MANY-BODY THEORY OF MULTIPLE CORE HOLE

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MANY-BODY THEORY OF MULTIPLE CORE HOLE

M. Ohno

To cite this version:

M. Ohno. MANY-BODY THEORY OF MULTIPLE CORE HOLE. Journal de Physique Colloques,

1987, 48 (C9), pp.C9-505-C9-508. �10.1051/jphyscol:1987982�. �jpa-00227403�

MANY-BODY THEORY OF MULTIPLE CORE HOLE

Institute of Theoretical Physics, Chalmers University of Technology, S-412 96 Gtiteborg, Sweden

La t h e o r i e a N corps du spectre de t r o u s m u l t i p l e s e s t formulee p a r l a methode des f o n c t i o n s de Green. Le spectre d'emission Auger e s t egalement formule en termes des f o n c t i o n s s p e c t r a l e s d'un simple t r o u i n i t i a l e t d'un double t r o u f i n a l . Une approache systematique u t i l i s a n t l ' o p e r a t e u r de masse e t l e s c o r r e c t i o n s au vertex e s t discutee.

Many-body theory o f m u l t i p l e core hole spectrum i s formulated u s i n g t h e Green's f u n c t i o n method. The Auger emission spectrum

i s a l s o formulated i n terms o f the spectral f u n c t i o n s o f t h e i n i t i a l s i n g l e h o l e and f i n a l double hole. A systematic approach u s i n g t h e self-energy and v e r t e x c o r r e c t i o n s i s discussed.

M u l t i p l e e l e c t r o n i c vacancies are o f t e n created i n a gas o f atoms o r molecules o r a s o l i d surface by m u l t i p l e i o n i z a t i o n o r secondary emission processes where a c o r e h o l e cascades up towards t h e Fermi l e v e l and causes f u r t h e r e j e c t i o n s o f e l e c t r o n s and photons. The r e s u l t i n g e l e c t r o n and photon spectra c a r r y i n f o r m a t i o n about the response o f the system t o t h e holes, e.g. r e l a x a t i o n s h i f t s o f t h e m u l t i p l e - h o l e l e v e l s , screening o f t h e hole- h o l e C~ulomb in t e r a c t i o n s and atom-solid energy s h i f t .

One o f t h e systematic approaches t o describe t h e m u l t i p l e h o l e spectrum i s t o use a s p e c t r a l f u n c t i o n and a self-energy o f m u l t i p l e h o l e . The center o f g r a v i t y o f t h e double h o l e spectrum i s given by t h e imaginary p a r t o f t h e double h o l e Green's f u n c t i o n . When o n l y the diagonal p a r t s o f t h e i r r e d u c i b l e diagrams a r e taken i n t o account w i t h i n t h e Hartree-Fock approximation, t h e double h o l e Green's f u n c t i o n i s given by [1,2]

Here Exand €?are t h e unperturbed (Koopmans s i n g l e h o l e energy o f t h e h o l e X and Y r e s p e c i i v e l y , ~ O ( x , y ) i s t h e bare Coulomb hole-hole r e p u l s i o n i n t e r a c t i o n ,

A

xy i s t h e screening c o r r e c t i o n f o r t h e Coulomb r e p u l s i o n i n t e r a c t i o n and&,qis t h e self-energy which can be i n general catagorized i n t o t h r e e p a r t s . namely t h e ground-state c o r r e l ation(6SC). non-hole hopping ( m a i n l y monopole) relaxation(M1 and hole-hopping (dominately d i p o 1 e ) r e l a x a t i o n ( D l o f t h e double h o l e [I]. The non-diagonal p a r t s o f t h e diagrams can be i n c l u d e d l a t e r by u s i n g t h e p e r t u r b a t i o n theory.

The double h o l e Green's f u n c t i o n can become much simpler when we use the t h e 1 in e a r response approximation f o r the mu1 t i p l e h o l e energies,e.g. one n e g l e c t s t h e e f f e c t s on t h e c o r r e l a t i o n and r e l a x a t i o n due t o presence o f an e x t r a h o l e . I n t h i s case t h e self-energy becomes a l i n e a r sum o f t h a t o f a s i n g l e hole. The w i d t h and energy ( c e n t e r o f g r a v i t y ) o f the double h o l e spectrum i s given by

("present address : Physikalisches Institut der Universitilt Bonn. Nussallee 12, D-5300 Bonn 1, F.R.G.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987982

JOURNAL DE PHYSIQUE

H e r e & a n d d y a r e t h e c o r r e l a t i o n and r e l a x a t i o n energy s h i f t o f a s i n g l e h o l e X and Y r e s p e c t i v e l y . When t h e holes are created on t h e same atomic sub-shell , eqs ( 2 ) and ( 3 become

Here E;(ASCF) i s t h e Hartree-FockASCF core h o l e energy and Ex i s t h e ' e x a c t ' s i n g l e h o l e energy. Eq(6) gives a f a i r l y good agreeement w i t h the4SCF r e s u l t s when t h e number o f holes i s l e s s than f o u r [ 4 ] . A s i m i l a r formula was d e r i v e d by u s i n g t h e argument t h a t t h e r e l a x a t i o n c o n t r i b u t i o n f o r a s t a t e w i t h two holes i n t h e s h e l l i should be f o u r times as l a r g e as t h a t f o r a s i n g l e h o l e i n t h a t s h e l l [ 3 ] . When non-hole hopping r e l a x a t i o n i s n e g l i g i b l e , both r e s u l t s are same.It was shown t h a t t h e change i n the e f f e c t i v e Coulomb i n t e r a c t i o n ( t h e Auger parameter) i s equal t o t h e two times o f the extra-atomic r e l a x a t i o n s h i f t ; t h e change i n t h e r e l a x a t i o n energy s h i f t due t o a change o f enviroment

[4]. T h i s i s a l s o v a l i d o n l y when t h e h o l e hopping c o r r e l a t i o n i s n e g l i g i b l e . When one o r both of t h e double holes are s t r o n g l y i n t e r a c t i n g , e . g . hole- hopping c o r r e l a t i o n becomes important, t h e l i n e a r approximation breaks down.

It i s i m p o r t a n t t o t a k e i n t o account the e f f e c t s o f presence o f an e x t r a hole.

The Auger spectrum can be o f t e n w e l l described as a number o f superimposed s p e c t r a l f u n c t i o n s o f t h e i n i t i a l s i n g l e hole, centered a t t h e r e s p e c t i v e f i x e d Auger energies and weighted by e s s e n t i a l l y the r a t i o o f t h e p a r t i a l t o t h e t o t a l Auger decay w i d t h [ 5 ] . This i s under assumption t h a t t h e two f i n a l s t a t e h o l e s a r e sharp. However, t h e Auger spectrum can be i n f l u e n c e d by t h e dynamics o f t h e f i n a l s t a t e double hole. It i s necessary t o extend t h e f o r m u l a t i o n t o a more general case. The formalism i s based on the idea t h a t t h e Auger e l e c t r o n s e m i t t e d i n an XPS experiment should be t r e a t e d i n a u n i f i e d manner together w i t h t h e photoelectrons as a one-step process, w i t h o u t d i s t i n g u i s h i n g between primary p h o t o i o n i z a t i o n and secondary Auger emission[5]. The Auger emission spectrum can be described i n terms o f t h e s p e c t r a l f u n c t i o n s o f t h e i n i t i a l s i n g l e and f i n a l double h o l e which a r e g i v e n by t h e imaginary par'ts o f t h e corresponcijng Green's f u n c t i o n s [ 6 ] . The spectrum f o r t h e Auger emission process i

+

^j-' k"' + EA i s given by

where t h e dresssed Coulomb i n t e r a c t i o n m a t r i x element

ci

i s obtained from

d r i ( ~ )

s t h e dressed p h o t o i o n i z a t i o n m a t r i x element v h i ch i s given by an i n t e g r a l equation o f t h e same type as eq. (10).

&

i s t h e k i n e t i c energy o f t h e photoelectron. i s t h e photon energy.

I . M ; ~ R

C E ) I ~ / L ~ E ~ [s!

i s t h e branching r a t i o o f t h e p a r t i a l l i n e - w ~ d t h t o t h e t o t a l l i n e w i d t h . Note t h a t dressing o f t h e i n i t i a l core h o l e

i n t r o d u c e s a complex energy and determines the t o t a l l i n e w i d t h through

Imzi

which appears i n t h e denominator o f t h e branchning r a t i o i n eq.

( 8 ) . The dressed Auger emission m a t r i x e l e m e n t s l ~ ; ~ ~ ( ~ j ? o n t h e o t h e r hand.

g i v e s r i s e t o a p a r t i a l l i n e widths and appear i n t h e numerator o f t h e branching r a t i o . The same dynamical processes t h a t dress up t h e core h o l e through t h e self-energy a l s o modify the emission v e r t i c e s

F a i l u r e t o renormalize t h e emission vertex means t h a t t h e p a r t i a l l i n e w i d t h s do n o t sum up t o t h e t o t a l l i n e width a t a given core h o l e energy E. This leads t o nonconservation o f t h e t o t a l emission p r o b a b i l i t y , i . e . t h e sum o f branching r a t i o s i s n o t equal t o one. The emission problem r e q u i r e s systematic i n c l u s i o n o f self-energy and vertex c o r r e c t i o n s . The self-energy describes t h e r e l a x a t i o n and decay processes and determines t h e m o d i f i c a t i o n o f t h e energy l e v e l spectrum o f t h e core hole. The vertex f u n c t i o n i n v o l v e s e s s e n t i a l l y t h e same components as t h e self-energy b u t describes t h e m o d i f i c a t i o n o f t h e wave f u n c t i o n o f t h e core hole[6].

When t h e i n i t i a l h o l e i s w e l l defined and t h e f i n a l s t a t e double h o l e a r e dressed, t h e Auger spectrum w i l l be influenced very much by t h e dynamics o f t h e f i n a l s t a t e , i . e . l o c a l i z a t i o n , d e l o c a l i z a t i o n , Coster- K r o n i g decay and f l u c t u a t i o n , e t c . [ 7 ] .However, t h e spectrum can be s t i l l analyzed i n terms o f t h e s p e c t r a l f u n c t i o n s o f t h e i n i t i a l and f i n a l s t a t e s . When t h e Auger emission processes i n v o l v e a s t r o n g l y dressed i n i t i a l core hole, t h e emission m a t r i x element w i l l be screened by p o l a r i z a t i o n o f t h e charge density.Coster-Kronig emission p r o b a b l i t i e s are p a r t i c u l l a r l y s e n s i t i v e t o super Coster-Kronig f l u c t u a t i o n and decay processes i n t h e i n i t i a l h o l e s t a t e . This r e l a x a t i o n leads t o prominent s a t e l l i t e s t r u c t u r e and a l s o t o l a r g e e l e c t r o n i c rearangement i n t h e i n i t i a l i o n i c ground s t a t e . I n p a r t i c u l a r , due t o r e l a x a t i o n , t h e emission i n t e n s i t y w i l l be reduced i n t h e i n i t i a l i o n i c ground s t a t e and enhanced i n t h e i n i t i a l i o n i c e x c i t e d s t a t e s [ 6 ] . This can be s t u d i e d as an i n t e r f e r e n c e e f f e c t between d i r e c t and induced emission. The emission m a t r i x elements

E/lijk (E)

can be w r i t t e n as

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M i j k (E) = Mijc F i j k (E)

where

i s a v e r t e x c o r r e c t i o n which e x p l i c i t l y expresses t h e energy dependence due t o t h e screening process and i s a response f u n c t i o n which describes t h e screening o f t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n i n Auger emission.

FijkfE)describes t h e i n t e r f e r e n c e between t h e d i r e c t and i n d i r e c t emission processes and shows resonant behavior i n t h e r e g i o n o f d i s c r e t e s a t e l l i t e s and continuum threshold[6].

The p r e s e n t approach f o r t h e Auger spectrum can be a p p l i e d e a s i l y t o any k i n d o f multi-channel r e l a x a t i o n and decay processes, e.g. a case where a d i s c r e t e monopole i n n e r - s h e l l shake-up l i n e i s degeneratewith an o u t e r - s h e l l shake-off continuum. The presence o f t h e d i s c r e t e e x c i t a t i o n

(resonance) i n t h e shake-off m a t r i x element guarantees t h e possi b i 1 i t y o f asymmetric shake-up l i n e s , s i m i l a r t o t h e Fano l i n e p r o f i l e s i n e.g.

p h o t o i o n i z a t i o n .

Acknowledgement T h i s work i s supported by t h e Swedish Natural Science Research Counci 1

.

References

[I] G.Wendin and M.Ohno, Phys.Scr. 14,148(1976) [ 2 ] L. S .Cederbaum, Phys. Rev A35,622( 1987) [3] O.A. S h i r e l y . Phys.Rev A7,1520(1973)

(41 M.Ohno and G.Wendin, J.Phys.C: S o l i d S t a t e Phys. , l 5 , 1787(1982) [5] M.Ohno and G.Wendin, J.Phys.6: Atom Mol.Phys.12,1305(1979) and Phys.

Rev. A31. 2318( 1985) [6] M.Ohno and G.Wendin, Z.Phys.0 5,233 (1987)

[7] M.Ohno and G.Wendin, S o l i d S t a t e Commun. 39,875(1981)