (pronounced maxima, minima, drastic changes in the rates of increase of the cross section), which have been called resonances [70]. Apparently, one of the first post-war studies which reported experimental observation of a resonance structure in the excitation cross section, initially called fine structure, was the pioneering research of excitation in the mercury atom due to Zapesochny and Frish [256,257]. Resonances in the energy dependence of the elastic scattering cross section were first observed by Schulz [258] in his studies of slow electron interactions with helium atoms. It should be noted that experimental detection of resonances in the cross sections of electron-atom collisions requires experiments with high electron energy resolution (AE < 0.1 eV). Zapesochny was the first to point out this fact in his paper [256]; he revealed a gradual smoothing of the fine-structure maxima in the excitation functions of the A = 5461 A line in mercury (6 3 p 2 - 7 351) and attributed this effect to an increase of the electron energy spread from 0.5 to 3.5 eV. It seems that this paper may be considered as a starting point for further experimental developments in investigations of resonance effects in elastic and inelastic electron-atom/molecule collisions by the use of electron beams with high energy resolution. With all this, a new branch of research in atomic physics, called spectroscopy of negative ions, appeared. An important point in studies of resonance processes was the establishment of the multichannel feature of the decay of negative ions, leading to the structure in the excitation cross sections of several final states simultaneously. This observation, combined with the graphical method suggested in ref.

I.I. Fabrikant et al., Electron impact formation of metastable atoms

58

[226] for the analysis of the measured cross sections versus energy, makes it possible to figure out, at least in principle, the main parameters of the resonances (their energy E r and width F) and often simplifies identification of the states of negative ions. In reviewing the physics of atomic resonances and the experimental developments involved one can hardly overestimate the contribution due to Schulz, whose pioneering studies (see refs. [70,259]) set the dominant experimental trends for many years. Theoretical studies had a considerable stimulating effect on the experimental developments in this field as well (see section 3.7). In the present review we have no possibility to dwell upon all the aspects of resonance phenomena involved in electron-atom collisions. Still it seems appropriate to discuss here the resonance phenomena occurring during production of metastable particles. 5.1. The hydrogen atom

The negative hydrogen ion is the simplest system for calculating the resonances in electron scattering by neutral atoms. Therefore, this system was considered in quite a few theoretical papers by the close-coupling method [260-263], its modification allowing for pseudostates [38,264], the method of complex rotation of coordinates [75,265,266] and the bound-state approach [267]. The energy range of special interest for us is that above the excitation threshold of the metastable 2s-level, here most attention was given to resonances between the n = 2 and n = 3 thresholds. Close to the n = 2 threshold there is a 1P shape resonance, while all other resonances are of the Feshbach type and are located below the n = 3 threshold. The theory developed by Gailitis and Damburg [87,268] predicts that for L <- 3 (L is the total orbital angular momentum) a sequence of dipole resonances should be observed below the n = 3 threshold whose position is governed by the exponential law e k = eoR-L k ,

k = O, 1, 2, 3 . . . ,

(5.1)

where ek is the energy of a resonance relative to the n = 3 threshold, and at n = 3 the values of R L are equal [159,179] to 4.8225, 5.164, 6.134, 9.323 for L = 0, 1, 2, 3, respectively (for any total spin S). The results of numerical calculations given in table 23 are well described by formula (5.1). (A comparison of the formula with the numerical data obtained prior to 1979 is given by Gailitis [159].) The applicability of formula (5.1) for higher values of k is limited by the magnitude of the relativistic splitting between the 3s-, 3p-, and 3d-levels in the hydrogen atom. While atomic hydrogen is a difficult object to deal with in experiments, still a number of papers reported on experimental studies of resonances in the formation of metastable states. The first study with good energy resolution was that of McGowan et al. [269], who revealed resonances in the excitation of the 2p-level between the n = 2 and n = 3 thresholds. The excitation of the 2s-state was investigated by the group of Fite [97]. Noteworthy are also the works of Cox and Smith [270] and Oed [271], who resolved the resonance structure in the excitation functions of the 2s-level. Thorough investigations of the resonance structure in the excitation functions of the 2s-state were carried out by Koschmieder et al. [272] with an energy resolution of 0.15eV. The best resolution (12meV) was attained by Williams [99]. Figure 38 and table 23 present the results of these works in comparison with other experimental and theoretical data. It can be seen that close to n = 3 the 1 S-, 1D-, 1P- an d 1S(3P)-resonances are well resolved. A certain resonance structure was also found by Williams [99] in the range 11.6-12.12eV (fig. 39), which was theoretically predicted by Callaway [38]. Noteworthy is

1.1. Fabrikant et al., Electron impact formation of metastable atoms

59

Table 23 Resonance energies E r (eV) and widths r (meV) of the hydrogen atom Theory

Experiment

Morgan et al. [264]

Callaway [38]

Complex rot. [381

[267]

Resonance

series

r

Er

r

Er

1~

Et

11.728 12.035

38.50 7.91

11.727

38.63

11.723 12.045 12.076 12.081

1S

11.727 12.032 12.077 12.081

38.91 8.31 2.108 0.916

3S

12.000 12.074

0.24 0.058

1p

10.220 11.898 12.011 12.072 12.083

19.99 32.51 0.243 0.059 1.86

11.756 12.042 12.070 12.083

0.24 0.122 10.221 11.898

32.92 0.122 0.299 2.204

44.76 8.31 0.114 1.65

11.758 12.043

46.25 8.67

11.757

11.810 12.058 12.087

44.48 6.57 0.73

11.811 12.06 12.09

4.40 6.65 0.57

11.810

3D

12.000 12.034 12.084

10.20 0.23 0.042

12.001 12.035 12.089

10.27 0.23 0.081

12.000

1F

12.066

0.136

12.067

0.136

3F

11.931 12.081

2.96 0.14

11.934 12.082

3.12 0.13

1D

[99]

[272]

11.65 11.77

11.74

11.7

E,

12.000 12.074

11.903 12.012 12.083 12.086

3p

[105] E,

14.14 31.83

12.0

11.908 12.011 12.078 12.084

43.23

11.89

12.06

11.813 12.058 12.088

10.3

11.85 12.0

11.750 12.041 12.071 12.083

43.53

E,

12.05

12.16

11.988 12.033 12.84 12.066

11.931

2.99

11.914 12.080

11.94

;i:" .o..] n =3

O

02

i).~

.

t .° "~"

-',

)

:!

0.25 -

=2

r

..

L~

"P

0.1

1D

0.05~

2/ 10

,

11

E(eV)

12

n=3

, 10

11

EteV)

I

12

Fig. 38. Excitation cross section for the 2s state of the hydrogen atom near the threshold. (a) From ref. [272]: - - - - - - , experiment of Oed [271]; . . . . , close-coupling calculations of Burke et al. [260, 261]; , theory folded with the experimental energy resolution [272]; $, ref. [272]. (b) From ref. [99]: O, experiment of KaupiUa et al. [97]; . . . . . , experiment of Williams [99]; - - - - - - , experiment of Koschmieder et al. [272]; experiment of Oed [271]; ,,, , close-coupling calculation of Burke et al. [260, 261].

60

1.1. Fabrikant et al., Electron impact formation of metastable atoms

0.5

I

I

I

I

I

I

i

0.4

I

:¢

12

0.-!

.,

t-

"x

i

C. '

/

C

N...~.~.~

\

\

/

i

i

i l

~I) ,I

0.1

0.855

sD4p sp'F I

'sl'Pl'oi I

0B65

I

'Pla I

0.875

? li li,;oli

/

:," i

//

;"

B"

~-

~.:-

I

,

0.885

k2 Fig. 39. Resonances in the 2s excitation cross section for tt [38]....., experimental points of Williams [991; , theory, Callaway [381. Vertical bars indicate the position of resonances.

:: I

I

20

21

20.0

l I

20.5 22

EleV)

Fig. 40. Cross sections for metastable helium formation [147]....., experiment of Brunt et al. [147]; - - - - - - , calculations of Berrington et al. [279]; - - . - - , calculations of Oberoi and Nesbet [59].

also the fact that some resonances given in table 23 were observed by Sanche and Burrow [273] in a transmission experiment. 5.2. The helium atom

The 2 p . and 2D-resonances are the most significant ones in the production cross section of the metastable helium atom close to threshold. The 2P-resonance, which gives rise to the first and main peak in the cross section (fig. 40), was first observed by Barranger and Gerjuoy [274] in the excitation cross section of the 2 3S-state and by Schulz and Fox [15] in the total cross section for the 2 3S- and 2 1S-levels. The first calculation of e - H e scattering carried out by the close-coupling method [144] resulted in a resonance energy value of 20.2 eV, while the FWHM was 0.52 eV; the 2D-resonance at E = 21 eV was found as well. Further experimental works [275 - 278] confirmed preliminary data on the symmetry of known resonances and helped to determine their widths more accurately. The Manchester group [147] reported that the energy of the 2p-resonance was 20.27 eV, while the width amounted to 0.77 eV. A recent theoretical result [46] gives 20.15 eV for the position of the ZP-resonance. However, the position of the peak in the calculated cross section is shifted to 20.32 eV owing to the contribution of other partial waves, which is in good agreement with experiment [147,148]. The calculations of refs. [71-73] showed that the ZP-resonance is of the Feshbach type supported by the 2 ~S-threshold. The threshold peak in the excitation cross section of the 2 tS-level at E = 20.62 eV (fig. 40) is caused by the virtual 2S-state. As shown in section 4.2, Phillips and Wong [141] reported that they found a new 2P-resonance with an energy of 20.8 eV and a width of 0.4 eV, which interferes with the known 2p. and 2D-resonances, thus resulting in the formation of a threshold peak in the excitation cross section of the 2 3P-level. However, the new 2P-resonance was not confirmed by the calculations. The next group of resonances is specified by the presence of threshold energies with n = 3 (fig. 41). Some of them were experimentally studied by Brunt et al. [147]. Theoretical calculations involving the target states with n = 3 were performed by Nesbet [59,153,280] with the use of the variational solution of the continuum Bethe-Goldstone equations, and by Freitas et al. [46, 47], who used the R-matrix

LI. Fabrikant et al., Electron impactformation of metastable atoms

61

0.10 0£)8 ('qo

i

i

i

i

I

i

!

i

He

~.~

--

H e ~

i'

i

m

0.0~

~

r-3

5

I

ii if} /!!i:

il','i"':'..',-.~.,.'.'-.'.': ':" •"' :'. i i!! / ;':-:"'. ii " ~' "

.... -:~".':.'-.~.

....... I " . l ' , . l " • '',.

~."

I! •

,j

~1

n=3

I

232

22

E(eV)

23

Fig. 41. A comparison of the eleven-state R-matrix calculation of Freitas et al. [46] on electron impact excitation of helium ( ) with the experimental cross section of Buckman et al. [80] ( . . . . ).

I

I

I

.

240

1

I

I

I

24.8

EleV) Fig. 42. Resonances in metastable helium yield, data of Buckman et

al. [283].

method. Table 24 presents the classification of found resonances and the comparison of experimental results with theory. The data of this table require some explanation concerning the 2S~-, 2po_ and 2D°-resonances in the energy range from 22.4 to 22.7 eV. Nesbet [153] found that the width of the resonances grows as one proceeds from S to D, while Brunt et al. [147] reported about almost equal widths for S and P and a considerably smaller width for D. As was shown by Andrick [281], the point was that in measurements of the total cross sections the resonance structure is obscured by the interference effects due to the proximity of the P- and D-resonances. Measurements of the energy dependences of differential cross sections made by Andrick [281] provide energies and widths close to the theoretical predictions. For the first time Buckman et al. [80] experimentally found resonances corresponding to two highly excited electrons (fig. 42). Each series of these resonances is labeled by the principal quantum number of the "lower" electron, whereas the values of the principal quantum number of the "upper" electron n b = n, n + 1 , . . . label the resonances within the series. Buckman et al. observed the series for n ranging from 3 to 7. According to Nesbet's classification [153], these resonances are of the valence type. In addition, there exist resonances of the nonvalence type, which are formed by the attachment of an electron in the polarization potential of the excited state (for example, ls3s~ 2S and ls3sO 2p according to Nesbet). The resonances observed by Buckman et al. [80] are connected with the highly correlated motion of two electrons in the field of the He + ion (discussed in section 3.7). The resonances ns 2 observed by Buckman et al. [80] are well described by the formula suggested by Read [79]: E(ns 2) =

I-

°'s)2 (n-Sns) 2 '

R(Z¢°r~ -

(5.2)

I.I. Fabrikant et al., Electron impact formation of metastable atoms

62

Table 24 Helium resonance energies (eV) and widths (meV) (in parentheses)

Feature

Oassification Brunt et al. [147]

Helium energy levels [284]

A

ls2s3S

19.820

Experiment e-He symmetry classif.

[285]

[275]

[2861

Theory [2811

[591

[1531

[46]

19.85

B

C

[147]

2po

ls(2s2p ip)2p 20.62

20.27 (780)

2Se

20.62

20.17 (330)

20.15 (430)

20.606

20.62

20.624 ls2p 3p

20.964

ls2p ~P

21.218

20.99 21.19

ls(3s2 tS)zS

ls(3s3p 3p)2p

2po

ls(3p 2 ID)2D

2De

ls3s 3S

22.719

20.955

20.85 (2O0)

20.82 (400)

21.209

22.34

22.44

22.42

22.44 (30-35)

22.43 22.47 (36)

22.44 22.441 22.442 (150) (25.2) (28)

22.60

22.67

22.60

22.60 (37-45)

22.69 (38)

22.45 22.608 22.614 (22) (39.9) (40)

22.66 (42-50)

22.70 (20)

2se

22.75

22.65

22.645 (44.7)

22.652 (50) 22.717

22.72

(1) 2pO

(ls3s3p 1p)2p

2De

22.73 22.81 22.85

22.86

2Se

ls(3p 2 iS)2 S 22.921

2po

22.79

2Se

22.867 (30.4)

22.885 (36)

22.89 (18)

22.~7 (6.7) 22.913 (18.7)

22.888 (6) 22.918 (5) 22.94 (30)

23.00 (10)

22.93

(20) 2po

U V

22.98

22.703 22.720 (23.3) (12)

~.~ ~.~ (18)

2FO T

22.79

23.01

2De 2FO 2p~, 2De 2Ge

23.05

22.97

22.99 (32)

23.05

23.05 (35)

22.938 (43.3)

23.01 (5) 23.01 (10) 23.02 23.06 23.07 23.07

1.1. Fabrikant et al., Electron impact [ormation of metastable atoms

63

where 3,, is the quantum defect of the corresponding Rydberg state, I is the ionization potential, and R is the Rydberg constant. In our case Zcore= 1. AS follows from the potential ridge model [76], o-~= 0.25. In this case, the error of formula (5.2) amounts to 1.6% at n = 2 and 9.5% at n = 7 [80]. Without dwelling upon the details (see, for instance, ref. [147]) we just provide a modification of this formula which gives the possibility to calculate the energies for n ranging from 1 to 6:

Enln" = ] -

R(Zcore - o-s) 2 (n 3.,) 2 -

R(Zcore - ors) 2 (5.3)

(n - ~nl,) 2

-

Calculations according to formula (5.3) make it possible to plot an energy level diagram for the observed resonances (see fig. 43). This figure shows how the resonances correlate with the energy positions of the levels of the negative ion [282]. Concluding our discussion of the structure in the excitation functions of metastable states in the helium atom we should mention the papers by Kuyatt et al. [285], Sanche and Schulz [286], and Heddle et al. [287-289] about observation of resonances in other channels of negative-ion decay (the channel of elastic scattering and excitation of optically allowed states [290]). In most cases their energy location is in good agreement with the data on excitation of metastable states (table 24). Of certain interest are the studies of the resonance structure of various channels of inelastic processes occurring in the range of double-excitation energies [146, 286,291-300]. In contrast to the aforementioned resonance structures, these are mainly caused by the presence of autoionization states of the atom. Here we will present the results of only one work [298] on the total production cross sections of metastable states in helium (fig. 44). The amplitude of the resonances amounts to 80% and 5% of the value of the total cross section. The resonances are well described by the formula of Fano and Cooper [291] and their positions properly correlate with the data obtained by other authors [286,292]. In ref.

n=2

n=3

He-*He -= obs pred

He = H~* He'* I~e*He-~ He-= obs pred obs pred 1~, V r--3p21fl Ip ====~l~ r"--4p 2 IS

'1~P, /

~--~- ;

~P ~

n=4

Is__T

y~

l r__3s3plp

__

~

1p

--

4s4p

~2p21 s -R=~/ 3-2 ID 2S2D1p Q_....///'-- P --3S /---.. 4p 2 1D ==2PZ'-ID3s---0 //---3s3p~ ~'/,---4s4p 3p .. 3s2 1S

3S - .---

2~ 2 is

z-J.,"

,"--- 4s2 1S "~"

I s2s- ~ 25

Fig. 43. A comparison of the positions of the energy levels, observed resonances and predicted resonances of neutral helium [from eq. (3.1)] [147].

64

I.I. Fabrikant et al., Electron impact formation of metastable atoms '1

1.95

i

.°°o. e..

E

•

°o

.o

...,oO

i

'\ i'"'..":'.... I°

57

I

EIeV}

I

58

I

59

Fig. 44. Total cross section for metastable-state formation for the helium atom in the doubly excited energy region [298].

[299] a group of new resonances was observed in the energy range above 57 eV, which are associated with the excitation of the (2s2)1S-, (2s2p)3p - and other autoionization levels of helium. The method of metastable-state spectroscopy was employed in resonance studies of the helium atom at different observation angles [300]. 5.3. The neon, argon, krypton and xenon atoms

The resonances observed in the excitation functions of metastable states of the Ne, Ar, Kr and Xe atoms are formed by two bound s-electrons [70] interacting with the field of the ion core. This was pointed out for the first time by Simpson and Fano [301,302], who managed to find the shape resonances 2p53s2, 3p54s2, 4p55s2 and 5p56s2 caused by splitting of the 2p3/2- and 2p1/E-components of the ion core. One should note that prior to 1976 the data on resonances for this group of atoms were much scarcer than for the helium atom. Noteworthy in this respect are the works of Sanche and Schulz [286] and the experiments by Pichanick and Simpson [275], which provided the basis for classification of the resonances. The contributions of the Manchester group [16, 170, 187, 303] and of Dassen et al. [171] to studies of the resonance structure in the region of a doubly excited state should be regarded as a final stage in the detection and classification of the resonances. Table 25 summarizes the data on the resonance energies in the neon, argon, krypton and xenon atoms, aswell as those on the doubly excited state region from ref. [171]. At present this is the most detailed information, which properly agrees with spectroscopic data and the reported location of energy levels in negative ions. Most of the resonances in excitation functions of metastable states belong to narrow doublet pairs whose splitting is very close to the separation between the 21)3/2- and 2P1/2-1evels of the ion core. These resonances were explained in the framework of the jLS-coupling scheme (grandparent model) advanced by Read et al. [303], according to which the two excited electrons couple together to form a total angular momentum L and a total spin S, and then the total angular momentum of the core j step-by-step couples with L and S to form the total angular momentum J of the negative ion. The positions of these valence resonances are well described by Read's formulae (5.2), (5.3) either close to the single electron excitation threshold [187] or in the vicinity of the double-excitation threshold [171] in inert gas atoms. Features of another type are due to the isolated resonances formed by the attachment of an electron to the j = 0 state of the neutral atom. These resonances are classified to be of the nonvalence type. An

I.L Fabrikant et al., Electron impact formation of metastable atoms

'

I

'

I

'

I

;

22

I

'

l~'

I

~

~

~

,..~ .~..r~

65

I

20

_

"-

-- 18

,

16

~ o,

14

°

~:,~P~ "e-~RU'a'~

,

.=5

12 ' ~ , ~ ~ ~ . . ' ~ ' = b.

7

e~

h,h='h~

9

11

13

et

el..

--SA~SC~Z

,leVI

21

Fig. 45. Plots of Feshbach resonance energies ENu, versus the appropriate ionization potential I for the Ne, Ar, Kr and Xe atoms [304].

explanation of all the observed features can only be given for each atom separately. A detailed analysis of possible experimental errors is very important for a precise interpretation of a specific resonance, since only exact knowledge of the energy location of a feature enables one to identify its origin with confidence. Soon after the publication of ref. [16] by Brunt et al., Spence [304] proposed a new classification of Feshbach resonances in inert gases. The main point of this classification is illustrated in fig. 45. Such an approach could not be formulated until the investigations of the Manchester group on the excitation functions of metastable states appeared. The suggested classification presents a rather simple method of calculation of resonance energies taking the structure of atomic shells into account. Figure 45 provides a more complete picture regarding this classification; good agreement is seen between the data of Brunt et al. [16] and the calculations according to the formula

ENn,(m ) = hNn,I(m ) + BNtt, ,

(5.4)

where ENtl,(m ) is the resonance energy for excited electrons with angular momenta l and l' in the mth atom and I(m) is the atomic ionization potential. The constants A NU' and BNt~, do not depend upon the atomic species. The difference between experimental and theoretical values for the resonance energies varies from 0.01 to 0.08 eV. Table 26 summarizes the data on positions of well-defined resonances obtained by different methods. It is seen that, despite different procedures applied to electron energy scaling, the disagreement in the resonance energies is rather small, ranging from 20 to 40 meV.

5.3.1. Neon Quite a few works deal with the investigation of the structure in the excitation cross sections of this atom [16, 18, 166, 171, 301, 303, 304]. Table 26 gives information about the energy positions of the observed features and their classifications according to the data taken from various works. From this

66

I.I. Fabrikant et al., Electron impact formation of metastable atoms Table 25 Energy positions of features in the metastable excitation cross sections for the Ne, Ar, Kr, and Xe atoms [187]. Parentheses: calculation flora eq. (5.4), data of Spence [304]. (*) Doubly excited energy region, data of ref. [171]; parentheses: calculation of ref. [305] Neon

Argon

Feature B1 b c c e d1 e d2 P nx n2 fl f2 N gt

g2 P hi it h2 i2 p

Ne + 2P3/a Ne + 2Pit2 (*) a b c d e f g h

Energy (eV) 16.619 16.906 (16.907) 18.34 (18.35) 18.464 18.580 (18.578) 18.626 18.672 18.965 19.498 19.598 19,686 (19.691) 19.778 (19.964) 20.054 (20.05) 20.120 20.150 20.369 20.636 20.693 20.737 20.798 20.890 20.952 21.153 21.565 21.662

Feature B1 b b c e e e d1 e e d~ e O P nI nz fl f2

gl P g2

42.11 (42.08) 43.10 (43.06) 43.67 (43.69) 43.86 44.07 (44.05) 44.36 (44.37) 45.27 (45.17) 45.45

Ar + 2ps:2 At + 2p1/2

Krypton Energy (eV) 11,548 11.638 (11.631) 11.850 12.76 (12,70) 12,925 (12.926) 12,942 12.99 13,057 (13.055) 13.163 13.190 13,217 13.282 (13.479) 13.475 13.864 13.907 14.006 14.054 (14.052) 14.209 14.434 14.478 14.530 14.568 14.634 14.711 (14.68) 14.735 14.81i (14.80) 14.859 14.983 15.064 (15.05) 15.120 15.160 15.288 15.364 15.429 15.477 15.760 15.938

(45.,*2)

i

46.53 (46.52)

Feature BI b a2 b cx e e d1 e 0 P c2 e d2 e P fl gl P f2

hi

g: i~ Ja kl 11

t12 iz i2 k~ 12 Kr + 2P3, ~

(*)a b

24.54 24.99

I ~ + 2pl/2

Xenon Energy (eV) 9.915 10.039 (10.027) 10.134 10.600 11.12 (11.06) 11.286 (11.278) 11.318 11.400 (11.388) 11.49 (11.644) 11.653 11.77 11.996 12.036 12.138 12.191 12.262 (12.324) 12.378 (12.669) 12.760 12.875 (12.93) 13.016 13.067 (13.10) 13.221 (13.19) 13.291 13.379 (13.32) 13.430 13.477 13.528 13.598 13.714 13.789 13.891 14.199 14.273 14.375 14.441 14.000 14.666

Feature B1 b b c~ c~ B2 e e d1 O e e e e nI c2 c2 e d: e n:

Energy (eV) 8.315 8.338 (8.42) 8.430 9.08 9.36 9.447 9.505 (9,495) 9.551 9.623 (9,619) (9.8%) 9.644 9.686 9.743 9.831 9.896 10.48 10.71 (10.76) 10.858 10.901 10.957 11.14 (11.18) 11.37 (11.27) 11.525 (11.47) 11.62 11.70 11.94 12.18 12.33 12.45 12.62 12.81 12.92 13.03 13.09

Xe* 2P3:2

12.130

Xe + 2P1/2

13.437

1.I. Fabrikant et al., Electron impact formation of metastable atoms

67

Table 25 (contd.) Neon

Argon

Krypton Energy (eV)

Energy Feature

Feature

(eV)

j

46.97

c x d e f g h i j k 1 n o p

(46.88) 47.38 46.64 (47.60) 49.02 (49.04)

k I m

26.45 27.08 27.57 27.80

28.30 28.57 28.98 29.44 29.86 30.33 30.90 31.73 32.12 32.40

Xenon

Feature

Energy (eV)

(*) a b c d e

22.79 23.27 24.08 24.51 24.74

f

25.04

g h i j k 1 m

25.59 25.92 26.20 26.55 26.89 27.19 28.80

Energy (eV)

Feature

table one may definitely draw conclusions about the progress reached in the experimental development since the work by Pichanick and Simpson [275]. Figure 46 shows the energy dependence of the excitation function of metastable states in Ne in the range of energies from the threshold up to 21.50 eV. The spectrum can be schematically divided into two parts. From the lower-level threshold (E = 16.619 eV) to E = 19.20 eV there exist two series of resonances, separated by an almost linear region (16.906-18.340eV) with no discernable structure at all; then, the first group of resonances (at E = 18.626eV) is followed by a developing structure. Dominant are here the doublet pairs of resonances denoted in ref. [187] by dl, d2, e 1, e2, nl, n2, f~, f2, g~, g2, hi, h2. As follows from fig. 46 and table 26, six resonance pairs were detected for the neon atom, which may be classified according to the/LS-coupling scheme. Moreover, there exist at least three "odd" resonances denoted by "P" with positions energetically coinciding with those of the 2pS(2p~/2)3p-, 4p-, and 5p-levels ( J = 0) in the neutral neon atom. Dassen et al. [171] investigated the resonances in the doubly excited region for E = 42-52 eV (fig.

5

4

...

! g~o, 3B23p

]')"

II

oE3

::

I1:

:::" ~' i.,,~ll

{ :~ nm

~

41U

/

~

u

I

,

17

.

.

18

,

,

19

,

VQ I'~ ! ~.

20

3s l

2

3s3p ,

3p 2 I

4e u

3823p o [2e2p6] ~S

4s4p u

!~pVJ

:,,.it../ll'

--%lU l l l U U'l'i •

3a3p 2

I

,

,

rl

r

\

,

21

EIeV)

Fig. 46. Resonance structure in the cross section of metastable-state formation for neon [187].

42

44

46

EIeV)

50

Fig. 47. Resonancesin the crosssectionof metastable-stateformation for neon in the doubly excited energy region [171].

68

I.I. Fabrikant et al., Electron impact formation of metastable atoms

Table 26 Energies of resonances in Ne, Ar, Kr and Xe Feature

Classification

Buckman et al. [187]

Sanche and Schulz [286]

Pichanick and

Kuyatt et al.

Swanson et al.

Simpson I275]

I285]

[182,183,306]

Neon

b c

2pS(2p3/2,1/2)3s3 p 3p 2pS(2p3/2.1:2)3s3 p Ip

16.906 18.34

16.85-16.91

dt

2p5( 2P3/2,1/2)3p21 S

18.580

18.55

18.58

d2 P ft f2

2pS(2P1/2)3p2 1S + 2pS(2P3/2,1/2)3p2 1D 18.672 18.964 2pS(2p3/2~s2tS 19.686 2pS(2Pl/2~s2 iS 19.778 2pS(2p3/2)3d21S 20.054 2pS(2pl/2)3d21S 50.150

18.65-18.70 18.95 19.57

18.66 18.97 19.69 19.83

gl g2

16.92 18.18 18.29 18.46 18.56

19.97-20.03 20.07-20.13 Argon

b b c e

3pS(2p312.1/2)4s4 p 3p 3pS(2p3/2,i/2)4s4 p 3p 3pS(Zp3/2 1/2~S4ptp 52 " 21 3P5(2P3/2)4p D

dx -

3p ( P3/2)4p2 IS

d2 e P fl f2

3pSs(2p3¢z)4p21s 3p (2Pl/2)4p2 tD 3pS(2p3)2)5s2 tS

3pS(2P112)5S2 18

11.638 II.850 12.76 12.925 13.057 13.217 13.282 13.475 14.054 14.209

11.71 11.91

11,72 11.88 12.80

12.89-12.92 12.95-13,06-13!11 13.22-13.28 13.33 13.45-13.50 14.03-14.07-14.11

11.67 11.85

13.08 13.24 13.55

Krypton b a1 b Cl C2 dI

4pS(2p3/2)Ss5 p 3p 4pS(2P1/2)5s 21S 4pS(2pl/2)5s5 p 3p 4p~(2p3, 2)SsSp tp 4p~(2p3/2)Sp 2 tD 4p°(ZP3/2)Sp21S

P d2

4pS(2pl/2)5p2 IS

10.39 10.134 10.600 11.12 11.318 11.400 ' 11.653 12.022

10.16--10.19 10.66-10.69 11.29 11.40 11.67 11.97-12.04-12.10

10.05 10.63 11.10

10.10

j=2 10.04 10.14

j=l

j=0

10,61 11.1 11.29 11.42 11.66 12.04

11.70 12.04

11.29 11.42 12.0

Xenon

2p3/26S j=2 b ct e e d1 n1 d2

5pS(2p3/2)6S6 p 31) 5pS(Zps/2)6s6p Ip 5pS(2p3/2)6p2 lD 5pS(2p3/2)6p 2 tD 5pS(2p3t2)6pZ IS 5pS(2pl/2)6p21S

8.338 9.08 9.505 9.551 9.623 9,.896 10.901

8.48 9.02 9.52 9.56 9.65 10.92

10.86

8.42 9.10 9.50 9.56 9.88 10.91

j=l

9.50 9.56

j=O

9.52 9.58 9.80

9.88

47). Thirteen resonances were observed, seven of which were identified as belonging to the system consisting of the core with charge Z and of three valence s- and p-electrons (table 25). The nonrelativistic R-matrix calculation of the excitation cross section of the 2p5 3s P-state carried out by Taylor et al. [42, 43] reproduces the b-, c- and d-structures in the excitation function reported by Brunt et al. [16].

I.I. Fabrikant et al., Electron impact formation 9f metastable atoms

69

5.3.2. Argon Figure 48 shows the excitation function of metastable states in the Ar atom, which can be schematically divided into three regions. The first region is well approximated by a smoothly increasing linear function, while in the interval of 11.9-12.9 eV, similarly to neon, no structure is observed. In the second region (13.6-14.3 eV) five features are revealed. The third region contains 17 features. All the observed resonances have been tabulated (table 25). The results given in ref. [187] considerably differ from the data of other authors. For instance, Sanche and Schulz [286] and Pichanick and Simpson [275] determined the energy positions of eight and six resonances, respectively; however, the number of resonances considered in ref. [187] amounts to 50, most of them identified (table 26). For argon, the number of resonance pairs in the range E > 13.5 eV is very great. There is also evidence of features associated with the j = 0 term of the 4p and 5p excited states. In the region of double excitation of the Ar atom (table 25) (E = 24-33 eV), Dassen et al. [171] observed 17 resonances, six of them having energies which agree well with the predictions following from Spence's classification [305]. The R-matrix calculation by Ojha et al. [45] reproduces the "b"-, "c"- and "d"-structures in the excitation functions reported by Brunt et al. [16,303]. It should be emphasized that the calculations due to Ojha et al. [45] are nonrelativistic; therefore, they cannot predict the splitting of the peaks (the resonance "pairs"). The first calculations in which the fine splitting in Ar was taken into account and a resonance pair was obtained were carried out by Scott et al. [56]. However, the results were obtained only for energies below the excitation threshold of metastable levels.

5.3.3. Krypton The existence of a large number of irregularities is a specific feature of the excitation function of metastable levels in the krypton atom. Figure 49 shows that, similarly to the neon and argon atoms, the spectrum obtained in ref. [187] can be conventionally divided into three regions which differ in the number of their features and their degree of discernability. A small resonance is observed in the near-threshold region of the curve. The data on the positions of resonances and their classification are given in table 25. One should note that much fewer works deal with krypton, while the number of resonances considerably exceeds that observed for lighter atoms. In addition to the works mentioned already also refs. [181, 182, 188, 308] were devoted to the study of the resonances in krypton. I

I

I

I ¢ l , dll.

20 i"

~d

".. v '

" ; ' ,I i . 0

"

"....I 7

lil.l

f'--I

IIh

I!!I~

~.A

~

.~

-.

ll, l_.r.r~:

'

,,r.,. • '.

%

111 I l l l

ooo

//~

8

•

!'

,%..

l

i-~.a " "~" I l l

I

12

I

I

1

I

I

1/.

I

i

15

0

i

16

EIeV) Hg. 48. Resonance structure in the cross section of metastable-state formation for argon [187].

P[/iL / ' ~,)",' -10

,.'..:

;;':.

I

I

~ J , .~,

I:

:::

:

,."

.

-'" ~:' " ' "

y, ".~ ./

.......

, ,

i

'

,ll

/

;/

z,

..

I'l

~ .:.'!i

12

I

*' "

..

i

12

I

.

g..?.~-7

7!

I

' ''<"

.....

~.,,

I

11

I

I

12

,,

I

I

13

,~

I

,1

14

E(eV) Fig. 49. Resonances in the cross section of metastable-state formation for krypton [187].

I.I. Fabrikant et al., Electron impact formation of metastable atoms

70

Within the double excitation region of the krypton atom (E = 22-32 eV) Dassen et al. [171] detected 13 resonances, five of which are covered by the classification of Spence [305] (table 25). 5. 3.4. Xenon Xenon, the heaviest atom in the group considered, exhibits the poorest (as regards the resolution) structure. The general similarity of the excitation function of metastable states in xenon with those observed in other atoms is seen in fig. 50. The energy positions of the resonances reported in ref. [187] and their identifications are given in table 25. 5.4. Classification o f resonances

Following the terminology given in ref. [187], one may consider separate groups of resonances. Pair resonances. The basic principles underlying the model have already been outlined, so here we will try to draw a generalized picture for the whole group of atoms which were discussed in section 5.3. Read et al. [303] were the first to classify the resonances observed in the excitation functions of metastable levels of the Ne, Ar, Kr and Xe atoms. They developed the "grandparent" model and introduced the concept of "parent-state effective energy". To calculate the binding energy of the outer electron coupled with the ion core the authors of ref. [303] advanced the concept of the center of gravity of the set of terms [309,310], independent of the angular momentum of the grandparent core. Figure 51 presents a diagram of the levels of the neutral Ne, Ar, Kr and Xe atoms together with their negative ions, and an analogy is drawn to the levels of alkali-earth atoms. The energies thus obtained are the "parent-state effective energies", which can be compared with those of the observed resonances [tables 25 and 26 and formula (5.4)]. Proceeding from this model, one may consider the interaction between the two excited electrons and the core as almost purely Coulombic if L = 0 and S = 0. Actually all resonances in the excitation cross sections of Ne, Ar, Kr and Xe appear to be due to this mechanism. Thus, it becomes possible to identify 21 doublets, whose apparent energy approximately corresponds to the state of the ion doublet np s 2p3/2,x/2. Table 27 provides the data on resonance doublets, the lowest of which is the doublet ala 2 (with the exception of xenon) for which L = S = 0. For the pair states with quantum numbers nLS I

I

I

I

I

I

I

b I

÷.[ ,"-t. C

. .V. b~,.-:.~-.'~

%

../% I: .'• i I . "~. ~&,.~.~.,-.-~".~;.~-.---;

/

LOE

"I~,

I:: I : \

ed.

"1II

b

n,

~ .,/

i 1 11

I~ ?sH

I

10

"

~:I

sdm

,

. '~'1

r~'l

I

"~"

I ~i :!`;- ~ 9

'

"

EleV)

11

12

10

I

Sdu/.u_l 'l~i~illdilUi ?pL~I BpeJ O~n I 8sU 9.r,l,l~ I I

l

11

I

12 EIeV)

Fig. 50. Resonances in the cross section of metastable-state formation for xenon [187].

I

I

13

i

I.I. Fabrikant et al., Electron impact formation of metastable atoms

np5( 2P3/2) o r Ne -1

Ne-

71

np6(IS) ~g

r--5s

os Ar

At-

Ca

Xr

Kr-

r-" 4~t

r-47

5p

e--- 3d ~ g

Sr

II Xe

Xe-

Ba

d2

~ - 4 s r--f -2 e

r--4p ~

5p

e

6p

d

6p 2 ID

e

-3

r'-" 3p ,,~

~":4"P2:~Sl--- b H,

[ ' ~ \ \ - - #P ~---4s4P 1p

._.~p23p -4

r

~

5p"-'1,/

~---Ss5p 2 ~ }

°111~d"--b"t'P." l

: - - 6 s 6 ] ''~P

4~, aF__/---4s 2 18

---~s3p Ip

a -~

r-- 3sb "'--'3s3p 3p

-5

a~ " , - - 3s 2 1S

-6 Fig. 51. Noble-gas atom energy levels, appropriate temporary negative ion energy levels and alkaline-earth energy level diagram. The reference energy is the first ionization energy of the neutral atom in the case of noble-gas atoms and their negative ions and the second ionization energy in the case of alkaline earths. The letters a to f represent the following configurations: a: np 5 2pl/2,3/2(n+ 1)s2 aS; b: nps 2pv2,3/2(n+ 1)s(n + 1)p 3p; c: np 5 2pl/2.3/2(n + 1)s(n + 1)p ap + (n + 1)p 23p (?); d: np 5 2pa/2,3/2(n + 1)p2 aS; e: np 5 2Pa/2,3/2(n+ 1)p 2 aD; f: nps 2pv2,3/2(n +2)s 2 aS.

formed by the field of the core with angular momentum ] the binding energy to the core has the form

E b = Ecore(]) -

E,~(nLSjJ).

(5.5)

The ratio of the degeneracy factors of the doublet states with j = 3/2 and j = 1/2 is 2:1. Allowing for this ratio, one may introduce the concept of the center of gravity of the doublet binding energy #b(nLS). The position of this doublet is described by a modified Rydberg formula similar to (5.2),

#b(nLS)= 2R(Z¢°re-0.25) 2 ( n _ 8)2

,

(5.6)

where 8 is the effective quantum defect of pair states and the value of the screening constant o-s = 0.25 was derived from the potential ridge model. In order to verify the model proposed it is useful to compare the effective quantum defect 8 with that averaged over spin for the ns-state of a single electron in the field of a grandparent ion, 8~s, which can be obtained from the following formula:

rzLo #ooro - #.s--

(n - $.s): '

ezLo

#ooro -

#.,-- (.

_ 4)2"

(5.7)

72

I.I. Fabrikant et al., Electron impact formation of metastable atoms

Table 27 Data on resonance doublets in Ne, Ar, Kr and Xe [187]. All energies are in eV, the numbers in parentheses give the errors in the resonance energy in the last significant figures Resonance doublet

AE

/~B

Tentative pair-state classification

t~ from

8nt from

eq. (5.6)

eq. (5.7)

Atom

AE~on

Ne

0.097

a,a 2 d~d2 nln 2 f,f2 gig2 h~h2 i~i2

0.097(1) 0.092(3) 0.100(5) 0.092(3) 0.096(5) 0.101(7) 0.105(7)

5.454(8) 2.987(10) 2.066(11) 1.881(10) 1.511(11) 0.928(11) 0.869(11)

382 tS 3p2 'S 482 lS ad s 'S 4p 2 ~S 5l 2 ~S 512 ~S

1.33(0) 0.74(1) 1.28(1) 0.15(1) 0.82(1) 0.94(2) 0.80(3)

1.33 0.86 1.26 0.01 0.84

Ar

0.178

a,a 2 d~d2 nxn 2 f~f2 glg2 h~h2

0.172(1) 0.160(3) 0.142(15) 0.155(3) 0.181(6) 0.172(6)

4.664(10) 2.709(10) 1.908(15) 1.714(10) 1.229(11) 0.951(11)

48~ ~S 4p ~ 'S 3d 2 ~S 5s2 ~S 5p 2 tS 6l 2 ~S

2.19(0) 1.62(1) 0.17(1) 2.01(1) 1.47(2) 1.99(2)

2.19 1.73 0.20 2.16 1.70

Kr

0.666

ala 2 c~c2 d,d~ f~f2 g~g2 b~h2 ili 2 hJ2 k~k2 IiI2

0.639(3) 0.650(50) 0.636~3) 0.638(15) 0.670(15) 0.670(15) 0.671(15) 0.675(15) 0.661(15) 0.652(15)

4.514(10) 2.885(50) 2.610(10) 1.631(15) 1.239(15) 0.778(15) 0.470(15) 0.399(15) 0.288(15) 0.216(15)

5s2 1S 584d tD 5p ~ ~S 682 ~S 6p 2 1S 7l 2 ~S 812 ~S 9l 2 ~S 10l 2 ~S 11l 2 tS

3.16(0) 2.70(2) 2.58(1) 2.94(1) 2.49(2) 2.56(4) 2.30(9) 2.81(11) 2.71(18) 2.58(28)

3.17 3.17,1.26 2.68 3.11 2.64

Xe

1.307

clc 2 c'~c2 d~d2 nln 2

1.400(50) 1.350(50) 1.278(5) 1.244(20)

3.019(50) 2.756(50) 2.517(11) 2.255(17)

685d 1D 6p" 3p 6p 2 ~S 5d 2 tS

3.75(2) 3.64(2) 3.53(I) 2.39(1)

4,11, 2.43 3.62 3.62 2.43

According to refs. [277,278], the difference in 8 and 8ns is not greater than 5% for all atoms with the shell configuration ns 2 1S, where n is the minimal value determining any charge state of the atom including resonance, autoionization state, neutral atom, positive ion of any multiplicity. The proximity of the values of 3, 8ns and 6nt provides the basis to draw conclusions about the membership of a given resonance doublet. This holds true for Ne, Ar, and Kr (figs. 46, 48, 49), since for all these atoms the difference ~ - 8nt does not exceed 0.10. This makes it possible to assign to them properties of the configuration nl 2 (table 27). Such a classification of the doublets ala 2 and did 2 in particular is supported by theoretical considerations [45, 56] of resonances in Ar. Thus, the-~S-configuration of the pair state subdivides the doublets intff-five classes: (n + 1)s 2, (n + 1)p 2, nd 2, (n + 2)s 2, and (n +2)p 2. With increasing energy the shell transforms into the configuration np 5 and for neon one has: (n + 1)s 2, (n + 1)p 2, (n + 1)d 2, (n + 2)s 2, and (n + 2)p 2. In krypton (fig. 47) the doublet nd 2 is not observed. The suggested interpretation seems to be quite comprehensive, yet it does not cover quite a few resonance doublets observed in experiment. Therefore, the effect due to the excited states of the

I.I. Fabrikant et al., Electron impactformation of metastable atoms

73

neutral atom may serve as an alternative explanation for the sharp maxima of higher doublet structures. For example, in the case of neon such states may be [2p5(2p3/2)4S]l/2-doublets, whose average energy overlaps the structures of fx, and [2pS(2p1/2)4s]o:, located close to the structure f2. In other words, the consideration presented above boils down to an important conclusion about the existence of cluster structures which may be caused by various physical phenomena in atomic objects. Structures "P". We have already mentioned the observation of maxima in the excitation functions of the metastable Ne, Ar, Kr, and Xe atoms which may be classified as P-resonances (figs. 46, 48-50). The energy positions of these resonances for all atoms coincide with the energies of the corresponding [npS(2p3/z)n'P]0 - and [np5(2px/2)n'p')]0-1evels. Excitation of these states occurs without changing the spin and parity; therefore, it can take place with s-waves in the entrance and exit channels. A similar case is presented by the thresholds of the helium levels ls2s 3S and ls2s ~S [157]. Structures b, c and e. The calculations of refs. !45, 42] have shown that b-resonances in Ne and Ar may be classified as npS(2P3/2.t/2)(n + 1)s(n + 1)p P. As regards the membership of c-resonances, the picture here is not so clear. For instance, Clark and Taylor [42] classified them as a 2pS(3s3p)lP resonance (for the neon atom). As far as the argon atom is considered, in ref. [45] the c-resonance is classified as a combination of two configurations, 3pS(4p2)lD and 3pS(3d)~D. For this case, there exists only a theoretical presumption, while for krypton and xenon the existence of the c~c:doublet has been proved experimentally. Without entering into the phenomenology of the discussion [187], the reader may refer to table 28 which presents the configurations and energy positions of the corresponding resonances. As regards the e-resonances, in most cases their origin is similar to the d-resonance since they are close in energy. This idea was suggested by Read et al. [303]. However, Ojha et al. [45] assumed that the structure "e" in argon is related to the 3pS4p-state of Ar and results from mixing of the 3pS(4s4p)Xp- and 3pS(4p34s)3S-configurations. As is seen from table 28, such a possibility does exist. If Table 28 Energies (eV) of the (np 5 2P3/2)(n+ 1)p and (np5 2P~/2)(n+ 1)p' states of the neutral atoms and the structures labelled "e" in the metastable excitation functions [187] Ne Neutral states 3p, J = l

Ar e structures

18.382

Neutral state 4p, J = l

18.464 3p 3p 3p 3p 3p' 3p' 3p 3p' 3p'

3 2 1 2 1 2 0 1 0

18.556 18.576 18.613 18.637 18.626 18.694 18.704 18.712 18.727 18.966 18.965~

4p 4p 4p 4p 4p 4p' 4p' 4p' 4p'

Kr e structures

Neutral states

12.925 12.942 12.99

5p, J--- 1

11.304

5p 5p

11.443 11.445

Xe e structures

12.907

Neutral states

e structures

11.286

3 13.076 2 13.095 1 13.153 13.163 2 13.172 13.190 0 13.273 1 13.283 13.282 2 13.302 1 13.328 0 13.480 13.475

11.318

5p 5p 5p 5p' 5p' 5p' 5p'

3 2

11.49 1 11.526 2 11.546 11.653t 0 11.666 11.996 1 12.101 1 12.141 12.138 2 12.144 0 13.257 12.262t

9.505 9.551 6p, 1 = 1

9.580

6p 6p

2 3

9.686 9.721

6p 6p 6p

1 2 0

9.789 9.821 9.934

6p' 6p' 6p' 6p'

10.858 1 10.958 10.957 2 11.055 1 11.069 0 11.141

9.644 9.686 9.743 9.831

74

1.1. Fabrikant et al., Electron impact formation of metastable atoms Table 29 Energy positions (eV) of features in the excitation function of metastable Hg [228]. The absolute calibration error of the energy scale is 10 meV, the error in the energy of peaks relative to each other is approximately 5 eV for the sharp features of high statistical accuracy, but is larger for other peaks The features with a subscript 0 correspond to neutral-state onsets, while the others correspond to resonance states Feature

Observed energy (eV)

Classification

Ao A B Co C D E F G H

4.660 4.702 4.9-5.4 5.452 5.59 6.3-6.8 6.702 7.50 7.6--8.1 8.367

5dl°6s6p 3Po 5dl°6s6p 2 ~P3:2 5d~°6s6p2 ZD3/2 5dl°6s6p 3P2 5dl°6s6p2 ZDsj2 5d~°6s6p2 2P1/2 5dl°6s6p 2 2Sl/2 5dl°6s7s 2 ~$1:2 5d~°6s6p2 2p3/2 5d96s:(2Ds:2)6p, J = 5/2

I J K Lo L M N O P Oo

8.508 } 8.560 8.650 8.792 8.854 9.439 9.593 9.988 10.356 10.54

Q R S T U1 U2 U3 V W Xo

10.563 10.598 10.90 11.05 11.317 11.364 11.43 11.59 11.780 12.043

X1 X2 Y Z A' B' C' D' E' F'

12.067 12.09'4 12.852 12.989 13.041 13.190 13.325 13.575 13.625 13.718

5d96s2(2Ds/z)7s7p 5d96s2(:Ds/2)7s7p 5d96s2(2Ds/2)7p 2 5d96s2(2Ds/2)7p 2 5dg0s2(2Ds/2)7p 2 5d96s2(2D~/2)7p2 5d96sZ(2D512)7p2 5d96s2(ZDs/2)8sSp 5d96s2(2Ds/2)SsSp

G' H' I'

13.881 13.944 14.034

5d96s2(2Ds:2)8p z 5d96s:(2Ds/2)8p 2 5d96s2(2Ds/2)9s9p

5d~°6s7p2 5d96sZ(2Ds:~)6p 3D~ 5d96sE(2Ds/2)6p 2 5d96s2(2Ds/;~)6p 2 5d96s2(2Ds/:~)6p 2 5d96s2(2Ds/2)6p 2 5d96s:(ZD3/2)6P(] ½)2

5d96s2(2Da/2)6p 2 5d96s2(2D3,E)6PZ

5d96s2(2Ds/E)7S 3D2

1.I. Fabrikant et al., Electron impact formation of metastable atoms

75

Table 29 (conL) Feature

Observed energy (eV)

J" K' L' M' N' O' P'

14.135 14.284 14.450 14.7 14.869 15.014 15.444

Q' R' S' T' U' V' W'

15.574 15.656 15.80 16.155 16.29 16.57 17.16

Classification

5d96sZ(2Ds/2)gsgp 5d96s2(2Ds/2)9P 2 5d96s2(2Ds/2)10p 2 5d96s2(2D3/2)Tp 2 5d9682(2D3/2)7p 2 5d96s2(2D3/2)8sSp

5d96s2(2D3/2)8P2 5d96s2(2D3:2)gp2 5d96s2(2D3:2)10p2

the difference between the excitation level energies and the position of the structure "e" does not exceed 11 meV, they coincide. However, the number of possible states is much greater than that of the observed structures "e", while their appearance in most excitation functions is rather symbolic. Therefore, further research in this field is necessary. 5.5. The mercury atom

Many works deal with the mercury atom and the resonances formed during its excitation. Investigation of the resonance structure in mercury is likely to have started from the experiments [285] of Simpson et al. on electron beam transmission through mercury vapor and from a series of studies, e.g. refs. [226,311,312], dealing with the optical excitation functions of the mercury atom. On the one hand, this helped to broaden the scope of our knowledge on resonances, and, on the other, stimulated experimental efforts in detection of resonances in studies of metastable states of the mercury atom. In most works [217, 223, 227, 257, 285,291,313-321, 10] the authors report on the dominant effect of the excitation of 6 3p-states upon the production of autodetaching states of the Hg- ion. Frish and Zapesochny [257] for the first time pointed to the existence of a metastable 6' 3D3-1evel of the 5d96s6p'-configuration of the mercury atom. The most reliable results on the detection of the resonance structure in the excitation cross sections of metastable levels in mercury were recently obtained by Koch et al. [223] and Newman et al. [228, 319]. A perfect experimental technique and, first of all, good electron energy resolution allowed them to determine and identify a number of maxima. Figure 52 presents the cross sections of metastable production in mercury atom. Well distinguished are the maxima corresponding to the states of the negative Hg- ion at 4.71 and 4.94 eV, which agree with the analysis made by Albert et al. [313], while the position of the first feature at E = 4.71 eV correlates properly with the data obtained by Shpenik et al. [226]. One should note that the position of this maximum was theoretically predicted by Scott et al. [321] and was considered as a symmetric resonance with j = 3/2. This resonance was experimentally confirmed by Newman et al. [228]. Five pronounced resonances were found in the energy interval ranging from the excitation threshold of the 6 1p~-level to the first ionization potential. Their positions

1.1. Fabrikant et al., Electron impact formation of metastable atoms

76

105

a

0=40*

I I ..,.'J:

¢:

I1'

:.J

a g5 =u(Z )

! ' '1' 'P,

,

i

i

'b' It

II '' "s,'s.

!/

•

I ,

,

~ ~ I

,

, .:,:. ~.

5

II

I •-.

•.

;;~ _t •-

•

5

b ~ 3

I

.:,:

f\

• I

i" i ['':"

I

":.t~..,t

'

i •. .

1 .a...a..,I..~';

4.5

z

~t

~

,

I

I

t

5.0

EleV)

5s

''~

s

8 ' ~'

EIeV)

1'o'

Fig. 52. Resonances in the formation cross section of metastable Hg; (a) 6 3p0-state, (b) 6 3P2-state. The positions of resonances and neutral atom energy levels are indicated. Top panel of (a): differential elastic scattering cross section at O = 40°.

are in good agreement with those obtained in the "transmission experiment" of refs. [285,323] and in refs. [226,227]. The work of Newman et al. [228] may be considered as one of the most detailed publications dealing with an experimental study of metastable-state excitation of the mercury atom by electron collisions with a high energy resolution (20 meV). So perfect experimental conditions made it possible not only to define the positions of known resonances with an accuracy up to three significant figures, but also to fred a number of new resonances in the interval 4.6-17.2eV. Table 29 gives their positions with an accuracy of up to 5 meV and the classification of their states. Of great interest is the fact that in addition to the already known metastable 6 3p2-, 6 3p0-, and 6' 3D3-states, a new state was recorded at E = 10.54 eV. The classification of resonances is similar to that made for the helium atom and inert gases in the work of the Manchester group. ill

26

• ! \

p-. ~.

.'

18

,

%

: :

•

"

k.,

-

d 10 I

'

~2

'

9.6

I

ld.o

'10.~

~0.8

Fig. 53. Resonances in the cross section of metastable-state formation for the mercury atom [228].

I.L Fabrikant et al., Electron impact formation of metastable atoms

77

The above-given list of works helps to eliminate the lack of adequate information on the mercury atom. Completing the discussion of resonances in the metastable excitation cross sections of the mercury atom we should note that most available data obtained by different methods allow one to ascertain the authenticity of resonances and to clarify the nature of their formation. As an example, fig. 53 presents the production cross section of the metastable mercury atom in the range 8.8-10.8 eV. The features known from the results given in refs. [291,321,322] are well resolved. It is interesting to note that the positions of the features in figs. 52 and 53 agree well with the resonances revealed in other channels of Hg- decay, elastic [313] or inelastic [217,226,227].

6. Drift-angle studies of metastable-state production In the previous sections (sections 4 and 5) we have analyzed the contribution of various channels to metastable-state population by electron excitation. For this purpose we considered the dynamics which affect the behavior of the metastable excitation cross section connected with the contributions due to higher energy levels of the neutral atom (cascades) or to negative ions (resonances). In this part we will dwell upon effects of another kind, namely the kinematic effects caused by the momentum transfer between electron and atom. We will analyze a number of recently performed experiments involving the metastable spectroscopy method [21,130, 131, 168, 324, 325, 300]. The experiments allow one to determine the contribution of kinematic effects in the process of metastable particle production by electron impact. Here we will also discuss the results of investigations concerning the metastable states of inert gas atoms obtained by the TOF technique [22, 142, 326-328]. Prior to discussion of these works we will deal with some aspects of the kinematics of electron scattering by atoms with reference to the technique of intersecting beams.

6.1. Analysis of the kinematics for the metastable spectroscopy method and the TOF method Let {0, q~} = to be the scattering angles in the center of mass (CM) frame, while {O, q~} = ,0 are the scattering angles of the atomic beam in the laboratory frame. The velocity and angular distribution I in the metastable beam is determined by the relation [142]

I(VA, ~9, alp)dV A dO = KOrcm(O)f(gA)d g A d o ,

(6.1)

where f(VA) is the velocity distribution in the initial beam, O'cm(0) is the differential cross section in the CM frame, and x is a constant factor allowing for the efficiency of detection of metastable atoms. The relations between to and/2 were obtained by Helbing [329]. For electron-atom collisions this relation is ambiguous, and one has to sum over two values of { O, q~} in formula (6.1) corresponding to the value of {0, q~). By analyzing the kinematics Zajonc et al. [22] determined the relation between dV A dO and dVA dto, which allowed them to find the differential cross section trcm(0) with an accuracy up to the constant factor x by use of the function I(V:~, O, ~). The function I(VA, O, q~) is directly related to the TOF spectrum J(t, O, qb) by the following relation:

J(t, ~9, ¢P)= ~ I(~/t, O, ~ ) ,

(6.2)

1.1. Fabrikant et al., Electron impact formation of metastable atoms

78

where ~ is the distance between the collision region and the detector (TOF base) and t is the time of flight. Due to the finite angular resolution of the detector and the electron energy spread one measures in the experiment not the I function but its integral over some finite region of t9 and 4. Therefore, one has to use an iteration method to solve the integral equation in order to find the differential cross section Crcm(0). However, in the case of good angular resolution of the detector two or three iterations are quite sufficient [22, 142]. The metastable spectroscopy method involves the measurement of the differential distribution function of metastable atoms only with respect to one angle ("the drift angle"). In particular, the measured value of the metastable-state current is integrated over a rather wide range of velocities VA, thus making good convergence of the mentioned iteration procedure impossible. Prior to demonstrating this difficulty let us consider the relation between the drift angle X and the scattering angle 0 in the CM frame (fig. 54). If we suppose that the detector only counts the particles scattered in the intersection plane of the beams and has an infinitely high resolution in the angle 4, then the angle 19 can be identified with the drift angle X. As a result, we obtain a simple relation between the drift angle and the scattering angle in the CM frame, 1 - A cos 0 tg )¢ = ~: -y-sin 0 '

(6.3)

where A = (1 - e/E) ~/z, e is the threshold excitation energy, ~: = MVA/mVe, E ~- mVZJ2 is the energy of the colliding particles in the CM frame, and Va and V~ are their velocities. The -Y- signs in (6.3) correspond to ~ = 0 and ~ = ~r in the CM frame. Hence it follows that there exists some range of X within which the scattering takes place: tg Xmin

----

y~y- +)t~: ,~,

tg ,)(max_ ysty+- A~ A'

(6.4)

where y = (1 + ¢2 _ ~2)1/2. Investigation of the Jacobian dto/dO shows that the main contribution to scattering arises from the range of angles close to X0 defined by 7

e /

D

,r,

S

I

f

y

\

Fig. 54. Schematic view of the collisional geometry in the metastable spectroscopy method [330]. The arrows indicate the directions of the colliding particles (A, atoms; e, electrons). The p vector lies in the O X Z plane.

I.I. Fabrikant et al., Electron impact formation of metastable atoms

tgXo=(1-a)/f.

79

(6.5)

The angle X0 is very close to Xmin; therefore the former is called the metastable drift angle in refs. [130, 131]. Its dependence on energy has the following form: tg

go(E)= 1

[El/2 _ (E

-

~,)1/2] ,

fl = M V A / ( 2 m ) '/2

(6.6)

Singularities should be observed in the function I at the angles Xmi, and Xmax" The better the resolution in X and in the beam energies, the sharper are the singularities. Similar singularities are observed in the TOF spectrum. These kinematic singularities (which are independent of the beam interaction dynamics) must also appear in the relation I(E) at fixed X. The corresponding values of the threshold energies E t at a given X may be obtained from formulae (6.4). The theoretical differential cross section in the laboratory frame in the "plane" geometry considered by us may be obtained from the CM cross section according to the following formula: d~ m

dX (X') =¢r _

d~

q(x+-(O)-X')~wsinOdO'

(6.7)

where q(x - X') is the normalized apparatus function of the detector. Strictly speaking, averaging over the energy distributions of atoms and electrons in beams as well as over the collision angle between the beams is necessary. As was already mentioned, averaging over the velocities of the atoms is most significant. In the metastable spectroscopy method the assumption that the detector registers all the particles in the plane perpendicular to that of the drift angle is more realistic. Although the corresponding range of angles is rather small, the effect of integration over this range may be rather great. Therefore, the analysis given in ref. [330] is based on a somewhat different kind of geometry. Suppose (according to fig. 54) the atomic beam moves along the Z-axis, and the electron beam along the X-axis. Let us accept that all atoms on line D, parallel to the Y-axis, are detected in the experiment. In this case, the differential cross section with respect to X has the following form: do"m ( cos(Lr, p) pdy O'L(O, ~ ) dx J r2

'

(6.8)

where p is the vector obtained by projection of r on the plane XOZ and r is the radius vector of the observation point. Substituting the variables {0, ~p}, we get t2

d' m dx

2 f

sin~

t)

[-~2_- t-~ _- t-~]~i- dt,

t1

where t = cos 0 and t2.1(s¢) are the roots of the equation tg2,1' (1 - tz) -

,~

t

-0

(6.9)

1.1. Fabrikant et al., Electron impact formation of metastable atoms

80

The domain of existence of the integral (6.9) gives the same limitations to the range of drift angles as formulae (6.4). Equation (6.9) can be averaged over the energy E A of the atomic beam. Introducing the Lorentz distribution function E0 fA(EA) = 7r[E~ + (E A - EA) z] '

EA = M V ~ / 2 ,

and integrating over E~,, we obtain 1

do'.; 2 f dt(1/;t-t) d X / = sin X cos X -I

~

(t)

x Im{[(az(t) - [3=)(al(t ) -/32)] -1/2 - [(%(0 -/31)(a~(t) -/3~)]-1;2},

(6.10)

where Oel,2(t ) = t -T- tg

/31 =

&=

X (1 -

t2) 1/2 ,

1 + (M/mEe)~/Z(EA + iE0) '/2 tg X a '

(6.11)

+2/a.

When taking the root of the complex expressions in (6.10) and (6.11), the value to be used lies in lower and upper half-plane, respectively. Naturally, the final result for (do'm/dg) greatly depends on the form of the function f(VA), thus making the solution of the inverse problem [i.e. finding o'cm(t) from (dtrm/dg} ] a rather unstable procedure. Therefore, when analyzing the results obtained by the metastable spectroscopy method, we only solved the direct problem involving the calculation of the theoretical dependence of (do'rJdX} according to formula (6.10) and the comparison with experimental data. 6.2. Differential cross sections Differential cross sections of production of metastable states of the He, Ne, Ar and Kr atoms are given in figs. 55 and 56. Here, the basic features of the given dependences are a small range of drift angles X especially for heavy atoms (Kr, Xe), the obvious appearance of limiting angles Xminand Xmax for each energy, the decrease of the drift angle with increasing electron energy, and the presence (in some cases) of two maxima on the curves. Let us discuss each atom more thoroughly. Helium. Figure 55 gives some experimentally measured curves of differential cross sections for excitation of the metastable 2 3'lS-states of the helium atom. The calculated [330] dependences allowing for the contribution of the 2 aS-, 2 1S- and 2 3p-levels are also presented here. In the calculation theoretical cross sections in the CM frame obtained by the close-coupling method including the 1 1S-, 2 aS-, 2 1S-, 2 3p., and 2 ~P-states [149,150,154] were used. The averaging over the energy distribution of the atomic beam was done with the use of a Lorentz distribution with a width of 0.06 eV. The dependence of the shape of the curve on the method of calculating dtr/dto is demonstrated in fig. 55.

I.I. Fabrikant et al., Electron impact formation of metastable atoms

1.2

'

iI

it

'

~

A V

i

i

81

i

E=29.6eV

II

I~A

// I

~-E 1.0

B '11

"

E:6OeV /

•

,i

I II

I I

8

42

II

46

2'o

'

7qdeg)

Fig. 55. Differential cross sections for metastable helium formation [330]. A, calculation using the data for dtr/dto of refs. [149, 150]; B, calculation using the data of ref. [154] averaged over EA; F, the same as B, with the data of refs. [149, 150]; Ex, experimental data of Shpenik et al. [131].

The contribution of additional channels of metastable atom production was also studied. In this case the data of Chutjian and Thomas [331] on the differential excitation cross sections of the 3 as-, 3 3p. and 3 3D-levels were utilized. The latter corrections produced a small effect upon the form of the curves. Figure 55 shows that two maxima are observed in the experimental curves at energies slightly exceeding the excitation threshold of the 2 3S-level, thus confirming the above discussion (section 6.1). Some discrepancies in the experimental and calculated values of gmin and gmax are due to finite energy resolution. Inasmuch as the absolute values of the cross section were not directly measured in the experiment, normalization of the curves using the calculated values was carried out in the range of angles X where the effect of kinematic singularities is small. Noteworthy is the fact that the discrepancy in the calculated and experimental curves for (d~m/dX)(X) is reduced as the energy of the electron increases. Shpenik et al. [131] emphasize the good agreement of the experimental dependence of the principal drift angle X0 on the energy with that calculated by formula (6.6).

Kr

Ne

Ar

F1

r

0

4

8

0

4

8

0

4

?C (deg} Fig. 56. Differential cross sections for metastabl¢ neon, argon and krypton formation, experimental data [131].

1.1. Fabrikant et al., Electron impact formation of metastable atoms

82

Neon. The neon atom is light compared with the other atoms of inert gases. Therefore, the right-hand singularity of the differential cross sections is fairly easily detectable at low energies (fig. 56). Furthermore, at E = 19.5 eV can one distinguish only this singularity, and this proves the relatively large contribution of large drift angles to the cross section. Starting from 20 eV the left-hand singularity in the cross section becomes more distinct, while the right-hand one diminishes and vanishes at E = 50 eV due to the gradual decrease of the contribution from large scattering angles and the increase of the contribution from small angles. On the whole, the positions of the maxima are in good agreement with the calculated data (table 30). A comparison with theory helps to come to the conclusion that the main contribution to the cross section at large angles X is due to the lower levels (most probably 3s[3/2]0 and 3s'[1/2]0), and that at small angles a large number of levels contribute. This is in accordance with Teubner et al. [165]. Argon. The same regularities as observed in the neon atom hold true for the argon atom. They involve a dynamic development of the singularities and the behavior of the cross section (dtrm/dX)(X) for energies near threshold and up to 50 eV (fig. 56). The agreement between the calculated and experimental values of the angles gmi, and Xmax(table 30) is somewhat worse than for the lighter atoms. This is probably explained by narrower ranges of drift angles for heavy atoms and by lower angular resolution of the detector. Krypton and xenon. The scattering kinematics is rather similar for these atoms, therefore, fig. 56 gives only differential excitation cross sections of the metastable states of the krypton atom. At E = 11 eV (threshold region) only the right-hand singularity related to the angle Xmaxis seen dearly, while the left-hand singularity is almost not observed. This is probably due to a low relative resolution in the angle X since the range of drift angles is rather narrow. On the other hand, as was mentioned above, the contribution from a large number of levels may result in overlap of the singularities

Table 30 Theoretical and experimental groin(left) and X~ax(right) values for He, Ne, Ar, Kr and Xe (deg).

E(eV) Atom

Level

He

23S 21S exp.

10

11

15

20 7.76 10.0

Ne

1 k exp.

3.03 5.15 2.50

Ar

1 k exp.

Kr

1 k exp.

1.66 1.99 1.82 1.82

Xe

1 k exp.

0.87 2.09 1.48 1.48 1.70

25 9.37 10.0 7.26 5.15 5.90

30

50

100

3.05 23.88 3.19 23.70 5.0

2.0 35.4 2.11 35.3 3.0

9 . 0 8 2.10 10.50 7.88 2.97 9.63 5.50 1.70 7.40

1.49 14.73 2.0 14.22 1.40

1.00 21.79 1.78 15.96 1.0

5.24 13.89 4.40 16.51 5.59 13.54 4.65 16.27 10.0 11.5 6.0 18.0 2.43 3.63 2.0

1.65 4.68 3.16 3.16 3.0

1.28 6.03 1.97 5.34 3.0 5.30

1.09 7 . 9 8 0 . 9 6 7.98 1.60 6.57 1.39 7.56 2.0 5.50 1.60 7.40

0.71 10.80 0.99 10.54 1.40

0.48 15.78 0.67 15.59 1.0

1.31 2.52 1.91 1.91 2.0

0.93 3.54 1.66 2.81

0.74 1.16 1.5

4.42 4.00 3.80

0.64 0.97

5.13 0 . 5 7 5.78 4.80 0.85 5.47 1.20 5.0

0.42 0.61 1.0

7.73 6.54

0.29 11.23 0.42 11.10 0.70

0.78 2.32 1.55 1.55

0.60 3.03 1.02 2.61 1.50 2.70

0.49 0.78 1.30

3.70 3.42 3.40

0.43 0.66

4.26 0.38 4.76 4.03 0 . 5 8 4.55 1.0 4.0

0.28 0.43 0.70

6.35 6.21

0.15 0.29 0.50

9.18 9.08

I.I. Fabrikant et al., Electron impact formation of metastable atoms

83

producing broad maxima. A sharp separation of the two singularities is observed only at E = 20 eV. At higher energies only the contribution of the main drift angle is detectable. 6.3. Energy dependences of differential cross sections

Figure 57 provides the most typical curves for (dO'm/dX)(E) versus E taken from ref. [131]. A common feature of the energy dependences is the complicated structure and distinct dynamics of its appearance with changing angle X (figures above the curves). The resonances correspond to maxima with positions on the energy scale independent of the angle X. On the other hand, the shape and the extent of the resonances depend on the observation angle. Noteworthy is the fact that the energy positions of the most clearly observed resonances are in good agreement with the data obtained for the total cross sections for metastable state production discussed in section 5.

Helium. The dependence of (dtrm/dX)(E) on E for helium atoms [131] at different observation angles is given in figs. 57 and 58, together with the calculated dependence obtained by use of theoretical data on differential cross sections in the center of mass frame [151] in the range of energies up to 23 eV. These cross sections were calculated by Fon et al. [151] for the 2 3S-, 2 IS-, and 2 3p-states only for four values of the angle 0, 0 = 30°, 60°, 90°, 120°. Therefore the angular dependence of (dtrm/dX)(X) was approximated with a fourth-order polynomial in cos/9, and the coefficients were obtained from the data on the differential and integral cross sections. In the range of energies E < 22.2 eV, the most important dynamical features are (figs. 57 and 58) the 2p_ (E = 20.27 eV) and 2D-resonances (E = 21.07 eV), observed in the excitation cross section of the 2 aS-level, as well as the threshold peak in the excitation cross section of the 2 1S-level caused by a virtual 2S-state. These singularities and their origin were discussed in section 5. At the observation I

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84

I.I. Fabrikant et al., Electron impact formation of metastable atoms 20

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angles X = 7.5% 8.5% and 9.5 ° the 2P-resonance is enhanced by a kinematic singularity due to the 2 3S excitation threshold. According to table 31, for these angles one has E t = 20.16, 19.82, and 20.08 eV, respectively. The discrepancy in experimentally measured and theoretically calculated positions of the 2p-resonance (20.27eV and 20.2eV, respectively) is probably due to inadequate accuracy of the theoretical calculations of differential cross sections [151]. More accurate calculations [46] for integrated cross sections gives the resonance energy in agreement with the experimental one. It is also worth noting that the calculated and experimental curves (fig. 58) were normalized to the maximum of the excitation cross section of the 2 3S-level. In this case angular averaging of the calculated curves for (dtrm/dX)(E) has been performed with a Lorentz distribution of width AX = 0.016 rad. The virtual state 2S appears rather weakly in the curves for (dtrm/dX)(E) as a result of a large contribution from higher partial waves to the excitation cross section of the 2 1S-level. At X = 6.5 ° and 8.5 ° one may notice a slight irregularity in the calculated curves in the range 20.65-20.75 eV. It is also seen in the experimental curves (fig. 57). The picture is not so clear in the case of the 2D-resonance. At X = 6 .5°, 7 .5°, 8 .5°, and 9.5 ° the resonance effect is masked by the effect of a sharp decrease of the cross section above the 2 3S threshold and a subsequent rise when approaching the 2 1S threshold (table 31). The resonance is well observed at X = 11°. For larger X it looks like a shoulder on the calculated curves at E = 21 eV, since its real form is masked by the increase of the cross section due to the approach of the electron energy to the 2 3S threshold. It is well pronounced in the experimental curves at X = 12-5°; however, the absence of an increase in the cross section at E = 21.5 eV for this case is hard to explain. Probably, the kinematic singularity due to the 2 3S excitation threshold is shifted considerably to lower energies (fig. 57) because of the energy spread of the beams• Of great interest are the studies of the functions (dtrm/dX)(E) in the range of energies from 22.4 to 23 eV near the excitation thresholds for the 3 3S-, 3 3p., and 3 3D-levels. A series of resonances appears

1.I. Fabrikant et al., Electron impact formation of metastable atoms

ou

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86

I.I. Fabrikant et al., Electron impact formation of metastable atoms

here (see section 5). However, an insufficiently well-defined energy of the electrons (AE = 0.08 eV) did not make it possible to detect these resonances in the experiments. Let us take, for example, the ls2s 2 2S-resonance. It is rather easy to observe because it has a large amplitude and makes an equal contribution for all scattering angles in the center of mass frame because of the S-wave isotropy. According to Nesbet [153], the resonance width is 0.025 eV. Therefore, in this case its amplitude is reduced by a factor 4.2 and becomes too small as compared with the nonresonance background. Instead of a stable peak at 22.5 eV (the resonance energy) only a slight irrl,gularity is observed in the experimental curves at some values of X. To interpret these irregularities (and similar ones at other energies) as resonances, one should have the calculated differential cross sections in the CM frame. However, not differential but integrated cross sections are given in the theoretical works [46,153] so far. As was mentioned above, the measured signal in the metastable spectroscopy experiment is the sum of partial cross sections for the 2 3S- and 2 ~S-levels of the helium atom. Nevertheless, it is possible to estimate their relative contribution to the cross section as well as the contribution of cascade transitions. For this purpose the results of theoretical calculations of excitation cross sections for the corresponding levels were used and their contribution to the cross section was analyzed. The contributions of the 2 3Sand 2 1S-levels greatly depend on the angle X. For instance, at small drift angles one can observe a hardly appreciable increase in the excitation cross section for 2 1S compared with that for 2 3S. Such a tendency is explained by the fact that the excitation of the triplet level is due to the exchange interaction, which decreases sharply at large angular momenta of the electron, thus corresponding to small-angle scattering. At large drift angles the ratio of cross sections does not increase so fast (table 32), while at intermediate drift angles it remains approximately constant. As regards the contribution of cascade transitions, the 2 3p-level contributes about one half of the 2 3S-level at small angles X, and an order of magnitude less at large angles X. In the region of intermediate angles the contribution of the 2 3p-level is comparable with (and in some cases exceeds) that of the 2 3S-level in a wide region of electron energies ranging from the threshold up to 100 eV. The total contribution of the 3 3S-, 3 3p_, and 3 3D-levels to the excitation cross section is smaller than that of the 2 3S-level by a factor of four at small and intermediate angles, and an order of magnitude smaller for large drift angles. Neon. The energy dependences of the differential cross sections for excitation of the metastable levels of neon were measured [131] in the range of observation angles from 2.5 ° to 10.5° with an interval

Table 32 Calculated partial 2 3S and 2 ~S differential cross sections for helium

E (eV) 21.0 22.0 26.5 29.6 40.1 50.0 60.0 80.0 200.0

dcr'/d~v

d~r/dx

at X = Xml, (aZosr -~)

at X = Xm,, (a~ sr -1 )

da'/dx daldx

5.40 6.82 6.13

2.91 4.12 1.61

L8 1.6 3.8

5.93

1

5.9

(da/dx)(2 iS)

(d~r/dx)(2 IS)

(da/dx)(2 3S) a t x = X=i,

(da/dx)(2 3S) at X = X=~

1 5

1 1

1

1

6O 1000

3 4

I.I. Fabrikant et al., Electron impactformation of metastable atoms

87

of 0.5o-2 ° (fig. 57). As in the case of He, the dynamics of appearance of each singularity in the excitation cross section depends on the angle X. Two maxima are suitable for observation, those at E = 16.91 and 18.67 eV, which according to the classification by Read et al. [303] correspond to the 2pS(2p3t2,1/2)3s3p 3p. and 2pS(2p3/2,1/2)3p2 1S + 2p 5 ( 2 P3/2,~/2)3P2 1D-states of the negative ion Ne-. Other resonances observed cannot be identified as a result of insufficient energy resolution.

Argon. The energy dependences of the differential cross sections for metastable level excitation of argon (fig. 57) were measured for observation angles g = 0.5°-6.5°. Here we have a complete picture of the appearance of the resonances corresponding to the 3pS(2P3/2,1/2)4s4p 3p_ and 3pS(2p3/2,1/2)4p2 1S_ states of the negative ion Ar- with energies of 11.85 and 13.22 eV, respectively. In general, the measured relations (dtrm/dX)(E) have a weaker dependence of the shape and slope of the curves on the observation angle, though for He and Ne the position of the first resonance remains unchanged. • • 5 2 Let us dwell upon the dynamics of production of the first resonance, n p ( P3/2 1/2)(n + 1)s(n + 1)p 3p. Consider the case of the helium atom. In the process of excitation of the 2 as-level of the helium atom an electron is attracted strongly to an atom at intermediate distances as a result of a strong dipole coupling 2 3S-2 3p and 2 1S-2 1E This situation is rather common [91] and similar resonances may always appear when there exists a strong dipole coupling between excited levels with a sufficiently small spacing and with a weak coupling to the ground level. In the case of excitation of the inert gas atoms, this condition holds for the level pairs (n + 1)s[3/212 , (n + 1)p[1/211 and (n + 1)s[3/212, (n + 1)p[5/213/2. The intensities of the (n + 1)s[3/212-(n + 1)p[1/2]land (n + 1)s[3/212-(n + 1)p[5/2]3/2-transitions are high, and thus the corresponding dipole matrix elements are large as well. Therefore, the threshold resonance is revealed in the excitation cross section of the (n + 1)s[3/2]E-level. As the atomic number increases the coupling between excited levels and the ground state becomes stronger, so the resonances in the excitation cross sections become less pronounced, a fact which is well observed in the experimental curves. The mechanism of formation of the second resonance in the cross section is more complicated. Provided the resonance is poorly distinguished from the background cross section, the dependence of the corresponding feature on X considerably hinders its detection. If the threshold energy E t is close to the resonance energy, such a dependence is easily explained by kinematic effects. However, for instance, in the case of excitation of the neon atom the threshold energy (table 32) is shifted to the region of high E with increasing X, while the resonance is not clearly observed. It is known that the differential cross section for resonance scattering in the CM frame vanishes at the points corresponding to zeros of the Legendre polynomials Pt(cos 0), where l is the electron angular momentum in the resonance state. However, given the metastable spectroscopy and as a result of the detector's shortcomings, the differential cross section should be integrated over some range of angles X, while the corresponding range of angles 0 is much broader; the heavier the atom, the wider this region is [this is due to a large value of s¢ in formula (6.4)]. Therefore, one should not expect a considerable variation of the resonance cross section when changing X. However, it is possible to explain this effect by variation of the interference contribution with X. Indeed, provided the resonance cross section is small compared with the cross section of direct excitation, the dependence of dtrm/d X on E near the resonance is mainly due to the interference term, which after integration rather strongly depends on X. Concluding the discussion of this aspect, we may say that the investigation of processes involving metastable-state production by means of metastable spectroscopy has some advantages compared with

88

1.I. Fabrikant et al., Electron impact formation of metastable atoms

other methods. First, there is no need for extrapolation to obtain the integral cross sections. They can be obtained by integrating the cross section as a function of the drift angle [168]. Secondly, the differential cross section gives the complete information about resonances, in contrast with methods measuring the total metastable production• 6.4. Some results of TOF metastable spectroscopy Defrance et al. [326], Perminov et al. [328] and Goodman and Wachman [332] demonstrated the feasibility of a TOF analysis for studies of metastable excitation• A distinctive feature of this method is that it allows a separation of different metastable states. Figures 59 and 60 present the TOF spectra of neon and argon atoms [328] at different drift angles obtained at energies near threshold. The curves f(t) for the neon atom are characterized by a structure caused by the limiting velocities of the metastable atoms at different drift angles (appearance of kinematic singularities). On the basis of the formulae given in section 6.1 the differential cross sections do-/dto can be reproduced from the spectra mentioned above, up to an angle of 180°. This is a considerable advantage of this method. The data obtained by Perminov et al. [328] also demonstrate another interesting feature. As is seen from fig. 59, using the neon atom as an example, as the drift angle changes, the maximum of f(t) as a function of time shifts over the time scale. As the angle X increases, the maximum moves to long times; in other words, the atoms with slower velocities are scattered (drifted) by large angles X. As was mentioned above (section 6.1), when one measures the TOF spectra by a detector with a narrow aperture which is placed at different observation angles, not all of the atoms are detected but only those whose velocities lie within the range limited by the maximum values for the given drift angle. Therefore, to obtain the integral TOF spectrum one should integrate the measured differential spectra over all values of momentum transfer or over all values of X. This integration procedure is correct only if in the measurements of the spectra the detector is not rotated relative to the center of the collision region, but radially shifted in the registration plane, as was the case in ref. [332]. However, we believe

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that the error introduced by rotating the detector does not lead to significant distortions in the obtained integral TOF spectra, since the transit base is rather large compared with the detector drift. Differential TOF spectra and the result of their integration for an argon beam are given in fig. 60. It is seen here that the positions of the distribution maxima at two excitation energies (E = 15 eV and 30 eV) are actually the same on the time of flight scale. The corresponding atomic velocities are equal to 523 m/s at E =15 eV and 512 m/s at E = 30 eV. It is interesting to compare the areas beneath the curves of the TOF spectra obtained at two excitation energies, since they are proportional to the cross section of metastable atom production. Indeed, the number of pulses in the maximum of the spectrum at 30 eV exceeds that in the maximum at 15 eV by a factor of more than 2.5. Inasmuch as the widths of both distributions are approximately the same the behavior of the f(t)-distributions corresponds to the behavior of the cross section for production of metastable argon atoms in electron collisions obtained in the experiments investigating the differential and total cross sections for excitation [131,168]. Another important conclusion drawn in ref. [328] is setting up the correspondence between the i

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Fig. 61. He(2s 2p) P-resonance shape obtained in an MTOF experiment by Defrance et al. [326].

90

I.I. Fabrikant et al., Electron impact formation of metastable atoms

velocity distribution function for metastable particles which are excited by a pulsed electron beam (20-30 ~s) at electron energies exceeding the threshold energy by a factor of 2-3 on the one hand, and the initial atomic beam distribution function on the other. Finally, let us give the results of Defrance et al. [326] on the cross section of metastable helium atom production by the time-of-flight procedure in the region of double excitation. Figure 61 presents the energy dependences of the differential cross sections. The change in the form of the (2s22p)Ep He-*** resonance at different drift angles is very pronounced.

7. Conclusion As we can see from the previous sections, at present there is rich experimental information concerning differential and integral cross sections for the production of metastable states of inert gases and of mercury atoms. Detailed studies of the energy dependences of the cross sections for these atoms including the investigation of resonance structures using electron beams with high energy resolution (AE _<0.02 eV) have been performed. The production processes of metastable states of the alkalineearth atoms and those with a high vaporization temperature (AI, Ga, In, TI, Cu, etc.) are not so well studied. As far as the production processes of metastable states of unstable atoms (oxygen, nitrogen, halogens) are concerned, the experimental information about them is rather scarce or does not exist at all. The desired accuracy (-<10%) in determining the absolute values of metastable particle production cross sections is not yet obtained. In many cases we have no data on the excitation of separate levels. In this connection we believe it expedient in future research to concentrate the efforts on further perfection of experimental techniques in order to obtain more accurate values of the absolute cross sections for excitation, widen the range of energies of bombarding electrons (up to 1000-1500 eV) and determine the excitation cross sections of separate metastable states (not only the sum of two or more states). In this respect the method of electron spectroscopy with its modifications remains quite promising, while the method of laser fluorescence is also far from having reached its limits. One may expect much from the use of metastable spectroscopy combined with the time-of-flight analysis of the products. Finally, we believe that new results can be obtained by precision experiments investigating the resonance structure in the cross sections of metastable state production with high energy resolution. At present we have rather powerful theoretical methods for investigation of the excitation of metastable states. However, only a small number of calculations for concrete systems have been carried out with these methods. For instance, the methods of pseudopotentials and pseudostates were employed mainly for the calculation of the excitation of hydrogen and helium, the R-matrix method for He, Ne, At, Hg, TI, and Pb and the continuum Bethe-Goldstone equations only for He, C, N, and O. For a more complete investigation of the formation of metastable states more calculations employing these methods are necessary. The R-matrix method and the method of the continuum Bethe-Goldstone equations seem to be most efficient for the investigation of resonance structure in the cross sections.

Acknowledgements The authors should like to thank Dr. R.J. Damburg, Dr. Yu.I. Ryabikh, Dr. E.E. Kontrosh, and Dr. V.D. Perminov for useful discussions, Prof. P.G. Burke for some useful remarks, and T.Yu. Popik for her help in the preparation of the manuscript.

LI. Fabrikant et al., Electron impact formation of metastable atoms

91

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Note added in proof After the completion of the present review there has appeared a series of works concerning the metastable excitation. The investigation of the excitation of noble gas atoms are still carried out by the laser-induced fluorescence method [1], TOF technique [2], by means of optical absorption method in a gas cell [3]. Using a new detection technique [4] the excitation of a new metastable state of Hg with excitation energy of 10.530 eV was observed [5]. A new branch of investigations is concerned with the simultaneous electron-proton excitation of He 23S-state near the threshold [6]. New data on the resonance structure due to the excitation of Ar [7] and Xe [8] were obtained. The differential and integral cross sections for metastable excitation in Xe were measured in ref. [9] together with an absolute normalization procedure. Finally, the first experimental data concerning the excitation of atomic oxygen metastable states has appeared [10]. Among the new theoretical works we mention the calculation [11] of the 3P1,2-1evel fine structure excitation in carbon atom, in which all the terms of the ground state configuration and three pseudostates have been considered. New data [12] on resonance structure in the excitation of He were obtained using the 19-state R-matrix calculations. New theoretical data [13] on the excitation of 2D°-state of Cu are of special interest. The results give a sharp peak near the threshold which is due to two shape resonances. The new cross section near the threshold is found to be an order of magnitude larger than the estimations previously used.

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