The radial wave functions of hydrogen are put into such a form that they form bases for irreducible unitary representations of an algebra isomorphic to that of $O(2,1)\mathrm{}$. Operators proportional to $r^k$ are found which form bases for the adjoint representations of this algebra. Matrix elements of these operators are evaluated, and selection rules are determined by considering Kronecker products of representations of $O(2,1)\mathrm{}$. Differences between this approach and one previously suggested are discussed.

Many-body perturbation theory is used to calculate the correlation energy of an atom with an inner-shell vacancy. Numerical calculations are given for neon with either a $1s$ or a $2s$ vacancy. For the case of the $1s$ vacancy, the energy has an imaginary part which is proportional to the Auger rate. Our results are compared with semiempirical determinations of the correlation energy and with experimental results for the fluorescence yield.

The zero-field level crossing and optical double-resonance techniques have been used to measure the lifetimes and $g$ factors of several rotational substates in the $A^2{\Sigma }^{+}$ excited state of the OH and OD free radicals. Optical excitation was provided by a molecular lamp and individual emission transitions were observed through a monochromator in the beam of resonantly scattered light. The measured $g$ factors are in agreement with the results expected from pure case b coupling for this state. The measured lifetimes are 660 ± 22 and 598 ± 20 nsec for OH and OD, respectively; the mean value 629 ± 22 nsec is suggested. The optical double-resonance experiments allow a tentative lower limit to be placed on the excited-state hyperfine interaction.

Relative intensities of $L$ x rays emitted during the radiative decay of $L_2$ and $L_3$ subshell vacancies are investigated by Si(Li) x-ray detectors, over a range of $65\le Z\le 94$. Intensities of transitions involving $N$, $O$,... electrons compared to those involving $M$-shell electrons are found to be slightly underestimated in Scofield's theoretical calculations. $L_1$ subshell yields for $Z=82\mathrm{}$ are measured by a high-resolution study of the Pb $L$-x-ray spectrum from the decay of ${}_{}{}^{207}{}_{}{}^{}\mathrm{Bi}$ with the following results: ${\omega }_1=0.09\mathrm{}\pm 0.02$, $f_{12}=0.17\mathrm{}\pm 0.05$, $f_{13}=0.61\mathrm{}\pm 0.08$, and $s_1=0.353\mathrm{}\pm 0.053$.

Exact, formal, and usable solutions of the Liouville equation corresponding to the problem of independent two-level atoms in an external field are presented. Two examples are worked out: calculation of the polarization, and a model of pumping or population inversion by a $\delta$-function external field. The relation of the solutions to other problems in statistical mechanics is also discussed.

All integrals needed to evaluate the wave function of the form

${\Psi }_{\mathrm{tot}}=\tilde{\mathfrak{a}}\left\{\left[\Pi \stackrel{}{j}\left[\Sigma \stackrel{}{s}{a}_{\mathrm{sj}} {\Phi }_s\mathrm{}\left(j\right)\mathrm{}\right]\right]\left[1+\Sigma \stackrel{}{j<i}?w_{\mathrm{ij}}{r}_{\mathrm{ij}}^n\right]\right\} \mathrm{for} n=1\mathrm{}$

and the Hamiltonian given are contained herein. For this form of the wave function, the integrals needed can be expressed as a product of integrals (involving at most four electrons). An indication of how to increase or decrease exponents of $r_{\mathrm{ij}}^n$ in steps of one or two is given. Some indication of how to proceed if the Hamiltonian contains $\frac{1}{r_{\mathrm{ij}}^3}$ terms or if the wave function is of the form

${\Psi }_{\mathrm{tot}}=\tilde{\mathfrak{a}}\left\{\left[\Pi \stackrel{}{j}\left[\Sigma \stackrel{}{s}{a}_{\mathrm{sj}} {\Phi }_s\mathrm{}\left(j\right)\mathrm{}\right]\right]\left[1+\Sigma \stackrel{}{j<l<k}?w_{\mathrm{jlk}}{r}_{\mathrm{jl}}^n r_{\mathrm{lk}}^v\right]\right\}$

is given. Consideration of all possible types of integrals (using picture-writing graph theory) involving $r_{\mathrm{ij}}^a{r}_{\mathrm{kl}}^b{r}_{\mathrm{mn}}^c$ with $a<\mathrm{}0\mathrm{}$, $b>\mathrm{}0\mathrm{}$, $c>\mathrm{}0\mathrm{}$; $\mathrm{}\left|a\right|=\left|b\right|=\left|c\right|=1\mathrm{}$, is given. Integrals corresponding to the graphs (numbers next to the individual diagrams indicate the number of the equation) are given in analytical form. These can be evaluated by numerical-integration routines.

The inelastic energy-loss distribution $Q$ has been measured for single large-angle collisions between argon atoms and incident projectile ions with atomic numbers $Z_1$ ranging from 13 to 25. The data are obtained by measuring the energy and scattering angle of the scattered incident particle. This method provides well-resolved $Q$ spectra, being almost undisturbed by thermal target motion. In all cases studied, the $Q$ value exhibits a sudden change and a triple-peaked structure in a narrow range of distance of closest approach $r_0$. The location of the three peaks $Q_{\mathrm{I}}$, $Q_{\mathrm{II}}$, and $Q_{\mathrm{III}}$ varies with $r_0$, but their energy separations remain independent of $r_0$. For the heteronuclear collisions reported here, the values of the $Q$-peak separations suggest that $Q_{\mathrm{II}}$ and $Q_{\mathrm{III}}$ correspond to the production of one and two $L_{2,3}$ vacancies, respectively, in one of the colliding particles, viz., the one with the lower atomic number. This is further supported by studies of Auger electrons emitted in the collisions.

A two-state semiclassical theory of elastic and inelastic scattering is formulated and compared to the Stueckelberg-Landau-Zener, distorted-wave, and close-coupled two-state methods. The semiclassical theory is shown to explain the physical origin of oscillatory structure on both the elastic and inelastic differential cross sections, and of a rainbowlike structure in the elastic scattering. Using simple and plausible interactions, theoretical elastic and inelastic differential cross sections are calculated and compared to the experimental measurements of the elastic and inelastic differential cross sections for ${\mathrm{He}}^{+}$ +Ne. The scattering is characterized by a ${}_{}{}^2{}_{}{}^{}\Sigma -{}_{}{}^2{}_{}{}^{}\Sigma$ crossing located at $r_x=2.02\mathrm{}$ a. u. with an energy $E_x=0.530\mathrm{}$ a. u., leading to the final state ${\mathrm{He}}^{+}$+${\mathrm{Ne}}^{*}$ ($2p^{5}3s$). The interaction energy at the crossing is found to be $V_{12}\left(r_x\right)=0.0082\mathrm{}$ a. u.

Radio-frequency spectroscopy of ions has been performed on the ${\left({\mathrm{Sr}}^{87}\right)}^{+}$, ${\left({\mathrm{Cd}}^{111}\right)}^{+}$, and ${\left({\mathrm{Cd}}^{113}\right)}^{+}$ ground states utilizing spin-exchange collisions with optically pumped rubidium atoms for both producing and detecting the orientation.

Measurements are presented of the total cross section for scattering of low-energy electrons by a mixture of ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$ metastable argon. The atom-beam-recoil method was used to make an absolute measurement. The cross section was measured from 0.35 to 6.75 eV with an angular resolution of 16.1° in the electron polar scattering angle at an electron energy of 1 eV. The velocity distribution of the atom beam was measured by time-of-flight spectroscopy. Electron beams with typical energy distributions of 0.3-eV full width at half-maximum were used. The over-all error in the cross section is estimated at ± 13% and the electron-energy determination is believed accurate to ± 0.150 eV.

Cross sections differential in angle and ejection energy for electron production by proton impact on nitrogen, oxygen, neon, and argon have been measured using electrostatic analysis and counting of individual electrons. The range of proton energies was 50-300 keV, the angles ranged from 10° to 160°, and the electron energies were measured from 1.5 to 1057 eV. Integrations over angle and/or electron energy yielded singly differential and total electron production cross sections. Our total cross sections for oxygen fall halfway between previous data of deHeer et al. and Hooper et al., but our argon cross sections agree better with deHeer et al. Cross sections for electron ejection in the backward hemisphere are much greater for these multishell targets than for hydrogen and helium. The momentum-energy conservation hump which was prominent in hydrogen is less conspicuous for these gases.

Angle and energy distributions of electrons ejected from single- and multishell atoms by 50-to 300-keV protons are compared with various theoretical treatments. Partial cross sections for each subshell are calculated by scaling according to the number of electrons and the binding energy and then are added. The Born approximation fails at large and at small angles but yields fair agreement at intermediate angles. The treatments of Salin and of Macek, which take account of the influence of the projectile after the collision, yield much improved agreement at small angles. The binary-encounter model is used to calculate cross sections which are differential only in electron energy. An analytical expression is given for the binary-encounter cross section averaged over the Fock hydrogenic distribution of orbital velocities. When the average orbital energy is calculated from Slater's rules, the results agree somewhat better with experiment than when this energy is set equal to the binding energy.

The population changes in various rotation-vibration levels of ${\mathrm{CH}}_3$${\mathrm{Cl}}^{35}$ were measured by microwave absorption in the presence of $P\left(26\right)\mathrm{}$ ${\mathrm{CO}}_2$-laser radiation. The transition involved in the laser absorption was identified as ${}_{}{}^R{}_{}{}^{}Q_3^{}{}_{}{}^{}\mathrm{}\left(6\right)\mathrm{}$. The polarities, strengths, and relaxation times of the double-resonance signals on various microwave transitions were measured. These data were evaluated for information on collisional transfer between rotational levels, vibrational levels, and on the equalization of the rotational temperature between adjacent $J$ levels. Collisional excitation of the isotopic species ${\mathrm{CH}}_3$${\mathrm{Cl}}^{37}$ was observed. Vibrational relaxation from the ${\nu }_6=1\mathrm{}$ state shows a smooth transition from diffusion to the walls at low pressure to thermal diffusion at high pressure. The conversion between ${\nu }_6=1\mathrm{}$ and ${\nu }_3=1\mathrm{}$ was shown to occur in either direction with a probability of about 1-in-300 per collision. The ${\nu }_6=1\mathrm{}$ vibrational satellite of the pure rotation spectrum in ${\mathrm{CH}}_3$Cl was shown to be irregular and the double-resonance method was used to assign the transitions $J_{5-6}$, $K=4\mathrm{}$, ${\nu }_6=1\mathrm{}$ and $J_{5-6}$, $K=l=+1\mathrm{}$, ${\nu }_6=1\mathrm{}$.

Two-body ion-ion neutralization cross sections $Q$ have been measured for ${\mathrm{NO}}^{+}$ + ${\mathrm{NO}}_2^{-}$ and ${\mathrm{O}}_2^{+}$ + ${\mathrm{NO}}_2^{-}$ and remeasured for ${\mathrm{O}}_2^{+}$ + ${\mathrm{O}}_2^{-}$. Using a superimposed-beams method, a range of barycentric energies between 0. 15 and 200 eV was covered. The product $v_{r}Q$ behaved somewhat similarly for all reactants when plotted against the relative speed $v_r$. For relative energies above 3 eV, $v_{r}Q$ varies slowly and is of a magnitude of ${10}^{-7\mathrm{}}$ ${\mathrm{cm}}^3$/sec or greater. Below about 3 eV, however, $v_{r}Q$ begins to increase rather rapidly with decreasing energy, approaching the expected $v_r^{-1\mathrm{}}$ dependence. The low-energy data have been fitted to four-term polynomials in powers of $v_r$. In this form the data can be extrapolated to the thermal range and rate constants obtained. Reaction rates (300 °K) are given for the above reactants, and also for ${\mathrm{H}}^{+}$ + ${\mathrm{H}}^{-}$, ${\mathrm{O}}^{+}$ + ${\mathrm{O}}^{-}$, ${\mathrm{N}}^{+}$ + ${\mathrm{O}}^{-}$, and ${\mathrm{N}}_2^{+}$ + ${\mathrm{O}}_2^{-}$, which were studied earlier.

Charge-state fractions have been determined for iodine, selenium, and bromine ions traversing oxygen, argon, krypton, xenon, carbon-foil, and gold-foil targets. Particular attention has been paid to high charge states, some containing as little as ${10}^{-6\mathrm{}}$% of the primary beam intensity. Despite the fact that heavy-gas targets produce a low mean charge, they were found to be more efficient producers of very high charge states than lighter targets. For the very highest charge states record (> 22+ for iodine ions at 12 MeV), the heaviest gas target Xe produced even more of these ions than a carbon foil. Measurements of the attenuation cross section and scattering for the ultrahigh charge states are also reported.

The binary-encounter collision model for ionization of atoms by proton impact proposed by Gerjuoy and Vriens is extended to the problem of proton-impact excitation of a neutral atom from the ground state. We have calculated the proton-excitation cross sections from the ground state of a helium atom to the $n{}_{}{}^1{}_{}{}^{}S$ ($n=2\mathrm{}-5$), $n{}_{}{}^1{}_{}{}^{}P$ ($n=2\mathrm{}-5$), and $n{}_{}{}^1{}_{}{}^{}D$ ($n=3,4\mathrm{}$) states using a $\delta$ function as well as a quantal-momentum distribution function for the target electrons. The calculated cross sections are compared with other theoretical calculations and the measured values.

A simple method for approximately calculating the effect of Bose statistics on the properties of liquid helium is given. It is shown that as the temperature is lowered, normal helium undergoes a transition to a state in which a finite fraction of the atoms are in a coherent state. Using only the experimental interatomic potential and pair-correlation function, the transition temperature for liquid helium in equilibrium with its vapor is calculated to be 2.15°K. The dependence of the transition temperature on pressure is calculated and found to be in good agreement with experiment. The specific heat is calculated and is found to diverge as the $\lambda$ temperature is approached. It is also shown that as the temperature is lowered, a density maximum will occur in liquid helium slightly above the $\lambda$ temperature.

Ultrasonic velocities in the melts of bismuth, tin, and a 50-at.% alloy of the two have been determined up to 8.5 kbar using pulse-echo techniques. For bismuth melt, one region exhibits increases in velocity with increasing temperature up to about 7.7 kbar, and from melting to the 270°C isotherm. The temperature behavior of the velocity goes through a maximum around the 275°C isotherm; and in the second region the velocity increases, but much less slowly, with increasing temperature. This behavior is related to a change in the liquid structure. Both regions show a normal increase in velocity with increasing pressure. Molten tin exhibits a normal decrease in velocity with increasing temperature over the range studied, and no structural changes in the liquid state were indicated. The 50-at.% alloy showed normal temperature behavior of the velocity over the limited pressure-temperature domain of 160-180°C and up to 2.5 kbar, and an anomalous pressure and temperature behavior of the velocity above 2.5 kbar. The possibility of structural changes in the liquid state, under pressure, and the increase in velocity with increasing temperature shown by molten bismuth lead to the conclusion that the completely molten state per se need not necessarily yield a decrease in velocity with increasing temperature.

The phonon spectrum of hcp ${\mathrm{He}}^4$ has been measured along the three principal symmetry axes by inelastic neutron scattering from a single crystal with a molar volume of 16.00 ${\mathrm{cm}}^3$ at 4.2 K. The results are in fair agreement with the theoretical calculations of Gillis, Koehler, and Werthamer, the maximum discrepancy being about 30%. Considerable phonon damping has been observed for the ${\Delta }_2\left(\mathrm{LO}\right)$ and ${\Sigma }_3\left({\mathrm{TO}}_{\perp }\right)$ branches. A modified axially symmetric force-constant model has been fitted to the dispersion curves and used as an interpolation scheme to calculate the frequency distribution function and other thermodynamic functions.

Helium-II heat-flow experiments have been carried out with the helium contained in small channels (tubes < 10 μ in diameter $d$) having simple geometries and with the temperature differences limited to ∼ 5 × ${10}^{-3\mathrm{}}$ K° or less. In this domain, the heat current density is found to be proportional to the temperature gradient and the differential thermohydrodynamic equations of London and Zilsel should be valid. Comparison with the London-Zilsel theory shows good agreement for most of our data with regard to the absolute magnitude of the thermal conductivity $\kappa$ and its dependence on temperature. The experimental $\kappa$ values have a weaker dependence on the channel size than the theoretically predicted square of $d$, and this size dependence is a function of temperature. Our data for the largest channels ($d\sim 8$ μ) and at the larger heat current densities depart significantly from the London-Zilsel predictions. This difference is not explainable on the basis of a Gorter-Mellink mutual friction, but is consistent with a dissipative flow for the superfluid depending linearly on its velocity. This superfluid dissipative flow is comparable to that reported in other helium-flow experiments.