SEMICLASSICAL TREATMENT FOR INELASTIC PROCESSES IN EXAFS

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SEMICLASSICAL TREATMENT FOR INELASTIC PROCESSES IN EXAFS

D. Lu, J. Rehr

To cite this version:

D. Lu, J. Rehr. SEMICLASSICAL TREATMENT FOR INELASTIC PROCESSES IN EXAFS.

Journal de Physique Colloques, 1986, 47 (C8), pp.C8-67-C8-70. �10.1051/jphyscol:1986810�. �jpa-

00225994�

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JOURNAL DE PHYSIQUE

Colloque C8, supplement au n o 12, Tome 47, d k e m b r e 1986

SEMICLASSICAL TREATMENT FOR INELASTIC PROCESSES IN EXAFS

D. LU and J. J. REHR

D e p a r t m e n t o f P h y s i c s , F M - 1 5 , U n i v e r s i t y of W a s h i n g t o n , S e a t t l e , WA 9 8 1 9 5 , U.S.A.

A b s t r a c t - Inelastic processes in EXAFS (extended x-ray absorption fine structure) are studied using an extension of the semiclassical model of dynamical screening of a core hole and a photoelectron. This treatment effectively includes both extrinsic and intrinsic inelastic losses as well as interference between them. A complex, energy-dependent potential which accounts for these inelastic effects is derived. This potential is compared with static final state potentials and with empirically determined mean free paths. An local relaxation method is developed to calculate the core hole Green's function in an inhomogeneous system. Application is made t o the EXAFS amplitude of the diatomic molecule Brz.

1. Introduction

In the X-rag absorption problem in atoms molecules and solids there are two basic inelastic processes following the creation of a core hole and a photoelectron. They are the 'intrinsic' process, which is the relaxation of the system due to the core hole, and the 'extrinsic' process.

which is due to the inelastic scattering of the outgoing photoelectron. In addition, the interference between these two processes becomes important at low photoelectron energies. VJe discuss here a method to calculate these effects in an approximate way.

To describe these inelastic processes one needs two different Green's functions: one is the core hole Green's function G,(t) = -%(Qo

/

~ ( b ( t ) b t (0))

/

Qo); the other is the photoelectron Green's function Ga(t) = -z(iPo I T(ck(t)ck(0))

t 1 ao) ,

where

1

Go) is the initial ground state of the system and b, bt, Ck,ek are the creation and annihilation operators for electrons in the core

t

state

I

b) and the outgoing state j k). Taking many electron effects into account, one can write the absorption cross section @(W) (in the dipole approximation) and the EXAFS spectrum ~ ( w ) as convolution integrais [ l ] ,

where p ( ' ) ( W ) = - (l/z)Im(b

/

Z. r'G(w)F'- r'

l

b) is the single particle absorption cross section, and X(l)(w) is the single particle EXAFS spectrum, i.e., the oscillatory part of p(')(w), normalized by the smooth atomic background po(w): x(l)(w) = ] @ ( ' ) ( W ) - po(w)]/po(w).

In this paper inelastic losses and interference effects are studied using the semiclassical model of dynamical screening 1'2). We have made two modifications to the original model which enable us to apply it to inhomogeneous electron systems such as atoms or molecules. The two Green's functions are then calculated using a complex energy dependent potential which effectively includes

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986810

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CS-68 JOURNAL D E PHYSIQUE

these two inelastic losses as well as interference. The results are compared with other theoretical model calculations and with experiment.

2. Semiclassical M e t h o d

The method we use here is based on the semiclassical model [2,3] of dynamical relaxation of electrons in a homogeneous system t o a suddenly created core hole and a photoelectron. In this model both the photoelectron and core hole are treated as point charges: One moves along a classical trajectory i'=

6,

while the other is at the center of the atom, r' = 0. _The charge density distributions associated with these two particles are p,(r',t) = -B(t)b(r'- kt), and ph(7;t) = B(t)6(F). Given the charge distribution po(< t ) = p, (7; t)

+

ph(< t ) , one can find the induced potential $rnd(F, t ) in the system using linear response theory,

The self-energy C(t) (i.e., the exchange-correlation potential) for the photoelectron is then

The static part has a simple analytical form C, = -wp/2k tan-' Xk/wp - iwp/4kIn(l

+

X2k2/w,2), which is very similar to other more sophisticated potentials such as that of Hedin-Lundquist. The dynamical part Cd(t) reduces the magnitude of C at low energies [2].

To apply these semiclassical results to an inhomogeneous electron system such as atoms or molecules. we use the following procedure:

J Local cler..~t? approxinlation ILDA) and classical trajectory The LDA has been used t o obtain the stat.c part of the final. excited state exchange-correlation potential in inhomogeneous rystems. based on calculat~ons of C for a homogeneous electron gas 141. We use the LDA for both C, and Ed

,

i.e., Cfnh(F,t) = CS(p(F), p(F))

+

C d ( ~ ( F ) , ~ ( f l , t ) . Here p(F) is the local electron density and p(F) is the local electron momentump(F) = (k2

+

k ; ( F ) ) 2 . To calculate the scattering phase shifts for the photoelectron using the time dependent exchange-correlation potential, we assume the photoelectron moves along a classical trajectory determined by the time function t (r) =

Sor

drl/p(r').

2. Independent local relaxation method for the core hole Green's function- We use a local relaxation method to calculate G,(t) in an inhomogeneous electron system; i.e., we treat the passive electrons in the system as a homogeneous electron gas in each local volume element d37

.

These local regions are assumed to relax independently so that Gc(t) can be written as product of local Green's functions, i.e., Gc(t) = n,-G!(p(F), t). Here G!(p(F), t) is the core hole Green's function in a homogeneous electron system with electron density p ( 3 , i.e.,

where f (r) = (sin qcr - qcr COS qcr)/(2n2r3), and q, = wp/kf is the plasmon cut-off wave vector 131.

3. Calculations a n d R e s u l t s

1. Electron Mean-Free-Path - Using the static part of the exchange-correlation potential C,, we calculate the electron mean-free-path X k = -I/Imk r -k/ZImC,(k) and compare the results

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with the Hedin-Lundquist exchange-correlation potential and with experiment [5,6]. These results are in good agreement.

2. Atomic Phase Shifts- The atomic scattering phase shifts for diatomic molecule Brz are calculated using the WKBJ method 171 with the time dependent exchange-correlation potential

CZnh(F, t ) .

We find that the dynamical corrections

&(t)

are only important for the central atomic phase shift 6,0Ut. Comparing the imaginary part of the phase shift 6,0Ut, with and without the dynamical part of the potential C,, shows that the dynamical effect reduces the extrinsic losses, particularly a t low energies, ck

1

300eV.

3. Core Hole Green's Function Calculation- We calculate the core hole Green's function for Brz using the local relaxation method given by Eq.(4). The results show that excitation spectra are suppressed due to the dynamical effects when the energy is below 500eV.

4. Effective Backscattering Amplitude of EXAFS for Br2- Taking inelastic losses into account the EXAFS spectrum for a diatomic molecule can be written 181,

The first two factors, S(k) = A(k) exp(-Im(6,0Ut

+

6in)), give a reduction in the overall EXAFS amplitilde due to inelastic losses. The function A(k), which is called the amplitude reduction factor, gives the reduction due to the intrinsic losses, and is related to the core hole Green's function by [g]

where kW = (k2 - 2w)

B.

In figure 1 A(k) is compared with Hartree-Fock calculations in which ad hoc corrections for dynamical effects are added [10]. We also find that for Bra the effect of the dynamical correction of central atomic phase shift almost cancels the amplitude reduction factor. i.e..

A(k) exp(-lmsldyn) E 1. These results for S ( k ) are in reasonably agreement with experiment for A- = 3.0 t o 7.0(a.u.). A more complete treatment of these results will be given elsewhere '11:.

Flq.1 Amplitude R e d u c t i o n F a c t o r A(k1

0.7

0.6

1. This Work

- 2.

H.F. ( S t a t ~ c )

-

3. H.F. (Dynurnlcl

" " I " " I " " I " " I " " I ' -

0 2 4 6 8 10

k la.11-l

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PHYSIQUE

References

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,

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(4, P. A. Lee and G. Beni, Phys. Rev. B15, (1976), 2862.

[S' G. D. Mahan, Many Parttcle Physics, (Plenum Press, New York

,

1981), Chap. 5.

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E.

A. Stern, R. L. Martin and

E.

R. Davidson, Phys. Rev. B17, (1978).

19; J.

J.

Rehr and S.-H. Chou, EXAFS and Near Edge Structure, Eds. A. Bianconi. L. Incoccia and S. Stipcich, (Spring Verlag. 1983).

110: S.-H. Chou, Ph.D Thesis, University of Washington, (1983).

ill! Dan Lu, Ph.D Thesis, University of Washington, (1986).