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.

Introduction

In many applications it is the rest distribution of the implanted (or primary) ions that is of principal importance, e.g. in the doping of semi-conductors (Sze, 1988). We examine this in detail because of its intrinsic importance and also because we can illustrate some modern statistical and numerical techniques applied to transport theory in a little more detail than described in the previous chapter. The penetration of ions into amorphous targets is described most simply by using a statistical transport model. The use of this model has the advantage that two methods exist for the prediction of the rest distribution of ions: the solution of transport equations (TEs) and Monte Carlo (MC) simulation. A statistical model is essential to the construction of TEs and the computational efficiency that it affords MC simulation is necessary in order to obtain good statistics.

In several ways the MC and TE methods are complementary. In direct form the MC method treats an explicit sequence of collisions, so the target composition can change on arbitrary boundaries (in space and time). The rest distribution is built up from a large number of ion trajectories, the statistical precision of which depends directly on this number. Hence, the use of the MC method is dependent on the necessary statistical precision being obtained in a ‘sensible’ amount of CPU time. On the other hand, Lindhard-type TEs assume a target that satisfies space (and time) translational invariance. The only target to satisfy this condition is infinite and homogeneous.

Introduction

Surface topographic changes occur as a result of all particle–solid interactions. For large doses > 1017 particles/cm2 of energetic (tens of kilo-electronvolts) particles, these changes are often visible with the naked eye. Many mechanisms give rise to these topographic features, for example bombardment-induced defects in a solid or differential sputtering yields across a surface, due to grain boundaries or impurity inclusions. Electron micrographs of such surfaces reveal a wide variety of features such as etch pits, ridges, facets, ripples, cones and pyramids. The article by Carter, Navinšek and Whitton (1983) illustrates some of the features that have been observed, but many examples can be found also in most recent issues of the Journal of Vacuum Science and Technology. Features that develop can be an unwanted artefact of the ion bombardment technique, for example, they can be responsible for considerable uncertainty in the depth resolution of surface analytical techniques such as dynamic SIMS (secondary ion mass spectroscopy).

On the other hand, surface engineering seeks to etch well-defined patterns on surfaces. Ion beam lithography uses focused beams to etch patterns directly into a substrate but for most technological applications broad beams with masks are used. The modelling of the development of surface shape as a result of particle bombardment is therefore important from the point of view of understanding the basic physical processes involved. In addition it also has important engineering applications.

Introduction. The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.