In 1912 von Laue proposed that X-rays could be diffracted by crystals and shortly afterwards the experiment which confirmed this brilliant prediction was carried out. At that time the full consequences of this discovery could not have been fully appreciated. From the solution of simple crystal structures, described in terms of two or three parameters, there has been steady progress to the point where now several complex biological structures have been solved and the solution of the structures of some crystalline viruses is a distinct possibility.

X-ray crystallography is sometimes regarded as a science in its own right and, indeed, there are many professional crystallographers who devote all their efforts to the development and practice of the subject. On the other hand, to many other scientists it is only a tool and, as such, it is a meeting point of many disciplines – mathematics, physics, chemistry, biology, medicine, geology, metallurgy, fibre technology and several others. However, for the crystallographer, the conventional boundaries between scientific subjects often seem rather nebulous.

In writing this book the aim has been to provide an elementary text which will serve either the undergraduate student or the postgraduate student beginning seriously to study the subject for the first time. There has been no attempt to compete in depth with specialized textbooks, some of which are listed in the Bibliography. Indeed, it has also been found desirable to restrict the breadth of treatment, and closely associated topics which fall outside the scope of the title – for example diffraction from semi and non-crystalline materials, electron and neutron diffraction – have been excluded.

Diffraction from a rotating crystal

It has been seen that methods of recording X-ray intensities usually involve a crystal rotating in the incident X-ray beam. We shall now look at the problem of determining the total energy in a particular diffracted beam produced during one pass of the crystal through a diffracting position. In order to do this we must make some assumptions about the geometry of the diffraction process; the configuration we shall take is that the crystal is rotating about some axis with a constant angular velocity ω and that the incident and diffracted beams are both perpendicular to the axis of rotation.

Let us first look at the situation when we have a stationary crystal in a diffracting position. Associated with the crystal, and fixed relative to it, there is a reciprocal space within which is defined the Fourier transform, Fx(s), of the electron density of the crystal. For a theoretically perfect crystal of infinite extent the value of Fx(s) would be zero everywhere except at the nodes of a δ-function reciprocal lattice, the weight associated with the point (hkl) being (l/V)Fhkl. However, if the crystal is imperfect in some way there may be non-zero Fx(s) well away from the reciprocal-lattice points and for a finite crystal there will be a small region of appreciable Fx(s) around each of the reciprocal-lattice points. The imperfect-crystal case we shall not consider here but we shall be concerned with the size of the crystal, for this is a factor which must be present in every diffraction experiment.

Consider a crystal completely bathed in an incident beam of intensity Io.

Trial-and-error methods

The object of a crystal-structure determination is to locate the atomic positions within the unit cell and thus completely to define the whole structure. Sometimes there are special features in the diffraction pattern, the space group or the suspected chemical configuration of the material under investigation which enable a guess to be made of the crystal structure or at least restrict it to a small number of possibilities. In the early days of the subject, when methods of structure determination were poorly developed, only the simpler types of structure could be tackled and trial-and-error methods based on such special features were commonly used. That is not to say that such techniques are now outmoded – no crystallographer would ignore the information from special features if it was available, but he does not rely on such information as much as hitherto.

One type of situation which is of great importance and is always sought by the crystallographer is when space-group considerations lead to the fixing or restricting of the positions of atoms or whole groups of atoms. If a centrosymmetric unit cell has only one atom of a particular species (or an odd number) then that atom (or one of them) must be at a centre of symmetry. In a case with an odd number of atoms in a cell with a diad axis one of the atoms would have to lie on the diad axis. Similarly, in some situations, an SO4 group may have to be symmetrically arranged on a triad axis as shown in fig. 8.1.

Since the first edition of this book was published in 1970 there have been tremendous advances in X-ray crystallography. Much of this has been due to technological developments – for example new and powerful synchrotron sources of X-rays, improved detectors and increase in the power of computers by many orders of magnitude. Alongside these developments, and sometimes prompted by them, there have also been theoretical advances, in particular in methods of solution of crystal structures. In this second edition these new aspects of the subject have been included and described at a level which is appropriate to the nature of the book, which is still an introductory text.

A new feature of this edition is that advantage has been taken of the ready availability of powerful table-top computers to illustrate the procedures of X-ray crystallography with FORTRAN® computer programs. These are listed in the appendices and available on the World Wide Web*. While they are restricted to two-dimensional applications they apply to all the two-dimensional space groups and fully illustrate the principles of the more complicated three-dimensional programs that are available. The Problems at the end of each chapter include some in which the reader can use these programs and go through simulations of structure solutions – simulations in that the known structure is used to generate what is equivalent to observed data. More realistic exercises can be produced if readers will work in pairs, one providing the other with a data file containing simulated observed data for a synthetic structure of his own invention, while the other has to find the solution.

The determination of unit-cell parameters

When a crystal structure is solved and refined the solution appears as a set of fractional coordinates from which can be determined bond lengths and angles, van der Waals distances, etc. However the accuracy with which these quantities can be determined will depend not only on the accuracy of the atomic coordinates but also on the accuracy of determination of the unit-cell parameters.

By the measurement of layer-line spacings or from Weissenberg photographs one can usually measure cell edges to about 1% and angles with an error of about ½°. The order of accuracy of cell dimension required to match that of coordinate determination is about one part in a thousand or perhaps a little better. This would correspond to less than 0.002 Å in a bond of length 1.500 Å and rarely is this order of accuracy really required.

For some other purposes more accurate unit-cell parameters may be required – for example for measurement of thermal expansion coefficients of crystalline materials or for investigating small changes in cell parameters with changes of composition of the material.

There has been a great deal of work in this field and it would be difficult to mention it all. What will be done is to select an example of each of the main types of method to illustrate the ranges of techniques and accuracy which are available.

The basic idea behind all the methods is to measure the Bragg angle for a number of reflections. This is related to the reciprocal-lattice constants as follows.

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.

The FORTRAN® listings given in these appendices relate to programs described and illustrated in the text and used for the solutions to examples. They are heavily interrelated, in that the output files from some of them become the input files for others. Readers are advised to examine the listings before use as they are well provided with COMMENT, C, statements which describe the workings of the programs. In addition, when running the programs users are guided by screen output and these should be carefully followed. In particular, it is important that data-file names should be correctly given and in all programs it is possible to designate the names of the input files if the default values are invalid.

A general description of the scattering process

To a greater or lesser extent scattering occurs whenever electromagnetic radiation interacts with matter. Perhaps the best-known example is Rayleigh scattering the results of which are a matter of common everyday observation. The blue of the sky and the haloes which are seen to surround distant lights on a foggy evening are due to the Rayleigh scattering of visible light by molecules of gas or particles of dust in the atmosphere.

The type of scattering we are going to consider can be thought of as due to the absorption of incident radiation with subsequent re-emission. The absorbed incident radiation may be in the form of a parallel beam but the scattered radiation is re-emitted in all directions. The spatial distribution of energy in the scattered beam depends on the type of scattering process which is taking place but there are many general features common to all types of scattering.

In fig. 2.1 the point O represents a scattering centre. The incident radiation is in the form of a parallel monochromatic beam and this is represented in the figure by the bundle of parallel rays. The intensity at a point within a beam of radiation is defined as the energy per unit time passing through unit cross-section perpendicular to the direction of propagation of the radiation. Thus for parallel incident radiation the intensity may be described as the power per unit cross-section of the beam.

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.