Within the framework of quasi equilibrium approximation, Josephson’s expression for current in a weakly coupled superconducting system is derived without the use of any specific microscopic model. It is based only on the existence of the off-diagonal long-range order in the two-particle reduced density matrix. Allowing for deviations from the quasi equilibrium approximation, generalised Josephson equations are obtained which include ohmic terms. The effect of relaxation and thermal fluctuations is examined in detail to emphasise the physical origin of various terms in the expression for current.

The problem of heating of a solid target to generate a nonequilibrium plasma by subnanosecond laser pulses is considered. For an appreciable absorption of energy from a Nd-glass laser, the critical density of the electrons in the plasma turns out to be 10²¹ cm⁻³. These electrons can be heated up to 10⁷ K or more by using pulses of about 10 picosecond duration and absorbed energy flux of the order of 10²¹ erg cm⁻² sec⁻¹. Starting from neutral atoms in a solid with a high atomic number, e.g., Z=26, for times in the picosecond regime the relevant rate equations are solved analytically to predict densities of the atoms at different ionization levels. It is shown that during such a short time the population density of the ions isoelectronic to neon builds up to a very large amount. This in turn leads to the population inversion in the 4s → 3p soft x-ray laser transition, via the electron-impact excitation of the 4s level of the isoelectronic neon ion. For the effective pumping times of the order of 5 picoseconds, a gain of the order of 10² db cm⁻¹ is predicted for the laser transition in Fe XVII, Co XVIII or Cu XX.

Simple approximation schemes are developed to calculate induced optical fields and local field corrections to the linear optical dielectric function in metals like aluminium and in insulators like germanium. In these calculations, the unperturbed electronic states in Ge are described within the framework of the bonding orbital approach, whereas the nearly-free-electron approximation is used for Al. As expected, explicit numerical calculations show that the contribution to secondary longitudinal induced fields is more appreciable in Ge. The second order susceptibility describing the non-linear mixing of an optical frequency with an x-ray frequency, which depends upon the magnitude of the microscopic induced optical charge density, is also calculated for these solids. For most relevant wavevectors of secondary optical fields, it is found to be of the order of 10⁻¹² esu in Ge and 10⁻¹⁴ esu in Al.

The problem of superconductivity in a metal-semiconductor system has been studied, using the dielectric formulation of superconductivity. The charge redistribution due to the quantum penetration of the metallic electrons to the semi-conductor side is approximated by a simple exponential function. The interface exciton modes are obtained within the framework of classical electrostatics, and their effect in modifying the effective electron-electron interaction near the interface is investigated. It is found that the strength of the excitonic term is small, and by itself, insufficient to lead to superconductivity. Nevertheless, it can alter the superconducting transition temperature of a metal, if it is already superconducting due to some other mechanism. This has been studied as a function of the various parameters entering in the problem.

Analytical methods to investigate the interaction of magnetic monopoles with known magnetic media have been developed. Trapping energies of monopoles inside ferro-magnetic or super onducting materials of size greater than about 10⁻⁶ cm are found to be of the order of several kiloelectron volts. These are two to three orders of magnitude higher than in paramagnetic materials. Thus if stable magnetic monopoles exist at all in the universe, they are perhaps trapped in these magnetic materials. The effect of the finite size of the magnetic bodies is taken into account explicitly in our calculations of the trapping energy.

An approximate method is developed for investigating the nature of interface exciton modes in a composite spatially dispersive medium. The method is general enough to be applicable to any composite system, in which each component is described by an arbitrary bulk dielectric functionε(q, ω). It is based on the extension of the usual electrostatic-image method of solving the Poisson’s equation, in the presence of an external point charge in the system. We have applied our general method to a composite system of a finite metal slab surrounded by a semiconductor on one side and the vacuum on the other side. Similarly, we have also considered the case of a metallic sphere of radiusR, surrounded by a semiconductor, with a spherical interface between them. With assumed spatially dispersive model dielectric functions for the bulk metal and the bulk semiconductor, the nature of the electron-electron interaction and the interface exciton modes in the metallic region are obtained in both the cases. For the relevant size of the metal large compared to the atomic dimensions over which the bulk dielectric functions are non-local due to the spatial dispersion, it is shown that one can obtain the interface exciton modes by first defining new effective dielectric functions for each of the media making the particular interface, and then using the usual expression which determines the modes in the non-dispersive case.

Approximate analytic solutions to the self similar equations of gas dynamics for a plasma, treated as an ideal gas with specific heat ratioγ = 5/3, are obtained for the implosion and subsequent reflection of various types of shock sequences in spherical and cylindrical geometries. This is based on the lowest-order polynomial approximation, in the reduced fluid velocity, for a suitable nonlinear function of the sound velocity and the fluid velocity. However, the method developed here is powerful enough to be extended analytically to higher order polynomial approximations, to obtain successive approximations to the exact self-similar solutions. Also obtained, for the first time, are exact asymptotic solutions, in analytic form, for the reflected shocks. Criteria are given that may enable one to make a choice between the two geometries for maximising compression or temperature of the gas. These solutions should be useful in the study of inertial confinement of a plasma.

An initial value problem is set up to describe propagation of a low-frequency wave-field interacting with two almost transparent wave-fields in a dispersive medium. With no linear loss, perfect phase-matching, and equal group velocities for the two high-frequency wave-packets, it is shown how the solution of the above problem can evolve to well-known soliton solutions of the sine-Gordon equation. Other attempts for solving the more general problem in which all the group velocities are different are also discussed.

The superconducting transition temperatureT_c for semimetals like bismuth has been calculated as a function of the densityn_c of the electron and hole charge carriers. A simplified two-band model for describing the longitudinal dielectric function for such a system has been used in our model calculation. We find that the attractive interaction responsible for the instability of the normal ground state comes not only from the exchange of lattice phonons, but also from the electronhole sound mode, provided the ratio of the averaged hole to electron mass,m_h/m_e ≠ 1. We have compared our theoretical values ofT_c(n_c) with experimental results for bismuth under hydrostatic pressure, and find reasonable agreement ifm_h/m_e is assumed to have a value which is only slightly larger than that at atmospheric pressures. A linear variation of the negative band gap as a function of pressure has been assumed for the sake of this comparison.

A general nonlinear response theory for the case of linear coupling of physical systems to arbitrary external fields is formulated for applications in different branches of physics. This is done within the framework of non-relativistic density matrix approach of quantum mechanics. Some simple properties of response functions and other related functions, which are introduced here for convenience, are studied to obtain suitable representations of the nonlinear response functions, including important sum-rules. As an example, the sum rule for the second-order response function is applied to electronic dipole nonlinearity at optical frequencies which includes both the Raman nonlinearity arising from perturbation to the electronic motion from external ionic displacement field and the usual optical sum, difference and harmonic generations. This immediately allows us to visualize a rigorous connection between these two types of non-linearities.

An explicit expression for the superconducting transition temperatureT_c in a paramagnetic material is derived, when the transition occurs just before a possible magnetic ordering. As first noted by Uspenskii, such a transition may arise from electronic mechanism itself, without the necessary role played by the usual phonon-exchange mechanism. The result is discussed in terms of some recent experimental observations on the binary alloy Y₉Co₇.

A general mathematical formulation is developed for calculating the effective electron-electron interaction in layered crystals like YBa₂Cu₃O_7−δ, and for finding the resulting superconducting transition temperatureT_c in such systems within the framework of the conventional BCS pairing arising from various possible excitations in the medium. This differs considerably from the usual case of an effective three-dimensional homogeneous system, and should be relevant in the calculation ofT_c for the new class of high-T_c perovskites in which oxygen deficiencies in Cu-O layers and their distribution in the crystal play a crucial role. The explicit form of the effective interactionV_jj(q_t,ω) in a given layerj in the unit cell of the crystal is found to be determined not only by the true polarization functionπ_j(q_t,ω) of that layer, but also of other layers. The exchange of electronic excitations of a nearby insulating layer by carriers in a conducting layer thus becomes possible to get highT_c, with or without the usual phonon exchange.

New oxide superconductors with layered structure are expected to have anisotropic energy gap in the generalized BCS pairing theory. The gap parameter $$2\Delta (\hat k)$$ can be quite different for $$\hat k$$ perpendicular to the plane of the layers as compared tok parallel to layer planes. Because of short coherence lengths ξ, quite small compared to the normal state carrier meanfree pathl, the effect of these anisotropies do not average out, as in many of the conventional superconductors. For a proper comparison of experimental results with the correct predictions of the pairing theory, a formulation is developed to obtain important physical quantities like specific heat and infrared absorption in the superconducting state of such anisotropic systems. This includes a brief account of the pairing theory generalized to layered crystals with arbitrary number of layers per unit cell, not necessarily equidistant. In an explicit model for the anisotropy of the gap parameter ink-space, with a simple form for the nonspherical Fermi-surface, it is shown that the low-temperature specific heat can have even a linear or a power-law temperature-dependence in the superconducting state. Even if the gap parameter does not vanish anywhere, its smeared-out exponential temperature-dependence may be difficult to be distinguished experimentally from a power-law behaviour. Similarly, it is shown that in the case of appreciable anisotropy, infrared absorption can take place much below the in-plane gap parameter $$2\Delta _t (\hat k_t )$$, wherek_t is the wavevector in the plane of the layers.

A theoretical framework for treating the effects of magnetic fieldH on the pairing theory of superconductivity is considered, where the field is taken in an arbitrary direction with respect to crystal axes. This is applicable to closed, as well as open normal state Fermi surface (FS), including simple layered metals. The orbital effects of the magnetic field are treated semiclassically while retaining the full anisotropic paramagnetic contribution. Explicit calculations are presented in the limits |H| → |H_c2(T)|,T ∼ 0 andT →T_c(|H|), |H| ∼ 0. Effects of weak nonmagnetic impurity scattering, without vertex corrections, have also been taken into account in a phenomenological way. The final results for the case of open FS and layered materials are found to differ considerably from those of the closed FS. For example, an important parameter,h(T=0)=|H_c2(0)|/[-Tδ|H_c2T|δT]_T{s0} for the case of a FS open ink_z-direction with thek_z-bandwidth, 4t₃, very small compared to the Fermi energy,E_F, is close to 0.5906, compared to 0.7273 for the closed FS, in the clean limit. Analytical results are given for the magnetic field dependence ofT_c and the temperature dependence of H_c2 for a model of layered superconductors with widely open FS. For a set of band structure parameters for YBa₂Cu₃O₇ used elsewhere, we find reasonable values for the upper critical fieldH_c2(0), the slope (dH_c2/dT)_Tc0, anisotropic coherence lengths ζ_i(T=0),i=x, y, z, and (dT_c/d|H|)_{|H| → 0}.