SOME ANALYTICAL PROPERTIES OF THE ONE-ELECTRON GREEN'S FUNCTION AT XANES RANGE

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SOME ANALYTICAL PROPERTIES OF THE ONE-ELECTRON GREEN’S FUNCTION AT XANES

RANGE

V. Kraizman, A. Novakovich, V. Popov

To cite this version:

V. Kraizman, A. Novakovich, V. Popov. SOME ANALYTICAL PROPERTIES OF THE ONE- ELECTRON GREEN’S FUNCTION AT XANES RANGE. Journal de Physique Colloques, 1986, 47 (C8), pp.C8-93-C8-95. �10.1051/jphyscol:1986816�. �jpa-00226063�

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JOURNAL DE PHYSIQUE

Colloque C8, supplkment au n o 12, Tome 47, dkcembre 1986

SOME ANALYTICAL PROPERTIES OF THE ONE-ELECTRON GREEN'S FUNCTION AT XANES RANGE

V.L. KRAIZMAN, A.A. NOVAKOVICH and V.I. POPOV

Institute of Physics, Rostov State University, Engels str. 105, 344006 ROS~OV-On-Don, U.S.S.R.

Abstract.-The interpretation of XANES sirqplarities is developed. O u r approach is based on the investi- gation of the analytical properties of the one- electron Green's function of the model cluster within the mulf;iple scattering formalism.

There are many attemptes to describe XANES in terms of linear combinations of atomic orbitals. Usually one transfers the ideas of MO X A O into the range of contfnious spectrum in a simple qualitative manner- In the present paper we develop the approach to the interpretation of XANES singularities which i s based an multiple scattering method. We consider the case when sharp shape

resonances i n photoelectron saatterLng occur. It is convinlent to deal with analytical properties of the one-electiron Green's function (GI?) because on the one hand photoabsorption intensity is propo&ional to the imaginary part of GF and on the other hand the last satisfy to the system of algebraic equations- If the basis of some irreducible representation of cluster symmetry group is used, these equations have the f o m

Jl

=

+ (I 1

k= d

BY

is free-electron GF, &-usual atomic t matrix,

t&

- f u U GI? [I

] .

If the shape resonance takes place, correspondiw t r@trix has a well known pole form

= - ( i +

1 ⁽²⁾

tua

^{2 i} ⁷

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986816

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C8-94 JOURNAL DE PHYSIQUE

where

Eo

i s the energy of resonance,

f

i s PWHH and

&,

^{i s a}

slowly varied function of e n e r a . Let us suppose some of the

t,

matrix namely

t

^{. c .}

tH

be equal t o

tw.

I n t h i s case one can show t h a t near the t matrix pole position GF maY be represented a s the sum of two terms

(3) where

M poles and M-1 zeros i n the v i c i n i t y of point

go - I;/Z

⁺ J u s t these poles a t t r i b u t e d t o t h e c l u s t e r resonances i s hatuxally t o c a l l l i n e a r combination of atomic resonances. Unfortunately it i s impossible t o predict the amom% of peaks appearing i n XILmES near t h e atomic resonance, because i f the pole and the zero of function

G-(E)

a r e located a t complex plane close t o each other the s i n g u l a r i t y i n

G-(E )

calculated a t r e a l values E would be pronounoed very slightly.

The function

C&(E)

may have poles a t compler plane

E

a s well. The nature of these s i n g u l a r i t i e s i s similar t o that of EXAFS maxima. They are due t o appearance of almost-standing photoelectron wave a r i s i n g a s a r e s u l t of interference of waves scattered from d i f f e r e n t neighbouring atoms. Their appearence i s not connected with sharp v a r i a t i o n of t matrix hence t h e i r positions a r e not

connected with the t matrix pole ones and. a r e determined by complex formulae. J u s t these s i n g u l a r i t i e s one may c a l l r e a l multiple

scatter* resonances.

To i l l u s t r a t e mentioned abwe l e t us consider the r e s u l t s of model calculations. We have calculated the i n t e t s i t y of 2p+& d t r a n s i t i o n s i n c e n t r a l atom of octahedral c l u s t e r f o r e irreducible representation, Central atom t matrix has no s i n g u l a r i t i e s i n t h e 2 energy range considered. Ligand p o t e n t i a l s have the f o m

V(~)+AV ,

where

V(Z)

i s Xd, p o t e n t i a l of fluorine atom and a value of

AV

i s constant i n ligand atomic sphere. For some positive values of

AV,

2s and 2p d i s c r e t e l e v e l s of P-atoms t u r n i n t o sharp resonances.

If the magnitude of

A V

becomes negative and grows i n i t s absolute value there appear 3p and 3d resonances which then tm i n t o diso- r e t e levels. The resonance energies a r e labelled i n Fig.'l by ver- t i c a l lines. A s one can see i n Pig.*l there a r e the singu&&,ties i n XANES a t the energies close t o the energy of t matrix resonance.

I n equation (I) ligand t matrixes te (here 1 i s o r b i t a l mo- ment) appear once f o r 14,l and twice f o r 1=2. I n accor-ce with it one can see tsro peaks a t t r i b u t e d t o 3d atomic resonance ff the energy of ³⁶resonance i s close t o threshold. If the value

11\/

i s

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great enough, the distance between zero and one of the poles becomes l e s s than the distance up t o the r e a l axis. (The positions of poles arid zeros of OF and t z a t different values of

AV

are shown i n Fig.2). So onLy a singLe peak corresponding t o the remaining pole i s kept i n spectrum.

In Fig.1 one can also clearly see the multiple scattering resonances whose energies are not d i r e c t l y connected with atomic resonance positions. I t must be emphasized t h a t these peaks a r e very narrow even i f %heir energies are a s high a s 3 ^Ry.

Fig. 1. I n t e n s i t i e s of 2p+td transitions a t &if

-

f e r e n t

AV

values. Vertical l i n e s mark atomic resonance positions.

Pig.2. Green's function

( -pole, A -zero) and t matrix t, ( 0-pole) positions a t differant A values.

References

1. Vedrinskii R.V. and Novakwich A.A., Piz. MetaSlov i metallwedenie 39 (1975) 7.