We have measured line shifts of the $3P-nD$ transitions in sodium induced by collisions with helium, argon, and krypton for $n=5\mathrm{} \mathrm{to} 18$. The shifts of the diffuse series transitions increase rapidly with principal quantum number and asymptotically approach the limit predicted by the Fermi model at high quantum numbers.

In the calculation of two-photon transition probabilities and related radiation-matter interaction problems, the "length" and "velocity" forms of the dipole interaction give results that can differe by many orders of magnitude if a complete set of exact eigenfunctions is not used. These differences have been illustrated in the case of the $1s-2s$ two-photon transition in hydrogen by Bassani, Forney, and Quattropani [Phys. Rev. Lett. 39, 1070 (1977)], and it was concluded by these authors that for this problem the length form of the interaction is far superior to the velocity form when an incomplete set of bound states is used. In order to remove the question of gauge and thereby to improve the accuracy of such calculations when a complete set of exact eigenfunctions is not available, this paper proposes a modified form of the second-order equation. The modified equation, which employs a unique average-frequency approximation, is derived through incorporation of a sum rule that is a direct result of gauge invariance. It contains the length and velocity forms as limiting cases, and, like those equations, is exact for a complete set. Conventional length or velocity equations violate this sum rule for an incomplete set, and therefore are subject to large errors. The use of this modified equation is illustrated for the $1s-2s$ transition of hydrogen. Using only the $2p$ state as the "set" of intermediate states, the modified equation yields results that for all photon energies are within 2% of the exact probability amplitudes. For the $1s-2s$ problem, the accuracy is comparable to that obtained from length- and velocity-form equations employing an ordinary average-frequency approximation and closure. Unlike the modified equation, however, these latter equations require a separate calcualtion of matrix elements of $r^2$ or $p^2$, which may be inconvenient for a many-electron system. The modified equation should therefore be useful when these matrix elements are not easy to calculate, or when the best gauge form for the problem is not known.

We have measured the blackbody photoionization rate from the $17d$ state of sodium as a function of temperature from 90 to 300 K. The experimental rates, which vary by a factor greater than one hundred over this temperature range, are in good agreement with our theoretical values.

An absorption cell containing low-pressure gas is placed inside a C${\mathrm{O}}_2$-${\mathrm{N}}_2$O infrared laser cavity and microwave resonances in the gas are detected through the off-resonant response of the infrared laser. The small variation in molecular population in certain rotational levels produced by a microwave pumping changes the bulk susceptibility of the gas through dispersion and thus changes the effective cavity length and the output power of the laser. This new mode of "double resonance" does not require exact coincidence between the laser lines and molecular transitions and is more widely applicable than the normal double-resonance method although its sensitivity is not as great. Characteristics of this method are theoretically studied and compared with experimental results obtained by using the simple molecules C${\mathrm{H}}_3$F, ${\mathrm{D}}_2$CO, ${\mathrm{D}}_2$O, and N${\mathrm{H}}_3$. Microwave resonances are observed through laser lines which are off from molecular infrared transitions by several ${\mathrm{cm}}^{-1\mathrm{}}$. Some unexpected asymmetry of the signals with respect to laser settings has been observed and discussed.

The influence of the ac-Stark shifts on the spin polarization (SP) of the photoelectron in a resonant multiphoton ionization is discussed. It is shown that the Stark shifts of various levels are different and this removes any initial fine-structure degeneracy of the excited states. This increased dynamic separation between the states increases the SP of the ejected photoelectron. Numerical results are presented to illustrate this effect. By expressing the equations of motion in the spin basis set the precise conditions necessary to observe nonzero SP are identified. It is shown on the basis of these conditions that although the removal of the fine-structure degeneracy occurs independent of whether the atom has any spin-orbit coupling, this does not induce any spin polarization if the atom has no spin-orbit coupling.

Equations of motion for a sample filled with totally inverted two-level atoms coupled to one-dimensional radiation field are solved in the linear regime without the slowly-varying-envelope approximation. The comparison with known results neglecting the coupling between left- and right-going waves is made.

The $4sns{}_{}{}^1{}_{}{}^{}S$ Rydberg series of calcium has been investigated with various levels of configuration-interaction (CI) calculations. The series has been found to be weakly perturbed by $4p^2{}_{}{}^1{}_{}{}^{}S$. The correlation effect is essential for $4p^2{}_{}{}^1{}_{}{}^{}S$ to act as a perturber. The calculated term energies relative to the inoized state by the largest CI calculation are -1.95, -1.02, -0.73, -0.61 eV, etc., whereas experimental values are -1.98, -1.07, -0.93, -0.62, eV, etc. The present assignments are $4s5s$, $4s6s$, $4p^2$, $4s7s,\dots$, in agreement with those of Risberg. The pure series $4sns{}_{}{}^3{}_{}{}^{}S$ has been calculated as a reference.

The two-atom fluorescence spectrum with finite detuning is calculated analytically. For large field strengths, a Lorentzian decomposition is presented which exhibits the presence of two-component peaks at the central peak and at the Rabi sidebands; thus finite detuning destroys the proportionality to the one-atom spectrum. The analytical solution for the incoherent part of the spectrum is also obtained and a peculiarity of its weak-field behavior is pointed out.

We calculate the steady-state probe absorption line-shape function for a strongly driven, Zeeman-degenerate molecular system. The probe laser is treated to lowest order while the pump laser is dealt with to all orders. We obtain the probe line shape for the cases of parallel and perpendicular linear polarization of the two lasers. As expected, the effects of $M$ degeneracy, as well as differences due to the relative laser polarizations, are most pronounced when Doppler broadening is not important. However, even in the presence of large Doppler broadening we find a narrowing of the population hole by including the Zeeman degeneracy and a further narrowing if perpendicular laser polarizations are used.

A theory of resonantly enhanced degenerate four-wave mixing in two-level systems including the effects of atomic and photon angular momentum is presented in the absence of pump-induced saturation of the transition. It is shown that there exist three distinct quantum-mechanical amplitudes leading to the third-order polarization density. These quantum-mechanical amplitudes are shown to be sensitive to the states of polarization of the incident fields. The quantum-mechanical transport equation in the $m$ representation is used to calculate the output signal in the various regimes of laser detuning, atomic linewidth, and polarization states of the radiation field for collinear interaction.

The semiclassical Lamb theory of gas lasers is extended to the case of lasers operating simultaneously on two coupled transitions. The theory is formulated in terms of the density-matrix equations for a three-level gaseous laser medium interacting with the field through electric-dipole transitions, so that there are two allowed transitions and one two-photon transition. It is assumed that there is a single resonator mode near each of the transition frequencies. Previous treatments of such three-level gas lasers either have used simple rate equations or have solved the density-matrix equations in an approximate fashion valid only for low intensities. Thus the numerical approach developed herein represents, to our knowledge, the first semiclassical theory of three-level gas lasers which is accurate for arbitrary detuning and intensity. Significant deviations from earlier, approximate treatments are found.

The amplitude equation for Taylor vortices with spatially varying amplitude in long but finite cylinders is derived for arbitrary gap width. The boundary condition to be satisfied by the amplitude at the end of the cylinders is obtained by a simple scaling argument. The same argument also yields a scaling property of the boundary-layer flow at the cylinder ends. The results for boundary conditions and correlation length are compared with recent measurements of Pfister and Rehberg, and excellent agreement is obtained.

We have measured dc electrical conductivity fluctuations in isobutyric-acid—water (I-W) and 2,6-lutidine-water (L-W) mixtures near the critical temperature $T_c$. The noise was measured in a capillary separating two reservoires initially filled to slightly different heights. Power spectra $S_V\mathrm{}\left(f\right)\mathrm{}$ encompassed the frequency ($f$) range $0.01\le f\le 100$ Hz and the temperature ($T$) range $1\le |T-T_c|\le 1100$ mK. Very close to $T_c$ the spectra for L-W and I-W (at early times after mixing the sample) are consistent with the assumption that the noise is generated by flow. The temperature dependence of the noise can be represented by a power law $S_V\mathrm{}(f,T)\mathrm{}\propto {|T-T_c|}^{-\gamma }$, with $\gamma =1.0\mathrm{}\pm 0.1$. This value of $\gamma$, being smaller than the Ising value of 1.25, suggests that the composition fluctuations are suppressed by the shear flow in the capillary. We cannot explain the appreciable noise which persists after the flow has ceased. An ac method of determining $S_V\mathrm{}\left(f\right)\mathrm{}$ yields results identical with those of the dc method. Finally, we demonstrate, through a direct test, that the noise in a binary mixture near $T_c$ has a negligible nonlinear component.

We consider stochastic differential equations for a variable $q$ with multiplicative white and nonwhite ("colored") noise appropriate for the description of nonequilibrium systems which experience fluctuations which are not "self-originating." We discuss a numerical algorithm for the simulation of these equations, as well as an alternative analytical treatment. In particular, we derive approximate Fokker-Planck equations for the probability density of the process by an analysis of an expansion in powers of the correlation time $\tau$ of the noise. We also discuss the stationary solution of these equations. We have applied our numerical and analytical methods to the "Stratonovich model" often used in the literature to study nonequilibrium systems. The numerical analysis corroborates the analytical predictions for the time-independent properties. We show that for large noise intensity $D$ the stationary distribution develops a peak for increasing $\tau$ that becomes dominant in the large-$\tau$ limit. The correlation time of the process in the steady state has been analyzed numerically. We find a "slowing down" in the sense that the correlation time increases as a function of both $D$ and $\tau$.. This result shows the incorrectness of an earlier analysis of Stratonovich.

Phase-transition properties of nematic liquid crystals aligned by a short-range, arbitrary-strength-substrate potential are examined in the framework of Landau—de Gennes theory. It is shown that the substrate potential, which can arise from surface treatment of liquid-crystal display cells, not only induces a boundary layer in which the order-parameter values can be significantly different from that of the bulk, but also introduces a new "boundary-layer phase transition" which occurs at temperatures higher than the bulk-transition temperature. This novel transition is found to take place only in a limited range of substrate potential strength. For 4-pentyl-4'-cyanobiphenyl (PCB), the limiting values of this range are computed to be ∼0.075 and ∼0.15 erg/${\mathrm{cm}}^2$. Calculations are performed for both the semi-infinite-sample case and the finite-thickness-sample case. Various phase diagrams are presented to show the effects of sample thickness and substrate potential on the bulk as well as the boundary-layer phase-transition temperatures. The paper concludes with a discussion of experimental possibilities.

In many plasma-magnetic-field configurations, the ion skin depth is comparable to magnetic field and density-gradient scale lengths and so inhomogeneity must play an important role for wave propagation in the ion-cyclotron range of frequencies. As an example, we consider ion-cyclotron heating in the Elmo Bumpy Torus-Scale (EBT-S) device, where the coupling of infinite and homogeneous fast and slow modes implies that fundamental ion-cyclotron heating can occur even if the waves launched are fast waves (which otherwise do not heat at the fundamental). We also show that singular, weakly damped, two-ion hybrid electrostatic solutions of the cold-plasma dispersion relation which include inhomogeneities are not consistent with the small ion-gyroradius basis for this dispersion relation. This suggests that two-ion hybrid-mode conversion requires an analysis which does not assume small ion-gyroradius and so, if conversion can occur at all in EBT-S, it must be different from that in idealized weakly inhomogeneous slab-geometry plasma models. For shorterwavelength higher-frequency waves in EBT-S and for ion-cyclotron range of frequency waves in larger bumpy tori, like the projected EBT-P (P for proof of principle), inhomogeneity should have a less profound, although not necessarily negligible, impact on wave propagation.

By the application of Mori's scaling method to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy equation of the classical electron plasma in thermal equilibrium, the static correlation functions are derived. These correlation functions are used to calculate the internal energy, which is found to be different from that of R. Abe [Prog. Theor. Phys. 22, 213 (1958)]. This difference comes from the short-range correlation function and its implications are discussed.

Reentrant behavior in nematic liquid crystals is described by the treatment of the system as a mixture of monomers and dimers interacting through attractive induced soft interactions as well as hard-core repulsions. It is shown that the induced forces between dimers, monomers, and between monomers and dimers, respectively, under suitable conditions of temperature and density lead to a smectic layer structure with a period equal to the length of the dimer. However, with decreasing temperature or increasing density, repulsive steric forces, due to the unfavorable packing of the dimers in the smectic planes, may take over and thus favor the nematic phase again. In agreement with experiment the best conditions to get a reentrant nematic phase are predicted for the ratio of the length of the dimer to that of the monomer between 1.3 and 1.4. The predictions of the model are in good agreement with currently available experimental data. In principle, the model applies also to discotic liquid crystals.

The velocity-autocorrelation function ${\rho }_D\mathrm{}\left(t\right)\mathrm{}$ for hard disks is computed for ten values of the reduced volume, ranging from 30 to 1.4 times the close-packed volume ($V_0$) for systems of as few as 168 and as many as 5822 particles, by a Monte Carlo molecular-dynamics technique. For values of the time greater than roughly 20 mean-free times ($t_0$), the results are compared with the predictions of a version of the mode-coupling theory for ${\rho }_D\mathrm{}\left(t\right)\mathrm{}$, modified to take into account the finite size of the system. Except at the highest densities, the data agree with the modified theory when one uses Knskog values for the transport coefficients in evaluating the theoretical ${\rho }_D\mathrm{}\left(t\right)\mathrm{}$, provided the comparison is limited to times beyond a value $s_i{t}_0$, with $s_i$ dependent on the density. The value of $s_i$ appears to increase from roughly 20 at the lowest densities to 40 at a volume of $2V_0$. At volumes of 1.6, 1.5, and $1.4V_0$, the theoretical ${\rho }_D\mathrm{}\left(t\right)\mathrm{}$'s are too large out to times as large as $320t_0$, unless we use values of the transport coefficients rather larger than the Enskog values in evaluating the theoretical ${\rho }_D\mathrm{}\left(t\right)\mathrm{}$. The velocity-autocorrelation-function results for $t<\mathrm{}20\mathrm{}{t}_0$ are presented as the difference relative to the Lorentz-Boltzmann-Enskog prediction, which has the exact slope at $t=0\mathrm{}$.

We consider a dynamical model containing a short-time scale ${\tau }_s$ and a long-time scale ${\tau }_l$ and exhibiting a continuous instability depending on a control parameter. We study how the threshold is shifted if the control parameter is noisy with a correlation time ${\tau }_c\ll {\tau }_l$ with the use of both qualitative and systematic methods of adiabatic elimination. We find a noise-induced increase of the threshold of instability depending on the ratio $\lambda =\frac{{\tau }_s}{{\tau }_c}$. If the fluctuations of the control parameter are Gaussian, and if $\frac{{\tau }_c}{{\tau }_l}\to 0$, $\frac{{\tau }_s}{{\tau }_l}\to 0$, on the scale ${\tau }_l$, the fluctuations act as a Stratonovich noise source for ${\tau }_c\gg {\tau }_s$ and as an Itô noise source for ${\tau }_c\ll {\tau }_s$. The intermediate regime ${\tau }_s=\lambda {\tau }_c$ with arbitrary $\lambda$ is analyzed and found to be observable by the noise-induced shift of the threshold associated with it.