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Fluctuations in atomic collisioncascades with power lawscatteringI. Pázsit a & R. Chakarova aa Department of Reactor Physics , ChalmersUniversity of Technology , S-412 96, Göteborg,SwedenPublished online: 20 Aug 2006.

To cite this article: I. Pázsit & R. Chakarova (1998) Fluctuations in atomic collisioncascades with power law scattering, Transport Theory and Statistical Physics, 27:5-7,477-497, DOI: 10.1080/00411459808205639

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TRANSPORT THEORY AND STATISTICAL PHYSICS, 27(5-7), 477-497 (1998)

FLUCTUATIONS IN ATOMIC COLLISION CASCADES WITH POWER LAW SCATTERING

I. Prizsit and R. Chakarova

S-412 96 G6teborg, Sweden Department of Reactor Physics, Chalmers University of Technology,

ABSTRACT In a preceding paper [l], fluctuations in a collision cascade were investi-

gated. The variance of the yield as well as the energy and depth correlations in sputtering and defects were calculated with constant cross sections and a hard-sphere scattering model. In a more real approximation, atomic coll- sion cascades obey power law scattering, which has a rather different collision kinematics. In particular, the energy transfer, and thus the energy distribu- tion within the cascade evolution, is very different from that of a hard-sphere model. Therefore the yield and mean distribution, as well as the variance and the two-point cascade correlations were calculated in the present paper in a power law scattering model. It was found that despite the differences in the cascade dynamics, most previous findings are still valid. Notably, the relative variance of defects is low in infinite media, whereas in half-spaces it is a monotonically increasing function of the energy of the bombarding ion. The two-point energy and depth correlations are also quite similar to their hard-sphere counterparts. Calculated values of the yield, mean distribution, variance and two-point correlations are shown in the paper for several cases. All calculations were made by using a binary collision Monte-Carlo algorithm.

I. INTRODUCTION

Low-fluence irradiation of a solid with an energetic ion beam leads to a collision cascade which can be described by linear transport theory. The cascade develops as the ion creates primary recoils, and the primary recoils generate further recoils. The sites of recoil atoms become vacancies and the energetic atoms that come to rest constitute interstitials. Some recoils will be ejected through the target surface, leading to sputtering.

Due t o recoil production, the collision cascade is a branching process. It is well-known tha t branching processes have non-trivial (non-Poisson) statistics. For this reason, cascade properties have been a mat ter of continuous interest ([Z] - [14]). Most works, however, deal with the calculation of the variance of integral quantities such as the sputtering yield or the total number of defects in the medium. It was pointed out recently in our previous work [I] t h a t more information is available on the cascade in the correlation functions or two-point

477

Copyright Q 1998 by Marcel Dekker, Inc. \ I W W dckkcr corn

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478 PAZSlT AND CHAKAROVA

densities that describe joint cascade statistics at two phase space points. In particular, the variances of the yields are obtained as integrals over the correlations. Thus, knowledge of the correlations help understanding the widely differing behaviour of the variance in seemingly similar collision cascade processes [I].

Although it is possible to give an explicit transport solution for both the variance and the correlations, the explicit evaluation of these formulae is extremely complicated and has not been achieved so far except in one very simple model case [12]. These solutions are based on an integral over first moment (i.e. ordinary) solutions for the Green's function of the transport problem. These solutions will be briefly described in the next Section. In addition, no general solutions for the Green's function are available with power law cross sections in half-spaces. Due to these difficulties, the variances and correlations were calculated in this work by Monte Carlo methods. The Monte Carlo model used is described in Section 111 and the results in Section IV.

11. THE TRANSPORT MODEL AND THE COLLISION KERNEL

The basic transport characteristics of atomic collisions lies in the scattering cross section and scattering function, describing the binary interactions between energetic ions and atoms of the host material. Unlike nuclear interactions, the collision dynamics is determined by long-range interatomic potentials. As a consequence, the total cross section, just as with Rutherford scattering, is infinite. This divergence does not cause though any major distur- bance in the particle behaviour in the cascade, and the number density or flux remains finite at any finite energy. The physical explanation of this is that the part of the differential cross section that causes the divergence is that belonging to large impact parameters. However, collisions with large impact parameters incur negligible energy transfer and deflection.

For this reason, in the transport theory treatment, only the differential cross section is used, and the total cross section does not appear in the equations. The differential cross stctioii is a smooth function of the projectile energy and the energy transferred to the recoil. Within a limited energy band this type of cross section can be described by a power law scattering cross section as

da(E, T ) = C,,,E-"T-'-"dT (1) where

Zle and Zae are the nuclear charges, MI and M2 are the projectile and target masses, respectively, E is the projectile energy, T is the energy transfer and A, is a dimensionless function of m [9]. 0 c: m < 1 is a paramcter dependent on the interatomic potential. In this mork a const,ant m will be used throughout.

With the above notations, du(E, T ) = a(B, T)dT ('4

is the probability per unit path length that an energetic atom with energy E will undergo a collision such that it will transfer an energy between (T , T+dT) to the recoil. This scattering event will incur a deflection of the projectile which is determined by the conservation laws. IIowevcr, in agreement with standard literature, notation on the angular variables in the scattering kernel will be omitted. As (1) shows, for the above rn values, the total cross section

/ d o = J O ( E , T)dT (3)

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES 479

is divergent. For this reason, it is impossible to factorize the scattering kernel into a (total) cross section and a normalized scattering function, as it is usual in the neutron transport theory.

Below we summarize the relevant transport equations for the mean value and the second moment to indicate the solution of the latter. To simplify notations, the fornmlae will refer to the case of identical projectile and host atoms. The full formalism for the multi-species case is found in e.g. [Il l . Further, azimuthal symmetry of both the source and the detector as well as a homogeneous semi-infinite medium will be assumed. I n such a. case slab geomclry can be used in both the forward and the backward equations. We shall use here the backward approach. Since we are only interested in quantities (yield, variance etc.), given rise by one projectile, the stationary (time-independent) equations can be used.

With these conditions, the backward transport equation for the total number oi‘sput,t.rrcd particles N ( z , E , q ) , given rise by a projectile starting with ( z , E . q ) reads as

a q z N ( z , E , 7) = Nt / d u [ N ( s , E , q ) - N ( T , E - T , 7)’) - N(r , T , f ) ]

+b(Z)O(E - Ul)qO(-q) (1)

where Nt is the atomic densit,y of the target. Similarly, the second moment of t,he sputlerctl part icles

M(s, E , q ) E ( N ( N - 1)) (5)

satisfies the equation

v z M ( ~ , E , q ) d = N t / d v [ M ( ~ , E , q ) - n / l ( ~ , E - T , q ‘ ) - nf(.L‘”,T/’’)]

-2Nt d a N ( z , E - T , q’)N(z, T , T I ” ) (6) J As it was shown in [ll] and [12], and can also be seen from (4) and ( G ) , if the proper boundary conditions are taken into account, the solution of ( G ) , taken at z = 0 w i ( l i I / > 0 (ion bombardment on the surface) can be given as

M(O,E,q) = 2 J d ~ o d ~ 0 d ~ 0 ? 1 ( 0 : E : 7 j -+ zo>EOr710).

.Nt J d m N ( 0 , Eo - T , q’)N(zo, T , 7”) ( 7 )

where n(0, E , q -+ z0, Eo, qo) is the Green’s function of the transport, operator iii (.I). I n a similar manner, the two-point density in energy and angle for the sputt,crctl particles. m(E1,ql, E2,77210,E,q) , in a single species medium can be given as

m(E1,71,E2,q210,E,q) = 2/dlodEod7107”(0,E,q + TO,EO,?IO).

.N J d u o n ( l o , Eh, + 0, El , ~ i ) n ( l o , E:, 7: + 0, Ez , ~ 2 ) (S) The transport theory treatment of the creat.ion of vacancies and interstitials is v t ~ y

similar to the above. In particular, the scattering kernel du(E , T ) remains the same. ‘I’lie

treatment of the defects requires introduction of a damage model, i.e. energetic conditions for an energetic projectile or recoil being trapped or a host atom being knocked out. ’ I lk is handled by introducing the damage function p ( E ) which gives the probability or llic

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480 PAZSIT AND CHAKAROVA

projectile or recoil to leave the collision site. The most commonly used damage model is that of Khinchin and Pease [15] and it is this model that will be used in the numerical work in Section IV.

By introducing the damage model, backward equations for the mean, variance and corre- lations of the vacancies and interstitials can be derived. A stochastic theory of vacancies and intcrstitials was given e.g. in [17], here we only quote the equation for the vacancy-vacancy correlation function as

+p’.p’’(f:.f;l+ f;..f;’) + 6(r - rz).(f: + f;’) + S(r - rI).(f; + f:) +6(r - rl)b(r - r2)]

1 da = 1 u(r,v --t v’, v”)dv’dv’‘ where:

and the followiiig notations and abbreviations are used:

f ( t - 1 , r z l r , v) = ~ I Z

f (n , r z l r , v’) = fL f(rIlr,v”) = f;’

p‘ = p(E’ )

(9)

et,c. p ( E ) is the damage function (probability of the projectile or the recoil to leave the collision site). As usual, r and v are the projectile (ion) coordinates (variables) whereas rl and r2 arc t,lie defect coordinates (parameters). In the forward form, the role of the variables arid parametcrs is exchanged, thus rl and r2 are operated upon, see e. g. [4] and (161.

Again, solutions ca.n be given in forms similar to (7) and (8), i.e. in form of a multiple integral over mean value Crecn’s functions. However, no analytical or semi-analytical solu- tions are currently available for the Green’s function n(a , E,9 + 2 0 , E0,qO) or the sputtering yield with power law cross sections in half spaces. Even if they were available, the evaluation of the formulae (7) and (S) would be a very complicated task. For this reason all work in this paper regarding the evaluation of the variances and correlations was made by Monte Carlo techniques. Some basic principles of these calculations are given in the next Section.

111. THE MONTE CARL0 ALGORITHM

Several Monte Carlo ion transport codes have been reported in the literature, among the most rrfcrcnced ones being MARLOWE [lS], TRIM [19] - [21] and BEST [22] - [24]. The cascadr niodrl and the corresponding Monte Carlo algorithm used in this paper are based on assumptions that are in principle equivalent to those of the BEST code. There are however certain extensions such as the tallying of the defects, calculation of two-point densities and parallelization of the code. The collisions are described by power scattering cross sections. The scattering kernel was already discussed in the preceding Section.

The damage model used is that of Khinchin and Pease [15]. A displacement energy, Ed , is defined as the threshold for both recoil production and projectile trapping. Thus if both E - T > Ed and T > Ed, a vacancy will be created, and if both E - T and T are below Ed, an interstitial atom will occur. The binding energy is taken to be zero. The surface is

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES 481

treated as a planar absorbing barrier of height G r s , where energy reduction and refraction of thc emitted particles take place. In the calculations rcported below Ed = Cl, is used.

The Monte Carlo algorithm is based on the concept of mean free path, 1 = l / ( N o f o f ) , where N is the target density and ofut = Jdu. Since the total cioss section divcrges, a fn i t r utot is artificially created by introducing a minimum of the transferred energy

T,,, = rnz?~(10-~E, u3) Details of this concept are discussed in [9], 1251 and [2?]. In the calculations, length scales

are expressed in units of [Z], with C, = C, for A41 =

The final results for each source particle are recorded, i. e. the total number of vacancies, interstitials and sputtered particles, as well as their distributions. In addition, the quantities y k = y3yk are calculated, where E: denotes the number of particles (defects or sputtered particles) in the 2-th depth or energy bin, Ax,. The mean and variance are estimated on the basis of 10" source particles comprising one statistical batch. The 23-value of the correlation function, C,, , for a particular batch is given by:

and Eo denoting the source particle energy.

where Axj # Ark and the brackets are mean value notations. The final values of the output quantities and the corresponding standard deviations are obtained using data from up to 20 statistical batches. The accuracy of the results presented below depends on the projectile energy and on the type of the estimated quantity. For example, the sputtering yield error is between 0.5 and 2 % for the energy region considered.

The program is arranged to analyse also outcomes due to source particle interactions only, e. g. the total number of interactions and the number of vacancies created, the primary recoil characteristics such as energy, angular and depth distributions, the probability of the source particle to become an interstitial at certain depth, etc. The code is written in FORTRAN and implemented on a SUN workstation and on an IBM SP2 parallel machine using the MPI library [26]. More details about the model are given in [27].

In this paper results based on a power scattering model are reported. Both half-spaces and infinite media have been considered (except for sputtering which was only calculated in half-spaces). The following quantities have been calculated for different projectile/target mass ratios:

a) Depth distributions, V ( z , E ) , Z(z, E ) and S(z, E ) , of the vacancies, interstitials and origin of the sputtered particles, respectively, induced by a fast atom with energy E ;

b) Mean yield and variance of the defects (vacancies, V ( E ) , and interstitials, Z(E) ) and sputtered particles Y ( E ) ;

c) Energy correlations for sputtered particles, c(E1, Ez) for a given projectile energy; d) Depth correlations, c(z1,zz) for vacancies and sputtered particles.

IV. RESULTS AND DISCUSSION

Atomic collision cascades initiated by particles of the same or different type as those in the target have been simulated. An infinite or semi-infinite medium was assumed, and

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v - l

1000

(I)

m - ._ 4- c .- $ 800 I .- U m .z 600 C

0 9 -

400. 0 .- c $

= 200- s n E

c

0 -

k

? -2 0 2 4 6 8 1 0 1 2

Depth

Zmax

, .** J - & I

I I I I I -2 0 2 4 6 8 10

(a) Depth

Fig.la. Depth distribution of defects i n a n infinite nwdiiini for Eo/Ed = 10" and Alz/:tll = 1. The depth units are shown in Eq.(lO).

1000

800 In

a,

0

I

r

5 600. c .- c n ._ L c

400-

5 a n

200 -

0-

. .. / \,

I

0 2 4 6 8 10 (b) Depth units

Fig.lb. Depth distribution of drfrc.t.s in a scnii-infinite rnedi~lm for E , I / & ~ = lo4 and Rl2/Ml = I. The depth units are shown in Eq.(lO)

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES 483

a monodirectional beam with a normal or oblique incidence. Depth and energy distribu- tions obtained for different cases will be described in section IV.A, whereas the variance behaviour and correlation function calculations will be discussed in sections 1V.B and IV.C, respectively.

A. Depth and energy distributions of defects and sputtered particles

Depth distributions obtained for vacancies and interstitials, created in an infinite medium, are presented in Fig. la. Source particles with energy ED/Ed = lo4 of the same type as the target atoms are assumed. Keeping m constant over four orders of magnitude of energy is of course not realistic. However, our purpose here is more a basic study rather than reproduction of measurements and taking a higher source energy was found to enhance the features of the scattering model. The depth variable is taken to be zero at the injection point. The source direction is normal to the depth scoring surfaces, pointing toward the positive values. It is seen that most of the defects are in the positive half-space. The proba- bility of a particle to move backward and cross the half-space boundary is low because of the dominance of the small angle scattering, so that the two half-spaces are nearly isolated from each other. In addition, the majority of the collisions takes place at low energies where the mean free path is short. Therefore the corresponding defect distributions in a semi-infinite medium, Fig. l b , follow closely the positive depth part of the infinite medium distributions.

In both figures, Fig. la and lb , the vacancies and interstitials are distributed close to each other. This is due to the fact that collisions with a small energy transfer dominate and the path length of the created low energy recoils is short. The main differences are around the ‘zero’ depth surface, as shown in the insets of the figures. In the case of infinite medium, the number of interstitials is large at small negative depths. This illustrates again that particles, entering the negative half-space, have a low energy and rather become interstitials instead of initiating cascades.

The fact that power law cross sections lead to asymptotically (i. e. with increasing ion energy) identical vacancy and interstitials distributions, even with hard sphere scattering, was shown in [4] by a moment expansion method. It was also shown there that, on the contrary, constant cross sections lead to different vacancy and interstitial distributions. The latter was shown by explicit calculations in [l] and [17]. The identical vacancy and interstitial distributions calculated in the present work show that the two distributions are equal even with power scattering, i. e. with strongly forward peaked scattering. The last fact implies that out of two different physical features of the power scattering, i.e. energy dependent cross sections and highly forward-peaked scattering, it is the power law dependence of the cross sections that has the dominating effect on the distributions. It was seen earlier with constant cross sections, that the forward biased scattering leads to larger differences between the vacancy and interstitial distributions than scattering that is isotropic in the laboratory system (see e. g. [l] and [li’]). Apparently, with power law cross sections, the angular dependence of the scattering operator does not play a role regarding the similarity of the defect distributions.

Defects created by impinging particles of different type than the target atoms have diverse distributions. The differences are due to different properties of the primary cascade, i.e. the part of the cascade where interactions between the source particle and target atoms take place. The rest of the cascade, including recoil-target interactions, develops under the same rules in all cases. The contribution of the primary cascade to the total outcomes depends

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484 PAZSIT AND CHAKAROVA

0 1 2 3 4 5 6 7 8 9 10 (4 E/Ed

Fig.2a. Energy distribution of sputtered particles for &/Ed = 10’’ and ,!l2/hl, = 1.

EO/Ed = 10000

M2fM1 = 1

0.0 0.05 0.1 0.15 0.2 0.25 (b) Depth of origin

Fig.2b. Distribution of the depth origin of the sputtered particles for Eo/Ed = 10” and M,/M, = 1. The depth units are shown in Eq.(lO)

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES 485

also on the source particle energy. According to the calculations, the number of the primary defects for &/Ed = lo4 constitutes less than 4 % of their total number, whereas it reaches about 40 % for Eo/Ed = 100.

The behaviour of the distributions of the sputtered particles displays well known features and is not so sensitive to the projectile mass or angle of incidence. Energy distributions for Eo/& = lo4 and different Mz/Ml ratios are shown in Fig. 2a. The values are normalized to the total sputtering yield for each case. The location of the maximum in the energy spectrum is in agreement with the analytical value U8/(2(1 - m)) [lo]. The distribution of the depth of origin of the sputtered particles is a monotonically decreasing function of the depth for different mass ratios and angles of incidence. The case for M*/M1 = 1 is shown in Fig. 2b.

B. Mean and variance of the sputtering and defect yields

The mean and the relative variance of the defects in an infinite medium, (Nd) and oi/(Nd), respectively, have been calculated as functions of the source energy for M*/MI = 1 (Fig. 3). Similarly to the hard sphere case [l], [3], with increasing the energy of the projectile, the relative variance approaches a constant value which is less than unity. The numerical value obtained with m = 0.25 is around 0.22, and it is higher than that of the hard sphere case, which is 0.15.

The corresponding functions for a semi-infinite medium, as well as those of sputtered particles, are shown in Figs. 4a and 4b. What regards the energy dependence of the mean and variance of the sputtering yield in half-spaces and with power scattering (Fig. 4a), such data are available from calculations with the BEST code [14]. We have used those data for benchmarking our calculations. It is seen, that the relative variance for both the sputtered particles and the defects diverges with increasing energy. A similar tendency was obtained and discussed in the hard sphere case. The suggestion, that the ‘separation’ of the two half spaces enhances the fluctuations, because the surface crossing of an energetic particle means losing of its whole progeny, can be accepted also in the case of power law scattering.

It is also worth noting that the relative variance of the defect yield is a monotonically increasing function of the energy, just as in the hard sphere case. In other words, even with power scattering, there is a significant difference between infinite media (less than unity, constant relative variance) and half-spaces (increasing, over-Poisson relative variance). Further, it was shown in [12] and [14], that with increasing projectile energy, the relative standard deviation of the sputtering yield becomes constant. As Fig. 4b shows, the same is true for the relative standard deviation of the vacancies in a semi-infinite medium.

The different behaviour of the variance of defects in infinite media and half-spaces can also be explained by energy considerations. In the present calculations, in absence of electronic stopping, the energy deposited in the medium, goes into recoil production. As a consequence, the number of defects created will be a nearly deterministic function of the energy deposited in the system. In an infinite medium, all energy is deposited in the system, i.e. the deposited energy is a constant, leading to small relative fluctuations in the number of defects. In a semi-infinite medium, some energy will escape from the system, and thus the deposited energy will be a random number. In turn, the fluctuations in the number of created defects will be large.

Next, the mass dependence of the mean and the variance will be considered. It was found that the number of vacancies and the relative variance for an infinite medium are independent of the M2/Ml ratio. Results in the case of a semi-infinite medium for &/Ed =

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486 PAZSIT AND CHAKAROVA

Infinite medium. Vacancies and interstitials

4000

3000

a, 3 m

m

- : 2000 2

1000

0

I I I I I

I rel.var. (vac)

0.4

.0.3

a C

'C

F 0.2 $

I?

.- - - 0.1

10 50 100 500 1000 5000 10000 Eo/Ed

Fig.3. Energy dependence of the mean and relative variance of the defects in an infinite medium for M,/M1 = 1.

lo4, are shown in Fig. 5 . As seen from the figure, the behaviour of the relative variance of the sputtered particles is similar to that of the mean value. The fluctuations are larger for M*/Ml between 2 and 6 and then decrease for heavier target atoms. This is a more complicated (non-monotonic) mass-dependence than the general trend suggested in [7] where a monotonic dependence of the magnitude of fluctuations on the M2/M1 ratio was predicted. The corresponding functions for the vacancies show another tendency. The mean value of the defects is monotonically decreasing, whereas the fluctuations (the relative variance and the variance) are monotonically increasing in the mass region considered.

The different behaviour of the relative variance of the vacancies and the sputtered par- ticles shown in Fig. 5 may appear contradictory. A proper explanation of this phenomenon should include consideration of the statistics of the reflected particles as well as of those absorbed due to the surface harrier. Such an analysis is outside the scope of this paper. however, some discussion can be found in [27].

C. Correlations of sputtered particles and defects

Depth correlations in an infinite medium

The depth correlation functions of the vacancies created in an infinite medium in the case of equal source-target masses and a normally incident source beam with energy Eo/Ed = lo4

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES 487

Ik1O2

5*10'

l*IO'

5'10'

l*lool 5*10-' <

1 '1 0.' 3

5*1 O-* $

Semi-infinite medium, sputtered particles

. . . - -. . r

// rel.st.bev.

r" 10 100 1000 10000

(4 EoEd

Semi-infinite medium, vacancies

I,<,,,O/I I , I , , , , , I ,,,,,,,, I , , , , , , , , I , , , , , , , , I

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PAZSIT AND CHAKAROVA

Eo/Ed = 10000, Sputtered particles

488

Eo/Ed = 10000, Vacancies

0 2 4 6 8 10 12 14 (b) MUM I

Fig.5. Mass dependence of the mean and relative variance of sputtered partiLicu a1.d defects for Eo/Ed = lo4.

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES 489

are shown in Fig. 6. Strong correlations, positive as well as negative, are obtained in the half-space towards which the initial particles are injected and where the cascades are mainly developing. The positive correlations are concentrated around depth layers close to each other (zl z zz). The niaxinium is near the source point. In this region, the probability of finding two vacancies belonging to the same branch of the cascade is the largest. It, is seen that with increasing the depth of one of the vacancies (Izl - 221 +large), the probability diminishes and the correlation function becomes negative. Thus regions of strong posi live correlations are combined with regions of strong negative ones.

Similar correlation features as mentioned above have been obtained in the hard sphere case as well [I]. By definition, the correlation functions are symmetrical, i. e. C(z,ib) = C(zkzj) . It is interesting to notice that the contour line sepamting the positive and negative correlations, i. e. when C(z1,zz) = 0, tends to a constant in 21 for negative iz values aiid

vice versa. This constant value is very close to the depth value z,,,, where the dcfcct dcpth distribution has a maximum. This is true for both hard sphere and power scattering (Fig. 7). Also from the contour plots, a more isotropic structure of the hard sphere correlations can be observed. The latter appear to be nearly symmetric around the separatrix zl N and zz x zmaz. Whether it is the anisotropy of the scattering or the energy dependent nicaii

free path which is the physical reason for this dependence is difficult to say.

Correlations in a semi-infinite med ium

Regarding the sputtered particles, energy correlations, irrespective of the depth of origin, and correlations in the depth of origin, irrespective of the energy, have been calculatcd. Results for equal source-target masses and a normally incident source beam with e n q y Eo/Ed = lo4 are shown in Fig. Sa and 8b. The behaviour of the correlation functioiis is similar to that in the hard sphere case. They are mostly positive with larger values a.t El M Ez M Ed and z1 N z2 M 0 . The maxima correspond to the ones in t.he energy and depth distributions, respectively. The analysis of the shape of the functions, given i n [ I ] , is valid here too.

The depth correlation function of the vacancies calculated for the case of equal source- target masses and a normally incident source beam with energy Eo/Ed = lo4 is plotted i n Fig. 9. The structure of the function is very similar to the corresponding one for an in fin it,^ medium (Fig. 6) within the quadrant 21 2 0 or zz 2 0 . This is in accordance with the previous conclusion that the two half-spaces of the infinite medium are nearly independcnt regarding the cascade development. However, a more detailed comparison between the two cases shows that the positive correlations are stronger and the negative ones weaker in tlie case of the semi-infinite medium. In addition, the quadrants z1 or z2 < 0 , which are excluded in the semi-infinite case, contain larger areas with negative tha.n with positive correlations. The prevalence of the positive correlations for the semi-infinite nicdium rcsults in over- Poisson variance.

In Fig. 10 correlation functions for M,/M1 = 10 are shown. The positive correla.tions cover a larger ( q z z ) domain compared to that for equal projectile- target masses; tiic negative correlations are weaker which causes a large relative variance. The sha.pe of the boundary between positive and negative correlations is more linear, compared to the previous cases and one cone-like part dominates the region of positive correlations. This also nieans that, unlike the case of equal masses when the correlation function was sharply concentratcd at .zl = zz, here the concurrent generation of defects at some distance apart is still likely. In other words, the decay of the correlations with increasing Izl - 221 is much slower thaii in the case of equal masses. The tendency is stronger for an oblique incidence of the source

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8 - Eo/Ed = 10000 I

M2IM1= 1

PAZSIT AND CHAKAROVA

-

Infinite medium, vacancy depth correlations.

EoEd = 10000

M2IM1 = 1

-1 0 1 2 3 4 5 6 7 8

(b) z1

Fig.6. Depth correlations of vacancies in an infinite medium for Eo/Ed = lo4 and h3,/M, = 1. The depth units are shown in Eq.(lO)

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HSCC model

Fig.i'a. Depth distribution of vacancies in an infinite medium for the hard sphere model (taken from Ref. [l]).

Vacancies deDth correlation, infinite medium

-10 i -10 -5 0 5 10 15

(b) z1

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EdEd = 10000

MUM1 = 1

Fig.8a. Energy correlations for sputtered particles for Eo/Ed = lo4 and itfz/hl~ = 1.

EdEd = 10000

M2/M1= 1

Fig.8b. Correlations of depth origin of sputtered particles for &/Ed = lo4 and M 2 / M 1 = 1. The depth units are shown in Eq.(lO)

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES

Vacancy correlations, semi-inf. medium

EdEd = 10000

MUM1 = 1

493

1 2 3 4 5 6 7

(bl 21

Fig.9. Depth correlations of vacancies in a semi-infinite medium for Eo/Ed = lo4 and hIz/hf,. The depth units are shown in Eq.(lO)

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494 PAZSlT AND CHAKAROVA

Vacancy correlations, semi-inf. medium

EdEd = 10000

MUM1 = 10

M2IM1 = 10

1 2 3 4 5 6 7 0 9

(b) z1

Fig.10. Depth correlations of vacancies in a semi-infinite medium for &/Ed = lo4 arid hf2/iW, = 10. The depth units are shown in Eq.(lO)

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FLUCTUATIONS IN ATOMIC COLLISION CASCADES

Vacancy correlations, semi-infmedium

M

Eo/Ed = 10000 MZM1 = 10

.i 1

Angle 75 deg

495

1 2 3 4 5 6 7 0 9

(b) z1

Fig.11. Depth correlations of vacancies in a semi-infinite medium for Eo/Ed = lo4, A4*/Ml = 10 and angle of incidence 75 degree.

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496 PAZSIT AND CHAKAROVA

beam as illustrated in Fig. 11 for a 75 degree angle of incidence and for M , / M 1 = 10. The dominance of the positive correlations over the negative ones can be seen, which leads to large relative variances in agreement with the results in Fig. 5. The areas of non-zero coirelations are close to the surface, which can be rrlated to effects of the oblique incidence on the vacancies depth distributions.

V. CONCLUSIONS

It appea.rs that the statistics of defects with power scattering are rather different in half-spaces and infinite media. An over-Poisson statistics was found in half-spaces, which shows the same asymptotic tendencies in energy in this simple model as the sputtered yield. Further, the structure of the two-point correlations is not very sensitive to the choice of the scattering model, (hard sphere or power scattering with hxed power), unlike most other cascade properties.

In coming work, similar calculations will be performed with a more realistic scattering model including the energy dependence of the power in the power scattering cross section. Also, other types of two-point distributions can be calculated, such as angular correlations in sputtering, vacancy-interstitials correlations etc.

VI. REFERENCES

[l] I. P k s i t and R. Chakarova, Transport Theory and Statistical Physics 27 (1997) 1. (Due to misprint of the figure set, the authors should be contacted to get the correct figures)

[2] P. Sigmund, Mat. Fys. Medd. Dan. Vid. Selsk. 36 No 14 (1978) [3] Von G. Leibfried, Nukleonik, Band 1, Heft 1 (1958) 57 [4] P. H. Dederichs, Phys. Stat. Sol. 10 (1965) 303 [5] J. Lindhard, V. Nielsen, M. Scharff and P. V. Thomsen, Mat. Fys. Medd. Dan. Vid.

[6] P. Sigmund, Bull. Am. Phys. SOC. 13 (1968) 464 [7] J. E. Westmoreland and P. Sigmund, Radiat. Eff. 6 (1970) 187 (81 K. B. Winterbon, Nucl. Instr, Meth. B18(1986)1 [9] P. Sigmund, Statistics of Charged-Particle Penetration, in NATO AS1 Series, Vol.

271: Interaction of Charged Particles with Solids and Surfaces, Editors Albert0 Gras-Marti, H. M. Urbassek, N. R. Arista and F. Flores, Plenum Press, 1991.

[lo] P. Sigmund, Topics in Applied Physics, Vol. 47: Sputtering by Particle Bombard- ment. Physics and Applications, Editor R. Behrisch, Springer Verlag, 1981

[ll] I. PAzsit, J. Phys. D 20 151 - 161 (1987) (121 I. Prizsit and A. K. Prinja, Phys. Rev. B52 (1995) 16877. [13] W. Eckstein, Nucl. Instr. and Meth. B33 (1988) 489. [14] U. Conrad and H. Urbassek, Nucl. Instr. and Meth. B48 (1990) 399. [15] G. H. Khinchin and R. S. Pease, Rep. Prog. Phys. 18 (1955) 1 [16] I. Prizsit, Ann. nucl. Energy, 14 (1987) 25. (171 I. Pizsit and M. M. R. Williams, J. Phys. D: Appl. Phys. 13 (1980) 351 [18] M. T. Robinson and I. M. Torrens, Phys. Rev. B9 (1974) 5008. [I91 J. P. Biersack and L. G. Haggmark, Nucl. Instr. and Meth., 174 (1980) 257. [20] J. P. Biersack and W. Eckstein, Appl. Phys. A34 (1984) 73. [21] J. P. Biersack, Nucl. Instr. and Meth. B27 (1987) 21. [22] M. Vicanek and H. M. Urbassek, Nucl. Instr. and Meth. B30 (1988) 507.

Selsk. 33 No 10 (1963)

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[33] U. Conrad and H. M. Urbassek, Nucl. Instr. and Meth. B61 (1991) 295 [24] H. Urbassek and U. Conrad, Nucl. Instr. and Meth. B69 (1992) 413. 1251 H. M. Urbassek, Low Energy Ion Penetration and Collision Cascades in Solids, in

NATO AS1 Series, Vol. 271: Interaction of Charged Particles with Solids and Surfaces, Editors Albert0 Gras-Marti, H. M. Urbassek, N. R. Arista and F. Flores, Plenum Press, 1991.

1261 Progress Report 1994/ 1995, PDC Center of Parallel Computers, Royal Institute of Technology, Stockholm, Publisher: Parallelldatorcentrum, 1996

[27] R. Chakarova and I. PBzsit, Fluctuations in atomic collision cascades - variance and correlations in sputtering and defect distribution, CTH-RF-123, 1997

Received: 1 October 1997 Revised: 20 January 1998 Accepted: 4 February 1998

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