The use and understanding of matter in its condensed (liquid or solid) state have gone hand in hand with the advances of civilization and technology since the first use of primitive tools. So important has the control of condensed matter been to man that historical ages – the Stone Age, the Bronze Age, the Iron Age – have often been named after the material dominating the technology of the time. Serious scientific study of condensed matter began shortly after the Newtonian revolution. By the end of the nineteenth century, the foundations of our understanding of the macroscopic properties of matter were firmly in place. Thermodynamics, hydrodynamics and elasticity together provided an essentially complete description of the static and dynamic properties of gases, liquids and solids at length scales long compared to molecular lengths. These theories remain valid today. By the early and mid-twentieth century, new ideas, most notably quantum mechanics and new experimental probes, such as scattering and optical spectroscopy, had been introduced. These established the atomic nature of matter and opened the door for investigations and understanding of condensed matter at the microscopic level. The study of quantum properties of solids began in the 1920s and continues today in what we might term “conventional solid state physics”. This field includes accomplishments ranging from electronic band theory, which explains metals, insulators and semiconductors, to the theory of superconductivity and the quantum Hall effect.

In the preceding two chapters, we have discussed various types of order that can occur in nature and how the ordering process can be quantified by the introduction of order parameters. We also developed a formalism for dealing with the thermodynamics of ordered states. In this chapter, we will use mean-field theory to study phase transitions and the properties of various ordered phases. Mean-field theory is an approximation for the thermodynamic properties of a system based on treating the order parameter as spatially constant. It is a useful description if spatial fluctuations are not important. It becomes an exact theory only when the range of interactions becomes infinite. It, nevertheless, makes quantitatively correct predictions about some aspects of phase transitions (e.g. critical exponents) in high spatial dimensions where each particle or spin has many nearest neighbors, and it makes qualitatively correct predictions in physical dimensions. Mean-field theory has the enormous advantage of being mathematically simple, and it is almost invariably the first approach taken to predict phase diagrams and properties of new experimental systems.

Before proceeding, let us review some simple facts about phase transitions. At high temperatures, there is no order, and the order parameter 〈φ〉 is zero. At a critical temperature, Tc, order sets in so that, for temperatures below Tc, 〈φ〉 is nonzero. If 〈φ〉 rises continuously from zero, as shown in Fig. 4.0.1a, the transition is second order.

Condensed matter physics

Imagine that we knew all of the fundamental laws of nature, understood them completely, and could identify all of the elementary particles. Would we be able to explain all physical phenomena with this knowledge? We could do a good job of predicting how a single particle moves in an applied potential, and we could equally well predict the motion of two interacting particles (by separating center of mass and interparticle coordinates). But there are only a few problems involving three particles that we could solve exactly. The phenomena we commonly observe involve not two or three but of order 1027 particles (e.g., in a liter of water); there is little hope of finding an analytical solution for the motion of all of these particles. Moreover, it is not clear that such a solution, even if it existed, would be useful. We cannot possibly observe the motion of each of 1027 particles. We can, however, observe macroscopic variables, such as particle density, momentum density, or magnetization, and measure their fluctuations and response to external fields. It is these observables that characterize and distinguish the many different thermodynamically stable phases of matter: liquids flow, solids are rigid; some matter is transparent, other matter is colored; there are insulators, metals and semiconductors, and so on.

Condensed matter physics provides a framework for describing and determining what happens to large groups of particles when they interact via presumably well known forces.

Large collections of particles can condense into an almost limitless variety of equilibrium and nonequilibrium structures. These structures can be characterized by the average positions of the particles and by the interparticle spatial correlations. Periodic solids, with their regular arrangements of particles, are more ordered and have lower symmetry than fluids with their random arrangements of particles in thermal motion. There are a number of equilibrium thermodynamic phases that have higher symmetry than periodic solids but lower symmetry than fluids. Typically interacting particles at low density and/or high temperature form a gaseous phase characterized by minimal interparticle correlations. As temperature is lowered or density increased, a liquid with strong local correlations but with the same symmetries as a gas can form. Upon further cooling, various lower-symmetry phases may form. At the lowest temperatures, the equilibrium phase of most systems of particles is a highly ordered low-symmetry crystalline solid. Nonequilibrium structures such as aggregates can have unusual symmetries not found in equilibrium structures.

In this chapter, we will investigate some of the prevalent structures found in nature and develop a language to describe their order and symmetry. We will also study how these structures can be probed with current experimental methods. Though tools such as scanning force and tunneling microscopes can now provide direct images of charge and particle density, at least near surfaces, most information about bulk structure, especially at the angstrom scale, is obtained via scattering of neutrons, electrons, or photons.

Introduction

It is easily seen from either the Sayre equation (3.52) or the tangent formula (3.60) that direct methods are likely to be much more powerful in phase extension and refinement than in ab initio phasing, since nothing can be known about the left-hand side of either the Sayre equation or the tangent formula without first putting into the right-hand side at least a small number of starting phases. One of the reasons why multi-solution procedures are so successful in practice is that they provide the possibility of having some trial sets with an initial pattern of phases able to converge to the correct point in the multi-dimensional phase space. On the other hand, if the phases of a sufficiently large number of reflections can be estimated in advance then direct methods will work even more efficiently. This gives the possibility of combining direct methods with other methods to tackle the phase problem in a number of special cases.

Fragment development

In the analysis of complex crystal structures, it is often the case that a fragment instead of the complete structure is first obtained. Hence fragment development plays an important role in crystal structure determination.

Recycling methods

Fourier synthesis with partial-structure phases has been a very efficient approach to obtaining the complete structure, especially when this is associated with weighting functions (Woolfson, 1956; Sim, 1960). A reciprocal-space alternative is a phase extension and refinement procedure based on partial structure information.

Multiple-beam scattering

A description of multiple-beam scattering

In fig. 1.12 there is shown a graphical means of determining the condition under which a particular diffracted beam will be produced. If the reciprocal lattice point at position s is the only one that touches the sphere of reflection then the associated diffracted beam will be the only one to occur – except of course for the straight-through diffracted beam, corresponding to the point O, which always occurs. Such a situation is described as two-beam diffraction. It is also possible to have more than two reciprocal lattice points on the surface of the reflecting sphere and in fig. 8.1 the points O, P and Q lie on the surface corresponding to the reciprocal-lattice vectors 0, h and k. This would be a case of three-beam diffraction. Four- and more beam diffraction is also possible but here we shall be restricting our attention to the three-beam case.

The line CQ in fig. 8.1 gives the direction of the k diffracted beam. Let us suppose that a beam of radiation was incident on the crystal from that direction. The crystal would still be in the same orientation, as would be the reciprocal lattice which is rigidly attached to the crystal, and the sphere of reflection would be displaced to put the new origin point, O, on the old point Q. It is now clear that after the displacement the reciprocal lattice point h – k will now fall at the same point as P and hence that this diffracted beam would occur.

When von Laue and his assistants produced their first smudgy X-ray diffraction photographs in Munich in 1912 they could not have known of the developments that would follow and the impact that these would have on such a wide range of science. Structural crystallography, the ability to find the arrangement of atoms inside crystals, has advanced over the years both theoretically and experimentally. Technical advances, such as the development of computers both for control of instruments and for complex calculations, and also the advent of high power synchrotron X-ray sources have all played their part.

In this book we bring together all the methods that have been and are being used to solve crystal structures. We broadly divide these methods into two main classes, non-physical and physical methods. In the first category we place those methods that depend on a single set of diffraction data produced by the normal Thomson scattering from the individual atoms. The Patterson methods and direct methods described in chapters 2 and 3 respectively are non-physical methods. In chapter 4 the basic principles are explained for two physical methods – isomorphous replacement, which combines the data from two or more related compounds to obtain phase information, and anomalous scattering, which uses data at wavelengths for which some of the atoms scatter anomalously, i.e. with an amplitude and phase differing from that given by the Thomson process. In chapter 5 the method of isomorphous replacement is explored in much greater depth and in chapter 6 the same is done for anomalous scattering.

Forming an image

The process of forming an optical image is one that is very well understood and frequently occurs. In the very act of seeing what is on this page the reader is forming a retinal image of its contents which is then conveyed to the brain in the form of electrical impulses for the complex task of interpretation and comprehension. What happens in the visual cortex is poorly understood but the formation of the retinal image via the lens of the eye is straightforward and can be followed by reference to fig. 1.1. The first stage in image formation is to direct towards the object some radiation (light in this case), part of which is scattered so that each point of the object becomes a secondary source of radiation which leaves in all directions. If we look in detail at what happens at a point of the object (fig. 1.1 (a)) we see that the radiation going off in different directions is not only coherent, because it derives from the same point source, but is also all in phase. Next the scattered radiation strikes a lens (fig. 1.1(b)). Because the speed of light in the lens material is different from that in air, rays travelling by different paths to the image point have the same optical path length and so undergo constructive interference there. The amplitude, and hence intensity, of each image point is proportional to that of the corresponding object point and consequently a true image is formed.

Introduction

The crystal structure analysis of a protein would be a routine procedure provided that two or more heavy-atom derivatives with good isomorphism to the native protein were available (chapter 4). Unfortunately, in practice this is often not the case. Usually there will be little difficulty in preparing one heavy-atom derivative that is isomorphous with the native protein. However, finding a second isomorphous derivative may not be straight forward so that the use of single isomorphous replacement (SIR) data is preferable if there is some way to resolve the intrinsic ambiguity of the method in the non-centrosymmetric case (§4.1.3). The double-phase method (§4.1.6) is one way of doing this but it usually gives a rather noisy map. There is now described a noise-filtering technique which can be used to develop better information in such a situation.

Resolving the SIR phase ambiguity in real space: Wang's solvent-flattening method

Protein structures are characterized by having large contiguous solvent regions surrounding other regions of somewhat higher average density within which the protein exists. The contrast between the ordered structure (protein, sometimes with some solvent molecules) and background (disordered solvent) is much less than for small-molecule structures and this is one of the reasons, additional to other factors including their size and complexity, which make protein structures difficult to solve.

The first critical step in Wang's method (Wang, 1981, 1985) is to define the molecular boundary from a noisy electron density map. Following that, the densities inside the protein envelope are raised by a constant value and then densities lower than a certain value are removed. Outside the protein region, the density is smoothed to a constant level.

The main focus of our previous discussion of MBE has been the identification of various universality classes. The models we discussed are expected to be valid on a coarse-grained level, at which the exact structure and form of an island does not matter. However, with the perfection of experimental tools, it is possible to observe the interface morphology at the atomic scale – leading to the discovery of rich island morphologies. In this chapter we focus on this early-time morphology, for which the coverage is less than one monolayer; this regime is usually referred to as submonolayer epitaxy.

The phenomenology is quite simple. Start with a flat interface, and deposit atoms with a constant flux. The deposited atoms diffuse on the surface until they meet another atom or the edge of an island, whereupon they stick. Thus if at a given moment we would photograph the surface, we would observe a number of clusters – called islands – with monomers diffusing between them. What is the typical size and number of the islands? What is their morphology? How do these quantities change with the coverage and with the flux? These are among the questions we address.

Model

Let us consider in more detail the deposition process outlined above. Consider a perfectly flat crystal surface with no atoms on it. At time zero we begin to deposit atoms with a constant flux F. Atoms arrive on the surface and diffuse (the deposition and diffusion processes take place simultaneously).