The role of the Boltzmann transport equation in radiation damage calculations

The purpose of this article is to discuss in some detail the uses of the Boltzmann equation in assessing displacement damage due to fast atoms in solids. We derive the appropriate equations for describing atomic motion in solids directly from the non-linear Boltzmann equation for mixtures. The basic assumption is that binary collisions are valid and that the solid can be regarded as amorphous or as a random lattice of atoms. Moreover, we assume that the density of fast atoms (i.e. those with energy greater than about 25 eV) is so small that they do not interact with each other. With these assumptions a set of linear, coupled Boltzmann transport equations are derived which describe the projectile and recoil atom distributions in space, time and velocity. Additionally, we show how electronic interactions may be accounted for in an accurate and analytically useful manner.The Boltzmann equations thus derived are in the so-called forward form, that is the final co-ordinates are the ones operated upon rather than the initial ones. We consider the corresponding adjoint equations and from the mathematical properties of the operators prove a reciprocity relation between the solutions of the forward equations and the adjoint ones. This enables either equation, forward or backward (adjoint), to be used to calculate the distribution function, a fact which is extremely useful in simplifying calculations for certain types of problem. It also demonstrates the inter-relationships between the works of various groups.The essentials of the scattering cross-sections are discussed and it is seen that two distinct problems arise: scattering in the C.M. system, which is determined by the interparticle force law, and scattering in the L-system which is governed only by the laws of conservation of energy and momentum. In both cases we discuss analytical simplifications which lead to useful approximations for solving the transport equation in closed form.One of the basic problems of radiation damage theory is that of sputtering and we spend some time explaining how binding forces at a vacuum-solid interface affect the solution of the transport equation.Several infinite medium problems are solved using backward and forward equations, thus we calculate the collision density of recoiling atoms due to a high energy source and also the time dependent behaviour of the energy spectrum following a pulse of fast atoms. The energy deposition and total number of moving particles at time t is calculated for various scattering models. The importance of electronic stopping in calculating the amount of energy eventually ending up in atomic motion is assessed via a time dependent forward equation. Its connection with the work of Lindhard is pointed out.We discuss displacement damage and show how the number of displacements per primary particle may be calculated via the collision density approach or through the backward equation with suitably modified energy transfer terms. The ideas of energy partitioning are discussed and methods for assessing its accuracy outlined.The spatial distribution of radiation damage receives considerable effort and we look at the various methods of dealing with the anisotropy of scattering, from Legendre polynomial expansions to the straight-ahead, or path-length, approximation. Analytical solutions are given for a variety of simple problems and in particular the vexed question of ion implantation in heterogeneous layered structures is discussed and various solutions proposed. For infinite, homogeneous media we discuss the method of moments using forward and backward equations and conclude that in many situations the forward form has much to commend it although in some situations the backward equation is not without merit. They are indeed complementary techniques.Sigmund's method for calculating the sputtering yield and efficiency is described via the backward equation and the approximations introduced by his “infinite medium boundary condition” are examined. We propose a more accurate theory which includes the half space nature of the system but which, because of this, leads to difficult mathematical problems. Certain simplified problems are solved and ideas for future progress are put forward.The concept of channelling is discussed and methods for including it directly into the Boltzmann equation are outlined. Essentially, these suggest the introduction of an angularly dependent cross-section which accounts in a phenomenological manner for the anisotropy of the microstructure of the solid.A final section is devoted to a stochastic formulation of the particle distribution function through a probability density. An equation is obtained for this density by probability balance and is then simplified by the introduction of a generating function. Successive differentiations of the generating function enable equations for the average value to be obtained, i.e. the conventional Boltzmann equation, and also higher order correlation functions which give a measure of the fluctuations in the number of particles in a cascade. Equations for vacancies and interstitials and their correlations are discussed in this respect.Finally, it should be noted that this paper shows a very personal and possibly biased view of radiation damage calculation. Indeed, it is not intended to discuss radiation damage as such, but rather to outline the difficulties and particularly to show how the Boltzmann equation has beeb and can be used in its forward and backward forms to understand atomic displacement problems.