SPHERICAL WAVE CORRECTIONS IN XAFS

HAL Id: jpa-00225988

https://hal.archives-ouvertes.fr/jpa-00225988

Submitted on 1 Jan 1986

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

SPHERICAL WAVE CORRECTIONS IN XAFS

J. Rehr, R. Albers, C. Natoli, E. Stern

To cite this version:

J. Rehr, R. Albers, C. Natoli, E. Stern. SPHERICAL WAVE CORRECTIONS IN XAFS. Journal de

Physique Colloques, 1986, 47 (C8), pp.C8-31-C8-35. �10.1051/jphyscol:1986804�. �jpa-00225988�

JOURNAL DE PHYSIQUE

Colloque C8, supplgment au n o 12, Tome 47, dhcembre 1986

SPHERICAL WAVE CORRECTIONS IN XAFS

J.J. REHR, R.C. ALBERS', C.R. NATOLI" and E.A. STERN Department of Physics, FM-15, University of Washington, Seattle, WA 98195, U.S.A.

'MST-5 MS G730, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.

*

Laboratori Nazionali di Frascati dell Istituto Nazionale di Fisica Nucleare, 1-00044 Frascati, Italy

Abstract- Spherical wave corrections in XAFS (x-ray absorption fine structure) are discussed using a new high-energy approximation based on asymptotic expansions. This approximation replaces the plane-wave approximation and is found to be in good agreement with exact treat- ments almost to the absorption edge. With this formulation the single-scattering XAFS formula is recovered in terms of distance-dependent back-scattering amplitudes. Multiple-scattering con- tributions in XAFS and spherical wave corrections in angular resolved photoemission are treated similarly.

l'! r -unl

approx~mations in EXAFS analysis'-3 are the small-atom. plane-wave approximation

\

and the restriction to single-scattering paths. It has generally been thought (cf. Ref. 14;)

h a t t

orrerting the PWA requires exact calculations. However, recently it has been shown that

r

h* appropriate high energy approximation in EXAFS is not the PWA, but rather an asymptotic theory which is almost as simple that includes spherical wave correction^.^ This theory is reviewed here. In particular, it is found that this approximation, termed the spherical wave approximation (SWA), yields excellent agreement with exact calculations almost to the absorption edge, thereby giving a unified high energy treatment of XAFS, i.e., both EXAFS and XANES.

2. Spherical Wave Approximation

The starting point for the SWA is the single-scattering XAFS formula2,

where G L , ~ , (2) is the dimensionless free propagator in an angular momentum L

( t , ^{m ) ,} and site

2 basis, and t

(l?)

ei6' sin is the dimensionless atomic t-matrix at site l?. Formally G L , ~ l is given by the expansion2, G L , ~ , (R) = 4xCL,, (YLY~,, JYL,) h:,, ( g ) , where h i ($1 i e h l

(R), h: is a spherical Hankel function, and p = kR. Carrying out the L sums in Eq. (1) using this expansion and 3j-symbol identities gives the Miiller-Schaich XAFS f ~ r m u l a . ~ Their result is considerably simpler than the original formulations, but it is still algebraically complicated.

The PWA theory can be derived by using an extreme high energy limit for the G's in Eq. (l);

however, this approximation is not strictly valid even at EXAFS energies. Our SWA consists

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986804

C8-32 JOURNAL DE PHYSIQUE

of replacing this form with a better high-energy approximation, based on rapidly convergent, asymptotic expansions of modulus and phase. This is similar to the WKB approximation and takes into account the centrifugal barrier seen by the photoelectron that is neglected in the PWA.

Explicitly, the SWA for the propagator G L , ~ 6 is erp

GL,U

( R ) = 4 a ~ e ' , ( k ) ~ e r o ( ~ ) - g j i ? ' ) ( p ) 6 , , , , , ^{= f RB.}

P (2)

For K-shell absorption, only G l m Y L and G L , i m are needed, whose correction factors are given by

where ce(p) are the correction factors to the limiting high-energy form of spherical Hankel func- tions, i.e., h:(p) i-e(erP/p)ce(p). Replacing the exact c&) with that from the leading terms of asymptotic expansions of the modulus and phase of h:(p), we obtain

where L2 e(e + 1). Note that when !-wave scattering is significant (i.e., t e non-negligible),

!/p < R m t / R < l , which implies the high-energy limit. Subsequent terms are smaller than those retained in Eq. (4) and can be neglected. The dominant spherical wave correction is seen to be the non-negligible phase-shift L2/2p. Alternatively, one can define the ce(p) recursively7, which is equivalent to obtaining the Hankel functions exactly: ce+l = cl-1 + i ( Z + I)ce/p, with c. = 1 and

c l

= 1

With these results, an exact single-scattering XAFS equation can now be obtained,

namely

Here the sum is over scattering sites R with effective backscattering amplitude j ( x , R) = j ( z , R ) ]

F ' @ . I;

= v7 is the photoelectron wavenumber (in atomic-Rydberg units, h = 2m = e2/2

1):

=

E - E, is the energy relative to the muffin-tin zero in Ry and (G), ^the l = 1 phase shift of an absorbing atom a t 2 ⁼ 6. The exponential decay factors are due to the mean free path X and to the Debye-Waller factor (assuming small disorder), respectively. This result has the same form as the convensional EXAFS formula except that the backscattering amplitude f (n) and phase @ are replaced by j ( a , R) and 6 (R), where

f i n , R) = 1 C ( - l ) ' l e ! ( l + l ) c ~ + , ( k ~ ) + (c:-,(kR)].

e

Note that as R

oo the term in brackets in Eq. (6) becomes unity, and J(n, R ) reduces to f ( a ) . A comparison between f ( a ) and f ( x , R ) is given in Fig. 1 below. We emphasize that Eq. (5) is exact. Our approximate (SWA) expression for ?(a, R ) results by replacing ce(p) in Eq. (6) with Eq. (4), and thereby avoids the explicit calculation of spherical Hankel functions.

Formulae sin~ilar to Eq. (3) can be developed for larger I or !'. but they involve an intermediate d

sum and hence become progressively more cumbersome. However one can also develop asymptotic

expansions for the modulus and phase of g i ~ ' ( p ) starting from a more general integral equation

for G I L . ~ , . One obtains

0 2 4 6 8 l 0

k ( 4

Fig. l ( a ) Modulus and (b) phase of the effective backscattering amplitude j ( x , R) versus wave number k beyond kF for the first (solid line) and fourth (dotted line) neighbors in Cu (R1 = 4.80 and R4 = 2R1 = 9.60 bohrs) and, for the standard backscattering amplitude, f(r) = f ( x , m ) (dashed curves). For other R, j ( n , R) is given approximately by interpolation in 1/R.

where Jo is the Bessel function of order zero. For non-shadowing multiple-scattering contributions to x ( E ) , one also needs propagators for general directions R # 2. Using rotation matrices, we find

.

G L , L ~

( k ) = ^{4 ~ (R)YL.(R) ~ 2 pgj:/(~). (8)}

The leading corrections to Eq. (7) are proportional to gj:,)(p). and not always negligible.

Finally we note that analogous SWA can be developed for other physical processes. For example,

ARPEFS (angular resolved photoemission fine s t r ~ c t u r e ) ' . ~ also exhibits EXAFS-like oscillatory

structure. For this case the PWA scattering amplitude must be replaced by5

JOURNAL DE PHYSIQUE

Our expression for !(B. R) differs from the PWA formula" by the factor a SWA results by evaluating (9) using Eq. (3) and (4).

3. Summary a n d (!onclusion

In summary we have rlisc~rssed a new, efficient high energy approximation of XAFS which replaces the PWA yet is accurate virtually to the absorption edge. Remarkably XANES can be treated as a high-energy phenomenon. This was not recognized previously because of the failure t o incorporate spherical wave effects correctly in the single-scattering expression a t high energies.

The appropriate limit leads to simple, analytical forms for the electron propagators, even when spherical wave effects are important. Compared to the exact formulae, our SWA is simpler and hence faster computationally with little loss of accuracy. These advantages are especially important for multiple-scattering contributions; for example, for the calculations presented in Ref. 5, which include shadowing, the SWA is more efficient by a factor of about 20 on a CRAY- 1A. Thus the exact formulations (e.g. Ref. 191) may be unnecessarily complicated. Also, due t o various uncertainties in the theory (e.g., muffin-tin corrections, and many-body effects) even the

"exact" treatments are really only approximate at low energies. The standard EXAFS formula can be recovered and extended by using, 'in place of f(r), the effective scattering amplitude

j(n, K) defined in Eq. (6). Because f ( n , R ) is an atomic quantity and is smoothly varying in l,'R, it is conveniently tabulated. We plan to do so for a number of elements. McKale et a1.I0 have also reported calculations of these backscattering amplitudes f(s, R), based on the Miiller- Schaich formula. Finally, the SWA yields a unified treatment of XAFS, valid for both EXAFS anti X ANES. Because it permits XAFS data to be analyzed all the way to the edge using standard tecllniques,' it should improve the utility of XAFS for determinations of local atomic structure, particularly for low-Z atoms and/or large disorder. Application of this theory to calculations of the XA YS of Cu shows that shadowing corrections alone cannot explain the descrepancy between the sir,gle scattering calculations and bandstructure" beyond lOOeV of threshold. Multiple-scattering calculations based on this theory are in progress.'2

ACKNO WLRDGMENTS: The authors thank J.J. Barton, C.S. Fadley, J.W. Wilkins and P.A. Lee for comments. This work was supported in part by the NSF, DMR-84-01781 (JJR),'DOE Contract 50. W-7405-Eng-36 (RCA). CNH-Italy (CRN). and NSF Grant DMR-80-22221(EAS).

1 For reviews. see E.A. Stern and S.M. Heald.

Handbook of Synchrotron Radzatron. E . E . Koch i ~ f i

Chap. 10 (Xorth-Holland. Xew York 1983) P.A. Lee. P.H. Citrin, P. Eisenberger, and B.hl Kincaid, Rev. Mod. Phys. 53,769(1981)

2. P.A. Lee and J.B. Pendry. Phys. Rev. B11,2795(1975); C.A. Ashley and S. Doniach.

Phys. Rev. %11,1279(1975); W.L. Schaich, Phys. Rev. B 8,4028(1973) 3. D.E. Sayers, E.A. Stern, and F.W. Lytle, Phys. Rev. Lett. 27,1204(1971)

4. G. Bunker and E.A. Stern, Phys. Rev. Lett. 52,1990(1984), 54,2726(1985); D.D. Vvedensky and J.B. Pendry, Phys. Rev. Lett. 54,2725(1985)

5. J.J. Rehr, R.C. Albers. C.R. Natoli and E.A. Stern, Physical Review B, in press (1986)

6. J.E. Miiller and W.L. Schaich. Phys. Rev. B27,6489(1983); W.L. Schaich, Phys. Rev. B29,6513

(1984): and W.L. Schaich, in EXAFS and Near Edge Structure III, K . O . Hodgson, B. Hedman,

and J.E. Penner-Hahn (eds.), p. 2. (Springer-Verlag. Berlin 1984)

;7.: J.J. Barton and D.A. Shirley, Phys. Rev. A32,1019(1985), Phys. Rev. B32,1892(1985), and Phys. Rev. B32, 1906(1985)

i10.j M. Sagurton, E.L. Bullock, R. Saiki, A. Kaduwela, C.R. Brundle, C.S. Fadley and J.J. Rehr, Phys. Rev. B33,2207(1986)

19.1

S.J. Gurham, N. Binsted and I. Ross, J . Phys. C l ? , 143(1984) 110.1 A.G. McKale, G.S. Knapp, and S.-K. Chan, Phys. Rev.B33,841(1986)

Ill.] R.C. Albers, A.K. McMahan and J.E. Miiller, Phys. Rev. B31,3435(1985); see also, J.E.

Miiller and J.W. Wilkins, Phys. Rev. B29,4331(1984)

112.1 R.C. Albers and J.J. Rehr, unpublished (1986)