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Submitted on 1 Jan 1986
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SPHERICAL WAVE CORRECTIONS IN ARPEFS
J. Rehr, J. Mustre de Leon, C. Natoli, C. Fadley
To cite this version:
J. Rehr, J. Mustre de Leon, C. Natoli, C. Fadley. SPHERICAL WAVE CORRECTIONS IN ARPEFS.
Journal de Physique Colloques, 1986, 47 (C8), pp.C8-213-C8-216. �10.1051/jphyscol:1986840�. �jpa-
00226162�
SPHERICAL WAVE CORRECTIONS IN ARPEFS
J.J. REHR, J. MUSTRE DE LEON, C.R. NATOLI* and C.S. FADLEY*"
Department of Physics, FM-15 University of Washington.
Seattle, WA 98195, U.S.A.
" ~ a b o r a t o r i Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, 1-00044 Frascati, Italy
*'~epartment o f Chemistry, University of Hawaii, Honolulu, HI 96822, U.S.A.
A b s t r a c t - A derivation of ARPEFS (Angular Resolved Photoelectron Spectroscopy Fine Struc- ture) which generalizes that based on the plane-wave approximation is reviewed. Spherical wave corrections in the theory are approximated using asymptotic expansions for the one-particle prop- agators. Single scatterjng 5 r m s are treated exactly in terms of a distance dependent, effective scattering amplitude, f (8. R). This spherical wave approximation (SWA) explains observed cor- rections to the plane-wave approximation and is found to be in good agreement with the full single scattering spherical wave treatment. The relation between this theory and XAFS (X-ray absorption fine structure) is discussed using a generalized optical theorem.
1 . t n t r o d u c t i o n
111 angle resolved photoelectron spectroscopy fine structure (ARPEFS) the oscillatory behavior of t h + differential cross section may be used to deduce informjtdon about the local enviroment of a particular atom, in analogy with EXAFS. In the plane-w&.e approximation to ARPEFS [I], the cross section d o / d n = I($zji, d$core)12 is readily evaluated and for a K shell core state gives
However the plane being necessary to This generalization
wave approximation is found to be inadequate even at fairly high energies, it take into account the spherical wave nature of the photoelectron final state.
is summarized here. When d a / d R is integrated over all possible directions of
-
the final photoelectron state one must obtain the total absorption cross section and consequently the correct XAFS formula. We demonstrate this reduction using a generalized optical theorem.
We show that the basic structure of Eq.(l) is preserved provided one generalizes the scattering amplitude f (8) to an effective scattering amplitude
j(8,Z).
2. P h o t o a b s o r p t i o n cross section in t h e p l a n e wave limit
The photoelectron final state in photoabsorption is conveniently viewed as a time reversed scatter- ing state121 consisting of incoming spherical waves and an outgoing plane wave in the observation direction
k,
($2) =) 1: +
4 ~ k ~ ~ ~ i ( i ) t i ~ l ~ - , 6 ) . The photoelectron state in the presence of ak -
scatterer at site R with t matrix t l = ez6[ sinbl is given by the Lipmann-Schwinger equation,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986840
C8-214 JOURNAL DE PHYSIQUE
I$;)
= (1+
G;tz)I$$). The dipole matrix element M; = ($,,,,li- d$;),
for a K-shell core state is then~ -= *t, ,i*; Y I O ( ~ )
+
e - " ' ~ Y L ( ( K ~ I G L , ~ o ( E )+ Y L ~ ( ~ ) ~ ~ , G L ~ , L ( - R ) ~ I G L , I O ( ~ ) ,
[
L Lf,L1
(2)where mlo = 4a($,,,,li- qqblo); we have used the identity ( L , ~ I L ' , ~ + ) = G L , ~ , ( R
- 2);
andIL,
2)
denote eigenstates of the free particle hamiltonian with origin at2.
The terms of this sum are the direct, single scattering afld double scattering-terms fespectively. In the plane wave limit the propagator becomes GL,LJ (R) = 4 r ( e z k R / R ) Y i (R)YL,(R),
leading to [I],where f (6') is the scattering amplitude of the scatterer and fc(6') is the scattering amplitude of the absorbing atom.
3. Spherical wave corrections
To recast Eq.(2) in a similar form to Eq.(3), we take as a starting point the expansion for the propagator propossed by Rehr et a1131
where DLnL = D;&,(O, -BR, - c $ R ) D ~ ~ ( o , - 6 ' ~ . - O R ) and D;,(O, -BR,
- 4 ~ )
is the rotation ma- trix corresponding to a rotation that takes the z-axis into the vector 2\51. Inserting this expansion in he singlr, art1 rm:: term of kc1 (2) W e firdl Ct\ l ( k ) t , l rY ) ~ [ , ( R )
-.=
1 ( c i k R / k i , ) f ( o ) I I I R )P t'l r<
f ( O ' = ~ t l ~ l ( C O S 8 ) ( 1 l)c1-, + l < ] - ] ( 5 )
I
Similarly, for m = *I, one has C L Y L ( ~ ) ~ ~ ( G ~ , ) ~ ~ ( S )
+ ~i;,b)(R))
= m ( e i k R / k ~ ) sin6R cosgl' f ( ' ' ( 6 , R ) , wheref (I)(@, R ) = x ( 2 1 + ~)tlP~(cosB)icl(kR)/kR.
I
( 6 ) Here PIm(%) is the associated Legendre polynomial of order 1 and degree m, 6' is the scat- tering angle,
4'
is the dihedral angle betweenk
and i through R, and el are the spherical correction factors of the plane wave limit of the Hankel functions hl(x) = (-i)'(ezz/x)cr(x).To approximate the double scattering term we use a separable Green's function, G L , ~ , ( $ )
=
4 r ( e z k R / k ~ ) ~ t ( ~ ) Y L ' ( R ~ ~ ~ ( ~ R ) C , I ( k ~ ) . Hence we finally obtain
where j(0, R ) = f to)(@, R)
+
tan 6 ' ~ cos 4'f(l)(8, R), fefi(6') = C1(21+ l)t~P~(cosB)el(kR) and fLff(6') = CltlP~(cos8)cl(kR)[(l+
l ) ~ ~ + ~+
Icl-I]. As an additional approximation, designed for computational speed and accurate to within a few eV above threshold, we introduce the asymp- totic formula for the spherical correction factors of the plane wave limit of the Hankel functions[3]cl(z) FZ
dl +
Z(l+
1)/2x2 exp(il(1+
1)/2x). In the same way a similar expression can be workedwhere
4. Numerical results
In figures 1 and 2 we present a comparison between the exact spherical wave result[6], and expressions using f , f (O) and for
x
= I d c l d f l - ( d ~ / d n ) ~ j / ( d o / d R ) ~ . We use Ni atomic phase shifts with a nearest neighbor distance R = 2.49A. We find good agreement between the expression using f (O) and the exact treatment at energies as low as 100eV. The term proportionalto
I ( ' )
is important at observation angles awayfrom the forward and backward directions when theenergy is low. The double scattering term is nonnegligible in geometries in which the photoelectron is backscattered.
Solid line exact, dotted line f ( O !
+
tan Orin d'f"',
,y dash-dot f ( O ) , dashed line plane wave.1 Figure 1: E=lOOeV,Bt = O0 F i ~ u r e 2: E=lOOe\'.Bc = 4 j 0
5. Generalized optical theorem and XAFS reduction
It is of interest to see how the XAFS formula can be obtained when d u / d n is integrated over all angles. The previous reduction, based on the plane wave approximation and- the optical theorem(l1, must be modified when spherical wave corrections are considered.The details of this derivation will be presented elsewhereI71. Integrating over all angles to get p we have p oc
d f l J M z 1 2
.
Evaluation of this expression yieldsHere ]*lo) = 1/2i(l+&) -
IG,)).
Since t i = ei61 sin6,, Imtl = Itr/' and the last two terms cancel.In the plane wave limit this is just a statement of the optical theorem[l], a = Imf ( O ) / 4 ~ k
.
In theC8-216 JOURNAL DE PHYSIQUE
present case this cancelation occurs 1 by 1, though there is not a simple optical theorem for j ( 8 : I?).
Using the expansion for the propagator given in Eq.(4): one recovers the XAFS expression[8].
References 'l]P.A. Lee. Phys.Rev.B13,5261(1976).
'21 G. Breit and H. Bethe, Phys. Rer. 93,888(1954).
;3] J.J. Rehr. R.C. Albers, C.R. Natoli and E.A. Stern. Physical Review B, to be published 1986.
141 M.Sagurton, E.L. Bullock, R. Saiki. A. Kaduwela. C.R. Brundle, C.S. Fadley and J.J. Rehr.
Phys. Rev. 33.2207(1986).
'51 Quantum Mechanics A. Messiah, Vol.2, p.1068 (Wiley. N.Y. ,1963).
'6! C.S. Fadley in Progress in Surface Sczence, S. Davison, (ed.). Vo1.16 p.275 (Pergamon, N.Y..1984).
171 J.J. Rehr and J. Mustre de Leon University of Washington preprint (1986) 181 J.J. Barton and D.A. Shirley, Phys. Rev.B32,1892(1985).
