A review of the empirical data on intrinsic states of odd-$A$ nuclei in the mass range $151\le A\le 193$ and $A\ge 221$ is presented. Global summaries of the data are presented in tables for each isotopic and isotonic chain, wherein the excitation energy, the $log\mathrm{ft}$ values, the moment-of-inertia parameter, and the decoupling parameter (for $K=\frac{1}{2}$ bands) are listed for single-particle, vibrational admixed, and pure vibrational states. Similar data are separately presented for three-quasiparticle excitations in the rare-earth region. Taking guidance from the systematics on nuclear deformation, the single-particle deformed potential for axially symmetric and reflection-symmetric shapes (the Nilsson model) modified by the hexadecapole deformation is used to interpret the data. Other variations of the Nilsson model, which include axially asymmetric shapes ($\gamma$ deformation) and especially reflection-asymmetric shapes (octupole deformation), have also been used to interpret the data in certain limited regions. Systematics for the intrinsic excitations are presented and discussed in terms of these models. The newly emerging regions of the octupole-quadrupole deformation and superdeformation are also discussed.

For many years the effects of a static or dynamic electric field upon electronic motion in a molecule have been studied. These effects have been described in terms of multipolar electronic polarizabilities and higher-order hyperpolarizabilities. Much less attention, however, has been paid to the effects of an electric field upon vibrational and rotational motion. It is the aim of this review to consider, in some detail, these effects. As in the electronic work, they too will be described in terms of polarizabilities and hyperpolarizabilities (the latter being particularly important for the study of nonlinear optics). The theory will be developed so as to bring together the different methods that have been used in various calculations. Examples drawn from the recent literature will be discussed and it will be seen that in many cases vibrational and rotational changes with an electric field are as important as electronic ones, if not more so. Examples of experimental work relevant to this review include research on the Kerr effect, electric-field-induced second-harmonic generation, and third-harmonic generation.

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Attempts to find a plausible naturalistic explanation of the origin of the solar system began about 350 years ago but have not yet been quantitatively successful. The period 1956- 1985 includes the first phase of intensive space research; new results from lunar and planetary exploration might be expected to have played a major role in the development of ideas about lunar and planetary formation. While this is indeed the case for theories of the origin of the moon (selenogony), it was not true for the solar system in general, where ground-based observations (including meteorite studies) were frequently more decisive. During this period most theorists accepted a monistic scenario: the collapse of a gas-dust cloud to form the sun with surrounding disk, and condensation of that disk to form planets, were seen as part of a single process. Theorists differed on how to explain the distribution of angular momentum between sun and planets, on whether planets formed directly by condensation of gaseous protoplanets or by accretion of solid planetesimals, on whether the "solar nebula" was ever hot and turbulent enough to vaporize and completely mix its components, and on whether an external cause such as a supernova explosion "triggered" the initial collapse of the cloud. Only in selenogony was a tentative consensus reached on a single working hypothesis with quantitative results.

The authors review the theory of semiconductor superlattice electronic structure. First a survey of theoretical methods is presented. These methods can be divided into two general classes: the supercell approach in which the superlattice is viewed as a material with a large unit cell, and the boundary-condition approach in which bulk wave functions in the constituent semiconductors are matched at the superlattice interfaces. Supercell approaches are essentially the same as conventional band-structure methods. They can only be applied to thin-layer superlattices because of numerical cost. The authors discuss problems of interface matching that occur in various boundary-condition methods and relate these methods to each other. A particular boundary-condition method is used to discuss the electronic structure of various III-V semiconductor superlattices. Emphasis is placed on discussing the qualitatively different behavior that can arise because of different energy-band lineups, strain conditions, and growth orientations. The authors compare the results of three commonly used boundary-condition methods and find generally good agreement.

In narrow-band systems electrons can interact with each other via a short-range nonretarded attractive potential. The origin of such an effective local attraction can be polaronic or it can be due to a coupling between electrons and excitons or plasmons. It can also result from purely chemical (electronic) mechanisms, especially in compounds with elements favoring disproportionation of valent states. These mechanisms are discussed and an exhaustive list of materials in which such local electron pairing occurs is given. The authors review the thermodynamic and electromagnetic properties of such systems in several limiting scenarios: (i) Systems with on-site pairing which can be described by the extended negative-$U$ Hubbard model. The strong-attraction limit of this model, at which it reduces to a system of tightly bound electron pairs (bipolarons) on a lattice, is extensively discussed. These electron pairs behaving as hard-core charged bosons can exhibit a superconducting state analogous to that of superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ II. The change-over from weak-attraction BCS-like superconductivity to the superfluidity of charged hard-core bosons is examined. (ii) Systems with intersite pairing described by an extended Hubbard model with $U>\mathrm{}0\mathrm{}$ and nearest-neighbor attraction and/or nearest-neighbor spin exchange as well as correlated hopping. (iii) A mixture of local pairs and itinerant electrons interacting via a charge-exchange mechanism giving rise to a mutually induced superconductivity in both subsystems. The authors discuss to what extent the picture of local pairing, and in particular superfluidity of hard-core charged bosons on a lattice, can be an explanation for the superconducting and normal-state properties of the high-$T_c$ oxides: doped BaBi${\mathrm{O}}_3$ and the cuprates.

The authors review the methods used before the operation of the high energy Stanford and CERN $e^{+}{e}^{-}$ colliders to determine the number of neutrino species $N_{\nu }$, or an upper limit on this number, within the framework of the Standard Model of light stable neutrinos interacting according to the SU(2)×U(1) universal couplings. The astrophysical limit based on the neutrino burst from supernova 1987A is discussed first, followed by a discussion of the cosmological constraint based on the observed He/H abundance ratio. Finally, the particle physics methods based on single-photon production in $e^{+}{e}^{-}$ collisions, on the production of monojets in $p\overline{p}$ collisions, and on the determination of $N_{\nu }$ from the ratio of the $W\to l\overline{\nu }$ to $Z^0\to l\overline{l}$ partial cross sections in $p\overline{p}$ collisions are discussed. The various sources of uncertainty and the experimental backgrounds are presented, as well as an idea of what may be expected on this subject in the future. There is a remarkable agreement between the various methods, with central values for $N_{\nu }$ between 2 and 3 and with upper limits $N_{\nu }<6\mathrm{}$. Combining all determinations, the authors obtain a central value $N_{\nu }={2.1}_{-0.4\mathrm{}}^{+0.6\mathrm{}}$ for $m_{\mathrm{top}}=50\mathrm{}$ GeV/${\mathit{c}}^2$ and $N_{\nu }={2.0}_{-0.4\mathrm{}}^{+0.6\mathrm{}}$ if $m_{\mathrm{top}}\ge m_W$. The results are perfectly compatible with the a priori knowledge that at least three families of neutrinos should exist. The observed consistency between this a priori knowledge, the laboratory determinations of $N_{\nu }$, and determinations from SN 1987A and cosmology represent an astounding success for the Standard Model and for the current descriptions of stellar collapse and the Big-Bang primordial nucleosynthesis. These results, however, severely limit the number of additional families. Although the consistency is significantly worse, four families still provide a reasonable fit. In the framework of the Standard Model, a fifth light neutrino is, however, unlikely. A noted added in proof summarizes the results recently obtained at the Fermilab $\overline{p}p$ and the Stanford and CERN $e^{+}{e}^{-}$ colliders which confirm these conclusions.

Thermodynamic states are assumed to be characterized by densities. Recent ergodic-theory results on the evolution of densities are used to give a unified treatment of the origin of classical nonequilibrium thermodynamic behavior. Asymptotic periodicity is sufficient for the existence of at least one state of (metastable) thermodynamic equilibrium and for the evolution of the entropy to a relative maximum that depends on the way the system is prepared. Ergodicity is necessary and sufficient for a unique state of thermodynamic equilibrium to exist. Exactness, a property of chaotic semidynamical (irreversible) systems, is necessary and sufficient for the global evolution of the entropy to its unique maximum for all initial states. Since all of the laws of physics are formulated as (reversible) dynamical systems, it is unclear why entropy is observed to approach a maximum. Setting aside the possibility that all of the laws of physics are incorrectly formulated, it is demonstrated that either observation of a subset of the complete dynamics (trivial coarse graining) or interactions with an external heat bath (addition of noise) may induce exactness with a consequent evolution of entropy to a maximal state.

Nonzero neutrino masses would provide new, unique information on particle physics beyond the standard model. Neutrino flavor oscillations provide the most sensitive method for directly testing for small neutrino masses. When the oscillations occur in matter, a resonance can occur that dramatically enhances the flavor mixing and can lead to conversion from one neutrino flavor to another. This is an attractive solution to the long-standing "solar neutrino problem." The phenomenon of neutrino oscillations in matter is reviewed. Analytic descriptions of the neutrino flavor survival probability in matter are considered in detail. A discussion is given for applications of neutrino oscillations in matter for the sun, Earth, supernovae, and the early universe.

The old question of "How long does it take to tunnel through a barrier?" has acquired new urgency with the advent of techniques for the fabrication of semiconductor structures in the nanometer range. For the restricted problem of tunneling in a scattering configuration, a coherent picture is now emerging. The dwell time ${\tau }_D$ has the status of an exact statement of the time spent in a region of space, averaged over all incoming particles. The phase times ${\tau }_T^{\varphi }$ and ${\tau }_R^{\varphi }$ are defined separately for transmitted and reflected particles. They are asymptotic statements on completed scattering events and include self-interference delays as well as the time spent in the barrier. Consequently, neither the dwell time nor the phase times can answer the question of how much time a transmitted (alternatively, reflected) particle spent in the barrier region. Our discussion of this question relies on a few simple criteria: (1) The average duration of a physical process must be real. (2) Since transmission and reflection are mutually exclusive events, the times ${\tau }_T$ and ${\tau }_R$ spent in the barrier region are, if they exist, conditional averages. Consequently, they must obey the identity ${\tau }_D=T{\tau }_T+R{\tau }_R$, where $T$ and $R$ are the transmission and reflection probabilities, respectively. The existence of this identity distinguishes tunneling in a scattering configuration from tunneling out of a metastable state. (3) Any proposed ${\tau }_T$ and ${\tau }_R$ must meet every requirement that can be constructed from ${\tau }_D$. On the basis of (2), the naively extrapolated phase times, as well as the Büttiker-Landauer time, must be rejected. The local Larmor times, as introduced by Baz', satisfy (2), but not every criterion of type (3). The local Larmor clock is therefore unreliable. The asymptotic Larmor clock shows the phase times, as it should. Finally, the inverse characteristic frequency of an oscillating barrier cannot always be defined. It is shown not to represent the duration of the tunneling process. This leaves the dwell time and the phase times as the only well-established times in this context. It also leaves open the question of the length of time a transmitted particle spends in the barrier region. It is not clear that a generally valid answer to this question exists.

A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given. Attention is focused on four classical nonlinear equations: the Korteweg-de Vries, nonlinear Schrödinger, sine-Gordon, and Landau-Lifshitz equations perturbed by various Hamiltonian and/or dissipative terms; a comprehensive list of physical applications of these perturbed equations is compiled. Systems of weakly coupled equations, which become exactly integrable when decoupled, are also considered in detail. Adiabatic and radiative effects in dynamics of one and several solitons (both simple and compound) are analyzed. Generalizations of the perturbation theory to quasi-one-dimensional and quantum (semiclassical) solitons, as well as to nonsoliton nonlinear wave packets, are also considered.

A scheme that reduces the calculations of ground-state properties of systems of interacting particles exactly to the solution of single-particle Hartree-type equations has obvious advantages. It is not surprising, then, that the density functional formalism, which provides a way of doing this, has received much attention in the past two decades. The quality of the energy surfaces calculated using a simple local-density approximation for exchange and correlation exceeds by far the original expectations. In this work, the authors survey the formalism and some of its applications (in particular to atoms and small molecules) and discuss the reasons for the successes and failures of the local-density approximation and some of its modifications.

This article reviews the commonly used methods for representing electronic potential-energy surfaces for small molecules and simple chemical reactions in terms of globally defined analytical functions. Four classes of methods are discussed: spline fitting methods, semiempirical methods, many-body expansion methods, and methods that represent global surfaces based on information determined along reaction paths. The application of these methods is examined in detail for four triatomic systems and one four-atom system: $\mathrm{O}\left({}_{}{}^3{}_{}{}^{}P\right)+{\mathrm{H}}_2$, Cl+HCl, H+CO, $\mathrm{O}\left({}_{}{}^1{}_{}{}^{}D\right)+{\mathrm{H}}_2\to {\mathrm{H}}_2\mathrm{O}\to \mathrm{OH}+\mathrm{H}$, and H+C${\mathrm{O}}_2$→OH+CO. These examples illustrate both the art and the pitfalls of representing surfaces. In addition, the consequences of different potential surface representations for the dynamics of collisions on these surfaces are discussed at length.

The thermal boundary resistance present at interfaces between helium and solids (Kapitza resistance) and the thermal boundary resistance at interfaces between two solids are discussed for temperatures above 0.1 K. The apparent qualitative differences in the behavior of the boundary resistance at these two types of interfaces can be understood within the context of two limiting models of the boundary resistance, the acoustic mismatch model, which assumes no scattering, and the diffuse mismatch model, which assumes that all phonons incident on the interface will scatter. If the acoustic impedances of the two media in contact are very different, as is the case for helium (liquid or solid) in contact with a solid, then phonon scattering at the interface will reduce the boundary resistance. In the limiting case of diffuse mismatch, this reduction is typically over 2 orders of magnitude. Phonons are very sensitive to surface defects, and therefore the Kapitza resistance is very sensitive to the condition of the interface. For typical solid-solid interfaces, at which the acoustic impedances are less different, the influence of diffuse scattering is relatively small; even for the two limiting cases of acoustic mismatch and diffuse mismatch the predicted boundary resistances differ by very little (≲ 30%). Consequently, the experimentally determined values are expected to be rather insensitive to the condition of the interface, in agreement with recent observations. Subsurface (bulk) disorder and imperfect physical contact between the solids play far more important roles and led to the irreproducibilities observed in the early measurements of the solid-solid thermal boundary resistance.

This article presents a general overview of the problems involved in the application of the quantum principle to a theory of gravitation. The ultraviolet divergences that appear in any perturbative computation are reviewed in some detail, and it is argued that it is unlikely that any theory based on local quantum fields could be consistent. This leads in a natural way to a supersymmetric theory of extended objects as the next candidate theory to study. An elementary introduction to superstrings closes the review, and some speculations about the most promising avenues of research are offered.