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Information on surface thermal vibrations from kev ion scattering
344
Surface
Science
North-Holland
INFORMATION ON SURFACE ION SCA’ITERING B. POELSEMA
THERMAL
133 (1983) 344-364
Publishing
VIBRATIONS
Company
FROM keV
and L.K. VERHEIJ
lnstirur filr Grenzfbchenforschung
und Vakuumphysik,
Kernforschungsanlage
Jiilich,
Posrfach 1913,
D - 5170 Jiilich, Fed. Rep. of Germany
and A.L. BOERS Kernfrsisch
Received
Versneller Instituut,
18 April
Rijksuniversileir
1983; accepted
for publication
Groningen. Groningen, The NerherlanL
7 July 1983
The possibilities of obtaining information on surface phonons from (quasi-)triple collisions in keV ion scattering (LEIS) are reinvestigated. Sensitivities to the ion-atom interaction potential can be elucidated with a semi-analytical three-atom model without the need for expensive computer simulations. The most reliable information is obtained from measurements along the low index (( 100) and (I 10)) azimuthal directions. A simple recipe is proposed for acquiring potential-independent information on surface thermal vibrations.
1. Introduction It has been shown that LEIS can be used to investigate thermal vibrations of surface atoms [l-6]. Since the ions are probing “ thermal structures” of only two or three atoms, information is obtained on the differences in thermal displacements of nearest neighbours or next nearest neighbours. Accordingly the results can be considered complementary to typical information extracted from diffraction of low energy electrons [7] 06 thermal energy He atoms [8], where characteristic transfer widths of 200 A are involved [9]. In previous papers, the authors have reported about an investigation of the influence of surface thermal vibrations on the backscattering of low energy (keV range) noble gas ions from monocrystalline surfaces. Under some specific conditions the shape of both the energy and the angular distributions of reflected ions is affected considerably by thermal effects [l-3]. At elevated temperatures and small scattering angles a new peak, resulting from scattering from three-atom thermal pits instantaneously present at the surface, was predicted by computer experiments [3,4] and observed experimentally [5]. It was shown that the 0039-6028/83/0000-OOOO/$O3.00
0 1983 North-Holland
B. Poelsema ei al. / Information on surface thermal vibrations
34s
appearance of this QT peak offered the possibility to estimate the surface Debye temperature 8, in a way not (or hardly) depending on intrinsic uncertainties of the method such as the unknown interaction potential and charge exchange effects [6]. For the scattering of 10 keV Kr+ ions over 30’ from a (310) and a (100) row of a Cu(100) face, a linear relationship was found between the QT peak height and the square root of the target temperature and this empirical relationship was reproduced by computer siumulations in a limited temperature range. This phenomenon allowed to give a reliable estimate of the surface Debye temperature 8,. Recently Walker and Martin (WM) derived a simple and more general relationship between the QT peak height and the temperature [ 10,111. When comparing this expression with Monte Carlo simulations (10 keV Kr+, $Jt= 30’9, they found good agreement for scattering along the (100) direction, but not for scattering along the (1 lo} direction [ 111.Therefore it is worthwhile to discuss in more detail the possibility to apply LEIS for investigating surface thermal vibrations. In the present paper, it will be shown that the method proposed by the authors (PVB), even in the simple form given by WM, can be used to obtain a reliable value for B,, provided that certain criteria are met. These criteria can be made explicit in terms of the scattering geometry and primary ion energy by employing a three-atom model [l&13], i.e. without the need to carry out expensive time consuming Monte Carlo calculations as claimed by WM [ 111 and it might even be possible to express these criteria in an analytical form. Moreover, an experimental procedure is given by which the applicability of the method is checked in a direct way. 2. Model
131: ~=expl-p-l(9:+9,2+ff,z)j,
(1)
346
B. Poelsema et al. / Information on surface thermal vibrations
or with 8, = $8, + 6, 8, = $9, - I - n and 6, = +I’$,+ I): E = exp{ -p-‘[@:
+ 2(.$2 + 577+ n2)]},
(la)
where p is the relative mass matom/mion. Thus the exact triple collision (4 = n = 0) energy is given by sr = exp( - fp-%:). Then one may define a quantity 6, relating 5 and n at some given energy E by the expression: 6,=+~(lnsr.-lns)=~2+[n+n2,
(2)
or n=
-1
2 [[*
(46,-
352)“2].
(2a)
So for scattering over 9, from the three-atom configuration a whole range of angles 6 (and n) is possible. We impose the condition that the closest collision takes place with the central atom (see ref. [ 131). Then only the positive root of I varies between $(3S,)‘/’ and f(38,)‘12. Following ref. [13], one obtains for the intensity at E:
where
f(5)= g(t)
=
NO exp i2aa2\/?- tan #
u2+uu+u2 3a2
(34
i’
(dp,/d9,)(dp,/d9,)(dp,/d9,)
(3b)
cos~cos(~-~~,-~)cos(~-~8,+7j)’
with u = x, - x2, u -x2 - x3 (xi is the thermal displacement of the ilh atom), No is the number of incident particles per unit length, pi and 6, are respectively the impact parameter and the scattering angle at atom i, # is the angle of incidence and (I is the root mean square atomic displacement. Expression (3) holds for the case of uncorrelared thermal vibrations. In the presence of correlations, eq. (3a) has to be modified [4,10,11,13]. Expression (3) can be simplified by carrying out the transformation 5 = ($&)‘I2 sin ‘p: (4) The integral
in eq. (4) can readily
1141. In the calculations V(r)
= Z,Z2e2r-’
For the screening
be evaluated
numerically
we apply a Thomas-Fermi
or approximated
potential
Cp(ra-I). function
(5) $a (x) we use Moliere’s
approximation
[ 151, where
B. Poelsema et al. / Information on surface thermal uibrations
347
the screening length a is given by [16]: a = 0.8853&(
2:‘3 + Z;L3)-“2,
with C being an adjustable constant of order unity. Z,, Z,, e and a, have their usual meaning. It has been shown that the three-atom model agrees well with the results of chain scattering simulations [ 131. It has the advantage that the computation times are two or more orders of magnitude smaller than those required for the corresponding Monte Carlo calculations [13]. However, by its nature, the three-atom model disregards essentially collisions with a higher multiplicity than a threefold one. Although this error is inherently present when employing this model, the implied errors are expected to be small in view of the slight probabilities for such highly multiple collision sequences [13]. This opinion is confirmed to some extent by the calculations of Walker and Martin (figs. 5 and 6 ref. fl I])_ The major difference between their spectra and those presented below, i.e. the sudden drop of the intensity at the triple collision energy, disappears in essence when convolutional broadening due to a finite angular and energy resolution is taken into account in the three-atom model calculations. 3. The PVB-WM
method
Since the interaction time of keV ions with the surface is negligibly small compared with the atomic vibrational periods, such ions probe essentially instantaneous surface structures influenced by thermal disorder. The QT peak is constituted by ions which are multiply scattered from accidentally present three-atom pit structures. In first approximation these three atoms contribute by an equal amount (14/3) to the total .scattering angle 9,. In that case, neglecting correlations, the probability for pits of the required depth h = d tan( S/6) is given by P, = (27re’J?r)-i
exp[-d2tan2(9,/6)/3a21,
(7)
where d is the interatomic distance. In a naive way the QT peak height is then given by a product of P, and some factor K, containing both charge exchange probabilities and the cross section. With the assumption that K, does not depend on the target temperature, one can easily show that the empirical relation found by PVB is an appro~mation of the WM relation in a restricted temperature range. For non-correlated vibrations and not too low temperatures, u2 (eq.(7)) is given by [1’7]: u2 = 3A’T(kmB;)-‘, where 8, is the surface Debye temperature,
(8) ?” the surface temperature,
m the
348
B. Poelsemo et al. / Information
on surface thermal vibrations
mass of a surface atom and A and k have their usual meanings. of eqs. (7) and (8), one obtains for the QT peak intensity: Iol- =
By combination
(9)
A plot of Ior versus fi is shown in fig. 1. The dashed line shows the tangent to the curve in its first point of inflection and indeed this straight line approximates well the actual curve in the low temperature range [6]. By rewriting eq. (9), Walker and Martin obtained the expression ln(Io,)+ln(T)=A
- BT-‘,
(10)
with A = ln[ KQkm8h(6rh2fi)-‘],
(104
B = (9A2)-‘kmB;d’
(lob)
tan2(9,/6).
Consequently a plot of. ln(Io,) + ln(T)versus T- ’ should yield a straight line and from the slope one may determine 8, independently from K, (and thus from the potential parameters). The advantage of this method is that its use is not restricted to a particular range of temperatures. It is clear from the discussion above that the correct value of 8, is obtained only if the temperature dependency of K, (eg. (9)) can be neglected, or, in other words, if only one specific scattering process contributes to the intensity at some relative energy over the whole temperature range. In general this assumption is not justified and Walker and Martin indeed showed that K, does depend on the temperature for specular scattering into 8, = 30” of 10 keV Kr+ ions along the (110) azimuth on Cu(100). We have reinvestigated this special case with the three-atom model described in section 2. The calculations were performed at a relative energy E = 0.875 (i.e. 1% below et.) and for potential parameters C = 1.0, 0.75 and 0.5 (see eq. (7)) and the results are shown in fig. 2. Indeed straight lines are obtained and from the slopes one may derive “apparent” Debye temperatures, which are for C = 0.5, 0.75 and 1 respectively 3% 25% and 62% too low with respect to the value of 8,, which was used in the calculations. From these results one may conclude that here too the three-atom model gives essentially the same result as the Monte Carlo calculations [ 111: Both calculations show that, for this special case, one cannot reliably estimate 8, because of a lack of knowledge of the potential parameter. From the Monte Carlo calculations, apparent surface Debye temperatures are obtained for E = 0.875 which are 33% (C = 0.75) and 75% (C = 1) too low with respect to the value used in the simulations. The agreement between the two calculations is reasonable and it may even be improved by correcting the three-atom model results for the neglect of collisions with a neighbouring atom on both sides.
B. Poeisema et al. / Information
10 keV
0
0.2
0.6 Th[h
0.6
0.8
1.0
349
on surface thermal uibrations
0
BD (m k)“2.3’/2h-‘]-
Kr-Cu
5
, E -0.875
10 ‘y
15 (K-l) -
Fig. 1. Comparison between the empirically found relationship between the intensity of the QT peak Ior and the square root of the temperature (dashed line) and eq. (9) (sohd line). Fig. 2. Calculated intensity of the QT peak as a function of the temperature, plotted according to eq. (IO), for keV Kr” ions scattered along the (1 IO) direction. The calculations were performed with three different screening parameters; C = 0.5, 0.75 and 1, E= 0.875, Q, = 30°, J, = 15”.
However, in the present context this aspect is a rather unimportant one since the method for estimating 8, is expected to be unreliable in those cases, where the latter corrections are of any importance.
4. Characteristics of the quasi-triple scattering process In the preceding section it was shown that the proposed method for measuring 8, is correct if a discrimination can be made between ions having experienced a triple collision sequence and those having participated in any other scattering process, i.e. whenever a clear QT peak can be resolved. In section 4.1 we will address this question by analysing the shape of the energy distributions of scattered ions near the QT collision energy. In section 4.2 we discuss in some detail the possibility of determining the surface Debye temperature, while in section 4.3 the influence of the potential parameter C on the calculated results is considered.
350
B.
Poekma
et a!. / Information on surface thermal vibrations
4.1. The QTpeuk In general, both experimental and calculated energy distributions reveal a QS and a QD peak, while in some cases a broad feature appears around the triple collision energy er. Although this feature has been designated as the QT peak [3,6], it has not been established yet whether it is a real peak or an essentially non-resolvable hump on the high energy flank of the QD peak. To clarify this phenomenon we analyse in detail the energy distribution near E = +. applying the three-atom model (eq. (4)). In order to be able to resolve a QT peak requires imposition of the condition
(11)
lim E-&Td’>o. de From eq. (4) one may derive (see also eq. (3)): d1 -= dt:
This integral can be calculated in the limit E -+ er as shown in appendix A. It appears that this derivative goes to infinity for E + Ed, its sign being given by: sign(g)
E=ET= sign[ d + 3 cosPJ6)
($$)
,+,I
(13)
(see eq. (A. 15)). This implies that a well-resolved QT peak may exist only if
d’ -3 c44’,/6) (~)_/3. Whenever condition (13a) is not fulfilled, the “QT peak” is essentially not resolvable. Accordingly, the important quantities governing the characteristics of QT scattering are the interatomic distance d, the total scattering angle &t and the primary energy E0 (determining (dp/d6),E,,), which can all be varied experimentally. Increasing d, 6, and E, improves the resolution of the QT peak, In order to illustrate the described behaviour, we have calculated energy distributions of 1, 10 and 100 keV Krf ions scattered specularly over 9; = 30’ from a Cu(100) chain (see fig. 3). For the screening parameter we used C = 0.8, 8, = 255 K and T = 196 K. For 100 keV multiple scattering is weak and a real QT peak appears, while at 1 keV multiple scattering is dominant and QT collisions show up as a shoulder on the high energy side of the QD peak. The 10 keV results rather represent some intermediate case. Although a QT peak can be resolved if inequality (13a) applies, this does not imply that this peak is well resolved in the whole attainable temperature range. This observation is illustrated in fig. 4a, where 100 keV Kr+ (lOO> spectra are plotted for surface temperatures of 196, 400 and 900 K. Although
B. Poelsema et al. / Information on surface thermal vibrations
351
even for T, = 900 K still a local maximum exists at E = cr, the resdution of the QT peak becomes increasingly poor with increasing q. This phenomenon is elucidated in appendix B. It may be noted that the opposite behaviour is
0.83
0.85
0.87
0.89
Fig. 3. Calculated energy distributions for 1, 10 and 100 keV Kr+ scattered specularly over 30” from a Cu (100) chain. Only the energy range between the double and triple collision energy is shown. The screening parameter C = 0.8, 8, = 255 K and T = 196 K.
observed if expression (13a) does not apply, as illustrated for 1 keV Kr+ ions in fig. 4b (see appendix B). In this case the QT peak seems to be resolved better at higher surface temperatures. 4.2. The surface Debye temperature
For exact triple collisions the surface Debye temperature eqs. (lob) and (10)): l/2
3h d tan( a/6)
with d B = - d(l/Z-)
is given by (see
ln(IT).
’
04)
B. Poelsema et al. / Informationon surfacethermal vibrations
352
a E,=lOOkeV
-.-T=196K ---T=LOOK
\ -.-Tz196K
'\
---TT-LOOK -T=900K
0.85
I
0.87
I
0.83
0.85
0.87
E-
0.89
E-
Fig. 4. Calculated energy distributions for three temperatures: T = 196,400 and 900 K for specular scattering over 30” from a Cu (100) chain; C = 0.8,8,, = 255 K; (a) 100 keV Kr+, (b) 1 keV Kr+.
Experimentally one never measures exclusively triple collisions due to the finite angular and energy resolution of the apparatus. Therefore we define an apparent surface Debye temperature Barr, according to eq. (14) and referring to “real” situations. In correspondence to section 4.1 we calculate Bar+,,or B, near E = Ed. The derivative dB/de goes to infinity for E -+ Ed as did dI/de and its sign is given (see appendix A, eq. (A.17)) by: ‘lgn .
dB ds
t-1
E=E7= -sign[d+3cos(W6)
(%,,=,,,I.
(15)
Consequently the slope of Bapp (E) changes sign for exactly the same experimental conditions where the characteristics of the QT “peak” change (see eq. (13)). In order to examine the behaviour of 13~~~ in a somewhat broader E range, we calculated &r,, (E) for 100, 10 and 1 keV Krf ions specularly reflected from Cu( 100) over 9, = 30’. The three-atom model (eq. (4)) was employed with C = 0.8 (eq. (6) and the results are shown in fig. 5. The general trends predicted by expression (15) are observed. However, the sudden change of sign of the slope of 13,,, (E), when (dB/dE)+ changes sign, is is not found. Probably this feature is very local at E = Ed and there appears to be a larger range where the derivative is effectively about zero. From a practical point of view one may therefore distinguish three cases: In this case the QT peak is reasonably (a) d Z+ - 3 cos(9,/6) (dp/d6),,,,,,.
B. Poelsema et al. / Information on surface thermal uibrations
353
well resolved and the apparent surface Debye temperature increases with decreasing relative energy E. In principle &,rr, derived from an experiment, may be slightly larger than the real 8,. However, this error is negligibly small
“X=“,X
=Xx.
xs, . .-.-.- l._._. _+++*+++ ++* . +* .. l
.
.
. +
1 keV 10 keV
a 100 keV
0 ’ 0.83
ED
ET
. 0.85
0.87
0.89 E-
Fig. 5. Calculated apparent Debye temperature for 1, 10 and 100 keV Kr+ scattered specularly K.
BaPP as a function of the relative scattered energy E over 30” from a Cu (100) chain; C = 0.8.8, = 255
since in this case the spectra exhibit at least a local maximum at E = or, where, according to the three-atom model, the exact Debye temperature should be obtained. In addition, the three-atom model applies well under this condition since the influence of neighbouring atoms is negligibly small. The 100 keV Kr+ curves in figs. 3 and 5 are typical for this situation. (b) d = - 3 cos(9,/6) The 10 keV Kr+ results plotted in figs. 3 (dp/d%!,,. and 5 are characteristic for this situation. Whether or not the QT peak may be (poorly) resolved does not really matter since 0,,, remains constant, and thus equal to 8,, in a rather broad E range when this condition applies. Therefore measurements will yield reasonably exact values for the surface Debye temperature in spite of the fact that the basic assumptions of the PVB-WM approach do not rigorously apply in this case. In this case the PVB-WM method works (c) d = - 3 cos(9,/6) (dp/d%,,,. rather poorly. The 1 keV Kr+ results are characteristic for this situation (see figs. 3 and 5). The apparent Debye temperature t9,,, is smaller than 8, for energies below er and large errors may be made when determining B,. Moreover, scattering processes with higher multiplicities than the triple ones will yield non-negligible contributions to the scattered intensity around E = er and therefore the three-atom model is less suited for the present purposes.
354
B. Poe&ma
et al. / fnformation
4.3. Influence of the screening parameter
on surface thermaf vibrations
C on the ~a~&u~at~~n
As mentioned in ref. [6] and in the introduction, the basic problem for obtaining reliable information on the surface Debye temperature is the lack of knowledge about the ion-atom interaction potential in the keV energy range. To circumvent this problem, PVB proposed their method which was found to be insensitive to the potential parameter C (see eq. (6)) for 10 keV Kr+ ions scattered over 30” from vibrating Cu( 100) rows. WM found a pronounced dependency on C for scattering along the (1 lo} rows [ll]. This dependency can easily be understood on the basis of the preceding discussion: By changing the interatomic distance d one may advance from d 2 - 3 cos(9;/6) where the method works, to d c - 3 cos(~,/6) (dp/d@),,,,, (dp/d%1,3, where a too small Bo value is obtained. In addition the (in)sensitivities to variation of C in both cases can easily be understood: A change of C to larger values implies a corresponding increase of (dp/d$),,,,. As long as the condition d =r - 3 cos(&/6) (dp/di?) 9,,3 approximately holds, a variation of C has no influence on the obtained surface Debye temperature 8,. If, on the always too small values for 19~ are contrary, d-c - 3 cos(19,/6) (dp/dB),,,,, obtained. By increasing C the difference of S,,, with the actual t9o value is still enhanced resulting finally in dramatic errors.
5. ~ornp~~n
of c~c~~tions with experiments
The only experimental results on QT scattering, which have been reported, were performed by the authors [6]. The only experimental parameter which is varied in these experiments is the azimuthal scattering direction. Therefore, a detailed comparison between calculations and measurements, involving variation of the scattering angle or the primary energy is not possible. Yet a comparison is interesting and in the following we will discuss three cases: Scattering of 10 keV Krf ions over 30’ along the (100) direction, along the (1 IO) direction and along the (310) direction. 5.1. (100) In fig. 6 the apparent surface Debye temperature Bapp is plotted as a function of the relative energy E, both for the experiment (fig. 6a) and for the calculations (fig. 6b). The experimental Bapp (E) curve is essentially composed out of three parts: below or, tiapp is constant in a relatively large energy range (> (al.-” en)/2), while close to the QD peak and above the QT peak Barr, (E) increases rather strongly with the relative energy. The behaviour for E > er is probably due to collisions with a higher multiplicity and to thermal broadening caused by the finite velocities of the vibrating surface atoms. For E < 0.855, one
355
B. Poelsema et al. / Information on surface thermal vibrations
500 _
I G -i tg LOO.
t
a
500. b
i 0 LOO.
C
x
0.5 ~*-.A.. )I
300.
x
xxxxxxx
‘-.075.++-H+ ++
I 200.
la.
300.
x
&..
.a l..- ..a .
.
200
1.0 **
(100)
100'
ED 0-t
.
.
0.83
0.05
0.87
(100)
100.
ET .'.
.
0.89
0.91 E-
ED 0,'.
0.83
.
0.85
.
ET' . Y
.
0.87
0.89
t-
Fig. 6. Apparent Debye temperature BaPP as a function of the relative energy E for 10 keV Kr+ scattered specularly over 30” from a Cu (100) chain: (a) experimental results; (b) calculations for C = 0.5, 0.75 and 1, 8, = 279 K.
approaches the QD peak and the behaviour of 6&r will have to change in this energy range to match the expected behaviour of 1oo(T) at E = eo. By putting 0app equal to 8, in the E range, where eaappis constant, and by assuming non-correlated vibrations, one obtains 8, = 279 K. In fig. 6b calculations are shown for three screening constants, C = 1, C = 0.75 and C = 0.5. It appears that Barr, does depend on C, though not very strongly, taking into account the large range of C values used. In fact, when taking the worst case: E = 0.855 and C = 1, a deviation of ear_, from 13, of 30% is found. However, the actual error made by setting 8, = e,,, is much smaller, since the energy resolution, both experimental and implicit (inelastic energy losses, influence of neighbour atoms), will not amount to more than 2%. According to ref. [ 181, C = 0.8 f 0.1, where an uncertainty twice that given in ref. [ 181 is taken into account. Taking these numbers, one may estimate the maximum error to be about 10%. Finally one may note that the curve for C = 0.75 is very similar to the experimental curve - both curves start to deviate from a horizontal line at E = 0.855 indicating that the error is more likely less than a few percent (see also ref. [6]) and that the estimation of C = 0.8 1 [ 181 is rather accurate.
356
3. Poelsema et al. / Information on surface thermal vibrations
5.2. (110) The experimental eaapp( E) curve for scattering along the (110) direction is shown in fig. 7a. The curve looks similar to the one obtained for scattering in the (100) direction though the energy range in which Barr = constant is much smaller and shifted to higher energies. Both these observations indicate already that multiple scattering effects are stronger in this case, causing an estimate of 8, to be less reliable. Moreover, when looking at the experimental energy distributions [6], a strong shift of the QD peak from en towards higher energies is observed and can be explained only by strong multiple scattering effects. From the experimental curve, one would estimate 8, = 244 K assuming no correlations, a value which is much lower than 279 K found for scattering along the (100) direction. The calculated t9=rapp ( E) curves for C = 1, 0.75 and 0.5 are shown in fig. 7b. From these curves it is directly clear that at least for C = 1 and 0.75 we are in the range where the PVB-WM method is not applicable (see section 4.2). Estimat~g the error made by applying this method gives - 108, close to er, which could be correct if a total resolution of 0.5% could be obtained to about - 50% for a resolution of 2% and C = 1.0. Accepting the reduced errors of the preceding section would yield an estimated value for 8,, which is about 20%
a t
r; coo
I 1
B
ap
300 -
Xx Ix 200
x
200
LOO
4
x
x
I ;; ::
x
300
b
x x
100
100
CD 13
x 0.85
f"T 0.87
0.89.
0.91
E------r
0 0.83
0.85
0.97
0.89
E---r
Fig. 7. Apparent Debye temperature BaPPas a function of the relative energy e for 10 keV Kr+ scattered specularly over 30" from a Cu( 1lo} chain: (a) experimental results; (b) calculations for C = 0.5, 0.75 and I, 8, = 279 K.
B. Poeisema et al. / Information on surface thermal vibrations
357
smaller than the actual one. This figure agrees quite well with the observed difference. 5.3.
(310)
Since the interatomic distance d in the (310) direction is larger than in the (100) direction, one would expect, at first sight, that in this case an energy range exists in which t$rP decreases with increasing energy. In contrast (see fig. 8), a behaviour is found which is rather similar to that for the (1 IO} case, i.e. a horizontal part in the BBPPcurve is found which extends over a slightly larger relative energy range than for the corresponding (110) case whereas it is also shifted to higher energies. As mentioned in the previous section, this feature indicates multiple scattering effects. In fact this is supported by the experimental energy distributions which also show a shift of the QS and the QD peak from respectively es and et, towards higher energies, though less than in the (1 IO} case [6]. These observations can be understood by considering the influence of neighbouring (310) rows. Geometrically one can easily calculate that atoms of neighbouring rows may give a non-negligible contribution to the scattering process implying a failure of any chain model in this case. The estimated 8, = 238 K is therefore also unreliable.
500
I too jz :: d 300
x
x
xXxXx
x
200
xx I
x x
<310>
tLD
p
0
( 33
0.85
0.87
0.89
0.91 E-
Fig. 8. Apparent Debye temperature 6&p as a function of the relative energy E for 10 keV Kr+ scattered specularly over 30° from a Cu(310) chain. Experimental results.
358
B. Poelsema et al. / Injormaiion
on surface thermal vibrations
6. Discussion In this paper we have shown that low energy ion scattering can be used to determine the surface Debye temperature with a good accuracy in the order of a few percent. The accuracy which can be obtained with a certain apparatus depends, among others, very strongly on the sensitivity of the system. The implicit error which is made due to the uncertainty in the screening parameter C is the smallest for d>
-3
cos(1’$/6)
(dp/dG)s,,j.
However, the intensity of the QT peak is under such scattering conditions also rather small. The experimental results reported in ref. [6] were not performed in an optimal configuration to measure the surface Debye temperature, yet the (100) result seems to be reliable and the uncertainty is only a few percent. Of course one has to keep in mind that the method described here does not give directly the surface Debye temperature, but the relative amplitude of neighbouring atoms. To obtain the Debye temperature, a model has to be assumed which takes into account the correlations between the thermal displacements of surface atoms and, for low temperatures, also the zero point vibrations. As an important result, the present paper shows that simple three-atom calculations are sufficient to check the validity of the method, so time consuming and expensive Monte Carlo calculations can be avoided. An even simpler check might be to calculate the function d + 3 cos(9,/6) (dp/d6),,,J which allows classification of the scattering condition into one of the three classes distinguished in section 4.2. However, to use this criterion more research on this subject is required. Another possibility is to apply an experimental criterion for the validity of the method. If eq. (10) is valid, one will measure the correct Debye temperature independent of the primary energy, whereas in the opposite case an apparent Debye temperature depending on the primary energy will be found. This requires, however, that the energy can be varied in a rather large range. As discussed in ref. [ll], information on correlations may be obtained at low target temperatures and small scattering angles. Such experiments require very low ion doses, since surface damage, which is created at lower temperatures (7’ < 400 K for Cu( loo)), will disturb the measurements. Recently Walker and Martin [l l] analyzed our experimental LEIS data [6] on the basis of eq. (10). They arrived at a final value for the surface Debye temperature of the Cu(100) surface of 8n1 = 276 + 6 K. This value is slightly higher than our initial estimate (255 f 15 K) [6,16]. The latter value was deduced from an experimental and a calculational comparison between (100) and (310) results [6]. Indeed the value of 238 K obtained for the (310) direction (see section 5.3) agrees well with the former result in view of the
3.
et
Poekern
ai.
/
information
on surface thermul oibrarions
359
scattering in the calculated and measured data points. (The original experimental data were no longer available.) As argued in section 5.3, however, the (3 10) result tends to overestimate the thermal vibrational amplitudes. In fact, this feature was already revealed in our early Monte Carlo calculations: The cut-off temperature for the (310) azimuthal direction was found to be 2.2 k 0.2 times higher than that for the (100) direction instead of the 2.5 expected on base of the chain model. Unfortunately, however, this difference was subsequently considered to be statistically insignificant [6]. Taking into account this difference would have led to a value for 8,& = 273 $- 25 in complete agreement with the present considerations.
7. summary We have reinvestigated the method, proposed by the authors, to measure the surface Debye temperature 8, [6]: - It was found that derivation of the relationship between the QT peak height and the temperature, given by Walker and Martin [ 111, ln(lo,)+ln(T)=A
-
kmd’ tan2( 9,/6) 9tt2
6; -tT
is correct for d B 3 cos(8J6) (dp/d8),t,,. - Although the derivation of this formula is basically not correct for d = computer simulations show that the formula can still 3 cos(a,) (dp/d%/j, be applied. This means that one can determine 8, also in this case without further detailed knowledge of the interaction potential. - Whenever calculations are required, to test the applicability of this formula, the employing of the general three-atom model suffices. Time consuming and elaborate Monte Carlo simulations are therefore superfluous for the present purpose. - Experimentally one may test the applicability of this formula by measuring 8, according to this formula, as a function of the primary energy. The obtained value of 8, is correct if it is constant with the energy. - Neglecting correlations, a surface Debye temperature of 279 + 6 K is found for the Cu( 100) surface. Additional experiments are required for a conclusive statement on correlated thermal displacements of the surface atoms.
Appendix A
In general it is not possible to calculate analytically the integral of eq. (4). Consequently it is impossible to calculate the derivative dI/de (eq. (4)) analytically. However, one can calculate these functions in the limit E -+ a,, In
360
B. Poelsema et al. / Information on surface thermal vibrations
this limit f(5)
is given by (see eq. (3a)):
f(E)=
N, 27ra2fi
tan(i+,/2)
with h = d tan(9,/6)
g(5) =
exp
i
--
h2
(A.1)
i’
3a2
and g(5) given by (see eq. (3b)):
P3 cos( &/2)
cos2 ( 6,/6)
with P = (dp/d6),,,,.
=
(A.2)
’
The integral
in eq. (4) reduces
to
%lp31P 18e,02
Its derivative
sin( 9,/2)
dZ/de
$ =r/6 f(E) -v/2
cos’( G,/6)
(A-3)
exp
in the limit E -+ E,- is given by (see eq. (12)):
g(t)
-+( f$ +;z - +)dq.
The function f(t) depends on the relative (3a)). According to ref. [ 131, u is given by: P2W
u=dtan(#-6,)+
energy
E through
u=h-d<[1+tan2(6,/6)]-
u=
-h+dq[l
(A.9
*
2E+V
uu +
one obtains
P+O(t2,tq,q2).
-
only
(A.6)
for u:
+tan2(6,/6)]
u2= h2
2) and keeping
co4 9,/6)
Thus u2 + uu + u2 becomes
u2 +
u and u (see eq.
-PdW
cos(qMl+,)
Changing variables from 9, and I?, to 5 and n (see section the first order terms in 5 and n yields
Similarly
(A.4)
&
cos2;8,6j
+
(A.7)
to first order in [ and n:
I
(~+,>[d+3Pcos2(~,,/6)]
+O(t2Am2)~
(A.81 and this depends on E only through tion 5 = ($E)‘/2sin IJI yields:
5 and n. Using eq. (2a) and the transforma-
(A.9)
il.
Combination
PoeLsem
et al.
/
r~~5r~iion
an surface
bmnai
vibrations
361
of eqs. (2), (3a), (A.8) and (A.91 results in:
(A. 10) For the function g we have (dp,/d%) (dp,/d%) (dp,/d%) g = cos( 9,,‘2) cost iIJ6 - <) cos( - 9J6 + q) * One can now easily calculate that
dp, dp, dp, =fP+P’$+P”p+
--1^_
d6, d9, d9,
. ..)[~-P’(rl+C.)fP”(rl+5)2+
x (P + P'?j + P'%J* + .+P3+0(v2&J,t2), and
[cog &/6
- t>
cog- &,A3 + 7))]4 t- sin 4 tan( aJ6) cos 7j + sin q tan( @,/6)]j-’
so g is in first order in { and q given by
and
.,.I
362
B. Poelsema et al. / ~n~arma~ion on surface thermal vibrations
remain finite. However, S, + 0 as E -+ E,,.and the integral represented by eq. (A.13) blows up, therefore, to plus or minus infinity. The sign is given by: d + 3P cos( 6,/6) - y
cos( &,/6) sin( 6,/6)
1 .
(A.14)
Since u2 < d2 (at the melting point u = O.ld) the last term in eq. (A.14) can be neglected and one obtains to a good approximation = sign[d + 3P cos( 9,,‘6)] a
(A.15)
Finally the derivative dB/de (eq. (15)) can be calculated for E -+ sr using eqs. (2), (8), (A.3) and (A.13). With x = l/7’, one obtains: = -1im lim E-‘ErdB de = -lim,,eTS~-“2 =
--
hl.c[ d -t- 3P cos( &/6)] 27+ co?( a&)
mke,:‘~ [d+ 3P~0s’'~6'1 fi2
limr_c,~~-
6%~~ cos2( 9,/6)
--d mk@x dx 3ti2 l/2
3
Consequently the function dB/de behaves in a similar way as df/de; approaches infinity for E --+sr. The sign of d B/d E is given by: sign(g)=
-sign(g)=
-sign[d+3Pcos(6,/6)].
(~.16) i.e. it
(A.17)
Appendix B On the temperature
dependency of the QT peak shape
As shown in fig. 4, the shape of the QT peak depends strongly on the surface temperature (see also discussion in section 4.1). It is obvious from eqs. (3) and (4) that this temperature dependence is determined by the term t=exp[-(U2+U2,+0*)/3e2].
03.1)
For a given scattering geometry and relative energy the relation between u and 0 is fixed. For E = 0.86 and C = 0.8 such relationships are shown in fig. 9 for 1, 10 and 100 keV Kr+ specularly reflected over 6, = 30” from a Cu(100) chain. (For comparison, ellipses denoting constant values of u2 + uo + v2 are drawn.) While performing the integration in eq. (4), these contours in the U, o plane are traversed. In fact, these contours determine what kind of scattering process will preferentially contribute to the intensity at some energy e. For instance, if
B. Poelsema et al. / information
a0
-01
-0.2
-0.3
-0.4 VlAl-----
on surface thermal vibrations
363
-o.!
Fig. 9. The solid lines give possible combinations of relative atomic displacements of the first (u) and third (0) atom for scattering of 1, 10 and 100 keV Krf over 30” from a Cu( 110) chain at a relative energy of e = 0.86, C = 0.8. The dashed lines represent the equations u* + M + uv* = U with U = 0.01, 0.04, 0.09 and 0.16.
w’-+ uu + u* is minimal along the diagonal (u = -v), then a symmetric collision process (4 = (tl) will be the most probable. This applies, e.g., for the 100 keV result. In order to look in more detail at the contributing collision processes, we have plotted for the three considered cases u* + uu + TV*as a
Fig. 10. Dependency of U ( = u* + ~lt,+ v*) on the asymmetry parameters a for 1, 10 and 100 keV IW+ scattered over 30” from a Cu( 100) chain at E= 0.86.
364
B. Poelsema et al. / Information on surface thermal vibrations
function
of an asymmetry
parameter
~=(5--‘)7)/(5+17).
(Ydefined
as (see fig.
i0) 03.2)
Again we distinguish two characteristic cases: 100 keV where the main contribution comes from symmetric collisions and 1 keV where asymmetric collisions are more probable. The 10 keV result shows again an intermediate case. It can be shown that again the sign of S: S=
[d+
3 co@,)
(dp,‘d%,,&
(B-3)
plays the decisive role (see appendix A). For positive S the symmetric scattering process is the most probable one. With increasing temperature, i.e. a2 (see eq. (8)), the preference of specific scattering processes decreases (see eq. (B.l)). Therefore in the case of 1 keV the relative contribution of symmetric collisions increases resulting in a better resolution of the QT peak (see fig. 4b) at higher temperatures. The opposite applies for the 100 keV case, where QD type contributions become more important at higher temperatures resulting in a worse resolution of the QT peaks. The 10 keV results represent an intermediate situation. The relative contribution of the various collision processes remains approximately constant.
References [I] L.K. Verheij and A.L. Boers, in: Proc. 5th Intern. Conf. on Atomic Collisions in Solids, Gatlinburg, 1973, p. 583. [2] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 47 (1975) 256. [3] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 55 (1976) 445. [4] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 60 (1976) 485. [5] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 64 (1977) 537. [6] B. Poelsema, L.K. Verheij and A.L. Beers, Nucl. Instr. Methods 132 (1976) 623. [7] R.J. Reid, Phys. Status Solidi 2 (1970) K109. [8] G. Armand, J. Lapujoulade and Y. Lejay, Surface Sci. 63 (1977) 143. [9] G. Comsa, Surface Sci. 81 (1979) 57. [IO] D.J. Martin, Surface Sci. 97 (1980) 586. [I I] R.P. Walker and D.J. Martin, Surface Sci. 118 (1982) 659 [12] B. Poelsema and A.L. Boers, Phys. Letters 64A (1977) 304. [ 131 B. Poelsema and A.L. Boers, Radiation Effects 34 (1977) 163. [ 141 B. Poelsema, unpublished results. [15] G. Moliere, Z. Naturforsch. 2a (1947) 133. [16] C. Foster, I.H. Wilson and M.W. Thompson, J. Phys. B5 (1972) 1332. [ 171 The correct expression relating (r2 to 8, should contain a factor 3, in contrast to the expressions we used so far. It should be noted that the omission of the factor 3 does not affect our determination of e*. However, our estimates for 8, [6] have to be multiplied by a factor offi. [ 181 B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 64 (1977) 554.
Surface
Science
North-Holland
INFORMATION ON SURFACE ION SCA’ITERING B. POELSEMA
THERMAL
133 (1983) 344-364
Publishing
VIBRATIONS
Company
FROM keV
and L.K. VERHEIJ
lnstirur filr Grenzfbchenforschung
und Vakuumphysik,
Kernforschungsanlage
Jiilich,
Posrfach 1913,
D - 5170 Jiilich, Fed. Rep. of Germany
and A.L. BOERS Kernfrsisch
Received
Versneller Instituut,
18 April
Rijksuniversileir
1983; accepted
for publication
Groningen. Groningen, The NerherlanL
7 July 1983
The possibilities of obtaining information on surface phonons from (quasi-)triple collisions in keV ion scattering (LEIS) are reinvestigated. Sensitivities to the ion-atom interaction potential can be elucidated with a semi-analytical three-atom model without the need for expensive computer simulations. The most reliable information is obtained from measurements along the low index (( 100) and (I 10)) azimuthal directions. A simple recipe is proposed for acquiring potential-independent information on surface thermal vibrations.
1. Introduction It has been shown that LEIS can be used to investigate thermal vibrations of surface atoms [l-6]. Since the ions are probing “ thermal structures” of only two or three atoms, information is obtained on the differences in thermal displacements of nearest neighbours or next nearest neighbours. Accordingly the results can be considered complementary to typical information extracted from diffraction of low energy electrons [7] 06 thermal energy He atoms [8], where characteristic transfer widths of 200 A are involved [9]. In previous papers, the authors have reported about an investigation of the influence of surface thermal vibrations on the backscattering of low energy (keV range) noble gas ions from monocrystalline surfaces. Under some specific conditions the shape of both the energy and the angular distributions of reflected ions is affected considerably by thermal effects [l-3]. At elevated temperatures and small scattering angles a new peak, resulting from scattering from three-atom thermal pits instantaneously present at the surface, was predicted by computer experiments [3,4] and observed experimentally [5]. It was shown that the 0039-6028/83/0000-OOOO/$O3.00
0 1983 North-Holland
B. Poelsema ei al. / Information on surface thermal vibrations
34s
appearance of this QT peak offered the possibility to estimate the surface Debye temperature 8, in a way not (or hardly) depending on intrinsic uncertainties of the method such as the unknown interaction potential and charge exchange effects [6]. For the scattering of 10 keV Kr+ ions over 30’ from a (310) and a (100) row of a Cu(100) face, a linear relationship was found between the QT peak height and the square root of the target temperature and this empirical relationship was reproduced by computer siumulations in a limited temperature range. This phenomenon allowed to give a reliable estimate of the surface Debye temperature 8,. Recently Walker and Martin (WM) derived a simple and more general relationship between the QT peak height and the temperature [ 10,111. When comparing this expression with Monte Carlo simulations (10 keV Kr+, $Jt= 30’9, they found good agreement for scattering along the (100) direction, but not for scattering along the (1 lo} direction [ 111.Therefore it is worthwhile to discuss in more detail the possibility to apply LEIS for investigating surface thermal vibrations. In the present paper, it will be shown that the method proposed by the authors (PVB), even in the simple form given by WM, can be used to obtain a reliable value for B,, provided that certain criteria are met. These criteria can be made explicit in terms of the scattering geometry and primary ion energy by employing a three-atom model [l&13], i.e. without the need to carry out expensive time consuming Monte Carlo calculations as claimed by WM [ 111 and it might even be possible to express these criteria in an analytical form. Moreover, an experimental procedure is given by which the applicability of the method is checked in a direct way. 2. Model
131: ~=expl-p-l(9:+9,2+ff,z)j,
(1)
346
B. Poelsema et al. / Information on surface thermal vibrations
or with 8, = $8, + 6, 8, = $9, - I - n and 6, = +I’$,+ I): E = exp{ -p-‘[@:
+ 2(.$2 + 577+ n2)]},
(la)
where p is the relative mass matom/mion. Thus the exact triple collision (4 = n = 0) energy is given by sr = exp( - fp-%:). Then one may define a quantity 6, relating 5 and n at some given energy E by the expression: 6,=+~(lnsr.-lns)=~2+[n+n2,
(2)
or n=
-1
2 [[*
(46,-
352)“2].
(2a)
So for scattering over 9, from the three-atom configuration a whole range of angles 6 (and n) is possible. We impose the condition that the closest collision takes place with the central atom (see ref. [ 131). Then only the positive root of I varies between $(3S,)‘/’ and f(38,)‘12. Following ref. [13], one obtains for the intensity at E:
where
f(5)= g(t)
=
NO exp i2aa2\/?- tan #
u2+uu+u2 3a2
(34
i’
(dp,/d9,)(dp,/d9,)(dp,/d9,)
(3b)
cos~cos(~-~~,-~)cos(~-~8,+7j)’
with u = x, - x2, u -x2 - x3 (xi is the thermal displacement of the ilh atom), No is the number of incident particles per unit length, pi and 6, are respectively the impact parameter and the scattering angle at atom i, # is the angle of incidence and (I is the root mean square atomic displacement. Expression (3) holds for the case of uncorrelared thermal vibrations. In the presence of correlations, eq. (3a) has to be modified [4,10,11,13]. Expression (3) can be simplified by carrying out the transformation 5 = ($&)‘I2 sin ‘p: (4) The integral
in eq. (4) can readily
1141. In the calculations V(r)
= Z,Z2e2r-’
For the screening
be evaluated
numerically
we apply a Thomas-Fermi
or approximated
potential
Cp(ra-I). function
(5) $a (x) we use Moliere’s
approximation
[ 151, where
B. Poelsema et al. / Information on surface thermal uibrations
347
the screening length a is given by [16]: a = 0.8853&(
2:‘3 + Z;L3)-“2,
with C being an adjustable constant of order unity. Z,, Z,, e and a, have their usual meaning. It has been shown that the three-atom model agrees well with the results of chain scattering simulations [ 131. It has the advantage that the computation times are two or more orders of magnitude smaller than those required for the corresponding Monte Carlo calculations [13]. However, by its nature, the three-atom model disregards essentially collisions with a higher multiplicity than a threefold one. Although this error is inherently present when employing this model, the implied errors are expected to be small in view of the slight probabilities for such highly multiple collision sequences [13]. This opinion is confirmed to some extent by the calculations of Walker and Martin (figs. 5 and 6 ref. fl I])_ The major difference between their spectra and those presented below, i.e. the sudden drop of the intensity at the triple collision energy, disappears in essence when convolutional broadening due to a finite angular and energy resolution is taken into account in the three-atom model calculations. 3. The PVB-WM
method
Since the interaction time of keV ions with the surface is negligibly small compared with the atomic vibrational periods, such ions probe essentially instantaneous surface structures influenced by thermal disorder. The QT peak is constituted by ions which are multiply scattered from accidentally present three-atom pit structures. In first approximation these three atoms contribute by an equal amount (14/3) to the total .scattering angle 9,. In that case, neglecting correlations, the probability for pits of the required depth h = d tan( S/6) is given by P, = (27re’J?r)-i
exp[-d2tan2(9,/6)/3a21,
(7)
where d is the interatomic distance. In a naive way the QT peak height is then given by a product of P, and some factor K, containing both charge exchange probabilities and the cross section. With the assumption that K, does not depend on the target temperature, one can easily show that the empirical relation found by PVB is an appro~mation of the WM relation in a restricted temperature range. For non-correlated vibrations and not too low temperatures, u2 (eq.(7)) is given by [1’7]: u2 = 3A’T(kmB;)-‘, where 8, is the surface Debye temperature,
(8) ?” the surface temperature,
m the
348
B. Poelsemo et al. / Information
on surface thermal vibrations
mass of a surface atom and A and k have their usual meanings. of eqs. (7) and (8), one obtains for the QT peak intensity: Iol- =
By combination
(9)
A plot of Ior versus fi is shown in fig. 1. The dashed line shows the tangent to the curve in its first point of inflection and indeed this straight line approximates well the actual curve in the low temperature range [6]. By rewriting eq. (9), Walker and Martin obtained the expression ln(Io,)+ln(T)=A
- BT-‘,
(10)
with A = ln[ KQkm8h(6rh2fi)-‘],
(104
B = (9A2)-‘kmB;d’
(lob)
tan2(9,/6).
Consequently a plot of. ln(Io,) + ln(T)versus T- ’ should yield a straight line and from the slope one may determine 8, independently from K, (and thus from the potential parameters). The advantage of this method is that its use is not restricted to a particular range of temperatures. It is clear from the discussion above that the correct value of 8, is obtained only if the temperature dependency of K, (eg. (9)) can be neglected, or, in other words, if only one specific scattering process contributes to the intensity at some relative energy over the whole temperature range. In general this assumption is not justified and Walker and Martin indeed showed that K, does depend on the temperature for specular scattering into 8, = 30” of 10 keV Kr+ ions along the (110) azimuth on Cu(100). We have reinvestigated this special case with the three-atom model described in section 2. The calculations were performed at a relative energy E = 0.875 (i.e. 1% below et.) and for potential parameters C = 1.0, 0.75 and 0.5 (see eq. (7)) and the results are shown in fig. 2. Indeed straight lines are obtained and from the slopes one may derive “apparent” Debye temperatures, which are for C = 0.5, 0.75 and 1 respectively 3% 25% and 62% too low with respect to the value of 8,, which was used in the calculations. From these results one may conclude that here too the three-atom model gives essentially the same result as the Monte Carlo calculations [ 111: Both calculations show that, for this special case, one cannot reliably estimate 8, because of a lack of knowledge of the potential parameter. From the Monte Carlo calculations, apparent surface Debye temperatures are obtained for E = 0.875 which are 33% (C = 0.75) and 75% (C = 1) too low with respect to the value used in the simulations. The agreement between the two calculations is reasonable and it may even be improved by correcting the three-atom model results for the neglect of collisions with a neighbouring atom on both sides.
B. Poeisema et al. / Information
10 keV
0
0.2
0.6 Th[h
0.6
0.8
1.0
349
on surface thermal uibrations
0
BD (m k)“2.3’/2h-‘]-
Kr-Cu
5
, E -0.875
10 ‘y
15 (K-l) -
Fig. 1. Comparison between the empirically found relationship between the intensity of the QT peak Ior and the square root of the temperature (dashed line) and eq. (9) (sohd line). Fig. 2. Calculated intensity of the QT peak as a function of the temperature, plotted according to eq. (IO), for keV Kr” ions scattered along the (1 IO) direction. The calculations were performed with three different screening parameters; C = 0.5, 0.75 and 1, E= 0.875, Q, = 30°, J, = 15”.
However, in the present context this aspect is a rather unimportant one since the method for estimating 8, is expected to be unreliable in those cases, where the latter corrections are of any importance.
4. Characteristics of the quasi-triple scattering process In the preceding section it was shown that the proposed method for measuring 8, is correct if a discrimination can be made between ions having experienced a triple collision sequence and those having participated in any other scattering process, i.e. whenever a clear QT peak can be resolved. In section 4.1 we will address this question by analysing the shape of the energy distributions of scattered ions near the QT collision energy. In section 4.2 we discuss in some detail the possibility of determining the surface Debye temperature, while in section 4.3 the influence of the potential parameter C on the calculated results is considered.
350
B.
Poekma
et a!. / Information on surface thermal vibrations
4.1. The QTpeuk In general, both experimental and calculated energy distributions reveal a QS and a QD peak, while in some cases a broad feature appears around the triple collision energy er. Although this feature has been designated as the QT peak [3,6], it has not been established yet whether it is a real peak or an essentially non-resolvable hump on the high energy flank of the QD peak. To clarify this phenomenon we analyse in detail the energy distribution near E = +. applying the three-atom model (eq. (4)). In order to be able to resolve a QT peak requires imposition of the condition
(11)
lim E-&Td’>o. de From eq. (4) one may derive (see also eq. (3)): d1 -= dt:
This integral can be calculated in the limit E -+ er as shown in appendix A. It appears that this derivative goes to infinity for E + Ed, its sign being given by: sign(g)
E=ET= sign[ d + 3 cosPJ6)
($$)
,+,I
(13)
(see eq. (A. 15)). This implies that a well-resolved QT peak may exist only if
d’ -3 c44’,/6) (~)_/3. Whenever condition (13a) is not fulfilled, the “QT peak” is essentially not resolvable. Accordingly, the important quantities governing the characteristics of QT scattering are the interatomic distance d, the total scattering angle &t and the primary energy E0 (determining (dp/d6),E,,), which can all be varied experimentally. Increasing d, 6, and E, improves the resolution of the QT peak, In order to illustrate the described behaviour, we have calculated energy distributions of 1, 10 and 100 keV Krf ions scattered specularly over 9; = 30’ from a Cu(100) chain (see fig. 3). For the screening parameter we used C = 0.8, 8, = 255 K and T = 196 K. For 100 keV multiple scattering is weak and a real QT peak appears, while at 1 keV multiple scattering is dominant and QT collisions show up as a shoulder on the high energy side of the QD peak. The 10 keV results rather represent some intermediate case. Although a QT peak can be resolved if inequality (13a) applies, this does not imply that this peak is well resolved in the whole attainable temperature range. This observation is illustrated in fig. 4a, where 100 keV Kr+ (lOO> spectra are plotted for surface temperatures of 196, 400 and 900 K. Although
B. Poelsema et al. / Information on surface thermal vibrations
351
even for T, = 900 K still a local maximum exists at E = cr, the resdution of the QT peak becomes increasingly poor with increasing q. This phenomenon is elucidated in appendix B. It may be noted that the opposite behaviour is
0.83
0.85
0.87
0.89
Fig. 3. Calculated energy distributions for 1, 10 and 100 keV Kr+ scattered specularly over 30” from a Cu (100) chain. Only the energy range between the double and triple collision energy is shown. The screening parameter C = 0.8, 8, = 255 K and T = 196 K.
observed if expression (13a) does not apply, as illustrated for 1 keV Kr+ ions in fig. 4b (see appendix B). In this case the QT peak seems to be resolved better at higher surface temperatures. 4.2. The surface Debye temperature
For exact triple collisions the surface Debye temperature eqs. (lob) and (10)): l/2
3h d tan( a/6)
with d B = - d(l/Z-)
is given by (see
ln(IT).
’
04)
B. Poelsema et al. / Informationon surfacethermal vibrations
352
a E,=lOOkeV
-.-T=196K ---T=LOOK
\ -.-Tz196K
'\
---TT-LOOK -T=900K
0.85
I
0.87
I
0.83
0.85
0.87
E-
0.89
E-
Fig. 4. Calculated energy distributions for three temperatures: T = 196,400 and 900 K for specular scattering over 30” from a Cu (100) chain; C = 0.8,8,, = 255 K; (a) 100 keV Kr+, (b) 1 keV Kr+.
Experimentally one never measures exclusively triple collisions due to the finite angular and energy resolution of the apparatus. Therefore we define an apparent surface Debye temperature Barr, according to eq. (14) and referring to “real” situations. In correspondence to section 4.1 we calculate Bar+,,or B, near E = Ed. The derivative dB/de goes to infinity for E -+ Ed as did dI/de and its sign is given (see appendix A, eq. (A.17)) by: ‘lgn .
dB ds
t-1
E=E7= -sign[d+3cos(W6)
(%,,=,,,I.
(15)
Consequently the slope of Bapp (E) changes sign for exactly the same experimental conditions where the characteristics of the QT “peak” change (see eq. (13)). In order to examine the behaviour of 13~~~ in a somewhat broader E range, we calculated &r,, (E) for 100, 10 and 1 keV Krf ions specularly reflected from Cu( 100) over 9, = 30’. The three-atom model (eq. (4)) was employed with C = 0.8 (eq. (6) and the results are shown in fig. 5. The general trends predicted by expression (15) are observed. However, the sudden change of sign of the slope of 13,,, (E), when (dB/dE)+ changes sign, is is not found. Probably this feature is very local at E = Ed and there appears to be a larger range where the derivative is effectively about zero. From a practical point of view one may therefore distinguish three cases: In this case the QT peak is reasonably (a) d Z+ - 3 cos(9,/6) (dp/d6),,,,,,.
B. Poelsema et al. / Information on surface thermal uibrations
353
well resolved and the apparent surface Debye temperature increases with decreasing relative energy E. In principle &,rr, derived from an experiment, may be slightly larger than the real 8,. However, this error is negligibly small
“X=“,X
=Xx.
xs, . .-.-.- l._._. _+++*+++ ++* . +* .. l
.
.
. +
1 keV 10 keV
a 100 keV
0 ’ 0.83
ED
ET
. 0.85
0.87
0.89 E-
Fig. 5. Calculated apparent Debye temperature for 1, 10 and 100 keV Kr+ scattered specularly K.
BaPP as a function of the relative scattered energy E over 30” from a Cu (100) chain; C = 0.8.8, = 255
since in this case the spectra exhibit at least a local maximum at E = or, where, according to the three-atom model, the exact Debye temperature should be obtained. In addition, the three-atom model applies well under this condition since the influence of neighbouring atoms is negligibly small. The 100 keV Kr+ curves in figs. 3 and 5 are typical for this situation. (b) d = - 3 cos(9,/6) The 10 keV Kr+ results plotted in figs. 3 (dp/d%!,,. and 5 are characteristic for this situation. Whether or not the QT peak may be (poorly) resolved does not really matter since 0,,, remains constant, and thus equal to 8,, in a rather broad E range when this condition applies. Therefore measurements will yield reasonably exact values for the surface Debye temperature in spite of the fact that the basic assumptions of the PVB-WM approach do not rigorously apply in this case. In this case the PVB-WM method works (c) d = - 3 cos(9,/6) (dp/d%,,,. rather poorly. The 1 keV Kr+ results are characteristic for this situation (see figs. 3 and 5). The apparent Debye temperature t9,,, is smaller than 8, for energies below er and large errors may be made when determining B,. Moreover, scattering processes with higher multiplicities than the triple ones will yield non-negligible contributions to the scattered intensity around E = er and therefore the three-atom model is less suited for the present purposes.
354
B. Poe&ma
et al. / fnformation
4.3. Influence of the screening parameter
on surface thermaf vibrations
C on the ~a~&u~at~~n
As mentioned in ref. [6] and in the introduction, the basic problem for obtaining reliable information on the surface Debye temperature is the lack of knowledge about the ion-atom interaction potential in the keV energy range. To circumvent this problem, PVB proposed their method which was found to be insensitive to the potential parameter C (see eq. (6)) for 10 keV Kr+ ions scattered over 30” from vibrating Cu( 100) rows. WM found a pronounced dependency on C for scattering along the (1 lo} rows [ll]. This dependency can easily be understood on the basis of the preceding discussion: By changing the interatomic distance d one may advance from d 2 - 3 cos(9;/6) where the method works, to d c - 3 cos(~,/6) (dp/d@),,,,, (dp/d%1,3, where a too small Bo value is obtained. In addition the (in)sensitivities to variation of C in both cases can easily be understood: A change of C to larger values implies a corresponding increase of (dp/d$),,,,. As long as the condition d =r - 3 cos(&/6) (dp/di?) 9,,3 approximately holds, a variation of C has no influence on the obtained surface Debye temperature 8,. If, on the always too small values for 19~ are contrary, d-c - 3 cos(19,/6) (dp/dB),,,,, obtained. By increasing C the difference of S,,, with the actual t9o value is still enhanced resulting finally in dramatic errors.
5. ~ornp~~n
of c~c~~tions with experiments
The only experimental results on QT scattering, which have been reported, were performed by the authors [6]. The only experimental parameter which is varied in these experiments is the azimuthal scattering direction. Therefore, a detailed comparison between calculations and measurements, involving variation of the scattering angle or the primary energy is not possible. Yet a comparison is interesting and in the following we will discuss three cases: Scattering of 10 keV Krf ions over 30’ along the (100) direction, along the (1 IO) direction and along the (310) direction. 5.1. (100) In fig. 6 the apparent surface Debye temperature Bapp is plotted as a function of the relative energy E, both for the experiment (fig. 6a) and for the calculations (fig. 6b). The experimental Bapp (E) curve is essentially composed out of three parts: below or, tiapp is constant in a relatively large energy range (> (al.-” en)/2), while close to the QD peak and above the QT peak Barr, (E) increases rather strongly with the relative energy. The behaviour for E > er is probably due to collisions with a higher multiplicity and to thermal broadening caused by the finite velocities of the vibrating surface atoms. For E < 0.855, one
355
B. Poelsema et al. / Information on surface thermal vibrations
500 _
I G -i tg LOO.
t
a
500. b
i 0 LOO.
C
x
0.5 ~*-.A.. )I
300.
x
xxxxxxx
‘-.075.++-H+ ++
I 200.
la.
300.
x
&..
.a l..- ..a .
.
200
1.0 **
(100)
100'
ED 0-t
.
.
0.83
0.05
0.87
(100)
100.
ET .'.
.
0.89
0.91 E-
ED 0,'.
0.83
.
0.85
.
ET' . Y
.
0.87
0.89
t-
Fig. 6. Apparent Debye temperature BaPP as a function of the relative energy E for 10 keV Kr+ scattered specularly over 30” from a Cu (100) chain: (a) experimental results; (b) calculations for C = 0.5, 0.75 and 1, 8, = 279 K.
approaches the QD peak and the behaviour of 6&r will have to change in this energy range to match the expected behaviour of 1oo(T) at E = eo. By putting 0app equal to 8, in the E range, where eaappis constant, and by assuming non-correlated vibrations, one obtains 8, = 279 K. In fig. 6b calculations are shown for three screening constants, C = 1, C = 0.75 and C = 0.5. It appears that Barr, does depend on C, though not very strongly, taking into account the large range of C values used. In fact, when taking the worst case: E = 0.855 and C = 1, a deviation of ear_, from 13, of 30% is found. However, the actual error made by setting 8, = e,,, is much smaller, since the energy resolution, both experimental and implicit (inelastic energy losses, influence of neighbour atoms), will not amount to more than 2%. According to ref. [ 181, C = 0.8 f 0.1, where an uncertainty twice that given in ref. [ 181 is taken into account. Taking these numbers, one may estimate the maximum error to be about 10%. Finally one may note that the curve for C = 0.75 is very similar to the experimental curve - both curves start to deviate from a horizontal line at E = 0.855 indicating that the error is more likely less than a few percent (see also ref. [6]) and that the estimation of C = 0.8 1 [ 181 is rather accurate.
356
3. Poelsema et al. / Information on surface thermal vibrations
5.2. (110) The experimental eaapp( E) curve for scattering along the (110) direction is shown in fig. 7a. The curve looks similar to the one obtained for scattering in the (100) direction though the energy range in which Barr = constant is much smaller and shifted to higher energies. Both these observations indicate already that multiple scattering effects are stronger in this case, causing an estimate of 8, to be less reliable. Moreover, when looking at the experimental energy distributions [6], a strong shift of the QD peak from en towards higher energies is observed and can be explained only by strong multiple scattering effects. From the experimental curve, one would estimate 8, = 244 K assuming no correlations, a value which is much lower than 279 K found for scattering along the (100) direction. The calculated t9=rapp ( E) curves for C = 1, 0.75 and 0.5 are shown in fig. 7b. From these curves it is directly clear that at least for C = 1 and 0.75 we are in the range where the PVB-WM method is not applicable (see section 4.2). Estimat~g the error made by applying this method gives - 108, close to er, which could be correct if a total resolution of 0.5% could be obtained to about - 50% for a resolution of 2% and C = 1.0. Accepting the reduced errors of the preceding section would yield an estimated value for 8,, which is about 20%
a t
r; coo
I 1
B
ap
300 -
Xx Ix 200
x
200
LOO
4
x
x
I ;; ::
x
300
b
x x
100
100
x 0.85
f"T 0.87
0.89.
0.91
E------r
0 0.83
0.85
0.97
0.89
E---r
Fig. 7. Apparent Debye temperature BaPPas a function of the relative energy e for 10 keV Kr+ scattered specularly over 30" from a Cu( 1lo} chain: (a) experimental results; (b) calculations for C = 0.5, 0.75 and I, 8, = 279 K.
B. Poeisema et al. / Information on surface thermal vibrations
357
smaller than the actual one. This figure agrees quite well with the observed difference. 5.3.
(310)
Since the interatomic distance d in the (310) direction is larger than in the (100) direction, one would expect, at first sight, that in this case an energy range exists in which t$rP decreases with increasing energy. In contrast (see fig. 8), a behaviour is found which is rather similar to that for the (1 IO} case, i.e. a horizontal part in the BBPPcurve is found which extends over a slightly larger relative energy range than for the corresponding (110) case whereas it is also shifted to higher energies. As mentioned in the previous section, this feature indicates multiple scattering effects. In fact this is supported by the experimental energy distributions which also show a shift of the QS and the QD peak from respectively es and et, towards higher energies, though less than in the (1 IO} case [6]. These observations can be understood by considering the influence of neighbouring (310) rows. Geometrically one can easily calculate that atoms of neighbouring rows may give a non-negligible contribution to the scattering process implying a failure of any chain model in this case. The estimated 8, = 238 K is therefore also unreliable.
500
I too jz :: d 300
x
x
xXxXx
x
200
xx I
x x
<310>
tLD
p
0
( 33
0.85
0.87
0.89
0.91 E-
Fig. 8. Apparent Debye temperature 6&p as a function of the relative energy E for 10 keV Kr+ scattered specularly over 30° from a Cu(310) chain. Experimental results.
358
B. Poelsema et al. / Injormaiion
on surface thermal vibrations
6. Discussion In this paper we have shown that low energy ion scattering can be used to determine the surface Debye temperature with a good accuracy in the order of a few percent. The accuracy which can be obtained with a certain apparatus depends, among others, very strongly on the sensitivity of the system. The implicit error which is made due to the uncertainty in the screening parameter C is the smallest for d>
-3
cos(1’$/6)
(dp/dG)s,,j.
However, the intensity of the QT peak is under such scattering conditions also rather small. The experimental results reported in ref. [6] were not performed in an optimal configuration to measure the surface Debye temperature, yet the (100) result seems to be reliable and the uncertainty is only a few percent. Of course one has to keep in mind that the method described here does not give directly the surface Debye temperature, but the relative amplitude of neighbouring atoms. To obtain the Debye temperature, a model has to be assumed which takes into account the correlations between the thermal displacements of surface atoms and, for low temperatures, also the zero point vibrations. As an important result, the present paper shows that simple three-atom calculations are sufficient to check the validity of the method, so time consuming and expensive Monte Carlo calculations can be avoided. An even simpler check might be to calculate the function d + 3 cos(9,/6) (dp/d6),,,J which allows classification of the scattering condition into one of the three classes distinguished in section 4.2. However, to use this criterion more research on this subject is required. Another possibility is to apply an experimental criterion for the validity of the method. If eq. (10) is valid, one will measure the correct Debye temperature independent of the primary energy, whereas in the opposite case an apparent Debye temperature depending on the primary energy will be found. This requires, however, that the energy can be varied in a rather large range. As discussed in ref. [ll], information on correlations may be obtained at low target temperatures and small scattering angles. Such experiments require very low ion doses, since surface damage, which is created at lower temperatures (7’ < 400 K for Cu( loo)), will disturb the measurements. Recently Walker and Martin [l l] analyzed our experimental LEIS data [6] on the basis of eq. (10). They arrived at a final value for the surface Debye temperature of the Cu(100) surface of 8n1 = 276 + 6 K. This value is slightly higher than our initial estimate (255 f 15 K) [6,16]. The latter value was deduced from an experimental and a calculational comparison between (100) and (310) results [6]. Indeed the value of 238 K obtained for the (310) direction (see section 5.3) agrees well with the former result in view of the
3.
et
Poekern
ai.
/
information
on surface thermul oibrarions
359
scattering in the calculated and measured data points. (The original experimental data were no longer available.) As argued in section 5.3, however, the (3 10) result tends to overestimate the thermal vibrational amplitudes. In fact, this feature was already revealed in our early Monte Carlo calculations: The cut-off temperature for the (310) azimuthal direction was found to be 2.2 k 0.2 times higher than that for the (100) direction instead of the 2.5 expected on base of the chain model. Unfortunately, however, this difference was subsequently considered to be statistically insignificant [6]. Taking into account this difference would have led to a value for 8,& = 273 $- 25 in complete agreement with the present considerations.
7. summary We have reinvestigated the method, proposed by the authors, to measure the surface Debye temperature 8, [6]: - It was found that derivation of the relationship between the QT peak height and the temperature, given by Walker and Martin [ 111, ln(lo,)+ln(T)=A
-
kmd’ tan2( 9,/6) 9tt2
6; -tT
is correct for d B 3 cos(8J6) (dp/d8),t,,. - Although the derivation of this formula is basically not correct for d = computer simulations show that the formula can still 3 cos(a,) (dp/d%/j, be applied. This means that one can determine 8, also in this case without further detailed knowledge of the interaction potential. - Whenever calculations are required, to test the applicability of this formula, the employing of the general three-atom model suffices. Time consuming and elaborate Monte Carlo simulations are therefore superfluous for the present purpose. - Experimentally one may test the applicability of this formula by measuring 8, according to this formula, as a function of the primary energy. The obtained value of 8, is correct if it is constant with the energy. - Neglecting correlations, a surface Debye temperature of 279 + 6 K is found for the Cu( 100) surface. Additional experiments are required for a conclusive statement on correlated thermal displacements of the surface atoms.
Appendix A
In general it is not possible to calculate analytically the integral of eq. (4). Consequently it is impossible to calculate the derivative dI/de (eq. (4)) analytically. However, one can calculate these functions in the limit E -+ a,, In
360
B. Poelsema et al. / Information on surface thermal vibrations
this limit f(5)
is given by (see eq. (3a)):
f(E)=
N, 27ra2fi
tan(i+,/2)
with h = d tan(9,/6)
g(5) =
exp
i
--
h2
(A.1)
i’
3a2
and g(5) given by (see eq. (3b)):
P3 cos( &/2)
cos2 ( 6,/6)
with P = (dp/d6),,,,.
=
(A.2)
’
The integral
in eq. (4) reduces
to
%lp31P 18e,02
Its derivative
sin( 9,/2)
dZ/de
$ =r/6 f(E) -v/2
cos’( G,/6)
(A-3)
exp
in the limit E -+ E,- is given by (see eq. (12)):
g(t)
-+( f$ +;z - +)dq.
The function f(t) depends on the relative (3a)). According to ref. [ 131, u is given by: P2W
u=dtan(#-6,)+
energy
E through
u=h-d<[1+tan2(6,/6)]-
u=
-h+dq[l
(A.9
*
2E+V
uu +
one obtains
P+O(t2,tq,q2).
-
only
(A.6)
for u:
+tan2(6,/6)]
u2= h2
2) and keeping
co4 9,/6)
Thus u2 + uu + u2 becomes
u2 +
u and u (see eq.
-PdW
cos(qMl+,)
Changing variables from 9, and I?, to 5 and n (see section the first order terms in 5 and n yields
Similarly
(A.4)
&
cos2;8,6j
+
(A.7)
to first order in [ and n:
I
(~+,>[d+3Pcos2(~,,/6)]
+O(t2Am2)~
(A.81 and this depends on E only through tion 5 = ($E)‘/2sin IJI yields:
5 and n. Using eq. (2a) and the transforma-
(A.9)
il.
Combination
PoeLsem
et al.
/
r~~5r~iion
an surface
bmnai
vibrations
361
of eqs. (2), (3a), (A.8) and (A.91 results in:
(A. 10) For the function g we have (dp,/d%) (dp,/d%) (dp,/d%) g = cos( 9,,‘2) cost iIJ6 - <) cos( - 9J6 + q) * One can now easily calculate that
dp, dp, dp, =fP+P’$+P”p+
--1^_
d6, d9, d9,
. ..)[~-P’(rl+C.)fP”(rl+5)2+
x (P + P'?j + P'%J* + .+P3+0(v2&J,t2), and
[cog &/6
- t>
cog- &,A3 + 7))]4 t- sin 4 tan( aJ6) cos 7j + sin q tan( @,/6)]j-’
so g is in first order in { and q given by
and
.,.I
362
B. Poelsema et al. / ~n~arma~ion on surface thermal vibrations
remain finite. However, S, + 0 as E -+ E,,.and the integral represented by eq. (A.13) blows up, therefore, to plus or minus infinity. The sign is given by: d + 3P cos( 6,/6) - y
cos( &,/6) sin( 6,/6)
1 .
(A.14)
Since u2 < d2 (at the melting point u = O.ld) the last term in eq. (A.14) can be neglected and one obtains to a good approximation = sign[d + 3P cos( 9,,‘6)] a
(A.15)
Finally the derivative dB/de (eq. (15)) can be calculated for E -+ sr using eqs. (2), (8), (A.3) and (A.13). With x = l/7’, one obtains: = -1im lim E-‘ErdB de = -lim,,eTS~-“2 =
--
hl.c[ d -t- 3P cos( &/6)] 27+ co?( a&)
mke,:‘~ [d+ 3P~0s’'~6'1 fi2
limr_c,~~-
6%~~ cos2( 9,/6)
--d mk@x dx 3ti2 l/2
3
Consequently the function dB/de behaves in a similar way as df/de; approaches infinity for E --+sr. The sign of d B/d E is given by: sign(g)=
-sign(g)=
-sign[d+3Pcos(6,/6)].
(~.16) i.e. it
(A.17)
Appendix B On the temperature
dependency of the QT peak shape
As shown in fig. 4, the shape of the QT peak depends strongly on the surface temperature (see also discussion in section 4.1). It is obvious from eqs. (3) and (4) that this temperature dependence is determined by the term t=exp[-(U2+U2,+0*)/3e2].
03.1)
For a given scattering geometry and relative energy the relation between u and 0 is fixed. For E = 0.86 and C = 0.8 such relationships are shown in fig. 9 for 1, 10 and 100 keV Kr+ specularly reflected over 6, = 30” from a Cu(100) chain. (For comparison, ellipses denoting constant values of u2 + uo + v2 are drawn.) While performing the integration in eq. (4), these contours in the U, o plane are traversed. In fact, these contours determine what kind of scattering process will preferentially contribute to the intensity at some energy e. For instance, if
B. Poelsema et al. / information
a0
-01
-0.2
-0.3
-0.4 VlAl-----
on surface thermal vibrations
363
-o.!
Fig. 9. The solid lines give possible combinations of relative atomic displacements of the first (u) and third (0) atom for scattering of 1, 10 and 100 keV Krf over 30” from a Cu( 110) chain at a relative energy of e = 0.86, C = 0.8. The dashed lines represent the equations u* + M + uv* = U with U = 0.01, 0.04, 0.09 and 0.16.
w’-+ uu + u* is minimal along the diagonal (u = -v), then a symmetric collision process (4 = (tl) will be the most probable. This applies, e.g., for the 100 keV result. In order to look in more detail at the contributing collision processes, we have plotted for the three considered cases u* + uu + TV*as a
Fig. 10. Dependency of U ( = u* + ~lt,+ v*) on the asymmetry parameters a for 1, 10 and 100 keV IW+ scattered over 30” from a Cu( 100) chain at E= 0.86.
364
B. Poelsema et al. / Information on surface thermal vibrations
function
of an asymmetry
parameter
~=(5--‘)7)/(5+17).
(Ydefined
as (see fig.
i0) 03.2)
Again we distinguish two characteristic cases: 100 keV where the main contribution comes from symmetric collisions and 1 keV where asymmetric collisions are more probable. The 10 keV result shows again an intermediate case. It can be shown that again the sign of S: S=
[d+
3 co@,)
(dp,‘d%,,&
(B-3)
plays the decisive role (see appendix A). For positive S the symmetric scattering process is the most probable one. With increasing temperature, i.e. a2 (see eq. (8)), the preference of specific scattering processes decreases (see eq. (B.l)). Therefore in the case of 1 keV the relative contribution of symmetric collisions increases resulting in a better resolution of the QT peak (see fig. 4b) at higher temperatures. The opposite applies for the 100 keV case, where QD type contributions become more important at higher temperatures resulting in a worse resolution of the QT peaks. The 10 keV results represent an intermediate situation. The relative contribution of the various collision processes remains approximately constant.
References [I] L.K. Verheij and A.L. Boers, in: Proc. 5th Intern. Conf. on Atomic Collisions in Solids, Gatlinburg, 1973, p. 583. [2] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 47 (1975) 256. [3] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 55 (1976) 445. [4] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 60 (1976) 485. [5] B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 64 (1977) 537. [6] B. Poelsema, L.K. Verheij and A.L. Beers, Nucl. Instr. Methods 132 (1976) 623. [7] R.J. Reid, Phys. Status Solidi 2 (1970) K109. [8] G. Armand, J. Lapujoulade and Y. Lejay, Surface Sci. 63 (1977) 143. [9] G. Comsa, Surface Sci. 81 (1979) 57. [IO] D.J. Martin, Surface Sci. 97 (1980) 586. [I I] R.P. Walker and D.J. Martin, Surface Sci. 118 (1982) 659 [12] B. Poelsema and A.L. Boers, Phys. Letters 64A (1977) 304. [ 131 B. Poelsema and A.L. Boers, Radiation Effects 34 (1977) 163. [ 141 B. Poelsema, unpublished results. [15] G. Moliere, Z. Naturforsch. 2a (1947) 133. [16] C. Foster, I.H. Wilson and M.W. Thompson, J. Phys. B5 (1972) 1332. [ 171 The correct expression relating (r2 to 8, should contain a factor 3, in contrast to the expressions we used so far. It should be noted that the omission of the factor 3 does not affect our determination of e*. However, our estimates for 8, [6] have to be multiplied by a factor offi. [ 181 B. Poelsema, L.K. Verheij and A.L. Boers, Surface Sci. 64 (1977) 554.