Qualitative theoretical treatments of the fullerene family of molecules can be used to count possible isomers and predict their geometric shapes, point groups, electronic structures, vibrational and NMR spectroscopic signatures. Isomers are generated by the ring-spiral algorithm due to D. E. Manolopoulos. Geometrically based magic number rules devised by the present author account for all electronically closed π shells within the Hückel approximation and these ‘leapfrog’ and ‘cylinder’ rules apply to the wider class of ‘fulleroid’ structures constructed with rings of other sizes. Extrapolations from the theory of carbon clusters are described for doped fullerenes, metallocarbohedrenes, fully substituted boron–nitrogen heterofullerenes and decorated-fullerene models for water clusters.

Introduction

The fullerenes offer a challenge to theoretical chemistry. They are large molecules and, even with modern computational methods, it would be expensive and often uninformative to perform full ab initio calculations on them cage by cage. In the first steps towards understanding these new materials a more qualitative approach is necessary and desirable. Methods based on little more than topology and elementary valence theory can provide information on isomerism, geometric and electronic structure, spectroscopic signatures, stability and selection rules for interconversion. This paper touches on developments in these areas but, in line with the ‘postbuckminsterfullerene’ theme of the meeting, concentrates on new ideas in the theory of exotic fullerenes, heterofullerenes and related heteroatom and molecular clusters.

Fullerenes: isomers and electronic structure

The discovery of C60 (Kroto et al. 1985) carried with it the implication of the existence of a whole series of similar molecules. In the original experiments strong C60 and C70 signals were found in association, and intensity was distributed over a wide range of even-numbered clusters.

The gaseous, solution and solid state experimental evidence for electron addition to the fullerenes is reviewed and it is shown that this class of molecules function as powerful electron acceptors. The topological character of C60 as described by Hiickel molecular orbital theory suggests that the molecule will undergo facile reduction, but comparisons with planar conjugated hydrocarbons show that this feature alone cannot account for the very low half-wave reduction potential of C60. Because of the curvature of the surface, fullerene hybridization falls between graphite (sp2) and diamond (sp3) and these new carbon allotropes are therefore of intermediate, and perhaps variable hybridization. According to POAVI theory the carbon atoms in C60 are of sp2.28 hybridization. It is concluded that rehybridization plays an important role in determining the electronic structure of the fullerenes and it is the combination of topology and rehybridization that together account for the extraordinary ability of C60 to accept electrons.

Introduction

The ability of the fullerenes to function as electron acceptors has been recognized since the first investigation of their chemistry (Haufler et al. 1990). Even before the isolation of bulk quantities of C60 (Kratschmer et al. 1990), a large electron affinity was demonstrated for this molecule in gas phase experiments (Curl & Smalley 1988).

This trend has continued with the development of the physics, chemistry and materials science of the fullerenes. In this paper some of the experiments that have thrown light on the ability of the fullerenes to accept electrons are summarized and qualitative explanations for their extraordinary electron affinity are discussed.

The theory presented above provides insight into the basic physical processes proceeding in the magnetosphere of a neutron star. Within this theory we can describe the principal details of the dynamics and evolution of pulsars, clarify the nature of their activity, establish the origin of the electron–positron plasma, and describe the coherent generation mechanism of observed radio emission. The main predictions of the theory are in agreement with observations.

There are, of course, many questions requiring further theoretical investigations. For example, it is necessary to carry out a reliable calculation of free electron energy on the neutron star's surface (Section 2.5), to investigate the wave excitation processes in the boundary layer near the light surface and in an extensive region behind it (Chapter 4), to study the transition layer structure on the boundary between the open and closed regions of the magnetosphere (Chapter 4), and so on. It is noteworthy that only first steps have been made in the study of the nature of non-stationary processes in the pulsar magnetosphere (Section 7.7). Nevertheless, the physical picture of the basic processes in the pulsar magnetosphere seems on the whole to be clear.

Pulsars, or more precisely radio pulsars – sources of pulsed cosmic radio emission – were discovered in 1967 and almost immediately identified with rotating neutron stars. Such stars must originate from catastrophic gravitational contraction (collapse) of ordinary stars that have exhausted the stores of their nuclear fuel. In neutron stars, the gravitational forces are brought to equilibrium not by plasma pressure, as in ordinary stars, and not by pressure of degenerate electrons, as in white dwarfs, but by pressure of strongly compressed neutron matter. They are, so to say, huge clusters of nuclear matter, and although their mass is of the order of the solar mass, their radius is only about 10 km.

An experimental discovery of pulsars, i.e. the neutron stars predicted by Baade and Zwicky as far back as 1934, is rightly regarded as one of the greatest discoveries in astrophysics. For this discovery, Hewish was awarded the Nobel Prize in 1974. There is a very large amount of literature devoted to radio pulsars – in a little more than two decades after their discovery, about 5000 papers have been published. The basic methods of study and the results of observations of radio pulsars are summarized in the excellent monographs by Manchester & Taylor, and Smith, both titled Pulsars and published in 1977, and in Pulsar Astronomy by Lyne & Smith published in 1990.

Radio pulsars are now under study in practically all of the biggest observatories of the world. In particular, the list of sources discovered is increasingly large.

We have determined the properties of electron–positron plasma in the pulsar magnetosphere. It is generated in the polar cap region and moves along a curvilinear magnetic field at a velocity close to the velocity of light. Particles of this plasma must give curvature radiation whose basic characteristics – the frequency range (3.31) and the direction (3.32) – are close to the observed pulsar radio emission. However, the wavelength of curvature radiation (3.33) is much larger than the distance between particles in a plasma flux. In this case an important role is played by a collective interaction between the particles and the field. In other words, the radiation must be coherent (Ginzburg et al., 1969), i.e. must be a fundamental mode of plasma oscillations. To find and to describe quantitatively the radio emission generation mechanism, it is therefore necessary to investigate the excitation of natural high-frequency oscillations of a relativistic plasma flux moving in the polar region of a magnetosphere. This chapter is aimed at solving this problem.

In Section 6.1 we investigate the dielectric properties of a relativistic electron–positron plasma placed in a homogeneous magnetic field and find the fundamental modes of electromagnetic oscillations. In Section 6.2 we take into account the effect of magnetic field inhomogeneity, i.e. the curvature which determines mode instability. The non-linear stabilization and mode transformation effects are studied in Section 6.3.

Electrodynamics of relativistic electron–positron plasma in a homogeneous magnetic field

Dielectric permittivity of a homogeneous plasma

We shall consider a relativistic electron–positron plasma placed in a strong homogeneous magnetic field B.

The theory we have constructed can be compared with observational results. But for the consideration of concrete processes, the theory in some cases should be detailed. In this chapter, we therefore make both a theoretical analysis, which allows us to obtain concrete calculational formulas, and a direct comparison of the theory with experiment. We analyse the structure of the active region (Section 7.1), the generation of electron–positron plasma and of high-frequency radiation (Section 7.2), the dynamics of a neutron star due to its current-induced deceleration (Section 7.3), the statistical distribution of pulsars (Section 7.4), the generation of radio emission (Sections 7.5 and 7.6) and, finally, nonstationary processes (Section 7.7).

The structure of the active region

The model of a partially filled magnetosphere

As shown in Chapters 3–5, the theory of plasma generation in the region of the double layer associates the pulsar rotation energy losses Wtot with the magnitude of the longitudinal electric current j∥ flowing in the magnetosphere (see (4.215)). This current is specified by the ‘compatibility relation’ (4.174) and by the pulsar ‘ignition’ condition (5.66). Owing to this, we can reconstruct the structure of the active region in the polar cap and determine important parameters of a neutron star as the magnitude of the magnetic field B0 from the observed quantities P and dP/dt.

It should be noted, however, that to be compared with observational data, the theory developed in Chapter 4 should be extended.