Influence of Dynamical Effects on Spectral Properties of Auger Electrons Emitted During Ion-Surface Scattering

INFLUENCE OF DYNAMICAL EFFECTS ON SPECTRAL PROPERTIES OF AUGER ELECTRONS EMITTED DURING ION-SURFACE SCATTERING

Zorari MiSkovic

Bor-Ls Kf dric Institute of" Nucte ar- Sciences P.O.B. 522, 11001 Belgr-ade , Yugoslavia

Kinetic e rie r-g y spectra of" e Iect.r-orrs emitted f r-om the Iorr-s ur-: f ace

collision pr-oces ses

contain impor-tant inf"or-mation about the

sur-face electr-on density of" states [1J. In the

pr-esent

wor-k

the

quantum mechanical t.be.or-y of" two-electron Auger- pr-ocesses is for-mulated [21 in or-del' to calculate the energy d i s t.r-Lbtrt.Lorrs of Augel' electr-ons and, p.art.i crrl.ar-Iy , to investigate the influence of collisional dynamics on such dfs't.r-Ibtrt.Iorrs. Using the eigen state expansion method of Tully [31, the coupled equations f or- amplitudes of" r-e Iev.arrt. states can be reduced to an integl'able f'or-m, 'We assume the single channel coupling and neglect all

collective

excitations

of

the

snrhst.rat.e.

The

projectile

ion

motion is appr-oximated by linear- t.r-aje-ct.or-y and we limit ourselves to the case of slow collisions (ion velocity being less than about 5 10 m/s) in or-de-r- to exclude kinetic electr-on emission. All one-electl'on t.r-arisd t.Lorus ar-e assumed to be ener-getically f or-bf dde ri, 'We consider the two-electron Augep transition leading to neut-

r-attz.at.Ion of the incident positive ion into the gr-ound state atom with the Auger electron being emitted into vacuum. 'Within the independent particle model, the initial states of the two active electr-oris ar-e described by two-electron state vect.or-s

Ij,k;t> (being the

usual antisymmetl'ized pr-odrrot.s of the cor-r-e sporsdf ng one-electron state vectors with eigen enepgies all

the

I': j

and

10

metallic one-electron states. The

electr-ons

ar'e

descr-ibed

by

the

state

k

) , Indices j and k span final states

vector-s

descr-Ibes fpee orre-r e lec t.r-orr states with errer-g y

If,a;t>,

of

active

wher-e

f

and a descr-ibes the f orre-r e Ie-ct.r-orr ground state of the atom with enel'gy I': Ct.), The total a IO

two-electl'ol' Hamiltonian is 1l(1,2;t)

=

11<1;0 + 11<2;0 + V<1,2)

wl re re one-electron Hamiltonian

(1)

IHi;t),

(i=1,2),

de scr-Ibe s

both

the

metal s urf ace and the atom COI'e in f r-orrt of it, while V(1,2) is domi n.at.e d by the Coulomb repulsion. Time dependence comes from the

motion of the pr-o jec t.i Ie ion, and the

image

interaction.

VIe

t~

a assume

(t) is shifted trpw ar-ds that

the

above

due to

intr-oduced

t.wo-ie lect.r-on basis set is pr-op er-Iy or-t.borror-malf ze-d and sufficient to de s cri be all Impor-t.arrt states.

Total

vector- can be expanded t(<:: c If,a;t> expl-U +<::f)dT] + fa a t + Ej,k c jk ( 0 'j,k;O exp[-U (£ j +£k )dT!

Er

11,2;0 =

Cat.omi c

two-electr-on state

as

follows

«»

units are

used).

Evolution of

11,2;t>

(2)

is gover-ned

by

time-dependent Scbr-odf ng er- equation with the Hanriltonian

By

(D.

pr-ojection onto the final and initial states one obtains t i ,\cfa(O = Ej,k Vfa,jk(O cjk(O eXP[ft(<::a+£f-£j-£k)dTl,

(3a)

Er

,\ c

the

(0 = Vjk,fa (0 cfa (0 exp[-if (lOa+£f-£ {£k)dTl, (3b) jk wher-e V' f (t)=
ons in

*

t

=-m as follows: c Ct. )=0 and OS means averaging over the ground state j, k'k of the metal and n(£) is the Fer-nu-Dtr-ac dtsrt.r-Ibtrt.Iors. In or-der- to solve Eqs.(3), we introduce the usual assumption of the time s epar-ab.i Ht.y of the where ur t, )=exp(-011t

I),

matrix

elenmt: V ik f' (t)=V' f . u(t), J , a Jk , a OI:=a' v (with v being the normal component of

collision velocity). St.arid ar-d pr-ocedur-e [2,3] reduces Eqs. (3) to an integl'able f'or-m, The final (t=+m) dts t.r-tbut.Iorr of f'r-e-e e.Ict.r-oris excited dur-Irig the

*

Augel'

rretrt.r alfz.at.Iorr

collision, given

=
-,J

+00

of

by

n

t

=

convolution

n = dz D Bf(d (4) f f(£) Her-e. D is the Auger- t.r-arrsf'or-m [1] of the e lect.r-orr populated f(£) p ar-t.s of metallic sarr-face bands having densities of states P and "z

t

D~(£)=rrfd",

• He-r-e

<0>

2 « IV"k ~ I ». . (5) J ,. a £+1.-',£-1.-' means double avepaging ove r- the degenel'ate band states n(£+",) n(£-I.-') P

i(£+"')

P

2

( £ - I.-')

having enel'gies £ j=£+1.-' and £k =£-1.-'. Function B de.scr-Ibe-s the inI(£) f ltrerrce 01 the dynamical ef'f'ect.s due to ion motion and is given by -1 +00 +00 2 = rr I_oof dt u(t) exp{-tJ dT[r + H£a+£1-2£}nl ' (6) B I(£} l/2 wheT'e r I(T) is time-dependent AugeT' t.r-ansf t.Ion r-at.e [21. Let 11 1(£) be the Augel' t.rarisf'or-m 01 the bands Pi and P (including both 2

populated and empty par-t.s 01 the bands), which is given by the RHS 01 Eq.(S) with n=1. II at least one 01 the bands has the width 'W

»

»

ex (which condition is cer-t.atrrly lullIilled 101' slow collisions), 2 then r I(t) can be appr-oxtmat.ed by r 1(t)~I{[£ a (t)+£1J/2)' u (0. By inspection 01 the time integr'al in Eq.(6) one can conclude

[21 that, in the low velocity limit (CX«I1;), the ellective corrt.r-Ibtr1In(I1;/CX), tion to the integl'al comes II'0m the instant t*=cxwhel'e 11;=11 a «, *)+£1J/2). II the ellective image shilt 01 the atomic 1{[£ gr-otrrid state level in the moment t* is V~«'W, one can evaluate the lunction B in the f'or-m of" Gaussian dfs t.r-Ibtrt.Iorr I(£) * 2 * 2 -1/2 * 2 2 * 2 BI(£}~(4/el1l)[4cx +(F I/a) 1 exp{-(£I-2£-I+V I) /[4ex +(F I/a) n.

(7)

Her-e I is the ionization potential 01 the atom f ar- away II'0m sUl'lace and F; is the effective image f'or-ce in the moment t=t*. In conclusion, the distl'ibution 01 excited Augel' electl'ons n

f is given by the Augel' t.r-ansf'or-m (5) which is convoluted with the Gaussian distl'ibution (7) whose:

n

peak position depends on colli-

sion velocity v t.br-o ugb

V~ (incl'eases with v), 2) width depends on v

F~ (both

Incr-e.asf ng with v), and 3) height depends on

t.hr-orrg h cx and

v t.hr-ough 11; (which is expected to be a weak function 01 v) [21. REFERENCES: [1)

Hag st.r-um H.D. in: Elect.r-ori and Ion Spe ct.r-oeoopy 01 Solids, Eds. L.Fiel'mans et al. (Plenum, N.Y. 1978);

[21 Miskovic Z.L. and Janev R.K.: SUI'lace Sci. 166 (1986) 480; [31 Tully J.C.: Phys. Rev. B16 (1977) 4324.