In addition to the fine-structure superplasticity (FSS) described in the previous chapters, there is another type of superplasticity known as internal-stress superplasticity (ISS). In these materials, in which internal stresses can be developed, considerable tensile plasticity can take place under the application of a low, externally applied stress. This is because internal-stress superplastic materials can have a strain-rate-sensitivity exponent as high as unity; i.e., they can exhibit ideal Newtonian viscous behavior. Such superplastic materials are believed to be deformed by a slip-creep mechanism.

There are many ways in which internal stresses can be generated. These include thermal cycling of composite materials, such as whisker- and particulate-reinforced composites, in which the constituents have different thermal expansion coefficients; thermal cycling of polycrystalline pure metals or single-phase alloys that have anisotropic thermal expansion coefficients; and thermal cycling through a phase change. In addition, pressureinduced phase changes have been cited as a possible source of superplastic flow in geological materials. For example, there is a phase transformation in the earth's upper mantle, because of pressure, from orthorhombic olivine to a spinel phase at a depth of about 400 km below the earth's surface. And it is believed that internal-stress superplasticity, arising from the transformation stresses through pressure cycling (analogous to temperature cycling), leads to a mixed-phase region of low effective viscosity.

Ordered intermetallic alloys and their composites generally have good high-temperature strength, low density, and environmental resistance and are, therefore, potential materials for high-temperature structures. However, ordered intermetallic alloys are also known to be brittle, have low toughness because of their ordered structure, and show a propensity for grain-boundary embrittlement. As a result, intermetallic alloys often either have poor fabricability and machinability or require a fabrication process that is complicated and tedious. The generic brittleness problem in intermetallics, particularly aluminides, has been studied extensively in recent years and some breakthroughs have been made. For example, polycrystalline Ni3Al that has an Ll2 structure exhibits almost no ductility, but Ni3Al containing a small amount (0.2 wt%) of boron exhibits room-temperature tensile ductility of up to 40%. Because of these technological breakthroughs, there is great interest in using these materials for engineering structures.

Superplasticity in intermetallics has only been recently demonstrated. Although large tensile elongations (∼100%) for an intermetallic (also known as Sendust, Fe–9.6 wt% Si–5.4 wt% Al) were indicated as early as 1981, true superplastic intermetallics were not observed until 1987. At present, several intermetallics of the Ll2 structure (e.g., Ni3Al and Ni3Si), iron aluminide, titanium aluminide (TiAl), and trititanium aluminides (Ti3Al) have demonstrated superplasticity. These intermetallics are being investigated for their structural applications.

Tests for the lack of a centre of symmetry

As we have seen in § 1.4 measurements of crystals with an optical goniometer can, in favourable circumstances, reveal the crystal class. Such measurements should be carried out on many crystals for there is a tendency for crystals to exist in a number of slightly different forms in each of which some facets may not be present. By-and-large, when incorrect conclusions are drawn from optical goniometry it is in the direction of assigning too high a symmetry to the crystal. However it is sometimes possible to be reasonably sure by means of such measurements that a crystal structure either has or does not have a centre of symmetry.

There are a number of physical properties of crystals, the measurements of which can be used unequivocally to detect the lack of a centre of symmetry in a crystal structure. We shall briefly consider these, the physical principles involved and the apparatus which may be used for the tests.

Piezoelectric effect

The first physical phenomenon we shall consider is that of piezoelectricity. This is the process whereby when a material is placed in an electric field it undergoes a mechanical strain and, conversely, when the material is mechanically strained it becomes electrically polarized and produces a field in its environment. Let us see what mechanism can produce this effect. In fig. 7.1(a) there is a schematic representation of a pair of atoms with their surrounding electron density. The two atoms are bonded together and the electron density is distorted from the configuration it would have for the superposition of that from two isolated atoms.

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.

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.