The phase shift for low-energy scattering by a potential consisting of a long-range and a short-range part, to first order in the long-range potential, is derived using the two-potential formula of scattering theory. For the $r^{-4\mathrm{}}$ and for the $r^{-6\mathrm{}}$ potentials and for their superposition, $tan{\eta }_0$ is calculated up to and including the $k^5\ln k$ terms, where ${\eta }_0$ is the zero-angular-momentum ($L=0\mathrm{}$) phase shift. For the $r^{-4\mathrm{}}$ potential and $L>\mathrm{}0\mathrm{}$, $tan{\eta }_L$ is calculated up to and including the $k^{2L+3}\ln k$ term. Finally, the long-range correction to the matrix element for photodissociation of negative ions is calculated.

The following theorem is proven: The $\mathrm{xx}$ and $\mathrm{zz}$ components of the electron energy-momentum tensor $t_{\mu \nu }$ and the magnetization $M$ of an electron gas in a constant magnetic field $B$ satisfy the relation $t_{\mathrm{zz}}-t_{\mathrm{xx}}=BM$. This relation is valid at any density, temperature, and magnetic field strength, if the system is in thermal equilibrium. Since the electromagnetic energy-momentum tensor $t_{\mu \nu }$ is anisotropic itself, this relation makes the total energy-momentum tensor $T_{\mu \nu }=t_{\mu \nu }+{\tau }_{\mu \nu }$ become a scalar. Because $t_{\mathrm{xx}}$ and $t_{\mathrm{zz}}$ can simply describe the electron pressure in these directions, the above theorem states that the difference $p_{\mathrm{zz}}-p_{\mathrm{xx}}$ equals twice the magnetic field energy, since $M=4\mathrm{}\pi B$.

Numerical solutions of the Dirac equation applied to low-energy electron scattering are presented. The calculated complex scattering amplitudes are used to discuss single, double, and triple scattering. Although knowledge of all three experiments leads theoretically to the explicit scattering amplitudes, the calculations show that the amplitudes can be determined, with a reasonable uncertainty only from high-precision experimental data. This is a consequence of the fact that $\left|g\right|$ is normally very small compared to $\left|f\right|$. However, for very selective combinations of the experimental parameters, a very strong influence of the spin on the scattering process is predicted. How this result can be utilized in studies of electron exchange scattering and in investigations of spin-flip processes is discussed. The important quantities for triple scattering from tungsten were calculated for different scattering potentials in an energy range 50 eV - 200 keV. The dependency of the observables on the incident electron energy and scattering potential is shown.

The cross section for the $1s-2s$ excitation of hydrogen atoms by incident hydrogen atoms is calculated for laboratory energies 1-56.25 keV. The calculation is made in the coupled-state impact-parameter approximation, including electron exchange. All exchange potentials are calculated as a function of internuclear distance and velocity. The cross section is much larger at low energies than that predicted by an unsymmetrized state expansion, and at higher energies shows a very smooth undulatory behavior as a function of velocity. The calculation establishes the importance of electron exchange for incident energies at least as high as 100 keV.

The Bethe procedure to obtain an asymptotic expression for the total cross section for inelastic scattering is extended to include three constants instead of the usual two. Within the first Born approximation, the third constant is essentially related to the total number of electrons in the target. By subtracting the sum of the cross sections for discrete excitations from the total cross section, the ionization cross section can also be given in an asymptotic form with three constants. The Mott formula is used to estimate the correction for electron exchange in the ionization cross section. The resulting cross sections compare favorably with experiments on the ionization of H, ${\mathrm{He}}^{+}$, ${\mathrm{H}}^{-}$, He, and ${\mathrm{Li}}^{+}$ by fast charged particles.

Absolute cross sections for the excitation of the Schumann-Runge dissociation continuum of molecular oxygen have been obtained from inelastic energy-loss spectra induced by 20-110-keV protons incident on gaseous targets of molecular oxygen. Apparent differential energyloss cross sections, ionization cross sections, and total inelastic cross sections are also obtained from the energy-loss spectra. The proton energy-loss spectra have an energy resolution of about 2 eV. The energy location of the first peak in the energy-loss spectra of ${\mathrm{O}}_2$ is in general agreement with the findings of photoabsorption and electron energy-loss-spectra measurements. The cross section for the Schumann-Runge dissociation continuum for 20-110-keV protons varies between 5 and 7 × ${10}^{-17\mathrm{}}$ ${\mathrm{cm}}^2$ and has a broad maximum at about 60 keV. Existing theoretical treatments show poor agreement with these reported cross sections for the excitation of the dissociation continuum with respect to both absolute magnitude and curve shape.

The Landau condition, $\overrightarrow{\nabla }\times {\overrightarrow{\mathrm{v}}}_s=0\mathrm{}$, for He II and the analogous Meissner effect in superconductors are shown to follow from the existence of persistent currents by the use of a new definition of the quasiequilibrium state. The novel aspect of this definition is the requirement that the effective Hamiltonian defining the density matrix is invariant under infinitesimal quasi-Galilei transformations. In applications to He II and superconductors, the existence of a condensate implies that the density matrix does not have the full symmetry of the effective Hamiltonian. By slightly relaxing the quasiequilibrium conditions we obtain the usual Landau equations for ${\overrightarrow{\mathrm{v}}}_s$ (for the case that ${\overrightarrow{\mathrm{v}}}_n=0\mathrm{}$).

We investigate the properties of a system of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atoms at the absolute zero of temperature interacting via a Lennard-Jones potential, and constrained to motion in two dimensions by an otherwise inert substrate. Results of a semiclassical calculation indicate that the ground state is that of a self-bound liquid. It further indicates that a close-packed-solid configuration is not self-bound. A quantum-mechanical variational calculation is performed to obtain an upper bound on the energy of the system. A result of -0.61 °K per particle is obtained at a density of 0.035 ${\mathrm{\AA }}^{-2\mathrm{}}$. The relationship between this calculation and recent experiments on ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ monolayers is briefly discussed.

Apparatus and experimental procedures suitable for a number of high-precision measurements of properties of liquid helium near the superfluid transition temperature $T_{\lambda }$ are described. Experimental determinations near $T_{\lambda }$ of the heat capacity at saturated vapor pressure are presented. On the basis of these measurements, the asymptotic temperature dependence of the heat capacity at constant pressure $C_p$ is examined and compared with current theories of critical phenomena. Although there is some latitude in the interpretation of the results in terms of the asymptotic behavior of $C_p$, no interpretation fully in agreement with the original formulations of the scaling laws for critical phenomena, and with a divergent $C_p$, is consistent with the measurements. It is expected that an extensive discussion of the thermodynamics of the $\lambda$ line at higher pressure, based on measurements of the heat capacity at constant volume $C_v$ will be presented in a later publication.

The equations of motion for the reduced density matrices of an interacting Bose system are treated using the concept of off-diagonal long-range order. The equations are decoupled on the one-particle level. With this approximation, the hydrodynamic equations for the densities and the velocities of the condensate and the depletion are derived. The interaction between the two fluid components depends on density variations and on the relative velocity. The latter interaction terms appear as kinetic pressure terms and a Magnus force term. Via the equation for the total momentum, a connection between the velocities of the condensate and the depletion, and the velocities in Landau's two-fluid model is derived. The excitation spectrum of the system is investigated in the phonon region. A comparison of the resulting velocities for the first and second sound with the experimentally determined ones shows that the functional dependence of the condensate density on temperature is similar to that of the superfluid component.

The total excluded volume for a gas of stable particles with repulsive interactions is calculated. From this, it is possible to define an effective negative-interaction number density which is formally identical to the previously derived expression for the equilibrium number density of unstable particles. Pointing out that this allows the unstable-particle formalism to be extended to the negative-lifetime case, it is argued that unstable particles with negative lifetimes behave effectively like a gas of unstable holes.

General formulas for the factorial cumulants of the photocounting distribution and the cumulants of the integrated intensity are obtained for chaotic light superposed with coherent light, for experiments involving $N$ different counting intervals. The chaotic light may be partially polarized, need not be time stationary or single mode, may differ in frequency from the coherent light, and may have any spectral distribution.

Drift velocities of electrons in fluid argon have been measured at temperatures from 90 to 160 °K and at pressures from 10 to 100 atm for applied electric fields in the range -25 to -200 V/cm. The electron drift velocity is found to be linear with respect to electric field strength only to -100 V/cm at temperatures from 90 to 125 °K and to become increasingly nonlinear at temperatures greater than 125 °K. Mobilities can be obtained from these data by extrapolation to zero field; maxima are found in the zero-field mobilities as a function of density, in the region of 0.81 g/${\mathrm{cm}}^3$. Using the model proposed by Lekner for electron scattering by a system of fluctuating potentials and assuming that the scattering length for electrons in fluid argon approaches zero at some density, it is possible to obtain a semiempirical relation for the zero-field mobility as a function of density. Excellent agreement between calculated and observed mobilities is found in the high-density range studied 1.0-1.4 g/${\mathrm{cm}}^3$. At densities less than 1.0 g/${\mathrm{cm}}^3$, several qualitative aspects of the experimental data are accounted for by the theory, but quantitative agreement is lacking. It is possible that at these lower densities, gas-like scattering is of dominant importance.

Isotope-shift measurements with a precision of two parts in ${10}^9$ of the optical transition frequency were made for the $2.02-\mu$ line of the three even isotopes of xenon: ${\mathrm{Xe}}^{132}$, ${\mathrm{Xe}}^{134}$, and ${\mathrm{Xe}}^{136}$. This shift is extremely small due to the large xenon mass, the proximity of the xenon isotopes to the neutron magic number $N=82\mathrm{}$, and the weak interaction of the nucleus with the $5d$ and $6p$ electron states. The measurements were 500 ± 300 kHz (0.17 ± 0.01 × ${10}^{-3\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$) for ${\mathrm{Xe}}^{132}$-${\mathrm{Xe}}^{134}$ and 1500 ± 300 kHz (0.050 ± 0.01 × ${10}^{-3\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$) for ${\mathrm{Xe}}^{134}$-${\mathrm{Xe}}^{136}$, where the lighter isotope had the larger transition frequency in each case. These two shifts are sufficiently different to suggest that nuclear field effects contribute to the results.

The propagation behavior of nontrivial $0\pi$ pulses [$\int {-\infty }^{\infty }\mathcal{E}\mathrm{}(t,z)dt=0\mathrm{}$, $\int {-\infty }^{\infty }{\mathcal{E}}^2\mathrm{}(t,z)\mathrm{dt}>\mathrm{}0\mathrm{}$] in attenuating media is examined. It is found that under appropriate conditions $0\pi$ pulses exhibit anomalously low absorption which is a direct consequency of atomic coherence. Specific analytical results valid for an unbroadened medium are presented, while the corresponding solutions for an inhomogeneously broadened resonance are calculated numerically. These two results are consistent in the limit of $T_2^{*}\to \infty$. Furthermore, the influences of $T_2$, $T_2^{*}$, and level degeneracy on the characteristics of propagation are determined. Finally, the behavior of more general $0\pi$ pulses is studied. The criteria for the production of $0\pi$ pulses are qualitatively established by numerical methods, and the influence of the relative phase of pulse pairs on propagation is examined. The latter indicates a method for the measurement of relative pulse phase.

The intensity and spectral width of light scattered by a critical mixture of phenol and water have been measured as a function of temperature both above and below the critical temperature $T_c$. The temperature dependence of the scattered intensity was fitted to $I\propto {(T-T_c)}^{-\gamma }$ for $T>\mathrm{}{T}_c$ and $I\propto {(T_c-T)}^{-{\gamma }^{\prime }}$ for $T<\mathrm{}{T}_c$. Also measured was the ratio $R_I\equiv \frac{I\left(\Delta T\right)}{I(-\Delta T)}$ of intensities scattered at a given temperature interval $\Delta T$ above and below $T_c$. The measured values of $\gamma$, ${\gamma }^{\prime }$, and $R_I$ were quite close to the predictions of the $3-D$ lattice gas model. The spectral width $\Gamma$ was measured using a photon correlation method, and the data were fitted to $\Gamma =D{K}^2 (1+K^2{\xi }_{\Gamma }^2)$, with the diffusion constant $D=D_0{|T-T_c|}^{\gamma *}$ and ${\xi }_{\Gamma }={\xi }_{0\Gamma }{|T-T_c|}^{{\nu }_{\Gamma }}$ The Fixman term $K^2{\xi }_{\Gamma }^2$ ($K$ being the photon momentum transfer) was observed only above the critical temperature. The value of $\nu$ and the values of ${\gamma }^{*}$ both above and below $T_c$ were in fairly good agreement with the theory of Kadanoff and Swift. The spectral width measurements also provided the ratio $R_D\equiv \frac{D(-\Delta T)}{D\left(\Delta T\right)}$, a quantity for which no theoretical prediction exists. Comparison of this work on phenol-water with that of Swinney and Cummins and others on ${\mathrm{CO}}_2$ near its gas-liquid critical point reveals remarkable similarities between the two systems.

The ground-state energy of a system of hard-sphere bosons of diameter $\sigma$ is calculated by a variational method using a Jastrow wave-function for the fluid phase and a "localized" Jastrow times a Gaussian wave function for the solid phase. The transition densities are ${\rho }_{\mathrm{fluid}}=0.23\mathrm{}\pm 0.02$ particles/${\mathrm{\sigma }}^3$ and ${\rho }_{\mathrm{solid}}=0.25\mathrm{}\pm 0.02$ particles/${\mathrm{\sigma }}^3$.

It has been shown that the presence of an $\alpha$-particle emitter causes a lowering of the ultrasonic-cavitation-noise threshold in helium II by as much as 50%. The lowest temperature investigated was 1.80 °K. There was no discernable effect on the threshold in helium I, so that the nucleation mechanism cannot be the formation of microbubbles at hot spots as in Seitz's theory. Neither the presence of a $\beta$ source nor electric fields affected the nucleation, so that ions, per se, are also ruled out as a nucleating mechanism. However, several similarities to the behavior of the threshold in rotating helium were observed. Evidence was obtained indicating that the cavitation noise was localized in the vicinity of the $\alpha$ source. It is concluded that the nucleation occurs on vortex filaments created by the $\alpha$ particles.

The spin-correlation functions of the one-dimensional $XY$ model are studied in the presence of a constant magnetic field. We find that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.

We present in this paper a new formulation of the kinetics of the Lorentz gas, based on an analysis of statistically independent collisional events. We give a method in which the general collision operator is expanded in terms of functions of the density, and we carefully treat the ${\mu }^s$ approximation ($\mu$ is the density parameter and $s$ is the dimensionality of the gas). The method is applied to the calculation of the self-diffusion coefficient.