The first observation of optical rotary echoes, the optical analog of rotary spin echoes, is reported. Rotary echoes are produced in a quantum-mechanical two-level system which is driven resonantly (nutation effect) by a coherent field that suffers a sudden phase retardation. The initial nutation transient dephases, because of an inhomogeneity in either the driving field or the transition frequency of the sample, and then rephases to form an echo following the phase-shifting pulse. Hence, optical rotary and photon echoes are similar processes—the former being an interference in nutation and the latter an interference in free precession. Perturbative solutions of the Bloch equations are derived for the case where inhomogeneous dephasing arises from Doppler broadening and the Rabi frequency exceeds the homogeneous damping rate. Observations of the optical rotary echo in ${\mathrm{I}}_2$ vapor are facilitated by the technique of laser frequency switching, which generates precise phase shifts in the optical field. The measurements corroborate detailed theoretical predictions.

The effect of field fluctuations on the motion of a two-level atom in resonant laser radiation is investigated theoretically. The internal motion of the atom is treated quantum mechanically (optical Bloch equations); the translational motion, which is coupled to the internal motion, is treated classically (Ehrenfest's theorem); and the fluctuating radiation is treated as a classical electromagnetic wave with stochastic phase (phase-diffusion model). An expression for the radiation force and equations of motion for the ensemble-averaged Bloch vector are derived and used to calculate (i) the radiation force in a plane running wave, (ii) the steady-state radiation force in a general standing wave, and (iii) the dissipative force in a plane standing wave in the limit of strong phase fluctuations. In case (iii) it is found that the atomic velocity is damped when the radiation is tuned above resonance, which is opposite to the case of a weak coherent standing wave, in which damping occurs for radiation tuned below resonance. It is shown that fluctuations have a substantial effect on the radiation force only when the spectral width of the radiation approaches or exceeds the spontaneous emission rate.

The problem of dispersive optical bistability has so far been treated only in the mean-field approximation. A rigorous justification of the mean-field theory can only be obtained from exact solutions of the steadystate Maxwell-Bloch equations which retain the spatial dependence of the field. In this paper exact analytic solutions are presented for these equations. The authors demonstrate that the mean-field equation connecting the input and the output fields follows naturally from these solutions in the limits $T\to 0$, ${\delta }_F\to 0$, and $\alpha L\to 0$ for the mirror transmission coefficient, the detuning of the field from the cavity resonance, and the linear absorption, respectively, with $\frac{\alpha L}{2T}$ and $\frac{{\delta }_{F}L}{2cT}$ remaining finite. The results are illustrated with the help of graphs showing the output versus input intensity for different values of the relevant parameters. The effect of these parameters on the phase shift of the output field is also displayed.

The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\xi$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.

A general analysis is presented of free-electron lasers in which a static periodic magnetic pump field is scattered from a relativistic electron beam. The steady-state formulation of the problem is fully relativistic and contains beam thermal effects. Growth rates associated with the radiation field, efficiencies, and saturated-field amplitudes are derived for various modes of operation. Effects of space charge on the scattering process are included and shown to play a dominant role in certain situations. Scaling laws for the growth rates and efficiencies at a fixed radiation frequency as a function of the magnetic-pump amplitude are obtained. The shear in beam axial velocity due to self-fields is discussed, and various methods of reducing it are suggested. Finally, a detailed illustration of a far-infrared ($\lambda =2\mathrm{}$ μm) two-stage free-electron laser using a 3-MeV electron beam and a 2-cm-wavelength magnetic pump field is presented.

The development of lasers in which the active medium is a relativistic stream of free electrons has recently evoked much interest. The potential advantages of such free-electron lasers include, among other things, continuous frequency tunability, very high operating power, and high efficiency. The free-electron laser (FEL) is characterized by a pump field, for example, a spatially periodic magnetic field which scatters from a relativistic-electron beam. The scattered radiation has a wavelength much smaller than the pump wavelength, depending on the electron-beam energy. The authors present a general self-consistent nonlinear theory of the FEL process. The nonlinear formulation of the temporal steady-state FEL problem results in a set of coupled differential equations governing the spatial evolution of the amplitudes and wavelength of the radiation and space-charge fields. These equations are readily solved numerically since the amplitude and wavelength vary on a spatial scale which is comparable to a growth length of the output radiation. A number of numerical and analytical illustrations are presented, ranging from the optical to the submillimeter-wavelength regime. Our nonlinear formulation in the linear regime is compared with linear theory, and agreement is found to be excellent. Analytical expressions for the saturated efficiency and radiation amplitude are also shown to be in very good agreement with our nonlinear numerical solutions. Efficiency curves are obtained for both the optical and submillimeter FEL examples with fixed magnetic-pump parameters. It is shown that these intrinsic efficiencies can be greatly enhanced by appropriately contouring the magnetic-pump period. In the case of the optical FEL, the theoretical single-pass efficiency can be made greater than 20% by appropriately decreasing the pump period and increasing the pump magnetic field.

The central regions of the plasma-broadened Balmer lines ${\mathrm{H}}_{\alpha }$, ${\mathrm{H}}_{\beta }$, and ${\mathrm{H}}_{\gamma }$ have been measured in a wall-stabilized arc over an electron density range between approximately 4×${10}^{14}$ and 2×${10}^{16}$ ${\mathrm{cm}}^{-3\mathrm{}}$. The experimental profiles exhibit much less structure in the line core than predicted by theories based on the static-ion approximations. These discrepancies increase towards lower electron densities and lower Balmerseries members, e.g., the ${\mathrm{H}}_{\alpha }$ half-width as well as the central minimum of ${\mathrm{H}}_{\beta }$ deviate by about a factor of three from current hydrogen-broadening theories at low electron densities (${10}^{-5\mathrm{}}$ ${\mathrm{cm}}^{-3\mathrm{}}$). Consistent with earlier experiments, the central minimum of ${\mathrm{H}}_{\beta }$ was found to depend on the reduced mass of the radiatorion perturber system. Also, the temperature dependence of the experimental data suggests that the central ${\mathrm{H}}_{\beta }$ minimum depends approximately linearly on the relative mean velocity between radiator and perturber. Extrapolation of the ${\mathrm{H}}_{\beta }$ results to the static case yields good agreement with the quasistatic calculations. The theoretically predicted "shoulder" in the ${\mathrm{H}}_{\gamma }$ line shape is nearly absent in the experimental profile. Besides ion dynamics, fine-structure (spin) effects can account for a considerable portion of the observed discrepancies in the case of ${\mathrm{H}}_{\alpha }$, and possibly a small part in the case of ${\mathrm{H}}_{\beta }$ at low electron densities (≲${10}^{15}$ ${\mathrm{cm}}^{-3\mathrm{}}$).

The author discusses the production of a secondary soliton when the wave equation describing the soliton undergoes a small perturbation. Discussion is limited to the Korteweg-deVries equation, where it is found that, for an appropriate sign of the perturbation, soliton production always occurs. The methods of inverse scattering are used where this production can be clearly demonstrated. Previous authors have ignored the production of secondary solitons and consequently have arrived at erroneous conclusions regarding the conservation laws. Two examples of perturbations are discussed in some detail.

Relativistic Hartree-Fock equations are proposed which incorporate radiative effects. Their derivation is based on functional integration methods and a new substitution rule inferred from the Pauli principle.

Starting from the concept of the pseudospinodal, the authors derive an explicit expression for the scaling equation of state applicable to liquid-gas [and universality class (3,1)] systems. The advantages of this equation over other phenomenological equations are: it has no adjustable, unphysical parameters; it is valid in the metastable region; and it can simultaneously predict PVT data and the critical amplitude ratio for specific heat. The equation also fits PVT data as well as other phenomenological equations, is easy to use, and accurately predicts other critical amplitude ratios. It is argued that the success of this equation supports the usefulness of the pseudospinodal, which has been questioned by certain theoretical considerations.

The problem of the momentum of the electromagnetic field near a homogeneous dielectric plane surface is studied. In the particular case of evanescent waves, our calculations are consistent with the experimental result of S. Huard and C. Imbert which points out that, during an interaction between a moving atom and a surface wave of pulsation $\omega$, the exchanged momentum is greater than $\frac{\hslash \omega }{c}$. This work leads at first to a straightforward procedure for quantizing the momentum parallel to the diopter of the field. The authors proceed to investigate the form of the component perpendicular to the interface which is transmitted by the field to an atom during an interaction. (In the case of an evanescent wave, this problem involves the interpretation of the imaginary part of the wave vector.) A general theory shows that the presence of a medium widens and shifts the $\overrightarrow{\mathrm{k}}$ levels of the field. This result agrees with the uncertainty principle.

In this paper we report the results of extensive numerical calculations on the equation of state of the square-well fluid in the immediate vicinity of its critical point, with the objective being the precise determination of the critical exponents $\beta$ and $\delta$. These determinations were carried out using data generated by solving the Yvon-Born-Green equation for the pair distribution function (with superposition approximation) over an extended range of integration (∼50 intermolecular diameters). When taken together with the value of the critical exponent $\gamma$ reported previously by the authors ($\gamma =1.24\mathrm{}\pm 0.04$), the results obtained, $\beta =0.330\mathrm{}\pm 0.008$ and $\delta =4.4\mathrm{}\pm 0.2$, suggest strongly that the essential physics of the gas-liquid transition in the neighborhood of the critical point is described correctly by an integral equation derived from the classical Bogoliubov-Born-Green-Kirkwood-Yvon theory of distribution functions.

An absolute upper bound of Debye-Hückel form, $U\left(r_s\right)\frac{e^{-r}}{r}$, is derived for solutions to the nonlinear, spherical, radial Poisson-Boltzmann equation for the problem of an isolated charged sphere, dimensionless radius $r_s$, in an infinite volume of electrolyte. For $r_s\ll 1$, $U\left(r_s\right)\propto r_s\ln \left(\frac{1}{r_s}\right)$. Thus, as $r_s\to 0$, $U\left(r_s\right)\to 0$ and the shielding cloud around a point charge shrinks to zero radius. From any finite distance away, however small, the point charge is electrically invisible.

The virial expansion for the inverse structure factor of a gas is known to involve coefficients which are a function of $q$ and $T$, and whose $q\to 0$ limits are the usual pressure virial coefficients. The authors have made careful measurements of the structure factor of krypton gas using improved neutron-diffraction techniques at 15 densities between 0.25 and 6.19×${10}^{27}$ ${\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}/\mathrm{m}}^3$ at $T=297\mathrm{}$ K. The methods used are described, and the experimental errors are analyzed and found to be less than 1%. These data are compared to virial-expansion calculations based on the Barker et al. pair potential, to extract the first coefficient depending on the triplet potential. Some difference is found between this result and that predicted by the long-range triple-dipole (Axilrod-Teller) term.

The one-particle hydrogenic Green's function has been calculated for a partially ionized plasma consisting of hydrogen atoms, electrons, and protons at high temperatures. The theoretical method extends a previous publication and involves an evaluation of the mass operator in the Dyson equation to include proper self-energy parts to "all orders" in the screened interaction. This mass operator characterizes the effective micropotential felt by the atom in the plasma and determines all of the one-particle properties and some two-particle properties associated with the atomic subsystem. The first-order mass operator is nonzero only for exchange scattering, which leads to a frequency-independent exchange shift. This temperature- and density-dependent theory of the exchange shift replaces the usual semiphenomenological schemes based on the Slater-Kohn-Sham type of theory. The exchange-shifted Green's functions are used in evaluating the higher-order contributions. Computer calculations and the resolution of the poles of the Green's function lead to level shifts, widths, and spectral functions. These are calculated within both the second-order and the all-order theory. The second-order theory, which may be valid at sufficiently high densities and in turbulent plasmas, overemphasises the atom-plasmon coupling and shows new structures. The inclusion of contributions beyond second order removes these structures and produces a more "conventional" spectral-intensity function. The effects of center-of-mass motion on the level shifts and level profiles are investigated and the onset of plasma instabilities touched upon. These calculations make contact with the work on "plasma-polarization shifts" and provide an approach to $q$,$\omega$-dependent plasma microfields.

The influence of local fluctuations on a symmetry-breaking chemical instability is considered in terms of concepts developed in the theory of equilibrium phase transitions. It is shown that for infinite systems some predictions of the mean-field theory are qualitatively incorrect. Indeed, the fluctuations destroy any structure in one- and two-dimensional systems and some of them in three dimensions. The stabilizing effect of the finite dimensions on these structures is also discussed, and it is shown that in this case the fluctuations induce a weak first-order transition near the critical point of the mean-field theory.

Ideas developed in problems of high-field electron transport in semiconductors have been extended to the case of gases. It is shown that nonlinearities in the electric current may occur at drift velocities comparable with the velocity of sound in the gas. This is in agreement with the recent experiments of Gee and Freeman on methane. Drift velocities much larger than the velocity of sound are needed for the nonlinearities to appear if the collisions are inelastic.

It is shown that in multiphoton free-free transitions the mean energy absorption per electron is independent of the exact laser energy distribution in space and time. It depends only on the average laser flux, averaged in time over the count time of the electron detector and in space over the electron-target scattering volume. This result is applied in the analysis of recent experimental data.

The atomic $N$-electron wave function of the independent-pair approximation is defined in terms of orbital configurations of one-electron functions and symmetry-adapted pair functions in the form of partial-wave (PW) expansions. This form is convenient for an extensive use of irreducible tensor operators. The formulation was used to set up a variational-perturbation scheme for closed-shell atoms in the case of the Rayleigh-Schrödinger perturbation theory with the symmetric sum of Hartree-Fock operators for the zerothorder Hamiltonian (RS-HFPT). A detailed study of the second- and third-order correlation energies of Ne is made in order to analyze the nature of various correlation effects. All PW's up to $l^{\prime }$,$l^{\prime \prime }\le 9$ are considered. Particular attention is given to the problem of eliminating the radial basis saturation errors. The upper bound to the second-order energy is determined by means of 13 880 nonoptimized configurations to be -0.38638 a.u., which represents 99.3% of the "experimental" correlation energy. Extrapolation of the pair energies for $l^{\prime }$, $l^{\prime \prime }>9\mathrm{}$ results in a second-order energy of -0.3879 a.u. (99.7% of the total correlation energy). The PW expansions for the ($\mathrm{ns}$,$n^{\prime }s$) pairs are compared for He, Be, and Ne. Remarkable regularities are observed, indicating that the PW formulation represents a convenient tool for the investigations of correlation effects. The third-order energy obtained for a shorter expansion of the first-order wave function amounts to 0.00245 a.u. A discussion of the relative importance of the diagonal and off-diagonal contributions is presented, and detailed comparisons with the results of many other methods are made. It turned out that the RS-HFPT approach in the present formulation has several advantages over other perturbation methods.

Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈L_z〉=-\hslash$ and $〈M_z〉=-\left|e\right|\mathrm{}\frac{\hslash }{2\mathrm{mc}}$, where the orientation of the $z$ axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.