AD-AO91 748 ROCHESTER UNIV NY DEPT OF CHEMISTRY Fig 7/4HIGH ;ENERGY POSITRON I ON IZATION OF ADSORBED SPECIES IN THE IMPU--ETC(U)OCT 80 K LAM, T F GEORGE 00O1I4_80!C-0'472

U N CLAASS I FIn T R-7 NL

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TECHNICAL REPORT NO. 7

High-Energy Positron Ionization of Adsorbed

Species in the Impulse Approximation

by

Kai-Shue Lam and Thomas F. GeorgeinII

The Journal of Physical Chemistry <2 >

University of RochesterDepartment of ChemistryRochester, New York 14627 V

October, 1980

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High-Energ Positron Ionization of Adsor d .1cesi the Impulse Approximation I Intr r

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A~UTNOR(s) I. C , Nu IBE --

Kai-Shue La * Thomas eorge

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Prepared for publication in The Journal of Physical Chemistry, in press (1981)

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AB 1STRACT (Continue en reverse side If necessary end identify by block tumahbe)

one-electron group-orbital linear-chain model and within the impulse ap-proximation.. Inter-orbital interference effects are found to be significant,

* leading to quenching (towards the low impact energy regime) and enchance-ment (at high impact energies) of ionization compared to positron gas-phaseionization.

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J. Phys. Chem., in press.

High-Energy Positron Ionization of AdsorbedSpecies in the ImpuIse Approximation

Kai-Shue Lam and Thomas F. George

Department of ChemistryUniversity of RochesterRochester, New York 14627

Abstract. High-Energy Positron Ionization of adsorbed speciesis treated using the one-electron group-orbital linear-chainmodel and within the impulse approximation. Inter-orbitalinterference effects are found to be significant, leading toquenching (towards the low impact energy regime) and enhance-ment (at high impact energies) of ionization compared to

positron gas-phase ionization.

s.5' V1 tp0 S:N

I1

I. Introduction

Recently, positron sources (available beam energy up to -400

eV) have come to be of value in the diffraction studies of solid1

surfaces. These studies complement the traditional low-energy

electron diffraction (LEED) ones and point to new directions in the

use of positrons as excitation or ionizing agents in the study of

chemisorption bond-characters. Unlike electrons, positrons lead

to no exchange effects during collision and thus allow for simplifi-

cations in theoretical analyses, provided the effects of positronium

formation can be neglected (which should be a good approximation for

beam energies of several hundred eV). Furthermore, in ionizing

situations, the distinguishability of the product particles (positrons

and electrons) also facilitates the measurement of ionization cross-

sections.

Here we report on a theoretical treatment of high-energy positron

ionization of adsorbed species based on the one-electron group-orbital

linear-chain (OEGOLC) model of the adatom-surface system (with only

2,3nearest-neighbor interactions), and the impulse approximation, 4

(IA) for the description of the collision process. Indirect effects

of the solid substrate on the ionization process, such as electron-

and positron-phonon interactions, positron diffusion within the solid,

positronium formation and positron annihilation, etc. will be ignored.

Such effects are either expected to be weak compared to direct posi-

tron-electron interaction, or are negligible under the conditions of

the high impact energies considered here (50-400 eV). Our main

purpose is to investigate the effects of orbital mixing in the chemi-

sorption bond on ionization. Hence we will focus on systems with

Mao i

2

a localized adatom-surface (chemisorption) bond (corresponding to

a localized state with energy lying outside a band). In this case,

if the surface coverage is large enough, the ejected electrons will

be mainly from near the surface, i.e., from the localized orbitals,

and these electrons would contribute predominantly to the ionization

cross-section compared to the "secondary" non-localized electrons.

In our model, the positron is accordingly assumed to interact only

with electrons in the localized bound state. The OEGOLC model,

though somewhat qualitative, is the simplest one capable of producing

a localized state. The IA, on the other hand, has been applied

with considerable success to complex collision events such as high-

energy gas-phase collisional ionization.5 Our calculations indicate

that inter-orbital interference effects in the group orbital picture

are significant in the ionization process: there is a slight quenching

towards the low energy regime (<90 eV) and considerable enhancement

at high energies, compared to gas-phase ionization.

II. Theory

(a) The ionization cross-section

The physical process considered here is the collision between

an incident positron (with beam energy considerably in excess of any

bound-electron energies) and an adatom, and the subsequent single-

ionization of the adatom. The IA assumes that, during collision,

the interaction between the bound electrons of the adatom-surface

system is "turned off". This implies that the incident positron

can only spend a very short time in the field of the bound electrons.

The electron interacting directly with the positron is "momentarily"

3

Ifree and has a momentum (k) distribution given by the Fourier

transform W of the one-electron chemisorption wave function.

The positron-electron system is then described by a positive-

energy Coulomb wave function6 weighted by V (0). If r is the

positron coordinate, and r2 and r3 are coordinates of the elec-

trons in the localized state, the positron-electron system wave

function in the IA is thus given by (in what follows, all quanti-

ties will be expressed in terms of atomic units)

T.l r,0 2' % 3 r ' 3)AT(P( r )fd k03 ) ~ilt 64* 1 r3 ) + q,0 ( 3 )fd k 2 0 ( 2 ) til (rl 1 1r 2 )}.

13 kI 2 ()

In (1), k is the wave vector of the incident positron, %0(r) is

the one-electron chemisorption wave function of the localized state,

(k) is its Fourier transform, referred to above, and C "i, r,

is the Coulomb wave function for a positron-electron pair given

by (on factoring into CM and relative coordinates)

' W ' x2 1 1 } (r-r (2)

2 l' 2 (27) 3 1

where- ew/2kr (1- l )e rFL;1;ikr- -r) .(3)

?'(r) is the wave function in the continuous spectrum for an attrac-

tive Coulomb potential; 1F1 is the confluent hypergeometric function.

The final states of the positron and ejected electron can be

approximated by plane waves, and the final-state wave function

for the positron-electron system can be written as

4

uf(rlr 2 ,r3)1 1 1 )3 {e i 0(r3) + e io(r 2 1}, (4)

where and are the wave vectors for the scattered positron

and emitted electron, respectively. The single-ionization scat-

tering amplitude f is then given by

f /2- -7r(T 2 T3 ) (5)

where Ti, the transition matrix elements, can be expressed asd3r 3 3 T (6)T I dr 2d r l3u(r 2 , 3) (_ _23 r rlr 2,r3);= rli

and r = ri - rj- . Using (1) and (4) in t6) and expanding, we

obtain a sum of eight terms for f, each of which represents different

initial and final configurations for the positron-electron system

coupled by a distinct /rij Coulomb force. Of these, in the Born

approximation, four vanish and four are equal to one another, say, T.

Based on this observation, we introduce the approximation that

T2 + T3 = 4T (7)

where T is still calculated within the IA:

r-ikj- r-ik'~ 43*T d 3_rdi3r) 1 1i.r-i2 2 i__- fdk 2 ( 2) (r IJr 2)"12 1 l d 3 re 122121, 2 )

Using (2) in (8), we obtain (8)

T2 3 e

--ei ()

S (2d) r

7The integral in (9) has been evaluated by several authors, and

gives4 - '! -- k'-

T(2 + Tp3=2_ 0(k1+k2 1)exp (1T/ I)

r(1-2-/iI 2 1 l .-j.i

r (1/ 2- {ei (Z1- 1-) (10)2 wV.

5

The wave vectors are required to satisfy the condition ofenergy conservation: of2 = k2 - 21 - k2 + k2, where -lei

is the bound energy of the localized electron. The total single-

ionization cross section a is then given by

d = dI I f1 2 )kl k<c 2

In (11), dQj designates the directions of the scattered positron.

Only "backward-scattered" directions of t need be considered since

these are the only observable ones. The scattering anplitude used

in (11) is then given by

2 2 (21r) 21 ; o +-1! 2exp-2"/I l )1f(li1 k) 2t Ez(1t.-! 2)4si2) €",1il-t{-tiI) (12)

(b) The chemisorption wave function

The chemisorption wave function in the OEGOLC model can be

written, using the LCAO approximation, asN-I.

4- N-1 A

VW = Cxoa(r) + E C m(r+ (+m)az), (13)MWO

A

where z is in the direction of the chain (outward from the surface),

a is the inter-atomic distance in the linear chain, Xa is the dis-

tance between the adatom and the first substrate atom, N+1 is the

total number of substrate atoms in the chain, and r is measured

from the nucleus of the adatom. We assume that only two types of

orbitals, c0a (corresponding to the free adatom) and * (corresponding

to individual substrate atoms) enter into 0" [This assumption

should be reasonable for most alkalis and some transition metals.)

The coefficients CX and Cm depend on the following parameters:

6

ea (bound energy corresponding to ca) , (bound energy correspond-

ing to 0), 8' (interaction between the adatom and the first substrate

atom) and 8 (nearest-neighbor interaction between the substrate

atoms) .

When (13) is used in the Schr8dinger equation with a one-

electron Hamiltonian, the vanishing of the secular determinant for

the coefficients CI and Cm leads to the following result. A

localized state (* damped in the crystal) exists if the equation

o- C a 82' 2(cose + sin8 cotNe)( + 2cose) - (.-) (14)

has at least one complex solution for e, (e- i& or w+i , E>O and

real), with energy given by e = a ± 28 cosh&. if 12 in (12) is then

given by1 12 2;2+ -2 2

X a + 2[-C2+2 Z C ,cmcos(akz(m,-m)}]MI>m m

(15)

+ 2C XaZCmcoS((X+m)at.z},m

where a and * are the Fourier transforms of Oa and 0, respectively,

and both are taken to be real. The secular equations for Cx and

Cm also lead to

cx'/±W C (16a)CA"{(a-E a )/ 8 -+2 cosh 0a

and Cm sinh{(N-m)&!C (16b)M sinh N& 0

For a localized (damped wave function, the series in (15)

terminates quickly. (15) displays clearly the inter-orbital inter-

ference effects.

,r-I

7

III. Results and Discussion

Calculations have been carried out for the H(adatom)/Li(sub-

strate) system with normally incident positrons. 8 and 8' are

estimated by the Wolfsberg-Helmholz formula, and the overlap

integrals required therein are obtained from tabulations from

Mulliken et al. Oa and 0 are taken to be the H ls and Li 2s

Slater AO's, respectively. The following parameters are then

obtained (in a.u.): 8 = -0.17, 8' = -0.35, e = -0.76. Table 1

gives the normalized coefficients Cx and Cm for N = 100 computed

using (16). To the accuracy stated, these results are identical

to those for N = 14. The energy gap between the localized state

and the bottom edge of the 2s band is 0.22 a.u.

The integrations in (11) are carried out numerically using

the 6- and 16-point Gaussian quadratures, and the ionization -cross

sections are compared with those for the positron ionization of the

H atom, also computed using the IA (Fig. 1). In the latter case,

only a simple orbital enters into (12), and inter-orbital inter-

ference effects are absent. [For purposes of comparison, a for

the H atom case only include "backward" ionization directions.]

Our results indicate that at high impact energies there is a dis-

tinct enhancement in ionization due to the interferencu ,ffects

of the group orbital, while at low energies these effects tend to

quench the ionization. The peak cross section, however, occurs at

a higher energy for the adatom case than for the gas-phase case.

Even though the low energy results are less reliable due to the

doubtful validity of the IA in this regime, the increasing prob-

ability of positronium formation, and hence positron annihilation,

.........

8IT

also tend to decrease the ionization cross section. Hence the

overall trend is expected to be quenching in the low-energy

regime and enhancement in the high energy regime. The enhance-

ment may be attributed to the fact that, at high energies, the

positron probes deeper into the crystal.

TABLE 1. Normalized coefficients for the localized H/Li chemi-

sorption wavefunction with bound energy e = -0.76 a.u.

m m-X 0 1 2 3 4 5 6 7 8

c 0.8005 0.5637 0.1917 0.0648 0.0220 0.0073 0.0025 0.0009 0.0003 0

.1m

9

Acknowledgments

This work was supported in part by the Office of Naval Research,and by the Air Force Office of Scientific Research (AFSC), UnitedStates Air Force, under Contract No. F49620-78-C-0005. The UnitedStates Government is authorized to reproduce and distribute reprintsfor governmental purposes notwithstanding any copyright notationhereon. One of us (TFG) acknowledges the Alfred P. Sloan Founda-tion for a Research Fellowship and the Camille and Henry DreyfusFoundation for a Teacher-Scholar Grant.

1. I. J. Rosenberg, A. H. Weiss, and K. F. Canter, Phys. Rev. Lett.

44, 1139 '1980), and references cited therein.

2. T. B. Grimley, Proc. Phys. Soc. (London) 72, 103 (1958).

3. J. P. Muscat and D. M. Newns, Prog. Surf. Sci. 9, 1 (1978).

4. For an application of the IA to gas-phase electron-atom scatter-ing, see, for example, R. Akerib and S. Borowitz, Phys. Rev. 122,1177 (1961).

5. H. Nakamura, T. Shirai, and Y. Nakai, Phys. Rev. A 17, 1892(1978).

6. See, for example, N. F. Mott and H. S. W. Massey, "The Theoryof Atomic Collisions," 3rd ed. (Oxford, London, 1965), p. 56.

7. See, for example, A. Nordsieck, Phys. Rev. 93, 785 (1954).

8. M. Wolfsberg and L. Helmholz, J. Chem. Phys. 20, 837 (1952).

9. R. S. Mulliken, C. A. Rieke, D. Orloff, and H. Orloff, J. Chem.Phys. 17, 1248 (1949).

Fig. 1. Positron single-ionization cross sections (a) for H/Liand H atom as a function of incident positrn energy (E).a 0 = 1 Bohr. , H/Li; ------, H atom. Ii both casesonly "backward" ionization directions are inc~tded.

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