MANY-BODY EFFECTS ON DEEP LEVEL SPECTRA OF METALS

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MANY-BODY EFFECTS ON DEEP LEVEL

SPECTRA OF METALS

L. Hedin

To cite this version:

L. Hedin. MANY-BODY EFFECTS ON DEEP LEVEL SPECTRA OF METALS. Journal de

JOURNAL DE PHYSIQUE _Colloque C4, supplkment au no 7, Tome 39, Juillet 1978, page C4-103

MANY-BODY EFFECTS ON DEEP LEVEL SPECTRA OF METALS

L. HEDIN

Department of Theoretical Physics, University of Lund, Lund, Sweden

RBsumC. - Les effets a N corps dam les metaux dus au changement du nombre d'electrons occu- pant les niveaw de cceur se manifestent dans de nombreux cas en spectroscopie UVL. On avait pris conscience et discutk depuis longtemps de l'importance possible du trou de cceur. La notion d'excitons dans les me'taux proposCe par Mahan avait fortement stimule ce sujet, et fut tout d'abord appliquke au cas des singularitks de seuil dans les mCtaux simples. Ce probleme, discute en detail dans une mise au point de Mahan en 1974, fut alors declare integralement rtsolu. Cette affirmation fut fortement remise en question par Dow et ses collaborateurs. I1 en a rtsulte une meilleure comprehension du phenomene, en particulier l'importance des effets statiques du trou de cceur, des phonons et des dkphasages de spin de la structure de bande electronique, du couplage d'tchange (effet Onodera) a 8ti: mise en evidence. Ainsi l'ancienne enigme du seuil-du lithium semble maintenant avoir une explication en termes de phonons et de structure de bande et non par l'effet MND. Le large seuil d'kmission peut Ctre expliquk par une relaxation incompl2te de phonon, et le large seuil d'absorption par une densite d'Ctat B un Clectron.

Abstract. - Many-body effects in metals due to a change in the core electron occupation number

appear in a large number of spectroscopies. The possible importance of the core hole has been realized and discussed for a very long time. Mahan's idea of excitons in metals gave the topic strong stimulus. First the interest was focused on threshold singularities in simple metals. The topic was thoroughly discussed in Mahan's 1974 review and declared solved and completed. This declaration was strongly challenged by Dow and collaborators. This has led to a deeper and more detailed understanding. In particular the importance of static core hole effects, of phonons, of electron bandstructure, of exchange coupling (the Onodera effect) and of spindependent phaseshifts has been realized. Thus the longstanding puzzle of the lithium edge now seems to have found its solution in terms of phonons and bandstructure and not in the MND-effect - the broad emission edge can be explained by

incomplete phonon relaxation and the broad absorption edge by the one-electron density of states.

1. Introduction. - Deep level spectra involve by definition a localized core electron. The core hole acts like an impurity which causes a range of different effects. Of these effects particularly the Mahan- Nozitres-De ~ o m i n i c i s (MND) singularity in X-ray emission and absorption has drawn a large interest in later years [l, 21. This has reactivated a general interest in the core hole impurity, and led to a greatly improved understanding of many questions. The original question of the MND singularity, however, seems to be the most difficult one and keeps defying a quantitative explanation. We will here concern our- selves mostly with core hole effects where a quan- titative understanding now is emerging.

At the Strathclyde Conference 1971 Mott [3] gave an introductory comment where he noted that theory could not yet tell whether the peak at the Fermi level in the Mg X-ray emission spectrum was a band- structure or an MND effect. In 1971 just two years after the Nozieres-De Dominicis paper [2] and the striking confirmation of the singularity provided by

the X-ray absorption measurements made at Desy [4], those were the days of strong many-body singularities. In 1973, however, Dow and collaborators started a crusade against the then prevalent views and stressed the possible influence of more conventional effects like phonon and Auger broadening and pointed out that the singularity phaseshifts had to satisfy certain compatibility relations [5]. The Dow crusade no doubt stimulated detailed work on effects which compete with the MND one in a quantitative evaluation of experimental data. If Mott asked his question about the Mg peak today I think the evidences (one is given in Ref. [6]) point in the direction of the peak being more of a density of states effect than of an MND one. The last few years have seen a rather heated debate on the relative sizeof the MND effect. I will not contribute much to that debate but rather attempt a rational description of the understanding we, to my mind, actually have reached today.

The removal of a core electron represents a strong perturbation, despite the fact that in a metal the core

C4-104 L. HEDIN

hole is more or less completely screened within the atomic sphere. The compact screening charge gives a large Coulomb potential of the order of a Rydberg at the core electron and thus has a large effect on the core electron binding energy. One should also expect strong effects on oscillator strengths and thus on spectral shapes. Surprisingly enough X-ray emission spectra show little evidence of such effects, while the emission satellites and the Auger spectra indeed can experience quite strong effects. These static effects are discussed in section 2.

The sudden switch-on of the core hole potential causes what is usually termed shake-up effects. The shake-up excitations are of boson type : electron-hole pairs, plasmons and phonons. We will discuss the shake-up of electron-hole pairs and plasmons in X-ray photoemission (XPS), which is a simpler case than X-ray absorption/emission, in section 3. In section 4 we discuss phonon effects on XPS and X-ray absorption, while in section 5 we discuss phonons in X-ray emission, where the interesting possibility of incomplete phonon relaxation appears. In section 6 , finally, we give concluding remarks.

2. Static core hole effects. - We will in this section discuss the effect of the core hole from a one-electron view point. If in a metal a core electron is removed, the core hole will be completely screened within a short distance. The screening descreases the binding energy by roughly half the screening potential [7, 81. A very precise study of screening in metals and core electron energy shifts has recently been made by Almbladh and von Barth [9], using the so called spherical solid model. They used the Kohn-Sham [l01 local density method to calculate total energies and obtained agreement with experiment within 0.5 eV for singly and 1 eV for doubly ionized core levels.

The core level energy can be approximated by [ 7 , 8 , 9 ]

=

&p

+

v;

+

₍₁₎

Here E? is the energy of the core electron when in a

free ion and VG the Coulomb-interaction between the core electron and the conduction electrons of the solid in its ground state. Vp(x) finally is the additional Coulomb interaction from the polarization cloud of conduction electrons formed in response to a fraction X

of a core hole. In addition to the terms in equation (I) there are a number of small correction terms from core-valence exchange,' readjustments of the core- electron wave functions etc. Vp(x) may be approxi- mated by xVp(l) and the last term in equation (1) then becomes (112) Vp(l). This is an overestimate while the linear response expression (112) Vk(0) is an under- estimate. In the case of two core holes the last term in equation (1) becomes

Vp(l)

+S

(Vp(x) - Vp(l)) dx = 1

TABLE I

Energies in eV for dzflerent contributions and

approximations oj E, in equation (1) jrom rqference [9]

One core hole Two core holes

- VG 1 0 )

1

'

V p d x (312) V@)

1:

X .. 0 Li, Is 11.4 6.5 7.4 19.4 26.0 Na, 2p 9.3 5.2 5.6 15.7 18.4 Al, 2p 40.6 8.2 8.6 24.5 27.6

In table I we quote some results obtained by Almbladh and von Barth [g] for V; and for the polarization energies where we give both the linear response and the full results. For one core hole the accuracy of the linear term is more or less within the accuracy of equation (1) itself, while for two core holes the linear expression is rather crude.

The spherical solid model takes detailed account of the ion that suffers a change in core electron occupa- tion. To have a model which is easier to evaluate one may represent the core hole by a potential. It is then tempting to choose a Coulomb potential. The screen- ing charge of a Coulomb potential actually gives reasonable numbers for the polarization energy, if linear response is used. This is however rather for- tunate in view of the fact that the non-linear result for the polarization energy differs strongly from the linear value [ll]. Thus one should refrain from using a Coulomb potential and instead take a pseudopotential where the non-linear effects only represent small corrections.

MANY-BODY EFFECTS ON DEEP LEVEL SPECTRA OF METALS

NO

( I 6 atoms per unit cell)

I

Local DOS for

A Na atom with a

2 p core hole

FIG. 1. -The local density of states for sodium in its ground state (upper part) compared with the same quantity when the atom has one core hole. Fromvon Barth and Grossmann (to be published).

good agreement with experiment led them to postulate that the one-electron functions of the Jinal state are those pertinent to describe experiment. This postulate is also in accord with the experimental results for lithium. Here the p rather than the s contribution enters and no distorsion of the satellite is observed. The von Barth-Grossmann postulate does not allow finer details like the MND singularity to be explained and it is a challenge to find its limitations and proper theoretical foundation in terms of a dynamical theory. An interesting hint how to do that starting from the ND theory is presented in the contribution by von Barth and Grossmann at this conference.

L2,ssatellite

-

theory

+ * + exp.(Hanson and

Arakawa, 1972) L,,- main band

FIG. 2. - Comparison between theory and experiment for the sodium L,, band. The upper part shows the main band and the lower part the satellite. From von Barth and Grossmann (to be

published). CI

-

--

S

ZI

-

L

2

t

g

-

Y C I . W V U

In connection with the L,, high energy satellites it should be noted that the earlier explanation of their shape involving multiplet stfucture, was disproved by Almbladh and von Barth [13]. They analyzed several experiments which measure the energy of the doubly ionized state (2p2). In particular they identified the three multiplet thresholds in the radiative Auger spectrum obtained by Aberg and Utriainen (Fig. 3). From energy considerations they could then conclude that in the emission spectra only the lowest energy level (,P) was involved ; the other multiplets decay away before emission takes place.

Strong influence of the core hole also finds support in recent experimental data on KLV Auger spectra obtained by Barrie and Street [14]. They report mystery peaks in the low energy part which corres- pond well with the S-resonances of figure 1. For KLV spectra there is no question about the pertinent number of core holes, there is one both in the initial and in the final state (von Barth and Grossmann, to be published).

L. HEDIN

-50 -55 -60 -65 -70 -75

Energy (eV1

FIG. 3. - KLL radiative Auger data for Mg and A1 obtained by Aberg and Utriainen [13]. The marked position for the 3P threshold is taken from the experimental L2, emission spectrum satellite and the 'D and 'S positions are given according to the term splittings

in the free ion.

3. Shake-up of electron-hole pairs and plasmons in X-ray photoemission spectra. - When the core hole occupation number changes there is a sudden change in the potential acting on the conduction electrons and also on the phonons. We then have an effect called shake-up. The shake-up of low energy satellites in X-ray Photoemission Spectra (XPS or ESCA) of free atoms is well-known and understood; similarly in F-center spectra there are the well-known multi- phonon cascades which is precisely the same type of shake-up effect [15]. This effect is particularly clearcut in XPS since the ejected photoelectron is effectively decoupled from the conduction electrons. The golden rule matrix element then factorizes into a one-electron oscillator strength and a many-body overlap function

where cp, and cp, are the one-electron wavefunctions for the ejected photoelectron and the core electron. The $: are state vectors for the conduction electrons and the phonons in the presence of the core hole, while $, is the initial ground state. The ejected photo- electron carries information on the state of the conduction electron and phonon system that it leaves behind, through the law of energy cbnservation.

There is a complication in this simple picture

-

the photoelectron is created inside the solid and has to get out before it can get energy-analyzed, and on its way out it will make inelastic collisions. The measured energy distribution will thus be a cor~volution of the

intrinsic spectrum with the loss spectrum. The cross sections of a fast electron for electron-hole and phonon productions are small and we mainly have to account for the plasmon losses. The plasmon cross-section is however quite large and the mean free path of a keV electron is only 20-40

A.

Thus most photoelectrons come from a region fairly close to the surface where there may be deviations from bulk conditions and where one may have interference with surface exci- tation~.

The intrinsic-extrinsic problem was recently dis- cussed by Penn [l61 using the three step model and a transport equation. Penn obtained very good agree- ment with a priori calculations of the strength of the intrinsic plasmon spectrum. Considering the uncer- tainty about the influence of the surface, Penn's work cannot be considered as quite conclusive. The three step model however usually works very well at lower energies (UPS) and it ought to work better the higher the energy.

We now turn to the asymmetry in the main XPS line which comes from shake-up of electron hole pairs. This effect can be obtained from the ND Hamil- tonian [2]

where the operators c and c, annihilate core and conduction electrons. V, is the Fourier transform of the core hole potential and can be approximated by a statically screened pseudopotential, V, = w,O/~(q, 0). The ND treatment of this Hamiltonian is quite complicated but gives an exact answer, which however is valid only infinitesimally close to the edge. A simple method to obtain an approximate answer was deve- loped by Schotte and Schotte [17], introducing quasi- bosons in the Tomonaga model. Their treatment can be extended if we brutally replace the electron-hole operators c,=, c, by operators a& which are taken to obey exact boson commutation rules, and if we introdrce particle-hole excitation energies

in a model Hamiltonian

R,= &C+ C

+

C'

_{a& akq}

+

_CC+

C'

_Vq(a&

+

_akq)

,

k4 k4

(4) where the prime indicates that the summation is restricted by the conditions E,,, > E,, E,

<

E,. This is

mathematically the same Hamiltonian as in the F-center problem and its solution is well-known. The core electron spectrum A , can thus be written on an explicit form which, however, involves a complicated Fourier transform. It is more convenient (cf. Ref. [18]) to use the equivalent integral equation

MANY-BODY EFFECTS ON DEEP LEVEL SPECTRA O F METALS

where o = E, - E and E, is the threshold energy. The quantity ~ ( w ) is given by

The particle-hole spectrum thus plays an important role in determining a(o). For small o the solution of equation (5) takes the ND form

which is valid for all o if a(o) is constant.

Langreth [l91 studied a more comprehensive pro- blem by adding the electron-electron interaction to the ND Hamiltonian of equation (3) and gave a formally exact solution. The screening is then pro- vided by the electron-electron interaction term and

V, = w;/s(q, 0) in END has to be replaced by W;.

Langreth's solution (to lowest order) has the same form as in equation (5) but a(w) is given by

Here P(q, co) is the polarization function connected with the dielectric function through ,

In RPA (- 1/71.) P(q, o ) is simply the particle-hole density of states

and we can see the close similarity between the result from the N D Hamiltonian in equation (6) and that from Langreth's Hamiltonian in equation (8). The essential effect of explicitly including the electron- electron interaction is thus to replace a statically screened pseudopotential by a dynamically screened one. The Langreth solution also follows from equa- tion (4) if we replace V, by

I

w:/s(q, a,,)

1.

The effect of dynamical screening is illustrated in figure 4. The full-drawn curve corresponds to the Langreth case (&(g, m)) and the dashed curve to the N D case (~(q, 0)). Since @(m) is almost constant in the Langreth case, A, closely keeps it power law shape right out to the first plasmon satellite. The third curve in figure 4 (dash-dot) gives results by Mahan [20] where Vq is replaced by a t-matrix in the ND Hamil- tonian. This improvement clearly has a marginal effect. The experimental data seem to give a power law dependence over a wide energy range and thus to favour Langreth's model.

The implication of these results is that the XPS asymmetries are determined both by the electron-hole

0.01 l I >

I _E

EF FIG. 4. - GI(w)/GI(~) for Na after Minnhagen [21]. The solid line is the result for Langreth's model (dynamic screen~ng), the dashed line for the N D model (static screening) and the dash-dot line for the ND model when the bare pseudopotential is replaced by

a t-matrix.

density of states and by dynamic screening [21]. Further, the results indicate that the ND Hamiltonian breaks down quite close to the edge, a fact which may be of importance for the quantitative evaluation of X-ray emission and absorption edges.

4. Shake-up of phonons. - In later years it has become quite clear from experimental data that phonons play an important role in the broadening of X-ray spectra, and often give a larger effect than the- finite lifetime of the core state. The broadening due to the temperature distribution of electron-hole pairs is on the other hand usually smaller.

The basic theory for phonon shake-up in the linear approximation is well established from the study of colourcenters, semi-conductors, Mossbauer spectra, etc. The problem is to find the correct coupling func- tion in specific cases. This problem has been addressed by many authors [22,23,24, 15,25,26]. The phonon broadening function discussed here applies to XPS and X-ray absorption, but not necessarily to X-ray emission, as we will discuss in the next section.

C4- 108 L. HEDIN

FIG. 5. - The derivative of the screened pseudopotential of Li,

w l ( r ) , in units of e2 kg, with r in units of Bohr radi~. Results are

given for an Ashcroft pseudopotential with different cut-off radii r, and with Singwi-Sjolander screening and for a Thomas-Fermi screened Coulomb potential. The long lines on the abscissa show the posltlons of the neighbouring ionic shells. The figure is taken

from reference [25].

choices of v,, the cut-off radius in the Ashcroft-type pseudopotential that was used. We see that in the limit of a Coulomb potential (r, = 0) the results for W' becomevery differenqand as far as we under- stand they are also quite wrong (cf. section 2). The phonon effects are strong as illustrated in figure 6. Thus already at T = 0 the no-phonon peak is negli- gible and the broadening curve is closely Gaussian.

FIG. 6 . - The phonon broadening function for Li at different temperatures. The no-phonon line should appear at - 0.11 eV but has such a small strength that it cannot be shown on the figure. Calculattons by Almbladh (unpublished) from the data in refe-

rence [25].

An elegant semi-empirical treatment of the phonon broadening problem has recently been given by Flynn [26]. He derived the coupling function by assuming that the atom with the core hole gives rise to the same interatomic forces as if it were replaced by the next atom in the periodic system. He further assumed that the equilibrium displacements around the impurity were spherically symmetric and given by

a dipolar field right down to the nearest neighbours. The coefficient for the dipole field was obtained from experimental alloy data. Knowing the equilibrium displacements and the dynamical matrix (or the phonon spectrum) the coupling function immediately followa This is clearly a crude method but the results nevertheless agree within about 50

%

with those of Hedin and Rosengren [25].

Flynn also took account of recoil effects and showed that they can be appreciable in the XPS case. Thus the expression for the phonon strength function g(@)

in reference [24 should be corrected by adding

I

U,, k . ~ ~ ~ .

I2

to

I

FqR

12.

Here k is the wavevector of

the ejected photoelectron ; the other symbols have the same meaning as in reference [25].

Finally we want to give a short comment on the work by Dow et al. [24], since it has been much cited and was the first paper in the Dow crusade. The basic argument in that paper, where the pressure dependence of the Knight shift is used to estimate the phonon coupling function, is in our opinion wrong (Almbladh and Hedin, to be published), simply because detailed calculations show that the charge density and the Coulomb potential at the core electron do not corre- late in the way that Dow et al. assumed. In addition there is the ambiguity pointed out in ref. [27]. Thus while the main thrust of that paper is, as far as we can see, completely misdirected, a side remark about the significance of the overlap of the emission and absorp- tion edges in lithium has been of great importance for the further discussion of the MND effect.

MANY-BODY EFFECTS ON DEEP LEVEL SPECTRA O F METALS C4- 109 and the intensity of the high energy satellites should

decrease. By varying the gas pressure we can vary the collision time and monitor the strength of the high energy satellites. We may then pictorially speak about

incomplete relaxation [15, 281.

If we turn to solids, the shake-up excitations created along with the core hole usually will be able to diffuse away to some extent before emission takes place. The amount of relaxation depends on the rate of diffusion as compared to the core hole lifetime. Electronic relaxation is quite fast and one normally assumes it to be complete. Effects of incomplete relaxation are however not always quite negligible [28]. The energy scale for phonons on the other hand is very small and since the diffusion time is of the order of the inverse bandwidth, phonon diffusion is comparatively slow and we are often in the opposite regime of no relaxa- tion 1151. Upper core levels may however have such long lifetimes that the possibility of incomplete pho- non relaxation appears.

For the quantitative evaluation of incomplete relaxation, sweeping semi-classical arguments of the type we just have made can easily lead to gross errors. The proper way to handle the problem is to consider the excitation and the subsequent emission as one quantum process.

Fortunately the case of incomplete phonon relaxa- tion allows a realistic detailed quantitative solution as has recently been presented by Almbladh [29, 301. We would here like to emphasize a few general features of Almbladh's study. Some of Almbladh's results have been obtained independently by Mahan [31].

The broadening is described by a function C(o,

T)

which should be convoluted with the no-phonon, infinite lifetime result for the emission spectrum. Here

r

is the FWHM value for the core level lifetime width. It is convenient to take out the Lorentzian corresponding to the core hole lifetime and instead study a function D(w, r ) . We first discuss the moments of the broadening function D, which are given by (h = 1)

<

a

)r = - 2

+

2

sow

~ u ) c o ~ ( u ) ) X

The first moment gives the Stokes shift, and the second the edge width. Here E , is the relaxation energy, and

g(o) and A(T) the phonon strength function and the width of the phonon distribution discussed in the previous section. The additional width A , is tempe- rature independent. The results in eqs. (1 1) and (12) may be parametrized to sufficient accuracy by

4&, U)

g(o) =

-

-

exp

J .

0; 0 0

Here U), turns out to be roughly 213 of the experi-

mental Debye energy. This parametrization leads to the universal curves shown in figure 7. We note the rather sharp maximum in the A: curve which occurs at

r

-

0.7 CO,. The maximum value of A , is roughly

equal to E,, which due to the strong phonon coupling

can reach values which are 5-10 times the Debye energy. The actual shape of the broadening function D is approximately Gaussian only in the limiting cases r - + O a n d r + m.

FIG. 7. -The solid curve shows ( A , / E , ) ~ and the dashed - ( o ),l2 e0 as functions of T/oo, cf. equations (11) and (12).

From Almbladh [29].

Typical shapes for C(U), T ) at intermediate values of

T

are shown in figure 8. The limiting case

r

= 0 corresponds to shifting the SXA curve to become centered at - 2 E ~ . It is interesting to note the double

peak structure of C(CO, T).

FIG. 8. - The broadening function C(o, r ) for Li at 80 K and different values of the core hole FWHM width

r.

Also drawn is the broadening function pertinent to soft X-ray absorption (SXA)

C4-110 L. HEDIN

The distance between the peaks is roughly 2 E,, which

is much larger than the Debye energy. The effect is thus probably not due to specific properties of the phonon spectrum but is instead easy to understand in terms of the conventional Franck-Condon figure with two parabolas. The effect of the double hump structure shows up in a break in the emission edge, see figure 9. The very nice agreement with experiment makes it likely that incomplete phonon relaxation is the solution to the longstanding puzzle of the broad Li emission edge. The small value of 16 meV used for the Auger width is consistent with available experimental data, but an independent verdication of this value would be very welcome.

-

r = 16 meV -.- r . 2 0 meV 2

-

_{... r = L O}

meV

..-*....

Callcott and Arakawa. SXE o o o o o Callcott and Arakawa. SXA

3

.-

C -

-2 -1 0 1

hw (eV)

FIG. 9. - Calculated X-ray emission and absorption edges for Li at 80 K and four different values of the core-hole lifetime. Experi- mental absorption and emission spectra (85 K), recorded with the same spectrometer (Callcott and Arakawa) are also given. The theoretical SXE/SXA edges for

r

= 16 meV have a Stokes shift of

320 meV. After Almbladh [30].

6. Concluding remarks.

-

We have avoided to discuss the classic MND problem, namely the SXA/SXE edges. It is now becoming clear that a detailed comparison with experiment for this problem is quite difficult. The simplest cases are perhaps the L,, SXE spectra of the monovalent metals. In those cases where an effect is distinguishable (Na, K), it is very weak. Nevertheless it seems quite clear that the MND effect exists, but its quantitative evaluation may not be easy. The absorption spectra are all complicated by bandstructure effects. Even the proto- type of a free electron metal, sodium, has a dip in the one-electron spectrum roughly one eV above the edge [32], which has to interfere with the MND effect.

To resolve the importance of bandstructure effects Petersen and Kunz [331 measured spectra from solid and liquid sodium, figure 10. They found that the peak at the Fermi surface remained in the liquid state. If there are remains, however, of the density of states effects in liquids, they should come from the first Brillouin zone which has the smallest reciprocal vector and thus stands the best chance to survive if given a chance by the presence of some short range order. It is also surprising if the MND peak were so much stronger in SXA than in SXE.

FIG. 10. - X-ray absorption of solid (LNT) and liqui6(390 K) Na in the region of the L,, edge. After Petersen and Kunz [33].

The interpretation of SXA edges is also complicated by the Onodera effect [34]. Onodera considered exchange coupling between conduction electrons and the spin orbit split core level and found large effects on the intensity ratios. These effects have to be pro- perly phased in with the bandstructure effects.

The effect of exchange coupiing has also been studied by Girvin and Hopfield [35] who found it to have a strong effect on the phaseshifts for Li. This study has recently been extended, and considerably more accurate values have been obtained by Almbladh and von Barth (to be published).

Another unsolved and difficult problem related to the MND singularity is the theory for Auger broadening in metals. If one-electron theory were applied one would run into an Anderson orthogo- nality block and there seems to be no simple way around that difficulty.

MANY-BODY EFFECTS ON DEEP LEVEL SPECTRA OF METALS C4-l l l

Acknowledgments. - Frequent and fruitful dis- G. Grossmann and P. Minnhagen are gratefully cussions over a number of years of the problems taken acknowledged. So is financial support to this activity- up in this paper with C . 0. Almbladh, U. von Barth, by the Swedish Natural Science Research Council.

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