A practical algorithm for estimating the Kolmogorov ($K$) entropy directly from a time series is presented. In all the cases studied, we verify the conjecture that $K$ is the sum of the positive Lyapunov exponents.

A generalization to two dimensions of the model given previously for the driven diode resonator is used to explain the main features of the actual system. Measurements of the characteristics of the diode are shown to justify the new assumptions. Thus, we have isolated the physical characteristic of the diode responsible for the multidimensional chaotic behavior of this system.

The three-particle correlation function $G_3(r_{12},r_{13},r_{23})$ for a fluid with a long-range pair potential is computed in two limits: (a) all $r_{\mathrm{ij}}\to \infty$, and (b) one distance, say $r_{12}$, fixed and $r_{13},r_{23}\to \infty$. In both cases, the pair potential times the square of the isothermal compressibility appears.

Power spectra of multistable systems have a low-frequency component corresponding to jumps among independent attractors. This situation is modeled with a one-dimensional cubic map disturbed by noise. Besides the dependence of the spectral slope on noise, we show that the Lyapunov exponent is made of two separate contributions corresponding to the attracting and repulsive regions, respectively, each weighted with its relative occupation time.

We report on the results of a magnetic birefringence study of the nematic-isotropic transition in the lyotropic liquid crystal cesium perfluoro-octanoate in water. The transition is weakly first order at $T_K$ with mean-field divergences and a virtual second-order transition temperature that can be within 20 mK of $T_K$.

The asymptotic behavior of the pair-correlation function in fluids with realistic long-range pair potentials is shown to give rise to a correction to scaling that has not been previously taken into account.

A molecular-beam experiment using TlF is described. The experiment searched for a $P$- and $T$-violating interaction of the form ${\mathcal{H}}_{\mathrm{PT}}=d{\overrightarrow{\sigma }}_{\mathrm{Tl}}·{\overrightarrow{\lambda }}_{\mathrm{TlF}}$, where $d$ is a coupling constant, ${\overrightarrow{\sigma }}_T$ is a unit vector parallel to the thallium nuclear spin, and ${\overrightarrow{\lambda }}_{\mathrm{TlF}}$ is a unit vector parallel to the internuclear axis. The experiment yields the null result $\frac{d}{h}=8\mathrm{}\pm 12$ mHz. Interpreting the interaction in terms of an electric dipole moment on the proton, $D_p$, or in terms of a possible weak neutral-current tensor interaction with a coupling constant $C_T$, this result corresponds to $D_p=(1.3\mathrm{}\pm 2.0)\times {10}^{-21\mathrm{}} e$ cm and $C_T=(6\mathrm{}\pm 9)\times {10}^{-6\mathrm{}}{G}_F$, where the errors incorporate unexplained systematic effects as well as statistical errors.

With the use of an accurate nonrelativistic wave function, the relativistic effects of the ${\mathrm{He}}^{-}$ $1s2s2p{}_{}{}^4{}_{}{}^{}P$ state are computed. These include the mass-correction term, Darwin term, and retardation and mass-polarization effects. The calculated result enables us to significantly reduce the uncertainty of the electron affinity given by Bunge and Bunge. To assist future experimental measurement on the fine structure of this ion, the expectation values of the spin-orbit, spin-other-orbit, and spin-spin operators are also calculated.

Wave functions, energies, and binding energies for the lowest singlet states of ${\mathrm{H}}^{-}$ and He in uniform magnetic fields $B<\mathrm{}{10}^9$ G are calculated using an adiabatic approximation in hyperspherical coordinates. In computing the angular part of the wave functions, a coupled expansion in one-electron oblate spheroidal angle functions is used. In addition to contracting the two-electron wave function radially, the magnetic field is found to distort the angular part of the wave function mainly by reducing the $\mathrm{ss}{}_{}{}^1{}_{}{}^{}S$ character of the state and replacing it with an $\mathrm{sd}{}_{}{}^1{}_{}{}^{}D$ character. Results for energies and binding energies are comparable with those obtained in variational calculations. In order to compute the binding energies of ${\mathrm{H}}^{-}$ and He we have also calculated the binding energies of the ground states of H and ${\mathrm{He}}^{+}$ in uniform magnetic fields $B<\mathrm{}{10}^9$ G using the adiabatic oblate-spheroidal-coordinate method of Starace and Webster.

Muonic Coulomb-capture probabilities are calculated assuming that electrons with binding energies less than a given limit (about 80 eV) contribute with a weight which is a function of the electron binding energy, the electron quantum numbers $n$ and $l$, and $Z$. This functional dependence is suggested by a quantum-mechanical calculation. By fitting few free parameters, good agreement with experimental capture probabilities is obtained; in particular, the periodicity with $Z$ of these values is well reproduced.

The axial oscillation frequency of a trapped particle in a Penning trap with hyperbolic electrodes has proven to be a very important observable for precision experiments. Shifts in this frequency have been used to measure magnetron frequencies, cyclotron frequencies, and $g-2\mathrm{}$ transitions for single trapped particles. Detection and damping of the axial oscillation were thought to be well understood except that a key constant which depends upon trap geometry had not been calculated. A relaxation calculation has thus been performed to provide a calculated value for this constant and several others, and to determine their dependence upon trap geometry. A surprising result is a discrepancy between the calculated value of the key constant and that deduced from measured values of the width of the axial oscillation frequency in the several cases where it has been measured.

The Schrödinger equation for two-electron systems is examined using hyperspherical coordinates. It is shown that the eigenvalues for the angular operator can be obtained through a variational principle. The definition of diabatic states is discussed and the transformation between adiabatic and diabatic states is found. The effect of the transformation on the off-diagonal elements of the interaction matrices $\mathit{U}$, $\mathit{P}$, and $\mathit{Q}$ is examined. It is shown that the behavior of reduced density can be used to characterize the adiabatic states for $L=1\mathrm{}$ along with the symmetry around $\alpha =\frac{\pi }{4}$. Also the effect of increasing the charge of the nucleus is examined.

Core-exchange effects in free atoms have been examined through the determination of the spin density at the nucleus of (${}_{}{}^2{}_{}{}^{}S$) Li, Na, Rb, and Cs atoms. The use of relativistic corrections and of different local-spin-density approximations is evaluated in both restricted and spin-polarized self-consistent-field wave functions.

Bound-continuum processes of the type $A^{*}+B\to A+B^{+}+e or {\mathrm{}\left(\mathrm{AB}\right)\mathrm{}}^{+}+e$, known as penning and associative ionization, respectively, are considered. The derivation of the partial-wave Schrödinger equation through the Feshbach projection-operator formalism is reviewed. The potential-energy operator for the general case is then examined carefully and found to be not only collision-energy dependent but also nonlocal. Strict locality is obtained only in the limit of infinite collision energy. Nonlocality due to bound-continuum interaction increases in importance as the collision energy is reduced, and in certain dynamical situations for associative ionization it is found to be predominant. thus, the common procedure of reducing the bound-continuum interaction to an imaginary local potential becomes suspect and the general validity of the attendant Franck-Condon approximation of vertical electronic transitions is also called into question. Two limiting situations—the high-energy ($E\to \infty$) and the low-energy ($E\to 0$) limits—are examined. In the former case, the local potential is isolated and the nonlocal part given explicitly to lowest order; in the latter, the predominant nonlocal part is given explicitly for the case of associative ionization.

The quantum-mechanical description of heavy-particle collisions is extended to the intermediate-energy range through the use of a second-order distorted-wave approximation. Cross sections are calculated using close-coupled solutions for low partial waves and first- and second-order unitarized distorted-wave solutions for higher partial waves where the distorted-wave method is accurate. The method is applied to the calculation of charge transfer cross sections for collisions of ${\mathrm{C}}^{3+}$ with H, and of ${\mathrm{O}}^{2+}$ with He over the energy range 0—5 keV/amu, and to investigate the influence of the charge transfer channel on ion-impact excitation of ${\mathrm{O}}^{+}$.

A relativistic formalism is used to include spin-orbit coupling in calculating cross sections and polarizations for one-photon free-free absorption of electrons scattering from mercury. A local exchange approximation and phenomenological polarization potential are used in the calculations. The scattering amplitudes are computed exactly. Off-shell effects are found to be important in determining the polarization of the scattered electrons. Polarizations in excess of 95% are found for some directions in the energy region where phase shifts are changing rapidly. Polarizations and off-shell effects are smaller where the phase shifts are slowly varying.

We show that the pronounced structure in the energy spectra of electrons ejected in the forward directions in ${\mathrm{H}}^{-}$ stripping depends upon the mean excitation energy of the target-gas molecules. The measured spectra for He, Ne, Ar, and C${\mathrm{H}}_3$Cl targets is in qualitative agreement with first-Born-approximation calculations.

Elastic angular differential cross sections for small-angle scattering of protons by atomic hydrogen have been measured. The technique utilized unambigously distinguishes the elastically and inelastically scattered ions. The cross sections fall monotonically by 3 orders of magnitude in the angular range from 0.5 to 3.0 mrad, in the center-of-mass system. The experimental data obtained are in very good agreement with a multistate calculation and in fair agreement with both our Glauber-approximation and classical-trajectory Monte Carlo results.

The unitarized distorted-wave approximation (UDWA) is applied to single-electron-capture processes in the collision of naked ions with a helium atom, $B+\mathrm{He}\left(1\mathrm{}{s}^2\right)\to \mathrm{}(B+e)+{\mathrm{He}}^{+}\mathrm{}\left(1s\right)\mathrm{}$, where $B={\mathrm{H}}^{+}, {\mathrm{Li}}^{3+}, {\mathrm{Be}}^{4+}, {\mathrm{C}}^{6+}, \mathrm{and} {\mathrm{O}}^{8+}$. We use the independent-electron model, in which the electron is assumed to move in an averaged Coulomb potential field with an effective nuclear charge. The UDWA cross section becomes more reliable as the charge of the incident ion increases, and good agreement with experimental data is obtained down to a few keV/amu for ${\mathrm{O}}^{8+}$+He.

A new theoretical approach named the exponential distorted-wave approximation (EDWA) is developed and applied to electron-capture processes of multicharged ions from a hydrogen atom. The EDWA is an improvement of the unitarized distorted-wave approximation (UDWA) in that the EDWA takes into account the couplings among the final states which are neglected in the UDWA. The difference between the cross sections of the two methods is small except for the low-energy region less than a few keV/amu, and it is confirmed that the mixing among the final rearranged states does not play an important role in the processes studied here.