ATOMIC INNER-SHELL THRESHOLD EXCITATION PHENOMENA

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PHENOMENA

B. Crasemann

To cite this version:

B. Crasemann. ATOMIC INNER-SHELL THRESHOLD EXCITATION PHENOMENA. Journal de

Physique Colloques, 1987, 48 (C9), pp.C9-389-C9-400. �10.1051/jphyscol:1987969�. �jpa-00227387�

JOURNAL DE PHYSIQUE

Colloque C9, supplgment au n012, Tome 48, dgcembre 1987

ATOMIC INNER-SHELL THRESHOLD EXCITATION PHENOMENA

B. CRASEMANN

Department of Physics and Chemical Physics Institute, University of Oregon, Eugene, OR 97403, U.S.A.

Photoionization of atomic inner shells in the threshold regime permits observation of the onset of multiple excitation processes and post-collision interaction. Some aspects of the interpretation of these phenomena in terms of atomic structure and dynamics are dis- cussed.

The study of threshold processes is among the most interesting and fruitful approaches to a deeper understanding of atomic and mole- cular structure and dynamics. This field has become accessible to detailed investigation through the advent of tunable, highly monochro- matic synchrotron radiation. Rapid progress is being made, as demons- trated by a number of reports at this Conference. I want to touch on two topics related to threshold phenomena that arise from Coulomb correlation: multiple photoexcitation and post-collision interaction.

1. Multi~le Photoexcitation 1.1 Introduction

Many-electron processes induced by photon impact, generally cate- gorized as shakeup and shakeoff, epitomize the limitations of the conventional most tractable models of atomic structure. In first-order perturbation theory, the photon-electron interaction is described by a one-electron operator. The traditional frozen-core, central-field model consequently does not predict shake processes, which involve a change of state of at least two electrons. The occurrence of shake processes is a direct result of electron-electron correlation, and their study is of fundamental importance in gaining an understanding of correlation effects in atoms.

The subject indeed has a venerable history. Gregor Wentzel alrea- dy in 1921 held double-electron excitation responsible for K x-ray satellites [I]. Druyvesteyn in 1927 showed convincingly that the [sa- tellite] "x-ray spectrum of the second kind" is emitted from the inner shells of doubly ionized atoms [2]. Richtmyer suggested two-electron- one-photon transitions as a source of x-ray satellite lines [3], and Felix Bloch calculated this process [4]. In the 19601s, Deslattes showed that double-vacancy satellites also appear in fluorescence, not only, as previously thought, under electron impact [5]; ~ 6 r b e r and Mehlhorn demonstrated the concomitant Auger satellites [61, and Krause et al. measured the first photoelectron satellite spectra and showed that specific satellites can be associated with specific double-hole configurations [7]

.

Studies of the threshold behavior of corelation satellites are particularly interesting because they reveal dynamical aspects of correlations due to relaxation. The basic physical process can be understood in terms of the response of the atomic electrons to the changing potential as the ionized atom relaxes. I will outline a cal- culation in terms of the traditional shake theory describing this response [8,9], and make comparison with experiment. It has become clear, to be sure, that valence satellite structure is not exclusively

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987969

governed by shake processes but is also affected by additional confi- guration interactions in the ground and final states, calling for multiconfiguration approaches [10,11] or many-body perturbation theory

[12,13]. A further interaction displayed in valence satellites is interchannel coupling among different satellite channels including the corresponding Rydberg series [14-161. Measurements of the angular- distribution asymmetry parameters in addition to partial cross sec- tions assist greatly in unravelling the contributions of such effects

[14,17,18]. Here I will limit myself to consideration of the threshold behavior of satellite intensities in terms of the independent-particle formulation of the shake theory, and give an example of its validity in at least one case in which these other effects have minor impact.

1.2 Shake probabilities in the sudden approximation

At large photon energies the effective charge seen by the atomic electrons changes suddenly during photoionization, and the sudden ap- proximation applies for shake processes. In a definitive study, %berg introduced atomic correlation in the sudden approximation treatment of excitation accompanying ionization of a many-electron system [8].

In the generalized sudden ap~roximation applied to the shake process in a many-body system [8], continuity at the instant of ioni- zation (t=O) and the closure property of the eigenfunctions of the system at t>O lead to an expression for the amplitude an(k) of the probability for a transition from the ground state of an N-electron system to a state n of the Hamiltonian H(N-I), with one electron in a continuum state associated with k. Integration over momentum space leads to the probability of finding the remaining system in its nth eigenstate.

If we neglect the correlation of the electron to be removed with the remaining electrons in the initial state, and take its original bound-state wave function to be orthogonal to the final-state eigen- functions, we arrive at the conventional sudden approximation or

fro-

zen structure approximation, which amounts to assuming that the re- moved electron leaves so quickly that the other orbitals have no

chance to relax. The unrelaxed core can then be expanded in eigenfunc- tions of H(N-1). The next step is to use central-field wave functions and represent all orbitals by Slater determinants; this leads to the independent particle formulation which is the stage at which the prob- lem becomes readily tractable with present-day computational resources

[19-211.

To illustrate, I will briefly summarize the sudden-approximation results computed by G. B. Armen in our group for 3p-electron shake processes that accompany Is photoionization of Ar [21]. We then turn to the dipole approximation and ask how large the photoelectron energy must be, theoretically, for the sudden limit to apply. Recent experi- ments provide sensitive tests of such predictions [22-241.

Independent-particle estimates of Ar 3p shake probabilities du- ring Is ionization are readily obtained. With the 3p orbital optimized in the Ar[ls] hole-state configuration and 3p' in the neutral atom we have [21], from Hartree-Fock wave functions, <3pt 13p>=0.9833484. Hence the probability that one or more electrons leave the 3p shell is

The probability that only one electron leaves the 3p state is R1 (3p) = 6[1-<3pt (3p> 2 ]/<3pt 13p>2 = 0.20, (2)

showing that single-electron excitation dominates.

How much of R1(3p) is due to lowest-order 3p-+4p shakeup? With the 4p wave function optimized outside the Ar[ls,3pJ frozen core, one finds <3pt(4p>=-0.1673134, whence [21]

pn theZother hand, if the 4p wave function is optimized for Ar[3p1(4p) S[ls] S (also with frozen-core wave functions) instead of the config- uration average, we have R1(3p+4p)=0.12. The large difference of this result from (3) is partly due to the strong dependence of the 4p wave function on the exchange potential; it illustrates the great sensiti- vity of overlap, and hence shake probabilities, to the fine details of the atomic model. We see that much of the single-particle excitation from the 3p shell is to the 4p state, but the result is not quantita- tive because the closure relation implicit in the independent-particle formulation does not necessarily apply: 4p wave functions generated outside the Ar[ls,Sp] core are not quite the same as those outside the Ar [ls] core.

The coupling scheme very much affects shake probabilities, as is apparent from the preceding example. For Ar[l$] sudden shake, monopole selection rules restrict the final states to S. One can show [21]

that the genealogical coupling scheme

applies for Sp+np if np is so far outside the core that its interac- tion is negligible, which is the case in Ar for n>6. If the coupling between the outer electrons is strong compared with their coupling to the core, the inverted scheme applies, which leads to the basis set

1 1 , 3 ~ > = Ar[Sp](np)1'3~[ls] 2 S ; ( 5 )

this is appropriate for n=4. For n=5, more or less equal mixing be- tween the two schemes obtains.

In the conventional sudden approximation, the independent-partic- le 3p shakeup amplitude is

where

r

indicates the coupling scheme. For fullrelaxation, both states are represented by determinantal wave functions. The Slater determinant

(Ar [ls] SMs> 2 = ] [ls ms=-Ms] )

is composed of the 18 (nkmQms) 2entral-field orbitals of the Ar ground state. The states Ar[ls,Bp](np) S are formed from linear combinations of

I

[ls]}, 1 [3p]}, and I (np)) determinants, the coefficients depending upon the coupling.

Armen [21] has shown that the resultant overlap determinant can be factorized into a 5x5 S determinant and a 12x12 P determinant. The former does not depend on n. The P matrix is sparse and can be manipu- lated into triangular form. Off-diagonal elements in the overlap ma- trix turn out to be relatively small, accounting for less than 1% in the amplitude; they can therefore be neglected. For a configuration Z, the probability that the core will remain in the same vacancy state is then given in partial relaxation terms as

-

where q(nk) is the number of electrons that occupy the (nk) shell of

2 . The independent-particle-formulation partial-relaxation shake amp-

litude then is

An(r) = ar~{[ls,3p]}1'2<3p'l~:np>, (9)

where the np orbital is optimized while coupled to the frozen core in scheme r , and a r is a coupling-dependent angular factor. With P{[ls]}

=0.767665, Armen [21] finds the shakeup probabilities listed in Table 1.

- - --

Table 1. Calculated relative Ar[ls,Sp](np) shakeup probabilities, in percent. From Ref. 21.

Coup 1 i ng n=4 n= 5 n=6 n>6(a) Genealogical singlet 11.24 1.37 0.37

Genealogical triplet 0.90 1.25 0.68 1.54 Inverted singlet 8.99 1.22 0.36

Inverted triplet 1.10 1.20 0.65 1.56 [Is, 3p] (np) conf ig. av. 14.87 2.84 1.09 1.52

&

(a)~rom quantum-defect theory.

For shakeoff of a 3p electron into the continuum, the coupling is the same as for shakeup; the relative probability is

per unit energy range. The factor

$

is 6 for the configuration aver- age, or 3/2 and 9/2 for singlet and triplet genealogical coupling

[21]; the continuum states are normalized per unit energy. Shakeoff electrons tend to be emitted with low kinetic energy (45% with &<I0 eV, and only "0.04% with s>1 kev). The total 3p shakeoff probability in the sudden approximation is 5.1% in genealogical coupling, or ap- proximately 1/4 of the 20.4% total 3p shake probability calculated with optimally coupled wave functions. Quite comparable results have been found by Dyall [20] with a shakeoff estimate based on a single- particle closure relation. Single 3p excitation accounts for two

thirds of the 30.3% total Ar[ls] shake probability [21].

1.3 Threshold behavior: =dipole approximation

The preceding discussion exemplifies how asymptotic shake proba- bilities, calculated in the sudden approximation, can serve to test delicate details of wave functions and coupling. The interesting questions that are experimentally accessible with tunable x rays are:

How high must the photoelectron energy be for the sudden limit to apply? How do shake probabilities depend on photoelectron energy in the threshold regime? For this purpose, it is necessary to calculate the cross sections as functions of incident-photon energy, for photo- ionization accompanied by excitation and for double photoionization.

Dipole transitions from an initial state i to a final state f:

are described by the formula

<firlczn/i>

I ,

⁽¹¹⁾

where E is the incident-photon energy, a is the fine-structure con- stant, and Q is either a singly or doubly differential cross section, depenfing upon the nature of f. In the case on hand, the initial state is Ar S and the final states are of the form Ar[ls,3p](xR',cR); for shakeup x denotes a bound state (x=n) and for shakeoff, a continuum state (x=E'). With central-field wave functions, Eq. (11) reduces to products of single-electron dipole and monopole matrix elements. Di- pole excitation of the Is electron and monopole selection rules for

the remaining orbitals calllfor R8=R=l, whence the pertinent final states are Ar [ls,3p] (xp,sp) P.

Aberg [9] has formulated the theory and performed the sum over From his work it follows that, neglecting terms that involve off-dia-

gonal overlaps, the ratio of the Ar[ls,3p](4p) shakeup to the Ar[ls]

(no-shake) cross sections is

where

and P { Z ) is as defined in Eq. (8). The continuum-electron energies are

~=E-I{[ls,3p](4p)) for shakeup and E'=E-I{[ls]} for the no-shake case, rlespectively. As these energies become large, the overlap matrix ele- ments decrease much faster than the dipole term. With increasing E, the shake probability R therefore approaches the sudden-approximation configuration-average limit, multiplied by an energy-dependent ratio of the Is excitation terms:

-

R ~ u d d e n I < alplrlls>("

lim R (E) = ^ci

At large E the continuum wave functions oscillate rapidly, except near the nucleus, and here the 1s wave function is insensitive to the outer configuration; the limit of the ratio of dipole matrix elements there- fore approaches unity as E increases. Moreover, since the dipole inte- grals monotonically decrease at large E, the shake probability R is expected to approach the sudden limit from below. These conclusions apply to shakeoff as well as shakeup.

The relative shakeoff or double-ionization cross section in the dipole approximation is found by replacing the 4p wave function in Eq.

(12) by the continuum wave function of the shakeoff electron with energy E (O(E(E~), SO that El= E2-E , where E2=E-I ( [Is, 3p] ) :

It is clear that the double-ionization probability must always start from zero at threshold 193

.

In summary, calculations in the dipole approximation with Har- tree-Fock wave functions for Ar[ls,3p](xp,~p) indicate that for shake- up the sudden limit is reached very quickly: within only 0.1 eV of threshold, the cross section reaches 64% of the sudden value, and 10 eV above threshold it has risen to 91%, reaching 100% at 95 eV. Shake- off, on the other hand, starts at threshold with zero intensity. It rises to 50% of the sudden limit at 20 eV excess energy, to 90% at 100 eV, and reaches the sudden limit 300 eV above threshold [21].

1.4 ExDeriments

The predicted behavior of shakeuw probabilities in the threshold region can be tested by measuring phoio;lectron satellite intensities over a range of kinetic energies down to zero. This difficult experi- mental task has been accomplished with high sensitivity and resolution with a "threshold-electron analyzer" in which a small electric field

extracts slow electrons from a synchrotron-radiation target, and time- of-flight analysis is used to measure the electron energy spectra 122,

181. The paper by Shirley at this Conference [23] contains a summary of some of the results attained by this technique.

Threshold measurements of cross sections for the production of multivacancy configurations by double photoionization or shakeoff, on the other hand, cannot readily be performed by photoelectron spectro-

metry because the emitted electrons share energy over a continuous range. Subsequent decay of the multiply excited atom by radiative or radiationless transitions, however, results in x-ray or Auger-electron satellites that reflect the probability of shakeoff as well as shakeup processes.

The potential of photon-excited fluorescence studies for double- vacancy satellite diagnostics was revealed by absorption and emission spectroscopy of the Ar Ka x-ray spectrum conducted by Deslattes et al.

in the Stanford Synchrotron Radiation Laboratory several years ago [251. Partial cross sections for the production of [ls,3s] and [ls,3p]

double-hole configurations and the most prominent features of their excitation functions could be determined. These studies have now been significantly extended with specially designed apparatus on the Natio- nal Bureau of Standard's x-ray beamline at the National Synchrotron Light Source in Brookhaven [26].

Auger satellites were used by our group to trace the near-thresh- old energy dependence of shakeup and shakeoff probabilities in Ar

1241. Highly monochromatized, hard synchrotron radiation was tuned through the thresholds for various multiple-excitation processes dur- ing 1s ionization of Ar, and the probabilities of accompanying 3s and 3p excitation were traced by measpring the intensities of pertinent satellites of the 2660-eV K-L2L3 D2 Auger line. The satellite spec- trum was calculated by computing the radiationless transition energies and rates of those initial doubly excited states which in the sudden approximation are expected to be significantly populated (Fig. 1).

While individual satellite lines were not resolved in the experiment, it is seen that 3s and 3p shakeoff satellites cluster in one peak near 2643 eV, while most 3s and 3p shakeup processes cause A u g e ~ satellites that fall within a peak near 2650 eV, unresolved from the SO diagram line. Analysis of these two Auger-peak intensities as a function of x- ray energy therefore permits one to deduce the near-threshold behavior of the 3s and 3p shakeup and shakeoff probabilities.

. ^.

FIG. 1. Calculated energies of Auger satellites caused by 3s and 3p electron excitation accompanying Is photoioniza- tion of Ar, with reference to

f

^{, , , , ,}

L,.

^,

,1

- the K-L23L23 Auger spectrum.

Relative satellite intensi- ties within each multiplet

r .

^{* * a 4} are indicated by the heights of the bars. From Ref. 24.

O 2640 2650 2660 2670

AUGER ENERGY ( eV

Results are compared in Fig. 2 with calculations in the dipole approximation (Sec. 1.3). Below the [ls,3p](4p) threshold, we see an indication of a resonance that may correspond to an x-ray feature 2 ascribed by Deslattes et al. [25] to the [ls,3p](4p ) bound-bound resonance, and recently identified in a calculation by J. W. Cooper as

[ls,3~1 (3d2) [271.

The [ls, 3p1 (4p) shakeup intensity [Fig. 2 (a) ] rises quickly with energy, but not quite as quickly as predicted: instead of reaching the calculated 64% of its asymptotic value 0.1 eV above threshold, it reaches 50% approximately 5 eV above threshold; this observation ag- rees with a photoelectron measurement by Kobrin et al. 1281. Satura- tion of [ls,3p](4p) and opening of the [ls,3s](4s) channel lead to a small gradual increase which levels off to a constant shakeup intensi- ty -60 eV above the [ls,3p](4p) threshold. The measured energy depen- dence of the shakeup probability is well predicted by theory. The calculated asymptotic configuration-average shakeup intensity is 14%, but with a more realistic coupled state it is lo%, in accord with measurement. The data also agree with the configuration-interaction calculation of Dyall [20], which predicts the same shape as ours, but some shifting of intensities to higher shakeup states.

- _{-+i - - - - - - -}

_{-'s diagram-}

_{- - - -} -

^lo

^-

> :

t :

a . (01 SHAKEUP

C i i : : ! ' : : : : : : : : : ' : : :

J 20 :

W .

a .

(0

-

( b ) SHAKEOFF 1

FIG. 2. (a) Intensity of the 2650-eV feature in the photoexcited Ar K-L23L23 Auger spectru~, with refe- rence to the D line inten- sity, as a function of x-ray energy. The d a ~ h e d line indicates the S diagram- line contribution. The solid curve is the normalized theoretical prediction from shake theory in the dipole approximation. (b) Photoex- citation-energy dependence of the 2643-eV Auger satel- lite-group intensity. The normalized theoretical rela- tive shakeoff probability is indicated by the solid curve. From Ref. 24.

.

. . . . . .

0 3200

[ .,.

3300

^,

^{3400 ,}

^-

5206

PHOTON ENERGY ( e V )

The shakeoff probability is given by the intensity of the 2643-eV peak less its 3 p 4 5 p admixture [Fig. 2(b)]. Its energy dependence agrees with the theoretical prediction. The asymptotic probability is 19%; our calculation predicts 25% and Dyallls 1201, 7.3%.

In summary, this experiment confirms the predicted difference in the photon-energy dependence of shakeup versus shakeoff close to threshold. The measured total shake probabilities are bracketed by the sudden-approximation values calculated from HF wave functions in a double-hole [ls3E] and a single-hole [Is] field; both procedures are somewhat inconsistent with regard to fulfilling the closure relation.

The inconsistency would be removed if many-electron wave functions were used [24].

The case of Ar [ls,3p] double photoexcitation on which we have dwelled is interesting because it is a prime example of the success that the shake theory in the independent-particle formulation can attain if realistically coupled wave functions are employed and confi- guration interaction plays a relatively minor role. The opposite ex-

guration interaction plays a relatively minor role. The opposite ex- treme is exemplified by very recent work by Krause and Caldwell in which beautifully resolved Auger satellites were used to measure the near-threshpg probability for conjugate shakeup of Be atoms to the

[ls,2s](2p) P states--a process that takes place up to 40% as often as [Is] single photoionization [29,30]! In this first photoionization study of the Be atom, a prototype for multiconfiguration interaction in the ground state, features appear in the shakeup partial cross sec- tions that obliterate any semblance of independent-particle predic- tions, including structure suggesting resonances due to quasi-discrete multiple excitation states.

2. Post-Collision Interaction 2.1 Role of the effect in inner-shell dynamics

Post-collision interaction (PCI) represents another correlational aspect of the complex dynamics of electron excitations which inevitab- ly accompanies the escape of a photoelectron from an inner shell of a many-electron atom. Considerable progress in the understanding of this phenomenon is currently being made. In the prototypal case, a hole state is created near threshold, i.e., a slow photoelectron is emit- ted. The vacancy is filled under emission of a fast Auger electron. In a semiclassical picture [31-331, as the photoelectron is passed by the Auger electron, a sudden reduction in screening alters the attractive ionic-core potential experienced by the photoelectron, from the Cou- lomb potential of a charge +e to that of a charge +2e. The energy lost by the slow photoelectron in this sudden transition is transferred to

the fast Auger electron. The mean energy of the Auger-electron distri- bution is shifted upwards and its shape is distorted.

Rather than being an isolated phenomenon, PC1 provides an impor- tant link between the sub-threshold regime and the asymptotic high- energy characteristics of inner-shell excitation-deexcitation dyna- mics. It has long been known that in threshold excitation of short-

lived, deep atomic hole states ionization and decay cannot be treated as distinct processes. Directly at threshold, photoionization and radiationless decay occur in a single step, the resonant Raman effect

1343. Only in the high-energy limit does a two-step model become va- lid, in which atomic excitation is followed by a relaxation phase, the ensuing deexcitation becoming decoupled from the excitation process and depending on properties of the isolated atom alone. Above thresh- old and below the high-energy regime, excitation and deexcitation are linked by PCI, in which the mutual Coulomb interaction between the ion and two electrons causes deviations from predictions of the two-step model. PC1 thus provides continuity in the energy evolution of inner- shell dynamics [35,36] (Fig. 3). A unified theoretical description of photoionization and Auger decay including resonant Raman and PC1 ef- fects is therefore an important goal.

2.2 Scatterina-theory treatment of PC1

The semiclassical picture sketched above provides a fairly good approximate description of PCI. A consistent description of the influ- ence of PC1 on total and differential inner-shell photoionization cross sections can be attained in terms of relativistic quantum theory [37,38]. A suitable ansatz is the resonant-scattering-theory approach to photon-induced inner-shell vacancy decay, developed by Wberg [39, 4 0 1 . The fundamental idea behind the scattering-theory treatment of PC1 is to generalize conventional multi-channel scattering theory to allow for two electrons in the outgoing channel [39]. A transition matrix element can then be derived which describes Auger decay as a one-step process which results from the photon-electron interaction--- a resonance in the double photoionization cross section [37]. The PC1 effect consequently appears as a property of the resonance behavior of

X-RAY ENERGY cu ( OV )

rn Z

rn ;D G )

<

U) FIG. 31 Energy of the Xe L3-MqMg ( G4) Auger-elec- -'tron peak, as function of bexciting x-ray energy. Near

-

threshold Auger satellites, zcaused by the spectator

photoelectron in bourid or- bits, exhibit linear Raman dispersion. The PC1 shift

(right-hand scale) vanishes only in the asymptotic li- mit. From Ref. 36.

double photoionization [37].

Following this approach we performed relativistic calculations of the resonance in the double ionization cross section that describes photoionization and Auger-electron emission [37]. The lowest-order

line-shape formula, corresponding to Read's "shake-down'' mechanism, emerges from approximations of the general multichannel transition matrix element:

where

H e r e , \ ~ ~ a n d

are

asymptotic continuum-electron wave functions without a short-range phase shift [37,46]. The relativistic intermediate-state PC1 function

I

^{r l >} has been computed for selected transitions by super- position of several thousand Dirac-Fock central-field continuum wave

functions, taking the dipole interaction between the ground state and outer-shell hole states into account [37,381. This function describes how far the photoelectron escapes before it encounters the change in the potential caused by the Auger decay. Simplification indicates that the leading term in the PC1 function is a damped spherical WKB wave;

the damping is determined by the intermediate-state lifetime.

Calculations for some cases that have been investigated experi- mentally show excellent agreement with regard to PC1 energy shifts of the Auger peaks. It also became clear that the non-Lorentzian shape of the resonant Auger-electron and photoelectron peaks is almost entirely determined by the asymptotic behavior of the wave functions of the ejected electrons in the ionic field and by the lifetime of the in- ner-shell hole state. The deformed line shape is insensitive to the structure of the residual ion and to exchange effects, and also to the angular momentum and to the non-Coulombic part of the potential energy of the escaping electrons. This fact explains the success of semiclas- sical models.

2.3 Final-state interaction

An important shortcoming of the shake-down model, even in its exact quantum-mechanical form, is that it does not account for the interaction between the photoelectron and the Auger electron in the final state: the only effect considered in this model is the response of the photoelectron to the change in the ionic core potential pro- duced by the Auger decay. The model works well when the photoelectron energy is small compared with the Auger-electron energy. It fails when this condition is not fulfilled. The reason for this failure, pointed out by Ogurtsov [41] and by Russek and Mehlhorn [33] in terms of the semiclassical model, is that account is not taken of the time required for the photoelectron to sense the potential change caused by the Auger decay. Consequently, at larger photoelectron energies, an excessive PC1 shift and line-shape distortion are predicted. In fact, the PC1 shift in photoexcited Auger transitions should disappear alto- gether if the energy of the photoelectron exceeds that of the Auger electron: in that case, semiclassically speaking, the Auger electron never passes the photoelectron and no change occurs in the ionic-core potential experienced by the latter. This "no-passing effect" was observed experimentally ₁ by Borst and Schmidt 142,431 in the Xe N5- 023023 ( S O ) Auger transition and later by our group [44] in Xe L2- L3N4 (J=3) Coster-Kronig decay (Fig. 4).

FIG. 4. Measured Xe L2-L3N4 (J=3) Coster-Kro- nig electron energies as a function of incident x-ray energy. The dashed curve

(I) represents the predic- tion of the simple shake- down theory (Ref. 37); the solid curve (NP) is calcu-

Photon excess energy (eV)

.- _{- -}

-

I I 1 I l l I I I 1 l I l l

A semiclassical PC1 moctel formulated by Russek and Mehlhorn [33]

correctly predicts both shift and line shape even for high excitation energy. How can the lowest-order quantum-mechanical theory be genera- lized to include final-state interaction among the continuum elec- trons, so as to incorporate the "no-passing" effect? In principle, this requires a generalization of the K-matrix theory of single-elec- tron photoionization [45]. This very elaborate approach has been deli- neated but not yet carried through [46].

Fortunately, a simpler way exists. In the lowest-order quantum theory [37], the PC1 results can be interpreted in terms of an analy- tical line-shape formula based on asymptotic Coulomb wave functions

[Eq. (16)l. Now the Coulomb line shape depends only on three parame- ters: the excess photon energy, the lifetime of the inner-shell hole, and the change of the ion-ic charge Q during Auger-electron emission.

Thus there is only one quantity which is exposed to the final-state interaction between the photoelectron and the Auger electron, viz., Q.

It can be shown [38,46] that the time dependent mutual screening of lated from the semiclassi- cal model of Russek and I Mehlhorn (Ref. 33). From

40 100 300 'OoO Ref. 44.

the two electrons in the final state can be translated into an ener- gy-dependent screening expressed by a dynamic charqe in the line-shape formula (16) :

s ¹

Q . = Q = 1 - - = I - ( € / E ) ~

d K " exc A

H

if the photoelectron is slower than the Auger electron (Ks<KA). If the photoelectron is faster than the Auger electron we have Q=0, and the Lorentzian form of the profile is recovered. The resultant quantum- mechanical Coulomb line shapes agree closely with the semiclassical results [33,47] and with experiment.

2.4 Angular-dependence

The analytic quantum-mechanical line-shape formula described here has been developed from angle-averaged first-order resonant scattering theory, leading to predictions that correspond to singles (non-coinci- dence) measurements. As the field advances, measurements of the angu- lar correlations between photoelectrons and Auger electrons are being envisioned [43]. In order to anticipate what such measurements might show, Armen [38,48] has drawn upon the close correspondence between the angle-averaged semiclassical model of Kuchiev and Sheinerman [47]

and the quantum-mechanical Coulomb line-shape formula, and applied the angle-dependent Kuchiev-Sheinerman model to the specific case of PC1 line shapes in photoexcited Auger decay. Striking effects are predic- ted, presaging the wealth of further insights into time-dependent inner-shell dynamics that may come from these studies.

Acknowledqments o

The writer thanks T. Aberg and G. B. Armen for many helpful dis- cussions. This work was supported in part by National science Founda- tion Grant PHY-85-16788 and Air Force Office of Scientific Research Grant AFOSR-87-0026.

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[4] F. Bloch, Phys. Rev. 48, 187 (1935).

[5] R. D. Deslattes, Phys. Rev. 133, A399 (1964).

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