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Inelastic aspect observed in quasi-resonant charge exchange
312
Nuclear Instruments and Methods in Physics Research B58 (1991) 372-378 North-Holland
Inelastic aspect observed in quasi-resonant charge exchange M. Kato ‘, A.D.F. Kahn and D.J.O’Connor Department
of Physics, The University of Newcastle, Newcastle, NS W 2308, Australia
We studied theoretically ion-yield oscillation, which is understood in terms of quasi-resonant charge exchange. From recent experiments, the ion-yield oscillation, observed in He+ scattered from metal surfaces, disappears as both collision energy and scattering angle decrease. We point out a possible mechanism to explain the disappearance of the ion-yield oscillations by making use of WKB and classical trajectory approximations. The present study suggests that the mechanism is the loss of coherency between two quantum states caused by inelastic scatterings.
1. Introduction At impact velocities less than the Bohr velocity, the ion-atom charge exchange collision is a powerful technique to study the collision complex. In particular, the ion-yield oscillation due to quasi-resonance is informative for the study of scattering potentials or, alternatively, adiabatic electronic states of the collision complex [1,2]. Our increasing understanding of the collision process at surfaces has come about through increasingly sophisticated ion-surface scattering experiments. In reverse, it is important to know, as accurately as possible, the scattering potentials and the electronic states of collision complexes when one analyses experiments. It seems that such knowledge is urgently required as experimental techniques are improving dramatically. For example, previously many researchers in atomic collision in solids used the simple interatomic potentials such as the Bohr, Born-Mayer and Moliere potentials for the energy range considered here, but now most of them frequently use the so-called universal potential (referred to as ZBL potential) [3] in their analysis. However, it is expected that in the next few years we will see that a correction to or an improvement of the ZBL potential is necessary. In this context, this tendency can be seen in this paper. Indeed, in quasi-resonant charge exchange collision, two different adiabatic (scattering) potentials play an important role at the same time. For appropriate combinations of projectiles and surfaces, e.g., He-Sn surface, ion-yield oscillation is observed as a function of both collision energy and scattering angle [4-111. This oscillation can be underi Present address: Department of Physics, Faculty of Science, Ehime University, Matsuyama, Ehime 790, Japan. 0168-583X/91/$03.50
stood in terms of quasi-resonant charge exchange scattering [7,12-151. When one studies the quasi-resonant charge exchange theoretically, the model of a single collision between the incident ion and target atom is applicable as a first approximation [7], so that throughout this paper we assume this model. A phenomenological interpretation of ion-yield oscillation is easy when we employ the adiabatic representation for electronic systems. As is well known, in the adiabatic representation, the electronic part of the total Hamiltonian is diagonalized at any instance by making use of a molecular-state basis set, so that the ion’s motion is given by the adiabatic potentials. Moreover, the non-adiabatic transition between the electronic states is often localized in a small region where the character of the electronic wave functions significantly changes from atomic-like to molecular-like states [16]. From this property, the adiabatic representation provides us a simple qualitative interpretation of the oscillatory behaviour. In the context of gas phase collisions [1,2], in which similar oscillations can be seen, the oscillation caused by quasi-resonance is one kind of the Stiickelburg oscillation [17]. We briefly discuss an interpretation based on ref. [7]. Slow ions, for which ion-yield oscillations are observed, are scattered in a different way depending on adiabatic potentials. Fig. 1 shows two distinct adiabatic potential curves, along which the individual scattering wave packets evolve. In fig. 1, we assume that the upper curve corresponds to the He ion state at a large R, and the He ion approaches the target. Thus, on the He inward trajectory, the wave packet approaches the target along the upper potential curve. As mentioned above, the non-adiabatic electronic transitions take place at a spatially localized region denoted by the square in fig. 1. When the scattering wave packet passes through the transition region, the wave packet splits into two parts
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
M.
Kato et al. /
Inelastic
by the electronic transition, and each part evolves independently. Reflecting at the classical turning point, the wave packets pass through the transition region again (outward trajectory). At the second passage, the electronic transitions lead to mixing of two wave packets, which have experienced different phase development. Thus, interference between them takes place, and, as a result, it manifests the oscillation in both ion and neutral yields. Olson and Smith [18] developed a theory in order to study the ion-yield oscillation, in particular, for the case of gas phase collision. By making use of their theory, we can carry out quantitative analysis about the peak position of ion-yield oscillation as a function of collision energy and scattering angle. As long as two individual phase developments are coherent until the second passage of the transition region, there is no reason why the oscillation disappears for a particular energy range and particular scattering angles. However, the recent experiment, which was made for metal surfaces, shows differences from the gas phase collision case. The remarkable difference is disappearance of oscillations at low collision energies and small scattering angles (see fig. 2). In fig. 2(a), most of the measurements were made by varying the incidence energy at a fixed scatte~ng angle, and then, monitoring the incidence energies corresponding to the He+ yield oscillation maxima, but for the small angle and low energy range, we carefully measured He+ yield as a function of scattering angle at a fixed incidence energy [see fig. 2(b)]. As shown in fig. 2, the oscillatory behaviour disappears at low energies and small scattering angles, but the He+ yield does not. Neither the re-ionization mechanism nor the finite resolution inherent in experiments explains this disappearance. Indeed, the re-ionization works only when the ion incidence energy exceeds a critical energy [19]. Therefore, two phase developments of quantum state
E
/
lR
Rc Fig. 1. The schematical figure for the adiabatic potentials. The upper and lower curves correspond to the antibonding and bonding states, respectively. The square indicates the transition region, where transitions take place from one potential curve to the other.
aspect
373
in chargeexchange 2000 1800
;
(a)
1
1600
* -
l
.
.
l i
800.
3
600
a
400 -
.
. .
.
l
.
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.
.
l
1000 s i=
.
l
1400 _
b t: 1200 I77
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-
e*
'
./
200 0
0
20
40
80
60 SCATTERING
ANGLE
100
120
140
ldegl
_. 2 i
0.7 -
;
06.
"
20
30
25 SCATTERING
ANGLE
35
40
ldegt
Fig. 2. (a) The observed positions of the oscillation maxima in He+ yield off Sn surface. The arrows indicate the positions corresponding to those in fig. 2(b). (b) Two sets of He+ yield (divided by scattering cross sections) off Sn at constant collisional energy Ea. There is a small peak in the E, =lOOO eV data as indicated by arrow a, and no peak above the noise for the data taken at E, = 800 eV. Arrow h indicates the expected angular position of the peak.
are incoherent for this range of energy and scattering angle. Here, it should be noted that there is a well-known mechanism to interfere the ion-yield oscillation. i.e. neu~ah~tion: in particular, the Auger neutralization [12-151 and, for some cases, the resonant neutralization between the conduction electrons of the target and the ion’s excitated state [13]. Roughly speaking, as the ion survival probability decreases, the amplitude of ion-yield oscillation also decreases [12-U] and, as a result, the oscillation would disappear. As a matter of fact, it has been expected that the ion survival probability decreases monotonically as the incidence energy decreases at a fixed scattering angle. Actually, however, this expectation is not generally realized, in particular, for a very low incidence energy [21]. Although the neutralization is one of the strong candidates for a mechanism to explain the disappearance of ion-yield oscillation, we will propose another mechanism. II. PARTICLE
SCATTERING
374
M. Kato et al. / Inelastic aspect in charge exchange
2. Proposed mechanism
For simplicity, we assume that the system contains only one electron, for which we employ the two-state approximation, and we assume that the interaction depends only on R. We use the adiabatic representation, and by making use of the Dirac notation we do not explicitly express the electronic coordinates. The time independent Schrodinger equation for the total system is
-$:+Y(R)+H,(R) G(R)=&(R), 1
(1)
where H,(R) is the electronic part of the Hamiltonian, V(R) is the internuclear potential, E is the collision energy and p is the reduced mass. In accordance with standard procedure, we expand the total wave function #(R) into 4(R)
=&CR)
lxa(R))
+ F,(R)
lxb(R)).
(2)
The expansion coefficient F,(R) describes the relative motion of nuclei. By substituting eq. (2) into eq. (1) and by multiplying the bra-vector, we have a coupled equation
;v;+
U,(R) F,(R)
-
-
1
&3Z(Xn(R) Iv~lxm(R))~~&m(R)
+(x,(R)
W,‘Ixm(R)>Fm(R>] =EF,(R)>
(3)
where n and m stand for a or b, and all V'S operate only to the nearest neighbour functions in the right-hand side. In eq. (3) we have used the property of the adiabatic electronic states, so that U,(R) = V(R) + E,,(R) is the adiabatic potential, where E,(R) is the electronic orbital energy. The second bracket of the left-hand side of eq. (3) causes non-adiabatic transition between electronic states. In the light of p + m (m is the mass of the electron), we may neglect (x,(R) 102 Ix, (R))&,(R)
and regard (x,(R)IG
as a perturbation to F,‘(R) -&$+Ufi(R)
1
Ixm(R))~vRG(R)
that satisfies
F,“(R)=EF,‘(R).
(4)
When the non-adiabatic transition is spatially well localized as usual in quasi-resonance, the relative motion of nuclei is well approximated by a scattering process by the potential U,(R), n = a or b, depending on the charge state. The coupled equation may be solved by several techniques. In any technique, as long as the non-adiabatic transition is well localized, it is permissible to think that each component of the total wavefunction, eq. (2), evolves along its own adiabatic potential curve. This was a key by which the interpreta-
tion [7] reviewed in section 1 becomes meaningful. Moreover, as long as the semiclassical description for the relative motion of nuclei is permissible, we may consider the ion’s motion in terms of classical mechanics with some care. Since two scattering potentials contribute to ion scatterings, the interference between two quantum states can be restated as the interference between two trajectories of ions with different impact parameters (one kind of the interference scattering [20]). Within the above model, no matter how large difference in scattering potentials exists, two quantum phases may be mutually coherent. However, actually, the surface causes inelastic effect on ion scatterings. In general, the probability of inelastic process depends on the ion’s trajectory, so that two phase developments are disturbed in an incoherent way. When a difference in the two ion’s trajectories become larger than the de Broglie wavelength of the relative motion of nuclei, the disturbance to the mutual coherency would become appreciable. As the difference in the two ion’s trajectories becomes larger, the disturbance to coherency becomes larger, and, as a result, the ion-yield oscillation would disappear because of the loss of mutual coherency In this paper, we will demonstrate that the condition under which no oscillations are observed corresponds to the region where a large difference in the two scattering processes is observed, and by doing so, we will show that the mechanism proposed above is promising in explaining the disappearance of ion-yield oscillations. This will be done by analysing the experimental data with the aid of WKB and classical trajectory approximations.
3. Analysis 3.1. WKB approximation
[18]
In the WKB approximation, the action plays an important role as it does in classical mechanics. The scattering amplitude f(0) is given by a summation of oscillatory terms with respect to quantum numbers, and each oscillatory term contains its own action as a phase factor of the exponential function. For massive particles, many oscillatory terms contribute to f(s). As a result, the actual ion’s motion is given by the trajectory that gives a stationary phase. Thus, in quasi-resonance, the interference between two such trajectories results in the ion yield oscillation. In order to solve the Schrodinger equation, we need the appropriate boundary condition. Here we assume, the plane wave state as the incident wave function, which is usually assumed for scattering problems. Then, we use the partial wave expansion method, and the action can be regarded as a function of the collision energy E and the angular momentum J. At the classical
M. Kato et al. / Inelastic aspect in charge exchange
limit, the condition that gives the stationary phase states that the scattering amplitude f (8) takes the form f(8)
= If(@) I exP{;a(E,J)}.
and the action ol(E,J) ff(E,J)
= 2
is given by
72p[Ej RO
+(71-
(5)
U(R)]
-fZ/R2
B)$/&zb,
tive energy positions of the oscillation peaks, the important quantity is 6. Therefore, in what follows, we do not consider p, and we treat # as a parameter. It should be noted that at some limit (Landau-Zener model), I$ is given by 7r/4 [18]. The explicit expression of the phase difference is
6=$z
dR (6)
-
where U(R) is the scattering potential, R, is the classical turning point given by the root of the integrand, and b is the impact parameter, which satisfies the relation J = b@. The scattering angle 8 is given by e=n-2J
1
5o I 41 \/2p[E-
U(R)]
dR -J2/R2
R2 ’
(7)
In our problem, two different scattering amplitudes contribute to the ion scattering amplitude. Phenomenological consideration about electronic transitions at R - R, (R, is the mixing distance, see fig. 1) permits us to write the ion survival scattering amplitude by [18]
r,,,=0(#)[1-4p(l
-p)
sir? +(a+#)],
(9)
where 6 is cyb- (Ye, and a(0) is the differential scattering cross section. u( r3) is not a rapidly varying function, so that yion exhibits oscillations and the relation Si#=2nn,
n=0,1,2,...,
(10)
gives the peak ion-yield. As introduced above, p is the transition probability between two different adiabatic electronic states, and r#~ is the additional phase associated with the electronic transitions. Thus, p and I#Jhave no direct relations to the phase development along the potential curves. Moreover, p and $I are expected to be slowly varying functions of both E and b [‘7], in the energy range of our interest. Hence, as we are concerned with the rela-
[
“f--d1 -
m/1 - U,(R)/E j R,
U,( R)/E
- b;/R2
dR
- b,2,‘R2 dR
01) where t&(R) and U,(R) are given by the potential curves a and b for R < R,, respectively; on the other hand, for R > R,, U,(R) and iJ,( R) are equal to the potential that corresponds to the He ion state (see fig. 1). R, and R, are the turning points for U,(R) and U,(R), respectively. h, and b, are the impact parameters that give the same scattering angle, so that the ions, scattered by U,(R) and U,(R), can be simultaneously detected by the detector. Therefore, h, and b, must satisfy 8=a-2baj0
where p is the transition probability from one electron state to the other, and + is the phase shift caused by the transition. (Y, ((Y,,) is the action that is calculated by integrating along the potential curve a (b) for the separations less than R,, and by integrating along the potential curve of the He ion state for the separations greater than R,. In other words, the scattering potentials are different only for the separations less than R, because both the incident and the detected particles are the He ion. Since ,f,( 8) -fi,( 8), we find that the ion survival probability q, takes the form
375
=?r-2b,
1
dR
R, 41 - Ua( R)/E
- b,2,‘R2 R=
co
IR, {l
1 - Ub( R)/E
- b;/R2
dR z R .
(121
In other words, two different ion trajectories contribute to the ion yield oscillation. We have to solve eqs. (10, (11) and (12) simultan~usly, but the current computers allow us to solve them numerically. For our convenience, we separate the adiabatic potential into the electronic orbital energy E,,(R), (n = a,b), and the other part V(R). In this separation, E,(R) is thought to be the energy level of the adiabatic electronic state, constructed from a d state of the target and He 1s state. There are two energy levels E,(R) and Eb( R), which correspond to the antibonding and bonding states, respectively. V(R) is the internuclear potential. For many-electron atoms, V(R) can be regarded as an average interatomic potential. We analyse the experimental data made for He-Sn surface. In this paper, we assume that E,(R) is given by the electronic orbital energy that was calculated by Tsuneyuki and Tsukada 1221, and we assume that V(R) + Eb( R) is equal to the ZBL potential (denoted by VZBL)
L31.
Fig. 3 shows the experimental data and the theoretical curves calculated by WKB approximation. The open circlesindicate the same set of the data as shown in fig. 2(a). For large angles (more than - 90”), the peak yield energies of each sequence of the data are essenII. PARTICLE SCATTERING
376
M. Kato et al. / Inelastic aspect in charge exchange He ---> Sn WKB Approximation
tween two ion trajectories, which lead to the same scattering angle in the missing oscillation range, is very large: in fact, as shown in table 1, the differences 4 - b, and R, - R, for those trajectories exceed 0.04 A. This value is one order of the magnitude larger than the de Broglie wavelength of the relative motion of nuclei. Unfortunately, the WKB framework is not convenient to visualize directly this large difference in ion trajectories. Instead, we will visualize it by using CTA in the followings.
2000 5‘ L
1800
&
1600
w"
1400
5
1200 1000
0 ii!
800
F
600
z
400
E
200
a
30
60
90
120
150
180
SCATTERING ANGLE (deg) Fig. 3. The experimental results (open circles) and the best fitting curves to the experiment by WKB approximation, in which we assumed R, = 1.43 A and + = n/4. n’s indicate the index number in eq. (10).
tially
constant.
As
the
scattering
angle
decreases,
points
deviate
If we approximate the ion’s trajectories by a single trajectory given by a single, with meaning of some average, scattering potential U,(R), Then, as a first approximation, eqs. (11) and (12) reduce to
U,(R) - h(R)
the
that constant value of energy. As mentioned in section 1, no oscillatory behaviour for the range of low energies and small scattering angles was observed. We refer to this range as missing oscillation range. The missing oscillation range starts from the region where the lines interpolated from the data points begin to bend significantly. Except for the missing oscillation range, we obtained the best fitting curves to the experimental data when we assume that R, = 1.43 A, 9 = n/4. This choice of parameters agrees with refs. [18,22]. As shown in fig. 3, the missing oscillation range coincides with the region where each theoretical curve is bent significantly. According to our detailed analysis, the difference bedata
3.2. Classical trajectory approximation
/2&E-
from
U,(R)]
dR
’
-J’/R*
(13)
and e=a-2J
1 sc / Ro /2~[ E - U,(R)]
q> R
- J2/R2
(14)
respectively. These expressions are identical with the classical trajectory approximation [7] (referred to as CTA). In CTA, the ion comes along a single classical trajectory. It is most reasonable to choose this classical trajectory as a solution at the classical limit of eq. (4). However, here, some questions arise because eq. (4) forms two equations for n = a and b. According to
Table 1 Values of variables for the solution of eqs. (lo), (11) and (12). n is the index appearing in eq. (10) and fig. 3; 6, (R,,) and b, (R,) are the impact parameters (turning points) corresponding to the bonding and antibonding state potentials, respectively. They satisfy eqs. (lo), (11) and (12); 8 and En are the scattering angle and the peak yield energy satisfying eqs. (lo), (11) and (12); X is the de Broglie wavelength of the relative motion of nuclei; column Yes/No indicates whether ion yield oscillations are observed (Yes) or not (No) n
b, (A)
b, (A)
R, (A)
R, (A)
fl (deg)
-% (eV)
x (A)
Yes/No
I 7 7 7 8 8 8 8 9 9 9 9 10 10
0.700 0.600 0.290 0.210 0.600 0.550 0.350 0.240 0.600 0.550 0.350 0.240 0.600 0.300
0.783 0.649 0.293 0.211 0.651 0.587 0.357 0.242 0.657 0.592 0.357 0.243 0.670 0.306
0.758 0.650 0.367 0.309 0.668 0.617 0.435 0.353 0.690 0.638 0.457 0.378 0.723 0.444
0.853 0.708 0.375 0.314 0.732 0.665 0.449 0.361 0.764 0.695 0.474 0.389 0.815 0.462
17.4 17.1 40.7 59.5 22.4 23.4 38.9 60.6 28.6 29.5 46.3 69.3 36.7 63.0
491 787 1503 1599 560 674 1040 1162 398 488 785 880 272 640
0.006 0.005 0.004 0.004 0.006 0.006 0.004 0.004 0.007 0.006 0.005 0.005 0.007 0.006
No No Yes Yes No No
Yes Yes No
No Yes Yes No Yes
M.
Kateet al. / Inelasticaspectin chargeexchange
He ---a Sn
s
25
z E E P ii F
20001 1800
"
Classical Trajectory Approximation *.
*
'.
3
"
*.
-
"
1
t
lSOOl1400. 1200 1000 -
377
reflects the fact that the difference in two ion’s trajectories leading to ion yield oscillations is not negligible in the missing oscillation range (see table 1). Speaking from the viewpoint of how CTA is a good approximation, this large departure means that the CTA approximation breaks down in the missing oscillation range.
800600 -
4. Discussions Y
I 0
30
60
90
120
150
180
SCATTERING ANGLE (deg)
Fig. 4. The results of the classical trajectory appro~mation, in which we assumed that R, = 1.43 w and ri,= ?r/4 (the same values as used in fig. 3). The solid lines show the result of choice-(i): the average of the bonding and antibonding state potentials used as a scattering potential. The dashed lines show the result of choice-(ii): the bonding state potential. The dashed-dotted lines show the result of choice-(iii): the antibonding state potential. For all cases, scattering potentials were determined by the assumption I’(R) + Eb( R) = VZaL(R). The experimental data are also shown by the open circles.
section 2, two equations for n = a and b should be treated in equal footing, and thus, two classical trajectories for n = a and b should be considered. However, within the CTA framework, we have to approximate two ion trajectories by a single ion trajectory. With this in mind, there would be three trial scattering potentials for the region R -C R,: (i) V(R) + [E,(R) + Eh( R)J/2, (ii) V(R) + Eb( R) and (iii) V(R) + E,(R) are regarded as the scattering potentials, respectively. Fig. 4 shows the result of CTA. In fig. 4, the solid, dashed and dashed-dotted lines correspond to choice-(i), (ii) and (iii), respectively. For each choice, in order to compare CTA with WKB, we used the scattering potential consistent with the assumption made in WKB, viz., V(R) + Eb( R) = F&(R). Therefore, for example, the scattering potential used in choice-(iii) is given by l’(R) + E,(R) = V,,,(R) + E,(R) - Eb( R). In the analysis, we assumed that R, = 1.43 A and $I = ~/4, i.e., the same values of parameters as used in the WKB analysis. It should be noted that choice-(i) is almost identical with the WKB results shown in fig. 3. This suggests that when we employ CTA, the actual scattering process can be approximated by the average potential between two adiabatic potentials, that is, the best choice of scattering potential is choice-(i). As shown in fig. 4, the missing oscillation range corresponds to the range where the three curves begin to depart from each other significantly. This departure is far larger than experimental error. With the WKB analysis in mind, we notice that this larger departure
From the CTA viewpoint, the missing oscillation range corresponds to the region where a large uncertainty in ion trajectories exists. On the other hand, from the WKB viewpoint, this uncertainty corresponds to a large difference in ion trajectories, which lead eventually to the same scattering angle. According to our analysis made for other sets of values of the parameters (R,, Cp) and several scattering potentials, the large difference between two trajectories can be seen generally in the range of low collision energies and small scattering angles, and. moreover, such a range just corresponds to a region where the theoretical curves bend significantly as shown in fig. 3. Even for the other surfaces (Ga [7], Pb [7,23]), the disappearance of oscillations appears to be seen in the low energy and small scattering angle range in which each sequence of the peak yield energies is bending significantly. These are arguments in favor of our proposed mechanism. In the proposed mechanism, the loss of mutual coherency is caused by inelastic processes. in which the excitation probabilities depend on ion trajectories. There may be several inelastic processes. They include the electron-hole pair excitations, the Auger and the resonant neutralizations. We should note that the meaning of neutralization mentioned here is different from that of neutralization mentioned in section 1. In this paragraph, we included the neutralization as one of the inelastic processes. An alternative explanation for the missing oscillation is that the model of single collisions would break down for grazing scatterings. However, the experiment was performed with a polycrystalline sample, so that the surface was not atomically flat. This means that most of the detected particles are those which have experienced single collisions with surface atoms. As mentioned briefly in Section 1, the reduction of ion survival probability due to neutralization is still a promising mechanism to explain the missing oscillation. On the other hand. the present study suggests that the proposed mechanism is also promising. At present, we do not have a clear answer to the question of which mechanism is the more important or whether both mechanisms are equally important. Further studies to clarify the question both experimentally and theoretically are underway. 11. PARTICLE SCATTERING
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M. Kafo et al. / Inelastic
References G. Lockwood, H. Helbig and E. Everhart, Phys. Rev. 132 (1963) 2078; E. Everhart. ibid. 2083. 121 D.C. Lorents and A. Aberth, Phys. Rev. Al39 (1965) 1017; R.P. March and F.T. Smith, ibid. 1025. 131 J.F. Ziegler, J.P. Biersack and U. Littmark, Proc. Int. Ion Engineering Congress, ISIAT’83 & IPAT (1983) 1861; D.J. OConnor and D.R. Biersack, NucI. Instr. and Meth. B15 (1986) 14. 141 R.L. Erickson and D.P. Smith, Phys. Rev. Lett. 34 (1975) 297. 151 H.H. Brogensma and T.M. Buck, Surf. Sci. 53 (1975) 649; Nucl. Instr. and Meth. 132 (1976) 559. 161 T.W. Rusch and R.L. Erikson, J. Vat. Sci. Technol. 13 (1976) 374. I71 N.H. ToIk, J.C. Tully, J. Kraus, C.W. White and S.H. Neff, Phys. Rev. Lett. 36 (1976) 747. 181 A. Zartner, E. TagIauer and W. Heiland, Phys. Rev. Lett. 40 (1978) 1259.
aspect in charge exchange
[9] W.L. Baun, Surf. Sci. 100 (1980) L491 [lo] F. Shoji, Y. Nakayama and T. Hanawa, Surf. Sci. 163 (1985) L745. [ll) R. Souda, T. Aizawa, C. Oshima, M. Aono, S. Tsuneyuki and M. Tsukada. Surf. Sci. 187 (1987) L592. [12] J.C. Tully, Phys. Rev. B16 (1977) 4324. (131 W. Bloss and D. Hone, Surf. Sci. 72 (1978) 277. 1141 R.K. Janev, P.S. Krstic and M.J. Rakovic, preprint. [15] S.I. Easa and A. Modinis, Surf. Sci. 161 (1985) 129. [16] E.E. Nikitin and S. Ya. Umanski, Theory of Slow Atomic Collision (Springer, Berlin, 1984). [17] E.G.G. Stiickelburg, Helv. Phys. Acta 5 (1932) 369. [18] R.E. Olson and F.T. Smith, Phys. Rev. A3 (1971) 1607. 1191 R. Souda and M. Aono. Nucl. Instr. and Meth. B15 (1986) 114. [20] N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions. 3rd ed. (Oxford University Press. 1965) p. 102. [21] H. Akazawa and Y. Murata, Phys. Rev. 839 (1989) 3449. [22] S. Tsuneyuki and M. Tsukada, Phys. Rev. B34 (1986) 5758. [23] D.J. O’Connor and R. Beardwood, NucI. Instr. and Meth. B48 (1990) 358.
Nuclear Instruments and Methods in Physics Research B58 (1991) 372-378 North-Holland
Inelastic aspect observed in quasi-resonant charge exchange M. Kato ‘, A.D.F. Kahn and D.J.O’Connor Department
of Physics, The University of Newcastle, Newcastle, NS W 2308, Australia
We studied theoretically ion-yield oscillation, which is understood in terms of quasi-resonant charge exchange. From recent experiments, the ion-yield oscillation, observed in He+ scattered from metal surfaces, disappears as both collision energy and scattering angle decrease. We point out a possible mechanism to explain the disappearance of the ion-yield oscillations by making use of WKB and classical trajectory approximations. The present study suggests that the mechanism is the loss of coherency between two quantum states caused by inelastic scatterings.
1. Introduction At impact velocities less than the Bohr velocity, the ion-atom charge exchange collision is a powerful technique to study the collision complex. In particular, the ion-yield oscillation due to quasi-resonance is informative for the study of scattering potentials or, alternatively, adiabatic electronic states of the collision complex [1,2]. Our increasing understanding of the collision process at surfaces has come about through increasingly sophisticated ion-surface scattering experiments. In reverse, it is important to know, as accurately as possible, the scattering potentials and the electronic states of collision complexes when one analyses experiments. It seems that such knowledge is urgently required as experimental techniques are improving dramatically. For example, previously many researchers in atomic collision in solids used the simple interatomic potentials such as the Bohr, Born-Mayer and Moliere potentials for the energy range considered here, but now most of them frequently use the so-called universal potential (referred to as ZBL potential) [3] in their analysis. However, it is expected that in the next few years we will see that a correction to or an improvement of the ZBL potential is necessary. In this context, this tendency can be seen in this paper. Indeed, in quasi-resonant charge exchange collision, two different adiabatic (scattering) potentials play an important role at the same time. For appropriate combinations of projectiles and surfaces, e.g., He-Sn surface, ion-yield oscillation is observed as a function of both collision energy and scattering angle [4-111. This oscillation can be underi Present address: Department of Physics, Faculty of Science, Ehime University, Matsuyama, Ehime 790, Japan. 0168-583X/91/$03.50
stood in terms of quasi-resonant charge exchange scattering [7,12-151. When one studies the quasi-resonant charge exchange theoretically, the model of a single collision between the incident ion and target atom is applicable as a first approximation [7], so that throughout this paper we assume this model. A phenomenological interpretation of ion-yield oscillation is easy when we employ the adiabatic representation for electronic systems. As is well known, in the adiabatic representation, the electronic part of the total Hamiltonian is diagonalized at any instance by making use of a molecular-state basis set, so that the ion’s motion is given by the adiabatic potentials. Moreover, the non-adiabatic transition between the electronic states is often localized in a small region where the character of the electronic wave functions significantly changes from atomic-like to molecular-like states [16]. From this property, the adiabatic representation provides us a simple qualitative interpretation of the oscillatory behaviour. In the context of gas phase collisions [1,2], in which similar oscillations can be seen, the oscillation caused by quasi-resonance is one kind of the Stiickelburg oscillation [17]. We briefly discuss an interpretation based on ref. [7]. Slow ions, for which ion-yield oscillations are observed, are scattered in a different way depending on adiabatic potentials. Fig. 1 shows two distinct adiabatic potential curves, along which the individual scattering wave packets evolve. In fig. 1, we assume that the upper curve corresponds to the He ion state at a large R, and the He ion approaches the target. Thus, on the He inward trajectory, the wave packet approaches the target along the upper potential curve. As mentioned above, the non-adiabatic electronic transitions take place at a spatially localized region denoted by the square in fig. 1. When the scattering wave packet passes through the transition region, the wave packet splits into two parts
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
M.
Kato et al. /
Inelastic
by the electronic transition, and each part evolves independently. Reflecting at the classical turning point, the wave packets pass through the transition region again (outward trajectory). At the second passage, the electronic transitions lead to mixing of two wave packets, which have experienced different phase development. Thus, interference between them takes place, and, as a result, it manifests the oscillation in both ion and neutral yields. Olson and Smith [18] developed a theory in order to study the ion-yield oscillation, in particular, for the case of gas phase collision. By making use of their theory, we can carry out quantitative analysis about the peak position of ion-yield oscillation as a function of collision energy and scattering angle. As long as two individual phase developments are coherent until the second passage of the transition region, there is no reason why the oscillation disappears for a particular energy range and particular scattering angles. However, the recent experiment, which was made for metal surfaces, shows differences from the gas phase collision case. The remarkable difference is disappearance of oscillations at low collision energies and small scattering angles (see fig. 2). In fig. 2(a), most of the measurements were made by varying the incidence energy at a fixed scatte~ng angle, and then, monitoring the incidence energies corresponding to the He+ yield oscillation maxima, but for the small angle and low energy range, we carefully measured He+ yield as a function of scattering angle at a fixed incidence energy [see fig. 2(b)]. As shown in fig. 2, the oscillatory behaviour disappears at low energies and small scattering angles, but the He+ yield does not. Neither the re-ionization mechanism nor the finite resolution inherent in experiments explains this disappearance. Indeed, the re-ionization works only when the ion incidence energy exceeds a critical energy [19]. Therefore, two phase developments of quantum state
E
/
lR
Rc Fig. 1. The schematical figure for the adiabatic potentials. The upper and lower curves correspond to the antibonding and bonding states, respectively. The square indicates the transition region, where transitions take place from one potential curve to the other.
aspect
373
in chargeexchange 2000 1800
;
(a)
1
1600
* -
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.
.
l i
800.
3
600
a
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.
. .
.
l
.
l
.
.
l
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.
l
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b t: 1200 I77
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Xo* t b*
-
e*
'
./
200 0
0
20
40
80
60 SCATTERING
ANGLE
100
120
140
ldegl
_. 2 i
0.7 -
;
06.
"
20
30
25 SCATTERING
ANGLE
35
40
ldegt
Fig. 2. (a) The observed positions of the oscillation maxima in He+ yield off Sn surface. The arrows indicate the positions corresponding to those in fig. 2(b). (b) Two sets of He+ yield (divided by scattering cross sections) off Sn at constant collisional energy Ea. There is a small peak in the E, =lOOO eV data as indicated by arrow a, and no peak above the noise for the data taken at E, = 800 eV. Arrow h indicates the expected angular position of the peak.
are incoherent for this range of energy and scattering angle. Here, it should be noted that there is a well-known mechanism to interfere the ion-yield oscillation. i.e. neu~ah~tion: in particular, the Auger neutralization [12-151 and, for some cases, the resonant neutralization between the conduction electrons of the target and the ion’s excitated state [13]. Roughly speaking, as the ion survival probability decreases, the amplitude of ion-yield oscillation also decreases [12-U] and, as a result, the oscillation would disappear. As a matter of fact, it has been expected that the ion survival probability decreases monotonically as the incidence energy decreases at a fixed scattering angle. Actually, however, this expectation is not generally realized, in particular, for a very low incidence energy [21]. Although the neutralization is one of the strong candidates for a mechanism to explain the disappearance of ion-yield oscillation, we will propose another mechanism. II. PARTICLE
SCATTERING
374
M. Kato et al. / Inelastic aspect in charge exchange
2. Proposed mechanism
For simplicity, we assume that the system contains only one electron, for which we employ the two-state approximation, and we assume that the interaction depends only on R. We use the adiabatic representation, and by making use of the Dirac notation we do not explicitly express the electronic coordinates. The time independent Schrodinger equation for the total system is
-$:+Y(R)+H,(R) G(R)=&(R), 1
(1)
where H,(R) is the electronic part of the Hamiltonian, V(R) is the internuclear potential, E is the collision energy and p is the reduced mass. In accordance with standard procedure, we expand the total wave function #(R) into 4(R)
=&CR)
lxa(R))
+ F,(R)
lxb(R)).
(2)
The expansion coefficient F,(R) describes the relative motion of nuclei. By substituting eq. (2) into eq. (1) and by multiplying the bra-vector, we have a coupled equation
;v;+
U,(R) F,(R)
-
-
1
&3Z(Xn(R) Iv~lxm(R))~~&m(R)
+(x,(R)
W,‘Ixm(R)>Fm(R>] =EF,(R)>
(3)
where n and m stand for a or b, and all V'S operate only to the nearest neighbour functions in the right-hand side. In eq. (3) we have used the property of the adiabatic electronic states, so that U,(R) = V(R) + E,,(R) is the adiabatic potential, where E,(R) is the electronic orbital energy. The second bracket of the left-hand side of eq. (3) causes non-adiabatic transition between electronic states. In the light of p + m (m is the mass of the electron), we may neglect (x,(R) 102 Ix, (R))&,(R)
and regard (x,(R)IG
as a perturbation to F,‘(R) -&$+Ufi(R)
1
Ixm(R))~vRG(R)
that satisfies
F,“(R)=EF,‘(R).
(4)
When the non-adiabatic transition is spatially well localized as usual in quasi-resonance, the relative motion of nuclei is well approximated by a scattering process by the potential U,(R), n = a or b, depending on the charge state. The coupled equation may be solved by several techniques. In any technique, as long as the non-adiabatic transition is well localized, it is permissible to think that each component of the total wavefunction, eq. (2), evolves along its own adiabatic potential curve. This was a key by which the interpreta-
tion [7] reviewed in section 1 becomes meaningful. Moreover, as long as the semiclassical description for the relative motion of nuclei is permissible, we may consider the ion’s motion in terms of classical mechanics with some care. Since two scattering potentials contribute to ion scatterings, the interference between two quantum states can be restated as the interference between two trajectories of ions with different impact parameters (one kind of the interference scattering [20]). Within the above model, no matter how large difference in scattering potentials exists, two quantum phases may be mutually coherent. However, actually, the surface causes inelastic effect on ion scatterings. In general, the probability of inelastic process depends on the ion’s trajectory, so that two phase developments are disturbed in an incoherent way. When a difference in the two ion’s trajectories become larger than the de Broglie wavelength of the relative motion of nuclei, the disturbance to the mutual coherency would become appreciable. As the difference in the two ion’s trajectories becomes larger, the disturbance to coherency becomes larger, and, as a result, the ion-yield oscillation would disappear because of the loss of mutual coherency In this paper, we will demonstrate that the condition under which no oscillations are observed corresponds to the region where a large difference in the two scattering processes is observed, and by doing so, we will show that the mechanism proposed above is promising in explaining the disappearance of ion-yield oscillations. This will be done by analysing the experimental data with the aid of WKB and classical trajectory approximations.
3. Analysis 3.1. WKB approximation
[18]
In the WKB approximation, the action plays an important role as it does in classical mechanics. The scattering amplitude f(0) is given by a summation of oscillatory terms with respect to quantum numbers, and each oscillatory term contains its own action as a phase factor of the exponential function. For massive particles, many oscillatory terms contribute to f(s). As a result, the actual ion’s motion is given by the trajectory that gives a stationary phase. Thus, in quasi-resonance, the interference between two such trajectories results in the ion yield oscillation. In order to solve the Schrodinger equation, we need the appropriate boundary condition. Here we assume, the plane wave state as the incident wave function, which is usually assumed for scattering problems. Then, we use the partial wave expansion method, and the action can be regarded as a function of the collision energy E and the angular momentum J. At the classical
M. Kato et al. / Inelastic aspect in charge exchange
limit, the condition that gives the stationary phase states that the scattering amplitude f (8) takes the form f(8)
= If(@) I exP{;a(E,J)}.
and the action ol(E,J) ff(E,J)
= 2
is given by
72p[Ej RO
+(71-
(5)
U(R)]
-fZ/R2
B)$/&zb,
tive energy positions of the oscillation peaks, the important quantity is 6. Therefore, in what follows, we do not consider p, and we treat # as a parameter. It should be noted that at some limit (Landau-Zener model), I$ is given by 7r/4 [18]. The explicit expression of the phase difference is
6=$z
dR (6)
-
where U(R) is the scattering potential, R, is the classical turning point given by the root of the integrand, and b is the impact parameter, which satisfies the relation J = b@. The scattering angle 8 is given by e=n-2J
1
5o I 41 \/2p[E-
U(R)]
dR -J2/R2
R2 ’
(7)
In our problem, two different scattering amplitudes contribute to the ion scattering amplitude. Phenomenological consideration about electronic transitions at R - R, (R, is the mixing distance, see fig. 1) permits us to write the ion survival scattering amplitude by [18]
r,,,=0(#)[1-4p(l
-p)
sir? +(a+#)],
(9)
where 6 is cyb- (Ye, and a(0) is the differential scattering cross section. u( r3) is not a rapidly varying function, so that yion exhibits oscillations and the relation Si#=2nn,
n=0,1,2,...,
(10)
gives the peak ion-yield. As introduced above, p is the transition probability between two different adiabatic electronic states, and r#~ is the additional phase associated with the electronic transitions. Thus, p and I#Jhave no direct relations to the phase development along the potential curves. Moreover, p and $I are expected to be slowly varying functions of both E and b [‘7], in the energy range of our interest. Hence, as we are concerned with the rela-
[
“f--d1 -
m/1 - U,(R)/E j R,
U,( R)/E
- b;/R2
dR
- b,2,‘R2 dR
01) where t&(R) and U,(R) are given by the potential curves a and b for R < R,, respectively; on the other hand, for R > R,, U,(R) and iJ,( R) are equal to the potential that corresponds to the He ion state (see fig. 1). R, and R, are the turning points for U,(R) and U,(R), respectively. h, and b, are the impact parameters that give the same scattering angle, so that the ions, scattered by U,(R) and U,(R), can be simultaneously detected by the detector. Therefore, h, and b, must satisfy 8=a-2baj0
where p is the transition probability from one electron state to the other, and + is the phase shift caused by the transition. (Y, ((Y,,) is the action that is calculated by integrating along the potential curve a (b) for the separations less than R,, and by integrating along the potential curve of the He ion state for the separations greater than R,. In other words, the scattering potentials are different only for the separations less than R, because both the incident and the detected particles are the He ion. Since ,f,( 8) -fi,( 8), we find that the ion survival probability q, takes the form
375
=?r-2b,
1
dR
R, 41 - Ua( R)/E
- b,2,‘R2 R=
co
IR, {l
1 - Ub( R)/E
- b;/R2
dR z R .
(121
In other words, two different ion trajectories contribute to the ion yield oscillation. We have to solve eqs. (10, (11) and (12) simultan~usly, but the current computers allow us to solve them numerically. For our convenience, we separate the adiabatic potential into the electronic orbital energy E,,(R), (n = a,b), and the other part V(R). In this separation, E,(R) is thought to be the energy level of the adiabatic electronic state, constructed from a d state of the target and He 1s state. There are two energy levels E,(R) and Eb( R), which correspond to the antibonding and bonding states, respectively. V(R) is the internuclear potential. For many-electron atoms, V(R) can be regarded as an average interatomic potential. We analyse the experimental data made for He-Sn surface. In this paper, we assume that E,(R) is given by the electronic orbital energy that was calculated by Tsuneyuki and Tsukada 1221, and we assume that V(R) + Eb( R) is equal to the ZBL potential (denoted by VZBL)
L31.
Fig. 3 shows the experimental data and the theoretical curves calculated by WKB approximation. The open circlesindicate the same set of the data as shown in fig. 2(a). For large angles (more than - 90”), the peak yield energies of each sequence of the data are essenII. PARTICLE SCATTERING
376
M. Kato et al. / Inelastic aspect in charge exchange He ---> Sn WKB Approximation
tween two ion trajectories, which lead to the same scattering angle in the missing oscillation range, is very large: in fact, as shown in table 1, the differences 4 - b, and R, - R, for those trajectories exceed 0.04 A. This value is one order of the magnitude larger than the de Broglie wavelength of the relative motion of nuclei. Unfortunately, the WKB framework is not convenient to visualize directly this large difference in ion trajectories. Instead, we will visualize it by using CTA in the followings.
2000 5‘ L
1800
&
1600
w"
1400
5
1200 1000
0 ii!
800
F
600
z
400
E
200
a
30
60
90
120
150
180
SCATTERING ANGLE (deg) Fig. 3. The experimental results (open circles) and the best fitting curves to the experiment by WKB approximation, in which we assumed R, = 1.43 A and + = n/4. n’s indicate the index number in eq. (10).
tially
constant.
As
the
scattering
angle
decreases,
points
deviate
If we approximate the ion’s trajectories by a single trajectory given by a single, with meaning of some average, scattering potential U,(R), Then, as a first approximation, eqs. (11) and (12) reduce to
U,(R) - h(R)
the
that constant value of energy. As mentioned in section 1, no oscillatory behaviour for the range of low energies and small scattering angles was observed. We refer to this range as missing oscillation range. The missing oscillation range starts from the region where the lines interpolated from the data points begin to bend significantly. Except for the missing oscillation range, we obtained the best fitting curves to the experimental data when we assume that R, = 1.43 A, 9 = n/4. This choice of parameters agrees with refs. [18,22]. As shown in fig. 3, the missing oscillation range coincides with the region where each theoretical curve is bent significantly. According to our detailed analysis, the difference bedata
3.2. Classical trajectory approximation
/2&E-
from
U,(R)]
dR
’
-J’/R*
(13)
and e=a-2J
1 sc / Ro /2~[ E - U,(R)]
q> R
- J2/R2
(14)
respectively. These expressions are identical with the classical trajectory approximation [7] (referred to as CTA). In CTA, the ion comes along a single classical trajectory. It is most reasonable to choose this classical trajectory as a solution at the classical limit of eq. (4). However, here, some questions arise because eq. (4) forms two equations for n = a and b. According to
Table 1 Values of variables for the solution of eqs. (lo), (11) and (12). n is the index appearing in eq. (10) and fig. 3; 6, (R,,) and b, (R,) are the impact parameters (turning points) corresponding to the bonding and antibonding state potentials, respectively. They satisfy eqs. (lo), (11) and (12); 8 and En are the scattering angle and the peak yield energy satisfying eqs. (lo), (11) and (12); X is the de Broglie wavelength of the relative motion of nuclei; column Yes/No indicates whether ion yield oscillations are observed (Yes) or not (No) n
b, (A)
b, (A)
R, (A)
R, (A)
fl (deg)
-% (eV)
x (A)
Yes/No
I 7 7 7 8 8 8 8 9 9 9 9 10 10
0.700 0.600 0.290 0.210 0.600 0.550 0.350 0.240 0.600 0.550 0.350 0.240 0.600 0.300
0.783 0.649 0.293 0.211 0.651 0.587 0.357 0.242 0.657 0.592 0.357 0.243 0.670 0.306
0.758 0.650 0.367 0.309 0.668 0.617 0.435 0.353 0.690 0.638 0.457 0.378 0.723 0.444
0.853 0.708 0.375 0.314 0.732 0.665 0.449 0.361 0.764 0.695 0.474 0.389 0.815 0.462
17.4 17.1 40.7 59.5 22.4 23.4 38.9 60.6 28.6 29.5 46.3 69.3 36.7 63.0
491 787 1503 1599 560 674 1040 1162 398 488 785 880 272 640
0.006 0.005 0.004 0.004 0.006 0.006 0.004 0.004 0.007 0.006 0.005 0.005 0.007 0.006
No No Yes Yes No No
Yes Yes No
No Yes Yes No Yes
M.
Kateet al. / Inelasticaspectin chargeexchange
He ---a Sn
s
25
z E E P ii F
20001 1800
"
Classical Trajectory Approximation *.
*
'.
3
"
*.
-
"
1
t
lSOOl1400. 1200 1000 -
377
reflects the fact that the difference in two ion’s trajectories leading to ion yield oscillations is not negligible in the missing oscillation range (see table 1). Speaking from the viewpoint of how CTA is a good approximation, this large departure means that the CTA approximation breaks down in the missing oscillation range.
800600 -
4. Discussions Y
I 0
30
60
90
120
150
180
SCATTERING ANGLE (deg)
Fig. 4. The results of the classical trajectory appro~mation, in which we assumed that R, = 1.43 w and ri,= ?r/4 (the same values as used in fig. 3). The solid lines show the result of choice-(i): the average of the bonding and antibonding state potentials used as a scattering potential. The dashed lines show the result of choice-(ii): the bonding state potential. The dashed-dotted lines show the result of choice-(iii): the antibonding state potential. For all cases, scattering potentials were determined by the assumption I’(R) + Eb( R) = VZaL(R). The experimental data are also shown by the open circles.
section 2, two equations for n = a and b should be treated in equal footing, and thus, two classical trajectories for n = a and b should be considered. However, within the CTA framework, we have to approximate two ion trajectories by a single ion trajectory. With this in mind, there would be three trial scattering potentials for the region R -C R,: (i) V(R) + [E,(R) + Eh( R)J/2, (ii) V(R) + Eb( R) and (iii) V(R) + E,(R) are regarded as the scattering potentials, respectively. Fig. 4 shows the result of CTA. In fig. 4, the solid, dashed and dashed-dotted lines correspond to choice-(i), (ii) and (iii), respectively. For each choice, in order to compare CTA with WKB, we used the scattering potential consistent with the assumption made in WKB, viz., V(R) + Eb( R) = F&(R). Therefore, for example, the scattering potential used in choice-(iii) is given by l’(R) + E,(R) = V,,,(R) + E,(R) - Eb( R). In the analysis, we assumed that R, = 1.43 A and $I = ~/4, i.e., the same values of parameters as used in the WKB analysis. It should be noted that choice-(i) is almost identical with the WKB results shown in fig. 3. This suggests that when we employ CTA, the actual scattering process can be approximated by the average potential between two adiabatic potentials, that is, the best choice of scattering potential is choice-(i). As shown in fig. 4, the missing oscillation range corresponds to the range where the three curves begin to depart from each other significantly. This departure is far larger than experimental error. With the WKB analysis in mind, we notice that this larger departure
From the CTA viewpoint, the missing oscillation range corresponds to the region where a large uncertainty in ion trajectories exists. On the other hand, from the WKB viewpoint, this uncertainty corresponds to a large difference in ion trajectories, which lead eventually to the same scattering angle. According to our analysis made for other sets of values of the parameters (R,, Cp) and several scattering potentials, the large difference between two trajectories can be seen generally in the range of low collision energies and small scattering angles, and. moreover, such a range just corresponds to a region where the theoretical curves bend significantly as shown in fig. 3. Even for the other surfaces (Ga [7], Pb [7,23]), the disappearance of oscillations appears to be seen in the low energy and small scattering angle range in which each sequence of the peak yield energies is bending significantly. These are arguments in favor of our proposed mechanism. In the proposed mechanism, the loss of mutual coherency is caused by inelastic processes. in which the excitation probabilities depend on ion trajectories. There may be several inelastic processes. They include the electron-hole pair excitations, the Auger and the resonant neutralizations. We should note that the meaning of neutralization mentioned here is different from that of neutralization mentioned in section 1. In this paragraph, we included the neutralization as one of the inelastic processes. An alternative explanation for the missing oscillation is that the model of single collisions would break down for grazing scatterings. However, the experiment was performed with a polycrystalline sample, so that the surface was not atomically flat. This means that most of the detected particles are those which have experienced single collisions with surface atoms. As mentioned briefly in Section 1, the reduction of ion survival probability due to neutralization is still a promising mechanism to explain the missing oscillation. On the other hand. the present study suggests that the proposed mechanism is also promising. At present, we do not have a clear answer to the question of which mechanism is the more important or whether both mechanisms are equally important. Further studies to clarify the question both experimentally and theoretically are underway. 11. PARTICLE SCATTERING
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M. Kafo et al. / Inelastic
References G. Lockwood, H. Helbig and E. Everhart, Phys. Rev. 132 (1963) 2078; E. Everhart. ibid. 2083. 121 D.C. Lorents and A. Aberth, Phys. Rev. Al39 (1965) 1017; R.P. March and F.T. Smith, ibid. 1025. 131 J.F. Ziegler, J.P. Biersack and U. Littmark, Proc. Int. Ion Engineering Congress, ISIAT’83 & IPAT (1983) 1861; D.J. OConnor and D.R. Biersack, NucI. Instr. and Meth. B15 (1986) 14. 141 R.L. Erickson and D.P. Smith, Phys. Rev. Lett. 34 (1975) 297. 151 H.H. Brogensma and T.M. Buck, Surf. Sci. 53 (1975) 649; Nucl. Instr. and Meth. 132 (1976) 559. 161 T.W. Rusch and R.L. Erikson, J. Vat. Sci. Technol. 13 (1976) 374. I71 N.H. ToIk, J.C. Tully, J. Kraus, C.W. White and S.H. Neff, Phys. Rev. Lett. 36 (1976) 747. 181 A. Zartner, E. TagIauer and W. Heiland, Phys. Rev. Lett. 40 (1978) 1259.
aspect in charge exchange
[9] W.L. Baun, Surf. Sci. 100 (1980) L491 [lo] F. Shoji, Y. Nakayama and T. Hanawa, Surf. Sci. 163 (1985) L745. [ll) R. Souda, T. Aizawa, C. Oshima, M. Aono, S. Tsuneyuki and M. Tsukada. Surf. Sci. 187 (1987) L592. [12] J.C. Tully, Phys. Rev. B16 (1977) 4324. (131 W. Bloss and D. Hone, Surf. Sci. 72 (1978) 277. 1141 R.K. Janev, P.S. Krstic and M.J. Rakovic, preprint. [15] S.I. Easa and A. Modinis, Surf. Sci. 161 (1985) 129. [16] E.E. Nikitin and S. Ya. Umanski, Theory of Slow Atomic Collision (Springer, Berlin, 1984). [17] E.G.G. Stiickelburg, Helv. Phys. Acta 5 (1932) 369. [18] R.E. Olson and F.T. Smith, Phys. Rev. A3 (1971) 1607. 1191 R. Souda and M. Aono. Nucl. Instr. and Meth. B15 (1986) 114. [20] N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions. 3rd ed. (Oxford University Press. 1965) p. 102. [21] H. Akazawa and Y. Murata, Phys. Rev. 839 (1989) 3449. [22] S. Tsuneyuki and M. Tsukada, Phys. Rev. B34 (1986) 5758. [23] D.J. O’Connor and R. Beardwood, NucI. Instr. and Meth. B48 (1990) 358.