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).

In Chapters 12–15, we discussed in detail the properties of growth processes dominated by deposition and surface diffusion. We saw that one origin of randomness is the stochastic nature of the deposition flux, which generates a nonconservative noise in the growth equations. A second component of randomness on a crystal surface comes from the activated character of the diffusion process. As we show in this chapter, this type of randomness generates a conservative component to the noise, leading to different exponents and universality classes.

While diffusive noise is certainly expected to be present in MBE, there is no experimental evidence for the universality class generated by it. For this reason, we separate it from the discussion of the simple MBE models. This chapter can be considered to be a theoretical undertaking of interest in its own right, investigating the effect of the conservation laws on the universality class.

Conservative noise

An important contribution to randomness on a crystal surface arises from the activated character of the diffusion process. Equation (13.5) results from a deterministic current that contributes to interface smoothing – i.e., only particle motion that aids smoothing is included. But, as discussed in Chapter 12, particle diffusion is an activated process in which all possible moves – each with its own probability – are allowed. Because the nature of the diffusion process is probabilistic, an inherent randomness is always present.

The notion of universality suggests that the exponents determined in the previous chapters are unique, since they belong to the only possible growth equation with a set of given symmetries. Any model or experiment will show these exponents if the hydrodynamic limit has been reached. But there is one term in all these equations that we have largely ignored thus far, one whose form is not fixed by symmetry principles: the noise η. We have assumed that this noise is uncorrelated – i.e., that it has a Gaussian distribution. But is this the only type of noise possible in a physical system?

What if the magnitude of the noise in a given point is not independent of the magnitude of the noise in a different point – i.e., what if there are spatial correlations in the noise? What if the magnitude of the noise at a given time is not independent of the noise at a different time – i.e., what if there are temporal correlations in the noise? In many experimental situations, we know little about the nature of any noise that may be present, so to consider it Gaussian is perhaps an unrealistically simple assumption.

One possibility is that the correlation length of the noise is finite, i.e., that the different events ‘know’ about each other only if they are within a finite spatial separation ξ or temporal separation τ.

In previous chapters, we focused on interfaces that grow and roughen due to thermal fluctuations, the origin of the randomness arising from the random nature of the deposition process. For a class of interface phenomena, however, we do not have deposition, but rather we have an interface that moves in a disordered medium. The experiment described in Chapter 1 in which a fluid interface propagates through a paper towel is one example. The velocity of the interface is affected by the inhomogeneities of the medium: the resistance of the medium against the flow is different from point to point; we call this quenched noise, because it does not change with time. Fluid pressure and capillary force drive the fluid, and disorder in the medium slows its propagation. If the disorder ‘wins’ the competition, the interface becomes ‘pinned.’ Conversely, if the driving forces win, the interface stays ‘depinned.’ This transition from a pinned to a moving interface – obtained by changing the driving force – is called the depinning transition.

In the following three chapters, we discuss how quenched disorder leads to interface pinning – and depinning. We will show that the same theoretical ideas describe interface motion in a random field Ising model, which is relevant to the problem of domain growth in a disordered magnetic material.

The problem of a moving interface in the presence of quenched noise is a new type of critical phenomena, arising from ‘quenched randomness.’

The primary focus of this book is on interface roughening generated by various nonequilibrium deposition processes. However, crystal surfaces may be rough even under equilibrium conditions – with no atom deposition. Consider, e.g., a flat surface in equilibrium at a very low temperature. Thermal fluctuations do not have an observable effect on the shape of the crystal, and all atoms remain in their appropriate lattice positions. As temperature increases, the probability that an atom will break its bonds with its neighbors increases. Some atoms hop onto neighboring sites, thereby generating roughness on the atomic scale. At first glance, one might expect a gradual transition to a rough morphology, since the higher the temperature, the more the atoms wander on the surface – until the surface melts. Indeed this is a correct description of the short-ranged correlations between neighboring atoms. However, as we shall see, on much longer length scales there is a distinct (higher order) thermodynamic phase transition that takes place at a critical temperature TR. For T < TR the crystal is smooth, corresponding to a flat facet, while for T > TR it is rough, implying a rounded crystal shape analogous to a liquid drop. This ‘roughening transition’ can be successfully described using ideas of statistical mechanics, the formalism being analogous to that developed in previous chapters.

Equilibrium fluctuations

The lowest energy state of the crystal corresponds to a flat surface.

In the previous chapter, we introduced a simple growth model that exhibits generic scaling behavior. In particular, the interface width w increases as a power of time [Eq. (2.4)], and the saturated roughness displays a power law dependence on the system size [Eq. (2.5)]. There exists a natural language for describing and interpreting such scaling behavior, and this is the language of fractals. In this chapter, we introduce the concepts of fractal geometry, which provide a language in terms of which to better understand the meaning of power laws. Isotropic fractals are self-similar: they are invariant under isotropic scale transformations. In contrast, surfaces are generally invariant under anisotropic transformations, and belong to the broader class of self-affine fractals. We will therefore also discuss the basic properties of self-affine fractals, as well as numerical methods for calculating the critical exponents α, β, and z.

Self-similarity

An object can be self-similar if it is formed by parts that are ‘similar’ to the whole. One of the simplest self-similar objects is the Cantor set, whose iterative construction at successive ‘generations’ is shown in Fig. 3.1. If we enlarge the box of generation 3 by a factor of three, we obtain a set of intervals that is identical to the generation 2 object. In general, at generation k we can enlarge part of the object by a factor of three and obtain the object of generation (k – 1).

Most of our life takes place on the surface of something. Sitting on a rock means contact with its surface. We all walk on the surface of the Earth and most of us don't care that the center of the Earth is molten. Even when we care about the interior, we cannot reach it without first crossing a surface. For a biological cell, the surface membrane acts not only as a highly selective barrier, but many important processes take place directly on the surface itself.

We become accustomed to the shapes of the interfaces we encounter, so it can be surprising that their morphologies can appear to be quite different depending on the scale with which we observe them. For example, an astronaut in space sees Earth as a smooth ball. However Earth appears to be anything but smooth when climbing a mountain, as we encounter a seemingly endless hierarchy of ups and downs along our way.

We can already draw one conclusion: surfaces can be smooth, such as the Himalayas viewed from space, but the same surface can also be rough, such as the same mountains viewed from earth. In general the morphology depends on the length scale of observation!

How can we describe the morphology of something that is smooth to the eye, but rough under a microscope? This is one question we shall try to answer in this book. To this end, we will develop methods to characterize quantitatively the morphology of an arbitrary interface.