Close encounter processes in Monte Carlo simulations of ion channeling

Nuclear Instruments North-Holland

and Methods

in Physics Research

RIUMI B

B 90 (1994) 142-149

Beam Interactions with Materials&Atoms

Close encounter processes in Monte Carlo simulations of ion channeling A. Dygo *, W.N. Lennard and I.V. Mitchell Department of Physics, University of Western Ontario, London, Ontario, N6A 3K7 Canada

The backscattering yield is analyzed in terms of the depth-dependent distribution of ion trajectories, characterized by instantaneous values of such parameters as the ion energy, the scattering angle, the angle between the ion direction and the surface normal, and the impact parameter with respect to the equilibrium site of the nearest lattice atom. The normalized nuclear encounter probability (NEP) is reformulated as the expected value of the normalized NEP for a single trajectory, and the definition is extended for directions tilted by a large angle with respect to a major axis. It is shown that the normalized NEP cannot be defined to correspond to a physically measurable quantity if the energy and angular spreads of the beam in the crystal are taken into account. The scattering yield measured in the energy spectrum - to be used in comparisons of simulations with experiment - is also expressed as the expected value of the respective yield associated with a single trajectory. Examples of the distribution of trajectories are given for 1.5 MeV 4He ions traversing a Si(100) single crystal.

1. Introduction Close nuclear encounter processes for energetic ions interacting with atoms in a crystal lattice, such as wide-angle Rutherford scattering or nuclear reactions, are of primary importance in channeling studies [l]. Monte Carlo simulation of these processes via a direct counting of the respective events is still impractical due to the small cross sections involved on the one hand, and an insufficient power of present-day computers on the other. An ingenious remedy to this situation has been proposed by Barrett over 20 years ago [2]. The essence of his idea was to simulate a representative distribution of ion trajectories, to be used to calculate the probability of the processes requiring close encounters between the projectile and target atoms - the so-called nuclear encounter probability (NEP) - rather than to look for the actual events that could be registered in the detector. If the NEP, simulated for a given direction in the crystal, is normalized to the respective probability in a random medium, it can be conveniently compared with the corresponding experimental yield measured relative to the random yield. In this way, generation of the energy spectrum can be avoided. While simplifying the analysis in many cases, this approach is only approxi-

mately valid. With a constantly improving accuracy available in both simulation and experiment (for recent references, see refs. [3-ll]), the methods to calculate the yield of the close encounter processes need to be re-examined in greater detail. For clarity, our attention is focused on a specific example of Rutherford (back)scattering. The distribution of ion trajectories is analyzed in terms of instantaneous values of parameters relevant for the scattering yield, such as the ion energy, the scattering angle, the angle between the ion velocity and the surface normal, or the impact parameter with respect to the equilibrium site of the closest lattice atom. The NEP as well as the yield in the energy spectrum are expressed as the expected values of respective quantities associated with a single trajectory. The distributions of the parameters of interest are calculated for 1.5 MeV 4He ions traversing a Si(100) single crystal.

2. Scattering yield in a random medium Let us consider a beam of ions incident on a random medium target, at an angle $ with respect to the surface normal. The scattering yield from a target layer of a thickness AZ is given by [12]: NAz

H,=uL’QNAl=u(E, * Corresponding author, on leave from the Soltan Institute for Nuclear Studies, Warsaw, Poland, Present address: Univ. of Florida, Dept. of Physics, 215 Williamson Hall, Gainesville FL 32611-2085, USA, fax + 1 904 392 8586. 0168-583X/94/$07.00 0 1994 - Elsevier XSDI 0168-583X(93)E1112-Y

Science

0)f2Qcos,

where cr(E, 0) is the differential scattering cross section, E is the ion energy immediately before scattering, 0 is the scattering angle, 0 is the detector solid angle,

B.V. All rights reserved

A. Qygo et al. /Nuci.

Ins&. and Meth. in Phys. Res. B 90 (1994) 142-149

Q is the number of incident ions, N is the atomic density of the target, and Al is the ion path length in the layer A z. There are a number of assumptions made in deriving the above equation: (a) The beam is monoenergetic and perfectly collimated, so the energy E and the angle J, are constant. (b) The scattering cross section, a(& 61, is constant over the solid angle ti and the layer thickness AZ (the ion path length Al). (c) A uniform atomic density characterizes the target, independent of the value for AZ. From the viewpoint of the channeling-backscattering technique, Eq. (1) along with the assumptions above can be adopted as a definition of a random medium. The yield per ion per unit solid angle (hereafter referred to as the scattering yield or simply the yield) is thus

where the area1 atomic density, as seen by the beam, has been introduced:

(3) One notes that the scattering probability, i.e. the yield per ion, is just h,fl. Thus, the terms “the scattering yield” and “the scattering probability” will be used interchangeably.

3. Scattering yield in a crystal lattice Consider the same beam of ions incident on a crystal lattice. When calculating the scattering yield in this case, the non-uniformity of the target must be taken into account. Let us assume that the atoms vibrate isotropically about their equilibrium positions. The 3-dimensional probability density for an atom is given by a Gaussian distribution 1131: P(r)

1

= PI

3/z4exp

i

$

’ 1

where r is the displacement from equilibrium, and ut is the l-dimensional vibrational amplitude. For a straight-line trajectory passing the atom with an impact parameter b with respect to its equilibrium position, the average area1 atomic density (area1 probabili~ density of the vibrating atom) sensed by the projectile is given by

p,(b)={J’(r)dl=&ev 1 where the integral is taken along the trajectory.

143

A beam of ions interacting in any medium - a single crystal or a random medium - acquires angular and energy distributions (i.e. spreads) that increase with the penetration depth. As a result of the angular spread, there is an associated depth-dependent distribution of the scattering angle. The vast majority of all collisions, however, result in small-angle deflections of the ion trajectory. Short segments of the trajectory where P(r) is significantly larger than zero can thus be well approximated by straight lines, and then Eq. (5) gives the average area1 atomic density sensed by the ion in successive “collisions”. Hence, the auerage scattering yield resulting from a single “collision”, in analogy to Eq. (21, is

Ah, =a(.& Q,(b). If there is more than one collision for a given trajectory within the slab AZ, all respective contributions, A.h,, have to be included: i

i

The scattering cross section (for a single trajectory) is assumed to be constant over the slab AZ, so a(E,, eil = a(E, e), and

By definition, for a close encounter event to occur, the actual impact parameter needs to be much smaller than the vibrational amplitude, ul. Eq. (8) can thus be applied to any close encounter process, provided a(E, 8) represents the respective cross section. Accordingly, h, will also be referred to as the nuclear encounter probability (or the nuclear encounter yield) for a single trajectory.

4. Distribution of trajectories Suppose we use a monoenergetic and perfectly collimated beam of ions, characterized by an incidence energy E, and an incidence angle @a. As a result of interactions between the ions and atoms in the crystal, the beam at a depth z will no longer be monoenergetic and per”Eectly collimated. The resulting distributions of E(z) and +(z> depend on the beam-target system studied, as well as on the crystallographic direction. Other parameters of interest are 6 and b. Although the scattering angle, 8, is usually defined to correspond to actual scattering events, it can be treated as a depth-dependent parameter describing a trajectory. Its value is uniquely determined by an instantaneous direction of the ion (i.e. the ion velocity) and the detector direction. Thus, 0 and I) are expected to be strongly correlated.

144

A. Llygo et al. /h&l. Insfr. and Mefh. in Phys. l&s. 3 90 (1994) 142-149

To simplify our analysis, let us assume that the depth slab AZ contains only one atomic plane (parallel to the crystal surface), i.e. AZ =d, where d is the interplanar distance. Then h, = Ah, = a(E, 6)p,(b), where b is the impact parameter with respect to the equilibria position of the closest atom in the atomic plane. As far as the yield is concerned, any trajectory within the slab At, at a depth z, is fully characterized by four parameters: E, 8, +, and b. For a randomly selected trajectory, these parameters may be considered to be random variables. Accordingly, the joint distribution function of the 4-dimensional continuous random variable T = (E, 8, I&,b) describes the distribution of all trajectories at a depth z. The corresponding probabili~ density function of the variable T, f(E, 8, $, b), fulfills the normalization condition: ////

f(E,

0, I/, b) dE dfl d+ db = 1.

1.38

1.40

1.42

1.36

1.40

1.42

0.06

0.06 s --T0.04

0.02

0.00

0.08

0.06

(9)

By definition, the expected value of a function y = y(E, 8, $, b) of the four random variables, with respect to their joint distribution, is (~)=flll~(E,e,JI,b>f(E,8,JI,b)dEdedd,db.

1.36

0.10

(10)

The functions of T that we are concerned with are: - h,(E, 8, b) = u(E, B)p,(b), which is the nuclear encounter probability (yield) for a given trajectory [incidentally, the NEP has nothing to do with the probability of the trajectory in question; the latter is f(E, 8, tfr, b) dE do d@ dbl; - h,fE, 8, I/J)= a(E, @)p,(Ji), which is the nuclear encounter probability for the trajectory (E, 8, $, b) in a random medium; and - x(#, b) = h,/h, =p,(b)/p,(+), which is the normalized nuclear encounter probability for the trajectory (E, 8, g, b), i.e. the NEP in the crystal, h,, relative to the corresponding probability in a random medium, h,. Based on Eq. (lo), one can calculate the expected values of the above functions: - (h,), which is the expected value of the NEP for trajectories generated in the crystal (it thus corresponds to the close encounter yield for a beam of ions interacting within the slab AZ); - (h,), which is the expected value of the NEP in a random medium for trajectories generated in the crystal; and - (x) = (h,/h,), which is the expected value of the normalized NEP for a trajectory. Let us take an example of 4He ions incident on a Si(100) single crystal. We consider the incident energy E0 of 1.5 MeV and 3 incident directions I& = 0, 6, and 12”. The case $,, = 0” corresponds to (100) axial channeling. The distributions for @a= 6” and 12” are taken as the average over all ~imuthal angles with respect to

s

2

0.04

0.02

0.00 1.36

ENERGY E (MeV)

Fig. 1. Energy distributions for (a) transmitted and (b) backscattered ions.

the (100) axis. As investigated in refs. [lO,ll], the azimuthally averaged yield for +!Q= 6” gives the closest possible approximation to the random yield for the system studied. The direction clto= 12” is taken for comparison. It is also a close approximation to “random” incidence, but it may be affected to some extent by planar channeling [lO,ll]. The distributions of trajectories have been generated by the CXX Monte Carlo simulation code [S,lO,ll]. A Hartree-Fock ion-atom potential [7,8], based on a solid-state charge density of Si 1141,and the vibrational amplitude of 0.08 A were used in the calculations. In each case, a total of 46< 10’ trajectories were followed to a depth of 4344.8 A, i.e. 3200 atomic planes from the surface. Let us note that the average path length travelled by the ions, Ii,, scales as z/cos &,. The detector is placed at an angle of 107” with respect to the incident beam. The ratio of outcoming and ingoing path lengths, l,JEin, remains constant for all directions of incidence. Figs. la, 2a, 3a, and 4a present the resulting probability densities - f,(E), fz(fJ), f&J/I, and f4(b) - associated with the respective marginal distributions of E, 8, $, and b. As expected, the distributions corresponding to axial channeling differ markedly from those for Jlo = 6” and 12”. The energy distribution, f&E), for

A. LIygoet al. /Nucl. Instr. and Meth. in Phys.Rex B 90 (1994) 142-149

$,, = 0” is shifted toward higher energies, and its shape is highly asymmetric. High-energy tails, observed in the distributions for I& = 6” and 12” indicate that a fraction of the total number of trajectories become channeled over the penetration depth, despite the apparent of these directions. The distributions “randomness” f2(0) and f3($) in the channeling case exhibit a substantially reduced angular spread as compared to those for I&,= 6” and 12”. At the depth studied, almost all ions in the channeling case are confined to directions within Lindhard’s critical angle, +I = 0~7, from the axis, with the maximum of the distribution f,(+> being shifted to 0.1” < t+Q < 0.2”. The observed shift corresponds to a ring-shaped pattern for the intensity of the ions when measured in a transmission experiment (assuming uniformity of the distribution in the azimuthal direction). Such patterns were indeed observed for 0” < & < tJl, although not for $,, = 0” [15]. Most probably the angular resolution in the experiments of ref. [15] was not high enough to detect this effect for I/J,,= 0”. The reduction of the small impact parameter collisions in the channeling case shows up very clearly in the respective distribution function fJb):,for b2 0 due to the central string, for b I 1.92 A due to first-neighbour strings, and for b I 2.72 w due to sec-

0

2

4

6

I

I

I

145

6 /

10

12

I

I

14 I

(a) 0.3

j

0

3 L

0.2

<

-

6’

-

lz”

I

,

i ‘:.

0.1

!

L-.

0.0

’ I

I

/

I

I

(b) 0.3

-

5 0.2

-

0.1

-5 :: : -..I. i.::

. .... .

00

-

6O

-

lz”

10

12

3=

%> ” -L_..,.,

0.0 0

2

I 4

6 ANGLE

0

14

.+ (deg)

Fig. 3. Angle JI distributions for (a) transmitted and (b) backscattered ions. 104

106

ond-neighbour strings. This reduction is accompanied by an enhancemett of f4(b) near the center of the channel (b = 1.36 A) as compared to the non-channeling distributions. Table 1 lists the expected values of the four parameters and their standard deviations, estimated from the distributions presented.

0.2 s < 0.1

1

0.0 I

I

I

I

I

1 (b) .

104

00

-

6’

-

lz”

106 SCATTERING

106 ANGLE

110

0 (deg)

Fig. 2. Scattering angle distributions for (a) transmitted (b) backscattered ions.

and

5. Normalized

nuclear encounter probability

The concept of the NEP was introduced by Barrett [2] to measure the yield of nuclear processes in single crystals with respect to the yield in a random medium. Basically, his analysis was restricted to the case of near-normal incidence (I,$~-=K1) and a shallow penetration depth. Consequently, the energy loss and straggling, as well as the angular spread of the beam were neglected. In such a case, one obtains h, = u(E,, B,)p,($,) = ho = constant, regardless of the trajectory. In general, however, the random yield depends on E, 6, and +, which are trajectory-dependent functions of z. To deal with the dependence of h, on trajectory, we introduce the normalized NEP for a single trajectory, x = h,/h, (cf. section 4). Then, the normalized NEP for the distribution of trajectories, Y(z), can be defined as II. CHANNELING

146

A. Dygo et al./Nucl. Instr. and Meth. in Phys. Res. B 90 j1994) 142-149

generated in a random medium, Y(z) = 1 at any depth 2. It is assumed in Eq. (11) that $ does not change significantly in successive collisions within AZ. If AZ = d, the normalized NEP can be explicitly written as [cf.

0.04

0.03

Eq. (1O)l

0.02

Y(z)=

2 c

0 f

r

0.01

-_ 0.00 0.0

0.5

1.0

f.5

2.0

2.5

3.0

0'1° -

I i 0.5 IMPACT PARAMETER

b (A)

Fig. 4. Impact parameter distributions for (a) transmitted and (b) backscattered ions. Note different scales in (a) and (b).

the expected value of x. Based on Eqs. (2) and (81, one obtains

PC(b)

Ml -f(E, Pr(JI)

0, Jl, b) dE d@ d@ db.

(12) In terms of the expected value with respect to the distribution of trajectories, Barrett’s definition of the normalized NEP corresponds to Yn(z) = (h,)/h,, where h, = a(E,, O,)p,(b). Assuming that the yield (h, > for a beam of ions can be determined experimentally as a function of depth into the crystal (cf. section 61, Y&l may be regarded as a physically measurable quantity (under the assumptions u(E, e) = tr(E,, 0,) and Jt = &,). When h, is no longer appro~mated by a constant, the normalized NEP as defined by Eqs. (11) or (12) ceases to be proportional to a measurable quantity. In fact, it is not possible to define the normalized NEP in such a way that Y(z) = 1 in a random medium, and at the same time Y(z) is proportional to (h,) through a constant, independent of z. Indeed, if one demanded that Y a (h,), it would result in the formula

(13)

(11) The resulting expression does not depend on the cross section. Obviously, for a distribution of trajectories

Table 1 Expected values and standard deviations of E, 0, 9, and 6 for 1.5 MeV 4He ions penetrating a Si(100) single crystal to a depth of 4344.8 A J10ldegl

0

6

12

(E) 1keVl s(E) lkeV1 (0) [de4 ~(8) BegI (9) ldegl A/J) Idegl

1412.16(4) 8.3%4) 107.000(1) 0.207(l) 0.186(l) 0.225(3)

1382.94(2) 7.84(Z) 106.999(2) 0.777(2) 6.062(2) 0.765(3) 1.466(2)

1380.98(4) 7.78(3) 107.001(2) 0.782(l) 12.017(2) 0.783(3)

(b) L% s(b) [A

1.434(l) 0.431(l)

0.545~1)

1.454(l) 0.542(l)

Here, the yield for all trajectories is normalized to the same value, (h,), and Y’(z) = 1 in a random medium. However, (h,) depends on z, and - more importantly _ it is not a physically measurable quantity itself, as it pertains to the distribution of trajectories generated in the crystal, not in a random medium. Consequently, Y’(z) does not correspond to a measurable quantity either. Additionally, Y’(z) depends on the cross section, and its evaluation would be much more involved. In Monte Carlo simulations of ion trajectories in single crystals, the expected value Y(z) can be estimated, based on a sufficiently large number (M) of simulated trajectories. Taking Eq. (121, for example, one obtains (14)

For a trajectory j, bj represents the impact parameter with respect to the equilibrium position of the closest atom in the given atomic plane.

6. Backscattering

energy spectrum

There is a distinct difference between the yield h,, given by Eq. (S), and appearing in the definition of the NEP (Eq. (ll)), and the yield that is measured in a backscattering energy spectrum. The difference comes from the fact that the NEF is a function of the penetration depth, while thie spectrum is a function of the outgoing ion energy, E,. ff the energy loss along the outgoing path is AE,,, then E, = KZ - A&.,,, where K is the kinematic factor. The energy loss AE,, is trajectory specific, similar to the parameters E, 8, 9, and b. The scattering yield corresponding to a single trajectory, registered in the spectrum at an energy Et, is given by with the dispersion AI&.,,, (keV/channel), h, = EAta,,, k

= IX&

(11%

W&Q?&

k

WhiX

E* -

4AECtI,5 Et,,

< G f $A&an.

Pc)

The index k counts all “collisions”, in which condition (15~) is satisfied. The individual contributions, Ah,,,, are not restricted to a fixed AZ. There is also no need to assume that the scattering cross section, Q(&, Qk), remains constant in all collisions that contribute to h,, Thus Eq. (3%) uses the form of Eq, (7) rather than that of Eq. (8). Taking into account that a given trajectory may give more than one contribertion to ia,, fet ns consider the distribution of all the contribtdms. In addition to the parameters E, 8, 9, and b, every contribution is now characterized by the depth, z, at which the scattering occurs. As the variable z refers to atomic planes, it is discrete in character. However, the standard deviation of the variable .z and/or the experimental depth resolution are usually much greater than the interpfanar distance. Therefore, to simplify the analysts, we treat z as a continuous variable, and describe the contributions to Ft, by tf”le Li-dimensional random variable S = (E, 8, $, b, z). ff g(E, 8, $* b, z) is the ~robabili~ density fnnction of the variable S, the expected value of tt, can be written as W

=

(

IX%

3= ~~~~~ x

W&W

k q%

>

@)JJ,(b)g(E,

dEdB dJI db dz,

8, Jr, b, q

W> where the summatian over K is replaced by the integration over z. Eq_ (16) gives the scattering yield actually measured in the spectrum, at the outguing energy

Et. Let us nate that the expected value (h,), where h, is given by Eq. (S), may be considered to be a physically measurable quantity inasmuch as it approximates (h,}. The density function g{E, 8, rfi,b, z> describes the distribution of all the contributions, regardless of their magnitude. In the case of ba~kscattering, more interesting is the weighted distribution

as it represents the distribution of the backscattered ions. The density function w has been estimated for the case considered in section 4, i.e. 1.5 MeV 4Wc ions incident on a Si(lO0)single crystal, and backscattered through an angle of 107”. The caiculations were performed for the same 3 directions of incidence with respect to the (100) z&s: @a= 0, 6, and 12”; the fatter two averaged over the ~~rnu~h~ angle, A total of 3.2 x 106 trajecto~jes in the (if@> axial channeling case, and 0.8 x IQ6 trajectories for each of the nonchanneling directions, were followed. The distributions were recorded for the outgoing energy E, of 433 keV, The probability densities wI(E), w,(B), w&I&)~w&b), and w,(z), associated with the respective marginal distributions of E, 0, $, 6, and z, are presented in Figs. Ib, 2b, 3b, 4b, ;and 5, respectively. The prominent difference between the miai channeling and the nonchanneting directions, observed in the distributions of all the trajeeto~es (section 4), is strftnggy subdued in the d~st~bnt~~ns of the tr~ecto~es that lead to back~tterin~ events. Especiahy the functions w,(a), ~~($1, and w,(b) show close similarity for all 3 directions studied. The energy and depth distributions w,(E) and w,(z) - in the case of (100) axial channeling exhibit somewhat larger spread than the respective non-channeling distributions. The mean energy loss corresponding to incidence along the (100) axis is very close to that for the non-channeling directions. Actu-

A. @go et al. /Nucl. Instr. and Meth. in Phys. Rex B 90 (1994) 142-149

148

Table 2 Expected values and standard deviations of E, 13, 1(1,b, and z for 1.5 MeV 4He ions backscattered to 107” in a Si(100) single crystal, recorded at E, = 433 keV & Bed

0

6

12

(E) lkeV1 s( El LkeVl

1376.8(4) 10.5(3) 106.94(5) 0.81(51 0.90(4) 0.72(71 0.107(2)

1380.5(2) 4.94(3) 106.97(l) 0.84(3) 6.07(2) 0.82(2)

1381.2(2) 5.4(l) 106.96(2) 0.83(5) 12.03(3) 0.7X5)

(0) [deal s(B) ldegl

($> Beg1 ~($1 ldegl (b) [Al s(b) ii%1

0.10001

0.059(2)

(2) [Al s(z) rz41

4345(4) 88(13)

7. Conclusions

0.09701 0.0519(3)

0.0528(2) 4350(2)

4261(3)

38(4)

87(5)

ally, the maximum of the distribution w,(E) for +,, = 0” is shifted towards lower energies, implying slightly higher (by = 4%) rather than lower mean energy loss for the backscattered ions in the axial channeling case. This result is consistent with experimental data (cf. Fig. 3 in ref. [16]). The expected values and the standard deviations for the distributions studied are given in Table 2. A natural normalization for the energy spectrum is provided by the random yield at the surface [cf. Eq. (211 ho=‘+(Eo,

~oM+o)

=dEo7

NAz, ~O)cos~

(181

where

A-&m

NAz, = -

LEOI’

and the stopping cross section factor [eo] is given by 1121:

fMEo) + 4Z=o) LeoI=-> cos$0

cos*mlt

(141. One notes that the NEP [i.e. Y(z)] is of no use in calculating Y(E,), although the same elementary components, p,(b), appear in both quantities.

(20)

where E is the stopping cross section and I),,“~ is the angle between the surface normal and the detector direction. The normalized yield is thus (cf. Eqs. (151 and (181)

The scattering yield for energetic ions moving in a crystal lattice has been analyzed in terms of the parameters E, 8, #, and b, characterizing individual ion trajectories. The normalized nuclear encounter probability for a beam of ions interacting in a single crystal, employed in Monte Carlo simulations of channeling, has been re-defined as the expected value of the normalized NEP for a single trajectory, with respect to the distribution of trajectories. The definition has been extended for directions making larger angles with the major axial direction in the crystal. The scattering yield registered in the energy spectrum has also been expressed as the expected value of the respective quantity associated with a single trajectory. The distributions of E, 0, I), and b were calculated for 1.5 MeV 4He ions traversing a Si(100) single crystal, and compared with the respective distributions for trajectories that lead to backscattering events. It has been shown that the normalized NEP cannot be defined in terms of a measurable quantity if the energy and angular spreads of the beam in the crystal are taken into account. This restriction limits the usefulness of the NEP concept: it is principally a measure of deviations of the nuclear encounter yield in a crystal with respect to that in a random medium. To compare with the experimental scattering yield, the yield registered in the energy spectrum, as given by Eq. (211, rather than the NEP, should be evaluated in the simulation.

Acknowledgements

We wish to thank Peter J.M. Smulders for discussions concerning the NEP question. A. Dygo wishes to acknowledge NSERC support during his tenure as a Postdoctoral Fellow at UWO.

(h,)

Y(4) = 7

References

0

(21) with conditions

given by Eqs. (15b) and (15~). In the Monte Carlo calculation, the expected value in Eq. (21) can be estimated based on a sufficiently large number of simulated trajectories, similar to Eq.

[I] See, e.g., D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129; L.C. Feldman, J.W. Mayer and ST. Picraux, Materials Analysis by Ion Channeling (Academic Press, New York, 1982). [2] J.H. Barrett, Phys. Rev. B 3 (19711 1527. [3] J.H. Barrett, Nucl. Instr. and Meth. B 44 (19901 367. [4] P.J.M. Smulders and D.O. Boerma, Nucl. Instr. and Meth. B 29 (1987) 471.

A. @go et al. /i%d.

Ins&. and Meth. in Phys. Res. B 90 (1994) 142-149

[S] A. Dygo and A. Turos, Phys. Rev. B 40 (1989) 7’704. [6] A. Dygo and A. Turos, Nucl. Instr. and Meth. B 48 (1990) 219. [7] A. Dygo, P.J.M. Smulders and D.O. Boerma, Nucl. Instr. and Meth. B 64 (1992) 701. 181 P.J.M. Smulders, A. Dygo and D.O. Boerma, Nucl. Instr. and Meth. B 67 (1992) 185. [9] A. Dygo, W.N. Lennard and I.V. Mitchell, Nucl. Instr. and Meth. B 74 (1993) 581. [IO] A. Dygo, W.N. Lennard, I.V. Mitchell and P.J.M. Smulders, Nucl. Instr. and Meth. B 84 (1994) 23. [Ill A. Dygo, W.N. Lennard, I.V. Mitchell and P.J.M. Smulders, these Proceedings (15th Int. Conf. on Atomic Collisions in Solids, London, Ontario, Canada, 1993) Nucl. Instr. and Meth. B 90 (1994) 161.

149

[12] WK. Chu, J.W. Mayer and M.A. Nicolet, Baekscattering Spectrometry (Academic Press, New York, 1978). [13] B.T.M. Willis and A.W. Pryor, Thermal Vibrations in Crystallography (Cambridge University Press, Cambridge, 1975). 1141J.F. Ziegler, J.P. Biersack and U. Littmark, The Stopping and Range of Ions in Solids (Pergamon, New York, 1985). 1151D.D. Armstrong, W.M. Gibson and H.E. Wegner, Radiat. Eff. 11 (1971) 241. [I61 J. Bottiger and F.H. Eisen, Thin Solid Films 19 (1973) 239.

II. C~NELING