Perturbative treatment of quasi-resonant proton neutralization at alkali-halide surfaces

Surface Science 161 (1985) 163-180 North-Holland, Amsterdam

163

PERTURBATIVE TREATMENT OF Q U A S I - R E S O N A N T P R O T O N NEUTRALIZATION AT A L K A L I - H A L I D E S U R F A C E S Franco BATTAGLIA * Department of Chemistry, Umverstty of Rochester, Rochester, New York 14627, USA T h o m a s F. G E O R G E Departments of Chemtstry and Physws, State Umverslty of New York at Buffalo, Buffalo, New York 14260, USA and Armando LANARO Department of Physics, Unwerstty of Rochester, Rochester, New York 14627, USA Received 1 February 1985, accepted for pubhcaUon 23 Apral 1985

A theoretical investigation of proton neutrahzation by proton scattering from several alkahhahde surfaces is presented These systems are suitable for a perturbatlve treatment since no hydrogenlc atonuc shell is embedded m the valence band of the sohd where the neutrahzang electron originates, which is a necessary condmon for fast convergence of the perturbaUve expansion for the neutrahzatlon probability The perturbatlve lnteractaon IS modeled by a Fano-Anderson effective potential, and the dependence of the results on the properues of the systems (namely, the width of the valence band of the sohd and its position relative to the discrete atomic level) and on the dynamics of the process (deterrmned here by a single parameter which controls the durauon of the mteracuon, 1.e, the colhslon energy) are critically discussed.

1. Introduction C h a r g e - e x c h a n g e processes arising f r o m m o n o e n e r g e t l c ion b e a m s scattered f r o m sohd surfaces have recently b e e n the subject of m u c h e x p e r i m e n t a l [2-4] a n d theoretical [5-14], interest. W h e n an ion impinges on a surface, a multiplicity of events are o f te n involved, of which 1on n e u t r a l i z a t i o n is p e r h a p s o n e of the m o s t studied. In fact, the c r e a t i o n of excited a t o m t c states by t r a n s m i s s i o n o f energetic ions t h r o u g h thin foils and by reflection on surfaces has i m p o r t a n t a p p l i c a t i o n s in b e a m - f o i l s p e c t r o s c o p y [14-16] and in the c h e n u c a l analysis of surfaces [17]. T h e m a j o r i t y of theoretical w o r k has been based on the * Permanent address Dlpartlmento dl Sclenze e Tecnolo~e Chmuche, Unlverslth degh Stud1 dl Roma, Vm Orazxo Raxmondo, 1-00173 Rome, Italy 0 0 3 9 - 6 0 2 8 / 8 5 / $ 0 3 . 3 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

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F a n o - A n d e r s o n effectwe Hamiltonlan introduced simultaneously by Anderson [18] and by Fano [19], who apphed it to sohd-state physics problems and to the study of atomic spectra, respectively. Subsequently, the F a n o - A n d e r s o n model has been applied by Newns [20] in the study of chemasorption on metals and recently has been used by other authors [7-14] mainly in connection with charge-exchange processes at surfaces. In a previous paper (hereafter called paper I) [13], we have apphed the Fano-Anderson effective Hamlltonian to the problem of neutrahzation of positive ions scattered from surfaces and have gtven, using diagrammatic techniques, a perturbative expansion of the reduced density matrix elements of the neutralized atom. In paper I, we had carefully analyzed the validity of the first-order approximation and have found that its contribution to the neutralization probability is found to be dominant prowded: (i) the discrete level of the atomic projectile is not embedded in the continuum of the valence band of the solid, and 0i) the duration of the interaction is sufficiently short, i e., the collision energy is high enough. In the present paper, we want to present calculations for a series of systems for which the requirement (i) is fulfilled, namely, hydrogen ions on alkali-hahde surfaces. From fig. 1, m which the relative position between the valence band

n=4 n=3

-3

m

n=2

- -

Iq=l

-6

> tlJ

NoBr LiBr

1F/#7 NoF

-~

-15

LiF

®

KBr

FXx;-/7 KF

NN

H

F i g 1 E l e c t r o m c s t r u c t u r e of the first four h y d r o g e m c shells a n d of the valence b a n d of some a l k a l i - h a l i d e sohds. N o t i c e that only L1F has a discrete a t o n u c state e m b e d d e d m its valence b a n d .

F Battagha et al / Proton neutrahzanon at alkah hahdes

165

of the alkali-halides and the hydrogenic level structure is depicted, we see that the first requirement is always fulfilled (except for LiF in whose valence band the ground level of hydrogen atom is embedded). We shall see that, m agreement with our results of paper I, calculations beyond first order are redundant for all the cases except for the neutrahzation of H ÷ into the ground state of H from LiF surfaces. For all other cases, we present a comparison among the various systems, pointing out how the neutralization probability depends on the valence band width of the solid, where the neutrahzing electron originates, and on the position of the atomic level relative to that band. Before presenting and discussing our results, we shall give in the following paragraph a brief summary of the theory whose details might be found in paper I.

2. Theory Let us consider the experimental situation in which the following process occurs: an ion H + is sent against the surface of a solid S detaching from it one electron, H + + S ~ H* + S +.

(1)

The asterisk signifies that the atom H can be formed, in general, in an excited state. The collision times, usually around 10-1~ s, are orders of magnitude less than the lattice vibration period. Therefore, the surface does not have time to respond collectively, and the picture reduces to a many-interacting-electron problem, whose most convenient treatment is provided by the formalism of second quantization. Moreover, since atomic radiative lifetimes, on order of 10 -8 s, are much longer than the collision times, here we need not be concerned with radiative de-excitation of possible excited atomic states during the collision, and we can treat the process (1) separately from any radiative decay process of atom H*. The total Hamlltonian is:

H( t) = EedPtdPa + • e k -- ~.,ekh~h k + H i ( t ) , d

k

(2)

k

where pdt is an electron creanon operator into the discrete state [ d > of energy ea of the hydrogen atom, and ht~ is a hole creation operator into the state [ k> of energy e~ of the valence band of the solid. The Fano-Anderson [21] time-dependent effective interaction is:

H t ( t ) = ~ - ~ [ V k d ( t ) hkPd+ V~d(t) ,"tht] ,~ ~ j

(3)

kd

We shall restrict ourselves to a single band and single energy level interaction, whereby the only non-zero interaction is between the states of the valence band of the solid (whose lower and upper edges are eL and eu, respectively) and one

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166

(possibly degenerate) empty level (of energy e) of H. Moreover, we assume that

[71

Vkd(t ) = e -xl/I Vk,

(4)

where X is a parameter measuring the duration of the interaction, i.e., the collision energy of the incoming ion, and Vk a real parameter independent of the state I d>. The maximum amount of information on a quantum-mechanical system is given by the elgenvalues of a complete set of commuting observables, and the state of the system is specified by a vector of a suitable Hdbert space. If the specified observables do not form a complete set, the state of the system is described by its statistical operator O. A typical situation in which a physical system is not completely specified occurs when we want to describe it after it has interacted with an unobserved system. In this case the system of interest is specified by a reduced density operator [22]. The reduced density operator elements of the hydrogen atom which, subJected to the interaction (3), has been neutralized according to the process (1), are [13]

(d,,o(m(t)ld,,)=~I (q'o'Pd'(t) hk(t)'gto)12

,

(5)

where I q'0 > is the (unknown) interacting ground state of the whole system, i.e., a vector state belonging to the Hilbert space tensor product of the electronic Hilbert space of the solid times the electromc Hdbert space of the incoming atom. In (5), the vectors and operators are in the Heisenberg picture, but for a perturbative treatment the interaction picture is more suitable because the matrix elements are written m terms of the (known) non-interacting ground vacuum state 10> [231,

(xgtolp(m) hk(o~)lVPo> <7"o1%)

(OITexp(-If~Ldt'e-Xlt'l[phkf/(t')]}lO > =


,

(6)

where T is the time-ordering operator, and the tilde denotes the interaction picture. In paper I, we have given a diagrammatic treatment by which we have shown that, applying Wick's theorem [24], the charge-transfer probability (assumed to be independent of orbital and magnetic quantum numbers) into the atomic level of energy e and degeneracy D is given by P = 4~r2DE k

~_,/¢2"+i)(k) 2, n=0

(71

F Battagha et al / Proton neutrahzatton at alkah hahdes

167

where the (2n + 1)th term of the sum is written according to the following rules: (1) Draw the (2n + 1)th diagram. It will have 2n + 1 dots, 2 external lines and 2n internal lines; n + 1 are particle lines and n + 1 are hole lines. (2) Number the dots from 1 to 2n + 1 so that the first and the (2n + 1)th have, respectively, a hole line and a particle line coming out of them. For example the fifth-order diagram is 1

2

3

4

5

(3) Write for each internal particle hne connecting the dots j and j + 1 a factor 1

1

+ [ j+l +

0] 2'

and for each internal hole line connecting the dots j and j + 1 1

~kj kj +1

(4) For the two external lines write 8kk~

1

6kkl x ~ + ( ~ - ~)~'

if

n
if

n --- 0.

(5) Write for the j t h dot a factor (l~/qr)Vk. (6) Integrate over all ~'s, o~1. . . . . o~2,, and sum over all kl's, k I . . . . . kz,+l. In table 1 is shown the correspondence between the elements of each diagram of the perturbative expansion and the analytical expression belonging to it. The relevant contractions of table 1 are given by the following:

~k,(tt) h~(tj)=Sk,k exp[iek,(tt--tj)] O(tl--t)),

(8a)

h ~h ~j ( tj ) = 6kk~ exp( -- iektj),

(8b)

(~('t) ~+(tj)=exp[-ie(t,-tj)] ppt(tj)=exp(ietj), all other contractions = 0,

~9(t,-tj),

(8c) (8d) (8e)

where O ( t z - tj) is the Heaviside step function. The counting of topologically distinct diagrams [25] is particularly easy here: at each order n there is only

168

F Battagha et a l /

Proton neutrahzatton at alkah hahdes

Table 1 Graptucal representauon of each contractton Dven by eq (8) Analytical expression

Graphical representauon

-kltjl

J

Vkj e

I a

I

hk

•

J

h~j(tj)

iCt~) iiCtJ)

J

@t,,

"

/

>

one diagram which contributes n! times. M a k i n g the substitution

k

(9)

7/.2 "10

where A = e v - e L lS the valence b a n d width, and m which the spin degeneracy of each I k ) state has been taken into account, the first-order contribution to the neutralization probability is P(1)(X, e)

4~/2~2 adE V2(E)v~

(lo)

Vk(e).

where V ( E ) = The upper limit to the next contribution to the neutrahzation probability is g~ven by

p(3)(~., e)

~.2

a()~, e)foadE(E-

e)V2(E) ~-

~P~(E) ~ }

(11)

F Battagha et al / Proton neutrahzauon at alkah hahdes

169

where o~(~k, e ) = 7~- foadE ~(E) ¢ ~ - [ ( E - e)3 + 7~k2(E- e)]

7

i77 5-: T

t~(a, e)= v/~ fadE ~( El ¢'~ [ 2 ( E -

e) 2 - 4X2]

(12) (13)

3. Calculations and discussion

We model the scattering from alkali-hahde surfaces of H +, for which the atomic energy levels are given by the usual Rydberg formula and whose position is assumed to be preserved during the scattering process. We also observe that conservation of energy would require that the 2~ parameter, which controls the ion kinetic energy, changes after the collision. However, although this woud be a trivial extension, we do not include it here because in the usual experimental s~tuation A)~/), << 1, that is, the energy exchanges are several orders of magnitude less than the ion kinetic energy. It is also clear from eq. (9) that the electrons in the valence band of the sohd have been considered "free". At the present, we are interested in exploring the following general features of the process: (1) its dependence on the position of the atomic level relative to the valence band of the solid; (2) how the neutralization probability varies with X, the only parameter which controls the dynamics of the process, in parncular, the duration of the interaction; (3) a qualitatwe comparison among the various alkali-halide systems, i.e., how the neutralization probability changes within a group of the periodic table; and (4) how important the valence bandwidth and the work function of the solid (see table 2) are for the neutralization process. The functmn V(E) has been considered constant in a

Table 2 Work function and bandwidth m eV for the alkah-hahde systems considered m the present paper, the data are taken from refs. [26,27] System

Work functmn

Bandwidth

L1F NaF KF L1Br NaBr KBr

12 50 11.00 11.40 9 00 8.20 8 30

2 10 1 70 1 50 1 20 0.75 0 55

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170

previous work [12,13], and the same will be done here. We have chosen V ( E ) = V = 0.1 a.u., a reasonable value for the strength of the interaction. From fig. 1 it is clear that, within the theory and approximations so far presented, the alkali-halide systems are suitable for a first-order perturbation calculation if the conclusions of our previous paper I are correct. We have computed both the first- and the third-order contributions to the neutrahzatlon probabdlty into the hydrogenic states of principal quantum number n = 1, 2, 3 and 4 for all the systems displayed in fig. 1. Indeed, for all the alkali-brolmde surfaces the third-order contribution to the neutrahzatlon probability into all hydrogenic bound states is, in absolute value, several orders of magnitude less than the first order. This is also the case for the neutralization into the hydrogenic excited bound states from all the alkali-fluoride surfaces. The situation is different for the neutralization into the ground state from the NaF, K F and, especially, LiF surfaces: p(3) is smaller than p(1) for N a F and KF, but only by one or two orders of magnitude. Although the atomic ground state is very close to one edge of the band, it is not in resonance with any state of the valence band of the solids, so that the first-order perturbation approximation can still be safely apphed. We also notice that eq. (11) gwes actually an upper limit for [p(3)I. This means that if p ( 3 ) a s computed is small m comparison w:th p(1), afortiori it would be so if eq. (11) were exact. In table 3 we display P(:) and e(3) as computed for eqs. (10) and (11) as a function of X for the neutralization into the hydrogenic ground state from the three alkali-fluoride systems considered. Clearly, for L1F where Ip(3) I can be several orders of magnitude larger than p ( 1 ) , the first-order perturbation approximation breaks down. We shall not pursue further the calculations for

Table 3 F i r s t - o r d e r ( p O ) ) a n d tturd-order (p(3)) n e u t r a h z a t l o n p r o b a i n h t l e s into the h y d r o g e m c g r o u n d state from the a l k a h - f l u o n d e surfaces as a function of X, for N a F a n d KF, I p(3) ] < ] p ( l ) ] and the first-order c a l c u l a u o n s c a n be c o n s i d e r e d satisfactory, for L~F a m o r e exact t r e a t m e n t is required, for all other cases not reported in tins t a b l e ( n e u t r a h z a t l o n from the a l k a h - b r o r m d e surfaces a n d i n t o h y d r o g e m c excited states), I p(3) ] << ] p(1) ] h

5 . 0 - - 4 a) 10-3 5.0--3 1.0-2 5.0--2 10 - 1 50 - 1 10

LF pO)

p(3)

NaF p(l)

p(3)

KF pO)

p(3)

034+1 017+1 034 017 0.24--1 0 75 - 2 0 33 - 3 082-4

030+3 077+2 035+1 096 027-1 0.27 -- 2 0.53 - 5 0.33-6

012-5 0.48-5 012-3 0.44-3 033 -2 0.27 - 2 0.23 - 3 0.58-4

-029-4 -030-4 --071-4 -0.17 -3 010-3 0.28 - 3 0 26 - 5 017-6

023-5 093-5 023-3 081 -3 0 4 1 --2 0.27 - 2 0.19 - 3 049-4

- 0 45 - 4 -048 -4 --014-3 - 0 34 --3 0 36 - 3 0.31 - 3 0.18 - 5 0.12 - 6

a) 5 0 - - 4 m e a n s 5 0 × 1 0 - 4

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Proton neutrahzatlon at alkah hahdes

171

this case, but shall be concerned only with cases in which the first-order approximation is satisfactory. Therefore, the neutralization from the LiF surface will be studied only for the hydrogenic excited states. This is tantamount to assuming that electron capture from different atomic shells is uncorrelated. This is physically reasonable because the spatial extension of the atomic orbltals increases as the squares of the principal quantum number. Finally , from table 3 we observe that the difference between the absolute values of p(1) and p(3) becomes greater as ?~ increases. This is easily understood in terms of the physical meaning of the parameter ?~: it represents the duration of the interaction, and the shorter it is, the better the first-order perturbatxve approximation becomes, and at very large ?, it applies even to the resonant case. The neutralization probabilities into various bound states of the hydrogen atom as a funtion of ?~ for the alkali-fluoride and alkali-bromide surfaces are shown in figs. 2 and 3, respectively, where two general features can be readily seen: the maxima of the neutrahzation probability and its rapid decrease for values of ~ larger than the value where those maxima occur. We expect the interaction time to become smaller as ?, increases, and therefore the electron to have less time to " j u m p " into the discrete atomic level. On the other hand, large ?~ can ease the resonance requirement because energy conservation can be violated for short-duration processes. Those two competing effects give rise to an optimal value at which the ionization probability reaches a maximum. The fact that in figs. 2 and 3 the maxima occur at greater values of ?, for greater values of principal quantum number n is also in agreement with what we have just mentioned: the smaller the value of n, the closer the atomic level is to the edges of the valence band, and the smaller is the value of ?~ at which those maxima occur given that the resonance requirement becomes less stringent. The neutralization probability is seen to increase with increasing quantum number n. There are two competing effects: as n increases, the atomic level moves away from the valence band of the solid, which decreases the neutralization probability, but also the degeneracy of the n shell increases as n 2, giving more states available to the neutralizing electron. The effect of the degeneracy of the atomic states is more dominant than that of the separation in energy between the atomic states and the valence bands of the solids in all cases except for the ground-state neutralization from the N a F and K F surfaces. In figs. 2b and 2c we see that the curves with n = 1 are well above all the others, in agreement with the proximity of the ground state to the valence bands of N a F and K F (see fig. 1). At greater values of ~, where the separation between the atomic excited states and the solid valence band states has less effect on the neutrahzatlon process, the degeneracy of these excited states becomes more important, and the neutralization probability into the ground state is, again, smaller than that into the excited states This also explains why the curves of fig. 3 cross each other while the ones of fig. 2 (except for n = 1) do not. The valence bands of the fluoride systems are so far away from the atormc excited

F Battagha et al / Proton neuiralrzatron af alkalr hahdes

172

a

l-f

F Battagha et a l /

Proton neutrahzatton at alkah hahdes

173

C

KF

X Q-

ll-

\

4

-I

0

I

LOG% Fig 2. Neutralization probablhtles P into various hydrogemc shells from the L1F (a), N a F (b) and K F (c) surfaces, as a function of log ?~. P is gwen by the s u m pO) + p(3) (eqs. (10) and (11)) even though there ~s very httle contribution from p(3) (see table 3) Notice that m (a) the hydrogenlc ground state (n = 1) IS missing, the neutralization into it from L1F needs a more exact treatment Notice also that the increase of the atonuc shell degeneracy is a more important factor m determining the neutrallzauon probablhty than the resonance factor, m all cases except for the neutrahzatlon into the aton'uc ground state form the N a F (b) and K F (c) surfaces.

states which, in turn, are comparatively so close to each other, that the quasi-resonant nature between the atomic states and the valence bands does not change appreciably going from n---2 to n = 4 (n = 1, as we said, is a special case). Therefore, the degeneracy increasing is, for all values of ?~, the dominant effect. The situation is different for the bromide systems. Their valence bands are located somewhere in between the n = 1 and n = 2 atomic shells. Hence, in the region of values of ~ where its increase can ease the resonant structure between the atomic levels and the valence bands, the competition between the resonant effect and the degeneracy effect is clearly seen by the crossing of the curves in fig. 3. At higher values of ?, where the short duration of the interaction makes the neutrahzation probability decrease with increasing of ~, the degeneracy effect is the dominant one. In figs. 4 and 5, we compare, for a given atomic shell, the neutralization probabihty as a function of ~ for scattering from different surfaces. In fig. 4

F Battagha et al / Proton neutrahzatlon at alkah halrdes

174

1.

c-l 4 x a .5

-1

0

1

175

F Battagha et a l / Proton neutrahzatton at alkah hahdes

c

KBn

/'X

1

.4

% X Gt-

.2

0

-I

1

L~ X

Fig 3 Neutrahzatlon probablhtyP into various hydrogemc shells from the DBr (a), NaBr (b) and KBr (c) surfaces as a function of log ~ Again, P = P ( 1 ) + P ° ) but now P is practically indistinguishable from p(l). Notlc the crossing of these curves due to the proramaty of the alkah-bromide valence bands to the excited atonuc shells

we show the n e u t r a l i z a t i o n p r o b a h i l i t y into the g r o u n d state f r o m the f l u o n d e surfaces (fig 4b). T h e L I F curve is missing f r o m fig. 4a b e c a u s e a c a l c u l a t i o n m o r e exact t h a n the f i r s t - o r d e r a p p r o x i m a t i o n w o u l d b e necessary to d e t e r m i n e it. T h e relative i m p o r t a n c e of the surface propertxes can b e seen here. T h e lower edge of the valence b a n d o f K F is closer to the a t o m i c K shell t h a n the o n e of N a F . Therefore, in the region of X where the r e s o n a n t effect is ~mportant, K F is m o r e easily i o n i z a b l e t h a n N a F . However, at higher values of X, the two curves cross each other, which is a m a n i f e s t a t i o n o f the larger b a n d w i d t h o f N a F c o m p a r e d to the one of K F (see fig. 1 a n d t a b l e 2). I n fig. 4b the n e u t r a l i z a t i o n p r o b a b i l i t y follows the t r e n d LiBr > N a B r > K B r for all values of X. I n this case, the r e s o n a n t effect a n d the b a n d w i d t h effect b o t h f a v o r an easier i o n i z a t i o n of LiBr t h a n N a B r a n d of N a B r t h a n K B r (see fig, 1). In fig. 5 we show the n e u t r a l i z a t i o n p r o b a b i l i t y i n t o the L shell f r o m f l u o r i d e surfaces (fig. 5a) a n d f r o m b r o m i d e surfaces (fig. 5b). F r o m fig. 5a we see that the curve relative to K F is always b e l o w the one relative to N a F : the

F Battagha et al / Proton neutrahzatron at alkall halrdes

176

a 4

, .-.

-l.u

^

LcGh

-.5

F Battagha et al / Proton neutrahzatwn at alkah hahdes

177

bandwidth of N a F is larger and its upper band edge closer to the L atomic shell than the bandwidth and upper band edge of K F (see fig. 1). The curve of LiF as above the other fluoride curves at large values of A, where only the bandwidth effect determines the trend of the neutralization probability. However, as X decreases, that curve crosses first the curve of N a F and then the curve of KF. This has to be so, given that the valence band of N a F is wider and its upper edge is closer than that for K F to the n = 2 atomic energy level. The behavior of the neutrahzation probabihty curves into the shells with n = 3 and n = 4 not presented here is qualitatively similar, for both the fluoride and bromide systems, to the behavior of the curves shown for n = 2. Only their quantitative behavior changes, and this has already been examined in figs. 2 and 3.

4. Conclusions We have performed calculations of the neutralization probability of protons impinging on various alkali-halide surfaces. Due to the ease with which single crystals of these ionic sermconductors m a y be grown, they should be hopefully very apt for experimental measurements. Moreover, they play an important role in our theory which has been summarized at the beginning of this paper and whose details might be found in paper I. In fact, their electronic structure is such that the width of their valence band and its locations in an energy scale do not allow any bound stationary state of the hydrogen atom to be in resonance with any stationary state of the valence electrons of the solid (with the exception of the ground state of the hydrogen which is indeed embedded into the continuum of the valence band of LiF). F r o m the conclusions of paper I, this is a most favorable situation for the rehabllity of first-order perturbation calculations. Under those conditions we are therefore in a p o s m o n of being able to explore how the neutralization probabihty depends on various properties of the systems considered. The decrease of the interaction time always plays a double role: it gives less time for a surface electron to " j u m p " into an atomic state but, also, enhances the resonance condition between the discrete atomic state and the continuum of the surface states. The increase of the atomic principal quantum number always enhances the neutralization probability due to the enhanced degeneracy of the energy levels. The proximity of an atomic state to the valence band always makes the neutralization into that state easier. Finally, the wider the valence band is, the easier the neutralization Fig. 4. Neutrahzation probablhty P into the hydrogemc ground state (n = 1) from the alkah-fluoride (a) and alkah-bronude (b) surfaces In (b), the resonance factor and the width of the valence b a n d s both favor the trend LIBr > NaBr > KBr for the ground-state neutrahzatlon probablhty (see fig 1), In (a), the interplay between those two factors causes the crossing of the curves as shown

178

F Battaglra et al / Proton neutraitzairon at alkalr halrdes

r I a

.4 t? 4 x a

.2

0

-1

1

LOG h

b

0

-1

l-!xlh

1

F Battagha et a l /

Proton neutrahzatwn at alkah hahdes

179

process occurs. It is the interplay of all these general rules that determines quantitatively the relative neutralization probability from valence bands of different systems into different atomic states. Our main goal has been to gain a qualitative idea of the behavior of the neutralization probability as a funcuon of all the above-mentioned properties. Therefore, although the parameterization chosen is an oversimplification for any quantitative-type calculation, it suffices to give the qualitative features we are after. For example, if it were possible to have, by other means, a realistic analytic expression for the function V~d (see eq. (3)) [14], 2, would then be the only truly undetermined parameter, and its dependence on the collision energy could be determined by comparing the first-order calculation with the neutralization probability expertmentally measured at high collision energy.

Acknowledgements This research was supported m part by the Air Force Office of Scientific Research (AFSC), United States Air Force, under Grant AFOSR-82-0046, the Office of Naval Research, and the National Science Foundation under Grant CHE-8320185. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. F.B. thanks the University of Rochester for an Elon Huntington Hooker Graduate Fellowship (1983-84) and the financial support of the Itahan National Research Council (CNR). T.F.G. Acknowledges the Camille and Henry Dreyfus Foundation for a Teacher-Scholar Award (1975-86). A.L. acknowledges the Department of Energy for support.

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Fig. 5. Neutralization probabihty P into the first hydrogemc excited state (n = 2) from the alkah-fluoride (a) and alkah-bromtde (b) surfaces.

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