We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential $V\left(r\right)=\frac{g}{r^2}$ with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant $g$, as a determinant of order $N\times N$. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.

A density-matrix analysis is given of some properties of photon echoes arising from nearly degenerate ground and/or excited states. Modulations of the echo intensity as a function of pulse separation time are predicted as well as more complicated effects near level crossings. In particular, the analysis is applied to the 2.06-kG level crossing in the ground state of ruby.

The lifetime of electrons trapped at the free surface of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ has been determined from the temperature dependence of the current flow in the liquid. These lifetimes are in substantial agreement with the values determined directly for the liquid-gas interface in ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. New results are presented for the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$-${}_{}{}^3{}_{}{}^{}\mathrm{He}$ interface which give lifetimes as long as ${10}^5$ sec.

The time dependences of the ion wall currents in a 76 °K helium afterglow were measured with a quadrupole mass-filter system. The species observed were ${\mathrm{He}}^{+}$, ${\mathrm{He}}_2^{+}$, ${\mathrm{He}}_3^{+}$, ${\mathrm{He}}_4^{+}$, and metastable atoms. The results of the present work are a zero-field mobility, ${\mu }_0\left({\mathrm{He}}^{+}\right)$, of 14.4 ± 0.3 ${\mathrm{cm}}^2$${\mathrm{V}}^{-1\mathrm{}}$${\sec }^{-1\mathrm{}}$, and a three-body conversion frequency of ${\mathrm{He}}^{+}$ into ${\mathrm{He}}_2^{+}$ of $(1.44\mathrm{}\pm 0.14)\times {10}^{-31\mathrm{}}\times N^2$ ${\sec }^{-1\mathrm{}}$, where $N$ is the density in atoms/${\mathrm{cm}}^3$. The diffusion coefficient that describes the late-time behavior of ${\mathrm{He}}_3^{+}$ was found to be 95.3 ± 8.6 ${\mathrm{cm}}^2$${\sec }^{-1\mathrm{}}$ Torr. The metastable helium atoms at 76 °K were found to have a diffusion coefficient given by $D_m{p}_0=146\mathrm{}$ ${\mathrm{cm}}^2$${\sec }^{-1\mathrm{}}$ Torr. Evidence is presented for the presence of helium negative ions in the 76 °K helium afterglow.

In recent years the utilization of "optical mixing" spectroscopic techniques in laser light scattering experiments has proved to be a successful method for the investigation of the dynamical properties of physical systems. By illuminating a photosensitive detector with the scattered light and measuring the spectrum or autocorrelation function of the resulting photocurrent, one can, in general, obtain the ensemble-average time dependence of specific collective excitations within the scattering medium. We present here a detailed quantitative analysis of the statistical errors inherent in such measurements due to the stochastic nature of both the scattering and photoemission processes. We determine the statistical errors on the optical-intensity correlation function as measured by two photocounting digital correlator models and on the intensity spectrum as measured by a "self-beat" optical mixing spectrometer. From these errors and a generalized least-mean-squares fitting procedure we calculate the uncertainty on the measured correlation time (linewidth) for the case of a Gaussian optical field with an exponential intensity correlation function (a Lorentzian spectrum). Scaling relationships are given which permit our numerical results to be applied to an arbitrary set of experimental parameters.

The Brandow boson linked-cluster expansion has been used to redo the Foldy calculation of the Bogoliubov theory for a charged boson gas. The particle-number conservation is treated exactly. The ring diagrams have been summed using the same techniques as used by Gell-Mann and Brueckner for the electron gas. The next-most-divergent diagrams (twisted rings) have also been summed. These diagrams contain a renormalized hole line and thus show explicitly the dynamics of the condensate which is treated only in an average way in the Bogoliubov theory. The results for the ground-state energy and the condensation fraction agree exactly with Foldy in the high-density limit.

The asymptotic time behavior ($\sim c{t}^{-\frac{d}{2}}$, where $d$ is the dimensionality of the system) of the velocity autocorrelation function and of the kinetic parts of the correlation functions for the shear viscosity and the heat conductivity is derived on the basis of a local equilibrium assumption and the linearized Navier-Stokes equations. The coefficients $c$ are expressed in terms of the transport coefficients and thermodynamic quantities. The physical mechanism responsible for the long-time tail is indicated, and the connections between the present work and investigations based on molecular dynamics and on kinetic theory are discussed.

The Mori memory-function formalism is used to derive systematically a hierarchy of approximations relating the dynamic structure factor $S(k,\omega )$ of a dense classical fluid to its self-part $S_s(k,\omega )$. The formalism is applied to a column vector of dynamical variables whose components include the self and distinct parts, ${\rho }_s$ and ${\rho }_d$, of the fluctuating density plus $N$ time derivatives of these variables. Increasing $N$ is analogous to a continued-fraction expansion of the memory functions, and builds in more short-time information about the correlation functions. In this manner approximations generated earlier by Vineyard, by Kerr, and by Ortoleva and Nelkin are concisely stated, and a new approximation which gives the first six frequency moments of $S(k,\omega )$ correctly is obtained. Since the sixth frequency moment of $S(k,\omega )$ is not known from molecular-dynamics calculations, it is used as a parameter to fit these calculations. Agreement for $S(k,\omega )$ to within a few percent is obtained for several $k$ values and two thermo-dynamic states. The deduced value of the sixth frequency moment has a reasonable dependence on $k$, and may give useful information about the three-particle static correlation function.

Optical third-harmonic coefficients ${\alpha }_{\mathrm{zzzz}}\left(3\mathrm{}\nu \right)$ for the inert gases have been calculated taking the $\mathrm{LS}$ coupling scheme for helium and $\mathrm{Jj}$ and $\mathrm{jl}$ coupling schemes for neon, argon, and krypton. Our calculated values agree reasonably well with the experimental values of set (i) of Ward and New.

The electron spin resonance (ESR) of manganese nitrate aqueous solutions was studied over the concentration range 3.5 × ${10}^{-3\mathrm{}}$-8.25 mole/liter. ESR and viscosity data were obtained from 20 to 90 °C. The spectrum exhibited six well-resolved hyperfine components at low concentrations which transformed to a partially exchange-narrowed singlet at high Mn (II) concentrations. Collapse of the hyperfine structure (hfs) was analyzed from computer-synthesized spectra whose component linewidths and positions were obtained from Anderson's theory of exchange narrowing. It was found that exchange-narrowed spectra of Mn (II) ions were described by Anderson's case (b), in which transitions occur only between adjacent hyperfine components.

The instability of the electromagnetic linearly polarized mode propagating perpendicular to the magnetic field is studied for a system composed of two colliding plasma streams, in each of which both the electrons and the ions are streaming at the same velocity. Based on the linearized Vlasov-Maxwell equations and allowing for anisotropic temperatures, it is found that in the presence of streaming ions the instability can occur in very low-$\beta$ plasmas. The higher the values of $\beta$, the streaming velocity, and the temperature ratios $\frac{T_{\parallel }e}{T_{\perp }e}$ and $\frac{T_{\perp }e}{T_{\perp }i}$, the more the plasma is susceptible to the electromagnetic instability. The ratio $\frac{T_{\parallel }i}{T_{\parallel }e}$ has negligible influence, except when the streaming velocity is smaller than the ion thermalspeed. It is found that for $\beta$ values of order unity, the streaming ions, while greatly enlarging the range of unstable wave numbers, contribute negligibly to the maximum growth rate, which is typically of the order of the electron cyclotron frequency. A comparison with the growth rates of the electrostatic two-stream instability is made.

Electrons drifting relative to ions across a magnetic field are found to drive a set of intermediate-frequency long-wavelength waves (${\omega }_{\mathrm{ci}}<\omega <{\omega }_{\mathrm{ce}}$) unstable for any temperature ratio $\frac{T_e}{T_i}$ One instability is caused by the coupling of a drift wave in a nonuniform plasma to either the ion plasma oscillation or a lower hybrid oscillation, depending on density and field strength. Another instability is a form of the two-stream instability, shown here to exist even for drift speeds less than electron thermal speeds. Growth rates are on the order of the ion plasma (or lower hybrid) frequency, depending on the density and field strength.

It is shown that the prediction of Garrett and McCumber that a light pulse's velocity may exceed the speed of light in a resonantly absorbing medium is due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light. It is also shown that in certain cases a pulse's maximum will propagate at the group velocity even when the pulse as a whole is distorted by dispersion.

It is shown that the classical form of the binary-collision expansion follows naturally from the multiple-scattering expansion for the $T$ matrix. The limits of applicability of this classical expansion are discussed.

A time-of-flight technique has been used to measure the radiative lifetime of the $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ state of helium. After passing from a cooled source, helium atoms are excited by a pulsed anti-parallel electron beam. The atomic beam then contains not only ground-state helium atoms but also metastable atoms in both the $2 {}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ and the $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ states. The metastable atoms are detected at two positions, 1.9 and 6.7 m from the electron gun, by Auger ejection of an electron from a copper target, the first of which is a 60% transmitting mesh. Ground-state helium atoms cannot cause an electron to be ejected and hence are not detected. The time-of-flight distribution for the metastable $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ state is separated from that of the $2 {}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ state by illuminating the beam with an rf-discharge helium lamp. The $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ state is quenched by resonant absorption of a 20 581 - ÅA photon, raising the atom to the $2 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ state, which then decays preferentially to the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ ground state; the $2 {}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ state remains unaffected because it is the ground state for the triplet system. The time-of-flight distribution for the $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ atoms is therefore obtained from the difference between the full beam and the quenched beam. A comparison of the number of $2 {}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ metastables within a given velocity interval at the two detectors determines the number which decay in flight and yields a value for the two-photon radiative lifetime. The value of the singlet lifetime for both ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ is 19.7 ± 1.0 msec, agreeing with the theoretical value of 19.5 msec, but disagreeing with the value 38 ± 8 msec measured by Pearl using a movable detector. The 1.0-msec error is an estimate of the remaining systematic errors in the experiment.

The ionization potentials, electron affinities, and term splittings of the lowest electronic configurations for first-row atoms for $Z$ from 5 to 10 have been computed. The method used is that applied previously to ground-state correlation energies. Variational calculations equivalent to solution of one- and two-particle Bethe-Goldstone equations are carried out, following approximate Hartree-Fock calculations for each state of each atom or ion considered. Computed quantities are in reasonably good agreement with experiment. Bethe-Goldstone equations as used here are defined in terms of individual orbital excitations. Calculations of three-particle effects are included in some cases, computed with less relative accuracy than the other results. The three-particle terms are not negligible. In particular, they substantially reduce the magnitude of computed electron affinities.

We have measured the $g$-factor anomaly of the free electron to an accuracy of ± 3 ppm (68% confidence level). The method (a refinement of that used by Wilkinson and Crane in 1963) is based on a direct observation of the difference between the spin precession and cyclotron frequencies of 100-keV electrons confined in a precisely measured magnetic mirror trap. An order of magnitude increase in precision over the Wilkinson-Crane experiment has been achieved by (a) a tenfold increase in the magnitude of the magnetic field (to 1 kG), and (b) a tenfold reduction in the relative depth of the magnetic trap (to 60 ppm). An extensive series of tests for systematic effects at the 1-ppm level has also been carried out. With $a=\frac{1}{2}\mathrm{}(g-2\mathrm{})\mathrm{}$, we find $a_{\mathrm{expt}}\left(e^{-}\right)=(1159657.7\mathrm{}\pm 3.5)\times {10}^{-9\mathrm{}}$. The difference between this result and the current theoretical expression for $a$, as calculated through second order in the fine-structure constant $\alpha$, is $a_{\mathrm{expt}}\left(e^{-}\right)-\left[0.5\mathrm{}\right(\frac{\alpha }{\pi })-0.32848\mathrm{}{\left(\frac{\alpha }{\pi }\right)}^2]=(1.68\mathrm{}\pm 0.33){\left(\frac{\alpha }{\pi }\right)}^3$. In the above, we have used ${\alpha }^{-1\mathrm{}}=137.03608\mathrm{}\pm 0.00026$, as recently recommended by Taylor, Parker, and Langenberg. The uncertainty in the coefficient is the rms sum of our experimental error and the uncertainty in ${\alpha }^{-1\mathrm{}}$. In this form, our result can be interpreted as an experimental determination of the sixthorder quantum-electrodynamic contribution to the anomaly. Current estimates of this quantity are $1.49{\left(\frac{\alpha }{\pi }\right)}^3$.