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Many-electron treatment of quasi-resonant ion neutralization at solid surfaces
State Communications, Vol. 71, ~Solid Printed in Great Britain.
No.
6, pp. 449-452, 1989.
0038-i098/8953.00+.00 Maxwell Pergamon Macmillan plc
MANY-ELECTRON TREATMENT OF QUASI-RESONANT ION NEUTRALIZATION AT SOLID SURFACES A. T. Amos and K. W. Sulston*t$
Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, U. K. and S. G. Davisont§
Departments of Applied Mathematics and Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI
(Revised Manuscript Received 23 May 1989 by A. A. Maradudin)
The many-electron theory of charge transfer in atom-solid surface scattering, previ-
ously used for resonant charge transfer, is applied to quasi-resonant ion neutralization. A simple expression for the ion neutralization probability is obtained and clearly demonstrates its oscillatory behaviour. To illustrate the method, results are presented for protons scattered from KBr and KF.
1. INTRODUCTION
two electrons are removed from the solid. As a consequence, the theory leads to a set of equations of very simple form and the fact that the core band is very narrow enables a good approximate solution to be obtained. These considerations apply, also, to the process of QRIN of protons scattering from alkali-halide surfaces, where the interaction is with the fully occupied valence band. Therefore, our theory is applicable to these systems and can be compared qualitatively to the perturbation method introduced by Battaglia et al [6].
Recently, there has been some interest shown in developing many-electron treatments of resonant charge transfer in the scattering of ions from solid surfaces [1-3]. We, ourselves, have introduced a many-electron theory of resonant charge transfer and have successfully treated Li + scattering from Cs/W [2]. In principle, our theory encompasses all types of charge transfer, and the purpose of this article is to demonstrate its particular application to quasi-resonant ion neutralization (QRIN). The best-known example of QRIN is the case of He + scattering from Pb, Ge, Bi and In surfaces [4], where oscillatory behaviour in the intensity of the backscattered ions, as a function of ion velocity, arises from quasi-resonant exchange between the vacant valence ion orbital and the core band of the solid (see, also, the scattering of noble gas ions [5]). Two circumstances combine to make this situation particularly easy to tackle using many-electron theory, since they reduce the number of many-electron wave functions we have to consider. Firstly, the solid band is completely full, so that there are no functions representing electron transfer from occupied to unoccupied band states and, secondly, the process is believed to involve the transfer of just a single electron to the ion, so that we need not include functions in which
2. FORMALISM Let us consider the situation in which a proton is scattered from a solid surface and acquires a single electron during the collision, thus becoming neutralized. The band from which the electron is removed is assumed to be either a valence band of some insclator (e.g., that of an alkali-halide, as treated in [6]) or the very narrow core-level band of a metal. In both these cases, the band in question is full. The time-dependent Hamihonian we wish to use is
,q(t) = E ekcLck~+Z eocJo~o~ k,o"
O"
+ ~ vkV(t)(cZocoo+C*ooCk,~), k, cr
* t $
§
NATO Science Fellow 1986-8 Work supported by Natural Sciences and Engineering Research Council of Canada Present address: Department of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104, USA. Guelph-Waterloo Program for Graduate Work in Physics
(1)
where the c's denote the usual annihilation and creation operators with spin index a, and the k(0) index refers to a band (ion) orbital ~bk(Zo) with energy ek(eo). For simplicity, we take e o to be constant and neglect image effects. The ion-solid interaction is assumed to be one-electron in nature, so that the vacant orbital Zo is coupled to each ~k by a term vkV(t), with a common time-dependent factor independent of k. 449
450
QUASI-RESONANT
ION NEUTRALIZATION
Note that (1) is the Anderson-Newns Hamiltonian with the Coulomb repulsion term U omitted, since we wish to discuss the transfer of one electron only. Consequently, the many-electron wave functions of interest are IP'o = 1~101 ""ON~NI
(2)
for the completely filled band with 2N electrons, and Iffi0 ---- ~ f ~ { l ¢ l ~ l ' " O i X 0 " "
] -b [ 0 1 ~ l ' - ' Z 0 ~ i " - l }
(3)
corresponding to the transfer to the ion of an electron from the band orbital ~i • (In the case of an ion, He + for example, where the valence orbital is already occupied by a single electron, the wave functions and the equations deduced from them will take a slightly different form.) The set of functions (2) and (3) define the restricted space within which we allow H(t) to operate, i.e., we are not using the full Hamiltonian (1), but rather its projection onto a smaller subspace. The time-dependent many-electron wave function can be written as N
~(t) = ao(t)groe-ie°t + ~, bi(t)~/io e-i(L,o+c°y,
(4)
i=1
where coi = e o - e l and E o = 2 ~ E i. Using the method of [2], the equations determining the coefficients in (4) are N
d o = -i~/'2V(t) ~ vibi e-ic°'t
(5)
i=1
l) i = - i ~ v i V ( t ) a o
ei~°'t,
i = 1 .... N
(6)
subject to the conditions that, at t = -,~,, a 0 = 1,
b i = O.
AT SOLID SURFACES
provide qualitative insight and are sufficiently accurate for most purposes. Here we discuss two simple approximations which should be valid for narrow bands, although we should caution that analyses of the range of validity of similar approximations [8,9] suggest that they will fail for very slow ions. (a) Perturbation Theory When c o is sufficiently far below the solid band, the ion-neutralization probability is small, i.e., ao(t) remains close to unity for all t, and perturbation theory is applicable. Thus, putting ao = 1 in (6) and integrating gives
where
bi(~) = - i ' ~ v i V(o)i)
(8)
V(co i) = fff~ V(t)e i°~'t dt
(9)
is the Fourier transform of V(t). Hence, P(,,~) = 2 Z v~ [V(O)i)[ 2
is the first-order expression for the ion-neutralization probability. It is a fairly simple matter to use the Picard method to extend this so as to obtain higher-order terms but, to conserve space, we shall not write them out in detail here. (b) Narrow-Band Approximation In the case of a narrow band, where the band energies differ by a small amount, it is an appropriate approximation to replace them by some single average value e, so that, as a consequence, the O)i are replaced by the single value co = e o - e . When this is done, the coefficients bi(t ) in (6) can be expressed in terms of a single function b(t) given by
b i = builD , (7)
i
While the overall form of (5) and (6) resembles that of the equations obtained in the Hartree-Fock theory of ion neutralization [3,7], their interpretation is completely different; most notably in that the dependent variables in (5) and (6) are the coefficients of many-electron functions, whereas in Hartree-Fock theory the analogous variables are coefficients of the one-electron molecular orbitals. In spite of this, (5) and (6) are, in fact, simpler, because only one set of equations has to be solved rather than N sets (i.e., one set for each occupied Hartree-Fock molecular orbital). Nevertheless, the similarity of form enables some of the techniques used in Hartree-Fock calculations to be applied here also. This is particularly the case when approximate solutions are considered, as will now be discussed. 3. APPROXIMATE SOLUTIONS Equations (5) and (6) can be solved numerically, if this is required. However, in the circumstances which give rise to quasi-resonance, approximate solutions can
(10)
i
The ion neutralization probability is given by
P(t) = 1 - l a o ( t ) [ 2 = Z ]bi(t)l 2.
Vol. 71, No. 6
(11)
where O =
Oi2
and b satisfies
D = - i ' ~ D V a o ei°)t.
(12)
Equation (5) now becomes
ho -= - i ' ~ D Vbe - i°~t
(13)
and the neutralization probability is
P(t) = [b(t)[ 2.
(14)
Eliminating ao between (12) and (13) gives a secondorder differential equation in b. For some choices of V(t), this equation can be solved in terms of standard functions [10, 11 ]. Perhaps a more useful approach [12] is to note that equations (12) and (13) are equivalent to those of the two-level system discussed by Rosen and Zener [13]. These authors introduced an approximate solution for the two-level system which is widely used and has proven
Vol. 71, No. 6
QUASI-RESON~qT
ION NEUTRALIZATION
AT SOLID SURFACES
very successful. When this Rosen-Zener approximation is applied to (12) and (13), it yields
0.6
p(oo) = [ 17(o9)sin [~2D17(0)] 2. V(O)
0.5
451
(15) 0.4
In the limit of small V(t), equation (15) becomes p(~o) = 2D 2[•(o9)[2,
(16) P 0.3
which agrees with the first-order perturbation theory result (10), when all the toi are set equal to the average value co. Up to this point we have not specified a value for o9. However, this comparison suggests that it may be most appropriate to choose it so that (16) reproduces the first-order perturbation result, i.e. so that [t7(o9) 12 = D -2
• Vi2]V(ogi)[ 2,
(17)
i
which amounts to taking an average of [17(o9i)12, weighted with respect to the squares of the magnitude of the coupling parameters 1)i, to determine [17(o9)12. When this is done, (15) becomes
p(~o) =
vi2117(o9i)[ 2
sin[
(0)]
.
(18)
Equation (18) is the main result of this paper. Although it is approximate, it should be sufficiently reliable for most purposes. In the case of small interactions, it gives the same results as perturbation theory, yet it is as easy to use. Moreover, it has a much wider range of applicability and can be used for large interactions. Most important of all, the sine term in (18) gives the oscillations in P, which correspond to the oscillations found experimentally in the intensity of back-scattered ions.
0.2
0.1 O _ _ -2
-1
0
Ioglo A Fig. 1. Neutralization probability, P()~), for protons scattered from KBr. Full line is calculated using first-order perturbation theory (eqn.ll). Broken line is calculated using the narrow-band and Rosen-Zener approximations (eqn.18). Crossesare the exact numerical values. 1.75
1.5
•.25
1.0
0.75
4. PROTON SCATTERING FROM KBr AND KF To illustrate the application of the methods of this paper, we have used equations (10) and (18) to compute neutralization probabilities in a simple model for protons scattered from KBr and KF. To test their accuracy, our results are compared with "exact' results, by which we mean values obtained from a numerical solution of (5) and (6). For the potential, we have taken the exponential form used in [6] and [7],
V(t) = Voe-~ltl,
.%
0.25
~,
oL .........x" k! -2
-1
0 log10 ,k
Fig. 2. As for Fig. 1, for protons scattered from KF.
(19)
with Vo = 0.I au. The valence bands of the alkali-halides were modelled as having uniform densities of states with N equally spaced energy levels lying between the band edges (as determined from work function and band width data taken from [6]), and with the coupling parameters all equal to N -½. With N = 100, numerically-stable results were obtained. Although this is rather simplistic, we believe it is adequate enough to give qualitatively meaningful results and we consider it to be more satisfactory, in the case of narrow bands, than the free-electron model.
We should note that the resulting probabilities are several orders of magnitude greater than would be obtained from a free-electron approach. The neutralization probabilities are plotted as functions of log ,,1. in Figs. 1 and 2. The parameter )!, is a measure of the duration of the proton-surface interaction and so should be related to the speed of the proton, with small values being equivalent to slow protons. For )1. > 0.6 the first-order perturbation and narrow-band
452
QUASI-RESONANT ION NEUTRALIZATION AT SOLID SURFACES
approximation results are in good agreement with each other and with the exact results, confirming the conclusion of Battaglia e t a l [6] that perturbation theory can be applied in the case of fast protons. As % decreases, the perturbative results become unreliable, giving results that are too large for KBr and physically meaningless for KF. However, eqn. (18) continues to give values which agree very well qualitatively and quite well quantitatively with the exact values. The oscillation in P as a function of 2 is shown clearly in the narrow-band and exact calcula-
Vol. 71, No. 6
tions on KF but is almost damped out for KBr. Overall the neutralization probability is larger for KF than for KBr and this is because the KF band is much closer to the ground-state energy of hydrogen. Acknowledgements- We wish to thank the SERC of the UK for financial support for the visit of SGD to Nottingham. SGD thanks the Department of Mathematics, University of Nottingham for the warm hospitality extended to him during this visit.
REFERENCES 1. 2. 3. 4. 5.
6.
K.L. Sebastian, Phys. Lett., 98A, 39 (1983); Phys. Rev., B31, 6976 (1985). K.W. Sulston, A.T. Amos and S.G. Davison, Phys. Rev., B37, 9121 (1988). A.T. Amos, K.W. Sulston and S.G. Davison, Adv. in Chem. Phys., in press. R.L. Erickson and D.P. Smith, Phys. Rev. Letters, 34, 297 (1975). T.W. Rush and R.L. Erickson, in: Inelastic IonSurface Collisions, Eds. N.H. Tolk, J.C. Tully, W. Heiland and C.W. White (Academic Press, New York) pp. 73-104 (1977). F. Battaglia, T.F. George and A. Lanaro, Surface Sci., 161, 163 (1985).
7
T.B. Grimley, V.C. Jyothi Bhasu and K.L. Sebastian, Surface Sci., 124, 305 (1983). 8. F. Battaglia and T.F. George, J. Chem. Phys., 83, 3847 (1985). 9. K.W. Sulston, A.T. Amos and S.G. Davison, Surface Sci., 197, 555 (1988). 10. S.G. Davison, K.W. Sulston and A.T. Amos, J. Electroanal. Chem., 204, 173 (1986). 11. S.I. Easa and A. Modinos, Surface Sci., 161, 129 (1985). 12. A.T. Amos, S.G. Davison and K.W. Sulston, Phys. Letters, A l l 8 , 471 (1986). 13 N. Rosen and C. Zener, Phys. Rev., 40, 502 (1932).
No.
6, pp. 449-452, 1989.
0038-i098/8953.00+.00 Maxwell Pergamon Macmillan plc
MANY-ELECTRON TREATMENT OF QUASI-RESONANT ION NEUTRALIZATION AT SOLID SURFACES A. T. Amos and K. W. Sulston*t$
Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, U. K. and S. G. Davisont§
Departments of Applied Mathematics and Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI
(Revised Manuscript Received 23 May 1989 by A. A. Maradudin)
The many-electron theory of charge transfer in atom-solid surface scattering, previ-
ously used for resonant charge transfer, is applied to quasi-resonant ion neutralization. A simple expression for the ion neutralization probability is obtained and clearly demonstrates its oscillatory behaviour. To illustrate the method, results are presented for protons scattered from KBr and KF.
1. INTRODUCTION
two electrons are removed from the solid. As a consequence, the theory leads to a set of equations of very simple form and the fact that the core band is very narrow enables a good approximate solution to be obtained. These considerations apply, also, to the process of QRIN of protons scattering from alkali-halide surfaces, where the interaction is with the fully occupied valence band. Therefore, our theory is applicable to these systems and can be compared qualitatively to the perturbation method introduced by Battaglia et al [6].
Recently, there has been some interest shown in developing many-electron treatments of resonant charge transfer in the scattering of ions from solid surfaces [1-3]. We, ourselves, have introduced a many-electron theory of resonant charge transfer and have successfully treated Li + scattering from Cs/W [2]. In principle, our theory encompasses all types of charge transfer, and the purpose of this article is to demonstrate its particular application to quasi-resonant ion neutralization (QRIN). The best-known example of QRIN is the case of He + scattering from Pb, Ge, Bi and In surfaces [4], where oscillatory behaviour in the intensity of the backscattered ions, as a function of ion velocity, arises from quasi-resonant exchange between the vacant valence ion orbital and the core band of the solid (see, also, the scattering of noble gas ions [5]). Two circumstances combine to make this situation particularly easy to tackle using many-electron theory, since they reduce the number of many-electron wave functions we have to consider. Firstly, the solid band is completely full, so that there are no functions representing electron transfer from occupied to unoccupied band states and, secondly, the process is believed to involve the transfer of just a single electron to the ion, so that we need not include functions in which
2. FORMALISM Let us consider the situation in which a proton is scattered from a solid surface and acquires a single electron during the collision, thus becoming neutralized. The band from which the electron is removed is assumed to be either a valence band of some insclator (e.g., that of an alkali-halide, as treated in [6]) or the very narrow core-level band of a metal. In both these cases, the band in question is full. The time-dependent Hamihonian we wish to use is
,q(t) = E ekcLck~+Z eocJo~o~ k,o"
O"
+ ~ vkV(t)(cZocoo+C*ooCk,~), k, cr
* t $
§
NATO Science Fellow 1986-8 Work supported by Natural Sciences and Engineering Research Council of Canada Present address: Department of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104, USA. Guelph-Waterloo Program for Graduate Work in Physics
(1)
where the c's denote the usual annihilation and creation operators with spin index a, and the k(0) index refers to a band (ion) orbital ~bk(Zo) with energy ek(eo). For simplicity, we take e o to be constant and neglect image effects. The ion-solid interaction is assumed to be one-electron in nature, so that the vacant orbital Zo is coupled to each ~k by a term vkV(t), with a common time-dependent factor independent of k. 449
450
QUASI-RESONANT
ION NEUTRALIZATION
Note that (1) is the Anderson-Newns Hamiltonian with the Coulomb repulsion term U omitted, since we wish to discuss the transfer of one electron only. Consequently, the many-electron wave functions of interest are IP'o = 1~101 ""ON~NI
(2)
for the completely filled band with 2N electrons, and Iffi0 ---- ~ f ~ { l ¢ l ~ l ' " O i X 0 " "
] -b [ 0 1 ~ l ' - ' Z 0 ~ i " - l }
(3)
corresponding to the transfer to the ion of an electron from the band orbital ~i • (In the case of an ion, He + for example, where the valence orbital is already occupied by a single electron, the wave functions and the equations deduced from them will take a slightly different form.) The set of functions (2) and (3) define the restricted space within which we allow H(t) to operate, i.e., we are not using the full Hamiltonian (1), but rather its projection onto a smaller subspace. The time-dependent many-electron wave function can be written as N
~(t) = ao(t)groe-ie°t + ~, bi(t)~/io e-i(L,o+c°y,
(4)
i=1
where coi = e o - e l and E o = 2 ~ E i. Using the method of [2], the equations determining the coefficients in (4) are N
d o = -i~/'2V(t) ~ vibi e-ic°'t
(5)
i=1
l) i = - i ~ v i V ( t ) a o
ei~°'t,
i = 1 .... N
(6)
subject to the conditions that, at t = -,~,, a 0 = 1,
b i = O.
AT SOLID SURFACES
provide qualitative insight and are sufficiently accurate for most purposes. Here we discuss two simple approximations which should be valid for narrow bands, although we should caution that analyses of the range of validity of similar approximations [8,9] suggest that they will fail for very slow ions. (a) Perturbation Theory When c o is sufficiently far below the solid band, the ion-neutralization probability is small, i.e., ao(t) remains close to unity for all t, and perturbation theory is applicable. Thus, putting ao = 1 in (6) and integrating gives
where
bi(~) = - i ' ~ v i V(o)i)
(8)
V(co i) = fff~ V(t)e i°~'t dt
(9)
is the Fourier transform of V(t). Hence, P(,,~) = 2 Z v~ [V(O)i)[ 2
is the first-order expression for the ion-neutralization probability. It is a fairly simple matter to use the Picard method to extend this so as to obtain higher-order terms but, to conserve space, we shall not write them out in detail here. (b) Narrow-Band Approximation In the case of a narrow band, where the band energies differ by a small amount, it is an appropriate approximation to replace them by some single average value e, so that, as a consequence, the O)i are replaced by the single value co = e o - e . When this is done, the coefficients bi(t ) in (6) can be expressed in terms of a single function b(t) given by
b i = builD , (7)
i
While the overall form of (5) and (6) resembles that of the equations obtained in the Hartree-Fock theory of ion neutralization [3,7], their interpretation is completely different; most notably in that the dependent variables in (5) and (6) are the coefficients of many-electron functions, whereas in Hartree-Fock theory the analogous variables are coefficients of the one-electron molecular orbitals. In spite of this, (5) and (6) are, in fact, simpler, because only one set of equations has to be solved rather than N sets (i.e., one set for each occupied Hartree-Fock molecular orbital). Nevertheless, the similarity of form enables some of the techniques used in Hartree-Fock calculations to be applied here also. This is particularly the case when approximate solutions are considered, as will now be discussed. 3. APPROXIMATE SOLUTIONS Equations (5) and (6) can be solved numerically, if this is required. However, in the circumstances which give rise to quasi-resonance, approximate solutions can
(10)
i
The ion neutralization probability is given by
P(t) = 1 - l a o ( t ) [ 2 = Z ]bi(t)l 2.
Vol. 71, No. 6
(11)
where O =
Oi2
and b satisfies
D = - i ' ~ D V a o ei°)t.
(12)
Equation (5) now becomes
ho -= - i ' ~ D Vbe - i°~t
(13)
and the neutralization probability is
P(t) = [b(t)[ 2.
(14)
Eliminating ao between (12) and (13) gives a secondorder differential equation in b. For some choices of V(t), this equation can be solved in terms of standard functions [10, 11 ]. Perhaps a more useful approach [12] is to note that equations (12) and (13) are equivalent to those of the two-level system discussed by Rosen and Zener [13]. These authors introduced an approximate solution for the two-level system which is widely used and has proven
Vol. 71, No. 6
QUASI-RESON~qT
ION NEUTRALIZATION
AT SOLID SURFACES
very successful. When this Rosen-Zener approximation is applied to (12) and (13), it yields
0.6
p(oo) = [ 17(o9)sin [~2D17(0)] 2. V(O)
0.5
451
(15) 0.4
In the limit of small V(t), equation (15) becomes p(~o) = 2D 2[•(o9)[2,
(16) P 0.3
which agrees with the first-order perturbation theory result (10), when all the toi are set equal to the average value co. Up to this point we have not specified a value for o9. However, this comparison suggests that it may be most appropriate to choose it so that (16) reproduces the first-order perturbation result, i.e. so that [t7(o9) 12 = D -2
• Vi2]V(ogi)[ 2,
(17)
i
which amounts to taking an average of [17(o9i)12, weighted with respect to the squares of the magnitude of the coupling parameters 1)i, to determine [17(o9)12. When this is done, (15) becomes
p(~o) =
vi2117(o9i)[ 2
sin[
(0)]
.
(18)
Equation (18) is the main result of this paper. Although it is approximate, it should be sufficiently reliable for most purposes. In the case of small interactions, it gives the same results as perturbation theory, yet it is as easy to use. Moreover, it has a much wider range of applicability and can be used for large interactions. Most important of all, the sine term in (18) gives the oscillations in P, which correspond to the oscillations found experimentally in the intensity of back-scattered ions.
0.2
0.1 O _ _ -2
-1
0
Ioglo A Fig. 1. Neutralization probability, P()~), for protons scattered from KBr. Full line is calculated using first-order perturbation theory (eqn.ll). Broken line is calculated using the narrow-band and Rosen-Zener approximations (eqn.18). Crossesare the exact numerical values. 1.75
1.5
•.25
1.0
0.75
4. PROTON SCATTERING FROM KBr AND KF To illustrate the application of the methods of this paper, we have used equations (10) and (18) to compute neutralization probabilities in a simple model for protons scattered from KBr and KF. To test their accuracy, our results are compared with "exact' results, by which we mean values obtained from a numerical solution of (5) and (6). For the potential, we have taken the exponential form used in [6] and [7],
V(t) = Voe-~ltl,
.%
0.25
~,
oL .........x" k! -2
-1
0 log10 ,k
Fig. 2. As for Fig. 1, for protons scattered from KF.
(19)
with Vo = 0.I au. The valence bands of the alkali-halides were modelled as having uniform densities of states with N equally spaced energy levels lying between the band edges (as determined from work function and band width data taken from [6]), and with the coupling parameters all equal to N -½. With N = 100, numerically-stable results were obtained. Although this is rather simplistic, we believe it is adequate enough to give qualitatively meaningful results and we consider it to be more satisfactory, in the case of narrow bands, than the free-electron model.
We should note that the resulting probabilities are several orders of magnitude greater than would be obtained from a free-electron approach. The neutralization probabilities are plotted as functions of log ,,1. in Figs. 1 and 2. The parameter )!, is a measure of the duration of the proton-surface interaction and so should be related to the speed of the proton, with small values being equivalent to slow protons. For )1. > 0.6 the first-order perturbation and narrow-band
452
QUASI-RESONANT ION NEUTRALIZATION AT SOLID SURFACES
approximation results are in good agreement with each other and with the exact results, confirming the conclusion of Battaglia e t a l [6] that perturbation theory can be applied in the case of fast protons. As % decreases, the perturbative results become unreliable, giving results that are too large for KBr and physically meaningless for KF. However, eqn. (18) continues to give values which agree very well qualitatively and quite well quantitatively with the exact values. The oscillation in P as a function of 2 is shown clearly in the narrow-band and exact calcula-
Vol. 71, No. 6
tions on KF but is almost damped out for KBr. Overall the neutralization probability is larger for KF than for KBr and this is because the KF band is much closer to the ground-state energy of hydrogen. Acknowledgements- We wish to thank the SERC of the UK for financial support for the visit of SGD to Nottingham. SGD thanks the Department of Mathematics, University of Nottingham for the warm hospitality extended to him during this visit.
REFERENCES 1. 2. 3. 4. 5.
6.
K.L. Sebastian, Phys. Lett., 98A, 39 (1983); Phys. Rev., B31, 6976 (1985). K.W. Sulston, A.T. Amos and S.G. Davison, Phys. Rev., B37, 9121 (1988). A.T. Amos, K.W. Sulston and S.G. Davison, Adv. in Chem. Phys., in press. R.L. Erickson and D.P. Smith, Phys. Rev. Letters, 34, 297 (1975). T.W. Rush and R.L. Erickson, in: Inelastic IonSurface Collisions, Eds. N.H. Tolk, J.C. Tully, W. Heiland and C.W. White (Academic Press, New York) pp. 73-104 (1977). F. Battaglia, T.F. George and A. Lanaro, Surface Sci., 161, 163 (1985).
7
T.B. Grimley, V.C. Jyothi Bhasu and K.L. Sebastian, Surface Sci., 124, 305 (1983). 8. F. Battaglia and T.F. George, J. Chem. Phys., 83, 3847 (1985). 9. K.W. Sulston, A.T. Amos and S.G. Davison, Surface Sci., 197, 555 (1988). 10. S.G. Davison, K.W. Sulston and A.T. Amos, J. Electroanal. Chem., 204, 173 (1986). 11. S.I. Easa and A. Modinos, Surface Sci., 161, 129 (1985). 12. A.T. Amos, S.G. Davison and K.W. Sulston, Phys. Letters, A l l 8 , 471 (1986). 13 N. Rosen and C. Zener, Phys. Rev., 40, 502 (1932).