The motion of a radiating atom immersed in a gas of other atoms, the bath, may be described by means of a nonunitary time-evolution operator $U\left(t\right)={\left({\mathrm{Tr}}_B\rho \right)}^{-1\mathrm{}}{\mathrm{Tr}}_B\rho e^{\mathrm{it}{H}^{\times }}$ acting in its Liouville space (${\mathrm{Tr}}_B$ is the trace over bath variables, $\rho$ is the density matrix, and $H^{\times }$ is the quantum-mechanical Liouvillian). In a previous paper, $U\left(t\right)\mathrm{}$ was written in the form $U\left(t\right)={\exp }_{\to }\left[i\int 0^t{\mathrm{dt}}^{\prime }{L}_1\left(t^{\prime }\right)\right]$ (${\exp }_{\to }$ denotes a time-ordered exponential), and the time-dependent effective Liouvillian $L_1\mathrm{}\left(t\right)\mathrm{}$ was expanded in powers of a "reduced density," or activity. In this paper the Fourier transform $U\left(\omega \right)=\int 0^{\infty }\mathrm{dt}{e}^{-i\omega t}U\left(t\right)\mathrm{}$ is written in the form $iU\left(\omega \right)={[\omega -L_2(\omega \left)\right]}^{-1\mathrm{}}$, and the frequency-dependent effective Liouvillian $L_2\left(\omega \right)$, or "memory operator," is expanded in powers of the reduced density. It is argued that in treating $L_2\left(\omega \right)$ to first order in the reduced density, one effectively allows the radiator to interact with only one perturber at a time, thus neglecting all multiple-collision effects. This is in contrast to performing the same-order approximation on $L_1\mathrm{}\left(t\right)\mathrm{}$, which is equivalent to treating different perturbers as uncorrelated, but still allows for multiple-collision effects. By adding the terms of higher order in the expansion of $L_2\left(\omega \right)$, one allows the radiator to interact simultaneously with two, three,... perturbers.

The molecular-dynamic method was used to simulate a fluid of 500 rigid diatomic homo-nuclear molecules interacting by a double Lennard-Jones potential. The equilibrium and time-dependent properties are calculated in the liquid phase. The computed pressure and the internal energy agree quantitatively to a few percent with experimental values for nitrogen. The reorientational and the velocity of the center-of-gravity self-correlation functions are also discussed. The memory-function formalism and the extended-diffusion models are used to interpret the reorientational self-correlation functions. The analysis reveals that these self-correlation functions have an exponential behavior for times larger than 5 × 1$0^{-13\mathrm{}}$ sec. In this model, considering present computing precision, there is no observable hydrodynamic-type relaxation in the reorientational self-correlation functions.

We develop the generalization of the Heisenberg near-neighbor-exchange Hamiltonian necessary to incorporate the effects of multiple-exchange processes. Many-body expressions for the multiple-exchange constants (pair exchange, triple exchange, and quadruple exchange) are derived. The physics that enters these exchange constants is discussed. For the most important of the pair and triple cases, these expressions are carefully evaluated using a Monte Carlo integration scheme. We show that the exchange Hamiltonian for solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is rapidly convergent, and that the near-neighbor pair, next-near-neighbor pair, and triple-exchange processes (involving two near neighbors and a next-near neighbor) are likely to be the only important exchange processes to the low-temperature thermodynamics of bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The magnitude of the triple-exchange process is such that the "effective" next-near-neighbor pair-exchange interaction in bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is ferromagnetic. This result provides qualitative and quantitative support to the explanation of the data of Kirk and Adams made by Zane.

Two chambers filled with superfluid helium to different levels were connected by a small orifice (10-15 μm). The flow of the liquid through the orifice was modulated by a sound field. With one experimental apparatus only a continuous flow was observed. With a modified apparatus it was possible to obtain stable states with no net flow at finite level differences between the baths. The spacing between these states is inversely proportional to the sound frequency. It depends on the total height of the liquid in one of the baths and not on the level difference between them. We conclude that the stable states of zero flow result from ultrasonic standing waves of the superfluid in one of the chambers. A phase coupling or Josephson effect can be excluded as an explanation for these states.

The $t^{\frac{-3\mathrm{}}{2}}$ long-time behavior of the Green's-Kubo autocorrelation functions is the first term in an infinite series of general order $t^{\frac{1}{2^n-2\mathrm{}}}$, $n$ integer ≥ 1. The coefficients of these series for the shear and bulk viscosity and for the heat conductivity are given in terms of linear recurrence relation. Similarly, after the usual Navier-Stokes order (in $k^2$), there exists an infinite expansion for the frequencies of the hydrodynamical modes with terms of general order $k^{3-\frac{1}{2^n}}$. The mean square displacement of a particle in a fluid is given, for large times, by an infinite series, the first term being the well-known Einstein displacement $6Dt$, and the following ones proportional to $t^{\frac{1}{2^n}}$. As an application of this expansion of the hydrodynamical theory beyond the Navier-Stokes order, the pressure pattern in a weak shock wave is computed.

The superradiant level shift of a system of $N$ two-level molecules, interacting with the radiation field, is calculated for pencil-like geometry. Introducing the self-energy ${\Sigma }^{\mathrm{}\left(N\right)\mathrm{}}\mathrm{}\left(E\right)\mathrm{}$ of the molecular systems due to its radiative interaction, we first discuss the trivial ($N=1\mathrm{}$) and familiar ($N=2\mathrm{}$) cases in more detail. The analytic properties of the self-energy ${\Sigma }^{\mathrm{}\left(N\right)\mathrm{}}\mathrm{}\left(E\right)\mathrm{}$ are used to derive dispersion relations for its real and imaginary parts ${\Delta }^{\mathrm{}\left(N\right)\mathrm{}}\mathrm{}\left(E\right)\mathrm{}$, ${\Gamma }^{\mathrm{}\left(N\right)\mathrm{}}\mathrm{}\left(E\right)\mathrm{}$. We then generalize to the case of $N\gg 1$, employing a self-consistency argument advanced by Arecchi and Courtens. The modified Dicke expression for the cooperative spontaneous decay rate is then used in the dispersion relation for the shift ${\Delta }^{\mathrm{}\left(N\right)\mathrm{}}$ and a value of ∼6 MHz calculated corresponding to typical molecular-gas parameters. The physical measurability of the shift is discussed and a Brewer-Shoemaker experiment in a molecular gas is proposed for the observation of the shift.

High-resolution studies of the depolarized Brillouin spectra of many molecular liquids have revealed a prominent doublet centered at the exciting frequency. The observed line shapes and intensities of the $I_H^V$ and $I_H^H$ polarized spectra, as well as the $\overrightarrow{\mathrm{k}}$-vector dependence of the doublet splitting on scattering angle, are all in quantitative agreement with the theories of Leontovich, Rytov, and Volterra on light scattering from orientational motions caused by heavily damped shear waves. Analyses of the spectra on the basis of these theories have given values of the shear-wave frequencies ${\nu }_T$ and relaxation times $\tau$. Values of $\tau$ in the range 3 × ${10}^{-10\mathrm{}}$ -1 × ${10}^{-11\mathrm{}}$ sec were obtained and $\tau$ was found to decrease with increasing temperature. Values of the shear modulus determined from ${\nu }_T$ and its dependence on $\theta$ were found to range from 0.6×${10}^8$ to 9×${10}^8$ ${\mathrm{d}\mathrm{y}\mathrm{n}/\mathrm{c}\mathrm{m}}^2$, and to be essentially independent of temperature. Of the 27 liquids investigated, only those composed of geometrically and electrically anisotropic molecules exhibited depolarized-doublet spectra, and the resolution of the intensity minima was limited by the magnitude of the relaxation time and by the instrumental linewidth.

The ratio of the nuclear $g$ factor to the electronic $g$ factor in the ground electronic state of free ${}_{}{}^{133}{}_{}{}^{}\mathrm{Cs}$ atoms and the ratio of the electronic $g$ factors of ${}_{}{}^{133}{}_{}{}^{}\mathrm{Cs}$ and ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ atoms have been determined using optical-pumping techniques in a magnetic field ≤50 G. The results are $\frac{g_I}{g_J}\left({}_{}{}^{133}{}_{}{}^{}\mathrm{Cs}\right)=-1.9917400\mathrm{}\times {10}^{-4\mathrm{}}\times (1\mathrm{}\pm 1.3\times {10}^{-6\mathrm{}})$; $\frac{g_J\left({}_{}{}^{133}{}_{}{}^{}\mathrm{Cs}\right)}{g_J\left({}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}\right)}=1.0001044737\mathrm{}\times (1\mathrm{}\pm 4.4\times {10}^{-9\mathrm{}})$. Using these values together with results of other researches yields $\frac{g_I\left({\mathrm{Cs}}_{\mathrm{aq}}^{+}\right)}{g_I\left(\mathrm{Cs}\right)}=1+(344.0\mathrm{}\pm 1.5)\times {10}^{-6\mathrm{}}$, the ratio of the nuclear $g$ factor of the Cs ion in aqueous solution to that of the free Cs atom. Finally, the value for the shielded Cs nuclear moment in units of the Bohr magneton is found to be $g_I\left({}_{}{}^{133}{}_{}{}^{}\mathrm{Cs}\right)=-0.39885395\mathrm{}\times (1\mathrm{}\pm 1.3\times {10}^{-6\mathrm{}})\times {10}^{-3\mathrm{}}$.

The Bradbury-Nielsen time-of-flight technique has been used to measure the longitudinal-diffusion coefficient $D_L$ for potassium ions in nitrogen in the $\frac{E}{N}$ range $2.83<~\frac{E}{N}\left(\mathrm{Td}\right)<~28.3$, where $E$ is the electric field strength and $N$ the gas number density. The results are consistent with the value predicted by the Nernst-Townsend (or Einstein) relation to within the claimed experimental error of 3%. The variation of $D_L$ with $\frac{E}{N}$ has been compared with the predictions of Wannier, based on a Maxwellian model.

An energy-scaling rule for atom-atom scattering is derived. It is shown to be approximately accurate to third order in the parameter $\frac{\Delta M}{M_{\mathrm{av}}}$, where $\Delta M$ is the difference in the reduced masses between two different atomic pairs. The energy scaling depends upon the energy itself and on the interaction potential. The method is illustrated by the He-He scattering problem using the isotopes ${\mathrm{He}}_3$ and ${\mathrm{He}}_4$ in their three different combinations.

Some results recently discussed by Chiu for interatomic coherence transfer are shown to have a simple physical interpretation, to be independent of collision model assumed, and to be applicable also to intra -atomic coherence transfer. A derivation using density matrices is presented which takes both depolarizing collisions and backtransfer of coherence into account.

Since two recent investigations revealed different structures for the one-electron reduced-density matrix (one matrix) of the Weiss wave function of the lowest (${}_{}{}^2{}_{}{}^{}P$) state of lithium, we have reanalyzed the Weiss wave functions for both the lowest ${}_{}{}^2{}_{}{}^{}S$ and ${}_{}{}^2{}_{}{}^{}P$ states of the three-electron ions. Although Weiss employed "${}_{}{}^2{}_{}{}^{}S$ restrictions" in the construction of his ${}_{}{}^2{}_{}{}^{}P$ wave functions, their natural spin orbitals are not as spatially symmetry adapted as they are for the ${}_{}{}^2{}_{}{}^{}S$ states.

This is an addendum to the author's previous paper [Phys. Rev. A 4, 1778 (1971)], where master equations describing the spontaneous emission from a collection of identical two-level atoms and oscillators were derived without the use of rotating-wave approximation. However, the terms corresponding to the frequency shifts were not adequately included. Here such terms are properly treated and the appropriate master equation is given. Explicit form of the frequency-shift terms is also given.

A system of $N$ two-level atoms interacting with a quantized field, the so-called Dicke model of superradiance, is studied. By making use of a set of Glauber's coherent states for the field, the free energy of the system is calculated exactly in the thermodynamic limit. The results agree precisely with those obtained by Hepp and Lieb, who studied the same model using a different method. The exhibition of a phase transition of the system is presented mathematically in an elementary manner in our approach. The generalization to the case of finitely many radiation modes is also presented.

In a first part of this work it is demonstrated how the impact theory of line braodening together with the interference mechanism for overlapping lines nicely fits the observed line shape of ammonia up to the values of pressure at which the impact approximation is expected to be valid. For higher pressures a new theory is developed for the relaxation mechanism of the dipole moment. It is shown that, because the frequency of collisions is much larger than the frequency of the absorbed wave, the latter performs a kind of average over many collisions. Relaxation is mainly due to the fact that the phase of the dipole moment after many collisions can be very different, depending on the random path that the absorbing molecule describes in walking between the perturbers. The result is a Lorentz shape for the Debye-type absorption at high pressure, which perfectly agrees with experiment. This theory applies more generally to Debye-type absorption, and ammonia provides a good test of it.

A configuration-interaction study of doublet autodetaching states of ${\mathrm{O}}^{-}$ has predicted the following levels: ${\mathrm{O}}^{-}\left[2\mathrm{}{p}^3\right({}_{}{}^4{}_{}{}^{}S_{}^0{}_{}{}^{}\left)3s3p\right({}_{}{}^2{}_{}{}^{}P\left)\right]$ at 9.50 eV, ${\mathrm{O}}^{-}\left[2\mathrm{}{p}^3\right({}_{}{}^2{}_{}{}^{}D_{}^0{}_{}{}^{}\left)3\mathrm{}{s}^2\right({}_{}{}^2{}_{}{}^{}D_{}^0{}_{}{}^{}\left)\right]$ at 12.05 eV, and ${\mathrm{O}}^{-}\left[2\mathrm{}{p}^3\right({}_{}{}^2{}_{}{}^{}P_{}^0{}_{}{}^{}\left)3\mathrm{}{s}^2\right({}_{}{}^2{}_{}{}^{}P_{}^0{}_{}{}^{}\left)\right]$ at 13.65 eV. These states can explain most of the structure, which results from collisions of the ground-state ${\mathrm{O}}^{-}$ ion with rare-gas atoms, as found by Edwards, Risley, and Geballe. An additional level ${\mathrm{O}}^{-}\left[2\mathrm{}{p}^3\right({}_{}{}^4{}_{}{}^{}S_{}^0{}_{}{}^{}\left)3\mathrm{}{p}^2\right({}_{}{}^2{}_{}{}^{}P_{}^0{}_{}{}^{}\left)\right]$ at 10.87 eV is found which has not yet been detected experimentally. The present formalism computes eigenvalues of $H_{\mathrm{QQ}}$ using a fixed ${\mathrm{O}}^{+}$ core with configuration mixing of states of the outer two active electrons. The shift contribution to the resonance energy is neglected.

Sternheimer shielding-antishielding parameters pertinent to the quadrupole perturbations outside an ion ($\lambda$), as well as within the ion owing to an unfilled 4f shell ($R$), have been calculated for rare-earth ions ${\mathrm{La}}^{3+}$, ${\mathrm{Ce}}^{3+}$, ${\mathrm{Pr}}^{3+}$, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Pm}}^{3+}$, ${\mathrm{Sm}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Gd}}^{3+}$, ${\mathrm{Tb}}^{3+}$, ${\mathrm{Dy}}^{3+}$, ${\mathrm{Ho}}^{3+}$, ${\mathrm{Er}}^{3+}$, ${\mathrm{Tm}}^{3+}$, ${\mathrm{Yb}}^{3+}$, ${\mathrm{Lu}}^{3+}$, ${\mathrm{Sm}}^{2+}$, and ${\mathrm{Eu}}^{2+}$. Hartree—Fock—Slater-type ionic wave functions have been used. The value of ${\lambda }_0$ varies smoothly from -65.82 for ${\mathrm{La}}^{3+}$ to -58.49 for ${\mathrm{Yb}}^{3+}$, whereas that of $R_0$ (without exchange) varies from 0.081 for ${\mathrm{Ce}}^{3+}$ to 0.128 for ${\mathrm{Yb}}^{3+}$. The calculation of Sternheimer parameters for ${\mathrm{Sm}}^{2+}$ and ${\mathrm{Eu}}^{2+}$ with ${\lambda }_0=-73.28\mathrm{}$ and $R_0=0.124\mathrm{}$ for ${\mathrm{Sm}}^{2+}$ shows that their dependence on the charge of the ions is appreciable. Sternheimer parameters are reported also for the core-electron sites.