INTRODUCTION

The sputtering of solids under ion bombardment is a complex physical phenomenon [1, 2]. When atoms collide inside a target, two particles are generated, each with its own energy and direction of motion, depending on the law of interatomic interaction. An explanation of the observed patterns of sputtering is possible mainly only with the help of computer simulation programs [3, 4].

For the theoretical treatment of cascade multiplication in [5, 6], an attempt was made to separate the analysis of the energy and angular distributions: first, to consider the energy distribution of a cascade of particles moving rectilinearly [5], and then to obtain the angular distribution as a correction to the rectilinear solution [6]. The results [5, 6] were obtained for a special type of scattering cross section corresponding to the interaction of electrons and photons. In this paper, the theory is generalized to the case of a power-law dependence of the scattering cross section on the transferred energy. A variation of the power parameter makes it possible to consider a wide class of interatomic interactions: from interaction according to the hard-sphere law to the Rutherford interaction characteristic of the Coulomb potential.

EQUATION OF CASCADE MULTIPLICATION

We consider the following problem. A primary ion of energy \({{E}_{0}}~\) enters the target. In the target, the ion undergoes elastic collisions with immovable atoms and knocks them out of their rest positions. The knocked-out atoms, in turn, suffer collisions with other target atoms, and so on, i.e., a cascade arises. The cascade multiplication of atoms continues until the moment when the energy of the atom decreases to a value of \({{E}_{{{\text{min}}}}},\) which is insufficient to break the atomic bond. Let us calculate the energy distribution and the number of cascade atoms depending on the target depth for various types of interatomic interaction. It is assumed that the masses of ions and atoms are equal, which corresponds to the case of so-called self-sputtering.

We will measure the energy of atoms \(~E\) in relative units \(u = {E \mathord{\left/ {\vphantom {E {{{E}_{0}}}}} \right. \kern-0em} {{{E}_{0}}}},\) \({{u}_{{{\text{min}}}}} = {{{{E}_{{{\text{min}}}}}} \mathord{\left/ {\vphantom {{{{E}_{{{\text{min}}}}}} {{{E}_{{0~}}}}}} \right. \kern-0em} {{{E}_{{0~}}}}}\) and denote by the function \(f\left( {x,u} \right)\) the number of atoms with the energy \(u\) at the normalized depth x. The transport equation for the distribution function \(f\left( {x,u} \right)\) has the form [7–11]:

$$\begin{gathered} \frac{{~\partial f\left( {x,u} \right)~}}{{~\partial x~}} + f\left( {x,u} \right) = \int\limits_u^1 {f\left( {x,u{\kern 1pt} '} \right)} \\ \times \,\,\left[ {K\left( {u{\kern 1pt} ',u{\kern 1pt} '\, - u} \right) + K\left( {u{\kern 1pt} ',u} \right)} \right]~du{\kern 1pt} '. \\ \end{gathered} $$
(1)

The first term on the left-hand side of equation (1) describes the variation of the number of atoms between collisions. The two terms on the right-hand side of Eq. (1) reflect the fact that as a result of the collision of an atom with the energy \(u{\kern 1pt} '\) with a stationary atom, two atoms are generated, the energy values of which are \(u\) and \(u{\kern 1pt} ' - u.\) The boundary condition to Eq. (1) has a delta-type form \(f\left( {0,u} \right) = \delta \left( {1 - u} \right)\) and indicates that the energy of the ion initiating the cascade is \({{E}_{0}}.\)

As the scattering cross section for the collision of two atoms, we choose the cross section that depends according to the power law on the transferred energy T :

$$K\left( {E,T} \right)dT = n{{E}^{{ - n}}}{{T}^{{n{\kern 1pt} - {\kern 1pt} 1}}}dT,$$
(2)

where E is the energy of the bombarding atom, and \(n > 0\) is the power parameter. In the cascade theory of electron showers [5], the authors used a combination of three scattering cross sections (2) with three different values of the power parameter: \(n = 1,2,3.\) In atomic collisions, the power parameter varies within \(0 < n \leqslant 1,\) which differs significantly from the case of electron showers. For small values of the ion energy, we have \(n = 1\) and the interaction of particles according to the hard-sphere law. At large values of the ion energy, the power-law parameter is small (\(n \ll 1\)), which corresponds to Rutherford scattering.

GENERAL SOLUTION

The substitution of (2) into Eq. (1) gives

$$\begin{gathered} \frac{{~\partial f\left( {x,u} \right)~}}{{~\partial x~}} + f\left( {x,u} \right) \\ = n\int\limits_u^1 {f\left( {x,u{\kern 1pt} '} \right)} \frac{{~{{u}^{{n\,\, - \,\,1}}} + {{{\left( {u{\kern 1pt} '\, - u} \right)}}^{{n - 1}}}~}}{{{{u}^{{'n}}}}}~du{\kern 1pt} '. \\ \end{gathered} $$
(3)

To investigate the equation instead of the variable \(u\) we introduce a new independent variable s using the Mellin transform:

$${{f}_{s}}\left( x \right) = \int\limits_0^1 {{{u}^{{s\,\, - \,\,1}}}f} \left( {x,u} \right)~du,\,\,\,\,\left( {1 \leqslant s < \infty } \right).$$
(4)

Function \({{f}_{s}}\left( x \right)\) is uniquely expressed through the function \(f\left( {x,u} \right)\) and vice versa. As a result of the transformations, we obtain

$$\frac{{~d{{f}_{s}}\left( x \right)~}}{{~dx~}} + {{f}_{s}}\left( x \right) = {{Q}_{s}}~{{f}_{s}}\left( x \right),$$
(5)
$$\begin{gathered} {{Q}_{s}} = n\int\limits_0^1 {{{\xi }^{{s\,\, - \,\,1}}}~\left[ {{{\xi }^{{n\,\, - \,\,1}}} + {{{\left( {1 - \xi } \right)}}^{{n\,\, - \,\,1}}}} \right]~d\xi } \\ = n\left[ {~\frac{1}{{~s + n - 1~}} + {\text{{ B}}}\left( {s,n} \right)} \right], \\ \end{gathered} $$
(6)

where \({\text{{ B}}}\;\) denotes the beta function [12]. The solution to equation (5) has the form

$${{f}_{s}}\left( x \right) = {\text{exp}}\left( {x{{Q}_{s}}} \right){\text{exp}}\left( { - x} \right).$$
(7)

Thus, the solution of the transport equation (3) is reduced to finding the inverse Mellin transform and calculating the integral in the complex plane [13, 14]:

$$f\left( {x,t} \right) = \frac{1}{{2\pi i}}\int\limits_{1\,\, - \,\,i\infty }^{1\,\, + \,\,i\infty } {{\text{exp}}\left( {st} \right){{f}_{s}}\left( x \right)~ds} ,\,\,\,\,t = {\text{ln}}\frac{1}{u}.$$
(8)

Function \(f\left( {x,t} \right)\) represents a differential characteristic, i.e., the energy distribution of cascade atoms at the depth x. Also of interest is the integral characteristic, i.e., the number of cascade atoms with energies in the range \({{u}_{{{\text{min}}}}} \leqslant u \leqslant 1\) at the depth x:

$$F\left( {x,{{t}_{{{\text{max}}}}}} \right) = \int\limits_0^{{{t}_{{{\text{max}}}}}} {f\left( {x,t} \right)~dt} ,\,\,\,\,{{t}_{{{\text{max}}}}} = {\text{ln}}\frac{1}{{{{u}_{{{\text{min}}}}}}}.$$
(9)

Another important integral characteristic is the total energy of the cascade at the depth x:

$$\bar {u}\left( x \right) = \int\limits_{{{u}_{{{\text{min}}}}}}^1 {uf\left( {x,u} \right)du} = \int\limits_0^{{{t}_{{{\text{max}}}}}} {{\text{exp}}\left( { - 2t} \right)f} \left( {x,t} \right)~dt.$$
(10)

SOLUTION FOR THE POTENTIAL OF POTENTIAL OF HARD SPHERES SPHERES

In the case of the interaction of particles according to the hard-sphere law, the solution can be presented in an analytical form. Substituting of \(~n = 1~\) into equation (6) gives \({{Q}_{s}} = {2 \mathord{\left/ {\vphantom {2 s}} \right. \kern-0em} s},\)

$${{f}_{s}}\left( x \right) = {\text{exp}}\left( {\frac{{~2x~}}{s}} \right){\text{exp}}\left( { - x} \right),$$
(11)

and from the tables [15] we find the distribution function:

$$f\left( {x,t} \right) = \left[ {\delta \left( t \right) + \sqrt {~\frac{{~2x~}}{t}~} ~{{I}_{1}}\left( {\sqrt {~8xt~} ~} \right)} \right]{\text{exp}}\left( { - x} \right).$$
(12)

Here, \({{I}_{1}}\) denotes the modified Bessel function of the first order [12], the first term in parentheses refers to the primary ion initiating the cascade, and the second term refers to the cascade itself.

From integral (9), we calculate the number of cascade atoms:

$$F\left( {x,{{t}_{{{\text{max}}}}}} \right) = {{I}_{0}}\left( {\sqrt {~8x{{t}_{{{\text{max}}}}}~} ~} \right){\text{exp}}\left( { - x} \right).$$
(13)

Differentiation of equation (13) with respect to \(x\) leads to the transcendental equation for determining the coordinate of the maximum number of cascade atoms \({{x}_{{{\text{max}}}}}{\text{:}}\)

$$\sqrt {\frac{{{{x}_{{{\text{max}}}}}}}{{~2{{t}_{{{\text{max}}}}}~}}} = \frac{{~{{I}_{1}}\left( {\sqrt {~8~{{x}_{{{\text{max}}}}}~{{t}_{{{\text{max}}}}}~} ~} \right)~}}{{{{I}_{0}}\left( {\sqrt {~8~{{x}_{{{\text{max}}~}}}{{t}_{{{\text{max}}}}}~} ~} \right)}}.$$
(14)

The asymptotic expansion of the Bessel functions in Eq. (14) shows that for large values of the ion energy, the curve \({{x}_{{{\text{max}}}}}\left( {{{t}_{{{\text{max}}}}}} \right)\) tends to a straight line:

$${{x}_{{{\text{max}}}}} = 2{{t}_{{{\text{max}}}}} - \frac{1}{{~2~}}~\,\,\,\,{\text{at}}\,\,\,\,{{t}_{{{\text{max}}}}} \gg 1.$$
(15)

RESULTS AND DISCUSSION

The distribution of the number of cascade atoms over the target depth is determined by function (9), the graph of which is shown in Fig. 1 for the power parameter \(n = 0.5\) and different ion energies. It is seen that the number of cascade particles at a given depth increases with increasing ion energy until the critical region is reached, after which a further increase in energy decreases the number of cascade particles. The curve has a maximum at a depth for which the derivative \(~{{dF} \mathord{\left/ {\vphantom {{dF} {d{{t}_{{{\text{max}}}}}}}} \right. \kern-0em} {d{{t}_{{{\text{max}}}}}}}\) vanishes.

Fig. 1.
figure 1

Dependence of the number of cascade atoms versus the target depth for the ion energy E0: 10Emin (1); 100Emin (2); 1000Emin (3). Power-parameter value \(n = 0.5.\)

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Figure 2 shows the dependence of the position of the maximum on the ion energy and the value of the power parameter. We can see that for the values \({{t}_{{{\text{max}}}}} \leqslant 0.5\) the distribution decreases monotonically and has no maximum. A maximum appears on the distribution at values of \({{t}_{{{\text{max}}}}} > 0.5,\) and its position shifts towards greater target depths with decreasing power parameter. This indicates that at high ion energies, particles undergo Rutherford scattering, which is characteristic of the Coulomb potential, and penetrate to a greater depth compared to low-energy ions scattered according to the hard-sphere law.

Fig. 2.
figure 2

Dependence of the coordinates of the maximum on the ion energy for the values of the power parameter n: 1 (1); 0.4 (2); 0.2 (3); 0.1 (4).

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Figure 3 shows the variation of the total energy of the cascade with the depth of the target, calculated by formula (10) for interaction according to the hard-sphere law. If the cutoff energy is zero, the total energy of the cascade at any depth would be equal to \({{E}_{0}},\) which corresponds to the straight line \(\bar {u} = 1\) in Fig. 3. The initial sections of all curves really coincide with the straight line \(~\bar {u} = 1.\) Taking the cutoff energy into account leads to the fact that at a certain depth the energy of atoms decreases so much that they leave the cascade, which leads to a decrease in the total energy. The lower the ion energy, the stronger the effect of the decrease.

Fig. 3.
figure 3

Dependence of the total cascade energy on the target depth for the ion energy E0: 10Emin (1); 100Emin (2); 1000Emin (3). Power-parameter value \(n = 1.\)

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The results obtained and the figures shown represent a generalization of previous theories of the cascade multiplication of particles for the case of various types of interatomic interaction. A change in the power parameter in the scattering cross section allows one to consider atomic potentials from the hard-sphere potential to the Coulomb potential. Analytical formulas (11)–(15), which can be used for testing computer modeling programs, as well as for educational purposes, are of individual importance.