Closed-form expressions for multiple-ionization cross sections are developed in the semiclassical Coulomb approximation at low velocities. The approximate expression for the Coulomb ionization probability, given by Brandt, Jones, and Kraner, is used to compute multiple-ionization cross sections. These approximate cross sections are evaluated in closed form. Results for multiple-$K$-, multiple-$L$ -shell-ionization cross sections are given explicitly and compared to the full semiclassical-approximation theory and to experiment.

The charge dependence of $K$-shell-electron vacancy production in argon by heavy-ion impact is found by adding charge-exchange and ionization cross sections computed in the binary-encounter approximation. The dependence of these cross sections on the nuclear charge, for fully stripped ions ($z=6,7,8,\mathrm{and} 9$), is in agreement with recent experimental results at 1.88 MeV/amu, where large deviations (>100%) from a $z^2$ dependence were observed. The dependence of atomic charge $q$ for ${\mathrm{F}}^{+q}$ at 1.88 MeV/amu ($q=9,7,8,6,\mathrm{and} 5$) is in reasonable agreement with experiment.

An explanation of the pretransitional scattering from liquid crystals is presented in terms of a dynamical concept of the soft mode.

The hyperfine structure of the $8{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ state of ${\mathrm{Cs}}^{133}$ has been measured by rf spectroscopy. The accidental coincidence between the Cs $8{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\to 6{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ transition and the helium 3888.65-Å line has been used to obtain a large pumping rate of the $8{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ state. We find, for the magnetic-dipole coupling constant, $A\left(8\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\right)=+42.97\pm 0.10$ MHz. The positive sign of $A$ has been ascertained by the decoupling technique.

Space-charge amplification of the direct and indirect ionization resulting from photoabsorption in alkali-metal vapor is shown to provide a useful complement to the conventional techniques of absorption spectroscopy. Because of its inherently greater sensitivity, this technique is particularly useful in spectroscopic applications involving weak transitions such as those to highly excited states or absorptive transitions from excited states of low population. This paper reports the use of this technique to extend prior measurements of the oscillator strengths of the ($6{}_{}{}^2{}_{}{}^{}S\to n {}_{}{}^2{}_{}{}^{}P$) series in Cs, the $n{}_{}{}^2{}_{}{}^{}D$ doublet separations, and the photoionization spectrum of ${\mathrm{Cs}}_2$.

The contributions from two important mechanisms to the hyperfine pressure shift (HPS) that are not included in restricted Hartree-Fock (RHF) calculations on interacting atoms are analyzed using the Li-He system as a specific example. First, it is shown that the influence of exchange core polarization, which is neglected in the RHF approximation, leads to about 40% of the direct contribution from distortions of the valence $2s$ orbital. Secondly, the long-range van der Waals polarization effect is missing in the RHF approximation and it is shown that this also can lead to a substantial contribution (about 20% of the direct contribution and of opposite sign) to the HPS. All of the three contributions to the HPS considered here—namely, from the direct polarization, exchange core polarization, and van der Waals mechanisms—are found, using two choices of intermolecular potentials, to be quite sensitive to the intermolecular potential, while their ratios are less sensitive. Thus, although the results for the total HPS from the present work are not to be taken as quantitative, nevertheless, they indicate that the exchange core polarization and van der Waals contributions, together with correlation effects, have to be included for a complete theory of HPS.

Transition probabilities for nine allowed transitions between the $5s^{\prime }{\left[\frac{1}{2}\right]}_1$ and $3p$ states of neon have been measured in side light emission from a discharge tube placed inside a He-Ne-laser radiation field. The laser was tunable at nine visible lines belonging to the $5s^{\prime }{\left[\frac{1}{2}\right]}_1-3p$ array. The gas mixture and the discharge current were adjusted so as to obtain a transparent plasma at the respective laser wavelength. Such a plasma is characterized by an absence of both resonance absorption and stimulated emission, creating a condition that $\frac{N_1}{N_2}=\frac{g_1}{g_2}$ between the upper- and the lower-energy levels of the laser transition. This condition immediately enables the intensity ratio of two transitions arising from two different energy levels to be reduced to the ratio of the transition probabilities. The relative $A$ values were put into an absolute scale using Bennett and Kindlmann's lifetime data on $3p$ levels. Nine independent measurements were made through the use of nine laser transitions. Excellent agreement was seen between these independent measurements. The estimated error in the relative measurements is 2%, and the best value of the absolute $A_{632.8}$ was found to be (0.339 ± 0.014) × ${10}^7$${\mathrm{s}}^{-1\mathrm{}}$ When compared with the data of Hansch et al. and Bychkova et al., who employed a similar method but only one laser transition at 632.8 nm, significant disagreement is seen. It was found that the self-absorption of $3p-3s$ lines is as large as 20%. Argon was added to reduce this effect to a negligible level. There is strong evidence that this effect was disregarded in other measurements, thereby resulting in the large disagreement with our data.

The great utility of the Thomas—Fermi (TF) theory of atoms lies in the fact that it gives a solution of the TF equation that is valid for for all neutral atoms. In contrast to this situation, the TF equation for positive ions has to be solved separately for each degree of ionization of each atom. Using a previously obtained approximate analytical solution of the TF equation for neutral atoms (which is based on a variational principle), and a series-expansion approach (which relates the parameters in the ionic solution to those in the neutral-atom solution), the present paper derives a universal approximate analytical solution of the TF equation for ions. This solution, in addition to being valid for positive ions, is also applicable for negative ions for which the original TF theory does not furnish a solution. Furthermore, the approximate solution obtained here is not associated with finite ionic radii, and gives an exponentially decreasing radial electron density, which is not the case in the original TF theory. For this reason, the electron densities and potentials of ions that result from the present work are expected to be in better agreement with the quantum-mechanical data for these quantities than the electron densities and potentials obtained with the original TF theory. To show that this is so, the accuracy of the universal approximate analytical solution is investigated by calculating the diamagnetic susceptibilities of singly and doubly charged positive and negative ions of noble-gas electron configurations. It is found that, in most cases, calculated and experimental values for ions of the Ar, Kr, and Xe electron configurations agree to within a factor of 2, while for ions of the Ne electron configuration the agreement is somewhat worse. This is about the same accuracy as that found using the Lenz-Jensen approximation to the TF theory which, like the universal approximation, also makes use of variational electron density but, unlike the universal approximation, has to be separately obtained for each particular ion. As a further check on the usefulness of the universal ionic solution, the diamagnetic susceptibilities of the isoelectronic ions ${\mathrm{Ga}}^{3+}$, ${\mathrm{Ge}}^{4+}$, and ${\mathrm{As}}^{5+}$ are also calculated and found to agree only slightly worse than within a factor of 2 with the experimental data. After further comparison with data obtained from more-refined statistical models, the conclusion is that the universal solution for ions may be useful in a variety of problems where quantum-mechanical accuracy may be traded for a simpler approach.

A variational approach to the approximate solution of a wide variety of self-consistency problems is described. Attention is focused on the variational formulation of the nonlinear integral equation, which determines the single-particle Green's function of a many-particle system; such an equation arises from a particular truncation of the hierarchy of equations which dynamically couple the complete set of Green's functions. To illustrate the method, we have applied it, with encouraging results, to the numerical determination of the self-energy function for the helium atom in the Hartree-Fock approximation. The applicability of this approach to the general problem of electron-atom scattering is pointed out.

After reviewing the properties of the photon considered as a quantized particle of zero mass, positive energy, and unit spin, the expansion of the unquantized and quantized electromagnetic fields and vector and scalar potentials in terms of the photon wave functions and creation and destruction operators is reviewed and extended. The most general vector and scalar potentials are obtained through the use of the eigenfunctions of the curl operator. The dichotomy between the photon and wave picture of electromagnetic radiation is discussed and resolved. The results are applied to the calculation of the exact electromagnetic matrix elements and transition probabilities (i.e., with retardation taken into account exactly) for hydrogenic atoms. The exact matrix elements are very simple in form. The notion of multipole radiation of the usual treatments is irrelevant. However, it is shown how multipoles appear as an approximation.

In this first article of work on radiation processes, a multiple-time-scale expansion method specifically for these processes is developed, based on the two distinct time scales corresponding to the natural frequency and the linewidth of the radiation. The method is then demonstrated by treating systems of one and two two-level atoms. It is seen that the exponential-decay behavior of the excited-state amplitude, assumed a priori by Weisskopf and Wigner, follows naturally from the mathematical procedure of eliminating the secularity in each order of the expansion. The results of two-atom systems also agree with those of previous workers.

The multiple-time-scale expansion method has been used to study a system of $N$ -two-level atoms with one of them excited initially. It is found that, in the limit of large time, the Schrödinger equation for the system reduces to a set of algebraic equations, and can thus be solved exactly. A detailed study of a four-atom system is given. The analysis is then extended to a regular lattice of $N$ atoms. It is found that in a large lattice, the probability for the excitation energy to be trapped is very large, even when the lattice spacing is comparable to the characteristic wavelength of the radiation.

Newly computed Hartree-Fock-Roothaan wave functions are reported for the atomic negative ions, ${\mathrm{B}}^{-}$, ${\mathrm{C}}^{--}$, ${\mathrm{N}}^{--}$, ${\mathrm{O}}^{--}$, ${\mathrm{Si}}^{--}$, ${\mathrm{P}}^{--}$, ${\mathrm{S}}^{--}$. They are obtained as self-consistently convergent solutions. The implication with regard to the stability of the restricted Hartree-Fock-Roothan method is also discussed.

The generalized oscillator strengths (GOS), which are essential factors in the Born cross sections, have been calculated for the lowest ${}_{}{}^1{}_{}{}^{}P$ transitions in Mg, Ca, Sr, and Ba from the multiconfiguration Hartree-Fock wave functions with two or three configurations each in the ground and the excited states. The optical limits ($f$ values) of the GOS agree with experimental values to within 12%. Theoretical data on zero minima and subsequent maxima in the GOS are presented. Our data show that the resonance transition in Ba is the most favorable one for experimental study of the extrema. Since the Born cross sections vanish at the minima, the experimental study of the minima would reveal valuable information on the validity of the first Born approximation, as well as on the magnitudes of higher-order effects not included in the theory. Data for the integrated Born cross sections are presented also. Recent electron-impact data on the resonance transition of Ca agree with the integrated cross section to within 5% for incident energies of 250 eV and higher. Our data show that the relativistic effects in the cross section for Ba are minor compared to the correlation effect between the two valence electrons.

The linked-cluster many-body perturbation theory with a $V^{\mathrm{N}-1\mathrm{}}$ potential has been applied to study the nuclear quadrupole interaction for the ${\mathrm{Be}}^9$ atom in the excited ($1s^{2}2s2p {}_{}{}^3{}_{}{}^{}P$) state. The first- and second-order shielding contributions to the field gradient are found to be 3.8% and — 2.0%, respectively. The former involves cancellations among $1s$ and $2s$ states and direct and exchange contributions with individual effects running as large as 12%. From these field-gradient results and the experimental quadrupole constant, the nuclear quadrupole moment of ${\mathrm{Be}}^9$ has been evaluated as 0.0525±0.0003 b.

Fine-structure transitions in the millimeter wavelength region and the $n\left(J\right)=1\mathrm{}\left(2\right)\mathrm{}\to 3\left(2\right)\mathrm{}$ submillimeter wavelength rotational transition have been measured with high precision. The submillimeter radiation was generated by a harmonic multiplier and detected by a silicon bolometer operated at 1.5 °K. The measured 1(2) → 3(2) frequency of ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}_2^{}{}_{}{}^{}$ is 378 831.51(20) MHz. A remeasurement of the 1(2) → 3(2) frequency for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}_2^{}{}_{}{}^{}$ yielded the value 424 763.12(10) MHz. Analysis of the results yielded the following values of the molecular constants for ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}_2^{}{}_{}{}^{}$: $B_0=38\mathrm{}313.721\left(2\right)\mathrm{}$ MHz, ${\lambda }_0=59\mathrm{}496.708\left(12\right)\mathrm{}$ MHz, ${\lambda }_1=0.0521\mathrm{}\left(3\right)\mathrm{}$ MHz, ${\mu }_0=-224.438\mathrm{}\left(3\right)\mathrm{}$, ${\mu }_1=-0.000\mathrm{}29\left(5\right)\mathrm{}$ MHz, $B_e=38\mathrm{}518.6\left(6.3\right)\mathrm{}$ MHz, and $r_e=1.20\mathrm{}743\left(10\right)\mathrm{}$ Å. The values agree well with those predicted from isotopic corrections of the ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}_2^{}{}_{}{}^{}$ values. The equilibrium constants were obtained with the aid of vibration-rotation interaction constants from optical spectroscopy.

A general analytical result has been obtained for the calculation of higher-order corrections to the momentum-translation approximation. Explicit analytical results are given for the calculation of higher-order corrections for bound-bound and bound-free transitions for hydrogenic atoms. Numerical results are given for $1S$ to $2S$ transitions in hydrogen.

The impact-parameter method used to calculate cross sections in high-energy proton—hydrogen-atom collisions is considered. By making use of the observation that the scattering is almost completely in the forward direction, a complete time-dependent impact-parameter model is obtained from the Lippmann-Schwinger equation. As a direct consequence of the proper treatment of the continuum, two equivalent Schrödinger time representations are found in which to describe the model system of a point charge moving along a specified trajectory, and thereby perturbing a hydrogen atom. The unambiguous identification of the dynamical states of this model system, and the unambiguous definition of transition probabilities are obtained. The specification of the dynamical states of the time-dependent model system can be conveniently incorporated into the customary impact-parameter method by associating an additional boundary condition with the usual time-dependent Schrödinger equation. The familiar traveling atomic orbitals are not dynamical states of the system, but the discrete dynamical states approach the traveling orbitals as $\mathrm{}\left|t\right|\mathrm{}\to \infty$; and hence, the set of traveling atomic orbitals provides a description of the system which becomes correct asymptotically. One Schrödinger representation is the natural representation in which to obtain the amplitudes for electronic excitation. The other representation is the natural representation in which to obtain the amplitudes for charge exchange. A noniterative technique is used to solve the integral equations which describe the evolution of the state vector in the two time representations, and expressions are derived for the amplitudes for excitation and charge exchange. The amplitudes for excitation can be expressed in terms of the usual Coulomb integrals, and the amplitudes for charge exchange can be expressed in terms of usual exchange integrals.