Tokamaks constitute the best plasma physics laboratory available today. The largest devices (e.g. JET and DIII-D) confine plasmas of considerable volume (many m3), high densities (ne ∼ 1020 m-3) and high temperatures (Te ∼ 10 keV) under quasi-stationary conditions (for an introduction to the general physics of tokamaks see Wesson, 1987). Tokamak plasmas exhibit a rich variety of MHD phenomena, being investigated by numerous diagnostic tools with high spatial and temporal resolution, which make theoretical interpretation a challenging task.

Particularly conspicuous MHD effects are the different kinds of disruptive events which affect global plasma confinement more or less severely. In this chapter we consider the three most important disruptive processes. Section 8.1 deals with the sawtooth oscillation, a quasi-periodic internal relaxation process, which is observed in most tokamak discharges. Their main effect is to limit the central temperature increase, generating a more uniform average temperature distribution. They also have the beneficial effect of preventing the central accumulation of impurity ions.

Section 8.2 considers major disruptions, which constitute the most violent processes in a tokamak plasma. Disruptions occur when certain limits in the plasma parameters are exceeded, causing loss of a large fraction of the plasma energy, which often leads to the termination of the discharge.

Plasma physics has sometimes been called the science of instabilities. In fact during the last three decades of plasma research, stability theory was probably the most intensively studied field. The reason for this widespread activity is the empirical finding that in general plasmas, especially those generated in laboratory devices, are not quiescent but spontaneously develop rapid dynamics which often tend to terminate the plasma discharge. MHD instabilities are considered as particularly dangerous because they usually involve large-scale motions and short time scales. Though a realistic picture of dynamic plasma processes requires a nonlinear theory, the knowledge of the basic linear instability is usually a very helpful starting point, in particular since linear theory has a solid mathematical foundation.

The organization of the chapter is as follows. Section 4.1 presents the linearized MHD equations. In section 4.2 we consider the simplest case of linear eigenmodes, waves in a homogeneous plasma. The energy principle is introduced in section 4.3. In section 4.4 we then derive in some detail the theory of eigenmodes in a circular cylindrical pinch, which contains many qualitative features of geometrically more complicated configurations. In section 4.5 this theory is applied to the cylindrical tokamak model. The influence of toroidicity, which most severely affects the n = 1 mode, is discussed briefly in section 4.6.

Magnetohydrodynamics (MHD) describes the macroscopic behavior of electrically conducting fluids, notably of plasmas. However, in contrast to what the name seems to indicate, work in MHD has usually little to do with dynamics, or at least has had so in the past. In fact, most MHD studies of plasmas deal with magnetostatic configurations. This is not only a question of convenience — powerful mathematical methods have been developed in magnetostatic equilibrium theory — but is also based on fundamental properties of magnetized plasmas. While in hydrodynamics of nonconducting fluids static configurations are boringly simple and interesting phenomena are in general only caused by sufficiently rapid fluid motions, conducting fluids are often confined by strong magnetic fields for times which are long compared with typical flow decay times, so that the effects of fluid dynamics are weak, giving rise to quasistatic magnetic field configurations. Such configurations may appear in a bewildering variety of shapes generated by the particular boundary conditions, e.g. the external coils in laboratory experiments or the “foot point” flux distributions in the solar photosphere, and their study is both necessary and rewarding.

In addition to finding the appropriate equilibrium solutions one must also determine their stability properties, since in the real world only stable equilibria exist.

Magnetohydrodynamics (MHD) is the macroscopic theory of electrically conducting fluids, providing a powerful and practical theoretical framework for describing both laboratory and astrophysical plasmas. Most textbooks and monographs on the topic, however, concentrate on two particular aspects, magnetostatic equilibria and linear stability theory, while nonlinear effects, i.e. real magnetohydrodynamics, are considered only briefly if at all. I have therefore felt the need for a book with a special focus on the nonlinear aspects of the theory for some time.

In contrast to linear theory which, in particular in the limit of ideal MHD, rests on mathematically solid ground, nonlinear theory means adventures in a, mathematically speaking, hostile world, where few things can be proved rigorously. While in linear stability analysis numerical calculations are mainly quantitative evaluations, they obtain a different character in the study of nonlinear phenomena, which are often even qualitatively unknown. Hence this book frequently refers to results from numerical simulations, as a glance at the various illustrations reveals, but consideration is focused on the physics rather than the numerics.

In spite of the numerous references to the literature the book is essentially self-contained. Even the individual chapters can be studied quite independently as introductions to or current overviews of their particular topics.

There is hardly a term in plasma physics exhibiting more scents, facets and also ambiguities than does magnetic reconnection or, simply, reconnection. It is even sometimes used with a touch of magic. The basic picture underlying the idea of reconnection is that of two field lines (thin flux tubes, properly speaking) being carried along with the fluid owing to the property of flux conservation until they come close together at some point, where by the effect of finite resistivity they are cut and reconnected in a different way. Though this is a localized process, it may fundamentally change the global field line connection as indicated in Fig. 6.1, permitting fluid motions which would be inhibited in the absence of such local decoupling of fluid and magnetic field. Almost all nonlinear processes in magnetized conducting fluids involve reconnection, which may be called the essence of nonlinear MHD.

Because of the omnipresence of finite resistivity in real systems resistive diffusion takes place everywhere in the plasma, though usually at a slow rate. Reconnection theory is concerned with the problem of fast reconnection in order to explain how in certain dynamic processes very small values of the resistivity allow the rapid release of a large amount of free magnetic energy, as observed for instance in tokamak disruptions or solar flares.

The study of linear stability of plasmas had for a long period been carried by the conception that only stable configurations can exist in nature, since instability would lead to destruction of the equilibrium and loss of plasma confinement, which would be the faster the larger the growth rate. Statements like: “all plasmas (meaning real inhomogeneous plasma configurations) are unstable”, sometimes pronounced by plasma theoreticians in the heyday of instability theory, seemed to imply that magnetic fusion research is basically a futile endeavor. The development in experimental plasma physics during the past two decades proved this conception thoroughly wrong. Tokamak discharges may exist, well confined, in spite of the presence of instabilities, which often lead only to a slight change of the plasma profiles and a certain increase of plasma and energy transport (and which may even have beneficial effects such as the removal of impurities by the sawtooth process). Thus in order to judge the effect of an instability it is evidently necessary to calculate or at least estimate its nonlinear behavior, in particular the saturation level. It will turn out that linear mode properties, in particular growth rates, often have little to say about the nonlinear behavior.

As a general rule an instability is found to be the more “dangerous”, i.e. its effect on the plasma configuration is the more detrimental, the longer the wavelength (global modes).

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.

Within the neighbourhood of the Sun, a number of highly evolved stars are carbonrich in the sense that they have more carbon than oxygen so their outer atmospheres contain molecules such as CN, CH and C2H2. These stars are cool with atmospheric temperatures near 3000 K and they are also luminous, typically 104 times more powerful than the Sun. The outer envelopes of these stars are tenuously bound, and they all are losing mass at a very high rate, in some cases more than 10−5MOa a−1 (where MO denotes the mass of the Sun). These high luminosity carbon stars remain in this phase for a time, very approximately, near 105 years. They exhibit a large amount of carbon in their atmospheres because the products of the nuclear burning that occurs in the very centre of the star, including the synthesis of carbon, appear on the surface.

In the extended envelopes around these stars, there is a very active chemistry, and the gas is sufficiently cool that nucleation of solid dust grains occurs. These solid particles may grow to sizes as large as 1 μm although a more typical size is near 0.05 μm. We therefore can identify both relatively small carbon-bearing molecules (for example HC7N) and much larger carbon-containing dust grains in the outflows. The amount of intermediate size particles or molecules, such as C60, and their possible role in the circumstellar chemistry is not yet well understood. At least in the envelope of the well studied carbon star IRC+10216, there appears to be more carbon in CO and solid grains than in poly cyclic aromatic hydrocarbons.