SPHERICAL WAVE APPROACH TO ELECTRON FOCUSING PROCESSES IN EXAFS

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SPHERICAL WAVE APPROACH TO ELECTRON FOCUSING PROCESSES IN EXAFS

R. Vedrinskii, L. Bugaev

To cite this version:

R. Vedrinskii, L. Bugaev. SPHERICAL WAVE APPROACH TO ELECTRON FOCUSING PROCESSES IN EXAFS. Journal de Physique Colloques, 1986, 47 (C8), pp.C8-89-C8-92.

�10.1051/jphyscol:1986815�. �jpa-00226052�

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JOURNAL DE PHYSIQUE

Colloque C 8 , supplement au no 12, Tome 47, decembre 1986

SPHERICAL WAVE APPROACH TO ELECTRON FOCUSING PROCESSES IN EXAFS

R.V. VEDRINSKII and L.A. BUGAEV

I n s t i t u t e o f P h y s i c s , R o s t o v S t a t e U n i v e r s i t y , E n g e l s str. 105, 3 4 4 0 0 6 R o s t o v - o n - D o n , U . S . S . R .

Abstract.

-

^Anovel method based on the spherical wave formalism is presented to account the low-angle electron scattering processes in EXAFS. The usually used plane wave approximation is shown to be inappro- priate when the electron forward scattering should be accounted. The developed formulae are used to obtain X-ray absorption spectra (XAS) of some crystals.

The effect of the low-angle electron scattering (wfocusingv processes) on the EX@S spectra is usually elLamlned by the approxi- mative approach based on the replacement of outgoing spherical electron waves by plane waves, as it was done for the backscattering processes

[I].

However one may expect plane wave approximation to be insufficient for the low-angle electron scattering in the linear system A-F-S (A

-

is the absorbing central atom, F

-

intervening atom, S

-

backscattering atom) due to the large size of spatial re- gions, where the low-angle scattering process takes place. Such idea was claimed recently in [2], but the exact solution of the problem was not carried out because of the difficulties appeared when the multiple-scattering processes A- F- S-I. A (b), A-S-F-A (c), A+F+S+F-cA (d) were taken into account in the spherical wave approach together with the single-scattering term A-S --cA (a).

Here we present a novel formulae for the processes ( b), (C )

,

^(d⁾

^,

which permit to obtain the normalized EXAFS-function without plane wave approximation:

-

x(E)=I~{[xI(E.+x~(E) +x&J

exp(2isA)} (I 1

where

s,

n is the phase shift for electron scattering with angular moment&

e,

from the n-th atom.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986815

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JOURNAL DE PHYSIQUE

estf+F W

6) (F)

(d) : x ~ ( E ) = N ' ~ (-1) (2&1)(2e~1~2P"+l)

tp t,

t pt x

efefie"

where K = ( E ~ ~ , & is the photoelectron energy (Ryd), S is the number of the backscattering shell that contains N'atoms, R,, is the dis- tance between the n-th and m-th atoms, he

+

is the Hankel function,

tT)=-exp

( i ~ t ) ) ~ i n

Sf) and the coefficients

c(.?el; e; ex)

are deter- mined as:

where

We have

4" ex ev

are the 3 j and 6 j symbols.

applied formulae (21, ( 3 1 , (4) to obtain the effective backscattering amplitude for the 4-th shell atoms in cooper, shadowed by the 1-st shell atoms. The crystalline El? potential was const- ructed according to 14, 5 1 with the nonlocal interaction within the atomic spheres, The r=lt is shown in Pig 1 (curve 1) in comparison with the effective amplitude, obtained by the plane wave approach

(curve 2). The backscattering amplitude, calculated by the single- scattering formulae (2) (curve 3 ) is also shown together with the same amplitude, calculated in plane waves (curve 4). As one can see the backscattering amplitude increases due to the low-angle electron scattering. But the effect becomes more than two times stronger if plane waves are used. !Phis makes the plane wave approximation in- apropriate for the low-angle electron scattering processes.

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We have applied the above f o m l a e (1)

-

⁽⁴⁾and the scheme of crystalline potential constructioh to obtain K X-Ray Absorption Cross Section (8) of Na in NaF. The low-angle electron scattering processes were accounted for the 4-th shell atoms, shadowed by the

1-st shelln Fig.2 shows the theoretical (curve 1) and experimental spectra, combined according to their first peak position.

I . . . , , , . . .

' ^EjR9-

Fig. 1 The effective backscattering Fig. 2 K X-Ray absorption amplitude for the 4-th shell in Cu. cross section of Na in Nal?

Spherical waves: 1- processes (a)-(a), Curve 1

-

theoretical

3- process (a) ; spectrum.

plane waves: 2- processes (a)-(&), 4- process (a).

Normali5ed EICBF.S function X(E) of crystalline Cu, calculated with the help of formulae (1)

-

⁽⁴⁾ ^and^{via plene}^waveapproximation, is compared with the experimental. one in Pig, 3.

2 Fig. 3 Normalized EXAFS function WE) of crystalline Cu: curve (1) - ^ex-

periment ; curve (2) - formulae (I) to (4) ; curve (3) -

plane wave approximation.

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C8-92 JOURNAL DE PHYSIQUE

As one can see the approach discussed above permit to obtain with high accuracy all the main features of the considered experimental spectra.

References

[I] Teo B.-K. J. Am. Chem. Soc. 103 (1981) 3990.

[2] Barton J.J., Sirley D.A. Phys. Rev. B. 32 (1985) 1892, [3] ~Gller J.E., Schaish W.L. Pbys. Rev. B. 27 (1983) 6489.

[4] Vedrinskii R.V., Bugaev L.A., et ,al. Sol.St. Comm.44 (1982) 1401.

[5] Bugaev L.A., Vedrinskii R.V. Phys. Stat. Sol. (b)132 (1985) 459.