High-resolution measurements of x-ray decay from excited states in highly stripped oxygen ions have yielded many transitions not attributable to the Lyman series in hydrogenic oxygen or to the analogous heliumlike oxygen series. These additional transitions are explained by considering the "screening" effects associated with adding a $2s$, $2p$, or $3p$ "spectator" electron to configurations governing the single-electron decays in excited heliumlike and lithiumlike oxygen. By comparing Hartree-Fock calculations for term energies, many of the observed transitions can be assigned to the doubly excited configurations $2pnp$ and $1smsnp$ in the heliumlike and lithiumlike sequencies, respectively.

The continuity equations for electrons and the various negative- and positive-ion species of oxygen (${\mathrm{O}}^{-}$, $\mathrm{O}_2^{}{}_{}{}^{-}$, $\mathrm{O}_3^{}{}_{}{}^{-}$, $\mathrm{O}_4^{}{}_{}{}^{-}$) are solved with the initial conditions and boundary conditions characteristic of electron avalanches in uniform fields. Account is made of the gain and loss reactions such as electron attachment, detachment, charge transfer, and ion conversion, as encountered in avalanches in oxygen over a wide range of pressures, and field strength over pressures (1-100 torr and 30-100 V/cm torr, respectively). Account is also made of the collision history of the velocity-distribution function of the atomic negative ion ${\mathrm{O}}^{-}$. Solutions are given for two cases of interest: first, for a study of the ion component of avalanches, with a time resolution of ∼ 0.5 μsec; second, for a study of the electron component, with a time resolution of ∼ 10 nsec. In this way measurements of "effective" rates of attachment, detachment, charge-transfer, and ion-conversion reactions can be given an interpretation in terms of the basic processes involved.

Energy shifts and thick target yields of the copper $\mathrm{Ka}$ and $K\beta$ x-ray lines have been produced with alkali-metal ions of a few MeV. Sodium, potassium, rubidium, and cesium give shifts that are nearly independent of energy and atomic number and slightly larger than the shift that would be observed for one hole in the $L$ shell. The yields are very strongly dependent on both energy and atomic number. Coulomb excitation with a binding-energy correction reproduces the yields for lithium and sodium but fails by many orders of magnitude for potassium, rubidium, and cesium, suggesting electron promotion as the excitation mechanism.

It is known that a nonabsorbing one-dimensional semi-infinite random medium is totally reflecting. In connection with this, when the medium is subject to monochromatic excitation with an arbitrary frequency, well-localized "stochastic resonances" may appear, at which the wave amplitude can exceed any given value, with nonzero probability in such a way as to make the mean energy density inside the medium infinite. The problem is considered both for continuous media (e.g., a random elastic bar) and discrete media (e.g., disordered linear atomic chains).

In ordinary fluids the angular time correlations between even irreducible symmetric tensors of rank $2l(l\ge 1)$ decrease at large times like $t^{\frac{3}{2}-5l}$ and $t^{-5l}$, according to whether one considers "bulk" or "self"-correlation. This slow decay is a new manifestation of the hydrodynamical phenomena that govern the asymptotic behavior of autocorrelation functions.

Laplace and Fourier transforms of the distribution function for orientation, and related correlation functions, for a spherical body undergoing rotational Brownian motion are calculated by solving partial differential equations governing their behavior. Also calculated is the Fourier transform of a correlation function involving both the orientation and angular velocity of a spherical body, which occurs in the theory of spin relaxation by spin-rotational interactions. The solutions obtained are the first few terms of infinite series, which appear to converge rapidly if the frictional retarding torque acting on the body is sufficiently large. The first term in each series is identical to results obtained by use of a rotational diffusion equation. From experimental values of the correlation times for reorientation of fairly small molecules in liquids, it is inferred that the solutions obtained here are probably applicable, and that the higher-order terms calculated here may be important.

We present here a simple calculation of electron-bubble radii in He Ii under pressure. An approximation for the two-particle density function is used which is identical to a scheme employed earlier for calculating the surface tension of ${\mathrm{He}}^4$ and surface states of ${\mathrm{He}}^3$ in solutions. There are no adjustable parameters, and the agreement with experiment is excellent. The approximation is proposed as a unifying model for all experiments involving helium surfaces.

The Dicke model of superradiance studied in a previous paper by Wang and Hioe is generalized (i) to include an interaction energy of the form $a^{†}{\sigma }^{+}+a{\sigma }^{-}$, where $a,a^{†}$ and ${\sigma }^{-},{\sigma }^{+}$ are the usual photon and spin operators, (ii) to include the spatial variation of the electromagnetic field, and (iii) to include the possibility of small classical motions of atoms from their equilibrium positions. The free energies of these generalized models are calculated exactly in the thermodynamic limit, and their phase-transition properties are presented.

The clustering of Ar to ${\mathrm{Li}}^{+}$ has been studied experimentally using a drift tube. The measurements were made at a gas temperature of 319 °K. The gas pressure was varied between 0.5 and 1.5 Torr, and $\frac{E}{N}$ was varied between 9 and 24 Td (1 Td = ${10}^{-17\mathrm{}}$ V ${\mathrm{cm}}^2$). Under these conditions, the major reaction is ${\mathrm{Li}}^{+}+\mathrm{Ar}+\mathrm{Ar}\rightleftarrows {\mathrm{Li}}^{+}\left(\mathrm{Ar}\right)+\mathrm{Ar}$. The rate coefficients and the mobility of ${\mathrm{Li}}^{+}$(Ar) were deduced by comparing measured arrival-time profiles of the ions with profiles generated by a three-dimensional, numerical drift-tube model. The predictions of this model are compared in some detail with those of two analytical, three-dimensional, drift-tube models. In all cases the agreement is satisfactory. We find that the association-rate coefficient $k_1$ is 1.8 × ${10}^{-31\mathrm{}}$ ${\mathrm{cm}}^6$/sec and remains constant over the range of $\frac{E}{N}$ used and that the collisional-dissociation-rate coeffiecient $k_{-1\mathrm{}}$ increases from 2.0 × ${10}^{-13\mathrm{}}$ ${\mathrm{cm}}^3$/sec at $\frac{E}{N}=9\mathrm{} \mathrm{Td}\mathrm{to} 3.2\times {10}^{-13\mathrm{}}$ ${\mathrm{cm}}^3$/sec at $\frac{E}{N}=24\mathrm{}$ Td. In addition, we find the reduced mobility of ${\mathrm{Li}}^{+}$(Ar) in Ar to be 2.0 ${\mathrm{cm}}^2$/V sec, measured at 319 °K. The amounts by which these values may be in error were determined as ± 24% for the association-rate coefficients, ± 22% for the collisional-dissociation-rate coefficients, and ± 16% for the mobility of ${\mathrm{Li}}^{+}$(Ar). The mobilities of ${\mathrm{K}}^{+}$ in ${\mathrm{N}}_2$ and of ${\mathrm{K}}^{+}$ in Ar were determined during checks for systematic errors in the new drift tube. The mobility of ${\mathrm{Li}}^{+}$ in Ar was also determined in the course of this work. Both random and systematic errors were considered in the determination of the standard deviations of these measurements. These reduced mobilities and their standard deviations are: ${\mathrm{K}}^{+}$ in ${\mathrm{N}}_2$, 2.55 ± 0.06 ${\mathrm{cm}}^2$/V sec, measured at 310 °K; ${\mathrm{K}}^{+}$ in Ar, 2.73 ± 0.06 ${\mathrm{cm}}^2$/V sec, measured at 310 °K; and ${\mathrm{Li}}^{+}$ in Ar, 4.57 ± 0.12 ${\mathrm{cm}}^2$/V sec, measured at 319 °K.

The density of methoxybenzylidene butylaniline has been measured for a 10°C temperature range about the nematic-isotropic transition temperature. The coefficient of expansion has been fitted to an equation of the form $\beta ={\beta }_0+Z{|T-T_c|}^{-G}$ both above and below the transition point. The values of $T_c$ found from above and below are interpreted as virtual transition temperatures. The power law is taken as evidence that fluctuations play a measurable role in determining the expansion coefficients.

Recent experimental shock-compression work on solid argon is analyzed to obtain an intermolecular potential. The intermolecular potential is used to compute the high-pressure melting curve. The author predicts that the shock-compression curve crosses the melting line at 50 kbar and just under 700 °K, but that the inherent experimental error is too large for the transition to be seen as a break in the $\mathrm{PV}$ or $U_s{U}_p$ curves.

The polarization of bremsstrahlung from electron-ion collisions is calculated for magnetically confined electrons in the Born approximation. The polarization at high temperature is found to be of opposite sign to the low-temperature result and of smaller magnitude.

Correlation-function expressions are derived for the linearized Burnett and super-Burnett-coefficients for a fluid. The results involve higher spatial moments of correlation functions such as occur in the Green-Kubo theory for the Navier-Stokes transport coefficients. The investigation is stimulated primarily by indications that the Burnett coefficients might actually be infinite, but this problem is not discussed here.

The propagation properties of a single picosecond pulse produced by a mode-locked Nd:glass laser are studied in several liquids by observing the optically induced birefringence and the changes in spectral content of the self-focused light. The spatial properties of the focal volume are studied using multiple second-harmonic (SH) probing beams which measure the induced birefringence from a single infrared pulse sequentially in many parts of the cell. These data allow a direct determination of the velocity of propagation of the high-intensity region. The formation and disappearance of high-intensity regions is also observed, as well as propagation lengths greater than 4 cm on a single pulse for the case of C${\mathrm{S}}_2$. Using temporally compressed SH pulses, we find that the induced birefringence is due primarily to the orientational Kerr effect in materials such as nitrobenzene and toluene. Results in C${\mathrm{S}}_2$ are also consistent with this mechanism and indicate that the duration of the infrared pulse in the focal volume is less than the duration of the incident pulse. The propagation properties of filaments which pass through a 1-mm glass slide are investigated, and the properties of beams which are focused within the cell by an external lens are studied. The occasional appearance of multiple filaments on some laser shots at different temporal positions in the pulse indicates the possibility of using this technique for beam wave-front diagnostics. The experimental spectra of the self-focused light were studied using beams of picosecond pulses in a number of materials, and they are all similar, showing a symmetric broadening about the laser frequency. The spectra are modulated with an interference structure which is characteristic of a phase-modulated pulse. The effective duration of the modulated pulse is consistent with the duration of the pulse in the high-intensity region as determined from the propagation data. Most of the results of the propagation and frequency-broadening experiments can be explained with a transient theory of self-focusing which results from numerical integration of the self-focusing equations with a relaxing nonlinear index. The theory, however, predicts an asymmetric frequency broadening, and therefore additional mechanisms are discussed which may account for the large amount of upshifted light which was observed.

It is known that the hard-sphere gas exhibits strongly stochastic properties. For it, most initially close phase-space-trajectory pairs separate exponentially with time, and Sinai has used this so-called $C$-system behavior to develop a rigorous proof of ergodicity and mixing in the hard-sphere gas. In this paper we numerically integrate the equations of motion for a Lennard-Jones gas, which has attractive as well as repulsive interparticle forces, and we demonstrate that this exponential separation of initially close trajectory pairs persists even in the presence of attractive forces over a fairly wide range of particle densities. In our calculations, this range extends from the dilute-gas region up to densities at which three-body and four-body collisions become significant. Moreover for this density range, we show that an expression for the trajectory-pair rate of exponential separation, rather crudely derived for a hard-sphere gas, fits the empirical Lennard-Jones data quite nicely. Our evidence thus indicates that a gas system with attractive forces can exhibit $C$-system behavior similar to that of the hard-sphere gas. Finally we point out that the exponential separation of trajectories as empirically observed here involves an unusual type of correlated collision sequence.

Extensive inelastic-neutron-scattering experiments have been carried out on single crystals of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ in the bcc and low-density hcp phases. The results of our present and previous measurements have established the over-all characteristics of the scattering function $S(Q,\omega )$ for a wide range of energy and momentum transfers. The general features of the scattering function can be described as showing well-defined, sharp, elementary excitations at energy transfers $\hslash \omega <1.0\mathrm{}$ meV, and single-particle excitations for scattering vectors $\mathrm{}\left|Q\right|>3.0\mathrm{}$ ${\mathrm{\AA }}^{-1\mathrm{}}$ and $\hslash \omega >5.0\mathrm{}$ meV. The changes in the structure of the scattering function to accommodate these two different excitations occur in an ill-defined "transition" region, wherein the spectral response of the sharp excitation gradually becomes more distorted and merges with the multiexcitation response at the higher energies. In the low-energy regime, the scattering cross section for sharp excitations observed in the [$0\overline{1}1$] zone for the bcc phase is found to be unusual in that the effective Debye-Waller factor constructed from the data via the Ambegaokar, Conway, and Baym sum rule displays oscillations. The data for the lowest transverse branch $T_1$, however, do not show oscillations. In our previous work, this anomalous cross section appeared to be associated with an energy transfer of ∼ 1.4 meV, and a concomittant distortion in the Gaussian-like spectral response. The present results show that this is not the case. The unexpected behavior of the cross section was observed for sharp, symmetric excitations whose energy distribution has no extraordinary contribution near 1.4 meV. The measurements in the hcp phase show that these anomalous features of the scattering function are similar in both phases. The high-resolution measurements on the sharp excitations at small wave vector were used to construct a self-consistent set of elastic constants. These elastic constants are considerably more accurate than those from previous measurements. They suggest that the dispersion of the $T_1$ mode in the [011] direction need not be anomalous, in contrast to recent theoretical results. Finally, at high energy and momentum transfers, single-particle excitations are observed in both phases. Their dispersion relation is isotropic, and is the same in both phases within the accuracy of the measurements. The profiles of these excitations are very similar to those recently observed by Woods and Cowley in the superfluid phase.

The dynamic form factor $S(k,\omega )$ for liquid ${\mathrm{He}}^4$ at $T=0\mathrm{}$°K is calculated by a straightforward perturbation technique, and the results are displayed graphically for $k\le 6.0$ ${\mathrm{\AA }}^{-1\mathrm{}}$. The dominant features of the single-excitation and multiphoton contributions are discussed and compared with those observed in neutron-scattering experiments. The reliability and accuracy of the computed form factor is studied with the aid of well-known sum rules and comparison with experimental data.

In this paper we attempt to unify the theoretical treatment of a variety of cooperative interaction phenomena occurring in resonantly interacting radiation-matter systems. The system consists of a multimode optical field interacting with $N$ resonant atoms in a region of large spatial extent. The field and atoms may or may not be coupled to reservoirs. We derive a set of coupled master equations for the reduced matter and radiation density operators of the composite system in the first Born approximation, using a method due to Bogoliubov. These equations describe within a single framework phenomena involving superradiant emission and coherent propagation. In the first place, our theory generalizes existing formulations of the superradiance problem by treating the radiation field as part of the dynamical system rather than as a reservoir. Hence, our equations describe stimulated interaction effects, which are entirely absent in the usual superradiance master equations. Moreover, our equations contain terms describing the self-consistent interaction of an average field with a set of polarizable atoms, as in the coupled Bloch and Maxwell equations of the theory of coherent pulse propagation, and enable us to describe fluctuations, arising from incoherent emission, about the average behavior described by the Bloch-Maxwell theory. We solve formally the dynamical equation for the radiation density operator and obtain a displaced multivariate Gaussian distribution centered on the solution of the Maxwell wave equation, which has as parameters the first and second moments of the matter density operator. Then we derive and discuss the field and atomic-moment equations, both in the wave-number and position representations. The moment equations enable us to show the relationship between descriptions of superradiant damping based on the Bloch equations and those based on the existence of correlated atomic states, including both as special cases. We introduce reduced matter moments, which allow us to separate incoherent- and coherent-emission effects and enable us to discuss the question of the closure of the moment equations. Then we consider cases where the behavior of the atomic system is dominated by a strong coherent pump wave or where the atoms are in a symmetric correlated state and show how the slowly-varying-envelope approximation simplifies the radiation-moment equations in these two cases. In the former case, we obtain an approximate theory which consists, in lowest order, of the Bloch-Maxwell equations with radiative-damping terms, plus an equation describing the time development of the correlation function for field fluctuations. Returning to our general formalism, we relate the usual parameters of the theory of radiative decay and of resonant interactions to the more general quantities appearing in our spatial analysis. Finally, we discuss the importance of the stimulated-emission terms appearing in our theory and describe several extensions, including the introduction of inhomogeneous broadening.

Absolute measurements have been made of total cross sections for the scattering of electrons by sodium over the electron energy range 0.5-50 eV; some new data for potassium at 10, 20, and 50 eV are also presented. The measured cross sections for sodium range from 352 ${\mathrm{\AA }}^2$ at 0.5 eV, to 57 ${\mathrm{\AA }}^2$ at 50 eV, with over-all uncertainty of 13%. At very low energies, the 250-meV energy spread in our experiment introduces additional uncertainty. As with earlier measurements from this laboratory using potassium, we find excellent agreement with close-coupling calculations.

The Weyl transform in relativistic quantum dynamics is formulated in terms of the coordinate eigenfunction and momentum eigenfunction that correspond to the ordinary and magnetic Wannier function and ordinary and magnetic Bloch function, respectively, in solid-state theory. Using this method, a rigorous but fairly simple dynamical derivation of the magnetic susceptibility of a relativistic Dirac electron gas (which includes the effect of an anomalous magnetic moment) is given. The terms coming from the Landau-Peierls formula and Pauli spin paramagnetism are cancelled by the terms arising from the second-order effect of spin and by an unfamiliar dynamical effect due to the inherent spread of the electron (approximately equal to its Compton wavelength). The very simple result reduces to the sum of Landau orbital diamagnetism and Pauli spin paramagnetism in the nonrelativistic limit. It is suggested that different physical processes dominate in the very-low-electron-density limit as compared to that in the very-high-electron-density limit. Striking similarities with the magnetic properties of the electrons in bismuth crystal are pointed out.