Exact duality transformations are used to study a three-dimensional lattice version of the de Gennes model of the smectic-$A$-nematic transition. It is shown that the partition function of this lattice model maps exactly onto that of a system of interacting dislocation loops. Analogies with other models are used to predict the behavior of this lattice model at several special points in the parameter space.

Scaling concepts and fractal statistics are used to assess the effects of intermittency on the important process of diffusion in fully developed turbulence. Sizable corrections to Richardson's "$\frac{4}{3}$ law" and related laws are found. Reexamination of existent data shows agreement with theory.

The effect of a periodic perturbation on a nonlinear dynamic system undergoing a sequence of period doublings is investigated. The results obtained from linear response theory and from numerical calculations resemble the observations made by Giglio on Rayleigh-Benard convection.

We derive a new Hamiltonian, for a charged particle in a time-dependent applied electromagnetic field, which depends on the fields $\overrightarrow{E}$ and $\overrightarrow{\mathrm{B}}$ directly rather than on the potentials $\phi$ and $\overrightarrow{A}$. The Hamiltonian is "nonlocal" in that it involves $\overrightarrow{E}$ and $\overrightarrow{\mathrm{B}}$ at all points in space, in contrast to the usual "local" Hamiltonian, which only involves $\phi$ and $\overrightarrow{A}$ at the position of the charge. The new and usual Hamiltonians are compared, and the canonical transformation which connects them is presented. We discuss the physical interpretation of the interaction terms appearing in the canonical momentum, angular momentum, and the Hamiltonian. A relativistic generalization is given.

Most irreducible-matrix representations in point-group symmetries can adopt monomial form. In that case all representational matrices have only one nonzero element in each row and column. The so-obtained standard basis choice contrasts with the conventional Wigner-Racah option. Monomial representations give rise to interesting properties of the corresponding Clebsch-Gordan series: All coupling coefficients are equal in absolute value and a natural intrinsic multiplicity separation is obtained. The concept is also useful in reaching a consistent solution of the multiplicity problem in the reduction of direct products, involving the fourfold $U^{\prime }$ representation of the octahedral spinor group. Several tables of basis transformations and coupling coefficients in octahedral and icosahedral symmetries are included.

The relationship between densities and density matrices is explored in the case of a finite-basis-set expansion. The space of one-electron density matrices can be divided into two orthogonal subspaces with elements in one of them in one-to-one correspondence with densities. A component in the other does not contribute to the density. The set of densities is convex but there may be densities which cannot be obtained from a density matrix. The matrix of a local potential has a component only in the first subspace, and any such matrix can be obtained from a local potential. It is possible for Hamiltonian matrices differing by the matrix of a local potential to have a common ground-state eigenvector, so a Hohenberg-Kohn theorem cannot always be established. When it can, the explicit local potential with a given ground-state density can be formally obtained when appropriate conditions are satisfied. The details of the decomposition of the space of matrices and of subsequent developments depend on linear-dependency relationships among basis-set products, and are thus basis-set dependent.

A hydrogenlike wave function is expanded in terms of a series of hydrogenlike wave functions referred to an arbitrary center in space. The expansion coefficient is given as a function of the distance between two centers. The coefficient for the radial part contains an integral of a spherical Bessel function with two Gegenbauer polynomials of different arguments. This integral is evaluated analytically to be a primitive polynomial multiplied by an exponential function of the distance. The coefficient for the angular part is represented in terms of the spherical harmonics.

The effects of vacuum polarization on the spin structure of the energy levels of $\overline{p}$ atoms are calculated from a new relativistic wave function which includes both recoil and the $\overline{p}$ anomalous magnetic moment. A new value for the $\overline{p}$ magnetic moment is derived, and the changes in the hyperfine structure of protonium are discussed.

The scaling-variational method is applied to anharmonic-oscillator models with the Hamiltonian $H=p^2+h{x}^2+g{x}^{2k}$ to enable the discussion of two important aspects not previously analyzed. First, it is shown that the introduction of a scaling factor which is variationally optimized assures us of the correct dependence of the approximate eigenvalue with $g$. Second, it is shown that quantities ${\overline{E}}_n=〈{\phi }_n|H{\phi }_n〉$ are very good approximations to the exact eigenvalues whenever the trial function ${\phi }_n$ satisfies the quantum virial theorem.

A new formula for the Fourier transform (FT) of a two-center RBF (reduced Bessel function) charge distribution permitting partial-wave analysis is derived with the use of Feynman's identity. This formula is valid for all quantum numbers. It is also independent of the orientation of the coordinate axes. A new representation for the two-center overlap integral (to which the kinetic energy, two-center attraction and Coulomb repulsion integrals can be readily reduced) and for the three-center attraction integral is obtained with the help of the FT. It is stable for all values of the orbital exponents. The method developed by Graovac et al. for computing repulsion integrals for $s$ states is generalized to include all states. Numerical test values of several one- and two-electron integrals are also reported. A strategy which should enhance the efficiency of computation by making maximal use of the FT's already computed is suggested.

Direct pair production in ∼5-MeV/amu heavy-ion-atom collisions with uranium target atoms is calculated with the plane-wave Born approximation and the semiclassical approximation. Briggs's approximation is used to obtain the electron and positron wave functions. Since pair production involves high momentum transfer $q$ from the moving projectile to the vacuum, use is made of a high-$q$ approximation to greatly simplify the numerical computations. Coulomb deflection of the projectile, the effect of finite nuclear size on the electronic wave functions, and the energy loss by the projectile exciting the pair are all taken into account in these calculations.

The spheroidal $s\sigma$-, $p\sigma$-, $p\pi$-, and $d\sigma$-wave phase shifts for the scattering of electrons from ${\mathrm{H}}_2^{+}$ are calculated at various internuclear distances by the Kohn variational method in the prolate-spheroidal coordinate system. The positions and widths of the two-electron excited resonance states are obtained for the symmetries ${}_{}{}^1{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}$, ${}_{}{}^{1,3}{}_{}{}^{}\Sigma _u^{}{}_{}{}^{}$, and ${}_{}{}^{1,3}{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$. Adiabatic quantum defects are also obtained by extrapolating the phase shifts to the low-energy limit. Importance of these quantities is discussed briefly in connection with the study of various dynamic processes of highly excited states of molecules.

Theoretical expressions for the vibrational excitation differential cross section (DCS) are given for the ${}_{}{}^2{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ and the ${}_{}{}^2{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ resonances. They fit some recent data on ${\mathrm{N}}_2$ and CO, respectively, at an electron impact energy of about 20 eV. These fits allow us to obtain individual rotational-vibrational DCS's. It is demonstrated that the discrepancy between the DCS data from different experiments is due to varying energy resolution resulting in selective partial exclusion of some rotational branches. Further, theoretical DCS's for each rotational branch in both gases are found, which are verifiable with the new experimental technique used to separate out the rotational branches at 2 eV.

The optical potential is computed exactly for a two-channel $1s$-$2s$ model in electron-hydrogen scattering. Exchange contributions in the $2s$ channel are calculated explicitly and compared to other terms in the nonlocal optical potential. It is concluded that exchange effects in excited states cannot be neglected in calculating optical potentials for low-energy electron-atom scattering.

Electron-ion crossed beams have been employed to obtain absolute cross sections for single ionization of ${\mathrm{Ne}}^{3+}$, ${\mathrm{Ar}}^{3+}$, ${\mathrm{Kr}}^{3+}$, and ${\mathrm{Xe}}^{3+}$. These data reveal a steady progression from excellent agreement with theory and semiempirical formulas for ${\mathrm{Ne}}^{3+}$ to significant departures from any simple models for ${\mathrm{Xe}}^{3+}$. For ${\mathrm{Ar}}^{3+}$ some small departure from the models is apparent and the measurements agree with previous experimental results of Müller et al. For ${\mathrm{Kr}}^{3+}$ and ${\mathrm{Xe}}^{3+}$ the measured cross sections in the energy range below four threshold units exhibit complex structure and exceed estimates of direct ionization by roughly factors of 2 and 3, respectively. These features are attributed to indirect ionization processes involving excitation of inner-shell electrons, and the observed cross-section shapes imply that the excitation-autoionization features cannot be simply attributed to dipole-allowed excitation transitions. Ionization rates for Maxwellian electron-temperature distributions are calculated from the data.

Electron detachment of ${\mathrm{H}}^{-}$ by impact of low-energy antiprotons and ${\mathrm{H}}^{-}$ is studied. Detachment cross sections for relative velocities $v\le 0.1\alpha c$ are evaluated with the use of several approximations. Results of different methods of calculation are consistent with each other. Cross sections of about ${10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$ are obtained just above the detachment threshold.

The effects of the spin-dependent interactions on the energy-level scheme of $\mathrm{U}\mathrm{}\mathrm{v} 5f^2$ were investigated. The interactions considered were the spin-spin, the spin-other-orbit, and the effective electrostatic-spin-orbit (EL-SO) interactions. The values of the appropriate radial integrals were determined by using both ab initio and semiempirical methods. It is shown that the introduction of these interactions greatly improves the fit between observed and calculated levels and that the interaction mainly responsible for this improvement is the effective EL-SO interaction. Various properties of all known $4f^2$ (in lanthanides) and $5f^2$ (in actinides) configurations are compared. It is found that the spin-orbit and the effective EL-SO interactions are stronger in $\mathrm{U}\mathrm{}\mathrm{v} 5f^2$ as compared to $\mathrm{Pr}\mathrm{}\mathrm{IV} 4f^2$, whereas the "real" electrostatic and the mutual magnetic interactions are weaker. Similarly, the ratio of the strengths of the spin-orbit and the electrostatic interactions is larger in $\mathrm{Th}\mathrm{}\mathrm{III} 5f^2$ than in $\mathrm{La}\mathrm{}\mathrm{II} 4f^2$. Thus, in the actinides, and particularly in U v, the coupling tends more to intermediate coupling than in the corresponding lanthanides.

The electron-impact ionization cross section for ${\mathrm{Be}}^{+}$ has been measured from threshold to 1600 eV with an absolute uncertainty of 8%. The cross section has a peak value of 46.5 × ${10}^{-18\mathrm{}}$${\mathrm{cm}}^2$ at an energy of ∼ 50 eV, and structure ascribed to the excitation autoionization of the $1s2s2p$ state at 118.5 eV is observed. Calculations using the semiempirical formula of Lotz for direct ionization and the semiempirical effective Gaunt-factor formula for excitation autoionization give summed cross-section values which agree well with the experiment over the entire energy range. Comparisons are also made with other calculations and cross sections in the literature.

Measurements were made of the cross section for electron-impact single ionization of the transition-element ions ${\mathrm{Ti}}^{3+}$, ${\mathrm{Zr}}^{3+}$, ${\mathrm{Hf}}^{3+}$, and ${\mathrm{Ta}}^{3+}$ for an electron-energy range from threshold to 1000 eV. The cross sections are enhanced by as much as a factor of 20 due to excitation autoionization primarily involving $\Delta n=0\mathrm{}$ transitions, $n{p}^{6}n{d}^m\to n{p}^{5}n{d}^{m+1\mathrm{}}$. Comparisons with recent theoretical predictions show reasonable agreement between measured and predicted positions of the autoionization states; however, the magnitudes of the theoretical cross sections are greater than the experimental values by a factor of approximately 2.5.

We have measured cross sections, ${\sigma }_{\mathrm{nL}}$, for the excitation transfer process $\mathrm{Na}\mathrm{}\left(3P\right)+\mathrm{Na}\mathrm{}\left(3P\right)\mathrm{}\to \mathrm{Na}\mathrm{}\left(3S\right)+\mathrm{Na}\mathrm{}\left(\mathrm{nL}\right)\mathrm{}$, where $\mathrm{nL}$ is the $4D$ or $5S$ level. Our results are ${\sigma }_{4D}=23\mathrm{} {\AA }^2\pm 35\%$ and ${\sigma }_{5S}=16\mathrm{} {\AA }^2\pm 35\%$ at $T\sim 600$ K. To obtain these cross sections we have used pulsed excitation and measured the intensities of $4D$, $5S$, and $3P$ fluorescence emissions, and the spatial distribution of excited atoms resulting from radiation diffusion, as well as the excited atom density as a function of time. Additionally, we have accounted for (time-dependent) radiation trapping of $3P$ and $\mathrm{nL}$ level radiation and for the resulting anisotropies of these fluorescence emissions. Comparisons of our results with theory have been made, and their relevance to other experiments is discussed.