Computer simulations of slowing down of ions in polycrystalline materials

Nuclear Instruments and Methods in Physics Research 218 (1983) 799-802 North-Holland, Amsterdam

COMPUTER MATERIALS

SIMULATIONS

OF SLOWING

DOWN

799

OF IONS IN POLYCRYSTALLINE

Mikko HAUTALA Department of Physics, University of Helsinki, Siltavuorenpenger 20 (\ SF-O0170 Helsinki 17, Finland

An efficient computer program using the binary collision approximation has been specially developed to study the slowing down of low energy heavy ions (in the keV-region) in polycrystalline materials. The program has been used to investigate the range distributions of 120 keV Pb in A1 and 20-100 keV A1 in Ta with a potential calculated assuming the total electron density to be the sum of the two atomic densities. The experimental RMS-vibration amplitudes were used. The displacements of the lattice atoms were uncorrelated and were distributed according to a Gaussian. The distributions are found to be sensitive to lattice vibrations and to the degree of polycrystallinity, but are less dependent on the potential and on the inelastic energy loss. Good agreement with measurements was found only if a partial preferred orientation of the microcrystals was assumed.

I. Introduction

In recent years considerable progress has been made in calculating the potential between colliding atoms and accordingly in calculating the nuclear stopping power of low energy ions in a m o r p h o u s solids. Wilson and Bisson [1] used a modified Wedepohl method [2] to calculate the potential between closed-shell atoms. In this method the charge distributions of the individual atoms are calculated using the Hartree Fock Slater-formalism. W h e n the interaction energy of the colliding atoms is calculated, the charge distributions are assumed to be undisturbed. The total energy of interaction between the two charge distributions is the sum of the Coulomb, kinetic and exchange energies. Wilson et al. [3] found these potentials to be suitable also for the calculation of nuclear stopping. We have calculated [4] the potentials by using Dirac Fock e l e c t r o n d i s t r i b u t i o n s in the original G o r d o n Kim m e t h o d [5], in which a small correlation term is added to the energy of interaction. We have tested the potentials by c o m p a r i n g with experiment the calculated ranges a n d range distributions of several ions in various a m o r p h o u s materials and found them to agree to within 10% over the reduced energy interval c -- 0.01-1 [4], [6-9]. However, when the experimental targets have been polycrystalline, the calculations assuming a m o r p h o u s targets fail, especially in explaining the tails of the distributions (e.g. refs. 9 a n d 10). The reason seems to be partly the crystalline structure of the targets. For simulations of the slowing down of energetic ions in crystalline solids, two methods have been used. In the first case one solves the force equations taking into account the n-body interactions (e.g. refs. 11-13). This m e t h o d is particularly useful at low energies when 0 1 6 7 - 5 0 8 7 / 8 3 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

n-body events are important; at higher energies the c o m p u t i n g time required precludes its use. This method is also generally restricted to studies where statistical information is unnessary. In the opposite case the other method, the binary-collision approximation, is superior, because of its rapidity. R o b i n s o n and Oen used this m e t h o d in the 1960s to study in detail the c o m m o n features of the slowing down in various crystals [14]. In the 1970s the code was extended (and n a m e d M A R L O W E ) to treat the displacement cascades [15]. M A R L O W E has primarily been used to treat fusion reactor first wall problems, i.e. sputtering, reflection and radiation damage a n d for light particles at low energies. There has been a limited interest in studying the basic stopping problem, i.e. the accuracy of the electronic and nuclear stopping. In the 1960s when channeling was discovered by R o b i n s o n and Oen using simulation [14] and was confirmed experimentally, the comparison between theory and experiment was only qualitative, because there were too m a n y u n k n o w n factors. In particular, the potential was uncertain. The situation has now changed, since the knowledge of the potential has improved considerably. Nevertheless, there is a considerable need to know better the stopping power in polycrystalline materials, because of their c o m m o n use in experiments; e.g., when measuring nuclear lifetimes using the Doppler-shift a t t e n u a t i o n (DSA) method, correction factors for nuclear a n d electronic stopping have to be included in the analysis of the lifetimes. In the present p a p e r we report a comparison between the results obtained from the new c o m p u t e r code with measurements made in our laboratory, in those cases where the assumption of an a m o r p h o u s target material has led to large deviations between calculated and measured range distributions.

M. Hautala / Computer simulations of slowing down of ions

800

2. The procedure A primary aim in developing the computer code C O S I P O was to make it as fast as possible without any serious loss of accuracy. A detailed description of the structure and the capabilities of the program, the accuracy of the approximations and a comparison with other calculations will be presented in a separate paper. The main features and assumptions in the present calculations are: (1) The incident particle makes only binary collisions with the target atoms. (2) The scatterer is chosen from the nearest and second nearest neighbours to be the one that is nearest and has an impact parameter less than 7r 1/2N 1/3 where N is the n u m b e r of atoms per unit volume. If there are no such neighbours, the one with the smallest impact parameter is chosen. If the distance to some other target atom deviates less than 0.4 ,~ from the distance to the primary scatterer, the collision with that target atom is treated simultaneously. The final direction of the ion after the simultaneous collisions is determined by adding the deflections vectorially. In a previous paper [16], the neighbour with the smallest impact parameter was always chosen and no simultaneous collisions were treated. The method above is now used, because (a) the computing time does not differ appreciably, (b) in some calculations in monocrystalline materials the results differ from each other and (c) the method is physically more correct. (3) The potentials between the colliders are those described above, i.e. the G o r d o n - K i m method was used. (4) The

scattering angle is evaluated from the standard scattering integral. (5) Thermal vibrations are included, assuming the displacements to be uncorrelated and distributed according to a Gaussian. The RMS-vibration amplitudes are taken from ref. 17. (6) The electronic stopping was assumed to be either a frictional force according to the LSS-theory ( d E / d x - - k E ]/2) [18] or was calculated separately in each collision according to the Firsov theory [19]. The statistics varied from 200 to 1000 ions. In the simulation, various degrees of polycrystallinity are allowed. When in the following the target is called "polycrystalline", the target crystal is randomly rotated before the simulation of the slowing down of each ion. The ions slow down in a single crystal. Accordingly, the calculations are valid, when the ranges of the ions are small with respect to the grain size. When partial polycrystallinity is assumed, the polar angle 0 of the incident ion is chosen randomly between some m i n i m u m and m a x i m u m angles with respect to a selected direction ((100) in the present simulations). The azimuthal angle ~o is random between 0 ° and 360 °.

3. Calculations and discussion The calculations should most accurately simulate experiment when low energy heavy ions slow down in polycrystalline stopping materials consisting of lighter atoms. Then the scattering angles in the laboratory coordinates are small. The effect of the electronic stopping, which is not well known, is small in this case due

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purposes, the calculations assuming an a m o r p h o u s stopping material are also included. It is seen that: (i) the distributions assuming an a m o r p h o u s target, differ appreciably from all the others; (ii) the distributions are sensitive to thermal vibrations; and (iii) in the distributions for the polycrystalline target, long tails have emerged but too m a n y of the ions still feel the target to be similar to the a m o r p h o u s case. This gives a strong indication that in the actual case the target material used in the experiment might have h a d partial preferred orientation of the microcrystals (texture). In fig. 3 it is shown how the range distribution of 120 keV Pb in A1 changes when partial orientation of the microcrystals is allowed. The calculated distributions are convoluted with the experimental detector resolution (o = 15 nm). Excellent agreement with experiment can be found by suitably choosing the orientation of the microcrystals. A remarkable point is that the most p r o b a b l e range now also agrees with experiment, whereas in the polycrystalline case the peak is too near the surface. It should be mentioned that, if a static lattice is assumed, the peak is even closer to the surface. The evident explanation is that, because the m a x i m u m scattering angle of Pb-ions from Al-ions is 7.6 °, Pb-ions may undergo sequentially close collisions, if they once come near an Al-ion. A comparison of the 20, 60 a n d 100 keV A1--* Ta integral distributions is presented in fig. 4. G o o d agreem e n t with experiment can be found already if the target is polycrystalline in the simulation, at least when comp a r e d with the " a m o r p h o u s " distributions. The calculated distributions lie only slightly below the experimental distributions. The Al-ion is lighter than the T a - a t o m a n d consequently large angle scattering can change its direction more easily than the Pb-ion in A1. It seems

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M. Hautala / Computer simulations of slowing down of ions

also that Ta is more polycrystalline than Al. Only slight preferred orientation is needed to fit the experimental distributions. The experimental maximum range, particularly for the 100 keV Al-ions, is shorter than the calculated one, because the effect of the electronic stopping is noticeable in this case. When the electronic stopping is multiplied by 1.5, good agreement near the maximum range is also achieved. A n o t h e r reason for the longer maximum range in simulation could be that the assumption of slowing down of the ions in a single grain is not valid, due to defects or the size of the microcrystals. In conclusion, the computer simulations presented here d e m o n s t r a t e that good agreement with experiment can be achieved, if texturing of the polycrystalline target is taken into account. Furthermore, the simulations indicate that even in the case of polycrystalline materials with very strong partial orientation, the effect of texture on the most probable range is small. The main effect of the crystallinity is to produce long tails in the distributions, leading to important asymmetries and consequently to higher mean ranges.

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[4] M. Bister, M. Hautala and M. J~.ntti, Rad. Eft. 42 (1979) 201. [5] R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 (1972) 312. [6] M. Hauta[a, Rad. Eft. 5l (1980) 35. [7] M. Hautala and M. Bister, Rad. Eft. 54 (1981) 191. [8] M. Hautala, R. Pahemaa, A. Anttila and M. Luomajfi.rvi, Nucl. Instr. and Meth. 209/210 (1983) 37. [9[ M. Bister~ J. Keinonen and A. Anttila, Phys. Lett. 74A (1979) 357. ]10] J. Keinonen, M. Hautala, M. Luomaj~.rvi, A. Anttila and M. Bister, Rad. Eft. 39 (1978) 189. [11] W.L. Gay and D.E. Harrison, Phys. Rev. 135A (1964) 1780. [12] H. Lutz, R. Schuckert and R. Sizmann, Nucl. Inst. and Meth. 38 (1965) 241. [13] J.B. Gibson, A.N. Goland, M. Milgram and G.H. Vineyard, Phys. Rev. 120 (1960) 1229. [14] M.T. Robinson and O.S. Oen, Phys. Rev. 132 (1963) 2385. [15] M.T. Robinson and I.M. Torrens, Phys. Rev. 9B (1974) 5008. [16] M. Hautala, Phys. Lett. 95A (1983) 436. [17] International Tables for X-ray Crystallography, Vol. llI (Kynoch Press, Birmingham, 1962). [18] J. Lindhard, M. Scharff and H.E. Schiott, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1963) no. 14; J. Lindhard, V. Nielsen and M. Scharff, Mat. Fys. Medd. Dan. Vid. Selsk. 36 (1968) no. 10. [19] O.B. Firsov, JETP 36 (1959) 1076. [20] J.L. Combasson, B.M. Farmery, D. McCulloch, B.W. Neilson and M.W. Thompson, Rad. Eft. 36 (1978) 149.