Introduction

From the point of view of polymer physics, globular proteins are reminiscent of certain polysoaps, where the hydrophobic part of the chain clusters in a central core, while the hydrophilic residues tend to lie on the outer surface (see Stryer, 1968). This conformation provides both stability and solubility. The crucial difference with polysoaps lies of course in the presence of specific receptors on the protein. A schematic representation for such a receptor is shown in Fig. 2.1. The active site directly involves a number p of aminoacids. These are linked together by comparatively long loops of the peptidic chain. As has been emphasized by Monod (1969), it is of some interest to estimate the minimum size required for each of these loops, when the conformation of the active site itself is prescribed. This should lead in particular to one lower bound for the molecular mass of a globular protein carrying one active site.

One possible approach to this problem would be to use Monte Carlo methods on a computer. This, however, (a) is expensive and (b) gives very little insight. Here we shall restrict ourselves to a much more modest, but explicit, calculation, based on rough statistical arguments. We deliberately neglect all the effects related to the hydrophilic/hydrophobic affinities of the aminoacids, although these effects are certainly very important. In the present chapter, we consider first a single loop, inserted in a dense proteic medium.

Observation and analysis of lattice defects

Point defects

The field ion microscope, with its spatial resolution sufficient to resolve atoms in most high index planes of an emitter surface and the capability of reaching into the bulk of the sample by removing surface layers with field evaporation, can be used to observe and analyze intrinsic and extrinsic lattice defects in the sample. The atom-probe, on the other hand can analyze the chemistry of a solid with comparable spatial resolution, and has been actively used to study compositional variations across phase boundaries, in precipitates, and across a grain boundary in impurity segregations, etc. In fact, atom-probe and field ion microscopy is currently the only technique capable of imaging lattice defects and simultaneously analyzing the chemistry of these defects in a solid on an atomic scale. Some of the most actively pursued and important applications in field ion microscopy have been in materials science and physical metallurgy. At least half the papers published in field ion microscopy are concerned with these subjects. As comprehensive reviews of these studies by experts in this field already exist, only selected topics will be briefly discussed here.

The simplest lattice defects as far as FIM observations are concerned are point defects, such as vacancies, self-interstitials and substitutional as well as interstitial impurity atoms. Vacancies invariably show up as dark spots in the field ion images. Other point defects may appear as either bright image spots or ‘vacancies’ in the image. Thus these defects can be identified from field ion images of high index planes where all the atoms in a plane are fully resolved.

Atomic structure of solid surfaces

The field ion microscope, when operated at low temperatures, has a resolution of better than 3 Å if the tip radius is less than a few hundred Å, which is sufficient to resolve the atomic structures of most solid surfaces. For closely packed surfaces, atomic spacings are usually less than 2.8 Å. But even such surfaces can be resolved if the tip temperature is below 20 K, the tip radius is less than 150 Å and the plane size is also very small so as to contain no more than, say, a few tens of atoms, as can be seen from images shown in Fig. 2.32. For a material to be able to be imaged in the field ion microscope, however, the low temperature evaporation field of the material has to be comparable to or higher than the best image field of the image gas used for the field ion imaging. In field ion microscopy, to eliminate the problem of contaminating a field evaporated clean surface and of destroying the surface by a field induced chemical etching from chemically reactive gases, one almost always uses an inert gas for the field ion imaging. The ionization energies of inert gases are higher than most chemically reactive gases, thus requiring a higher field ionization field, or an image field. This image field can serve to protect the clean, field evaporated emitter surface from contamination and from a field induced chemical etching by chemically active residual gases inside the vacuum chamber.

Field ionization

Basic mechanisms and field ionization rate

Field ion images are formed by field ionization. Without this process there would simply be no field ion microscope. Field ionization is therefore the most important physical process in field ion microscopy. Theories of field ionization actually preceded field ion microscopy by over twenty years. In 1928, when quantum mechanics was still in its infancy, Oppenheimer found by detailed quantum mechanical calculations that under a field of about 2 V/Å the electron in a hydrogen atom could tunnel out of the atom into the vacuum. This is one of the earliest theories of quantum mechanical tunneling phenomena. In the same year Fowler & Nordheim used the quantum mechanical tunneling effect to explain field emission phenomena, and Gamow, Gurney and Condon also explained α-particle decay by tunneling of helium nucleus out of a heavy nucleus. These theories represented early triumphs of quantum mechanics outside of atomic systems. Tunneling from excited states of hydrogen atoms was treated by Lanczos to explain field induced quenching of spectral lines in Stark effect at a field near 106 V/cm, the highest attainable in gaseous discharge tubes. Gurney considered inverse tunneling in the much higher fields that were encountered in the cathodic neutralization of H+ ion in an electrolyte, and was already discussing how the atomic level would line up with the electronic levels, including the Fermi level, of the electrode at different distances. Although these studies were probably known to some of the early pioneers in field ion microscopy, there is no evidence that they directly influenced the early development of the field ion microscope.

The field ion microscope (FIM) was invented by Erwin W. Müller in 1951. By 1957 he was able to show atomically resolved images of surfaces of tungsten and other refractory metals. The FIM is very simple in design and it is hard to imagine that one can image atomic structures of solid surfaces routinely with this instrument. Of course, the simplicity is somewhat deceptive. While atomic images can be routinely obtained for refractory metals with the FIM, to widen its application to other materials has not been easy. It is no wonder that even now, after the FIM has been in existence and in active research for nearly forty years, I am still often asked by scientists outside the field, some of them are very prominent and are also very knowledgeable in science, whether or not FIM can now be used to image materials besides tungsten. The truth of the matter is that the field ion microscope has long been successfully used to study most metals and many alloys, and recently good field ion images of some semiconductors and even ceramic materials, such as high temperature superconductors and graphite, etc., have also been successfully obtained.

The atomic resolution aspect is no longer unique to field ion microscopy. Other microscopies have now also achieved atomic resolution. Some of these microscopes promise to have a great versatility and are currently very actively pursued. On the other hand, there are still some experiments unique to field ion microscopy. An example is the study of the behavior of single atoms and single atomic clusters on well characterized surfaces where quantitative data on various atomic processes can be derived routinely with the field ion microscope.

Early developments

The field ion microscope (FIM) is the first microscope to have achieved atomic resolution. It is a surprisingly simple instrument (though the simplicity is somewhat deceptive), consisting of a sample in the form of a tip and a phosphorus screen some 10 cm away. To obtain an atomic image, one has only to introduce about 10−4 Torr of an image gas such as He into the system, cool down the sample tip below liquid nitrogen temperature and apply a positive voltage to the tip and raise the tip voltage gradually. When the field at the tip surface reaches about 4 V/Å, an atomic image will start to appear. This simple instrument is an outgrowth of field emission microscope, invented by Miiller in 1936. In a field emission microscope, the sample is also a sharp tip. In ultra-high vacuum, a negative voltage is applied to the tip. When the field at the nearly hemispherical tip surface reaches a value above ˜0.3 V/Å, electrons are emitted out of the surface by the quantum mechanical tunneling effect. These electrons are projected onto a phosphorus screen some 10 cm away to form an image of the surface. As the electron current density depends very sensitively on the work function of the surface, the greatly magnified, radial projection image of the field emitted electrons represents a map of the work function variation of the surface. These field emitted electrons originate mostly from the vicinity of the Fermi level. They have a relatively large kinetic energy and therefore a relatively large velocity component in the lateral direction of the surface.

The vibrations of regular polymer chains

Introduction

It has been pointed out already, in section 1.3, that the spectra of polymers are a great deal simpler than might at first be expected, because of the repetitive nature of the structure of the polymer chain. In this section we wish to make the arguments presented earlier more precise by first considering the vibrations of a single perfectly regular polymer chain.

We define a perfectly regular chain to be one with which a straight line can be associated in such a way that all translations of the chain parallel to this line by integral multiples of some basic distance are symmetry operations, i.e. they bring the chain into exact coincidence with itself when the indistinguishability of all atoms of the same type is taken into account. Such a polymer chain cannot exist in reality, since it would have to be of infinite length, but in polymer crystals there are considerable lengths of chain for which such translations may be considered to be symmetry operations to a first approximation. Even in the non-crystalline regions of solid polymers there can sometimes be appreciable lengths of such straight chains. Since the interactions between the atoms in the same chain are often, but not always, considerably greater than those between the atoms in adjacent chains, the isolated perfectly regular polymer chain is a good starting point for the discussion of the normal vibrations and vibrational spectra of polymers.

The aim of this book is to present a coherent introductory account of the theory of vibrational spectroscopy and of its application to the study of synthetic organic polymers. The level of presentation is intended to be suitable for the research student who has previously obtained a degree in either physics or chemistry and who is embarking on research in this area. Such a student would, we hope, read the book in its entirety and then be equipped with sufficient background knowledge and understanding to tackle the specialised literature of the subject with some confidence. We hope that in addition the book will fulfil a similar function for any research worker new to the subject and that parts of it may also be found useful by undergraduate students studying either vibrational spectroscopy or polymer science. To make the book accessible to those new to the study of polymers we have given a brief introduction to the subject in chapter 1. This chapter also contains a brief description of experimental methods in vibrational spectroscopy which is intended to give the reader the minimum amount of information required to follow the rest of the book and to feel that his feet are firmly on the ground. Although we regard the book primarily as one to be read rather than simply referred to, we have provided detailed indexes, since the nature of the subject is such that the spectroscopist needs to master a large number of ideas and facts and easy reference to these is vital if that mastery is to be quickly obtained.

Semiclassical treatment of origins of spectra

Polarizability tensors and dipole moments

In this section we explain the origins of infrared absorption and Raman scattering, using as far as possible the ideas of classical physics. Quantum mechanics is introduced only where its use makes the treatment easier or where classical physics is inadequate for explaining a phenomenon or deriving a result. In addition to an explanation of the origins of the effects, an explanation is also given of the selection rules which determine which modes of vibration are actually active for infrared or Raman spectroscopy. Since infrared absorption and Raman scattering involve the interaction of molecules with electromagnetic waves, in particular with the oscillating electric field of the waves, it is necessary to consider the electrical properties of the molecules. We begin by defining the molecular polarizability tensor.

Imagine first a non-vibrating molecule which belongs to one of the point groups D2, D2h, D2d or C2v. For such a molecule the intersections of the symmetry elements define a unique set of three mutually perpendicular axes, which will be called Oxyz. The molecule consists of a large number of charged particles, the nuclei and the electrons, but is electrically neutral overall. If a steady electric field E is applied to the molecule in a direction parallel to one of the reference axes it will cause a redistribution of the charges in the molecule so that the molecule attains a net electric dipole moment, or undergoes a change in an already existing dipole moment, of amount µ.