A model for cholesteric liquid crystals with biaxial molecules is introduced. It is found that the biaxial order parameter for such systems is of order ${\left(q_{0}a\right)}^2$ where $a$ is a molecular length and $q_0=\frac{2\pi }{p}$ where $p$ is the helical pitch. The generalized susceptibility is calculated and is found to predict a fluctuation instability as in uniaxial cholesterics.

The molecular theory of the smectic $A$ and $C$ phases is extended to the smectic $B$ and $H$ phases by including a soft-core repulsive interaction in addition to the dipole-dipole interaction used previously. The possibility of translational order (a two-dimensional hexagonal lattice in each smectic plane) is included in addition to the orientational order of the molecular dipoles. Four phases are found: (1) a disordered phase (smectic $A$), (2) an oriented phase with no translational order (smectic $C$), (3) a translationally ordered phase with no orientational order (smectic $B$), and (4) a phase with translational and orientational order (smectic $H$). In the oriented phases (smectic $C$ and $H$) the director is tilted with respect to the plane normal. The smectic-$A$-smectic-$C$ phase transition and the smectic-$B$-smectic-$H$ phase transition are second order; the other transitions are first order. The temperature dependence of the order parameters, the entropy, and the specific heat are computed.

The quantum second virial coefficient of two-dimensional systems interacting with a hard-core or Lennard-Jones 6-12 potential are calculated together with the contribution this coefficient makes to the specific heat of such systems. The results are compared with experimental data from low-density ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ monolayers adsorbed on graphite. The calculation provides a qualitative and quantitative understandiang of the experimental measurements. In particular it illuminates the manner in which the interaction and the difference in statistics cause the very different specific-heat measurements in the two isotopes. The magnetic susceptibility of low-density ${\mathrm{He}}^3$ monolayers is predicted.

The motion of charged particles owing to an electromagnetic plane-wave pulse propagating in a dispersive region is considered. The calculation includes nonlinear and relativistic effects. It is argued that this solution may be viewed as a correction (for the motion of the particles) to the linear theory of propagation of an electromagnetic pulse within a plasma. The importance of such corrections is also considered.

A theory treating the space-time structure of nonequilibrium processes is developed. The basis of the theory is founded upon the validity of: (i) the kinetic form of the Gibbs equation and consequently, the generalized thermodynamic potentials, and (ii) the balance equation of entropy, the continuity equation, and the conservation of energy. The dissipation function is derived from the kinetic form of the Gibbs equation. The only additive invariant remaining in nonequilibrium thermodynamics is the energy; within the entropy representation, it can be used to establish a local energy conservation or power equation. The first and second variations of the power equation determine the stationary-state conditions and the stability of the stationary state, respectively. It is impossible to characterize nonequilibrium stationary states in terms of only one generalized thermodynamic potential; a free extremum of the potential does not exist since the stationary-state values of the forces enter in the first variation of the potential. A constrained variational principle of least dissipation of energy is applicable to certain classes of nonequilibrium stationary-states, which may or may not be spatially homogeneous. Distinction is made between variational principles (e.g., least dissipation of energy) and extremum principles (e.g., minimum production of entropy). A kinetic criterion of mechanical equilibrium is derived from the direction of the entropy flux.

The kinetic theory of gases and liquids is discussed in terms of the phase-space density $f\left(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\right)$ and its autocorrelation function. Associated with the latter is the memory function ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}z\right)$ troduced in an earlier paper. Here, we analyze those properties of ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}$, for a classical system of structureless interacting particles, which can be obtained without approximation. Apart from symmetry- and stability-related properties, and those that express the conservation laws, we give microscopic derivations of (i) a new sum rule for ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}z\right)$ which involves only the static-pair-correlation function and (ii) two new relations between ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}$ and thermodynamic derivatives, namely the specific heat $c_v$ and the pressure derivative ${\left(\frac{\partial p}{\partial T}\right)}_n$. These exact relations, if fulfilled by an approximate model for ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}$, guarantee that the approach to equilibrium is described in a thermodynamically and dynamically consistent manner. It is also shown how, for such a model, the transport coefficients are obtained.

We derive an identity which determines the infinitesimal response of the cumulant average of the product of $m$ arbitrary local fluctuating quantities (connected $m$-point functions) to changes in its arguments engendered by a member of an arbitrary continuous group of point transformations. We illustrate the use of the identity in a variety of cases, and in particular we show that there is an intimate connection between the covariance of the connected $m$-point functions under the special conformal group and under the group of dilations. By combining our identity with the assumptions of the operator algebra, we find that at the critical point the connected density $m$-point functions are covariant under both groups. We also give another derivation for an expression for the exponent $x=\left(\frac{1}{2}\right)(5-\eta )$, which was previously found by Green and Gunton.

A simple formula for $S(k,\omega )$ which was originally established only as an approximation applicable to a boson system is found to hold exactly for any uniform macroscopic system of bosons or fermions. This is deduced by making elementary substitutions in equations written down by Huang and Klein several years ago. A general formula for moments of $S(k,\omega )$ is derived, and examined for several special cases. The utility of some of these relations in specific perturbation treatments of $S(k,\omega )$ for liquid ${\mathrm{He}}^4$ is discussed. They are also used here to derive a new approximate formula for the kinetic energy per particle in ${\mathrm{He}}^4$ in terms of the density and ${\mathrm{He}}^4$ liquid-structure fnction. A procedure for applying the formulas in semiphenomenological theories is also outlined.

Preliminary observations of a new effect in photoelectron statistics is reported. By utilizing an AND-gate system, the joint probability of emission of two photoelectrons from a photocathode at times $t_1$ and $t_2$ has been measured over a range of intervals, $\tau =t_2-t_1$, lying between 4.0 and 50 nsec with a resolution $\Delta \tau$ at 1.0 nsec, under illumination with a single-mode He-Ne laser and a tungsten lamp. The joint probability of emission as a function of $\tau$, under both conditions of illumination, rises to a large peak in the neighborhood of $\tau =15\mathrm{}$ nsec, and when $\tau$ exceeds 25 nsec, it drops rapidly to the constant value predicted by the classical theory of Mandel. In addition, there is a distinct fine structure which becomes more pronounced in the "dark" emission. Significantly, the amplitude of the peak varies linearly with the mean intensity of the incident light, while the constant value obtained at long intervals varies quadratically. To explain the observed phenomena, a generalization of the Mandel theory is formulated on the basis of the axiomatic theory of probability and the following hypotheses: (1) The emission of two photoelectrons separated by a time interval are statistically dependent events owing to intrinsic processes within the cathode as well as extrinsic processes of the radiation field; (2) If the instantaneous intensity of the radiation field is proportional to the square of the field amplitude, then the probability of emission is a function of this amplitude. Both hypotheses are found to be necessary to provide a base for interpreting the linear dependence of the joint probability peak on field intensity. Moreover, the generalized formulation reduces to Mandel's expressions when the following restrictions are imposed: The emission process is statistically independent; the probability of emission is proportional to the intensity of radiation, and the radiation field is a Gaussian random process.

The quenching mechanism of positronium in N${\mathrm{O}}_2$ was investigated by measuring both positron lifetimes and two-photon angular-correlation curves of annihilating positron-electron pairs in N${\mathrm{O}}_2$ adsorbed on silica gel. It is found that instead of a conversion reaction by N${\mathrm{O}}_2$, chemical quenching is responsible for the quenching of positronium. The quenching cross section for N${\mathrm{O}}_2$ is compared with the conversion cross sections of ${\mathrm{O}}_2$ and NO gases and also with the chemical quenching cross sections of ${\mathrm{Br}}_2$ and ${\mathrm{I}}_2$ molecules. The cross sections for chemical quenching are at least two to three orders of magnitude larger than the cross sections for conversion quenching of Ps. The bond strengths of these molecules are found to play an important role in Ps quenching process.

Calculations of the trimer ground-state energies of three identical bosons interacting via molecular pair potentials are discussed. The accuracies of variational calculations with a Jastrow trial function and of upper and lower bounds in terms of dimer ground-state energies are determined in a harmonic approximation.

In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the ${\mathrm{He}}^{+}$-Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the ${\mathrm{He}}^{+}$-Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant $a_4$ by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-Stückelberg theory, and is found to behave correctly in the strong-coupling regime as $a_4$ increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-Stückelberg theory over all values of $a_4$ for the system studied.

The results of Haake and Glauber concerning the solution of the superradiant master equation of Bonifacio et al., are compared with those of Degiorgio and Ghielmetti. This is accomplished by using the atomic coherent-state representation of Arecchi et al., which provides a convenient mathematical formalism for the comparison. The relationship of these results to the recently presented exact Fokker-Planck-equation representation of Narducci, Coulter, and Bowden is discussed. This representation is used to prove the equivalence of the main conclusions derived by Degiorgio and Ghielmetti to a special case of the results of Haake and Glauber. In addition, it is shown that the expectation values calculated by Haake and Glauber are particular solutions of a coupled hierarchy of equations of motion constructed from the exact Fokker-Planck equation, neglecting the noise and diffusion terms.

The vuv transition probabilities of C I have been measured in emission with a high-current, wall-stabilized are operated in a mixture of argon, carbon dioxide, and hydrogen at atmospheric pressure. A radiometric technique has been performed utilizing synchrotron-calibrated gas discharge lamps (deuterium lamps) as spectral radiance standards. At present, the number density of the emitting carbon atoms of the used multicomponent plasma cannot be determined by plasma-spectroscopic means with satisfactory reliability. Therefore, the $A_{n m}$ values have been measured relative to that of the longest-wavelength resonance multiplet ($\lambda =165.7\mathrm{}$ nm) and subsequently have been normalized to an absolute scale by recent results of lifetime measurements of the upper levels $3s{}_{}{}^3{}_{}{}^{}P_{}^{\circ }{}_{}{}^{}$ of this multiplet. Because the $3s{}_{}{}^3{}_{}{}^{}P_{}^{\circ }{}_{}{}^{}$ levels decay with a significant transition probability only to the ground-state level, branching ratios need not be considered when converting the lifetime to the absolute $A_{n m}$ value. The transition probabilities determined on this basis disagree in some cases with previous experimental and theoretical results presently adopted as the most-reliable values. A maximum deviation by a factor of approximately 2 has been observed. Using the synchrotron-calibrated spectral radiance standards, the ratio of the $A_{n m}$ values between the longest-wavelength resonance multiplet $\lambda =165.7\mathrm{}$ nm and the prominent visible line $\lambda =505.21\mathrm{}$ nm was determined. In this manner a set of absolute ${\mathrm{C}}_{\mathrm{I}}{A}_{n m}$ values in the visible range was established based also on the lifetime of the $3s{}_{}{}^3{}_{}{}^{}P_{}^{\circ }{}_{}{}^{}$ levels. $A_{n m}$ was found to be (1.4 ± 0.3) × ${10}^6$ ${\mathrm{s}}^{-1\mathrm{}}$ for $\lambda =505.21\mathrm{}$ nm.

Variational principles for the estimation of the matrix element $W_{\mathrm{nn}}\equiv ({\phi }_n,W{\phi }_n)$ for an arbitrary operator $W$ are of great interest. The variational estimates are constructed from a trial wave function ${\phi }_{nt}$, an approximation to the nth normalized bound-state eigenfunction ${\phi }_n$, and of a trial auxiliary function $L_t$, an approximation to $L$ which satisfies $(H-E_n)L=(W_{nn}-W){\phi }_n\equiv q\left({\phi }_n\right)$. Variational-principle applications have been limited by the difficulty of obtaining a reasonable $L_t$, among other things, one demands that $L_t$ approach $L$ as ${\phi }_{nt}$ approaches ${\phi }_n$. The equation $(H-E_{nt})L_t=q\left({\phi }_{nt}\right)$, where $E_{nt}=({\phi }_{nt},H{\phi }_{nt})$, is known not to provide such an $L_t$. A practical procedure for handling complicated systems given a reasonably accurate Rayleigh-Ritz trial function ${\phi }_{nt}$ is called for. This paper provides such a procedure using techniques developed in the establishment of variational bounds on scattering lengths. Given $H$ and ${\phi }_{nt}$, we define $L_t$ by $A{L}_t=q\left({\phi }_{nt}\right)$, where $A$ differs from $H-E_n$ in that the influence of states 1 through $n$ has effectively been "subtracted out"; the operator $A$ is non-negative. A functional $M\left(L_{tt}\right)$ is constructed which is an extremum for $L_{tt}=L_t$. Variational parameters contained in $L_{tt}$ can be determined by extremizing $M\left(L_{tt}\right)$, thereby providing an approximation to $L_t$. The method is analogous to the determination of parameters in ${\phi }_{nt}$ by the minimization of $\frac{({\phi }_{nt},H{\phi }_{nt})}{({\phi }_{nt},{\phi }_{nt})}$. The method is immediately applicable to the variational determination of off-diagonal matrix elements $W_{nm}$ and of diagonal matrix elements of normal and of modified Green's functions.

Rigorous upper and lower bounds to dipole oscillator strengths have been calculated for 17 of the low-lying singlet and triplet transitions in He and ${\mathrm{Li}}^{+}$ using accurate Hylleraas-type wave functions of some 135-138 terms. About a third of the calculated $f$ values can be guaranteed to 1% or better, and in the best case (${\mathrm{Li}}^{+} 2{}_{}{}^3{}_{}{}^{}S\to 2{}_{}{}^3{}_{}{}^{}P$) the bounds tighten to about 0.05%, some two orders of magnitude better than corresponding experimental determinations. By comparing with an extensive compilation of experimental results, we find the quality of the rigorous bounds to generally match or exceed that of experimental determinations, and about a dozen experiments can be ruled out entirely. Trends with respect to nuclear charge, spin multiplicity, and principal quantum number are examined, and special emphasis is given throughout to sources of uncertainty in the calculated oscillator strengths which are revealed by the error-bound formulas.

The rotation-inversion spectrum of ${}_{}{}^{14}{}_{}{}^{}\mathrm{NT}_3^{}{}_{}{}^{}$ in the ground vibrational state has been measured with high-resolution microwave techniques in the frequency range of 210-633 GHz. Components of the $J=0\mathrm{}\to 1, 1\to 2, \mathrm{and} 2\to 3$ transitions were observed. Analysis of the results yielded the following rotational constants: $B_0=105565.373\mathrm{}\pm 0.034$ MHz, $D_J=2.5981\mathrm{}\pm 0.0024$ MHz, and $D_{JK}=-4.472\mathrm{}\pm 0.006$ MHz, with the ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$ nuclear quadrupole coupling $eqQ=-4.170\mathrm{}\pm 0.049$ MHz. The inversion frequency, corrected for rotational distortion, was found to be 305.89 ± 0.11 MHz. Molecular substitution structures obtained were bond length 1.0132 ÅA and bond angle 107° 13' for the combination ${}_{}{}^{14}{}_{}{}^{}\mathrm{NH}_3^{}{}_{}{}^{}$: ${}_{}{}^{15}{}_{}{}^{}\mathrm{NH}_3^{}{}_{}{}^{}$, ${}_{}{}^{14}{}_{}{}^{}\mathrm{NT}_3^{}{}_{}{}^{}$; bond length 1.0128 ÅA and bond angle 107° 2' for the combination ${}_{}{}^{14}{}_{}{}^{}\mathrm{ND}_3^{}{}_{}{}^{}$: ${}_{}{}^{15}{}_{}{}^{}\mathrm{ND}_3^{}{}_{}{}^{}$, ${}_{}{}^{14}{}_{}{}^{}\mathrm{NT}_3^{}{}_{}{}^{}$.

Compton profiles are calculated for $L$-shell electrons within the Born approximation by use of nonrelativistic "exact" hydrogenic bound- and continuum-state wave functions. The wave functions are expressed in parabolic coordinates and the resulting matrix elements are evaluated following the method of F. Bloch (1934). Impulse-approximation profiles are calculated, and comparisons with the "exact" profiles show that, as expected, the two profiles lie very close to one another for weak binding and high incident photon energies. However, even when these conditions are not fulfilled, the two curves have a tendency to cross one another in the neighborhood of the profile center. This tendency has been observed in previous studies of $K$-shell profiles. However, unlike $K$-shell profiles, the $2S$ Compton profiles exhibit a secondary maximum. The secondary maxima occur at approximately the same region in $q$, where the impulse-approximation profiles exhibit a plateau. The location of this plateau is shown to be related to a node in the bound-state wave function and is around $q=\frac{Z}{2}$ for the $2S$ profiles. The impulse approximation, being a monotonic decreasing function in $\mathrm{}\left|q\right|\mathrm{}$, cannot exhibit the secondary-maximum structure appearing in the "exact" hydrogenic profiles. The intensity of the secondary maximum in the $2S$ profiles is reduced by over an order of magnitude from the central peak. The $2P^{\mathrm{}\left(0\right)\mathrm{}}$ Compton profiles also exhibit structure, however, unlike the relatively small secondary maximum in the $2S$, the two maxima in the $2P^{\mathrm{}\left(0\right)\mathrm{}}$ profiles are of the same order of magnitude. Integrated profiles (incoherent-scattering factors) are calculated and the impulse-approximation results agree with the "exact" results over a wide range of binding energies owing to profile crossover near the center. Waller-Hartree incoherent-scattering factors give a closer agreement with the "exact" results than observed for $K$-shell electrons; however, for low-momentum transfer the Waller-Hartree results can differ from "exact" results by more than 50%. In such regions, impulse-scattering factors represent a considerable improvement over the Waller-Hartree factors.

The inelastic collisions of a heavy fast particle of charge $Z_{1}e$, incident at an impact parameter $b$, with an electron bound in an atom isotropically and harmonically is considered. The "atom" is treated by quantum mechanics, extending an earlier classical calculation by Ashley, Ritchie, and Brandt. A multiople of the Coulomb interaction for distanct collisions is used. Two methods are employed to calculate $Z_1^3$ contributions to the mean energy loss and the excitation probabilities due to dipole and quadrupole terms in the multipole expansion. In one method the dipole interaction is taken into account exactly by applying the quantum formalism of the forced harmonic oscillator, and the quadrupole interaction is treated as first-order perturbation. The second method of calculation employs both dipole and quadrupole interactions as time-dependent perturbations to second order on the free harmonic oscillator. The energy loss calculated for a particle incident on a straight-line trajectory agrees with the classical calculation. Reasons for this agreement are goven. Simplifications of the calculation owing to time-reversal symmetry considerations and selection rules for the harmonic oscillator are discussed. Comments are made concerning the relative importance of distant and close collisions at high and low velocities.

Cross sections have been determined for electron capture into the $3s$ and $4s$ states of atomic hydrogen by proton and deuteron (1.2-8.2 keV) impact on Ar, He, and Kr. The He cross sections both show an exponential increase with energy. Previous structure is confirmed and additional structure is indicated in the $3s$ cross sections for impact on Ar. These data represent the first measurements of the $3s$ and $4s$ cross sections for impact on krypton.