The energy of the Ta $K\alpha$ x rays emitted after electron capture of ${}_{}{}^{181}{}_{}{}^{}\mathrm{W}$ has been observed to be lower by 0.79±0.06 eV compared to the fluorescent Ta $K\alpha$ x rays. Similarly, it was found that the Ta $K\alpha$ x rays emitted after internal conversion following $\beta$ decay of ${}_{}{}^{181}{}_{}{}^{}\mathrm{Hf}$ were higher by 0.7±0.1 eV in energy compared to the same fluorescent Ta x rays. The observed effects cannot be explained by the known mechanisms of chemical shift, hyperfine effects, isomer shifts, and shake-off effects alone. Similar observations of the $K$-x-ray energy shifts in the electron-capture decays of ${}_{}{}^{153}{}_{}{}^{}\mathrm{Gd}$ and ${}_{}{}^{175}{}_{}{}^{}\mathrm{Hf}$ with the corresponding fluorescent x rays in Eu and Lu, respectively, can be accounted for by the hyperfine effects, isotope shifts, and chemical shifts.

Absolute intensity measurements of resonant third-harmonic generation in a Sr-Xe mixture have been performed using a flash-lamp-pumped dye laser. The third-harmonic power increased by about 3 orders of magnitude with respect to the pure Sr system by adjusting a pressure ratio of $\frac{p_{\mathrm{Xe}}}{p_{\mathrm{Sr}}}=53\mathrm{}$. Excellent agreement with theory is obtained provided the absorption cross section ${\sigma }_3=3.5\mathrm{}\times {10}^{-18\mathrm{}}$ ${\mathrm{cm}}^2$ of Sr at the third-harmonic wavelength of 191.97 nm has been taken into account. The absorption cross section limits the conversion efficiency to 2 × ${10}^{-5\mathrm{}}$ for an input intensity of 5 × ${10}^6$ W/${\mathrm{cm}}^2$. Up to these intensities no saturation effects were observed.

The phenomenon of nonlinear conical refraction in a noncentrosymmetric biaxial crystal is investigated experimentally. Both free and forced second-harmonic cones are observed, depending on whether the incident laser pulse has wave normals encompassing the optic axis at the second-harmonic or the fundamental frequency, respectively. The photographically recorded second-harmonic intensity patterns at the wavelengths of 0.53 and 0.42 μm are in good agreement with theoretical predictions.

The optical analog of the nuclear magnetic resonance rotary saturation effect is demonstrated and analyzed. A 9.6-μm C${\mathrm{O}}_2$ laser resonance excitation of vibrational transitions in gaseous C${\mathrm{H}}_3$F creates an effective splitting in the interaction representation of the optical two-level system. A radio-frequency standing-wave Stark electric field is tuned in rotary saturation resonance with the splitting. The change in transmitted laser beam intensity, because of rotary resonance, is measured in terms of transient optical signals obtained by the Stark switching method of Brewer and Shoemaker. The measured single and multiphoton rotary saturation resonance signals are compared with perturbation and computer solutions of nonlinear Bloch-Maxwell macroscopic equations. Doppler-free rotary saturation spectra are obtained in the presence of inhomogeneous Doppler broadening by a special scheme of Fourier analysis. A discussion is presented of the coupling of energy between two traveling waves that are in double resonance with the optical two-level system. The concept of spin temperature is discussed in relation to a coherently excited optical level system with a long fluorescence lifetime. The possibility of a cross-relaxation double resonance experiment between an optical system and a spin system is outlined.

Two previously given theories are unified, facilitating the discussion of ray tracing in nonlinear and absorbing media. The media may be fairly general: dispersive, absorbing, anisotropic, inhomogeneous, and time varying. Self-focusing is demonstrated on a simple example, showing that absorption will slow down the buildup of high intensities expected from the convergence of the rays.

We present a formalism to analyze the effects of collisions and intramolecular energy transfer and dephasing processes ($T_1$ and $T_2$ relaxation) on level-population dynamics and unimolecular decay of laser-driven gas molecules. Utilizing a Boltzmann or optical Bloch equation for the density matrix to describe laser excitation of a relaxing multilevel absorber, we develop a generalized master equation (GME) to describe the dynamics of level population evolution for multilevel absorbers having arbitrary numbers of quantum levels $N$. Because the resulting level-population equations explicitly retain phase memory, the effect of complete or partial phase-coherent driving or incoherent driving by the laser field are all included within the GME solutions. The formalism provides straightforward means to calculate time-dependent solutions for arbitrary $N$, where a single expression is valid on transient, post transient, approach to steady state, and steady-state time scales. We illustrate the behavior of these solutions for the special case of three- and eight-level systems, but the method is not restricted with regard to number of levels. In both this and the following paper, we use specific examples and general arguments to show that an ordinary master equation (OME) is invalid even under apparently favorable conditions including saturating field strengths. We demonstrate that, for an $N$-level absorber with given coupling elements, there are $\frac{N}{2}$ [or $\frac{\mathrm{}(N+1\mathrm{})\mathrm{}}{2}$] independent saturation solutions and that special conditions on the relaxation times are required to produce the ordinary equal-population solution. Explicit forms for the steady-state radiative-transition matrix $W(\lambda =0)$ are derived for arbitrary $N$, and general arguments are given to show that the correct description of the dynamics given by the GME results from matrix singularities in $W(\lambda =0)$ which can never occur in an OME or similar rate equations.

Using the generalized master equation (GME) formalism, we analyze several approximations for the solutions of Boltzmann or optical Bloch equations. Included in our analysis are the Wilcox-Lamb equations, ordinary master equation (OME), and a new approximation which we designate as the renormalized small-eigenvalue (RSE) approximation. We demonstrate that in multilevel systems, the GME formalism allows direct systematic calculation of radiative-transition coefficients required in the Wilcox-Lamb equations, including all multilevel phase-coherent transitions. However, serious limitations in the applicability of the Wilcox-Lamb approximation are noted both in the high-pressure domain, as an asymptotically exact equation, and for low pressures, as a coarse-grained approximation. The Wilcox-Lamb equations, generally assumed to be valid in a high-pressure domain, are shown to fail in the presence of very efficient inelastic ($T_1$) relaxation processes regardless of the pressure. Under similar conditions the presence of efficient inelastic processes is also shown to invalidate the use of an OME even when inelastic cross sections are somewhat smaller and the Wilcox-Lamb equations are valid. In particular, we show that when energy losses to the heat bath are severe, an ordinary master equation, traditionally believed to provide a valid description of radiative contributions to level-population evolution when the ratio of Rabi to elastic collision frequency is small, is not valid under any conditions. On this ground we strongly question the recent use of a multilevel OME description to explain the existence of an energy threshold for laser-induced decomposition of S${\mathrm{F}}_6$. Finally, we show that for low gas pressures where Rabi oscillations are a prominent feature of the exact solutions, neither the OME nor the Wilcox-Lamb approximation provides an adequate coarse-grained solution. We show that the Wilcox-Lamb solution is in error primarily because of a circumstance not previously recognized, namely, a systematic large error in the way exponential terms contributing to the solution are normalized. A closely related approximation, the RSE approximation, which we present here for the first time, is shown to provide an accurate coarse-grained approximation to the fully coherent GME solutions.

The use of variational wave functions that are products of overlapping locally correlated functions for studying the ground state of lattice spin theories is described. It is found that very accurate results can be obtained for very simple trial functions. For determining the energy, the method is superior to methods based on blocks.

We consider the field equation obtained by minimizing the free energy of nematics with only the splay term different from zero. Solutions of this nonlinear equation are found by use of a geometric method. This approach makes it possible to generate solutions that would be very difficult to obtain in a more direct way. The method is illustrated by several examples. Special emphasis is put on configurations containing singular structures.

In the Van der Waals theory of liquids, very short-ranged intermolecular repulsions are approximated by hard-particle exclusions and somewhat longer-ranged intermolecular attractions are subject to a self-consistent mean-field treatment. This approach, which has recently been applied to pure nematogens, is here extended to nematogenic solutions. The molecular hard cores are taken to be spherocylindrical or spherical in shape and their contribution to the free energy of the system is evaluated using scaled-particle theory. The general theory, applicable to mixtures of any number of rodlike or effectively spherical molecules is derived and its applications to various types of nematogenic systems are discussed. Finally, the way in which the theory could be modified to treat species whose molecular shapes cannot satisfactorily be approximated by spheres or spherocylinders is indicated.

The Van der Waals theory of nematogenic solutions derived in Paper I of this series is here applied to binary mixtures with effectively spherical solute molecules and rodlike solvent molecules. Predicted temperature-mole-fraction phase diagrams are shown to be in rather good agreement with experimental data for the systems C${\mathrm{Cl}}_4$-$p$, $p^{\prime }$-dihexyloxyazoxybenzene (DHAB), (C${\mathrm{H}}_3$)${}_4{}^{}{}_{}{}^{}\mathrm{Sn}$-methoxybenzylidene $n$-butylaniline (MBBA), (${\mathrm{C}}_2$${\mathrm{H}}_5$)${}_4{}^{}{}_{}{}^{}\mathrm{Sn}$-MBBA, (${\mathrm{C}}_3$${\mathrm{H}}_7$)${}_4{}^{}{}_{}{}^{}\mathrm{Sn}$-MBBA, and (${\mathrm{C}}_4$${\mathrm{H}}_9$)${}_4{}^{}{}_{}{}^{}\mathrm{Sn}$-MBBA. On the basis of these calculations, it is argued that when a more or less spherical solute is added to a nematic mesophase, the onset of the nematic→isotropic phase transition is controlled primarily by excluded-volume effects and the overall strength of the intermolecular attractions in the system.

We have measured the coexistence curve of a high-molecular-weight ($M_w=1.56\mathrm{}\times {10}^6$) polystyrene in cyclohexane over a temperature range of $4\times {10}^{-5\mathrm{}}<\frac{(T_c-T)}{T_c}<4\mathrm{}\times {10}^{-2\mathrm{}}$. The concentration difference $X^{+}-X^{-}$, in volume fraction of polystyrene, was well represented by the expression $X^{+}-X^{-}=B{\epsilon }^{\beta }+B_1{\epsilon }^{\beta +\Delta }+B_2{\epsilon }^{\beta +2\mathrm{}\Delta }$ based on the Wegner expansion, where $\beta =0.327\mathrm{}\pm 0.004$ and $\Delta$ is fixed at 0.50. The concentration difference could also be well represented by the expression $X^{+}-X^{-}=B{\epsilon }^{\beta }+B^{\prime }{\epsilon }^{{\beta }^{\prime }}$, where a somewhat higher value of $\beta$ was required. The diameter in the temperature range $4\times {10}^{-5\mathrm{}}<\frac{(T_c-T)}{T_c}<{10}^{-2\mathrm{}}$ was represented by $\frac{(X^{+}+X^{-})}{2}-X_c=A{\epsilon }^{\mu }$, where $\mu =0.860\mathrm{}\pm 0.006$. We have established, for the first time, the value of the asymptotic critical exponent $\beta$ of even a polymer system to be like that presently calculated for the Ising model. The observed exponent $\mu (=1-\alpha )$ is consistent with results of specific-heat measurements.

Modulational instability of the whistler-wave propagating oblique to the static magnetic field is discussed for a low-$\beta$ plasma. The instability conditions and the maximum growth rates are derived for both of the longitudinal and perpendicular modulation relative to the propagation direction of the whistler wave.

The photoabsorption coefficient of high-temperature (${10}^2$-${10}^5$ eV) and high-density (${10}^{20}$-${10}^{22}$ ${\mathrm{cm}}^{-3\mathrm{}}$) aluminum plasma in the range of $h\nu ={10}^2-{10}^5$ eV is calculated. The calculation uses detailed configuration accounting. The photoabsorption mechanisms included are the photoelectric effect, inverse bremsstrahlung, Compton effect, and bound-bound photoexcitation. The Stark broadening of the sharp-line and the absorption-edge structure is properly taken into account and its influence versus ion density is shown. The sensitivity of the results to changes in the various parameters is discussed.

Using the quantum-mechanical hard-sphere system as a reference system and the attractive interaction as a perturbation, we present computationally convenient theoretical methods for calculating the structural and thermodynamic properties of a fluid in the semiclassical limit. This approach is based on the assumption that, for a dense fluid, the quantum effects are largely determined by the repulsion due to the hard core and the attractive interactions play a minor role. The exponential approximation for the radial distribution function and "(optimized random-phase approximation) + $B_2$" for the excess Helmholtz free energy have been applied to calculate the structural and thermodynamic properties of the Yukawa-tail fluid in the semiclassical limit.

Comprehensive measurements of ultrasonic absorption in sodium-ammonia solutions near the critical mixing point and passing through the metal-nonmetal transition are presented and analyzed in detail. A few ultrasonic-velocity measurements in the same system are also reported. The observed critical divergence of the absorption (all the data being in the hydrodynamic regime) is shown to obey the Fixman-Kawasaki multiple-relaxation calculation by the adjustment of three fit parameters. The measured absorption and the characteristic frequency and amplitude parameters all appear to undergo a scaling crossover at a reduced temperature of about ${10}^{-2\mathrm{}}$. The presence of large-amplitude fluctuations in the solutions near the critical point is inferred as well as the presence of a distribution of relaxations related to the diffusive decay of these fluctuations. The results are also discussed in terms of the microscopic inhomogeneities or clusters proposed by Cohen and Jortner for metal-ammonia solutions.

The effect of backflow of quasiparticles on the dynamics of classical systems is investigated using the mean-field expression for the density response function proposed recently by Aldrich et al. It is observed that the overall results for classical liquids (e.g., liquid argon and rubidium) are qualitatively of the same nature as obtained from earlier mean-field expressions, whereas qualitative and quantitative improvement is achieved for one-component classical plasma.

The fluctuation-dissipation theorem gives the spectrum of thermal fluctuations in a physical variable for a system maintained at a given temperature in equilibrium. A more general theorem is derived here for the spectrum of fluctuations in a nonlinear system subjected to a stationary driving force under a stated set of assumptions. The method of derivation is thermodynamic and the theorem expresses the fluctuation spectrum in terms of the magnitude of the driving source as well as the nonlinear characteristic of the system.

In this Comment it is established that the numerical results for the nonlinear evolution of the $m=2\mathrm{}$ tearing mode are affected by the density of the poloidal grid. A coarse grid gives results that are quantitatively and qualitatively incorrect. The important qualitative error is that with a coarse grid the growth of the width of the associated magnetic island is exponential rather than algebraic.

We maintain that there was no qualitative error made in our calculations. As originally reported in our Letter [Phys. Rev. Lett. 39, 1618 (1977)], the width at which exponential growth ceases and algebraic growth starts depends on the values of $S$. As $S$ decreases, the transition from exponential to algebraic growth occurs at a larger island width. The coarse grid gave the right qualitative result, but gave a small error in the critical value of $S$ for which the growth is always exponential. Since the comparison with experiment was made using large-$S$ results and algebraic growth, this error has no effect on the conclusions of our orginal Letter.