In this chapter we examine, as before, electromagnetic fluctuations in a homogeneous, magnetized, collisionless plasma. In contrast to the previous chapter, however, we admit anisotropies in the distribution functions. In particular we consider a two-temperature bi-Maxwellian zeroth-order distribution function; this permits the growth of temperature anisotropy instabilities. Section 7.1 outlines the derivation of the dispersion equation; Section 7.2 discusses the properties of modes driven unstable by a proton temperature anisotropy, whereas Section 7.3 discusses the properties of electron temperature anisotropy instabilities. Section 7.4 is a brief summary.

Our emphasis in this chapter is on instabilities driven by T⊥j > T‖ a condition that is observed more often in space plasmas than the converse T‖ > T⊥. The reason for this discrepancy is simple: although space plasmas do not necessarily exhibit a bias toward perpendicular heating processes, perpendicular heating does not much change the mobility of the heated particles, whereas parallel heating enables the particles to move more rapidly along B0. Thus parallel-heated particles may leave the region of energization more quickly, implying that T‖ > T⊥ should be a less frequently observed condition. Of course, parallel-heated particles may appear elsewhere as a magnetic-field aligned beam streaming against a cooler background plasma; the electromagnetic instabilities driven by such configurations have quite different properties from temperature anisotropy instabilities, and are studied in detail in the next chapter.

Plasma instabilities are normal modes of a system that grow in space or time. Thus the word “instability” implies a well-defined relationship between wavevector k and frequency ω; this in turn implies that the associated plasma fluctuations are relatively weak so that linear theory is appropriate to describe the physics.

This book uses linear Vlasov theory to describe the propagation, damping and growth of plasma modes. Linear theory cannot describe the ultimate fate of a plasma instability, nor its interactions with other modes. Of course the questions of how an instability reaches maximum amplitude, whether and how it contributes to plasma transport and whether such transport affects the overall flow of mass, momentum and energy at large scales are crucial for establishing the relevance of microphysics to large scale modelling of space plasmas. But these questions must be addressed by nonlinear theory and computer simulation, which are beyond the purview of this book. Our relatively modest goal is to use computer solutions of the unapproximated Vlasov dispersion equation to firmly establish the properties of plasma normal modes; our hope is that this information will provide a useful foundation for the interpretation of computer simulations and spacecraft observations under conditions of relatively weak fluctuation amplitudes.

Micro- vs macro-

The most general classification of growing modes in a plasma divides them into two broad categories: macroinstabilities at relatively long wavelengths and microinstabilities at shorter wavelengths.

This chapter begins our consideration of instabilities, plasma modes that grow in time or space. The source of growth of a plasma microinstability is what is imprecisely called “free energy:” an anisotropy or inhomogeneity in the zeroth-order velocity distribution function. In this chapter we consider free energy sources associated with the relative drifts of plasma components, and find that different types of relative drifts each can give rise to several different unstable modes.

As in Chapter 2, we consider uniform collisionless plasmas in which the evolution of the distribution function of the jth component is described by the Vlasov equation (1.3.2). Again we restrict ourselves to electrostatic fluctuations; that is, we assume there are no fluctuating magnetic fields and the fluctuating electric fields are derived from Poisson's equation (1.2.4). In Section 3.1 we state zeroth-order distribution functions representing several different free energy sources. Section 3.2 considers electrostatic instabilities driven by component/component relative drifts in unmagnetized plasmas and Section 3.3 considers electrostatic instabilities driven by the same free energy sources in magnetized plasmas. Throughout this chapter, we assume the plasma to be charge neutral and to bear no steady-state electric field.

Although virtually all space plasmas bear ambient magnetic fields, some waves and instabilities have properties that are essentially independent of B0. In Section 3.3 we demonstrate this by showing that the magnetized electrostatic dispersion equation at k × B0 = 0 reduces to the unmagnetized form.

If a charged particle species of a collisionless plasma possesses a non-Maxwellian velocity distribution function, a short wavelength normal mode of the system may grow in amplitude. This is a microinstability; its theory is well described by the Vlasov equation. The purpose of this monograph is to describe in an accurate way the theory of damped normal modes and a limited number of microinstabilities that may arise in various space plasma environments.

The two words that best characterize the work described in this book are “limited” and “accurate.” In order to keep the discussion limited, I have chosen idealized, not observed, distribution functions. Many spacecraft have provided excellent observations of electron and ion distributions in the Earth's magnetosphere and nearby solar wind. The tremendous variety of these distributions makes it difficult to select a few for special representation. My choice here has been to use Maxwellian or bi-Maxwellian distributions with field-aligned drifts to represent some of the more important general free energy sources. Although the resulting instabilities may not correspond to any particular data set, I hope that each one represents the general properties of a very broad class of data.

To provide accuracy, I have followed the same procedure for each distribution function and plasma model. After assuming a zeroth-order distribution, I derive (or at least explicitly state) the associated dispersion equation without approximation. Because I deal with linear theory throughout this book, it is always straightforward to do this, although the algebra gets tiresome at times.

The structural form of geodesic domes, composed of pentagons and hexagons, played an important role in understanding the structure of carbon clusters. In this paper an analogy between geodesic domes and fullerenes is investigated. A brief survey is given of the geometry of geodesic domes applied in engineering practice, in particular of the geodesic domes bounded by pentagons and hexagons. A connection is also made between these sorts of geodesic domes and the mathematical problem of the determination of the smallest diameter of n equal circles by which the surface of a sphere can be covered without gaps. It is shown that the conjectured solutions to the sphere-covering problem provide topologically the same configurations as fullerene polyhedra for some values of n. Mechanical models of fullerenes, composed of equal rigid nodes and equal elastic bars are also investigated, and the equilibrium shapes of the space frames that model C28, C60 and C240 are presented.

Introduction

From visual inspection one can easily discover an analogy between the structure of C60 and the inner layer of the structure of the great U.S. pavilion of R. B. Fuller at the 1967 Montreal Expo. This analogy and other geodesic structures of Fuller were responsible for the name of C60: Buckminsterfullerene (Kroto et al. 1985). This is not the first time that Fuller's geodesic domes have helped researchers to understand the structure of matter. In the early 1960s Fuller's geodesic domes, especially his tensegrity spheres, inspired Caspar & Klug (1962) to develop the principle of quasiequivalence in virus research.

Small carbon grains are assumed to be the carrier of the prominent interstellar ultra violet absorption at 217 nm. To investigate this hypothesis, we produced small carbon particles by evaporating graphite in an inert quenching gas atmosphere, collected the grains on substrates, and measured their optical spectra. In the course of this work – which in the decisive final phase was carried out with the help of K. Fostiropoulos and L. D. Lamb – we showed that the smoke samples contained substantial quantities of C60. The fullerene C60 (with small admixtures of C70) was successfully separated from the sooty particles and, for the first time, characterized as a solid. We suggested the name ‘fullerite’ for this new form of crystalline carbon.

Introduction

The production of laboratory analogues of interstellar grains was the initial aim of our research. In the autumn of 1982 while one of us (D.R.H.) was a Humboldt Fellow at the Max Planck Institute of Nuclear Physics in Heidelberg we decided to study the optical spectra of carbon grains. We felt challenged by the intense, strong interstellar ultra violet (uv) absorption at 217 nm which it had been proposed was due to graphitic grains (see, for example, Stecher 1969). The arguments in favour of such carriers are based primarily on calculations of the absorption of small, almost spherical, particles which exhibit the dielectric functions of graphite (for more recent literature see, for example, Draine 1988). There had already been very early experimental attempts to produce graphitic smoke particles by almost the same technique that we later applied to C60 production (see, for example, Day & Huffman 1973).

The chemistry by which the closed-cage carbon clusters, C60 and C70, can be formed in high yield out of the chaos of condensing carbon vapour is considered. Several mechanisms for this process that have been proposed are critically discussed. The two most attractive are the ‘pentagon road’ where open sheets grow following the alternating pentagon rule and the ‘fullerene road’ where smaller fullerenes grow in small steps in a process which finds the buckminsterfullerene (C60) local deep energy minimum and to a lesser extent the C70 (D5h) minimum. A clear choice between the two does not seem possible with available information.

Introduction

The observation (Kroto et al. 1985) that the truncated icosahedron molecule, CBF60 (buckminsterfullerene), is formed spontaneously in condensing carbon vapour was greeted by some in the chemical community with some doubt. It seemed incredible that this highly symmetrical, closed, low entropy molecule was forming spontaneously out of the chaos of condensing high-temperature carbon vapour. We still believe that the formation of CBF60 in supersonic cluster beam sources must be a relatively minor channel, probably accounting for less than 1% of the total carbon. Thus when CBF60F was finally isolated from graphitic soot (Krätschmer et al. 1990), it came as a surprise that the CBF60 plus C70 yields were as large as 5%. Later yields have improved substantially, for example Parker et al. (1991) obtained a total yield of CBF60 of about 20% with total extractable fullerene yields totalling 44% from a carbon arc soot. Thus conditions can be found where CBF60 and fullerene formation in carbon condensation can hardly be called a minor channel.

C60 has not yet been detected in primitive meteorites, a finding that could demonstrate its existence in the early solar nebular or as a component of presolar dust. However, other allotropes of carbon, diamond and graphite, have been isolated from numerous chondritic samples. Studies of the isotopic composition and trace element content and these forms of carbon suggest that they condensed in circumstellar environments. Diamond may also have been produced in the early solar nebula and meteorite parent bodies by both low-temperature-low–pressure processes and shock events. Evidence for the occurrence of another carbon allotrope, with sp hybridized bonding, commonly known as carbyne, is presented.

Introduction

At the same time that buckminsterfullerene was being conceived as a molecule of possible astrophysical significance, a number of much older forms of carbon were about to enjoy a new lease of life because of their discovery as presolar grains in primitive meteorites. Ever since the 1960s, it has been recognized that carbonaceous chondrites were a host for noble gases of anomalous isotopic composition (Anders 1981). The carriers of a litany of components, enjoying names such as Xe(HL) (also called CCF-Xe), s-Xe, Ne-E(L), Ne-E(H), etc., were believed to be unidentified carbon species called C∂, Cβ, Cα and C∈ respectively, themselves exhibiting unusual or exotic isotopic compositions (Swart et al. 1983a; Carr et al. 1983). In 1987, C∂ was shown to be diamond (Lewis et al. 1987) the meteorite mineral which contained Xe(HL) and nitrogen whose isotopic composition was greatly enriched in the light isotope 14N (Lewis et al. 1983).

The early prediction of hollow graphite molecules suggested that they should be supercritical under ambient conditions. This is not true of C60, but might still be true of higher fullerenes and graphite nanotubes of large diameter.

Introduction

My title refers to the celebrated vision of Kekulé, one of the founders of the concept of chemical structure. In 1865, staring drowsily one evening into the fire, he saw in a dream the cyclic structure for benzene, that fundamental unit of all aromatic molecules, and of graphite and the fullerenes. In his reverie, he imagined the atoms gambolling before his eyes… ‘one of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes’ (Kekulé 1890). In this paper I deal, not so much with the recent triumphs of the identification and bulk preparation of buckminsterfullerene, as with its imaginative prehistory. This begins with Dalton's atomic theory, elaborated from 1803 onwards. Despite a very promising start, atomic theory languished for decades as merely a sort of useful metaphor. One good reason was its failure to come up with consistent atomic weights for the elements and formulae for their compounds. Whether, for example, the atomic weight of oxygen was 8 and water was HO, or whether it was 16 with water as H2O, remained uncertain for half a century.

And yet shortly after Dalton proposed his theory, the whole problem had been solved (Avogadro 1811).

By chance in 1970, we conjectured the possibility of the football-shaped C60 molecule, now known as buckminsterfullerene, while considering superaromatic molecules having three-dimensional π-electron delocalization. A translation of the original description, initially written in Japanese, is given. The processes leading to scientific discoveries are analysed in the light of our missed opportunity.

Introduction

The timescale of scientific and technological advance is becoming shorter and shorter in modern society, partly as a consequence of the rapid advance of technology and improving information transfer. It is no wonder then that the time has come to look back and discuss the future of fullerene science after less than a decade since its discovery by Kroto et al. (1985) and after only two years since it was isolated by Kratchmer et al. (1990). The purpose of this paper is to recount the story of original early proposal of the football-shaped C60 molecule back in 1970, and refer to other interesting ‘prehistoric’ events and analyse the process of scientific discovery.

Background

In the 1960s and 1970s, non-benzenoid aromatics were favourite targets for organic chemists. There was a prevailing dogma that aromaticity, due to the delocalization of π-electrons, is best realized in planar molecules. Everyone wished to constrain their molecules to be as planar as possible and for this reason aromaticity tacitly remained a two-dimensional concept. [18]Annulene (1), synthesized by Sondheimer et al. (1962), can be regarded as the masterpiece of planar aromaticity for its symmetric beauty (D6h) and high level of π-electron delocalization. In view of the wide availability and its perfect aromaticity, however, benzene remains the archetypal superstar of aromatic molecules.