The $3p\to 3d$ absorption spectrum of Mn vapor measured by Connerade, Mansfield, and Martin is compared to calculations based upon single-configuration Hartree-Fock multiplet theory and the formalism of Fano for the interaction of discrete and continuum states. It is concluded that the principal decay is through super-Coster-Kronig transitions. The observation of sharp structure associated with the excited states $3p^{5}3d^{6}4s^2\left({}_{}{}^6{}_{}{}^{}D\right)$, for example, is shown to be consistent with this mechanism. Decay selection rules are found to be a consequence of rotation and inversion symmetry. Spin-orbit interaction in the $3p$ subshell is treated exactly to obtain the sharp structure below the giant ${}_{}{}^6{}_{}{}^{}P$ resonance. It is necessary to include the off-diagonal matrix elements of the decay matrix $\Gamma$ to obtain the correct line shape since the resonances overlap. The resonance shift of the Hartree-Fock multiplet energy of the ${}_{}{}^6{}_{}{}^{}P$ state is computed to be -2.2 eV. The effects of the energy dependence of the parameters entering the Fano formalism are examined and the strength of other transitions (e.g., $3p\to 4d$) is estimated. The importance of the interaction with continua of the form $3p^{5}3d^{m}4s^n\epsilon l$, $m+n=7\mathrm{}$, (the mechanism of Dehmer, Starace, Fano, Sugar, and Cooper) is assessed. The relationship of this work to calculations of the nonradiative lifetime of $3p$ holes in transition metals is discussed.

We have measured the angle and concentration dependence of the first cumulant of the field-correlation function for multiply scattered light from polystyrene latex particles in water. The experimental results are analyzed in terms of a model based on the assumptions that the scattering particles are pointlike and that the number of renormalized scattering events is Poisson distributed. The agreement between theoretical and experimental results is satisfactory.

We consider the total multiple-scattered light field from a system of Brownian diffusing particles. We assume the correlation function is due to contributions from all the independent single-scattering events leading to the multiple-scattered field. We then assume the relative magnitudes of the various orders of the multiple-scattered light are given by the Poisson distribution. For particles of radius $r\ll \lambda$, $\lambda$ being the wavelength of light, we arrive at expressions for the depolarization ratio and the polarized and depolarized linewidths as functions of the average number of scattering events $\overline{n}=\sigma \rho x$, where $\sigma$ is the single-particle scattering cross section, $\rho$ is the particle concentration and $x$ is a distance related to the scattering volume size. We give experimental data to support our contentions.

A complete scheme for a scaled particle theory for hard spheres with nonadditive diameters is developed. As macroscopic representation of the nonadditive hard spheres a cavity surrounded by a family of concentric selectively permeable membranes is used. The resulting equations are solved explicitly for a binary mixture with positive nonadditivity. Independent of the scheme used, scaled particle theory predicts a demixing phase transition at high density and sufficient nonadditivity. The critical value for the nonadditivity parameter is found to be ${\Delta }^c=0.215\mathrm{}$. Agreement with the few Monte Carlo results available is good.

Using the formalism of irreducible spherical tensors, we derive a general expression for the light intensity scattered in a process of any order in a low-density fluid; elastic (both coherent and incoherent) and inelastic (incoherent) scattering are considered. The relevant physical quantities involved are the scalar invariants associated with the irreducible parts of the various scattering tensors; the polarization dependence is governed by $P_0\left(cos\eta \right)$ and $P_2\left(cos\eta \right)$ only, where $P_k$ is the Legendre polynomial of order $k$ and $\eta$ the angle between the polarizations of the scattered and incident light. Depolarization ratios and molecular selection rules are also given for up to four-photon processes.

The critical slowing down of the optical soft mode, or the order-parameter fluctuation, in the nematic phase of methyl-deuterated $p$-azoxyanisole is detected for the first time by combining the proton spin-lattice relaxation times in the laboratory and in the rotating frames. Freed's recent theory of spin relaxation by quasicritical order-parameter fluctuations is used to satisfactorily interpret our NMR measurements. The present study confirms the mean-field prediction of symmetry in critical order-parameter fluctuations at isotropic-nematic transitions.

In this paper Onsager's theory of the orientational order in a three-dimensional system of hard rods is reanalyzed as a nonlinear eigenvalue problem. Bifurcation is found and the equation of state is calculated from the orientational distribution function for a nematic phase. We also investigate the corresponding twodimensional system of hard lines. The existence and order of a phase transition are shown to depend on both the direction of bifurcation and on properties of the global solutions. The analysis can be adapted to other nonlinear equations obtained in theories of liquid crystals.

A simple Thomas-Fermi shell model is proposed, in which the discrete natures of the bound states and the resonance states are considered. The features of the shell oscillations of the equation of states along the Periodic Table are examined and discussed.

A recent x-ray study, by De Jeu and De Poorter, of the smectic phases of heptyloxybenzylidene-pentylaniline shows that the individual molecules are tilted with respect to the layer normal in both the smectic-$A$ and the smectic-$C$ phases. We demonstrate here that a system of long molecules, moving within a planar layer, and making a fixed angle $\epsilon >0\mathrm{}$ with the layer normal, has a second-order transition from a phase with disordered tilt directions to a phase with the tilt directions aligned along some axis. It is suggested that this model phase transition underlies the smectic-$A$—smectic-$C$ transition, at least in those cases where the smectic-$A$ layer thickness is significantly smaller than the molecular length. In this model the effective tilt angle $\alpha$, as measured in optical and magnetic resonance experiments, grows continuously with increasing order of the tilt directions, and saturates at the value $\alpha =\epsilon$. On the other hand, the biaxiality ($\Delta$) in the molecular orientational order passes through a maximum and remains small at all temperatures; e.g., we find $\left|\Delta \right|<\mathrm{}6\mathrm{}\times {10}^{-2\mathrm{}}$ for $\epsilon =30\mathrm{}°$. The phase transition is driven by the gains in translational entropy and total attractive energy which accompany an increase in order of the tilt directions. (At the transition temperature these effects balance the loss in orientational entropy.)

We apply an approximate form of the exact one-particle-irreducible renormalization-group generator to the calculation of the equation of state. Several approaches are explored. (i) Global nonlinear trajectories for the Hamiltonian parameters are exploited to give nonlinear crossover equations of state. Logarithms of the reduced temperature $t\equiv \frac{(T-T_c)}{T_c}$ are automatically exponentiated to give power-law behavior. (ii) The form and asymptotic properties of the equation of state are described for those systems whose complete nonlinear trajectories cannot be explicitly obtained. (iii) Nonlinear trajectories for the irreducible Green's functions are solved to give a fully exponentiated equation of state containing no logarithmic terms. (iv) Simple critical (noncrossover) equations of state are obtained by iteration of the generator with linear and quasilinear trajectories. (v) Operator nonlinear equations are solved to give the crossover equation of state for an arbitrary order $\theta$ critical system, both in an $\epsilon$ expansion and at the borderline dimension. (vi) Combination of the Green's functions are formed into Green's eigenfunctions or operators and their nonlinear trajectories used to calculate fully exponentiated equations of state for order $\theta$ Ising systems. Systems considered include the usual Wilson-Fisher Hamiltonian [methods (i)-(iv)], the Sak-model compressible ferromagnet [(i)-(ii)], the $\mathrm{nm}$-hypercubical model (of which the dilute quenched random ferromagnet forms the $n\to 0$ limit) [(ii)], and the isotropic $n$-component order $\theta$ ferromagnet [(iv)-(vi)]. The tricritical ($\theta =3\mathrm{}$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.

We describe radiative-decay-induced four-level crossover signals in Doppler-broadening-free saturation spectroscopy. A four-level crossover signal occurs when atoms in a segment of the velocity distribution absorb light from the saturating beam, radiatively decay in part to a different lower level, and cause enhanced absorption from the probe beam on a transition that shares neither a common upper nor a common lower level with the transition of the first absorption. We interpret a signal in a saturation spectrum published by Hänsch, Shahin, and Schawlow as a four-level crossover signal induced by radiative decay.

Recent stopping-power data of solids with atomic number $Z_2$ ranging from 13 to 79 for ions ranging in atomic number $Z_1$ from 1 to 9 and in velocity from $7v_0$ to $12v_0$ are analyzed in terms of $Z_1^3$ effects and effective ion charges. They are found to be consistent with current theory. High-velocity forms of the $Z_1^3$-proportional stopping-power contributions are calculated by the method of moment integrals.

We extend our recent work [Phys. Rev. A 16, 1423 (1977)] on total elastic electron scattering cross sections for ${\mathrm{N}}_2$ to include differential cross sections from 0 to 30 eV. Extensive comparisons with the results of experiments and other theories are made. The overall agreement is good, indicating that the multiple-scattering method is capable of providing a realistic first approximation to the electron-molecule interaction. At the same time, small systematic disagreements with experiment indicate clearly the need and objective of future improvements.

In order to examine the adiabatic invariance of the magnetic moment of a charged particle gyrating in electromagnetic fields, Kulsrud (1957) studied the following problem: $\frac{d^{2}x}{d{t}^2}+{\omega }^2\left(\tilde{t}\right)x=0\mathrm{}$, $\tilde{t}=\epsilon t$, $\epsilon \ll 1$. Here, a systematic procedure is adopted and a different choice of independent variables made, giving better results. This systematic procedure involves a combination of the method of multiple scales and Poincare-Lighthill-Kuo method to treat this problem.

The physics that determines the shape of the energy spectrum of stopping negative mesons is discussed. We note that a white energy spectrum, for arrival probability $F\left(E\right)\mathrm{}$, not flux $P\left(E\right)\mathrm{}$, follows for a large class of models. The importance of distinguishing between $F\left(E\right)\mathrm{}$ and $P\left(E\right)\mathrm{}$ is stressed: $P\left(E\right)\mathrm{}$ reflects the energy ($E$) dependence of the total inelastic and capture cross sections, while $F\left(E\right)\mathrm{}$ is independent of these. The $F\left(E\right)\mathrm{}\propto E$ behavior found in some models is due to meson capture at positive energies, i.e., capture behind a positive centrifugal barrier. We explore the possibility of a concentration effect on capture ratios in binary mixtures, and conclude that, as long as the energy losses in meson-atom collisions are not much smaller than the barrier height, such concentration effects are negligibly small.

The $1s2p^2{}_{}{}^2{}_{}{}^{}P$ state of lithium can autoionize by decaying radiatively into the adjacent continuum. The probability is smaller than the discrete transition probability to the ground state $1s^{2}2p{}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ and equal to the decay probability to the ${}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ Rydberg series. Our theoretical lifetime, $\tau =0.041\mathrm{}\times {10}^{-9\mathrm{}}$ sec, differs from the experimental value of $\tau =0.015\mathrm{}\pm 0.01\times {10}^{-9\mathrm{}}$ sec. The mechanism of radiative autoionization affects the deexcitation dynamics of many valence excited states of neutral species.

The sign reversal of the wave function under a $2\pi$ rotation ("spinor character") is demonstrated in a spin-1 system (deuterium) when only a two-level subsystem is irradiated. Under irradiation of all three levels of the system $2\pi$ symmetry is recovered, and a continuous variation from $4\pi$ to $2\pi$ symmetry is demonstrated by irradiation with two different, continuously adjustable asynchronous rf fields.

Self-consistent-field (SCF) calculations have been used to determine the ground negative-ion and ground neutral potential-energy curves for HeH, ArH, and ArCl. In the systems studied, the $X{}_{}{}^1{}_{}{}^{}\Sigma$ potential of the negative-ion state is more repulsive at intermediate internuclear separations than the $X{}_{}{}^2{}_{}{}^{}\Sigma$ potential of the neutral partners, leading to a curve crossing of the negative-ion state into the continuum. However, at small internuclear separations the negative-ion potential becomes less repulsive than the neutral potential, leading to a second curve crossing out of the continuum. The calculated potential-energy curves are used to provide estimates of the energy thresholds and cross sections for electron detachment in the collisions ${\mathrm{H}}^{-}$ + He, ${\mathrm{H}}^{-}$ + Ar, and ${\mathrm{Cl}}^{-}$ --Ar. The angular thresholds for electron detachment in differential-scattering measurements are also given and compared to experimental data.