This article is a review of recent developments in the phenomenological description of unconventional superconductivity. Starting with the BCS theory of superconductivity with anisotropic Cooper pairing, the authors explain the group-theoretical derivation of the generalized Ginzburg-Landau theory for unconventional superconductivity. This is used to classify the possible superconducting states in a system with given crystal symmetry, including strong-coupling effects and spin-orbit interaction. On the basis of the BCS theory the unusual low-temperature properties and the (resonant) impurity scattering effects are discussed for superconductors with anisotropic pairing. Using the Ginzburg-Landau theory, the authors study several bulk properties of such superconductors: spontaneous lattice distortion, upper critical magnetic field, splitting of a phase transition due to uniaxial stress. Two possible mechanisms for ultrasound absorption are discussed: collective modes and damping by domain-wall motion. The boundary conditions for the Ginzburg-Landau theory are derived from a correlation function formulation and by group-theoretical methods. They are applied to a study of the Josephson and proximity effects if unconventional superconductors are involved there. The magnetic properties of superconductors that break time-reversal symmetry are analyzed. Examples of current and magnetic-field distributions close to inhomogeneities of the superconducting order parameter are given and their physical origin is discussed. Vortices in a superconductor with a multicomponent order parameter can exhibit various topological structures. As examples the authors show fractional vortices on domain walls and nonaxial vortices in the bulk. Furthermore, the problem of the possible coexistence of a superconducting and a magnetically ordered phase in an unconventional superconductor is analyzed. The combination of two order parameters that are almost degenerate in their critical temperature is considered with respect to the phase-transition behavior and effects on the lower and upper critical fields. Because heavy-fermion superconductors - which are possible realizations of unconventional superconductivity - have been the main motivation for the phenomenological studies presented here, the authors compare the theoretical results with the experimental facts and data. In particular, they emphasize the intriguing features of the compound UPt3 and consider in detail the alloy U(1-x)Th(x)Be13.

Magnetic materials research has entered a new and exciting period with the advent of the ternary rare-earth-iron-boron compounds, R2Fe14B. From the fundamental physics perspective the R2Fe14B series and its isostructural relatives comprise a rich, fascinating area for the investigation of many intrinsic properties, including magnetic structures, magnetocrystalline anisotropy, and rare-earth-transition-metal exchange interactions. Intense interest in the technological aspects of these compounds has been ignited by the fact that energy products eclipsing all previous values have been realized in practical magnets based on Nd2Fe14B, the prototypical representative; these magnets also feature economic advantages over the earlier samarium-cobalt materials. Both facets of the R2Fe14B systems are considered in this review.

The spin-1/2 antiferromagnetic Heisenberg model on a square lattice is used to describe the dynamics of the spin degrees of freedom of undoped copper oxides. Even though the model lacks an exact solution, a solid, accurate, and rather conventional picture emerges from a number of techniques-analytical (spin-wave theory, Schwinger boson mean-field theory, renormalization-group calculations), semianalytical (variational theory, series expansions), and numerical (quantum Monte Carlo, exact diagonalization, etc.). At zero temperature, the effect of the zero-point fluctuations is not strong enough to destroy the antiferromagnetic long-range order, despite the fact that we are dealing with a low-spin low-dimensional system. The corrections to the spin-wave theory, which treats perturbatively the effect of such fluctuations around the classical Neel ground state, appear to be small. At any nonzero temperature the order disappear and the correlation length at low temperature T(k(B)T/J << 1, where J is the antiferromagnetic coupling) follows the singular form zeta(T) = C exp(alpha-J/k(B)T). In the long-wavelength limit and at low T, the model has the same behavior as the quantum nonlinear-sigma-model in two spatial dimensions and one Euclidean time dimension, which we also study with available analytical and Monte Carlo techniques. The quasiparticles of the theory are bosons; at low T and for wavelengths shorter than the correlation length they are well-defined spin-wave excitations. The spectrum of such excitations and the temperature-dependent correlation length have been determined by neutron and Raman scattering experiments done on La2CuO4. The good agreement of the experimental data with the predictions of this theory suggests that the magnetic state of the undoped materials is the conventional ordered state. We discuss, within a simple mean-field theory, the effect of weak three-dimensional antiferromagnetic coupling and the role of an antisymmetric term, introduced to explain a hidden ferromagnetic behavior of the uniform susceptibility. We find that understanding the copper-oxide antiferromagnetic insulator is only the first essential step towards the development of a theory of the superconductor created upon doping such materials.

The concept of boson realization (or mapping) of Lie algebras appeared first in nuclear physics in 1962 as the idea of expanding bilinear forms in fermion creation and annihilation operators in Taylor series of boson operators, with the object of converting the study of nuclear vibrational motion into a problem of coupled oscillators. The physical situations of interest are quite diverse, depending, for instance, on whether excitations for fixed- or variable-particle number are being studied, on how total angular momentum is decomposed into orbital and spin parts, and on whether isotopic spin and other intrinsic degrees of freedom enter. As a consequence, all of the semisimple algebras other than the exceptional ones have proved to be of interest at one time or another, and all are studied in this review. Though the salient historical facts are presented in the introduction, in the body of the review the progression is (generally) from the simplest algebras to the more complex ones. With a sufficiently broad view of the physics requirements, the mathematical problem is the realization of an arbitrary representation of a Lie algebra in a subspace of a suitably chosen Hilbert space of bosons (Heisenberg-Weyl algebra). Indeed, if one includes the study of odd nuclei, one is forced to consider the mappings to spaces that are direct-product spaces of bosons and (quasi)fermions. Though all the methods that have been used for these problems are reviewed, emphasis is placed on a relatively new algebraic method that has emerged over the past decade. Many of the classic results are rederived, and some new results are obtained for odd systems. The major application of these ideas is to the derivation, starting from the shell model, of the phenomenological models of nuclear collective motion, in particular, the geometric model of Bohr and Mottelson and the more recently developed interacting boson model of Arima and Iachello. A critical discussion of those applications is interwoven with the theoretical developments on which they are based; many other applications are included, some of practical interest, some simply to illustrate the concepts, and some to suggest new lines of inquiry.

The theory of bifurcation from equilibria based on center-manifold reductio, and Poincare-Birkhoff normal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and recent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest generic bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation occurring in a model of three-wave mode coupling and steady-state bifurcations occurring in the real Landau-Ginzburg equation. The former provides an example of the importance of degenerate bifurcations in problems with more than one parameter and the latter illustrates new effects introduced into a bifurcation problem by a continuous symmetry.

Recent experimental progress in the search for atomic electric dipole moments (EDMs) d(A) of cesium and thallium leads in particular to a substantially increased sensitivity to a possible electron EDM d(e) compared with existing upper bounds. Further considerable improvement in the measurement of d(Tl) is likely. After a brief synopsis of the theory of atomic EDMs, the authors discuss-in view of the expected experimental sensitivity to d(e) - the predictions for the electron EDM in various models of CP violation.

Results concerning the rigorous justification of the effective Hamiltonians for band electrons in the presence of weak homogeneous electric and magnetic fields are reviewed. In the electric-field case the existence, in the sense of spectral concentration, of the Stark-Wannier resonances is proved. In the magnetic-field case, the existence of exponentially localized magnetic Wannier functions is established. As a consequence the Peierls-Onsager effective Hamiltonian is obtained.

For more than 40 years it was thought that polaron- and exciton-phonon systems exhibited unexpected localization properties. Particular attention was paid to the so-called phonon-induced self-trapping transition, which, it was believed, should manifest itself as a point of nonanalyticity in the ground-state energy as a function of the electron-phonon coupling parameter. It will be demonstrated for a large class of (generalized Frohlich) models that no such transition exists. The dimensionality of space has no qualitative influence; insofar, an application of the authors' results to problems in lower dimensions (e.g., polarons in quantum wells) is straightforward. The same holds true if homogeneous external fields are involved; for example, a discontinuous mass stripping for magnetopolarons can be excluded. On the other hand, a phase-transition-like behavior will be found, if a polaron or exciton is exposed to a short-range potential, allowing a so-called pinning transition. The authors emphasize, however, that even in this case the transition is only modified, and not induced, by phonons.

In the more than half century since the semiclassical Thomas-Fermi theory of the atom was introduced, there have been literally thousands of publications based on that theory; they encompass a broad range of atomic bound-state and scattering problems. (The theory has also been applied to nuclear physics and solid-state problems.) We will concentrate here on the essence of the theory, namely, its implementation of the uncertainty and exclusion principles and of the Coulomb or Newton force law. Since we are often far more interested in physical concepts than in numerical accuracy or rigor, we will sometimes consider the implementation in a qualitative rather than quantitative fashion. The theory is then capable of giving only qualitative information about a system-one obtains the dependence of the total ground-state binding energy E and radius R of an atom on the nuclear charge Z, for example, but one obtains only rough estimates of the numerical coefficients; in compensation, the calculations are often literally trivial, very much simpler than the already simple Thomas-Fermi calculations. A point to be emphasized is that in the course of obtaining an estimate of E and R of an atom in a Thomas-Fermi approach, one also obtains an estimate of the electronic density, and, particularly if the analysis is more than simply qualitative, an electronic-density estimate can be very useful in a wide variety of problems. We include a short comment on alternative formulations of Thomas-Fermi theory in a D-dimensional space. We will review the applications of the theory, from both qualitative and (Thomas-Fermi) quantitative viewpoints, to heavy atoms, where we are concerned with a Coulomb interaction, and to neutron stars and white dwarfs, where we are concerned with a gravitational interaction and with gravitational-plus-Coulomb interactions, respectively. In the latter case, the first two Coulomb corrections are evaluated. Very rough (relativistic) estimates are made of the conditions under which heavy atoms, neutron stars, and white dwarfs collapse. A one-dimensional Thomas-Fermi-like theory also exists for heavy atoms in a uniform strong magnetic field B, of the order of the field believed to exist at the surface of a neutron star. Here, too, the qualitative picture immediately gives some of the main results, namely, the dependence of E and R on B and Z. We also comment briefly on some relatively recent and very recent developments in Thomas-Fermi theory. These include a proof of the stability of matter. Though it was first proved by Dyson and Lenard, we consider the Lieb-Thirring proof, both because it is much simpler and because it makes extensive use of Thomas-Fermi theory, including a no-molecular-binding theorem that follows in the Thomas-Fermi approximation: Teller proved that, in that approximation, atoms could not form molecular bound states. These developments also include (a) the Lieb-Simon proof that the prediction of the theory that E = -c7Z7/3, with c7 a specified coefficient, becomes exact at Z approximately infinity, (b) the Scott-c6Z6/3 correction term, with c6 specified and now known also to be exact, and (c) the Schwinger estimate of the coefficient c5 of the Z5/3 term, which there is good reason to believe is exact. The many digressions include comments on QED, on lower bounds on the ground-state energy of a system, and on mini-boson stars.

The electronic transport theory of semiconductors is not, from a first-principles point of view, as well understood as is that of metals, where the degeneracy of the Fermi system leads to a simplified but comprehensive theory. In the case of semiconductors, degeneracy usually plays no simplifying role at all. However, in many transport problems of current interest one is effectively dealing with the equivalent of a single particle interacting with an environment, e.g., a heat bath or a random potential. In view of this, the author presents a simple formalism for the quantum dynamics of a single continuous degree of freedom. The quantum-statistical description is in terms of the density matrix, and the Feynman rules for a standard treatment of the density matrix are presented and illustrated by applications to problems of current interest. It is shown that such an effect as, for example, the intracollisional field effect, which in the past has been dealt with using complicated formalisms, in the present treatment is described in an elementary way. The single-particle approach conveniently displays the interference aspect of quantum-mechanical transport, as is discussed in a treatment of the weak localization effect in disordered conductors. The real-space representation of quantum transport is stressed, as is appropriate for a proper discussion of mesoscopic physics. The author treats the connection between the linear-response formalism and the Landauer approach by expressing the conductance in terms of the scattering properties of a sample. He also discusses the conductance fluctuations of mesoscopic samples.

The basic physics regarding self-trapping of light particles in simple fluid hosts is reviewed pedagogically. Electron and positronium self-trapping in fluid helium is taken as a historical starting point. The theoretical context consists of simplified continuum models with averaged interactions, but required improvements are discussed. Experimental examples are chosen to illustrate bulk, surface, and impurity effects. Equilibrium and dynamical aspects of the field are illustrated. In noting applications to more complex systems, reference is made to recent developments using path-integral and computer simulation methods. The article spans certain aspects of studies in this fascinating area over the last 30 years.

We compute all the three-dimensional quasicrystallographic space groups with n-fold axial point groups and standard lattices by a method that treats crystals and quasicrystals on an equal footing. We do not rely on projecting higher-dimensional crystallographic space groups, our results are valid for arbitrary n, and our analysis is elementary. We regard space groups as a scheme for classifying diffraction patterns to be carried out in three-dimensional reciprocal space. The familiar three-dimensional crystallographic space groups with axial point groups emerge simply and directly as special cases of the general n-fold three-dimensional quasicrystallographic treatment with n = 3, 4, and 6. We give a general discussion of extinctions in quasicrystals and give a simple (three-dimensional) geometrical specification of the extinctions for each axial space group. The paper is intended both for people trying to systematize quasicrystal diffraction patterns and for people interested in a simple alternative approach to the computation of crystallographic or quasicrystallographic space groups.

At the thirty-year anniversary of the introduction of the technique of computer-generated random-dot stereograms and random-dot cinematograms into psychology, the impact of the technique on brain research and on the study of artificial intelligence is reviewed. The main finding -that stereoscopic depth perception (stereopsis), motion perception, and preattentive texture discrimination are basically bottom-up processes, which occur without the help of the top-down processes of cognition and semantic memory - greatly simplifies the study of these processes of early vision and permits the linking of human perception with monkey neurophysiology. Particularly interesting are the unexpected findings that stereopsis (assumed to be local) is a global process, while texture discrimination (assumed to be a global process, governed by statistics) is local, based on some conspicuous local features (textons). It is shown that the top-down process of "shape (depth) from shading" does not affect stereopsis, and some of the models of machine vision are evaluated. The asymmetry effect of human texture discrimination is discussed, together with recent nonlinear spatial filter models and a novel extension of the texton theory that can cope with the asymmetry problem. This didactic review attempts to introduce the physicist to the field of psychobiology and its problems - including metascientific problems of brain research, problems of scientific creativity, the state of artificial intelligence research (including connectionist neural networks) aimed at modeling brain activity, and the fundamental role of focal attention in mental events.

The introduction of a powerful new microwave source, the free-electron laser, provides new opportunities for novel heating and current-drive schemes to be used in toroidal fusion devices. This high-power, pulsed source has a number of technical advantages for these applications, and its use is predicted to lead to improved current-drive efficiencies and opacities in reactor-grade fusion plasmas in specific cases. The Microwave Tokamak Experiment at the Lawrence Livermore National Laboratory will provide a test for some of these new heating and current-drive schemes. Although the motivation for much of this research has derived from the application of a free-electron laser to the heating of a tokamak plasma at a frequency near the electron cyclotron frequency, the underlying physics, i.e., the highly nonlinear interaction of an intense, pulsed, coherent electromagnetic wave with an electron in a magnetized plasma including relativistic effects, is of general interest. Other relevant applications include ionospheric modification by radio-frequency waves, high-energy electron accelerators, and the propagation of intense, pulsed electromagnetic waves in space and astrophysical plasmas. This review reports recent theoretical progress in the analysis and computer simulation of the absorption and current drive produced by intense pulses, and of the possible complications that may arise, e.g., parametric instabilities, nonlinear self-focusing, trapped-particle sideband instability, and instabilities of the heated plasma.

Non-neutral plasmas, like electrically neutral plasmas, exhibit a broad range of collective properties, such as plasma waves and instabilities, and the ability to support long-lived, large-amplitude coherent structures. This paper reviews the equilibrium and stability properties of intense non-neutral electron flow in crossed electric and magnetic fields. Following a description of equilibrium properties for magnetically insulated electron flow in planar geometry, extraordinary-mode stability properties are investigated for relativistic non-neutral electron flow between planar conductors. Particular emphasis is placed on the magnetron and diocotron instabilities, and detailed stability behavior is shown to exhibit a sensitive dependence on the self field intensity (as measured by the dimensionless parameter s(e) = gamma-e0-omega-pe2/omega-ce2) as well as on the shape of the equilibrium profiles. The influence of cylindrical effects (such as the centrifugal and Coriolis accelerations of an electron fluid element) on stability behavior is then investigated for rotating electron flow in cylindrical geometry. Finally, the properties of large-amplitude coherent structures in non-neutral plasmas with circulating electron flow are investigated. Topics covered in this area include particle-in-cell computer simulations of dense (s(e) approximately 1) electron flow in relativistic magnetrons which show large-amplitude spoke formation in the circulating electron density, and application of a cold-fluid guiding-center model to investigate large-amplitude vortex structures in low-density (s(e) << 1) non-neutral plasma. The accessibility and stability of such stationary structures (in the rotating frame) remain important topics for future investigation.

A systematic review of theoretical results for the longitudinal and transverse impedances obtained by different methods is presented. Definitions, general theorems, modal analysis, diffraction model, and analytical results comprise the content of the paper. Several new results are included. In particular, necessary and sufficient conditions for the independence of the impedance on the beam longitudinal direction are given. The impedances of two basic simple structures - that of a cavity and that of a step - are studied in detail. The transition from the regime of a cavity to the regime of a step is explained, an approximate formula describing this transition is given, and the criterion for determining the applicability of each regime is established. The asymptotic behavior of the impedance for a finite number M of periodically arranged cavities as a function of M is studied. The difference in the behavior of the impedance for a single cavity and that for an infinite number of cavities is explained as the result of the interference of the diffracted waves. A criterion for determining the transition in the impedance behavior from small M to large M is presented.