Green's Operator Approach to Multiple-Scattering Photo-electron and Auger-Electron Diffraction

$Page 1: Green's Operator Approach to Multiple-Scattering Photo-electron and Auger-Electron Diffraction$
Review Article

phys. stat. sol. (b) 182, 9 (1994)

Subject classification: 79.20 and 79.60; 61.14

Department of Theoretical Physics, University of Trieste and International Center for Theoretical Physics, Trieste‘)

Green’s Operator Approach to Multiple-Scattering Photo- electron and Auger-Electron Diffraction ’)

BY L. FONDA

Contents

1. Introduction

2. Multiple-scattering theory of core level electron diyfraction

2.1 The single-particle approach 2.2 The multiple-scattering theory 2.3 General structure of the emission process

3. Evaluation of the T-matrix

3.1 Evaluation of the direct (free atom) term 3.2 Scattering from a spherically symmetric potential 3.3 Evaluation of the multiple-scattering terms 3.4 The complete (non-perturbative) solution

4. Single-scattering and plane-wave approximations to core level electron dij’fraction

4.1 Single-scattering approximation (SSA) 4.2 Plane-wave approximation (PWA) 4.3 Combined SSA and PWA approximation

5. Electric dipole interaction in the photoemission process

6. Auger transition probability amplitude

References

’) POB 586, 1-34100 Trieste, Italy. ’) Supported in part by the Istituto Nazionale di Fisica Nucleare

10

1. Introduction

L. FONDA

Within the framework of the Green’s operator approach, in this paper we review the multiple-scattering process underlying electron diffraction as resulting from photo- or Auger-electron core level emission from an excited atom of a solid system. The relevance of this field of research stems from the fact that experimentally with this technique one is able to determine short-range order in surfaces. Surface local atomic geometries can be determined, typically within three to five layers from the surface [l].

Several theoretical papers have appeared on the subject but a formalism covering in a comprehensive manner, from the very beginning, all the mathematical techniques of the multiple-scattering process underlying the case under examination is up to now missing.

The technique has a wide range of applicability, particularly in cases where translational symmetry is broken. It is atom specific as it involves the emission of a core level photo- or Auger-electron following the excitation of an atom embedded in a solid system. The emitted electron is detected in the direction of the unit vector kJk, characterizing the final momentum. One measures the differential cross section either by varying the exciting energy keeping fixed the emission direction kJk, (a procedure which applies of course only to photoelectron diffraction), or by varying the azimuth and/or the polar angle with the energy kept fixed.

A variety of acronyms have been coined to describe these spectroscopies. Among others, the scanned-energy type has been referred to as angle resolved photoemission .fine structure (ARPEFS). This technique has of course as its close relative the extended X-ray absorption

f ine structure (EXAFS) which appears, as a function of the energy, in the total X-ray absorption cross section (absorption coefficient) above the edge for the excitation of a deep core level.

The scanned-angle measurements of soft X-ray excitations are termed X-ray photoelectron diffraction (XPD), while the Auger type is called Auger-electron diffraction (AED).

In the scanned-energy ARPEFS technique, one finds experimentally large oscillations which are not present when the electron emission occurs from the free (isolated) atom. These are in principle analogous to the oscillations which appear in the case of EXAFS. According to the intuition of Kronig [2] and Petersen [3] back in the thirties, the explanation of the phenomenon lies in the existence of final-state interactions; in particular, the final state of the electron is modified by the presence of an environment. This is a typical quantum-mechanical interference effect due to the superposition of photoelectron waves which originate coherently from the excited atom A and from its neighbours as a result of the multiple scattering of the electron in the condensed system. The phenomenon has been tackled systematically by quantum-mechanical methods only in the seventies both for the case of EXAFS [4 to 81 and for the case of core level photo- and Auger-electron diffraction [9 to 131. Several relevant papers have since then appeared on core level electron diffraction [14 to 301 (see also [l] for a complete list of references, particularly on experimental work). This technique yields a very powerful tool of investigation since, as it goes through the multiple-scattering process, the emitted electron collects information on the local environment of A, in particular on physical observables such as bond distances, bond angles, coordination numbers, site symmetries. For typical directions of emission, the interference pattern of ARPEFS exhibits, as a function of the exciting energy, modulations much larger, even though the overall intensity is weaker, than the corresponding ones in EXAFS, as can be understood from the fact that in the latter experiment the integration over the 47c solid angle averages out the effect. The problem will of course be that of obtaining a good signal

Multiple-Scattering Photoelectron and Auger-Electron Diffraction 11

to noise ratio. At the new high brightness synchrotron radiation facilities, such as ELETTRA (Trieste) and ALS (Berkeley), this method of investigation will then receive greater momentum.

Scanned-angle measurements, such as XPD and AED, are able to determine local atomic geometries especially relevant for the study of surface chemistry and catalysis. In fact, the final-state interactions produce interference peaks in certain well-defined directions, and this allows to reveal information on the geometric structure of the surface and on the adsorbate/substrate relationships.

Section 2 summarizes the multiple-scattering theory in the Green’s operator approach. In Section 3 the 7’-matrix is evaluated without making any approximation. Section 4 deals with the single-scattering and plane-wave approximations. The dipole photoemission expressions are discussed in Section 5. Auger emission is discussed in Sections 6, where it is seen that, by properly changing the initial conditions, the same formulas apply both to photo- and Auger-electron diffraction.

2. Multiple-Scattering Theory of Core Level Electron Diffraction

In order to fix the formalism, we start the discussion by tackling the problem of core level photoemission. We shall discuss core level Auger emission in Section 6. The standard single-particle approach will be used. The formalism follows the Green’s operator approach extensively discussed in the literature [31 to 381.

2.1 The single-particle approach

Within the standard single-particle approach, we shall consider the photoemission 7’-matrix, to be evaluated on the energy shell, to the first order in the incoming photon field,3)

T,+i = <$)I (01 HI l l i ) Iw,> 3 (2.1)

where the initial state vector is the product of the one-photon state vector /li) describing the incoming photon (of energy hw; here the subindex i symbolizes the initial photon momentum and polarization) times the normalized vector Iw,) which represents the initial single-electron bound state relative to the core level c (of energy E,). Thefinal state is given by the product of the normalized photon vacuum 10) times the single-electron scattering state Iyf)) describing the photoemitted electron in interaction with

(i) the ionized absorbing atom A, (ii) the neighbours of A in the condensed material. If the excitation is from the ground state, then the completely relaxed state (i.e. lowest

energy configuration of the passive Z - 1 electrons in the presence of the core hole) will be the appropriate configuration for the ionized atom.

The interaction of the quantized photon field A(r) (evaluated in the Coulomb gauge V A = 0) with the active emitted electron is given by

~

3, For the definition of the T-matrix and cross sections, see [39,40]. A factor F , = (I&~;/’ I v$;:)) is actually missing in front of the r.h.s. of (2.1). F , represents the Z - 1 passive electron overlap integral; i t shall be understood in what follows. Since its modulus is less than one, it correspondingly decreases the final flux. IF,JZ has normally values between 0.6 and 0.9 [41, 421.

L. FONDA 12

where m and p are the electron mass and momentum operator, respectively. The evaluation of the matrix element of H , is performed within the standard quantum theory of radiation [43, 441 (see Section 5).

The single-particle approach (2.1) works nicely at energies away from the threshold where the active electron scarcely intermingles with the passive electrons of the atom absorber or with the valence electrons [45 to 471. Close to threshold, X-ray absorption from deep core levels is still dominated by single-electron transitions; however, some many-body effects may appear at energies near the edge (for a brief discussion, see Section 2 of [38]), and the experimentalist will have to distinguish them from single-particle contributions proper.

For simplicity we shall deal only with spin independent interactions so that our photoelectron can be considered as spinless. The generalization to include the photoelectron spin is straightforward [48 to 501. Furthermore, we shall consider only elastic scattering from the ionized atom A and from its neighbours. In order to include the effect of all types of inelasticities (such as inelastic electron-atom scattering, electron-electron scattering, decay of the core hole, etc.) and of thermal vibrations, one usually introduces proper damping factors in the final formulas to simulate their effects. (See, for example, [lo, 13, 21, 23, 241. A correct taking into account of the inelasticities would involve also the introduction of a many-channel treatment [51].)

In (2.1), the vector It&)) is a “mathematical” state, eigenstate of the total final state (hermitian) effective single-particle electron Hamiltonian H,, which satisfies incoming wave boundary conditions (plane wave + spherical incoming wave). It is connected to the “physical” state Iy$’)), which obeys outgoing wave boundary conditions (plane wave + spherical outgoing wave), through time reversal invariance (see, for example, [52], Section 2.7),

$)(kf> 4 = [W‘+’(--k,, 4l* 7

where hk, is the momentum of the final emitted electron. The states l ~ ~ ( ’ ) ( k ) ) are normalized exactly to a three-dimensional Dirac &function

( ~ ‘ “ ( k ) I y ( ’ ) (k ‘ ) ) = Zi3(k - k’) .

The differential cross section, for photoemission in the direction defined by the unit vector k,/k,, is given by

dQ mi C

where mi labels the orbital angular momentum degeneracy of the completely filled core level c and the factor 2 represents the two electrons lying on this level (Ec, li, mi).

Note however that, in order to compare theoretical cross sections with experimental data, one must correct the polar emission angle for the refraction of the electron at the surface barrier of the s01id.~)

The (hermitian) Hamiltonian H , can be written as

Hf = K + U , + U , , n * a

(2.4)

“) Under the approximation of a sharp surface barrier, this correction can be obtained through the simple-minded hell’s law (A, sin O 2 = i2 sin 0,) linking electron wavelengths and polar angles inside and outside the surface [l, 211.


where K is the photoelectron kinetic energy, U , the potential of the ionized atom A, and U , is the sum of all the remaining terms of the lattice potential. Since the photoelectron

has a finite (energy dependent) mean free path in the material, this sum will cover in practice only a cluster of atoms around the atom absorber A.

We shall deal with atoms embedded in an interstitial space of constant potential. This (muffin-tin zero) level determines our zero of energy. The atoms are supposed to be fully screened so that the photoelectron feels finite-range spherically symmetric non-overlapping local potentials U,. This is the so-called muffin-tin potential model. (Note, however, that all formulae in this section are valid irrespective ofwhether or not the potentials overlap.)

n + a

2.2 The multiple-scattering theory

As the photoelectron is emitted, it first feels the potential U,. It is then convenient to separate this contribution from the final state vector. Introducing the scattering state I&)(")) from the isolated ionized atom A (Hamiltonian K + U,) for the state vector I$)) one then obtains

I&)) = + G(-) 1 u, I&)(")), (2.5) n C a

where G(-) is the incoming wave Green's operator belonging to the Hamiltonian H,, which is equal to the hermitian conjugate of the complete (outgoing wave) Green's operator: G(-) = G'. The Green's operator G is given by

1 G =

E + iE - H ~ '

where E = hw + E , is the total energy. Here, and in what follows, the limit E + 0' is understood.

Using the integral equations defining the complete propagator G, one can easily prove that G is given by [38]

G = G, + G,T,,G,, (2.7)

where G, is the Green's function belonging to the potential U,,

1 Gn =

E + iE - K - un' and the operator Ka is given by

n + a

Taking the hermitian conjugate of Gi-) on the left, one obtains

[ Z a "-1 (2.9) and then multiplying the resulting expression by

Using this expression, for the wave function (2.5) we can finally write

I ~ I - ) ) = ly i-) (a)) + Gi-)Ti, Iyj-) (a)) (2.10)

14 L. FONDA

and correspondingly the T-matrix (2.1) reads as follows:

Tf - 1 . = - (I&) I A y ) = ($'(a) I A?) + (y$- ) (a ' l T,,G, 1 (2.11)

where the photoelectron state vector ]A:hE) is the product of the core state vector \w,) and the electromagnetic interaction,

IArhE) = (01 HI I l i> I ~ l c ) 9 (2.12)

(y(p''"' 1 A!hE) is the 7'-matrix for the direct emission from the free absorbing atom A. All the multiple-scattering effects are included in the second term at the r.h.s. of (2.10)

and (2.11), in particular they are contained in the operator qa. (Note that, due to the presence of the complete Green's operator at the r.h.s. of (2.9), this term includes also the possibility for the photoelectron to be rescattered by the emitter A after having undergone multiple scatterings from other sites.)

We want to express the operator T,, as a function of the various T-matrices describing the complete scattering from each isolated atom N. One must then consider the operators T k j defined as follows [38]:

(2.13) n + j

For k = j = a, qj reduces to x,. The qj operators satisfy the following integral equations (for the proof see [38], Section 2):

(2.14)

n + j

where tn is the above-mentioned complete T-operator for the scattering of the photoelectron from the isolated atom N,

tn = U, + UnGnU, (2.15)

and Go is the free space Green's operator,

1 E + i E - K K '

Go = - (2.16)

In order to get (2.14), use has been made of the identity

U,G, = tnGo (2.17)

which holds good under our assumption of complete screening (i.e., absence of a long-range (Coulomb-like) tail).

If we write

(2.18)


we see, using (2.14), that the z-operators satisfy the integral equations

z q p = t$q, + c tqGozmp7

z p q = t p h , , + c ZpmGOtq '

m * q

m * q

(2.19)

These z-operators are called scattering path operators in the literature [34]. Introducing the second of (2.19) at the r.h.s. of the first, we get

z q p = tphqp + tqGot,(l - d q p ) + tqGoT,,Gotp 7 (2.20)

which for p = q = a reads

zaa = t , + taGOKaCOta. (2.21)

Finally, (2.21) can be inverted to find T,, in terms of z,,,

Ta = G;'la-'(zaa - t,) ta-'G,' . (2.22)

To find the complete analytic solution of the integral equations (2.14) and (2.19) is a rather formidable task. We shall discuss it in Section 3.4. A perturbative solution is, however, easily obtained,

m * y

+ tqGo c tnGo c tmGOtp + ... , n * q m + p

m + n

K a = c t n + tnGO 1 'm + c tnGO tmGO c tk n + a n*a m + n n * a m * n k+m

m * a k * a

(2.23)

(2.24)

The connection between (2.23) and (2.24) is of course given by (2.18) with k = j = a,

T,, = c TQD' (2.25) q * a p + a

The perturbative expansions (2.23) and (2.24) converge usually at high photoelectron energies, where the electron scattering is weak. (The relevance of multiple scattering ver- sus single scattering (first term at the r.h.s. of (2.23) and (2.24)) is discussed in Sections 3.3 and 4.)

Of course, in the energy region where the expansions fail to converge, the full (non-perturbative) solutions must be used. (This is discussed in Section 3.4.)

Note that in the muffin-tin potential model, due to the non-overlapping structure of the potentials, in the final evaluation of the T,+, matrix all the t-operators appearing in (2.1 l ) , (2.23) and (2.24), will turn out to be evaluated on the energy shell.

16 L. FONDA

2.3 General structure of the emission process

In the next section we shall come to the problem of evaluating explicitly the T-matrix. It is therefore important to understand the general structure of the emission process. Let us then go back to (2.11),

TCi = (~i-""' I A:hE) + (yj-""') T,,G, (2.11)

where the separation of the direct wave (i.e., the one originating from the free absorbing atom A) from the complete T-matrix has been performed.

The second term at the r.h.s. of (2.1 1) represents all multiple-scattering processes. Let us work on it by splitting as follows:

lv,g-) (a) ) = lkf) + G:-'U, 14) > (2.26)

where Jk,) is the free electron wave. Using the hermitian conjugate of (2.17),

(2.27)

(a) ) = Ik,) + GL-'tf; I k f ) .

Using (2.28), for the T-matrix (2.11) we finally obtain

(2.28)

T f C i = Ti?i + Thyi + T f Y i , (2.29)

where

T p i = (&' (a) I AYE) ,

T[,?i = (kfl T,,G, I A Y ) ,

Ti?i = (k , ( taGOqaGa IAzhE).

The general graphic structure, as resulting from (2.29), is shown in Fig. 1.

(2.29a)

(2.29 b)

(2.29~)

Fig. 1. The graphic representation of the electron emission process. The double line in the case TIo] reminds the reader that the final electron waves are distorted by the potential U ,


The term T:Yi represents the direct emission, which takes properly into account the distortion ofthe final electron wave due to the potential U , ofthe ionized atom absorber.

Ti?i and represent all the multiple-scattering processes: Ti?i represents all the multiple-scattering processes which do not have A as the last scatterer; Tc?i represents all the multiple-scattering processes where the atom emitter A is the last scatterer.

3. Evaluation of the T-matrix

We now evaluate the T-matrix, as given by (2.11) and (2.29), describing the core level photoemission of the electron from the excited atom A.

As exhibited by (2.29), the T-matrix contains three terms. The first represents the contribution of the direct (free atom) photoemission of the core level electron, which reaches the detector with momentum hk, as if no environment were present around the atom absorber A. The second and third terms are the result of the multiple scrrttering suffered by the photoemitted electron from the neighbours of A, with the electron emerging from the material always with final momentum tik, (the detector lies at infinity). (Remember also the correction to be applied a posteriori for refraction at the surface!)

3.1 Evaluation of the direct (free atom) term

The expansion of the final electron scattering wave function in spherical harmonics is given by 5 ,

&'(k, r ) is the physical radial wave function of the photoelectron in the field of the potential U,. As well known, it satisfies a regular boundary condition at the origin.

Introducing (3.1) in the direct free atom term (2.29a) of the T-matrix, we get

where the photoelectron state IA!hE) is given by (2.12).

3.2 Scattering from a spherically symmetric potential

Before proceeding to evaluate the second and third terms of the T-matrix (2.29), we would like to summarize the relevant formulas we shall use for the elastic scattering from a spherically symmetric local potential.

') For the spherical harmonics we follow the notations of [52], Appendix to Chapter 2. By the shorthand Y,,(k) we of course mean Y,,(Q(k)), where Q(k) defines the polar angles of the vector k . Thc phase convention is: Yf,(k) = (-)"' Y,, -"'(k). We shall often use the compound index L = ( I , m). From now on we shall drop the subindex f from the modulus of the electron final wave vector lkfl = k.

2 physica (b) 182/1

18 L. FONDA

Let us write down the expressions of the free space Go and of the complete G, Green’s functions in the coordinate representation. We shall use the notation I Y ) ~ for the eigenket of position used by the “observer J”, i.e. the one who has placed the origin of coordinates in the centre of the atom J.

For the free propagator Go we get (see p. 269, 298 of [39] where units h = 1 are used)

where

(3.6)

r , indicates the smaller of r and r‘, r > the larger. j,, n,, and hif) = h‘l) , = jZ ‘ + in, = [hi2)]* = [hi-)]” are the usual spherical Bessel, Neumann, and Hankel functions (see [39], p. 38, and [53], p. 437).

2 m . h

Go,(r, r’) = -ik j , ( k r , ) hi+)(kr,),

For the complete propagator G, we get (see [39], p. 374)

(3.7a)

(3.7b)

As mentioned above, &)(k , r ) is the physical (plane wave + outgoing wave) scattering eigenfunction belonging to the Hamiltonian K + U , (note that our vi” differs from that

f l : ) ( k , r ) is a purely outgoing wave eigenfunction of K + U,, in general irregular at the

We list here the properties of these eigenfunctions for the scattering from a generic site:

of [39]: vi+) = 471Wi+)(Newton) ).

origin of the atom A, determined by a boundary condition at infinity.

4.n lim (kr) - ’ - ’ vf+)(k, r ) = r - 0 f , ( k ) (21 + l)!! ’

(3.9)

(3.10)

tpf+)(k, r ) = [&’(k, r)]* = e2ibl yf-)(k, r ) , (3.11)

(3.12)

(3.13)

where R,, is the muffin-tin radius of the potential and 6, the corresponding phase shift.


f , ( k ) = Ifi(k)l exp (-id,) is the so-called Jost function which, for the scattering from the site P, in terms of the purely outgoing wave eigenfunction (3.12), is given by (see [39], p. 374, and [54], p. 38)

(3.14)

For our photoelectron diffraction problem, we want to obtain final formulas where all (on the energy shell) t,-matrix elements contained in T,, are written in the following angular momentum representation:

,GYLI t , Lil’YLOP. (3.15)

The representation basic vectors LilyL), are defined, for fixed klm, relative to the site P. They are the product of a spherical harmonic (with angles originating from P), and a spherical (regular in P) Bessel function. Whenever we shall not specify the k-dependence in the ket and bra symbols, as in (3.15), it will mean that the matrix element is given on the energy shell.

Their coordinate and momentum representations are

(v I jdk) YL) = jI(kr) Y I ( 4 (3.1 6 a)

while the normalization and completeness relations are as follows:

t J k2 dk b,(k) YL) ( j , (k) Y,I = 1 . L 7 1

(3.16b)

(3.17a)

(3.17b)

0

However, most of the matrix elements we shall find on the way will be expressed through waves of the type hiz)YLp, rather than through our representation waves jI,YL,. We must then transform those waves h{J)YLp, outgoing from the site P, in waves of the type jI,YL, converging to a site Q (and regular there). This so-called re-expansion is readily obtained [35, 51, 381,

h!p+)(kr’) yLP(y’) = CjIq(keq) YL,(eq) igLqLp(Rqp) ; Q , < R,, (3.18a) L ,

where the g-propagator is given by

g is usually referred to as structure factor in the literature, since it bears information on the geometric order in the solid.

2*

20 L. FONDA

Fig. 2. The vectors defining the Green's function in coordinate representation. Y' joins the centre of the site P with the muffin-tin region Q =k P. The Greek letter e always indicates a vector joining the centre of the site to a point within the same muffin-tin region

The various vectors are defined in Fig. 2. Note that both (3.18a) and (3.18b) are valid for eq < Rpq. But this condition is satisfied since we are dealing with muffin-tin potentials and eq lies within the muffin-tin region Q.

In our notations, the (Gaunt) coefficients ( Y L I Y,,YL,) are real [52], r

C(Il/,/; mlrn2m) C(l1I,/; 000). (3.20) 4n(21 + 1)

Note that they vanish for 1 + I, + 1, = odd, as it follows from parity conservation.

invariance yields the Kronecker's aLL,), The matrix element (3.15) defines the I-th partial wave scattering matrix tf (here rotational

(3.21) h2

2mk ptiiYLI t , Iji,YL'), = ~ d L L , t f ,

tf is given in terms of the l-th partial wave phase shift for the elastic scattering from site P,

tf = -@?sin 8; . (3.22)

The connection with the momentum space scattering matrix on the energy shell and the scattering amplitude f, is then given by

(3.23)

where 6, the angle formed by the vectors k and k', represents the polar angle of scattering from the site P.6)

6 , The modulus square of the scattering amplitude &(O) is exactly equal to the differential cross section for the elastic scattering of the electron from the atom P: dcr/dQ = If(0)lz. In terms of phase shifts, f(0) is given by

1 f ( ~ ) = c YT(~,); eia1 sin s,Y,(~,) = - (21 + 1) ~ , ( c o s H ) ei61 sin S, ,

L k i


3.3 Evaluation of the multiple-scattering terms

The matrix element ,(rl G, ]yoa, as it appears in the T-matrix (2.11) and (2.29), is evaluated for r‘ lying in the muffin-tin region of the atom absorber A and for Y lying in the muffin-tin of some neighbour of A. It follows that r > r‘, so that f b ; ) is evaluated at Y and I&’ at r‘. ForfQ:)(k, r) we can actually use the free solution (3.13) since r lies outside the muffin-tin region A.

For the multiple-scattering part of the 7’-matrix we therefore get

( vj - - ) (a) I T,,G, = C .T~1+21M C L (3.24)

always given by (3.4) and

p + 2 1 = - ____ imk s d ’ r ( 1 & ) ( ~ ) 1 T,, Ir), YL(r) hff ) (kr ) .

L

with

(3.25)

according to (2.28), where Ik,) is the free electron wave

2nh2

Now, in (3.25) we split function normalized according to

( r I k , ) = (2n)-3’2 eikrr = 14ni’j,(kr) YZ(k,) YL(r) L

(3.26)

The matrix element .TLlt2] of (3.25) is then split into two pieces, p + 2 1 = 9-11] + p 2 1 L 7 (3.27)

where S5_f1(S~21) corresponds to the term T ~ ~ i ( T ~ ~ i ) of the T-matrix (2.29),

g r 1 1 = imk s d ’ r (k,l T,, Ir ) , YL(r) hf ”(kr), (3.27a) 2nh2

y [ 2 1 = - ____ d3r (k,l &GOT,, Ir ) , YL(r) h$+)(kr). (3.27 b) 2nh2 imk s

Let us first evaluate .Ti1] ; using (2.25) and (3.18a), we get

(3.28)

The matrix element a(r’l zqp Ir), can be written in terms of @-vectors originating from the centres of Q and P, respectively (see [38], (4.7)),

a(r’l T q p l r > a = q(@ql T q p l e p ) p . (3.29)

Let us now transform ( k , 1 r ‘ ) , ,

(2n)3/2 ( k , I = e - ik r r ’ = , - ikr(eq+RqaI = e- ikrRqa (2n)3/2 ( k , I Q q ) q .

Using the completeness of the coordinate representation basic set {I@,),}, for (3.28) we obtain

(3.30)

22 L. FONDA

In order to evaluate the matrix element (itf/ zqp Ijl,YL,)p we introduce the completeness relation (3.17b) at the Q-site. Using then (3.16b) we obtain

cg

(k,I zqp Lil,yL,)p = c 2 1 k'2 d k <kfl jlq(k') yLq)q qt i lq (k) yLql zqp I ~ ~ , Y ~ , ) ~ L, 71

0

(3.31)

We next define the matrix z&, by

(3.32)

(3.32) exhibits the same factor as that appearing in the definition (3.21) of the l-th partial wave scattering matrix t:. Using (3.31) and (3.32), for F p l we finally obtain

h2 2mk , t i l ,YL,I z q p Lil,yL,)p = ___ G:L, '

(3.33)

x a w l T q p IV>, YL(4 h!+'(kr) . (3.34)

Since the operator zqp shows a t, (q =+= a) to its left, there follows that Y' lies in the muffin-tin region Q =I= A. On the other hand, Y" lies in the muffin-tin region A; as a consequence, we have that r" < r'. Using ( 3 4 , (3.6), (3.18), and (3.29), we obtain

x g t i l , Y L , I z q p Lil,YL,)p g L p L ( R p a ) . (3.35)

We next use (3.31) with tadba replaced for zha, (3.21), (3.32) and finally obtain

p i L - - (2n)-312 c c (-V Y L W t?,gL'Lq(Raq) 4:L,gLpL(Rpa) ' (3.36) p e a L'L,L,

Collecting together (3.3), (3.33), and (3.36), we can write the T-matrix Tf+i in its final form,

Tfci = (Fi01 + sp + Tf?.') AT,L = C T L A L L , (3.37a) L L

yL = 1 (271)-3/2 ( - i ) l ' yLr(kf) L'


Equation (3.37) is the correct single-particle expression of the T-matrix q+i. It takes correctly into account the contributions of all multiple scatterings. Any approximation to the single- particle description of photoelectron diffraction should start from (3.37).

Several perturbative calculations, using the multiple scattering expansion (2.23) for the operators ~(tp4~,, have been performed. However, due to the large number of angular momenta occurring at each step of the multiple-scattering process, calculations performed with the perturbative expansion of the correct T-matrix (3.37) might take long computing times.

In the literature, one therefore finds calculations performed by using simplified approaches several of which try to treat properly the spherical waves present in the multiple-scattering process [21 to 30, 55 to 601.

Most of the calculations (performed as usual by taking into account a cluster of atoms within few layers from the surface) have been confined to collinear chains of atoms radiating away from the emitter. It has then been found that, if the chain of collinear atoms is short, multiple-scattering focusing effects appear obtaining an enhancement of the differential cross section in that direction. However, with increasing chain length, a multiple-scattering defocusing effect, apparently increasing with energy, sets in suppressing in part (and narrowing) the forward peak [20, 25, 26, 28, 61, 621.

Calculations performed in the case of non-collinear bent chains, show that multiple scatterings are important at relatively lower energies [28], due to the fact that, as the energy gets higher, finite angle electron-atom scattering becomes weaker (see for example [I, 251).

3.4 The complete (non-pevtuvbative) solution

The operators z&, satisfy integral equations. By projecting for example the first of (2.19) on the basis lilyL), we get

(3.38)

The perturbative expansion (2.23) corresponds of course to the expansion of (3.38)

t&,, = t f p d q P d L s L , + tfb[gLqLp(Rqp) (1 - dqp)l tP,

+ C t f , [gL ,L , (Rqm) (1 - 8qrn)I t z [ g L , L , ( R m p ) (1 - d m J I tf, + ... . (3.39) m

The structure of the perturbative expansion (3.39) is very simple. At the interaction of the Ij-th photoelectron partial wave with an atom J there corresponds a T-matrix element LiJ, while the travelling of the photoelectron from atom P to atom Q (+ P) is provided by the propagator (structure factor) gLqLp(Rqp).

Equations (3.38) and (3.39) can be written in a compact form. Define the matrices

(4&., = Q$, 1 (t)l:L, = tfp41pfiL,L, (g)l:Lp = gLqL,(RqP) (1 - d q p ) ' (3.40)

For (3.38) one gets the matrix equation

z = t + t g 7 , (3.41)

which is easily solved for z,

7 = (1 - tg)-' t = (t-' - g ) - l . (3.42)

24 L. FONDA

This is the complete (non-perturbative) solution of our problem expressed in the combined angular momentum -atomic site discrete Hilbert space.

The perturbative expansion (3.39) is, on the other hand, written as

(3.43)

In order to discuss the absolute convergence of this expansion, let us diagonalize the matrix K = t g : K , = BKB- ' , where of course the matrix B diagonalizes also the inverse matrix

T

appearing in (3.43): B(l - K)-' B-' = (1 - BKB- ' ) - ' . The geometric series K" - n = O - converges absolutely if the series 2 (K,)" does so. The diagonal matrix K , has for diagonal

n = O

elements just its eigenvalues A,. The general criterion for the absolute convergence is then to require that all eigenvalues I,, be in modulus less than 1,

max l i o l < 1 . (3.44)

The eigenvalues 1, of K , are of course also eigenvalues of K = t g . They are energy dependent, so that the energy axis is divided into regions where (3.44) does or does not hold. These regions are obviously also system dependent. The condition of convergence (3.44) will be satisfied when the coupling is weak, e.g. at high enough photoelectron energies. At low energies, i.e. near the edge, the convergence will in general fail and the full solution (3.42) will have to be used.

We emphasize that dampings due to inelasticities and disorder, when properly introduced, will help the convergence of the perturbative expansion.

4. Single-Scattering and Plane-Wave Approximations to Core Level Electron Diffraction

In the literature, many applications have been performed by making some approximations in the calculation of the 7'-matrix. In particular, some papers have considered only contributions from single scatterings of the electron from the environment of the atom A, others made the approximation of taking in each g-propagator (structure factor) the asymptotic form of the Hankel function (plane-wave approximation), still others applied both these approximations. We shall consider briefly here these two types of approximations. For other approaches, which are able to take into account spherical-wave effects, we refer the reader to the literature [21 to 30, 55 to 601.

4.1 Single-scattering approximation (SSA)

This approximation consists in applying to (3.37) the substitution

Therefore, only one scattering from each atom of the neighbourhood of the atom emitter A is taken into account.


For (3.37b) one gets

(FL)SSA = (2n)- 3'2 ( - i)" YL'(kf) L'

In the literature, the customary definition of single scattering approximation omits from (4.2) the second term in square brackets at its r.h.s., which actually describes a double- scattering event of the type emitter + scatterer + emitter + detector. The presence of this term is, however, necessary in order to guarantee flux conservation [24].

SSA works nicely at rather high energies where multiple scatterings, unless occurring in a chain of collinear atoms, are damped due to the fact that finite angle electron-atom scattering is generally weak (see, for example, [l, 251).

In the case of a collinear chain of atoms long enough to have defocusing effects at work, calculations performed in the SSA should at least predict rather accurately the position of the forward peak; it will then take the skill of the experimentalist to unravel a small bump over the background.

On the other hand, calculations performed for peaks appearing for two-atom chains show that the SSA works nicely, since multiple scatterings do not apparently add appreciable contributions in these cases [28].

4.2 Plane-wave approximation (P WA)

The so-called plane-wave approximation (PWA) consists in replacing, in each g-propagator, the Hankel function hi+)(kr) with its behaviour for large kr : (-)'+' eikr/kr. Note that here we perform this approximation on the angular momentum representation of the T-matrix. Historically the PWA has first been applied to the coordinate representation of the T- matrix, and there the approximation included: the passage from eikr to eikv (plane wave!) and the small atom approximation (see, for example, Section 111 of [S], [13], or Section 4 of [38]).

By performing the above-mentioned substitution, one can make a summation on the relevant spherical harmonics appearing in the g-propagator. Using the completeness of these functions, we in fact get

For the g-propagator, we finally obtain

(4.3)

26 L. FONDA

Using (4.3), for the T-matrix (3.37b) we get

( Y L ) p W A = (2n)- 3'2 (- i)' Y L ( k f ) f (271)- 3'2

where we have made use of (3.23) for the scattering arnplitudef,(fI,,,), which describes the electron as being shot from the atom Q, scattered from the atom A, and emerging in the final direction f (i.e., in the direction of the final momentum k,). OfaY is of course the corresponding polar angle of scattering.')

Let us now define the amplitude

47c FFY(kx, ; kpa) = - - C W' ' Y L q ( k x y ) ( ~ I ~ L J F w A YL*,(kpa) . (4.5)

k LPL,

F:Y(k , , ; kpa) constitutes the generalization of the scattering amplitude from a single atom. As will be evident in a moment, FIFA(kx,;k,,) is in fact responsible for the following scattering process: The electron, outgoing from the atom A with momentum kpa, gets scattered from the atom P and then it undergoes all possible intermediate multiple-scattering processes in the condensed system until, after getting scattered from atom Q, emerges with momentum k,, pointing, from Q towards X, which may be another atom or the detector (in which case k,, = k,, = k,). The modulus squared of F i Y ( k , , ; kpa) is, in the PWA, the differential cross section for the process just described.

Using (3.38), with the g-propagator given by (4.3), we get for F : y ( k , , ; kpa) the following matrix equation:

eikRqm

F:yA(kxq; 'pa) = hypfp(kxp; kpa) + C&(kxq; kym) ~ (1 - ' q m ) F:y(kqm; k p a ) 3

m R Y m

(4.6)

with its perturbation expansion

eikRq, eikRmp

+ C.Ly(kxy; kym) ~ (1 - ' ym)fm(kym; k m p ) ~

x (1 - a m p ) . f p ( k m p ; kpa) + ... 1

m R w

(4.7)

where fp (kxp; kpa) is synonymous of fb(Oxpa), but the former notation gives a more direct visualization of the physical process which, as one understands from (4.7), just agrees with the above described interpretation of FIFA(kXq; kpa) .

') Note that formulae must be read from right to left in order to follow the correct time arrow of the process.


Using (4.5) in (4.4), we finally get the PWA expression of the T-matrix,

(9-JPWA = ( 2 ~ ) - ~ / ~ (-i)' Y L ( k f )

eikRpa

R p a

x ~ ( - i)' YL(Rpa) .

4.3 Combined SSA and PWA approximation

If we combine the SSA with the PWA, by replacing in (4.8) 6,,fP for F:YA, one obtains

(YL)pwA = ( 2 ~ ) - ~ / ' (-i)' Y L ( k f ) eikR,,(l - cos8rp,) eZikR,,

.fp(Qfpa) + f a ( 6 f a p ) ~

SSA

+ (271)-3/2 c [ x ( - i)' Y L ( R p a ) ,

p*a 'pa R i a

(4.9)

where of course f(Qapa) = f ( ~ ) is the amplitude for backscattering from atom P. As already pointed out after (4.2), in the literature the second term in square brackets is usually omitted. As a matter of fact, at energies high enough so that backscattering becomes unimportant [l, 251, this term turns into a small correction.

In applications, the structure of (4.9), where simple k-dependent exponentials appear, allows the introduction, for the corresponding differential cross section, of Fourier transform methods in order to determine relevant quantities such as bond lengths R,, and bond directions Ofpa [13, 16, 17, 231.

The reader is referred to [l] for a discussion on the limitations of the single-scattering and plane-wave approximations.

5. Electric Dipole Interaction in the Photoemission Process

We need now to evaluate the matrix element (01 HI lli) appearing in (2.1) and (3.37a), in particular in McL (see (3.4)). The evaluation follows very well-known quantum radiation theory methods [43, 441 which we like anyway to summarize here.

For this purpose, let us write down the explicit expression of the quantized electromagnetic vector field

where o = kc and the sum extends over the two possible (unit) polarization vectors zk (in the Coulomb gauge one has k . zk = 0 (transversality condition)).

The operators uk and ul annihilate and create, respectively, a photon with momentum k and polarization zk. They satisfy the following commutation relations:

L. FONDA 28

In (2.1) Ili) =_ aLi (0) so that (01 ul Ili) = 0. Using the commutation relations, one also

For (01 H , Ili) one then obtains has (01 uk lli) = ?i3(k - ki) a,,+.

( E . p ) eikPh'r (01 HI I l i ) = - - ~

m "c 2(27~)~ o I" (5.3)

where k,, and E are the momentum and polarization of the initial photon. In our matrix elements, the electromagnetic vector potential A(Y) is evaluated for r < R,,,

inside the muffin-tin region A. If lkph(r - v0)l < 1, where yo measures the centre of the absorbing atom A, we can expand the exponential appearing in (5.3) and retain only the first term,

(5.4) e i k ~ h ' V eikph'Vo

When (5.3) is bracketed, as in (2.1), between the two eigenstates (I$$ and Iv,) of the effective single-electron total Hamiltonian, in the absence of velocity dependent forces one readily obtains the dipole expression (usep = i(rn/h) [H,, Y], E, - E , = ho and put yo = 0)')

Defining the core bound state eigenfunction as

for E . Y Iw,) we can then write

J ml

For JVL, as given by (3.4) we therefore get

(5.5)

(5.7)

(5.8)

with given by

Note that the Gaunt coefficient in (5.8) is responsible for angular momentum and parity conservation in the photoemission process. Rotational invariance allows 1 = Ei + 1, li, li - 1 as possible values for the photoelectron angular momentum. On the other hand, space reflection invariance forbids the value 1 = li.

') If dipole transitions are suppressed, one must take the next term of the expansion of the vector potential. The resulting pure electric quadrupole transition is then described by replacing (8. v ) in (5.5) with (i/2) (kPh r) ( E . r).


A simple expression for the Tmatrix is obtained if the initial core level is in an s-wave li = 0 and if the PWA is applied. The Gaunt coefficient in (5.8) collapses then to

1

allowing only a p-wave final photoelectron. Using (4.8), we immediately get

( 5 . 1 0 )

The most commonly used version of this expression is the single-scattering approximation,

eikR,,(l -cosf3fpa) eZikR,,

(5.11) +I-[ ' R,, f p ( o f p a ) + fa (ofap) T f p ( o a p a ) ] ] >

N - ( 8 . :)-I :a ( 8 . K) -7-

p + a Rpa ' p a

where again Oapa= IT. If moreover the scattering amplitudesfJ8) are small, so that we can neglect all second-order

termsf,(@)f,*(H') ( p and q any), using (5.11) we easily get for the differential cross section

do (k,)/dQ - do'") @,)/do 'pa 2 l fp(ofpa)l

dota) (k,)/dQ

x cos [ k R , , ( 1 - cos ~ f , , ) + cP(&,)l > (5.12)

where do(")(k,)/dQ is the differential cross section for photoemission from the isolated excited atom A, and the phase q(Ofpa) is defined by

fp@fpa) = Ifp(ef,,)l eim(Bfpa) . ( 5 . 1 3 )

Apart from the non-appearance of the central atom phase shift, the (approximate) formula (5.12) is similar to the standard kinematical EXAFS expression usually employed in applications. Of course, due to its simplicity, (5.12) has been applied rather heavily in applications, particularly using Fourier transform methods.

6. Auger Transition Probability Amplitude

Let us now consider the modifications that one has to introduce in the above formalism in order to describe the electron diffraction following an Auger emission process.

Assuming that a core hole c has somehow (e.g. by the impact of a photon or electron) been created in the atom A, the Auger transition probability amplitude is given by

n

L. FONDA 30

It is essential for the applications that the electron is emitted from a localized atom; we must therefore consider Auger emission from a core level. The process is one in which an electron of the ionized atom A makes a transition from the core level 1 (core level 2) to the empty core level c, while an electron from the core level 2 (core level 1) is ejected from the atom A. This ejected Auger electron, represented in (6.1) by the usual wave function yi-), propagates then in the material and suffers multiple scatterings from the atoms surrounding the doubly ionized atom A until, after having finally assumed the momentum hk,, it reaches the detector.

The Auger matrix element (6.1) can be written as

where the Auger state vector IAtF;") is defined by

Comparing (6.2) with (2.11) and realizing that both state vectors IA$'jer) and IApE) are localized in space within the atom A - a fact which is crucial for the mathematical follow up of the theory - we see that the very same equation (3.37), with its counterparts (4.2), (4.Q (4.9) in SSA and PWA, holds good also for T$'ter, provided we replace the photoemission form factor A:,, with the corresponding Auger factor Jft?;? given by

We conclude then that for scanned-angle measurements the same information, as that obtained from photoelectron diffraction, can be extracted from Auger-electron diffraction.

Acknowledgement

I would like to thank C. S. Fadley for helpful comments.

References

[1] C. S. FADLEY, in: Synchrotron Radiation Research: Advances in Surface and Interface Science,

[2] R. DE L. KRONIG, Z. Phys. 75, 468 (1932). [3] H. PETERSEN, Z. Phys. 76, 768 (1932), 80, 258 (1933). [4] D. E. SAYERS, F. W. LYTLE, and E. A. S n x N , in: Advances in X-Ray Analysis, Vol. 13, Ed.

B. L. HENKE, J. B. NEWKIRK, and G. R. MALLETT, Plenum Press, New York 1970 (p. 248); Phys. Rev. Letters 27, 1204 (1971).

Vol. 1, Techniques, Ed. R. Z . BACHRACH, Plenum Press, New York 1992 (p. 421).

[S] W. L. SCHAICH, Phys. Rev. B 8, 4028 (1973). 161 L. N . MAZALOV, F. KH. GELMUKHANOV, and V. M. CHERMOSHENTSEV, J . Structure Chem. 15, 975

[7] C. A. ASHLEY and S. DONIACH, Phys. Rev. B 11, 1279 (1975). [8] P. A. LEE and J. B. PENDRY, Phys. Rev. B 11, 2795 (1975). [9] A. LIEBSCH, Phys. Rev. Letters 32, 1203 (1974); Phys. Rev. B 13, 544 (1976).

( 1 9 74).

[lo] L. MCDONNELL, D. P. WOODRUFF, and B. W. HOLLAND, Surface Sci. 51, 249 (1975). [11] J. B. PENDRY, J. Phys. C 8, 2413 (1975). [12] B. W. HOLLAND, J. Phys. C 8, 2679 (1975). [I31 P. A. LEE, Phys. Rev. B 13, 5261 (1976).


[I41 S. KONO, S. M. GOLDBERG, N. F. T. HALL, and C. S. FADLEY, Phys. Rev. Letters 41, 1831 (1978);

[15] T. FUJIKAWA, J. Phys. Soc. Japan 50, 1321 (1981), 51, 251 (1982), 54, 2747 (1985). [I61 S. Y. TONG and J. C. TANG, Phys. Rev. B 25, 6526 (1982). [17] P. J. ORDERS and C. S. FADLEY, Phys. Rev. B 27, 781 (1983). [I81 W. F. EGELHOFF, JR., Phys. Rev. B30, 1052 (1984). [I91 R. A. ARMSTRONG and W. F. EGELHOFF, JR., Surface Sci. 154, L225 (1985). [20] S. Y. TONG, H. C. POON, and D. R. SNIDER, Phys. Rev. B 32, 2096 (1985). [21] J. J. BARTON, S. W. ROBEY, and D. A. SHIRLEY, Phys. Rev. B 34, 778 (1986). [22] A. CHASS~ and P. RENNERT, phys. stat. sol. (b) 138, 53 (1986), 141, K147 (1987). [23] M. SAGURTON, E. L. BULLOCK, and C. S. FADLEY, Surface Sci. 182, 287 (1987). [24] J . MUSTRE DE LEON, J. J. REHR, C . R. NATOLI, C. S. FADLEY, and J. OSTERWALDER, Phys. Rev.

[2S] M.-L. Xu. J. J. BARTON, and M. A. VAN HOVE, Phys. Rev. B 39, 8275 (1989). [26] M.-L. Xu and M. A. VAN HOVE, Surface Sci. 207, 215 (1989). [27] A. P. KADUWELA, G. S. HERMAN, D. J. FRIEDMAN, C. S. FADLEY, and J. J . REHR, Physica Scripta

[28] A. P. KADUWELA, D. J. FRIEDMAN, and C. S. FADLEY, J. Electron Spectroscopy related Phenomena

[29] R. V. VEDRINSKII and L. A. BUGAEV, Physica Scripta T41, 56 (1992). [30] A. SAITO and T. FUJIKAWA, Surface Sci. 269/270, 49 (1992). [31] J. L. BEEBY and S. F. EDWARDS, Proc. Roy. Soc. A274, 395 (1963). [32] M. L. GOLDBERGER and K. M. WATSON, Collision Theory, J. Wiley & Sons, New York 1964

[33] J. L. BEEBY, Proc. Roy. Soc. A302, 113 (1967). [34] B. L. GYORFFY and M. J. STOTT, Solid State Commun. 9, 613 (1971); in: Band Structure

Spectroscopy of Metals and Alloys, Ed. D. J. FABIAN and L. M. WATSON, Academic Press, New York 1973 (p. 385).

Phys. Rev. B 22, 6085 (1980).

B 39, 5632 (1989).

41, 948 (1990).

57, 223 (1991).

(p. 749).

[35] P. LLOYD and P. V. SMITH, Adv. Phys. 21, 69 (1972). [36] J. S. FAULKNER and G. M. STOCKS, Phys. Rev. B 21, 3222 (1980). [37] P. PAVONE, R. GIRLANDA, G. MARTINO, and E. S. GIULIANO,

Vol. 18, Ed. G. CUBIOTTI, G. MONDIO, and K. WANDELT, S [38] L. FONDA, J. Phys.: Condensed Matter 4, 8269 (1992). [39] R. G. NEWTON, Scattering Theory of Waves and Particles, McGraw-Hill Publ. Co., New York

[40] L. FONDA, in: Scattering Theory, Ed. A. 0. BARUT, Gordon & Breach, New York 1969 (p. 129). I411 E. A. STERN, B. A. BUNKER, and S. M. HEALD, Phys. Rev. B 21, 5521 (1980). [42] J . J. REHR. E. A. STERN, M. L. MARTIN, and E. R. DAVIDSON, Phys. Rev. B 17, 560 (1978). [43] W. HEITLER, The Quantum Theory of Radiation, Oxford University Press, London 1954. [44] H. A. BETHE and E. E. SALPETER, Quantum Mechanics of One- and Two-Electron Systems,

[45] U. FANO and J. W. COOPER, Rev. mod. Phys. 40, 441 (1968). [46] L. HEDIN and S. LUNDQVIST, Solid State Phys. 23, 1 (1969). [47] P. A. LEE and G. BENI, Phys. Rev. B 15, 2862 (1977). [48] C. BROUDER and M. HIKAM, Phys. Rev. B 43, 3809 (1991).

Springer Series in Surface Sciences, nger-Verlag, Berlin 1989 (p. 63).

1966.

Springer-Verlag, Berlin 1957.

[49] R. FEDER, Polarized Electrons in Surface Physics, World Scientific Publ. Co., Singapore 1985 (p. 125).

[50] P. STRANGE, H. EBERT, J . B. STAUNTON, and B. L. GYORFFY, J. Phys.: Condensed Matter 1,2959

[51] C. R. NATOLI, M. BENFATTO, C. BROUDER, M. F. RUIZ LOPEZ, and D. L. FOULIS, Phys. Rev. B 42,

[52] L. FONDA and G. C. GHIRARDI, Symmetry Principles in Quantum Physics, Marcel Dekker, New

[53] M. ARRAMOWITZ and I. A. STEGUN, Handbook of Mathematical Functions, Dover Publ., Inc.,

1541 V. DE ALFARO and T. REGGE, Potential Scattering, North-Holland Publ. Co., Amsterdam 1965. [55] J . BARTON and D. A. SHIRLEY, Phys. Rev. B 32, 1892, 1906 (1985).

( I 989).

1944 (1990).

York 1970.

New York 1965.

32 L. FONDA: Multiple-Scattering Photoelectron and Auger-Electron Diffraction

[56] V. FRITZSCHE and P. RENNERT, phys. stat. sol. (b) 135, 49 (1986). [57] J. J. REHR, R. C. ALBERS, C. R. NATOLI, and E. A. STERN, Phys. Rev. B 34, 4350 (1986). [58] J. J. REHR and R. C. ALBERS, Phys. Rev. B 41, 8139 (1990). [59] V. FRITZSCHE, J. Phys.: Condensed Matter 2, 1413 (1990). [60] D. K. SALDIN, G. R. HARP, and X. CHEN, Phys. Rev. B 48, 8234 (1993). [61] W. F. EGELHOFF, JR., Phys. Rev. Letters 59, 559 (1987); Crit. Rev. Solid State Mater. Sci. 16,

[62] S. A. CHAMBERS, I. M. VITOMIROV, S. B. ANDERSON, H. W. CHEN, T. J. WAGENER, and J. H. WEA-

(Received July 19, 1993; in revisedform November 24, 1993)

213 (1990).

VER, Superlattices and Microstructures 3, 563 (1987).

Documents

Green's Operator Approach to Multiple-Scattering Photo-electron and Auger-Electron Diffraction