This introductory chapter begins with a review of uncorrelated classical probabilities and then extends these concepts to correlated quantum systems. This is done both to establish notation and to provide a basis for those who are not experts to understand material in later chapters.

Probability of a transition

Seldom does one know with certainty what is going to happen on the atomic scale. What can be determined is the probability P that a particular outcome (i.e. atomic transition) will result when many atoms interact with photons, electrons or protons. A transition occurs in an atom when one or more electrons jump from their initial state to a different final state in the atom. The outcome of such an atomic transition is specified by the final state of the atom after the interaction occurs. Since there are usually many atoms in most systems of practical interest, we can usually determine with statistical reliability the rate at which various outcomes (or final states) occur. Thus, although one is unable to predict what will happen to any one atom, one may determine what happens to a large number of atoms.

1 Single particle probability

A simple basic analogy is tossing a coin or dice. Tossing of a coin is analogous to interacting with an atom. In the case of a simple coin there are two outcomes: after the toss one side of the coin (‘heads’) will either occur or it will not occur.

There are, in general, many possible transitions of electrons in atoms. In some processes of practical interest more than one electron may undergo a transition. Such multiple electron transitions are the topic of the next and subsequent chapters. In this chapter the simpler topic of single electron transitions is considered, where the activity of a single electron in an atom is the focus of attention. Even this relatively simple case may be impossible to fully understand if the electron of interest is influenced by other electrons in the atom. So in this chapter the interdependency of electrons in the system is ignored. That is, the electrons are treated independently. Typically, such an independent electron is regarded as beginning in an initial state characterized by some effective nuclear charge ZT and a set of quantum numbers n, l, m, s, ms from which all possible properties (e.g., energy, shape, magnetic properties, etc.) may be determined. Interaction with something else, (usually a particle of charge Z and velocity, v), may cause a transition to a different final state of the atom.

The simplest transition occurs in interaction of atomic hydrogen with a structureless projectile. There are various ways to evaluate the transition probability for such a system. Exact calculations usually require use of a computer. Approximate calculations may be done more easily. Calculations for many electron systems are often done approximately using single electron transition probabilities.

Correlation

There is a significant difference between complex and merely large. This difference is related to the the notion of correlation which defines the rules of interdependency in large systems. The relevant question is: how may one make complicated things from simple ones? Biological systems are complex because the atomic and molecular subsystems are correlated. From the point of view of atomic physics correlation in condensed matter, chemistry and biology is determined at least in part by electron correlation in chemical bonds and the complex interdependent structures of electronic densities. Understanding correlation in this broad sense is a major challenge common to most of science and much of technology. This is sometimes referred to as the many body problem. In a general sense correlation is a conceptual bridge from properties of individuals to properties of groups or families.

The concept of correlation arises in many different contexts. ‘Individual’ may mean an individual electron, an individual molecule or in principle an individual person, musical note or ingredient in a recipe. In this book individual refers to electron for the most part. In this case the interaction between individuals is well known, namely l/r12. However, that does not mean that electron correlation is well understood in general. Although much has been done to investigate correlation in various areas of physics, chemistry, statistics, biology and materials science, in many cases little is well understood except in the limit of weak correlation.

In previous chapters interactions with structureless point charge projectiles have been considered. There are many interactions, however, which involve at least two atomic centers with one or more electrons on each center. In such cases the projectile is not well localized and there is a need to integrate over the non localized electron cloud density of the projectile. Evaluation of cross sections and transition rates for such processes requires a method for dealing with at least four interacting bodies. If multiple electron transitions occur on any of the atomic centers, then some form of even higher order many body theory is required. In general such a many body description is difficult.

In this chapter the probability amplitude for a transition of a target electron caused by a charged projectile carrying an electron is formulated. This probability amplitude may be used for transitions of multiple target electrons if the correlation interaction between the target electrons is neglected. Unless the projectile is simply considered as an effective point projectile with a charge Zeff, the interaction between the target and the projectile electrons may not be ignored. Since this interaction is between electrons on two different atomic centers, the effects of this interaction have been referred to as two center correlation effects (Cf. section 6.2.4).

Introduction

In this chapter interactions of photons with atoms are considered. Here the emphasis is on systems interacting with weak electromagnetic fields so that a single atomic electron interacts with a single photon. Initially interactions with a single electron are considered. In this case the photon tends to probe in a comparatively delicate way the details of the atomic wavefunction (e.g. effects of static correlation in multi-electron atoms). Later two electron transitions are considered. Because these two electron transitions are often negligible in the absence of electron correlation the two electron transitions are usually a direct probe of the dynamics of electron correlation.

In previous chapters the impact parameter (or particle) picture has been used wherever possible in order to recover the product form for the transition probability in the limit of zero correlation. However, here the likelihood of interacting with more than a single photon is quite small since the electromagnetic field of a photon, even for strong laser fields, is almost always small compared with the electric field provided by the target nucleus. Consequently, this independent electron limit is not often useful. Also, photon wavepackets are usually much larger in size than an atom. Consequently the wave picture is used where the electric and magnetic fields of the photon are considered to be plane waves. Transformation to the particle picture may be done using the usual Fourier transform from the scattering amplitude to the probability amplitude (Cf. section 3.3.3).

The purpose of this book is to give an introduction to some of the non-experimental techniques available for studying the interaction of energetic particles with solid surfaces. By energetic we mean particles with energies from <1 eV up to the mega-electronvolt range. The word non-experimental is chosen carefully because much of the book focuses on computer simulation in addition to basic theory. Simulation is a relative scientific newcomer, which contains elements both of theory and of experiment within its borders. A simulation is not a theory but a numerical model of a system. If it is a good model one may explore the behaviour of the real system by changing the numerical value of its input parameters and noting the changed responses. Simulations enable one to determine which are the important factors in a physical system that control its behaviour without the need necessarily to perform complex and expensive experiments. Sometimes we can probe areas that no experiment can determine, for example, the displacement and mixing of identical atoms in an atomic collision cascade. Usually, in performing the computational experiments on a model, the important parameters should be identified and need to be fixed at the start of the calculations. Usually we perform a sensitivity analysis by varying one parameter at a time.

This book is intended to describe methods that will be applicable both to hard collisions between nuclear cores of atoms and to soft interactions in which chemical effects or long-range forces dominate.

Introduction

The energetic interaction of a particle beam with a solid cannot be described fully by the path of a single projectile. The path a particle takes and the paths of the subsequent recoils are dependent upon the initial impact point on the surface. Thus, to get a clear description of the effects of particle interaction with a solid, many such paths must be followed. A typical ion beam experiment would entail the interaction of 1011–1020 particles per cm2 of the target.

Trajectory simulations obtain an ensemble – or set – of independent particle solid impact histories. Each history is followed from a different starting point on the solid to simulate the arrival of many particles at random points on the surface.

Conceptually the molecular dynamics (MD) simulation method (see Chapter 8) is the simplest and most complete simulation method to model the behaviour of a solid undergoing energetic particle bombardment; in particular, for calculating the displacement of particles in the solid during a single particle impact. In principle, the development of the ensuing collision cascade is followed chronologically in time as the energy of the ions propagates through the target system. The complexity comes from the solution of the many-body equations of motion which must be performed at successive time steps.

The Boltzmann transport equation

Introduction

Atomic particles are both deflected and slowed down after scattering by a target atom. This process is fundamental to the study of the penetration of ions in solid targets. A typical ion–solid experiment would involve many ion trajectories comprising several scatterings. Computer models tackle the problem head-on by calculating entire collision cascades from a representative set of trajectories. These results can then be used to evaluate average values such as the mean penetration depth and the mean number of particles ejected within a certain angle or energy range. However, the computer models often contain details that are not accessible to experimental observation and vast amounts of computing time can often be expended in generating these average results.

Computational techniques are discussed in more detail elsewhere in this book. In this chapter a probabilistic description amenable to analytic methods is described.

The mathematical means to tackle problems such as those in ion–solid interactions were introduced in the last century, in the context of kinetic theory. This theory allows the determination of macroscopic properties of matter from a knowledge of the elementary atomic interactions. One of the most outstanding results of this theory is the Boltzmann transport equation and we will discuss in this chapter the derivation of the equation and how it may be used to solve a variety of problems concerning the penetration of ions in solids.

In this section the Boltzmann transport equation in the so-called forward form is derived.