The method of reduced density matrices for obtaining the ground-state energy of an atomic system is developed, making full use of the symmetry relations for orbital and spin angular momentum. These, together with an extensive set of Hamiltonian-dependent identities, serve to decrease the number of parameters which must be varied in the density-matrix variational principle. With only a small number of parameters required, inequalities such as the Pauli restriction can then be enforced. Numerical calculations for ${\mathrm{C}}^{++}$ show the relative ineffectiveness of certain low-lying geminals (in the ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$ expansion) in reaching the Pauli restriction limit, and thus point the way to significant improvement.

The hyperfine structure and $g_J$ values of the 17 lowest-lying atomic levels of ${\mathrm{Tb}}^{159}$ have been measured by the atomic-beam magnetic-resonance technique. Two of the three levels studied in the $4f^{9}6s^2$ configuration had not been seen before, and the present work in $4f^{8}5d6s^2$ provides the first hyperfine-structure information obtained for 11 of the 14 levels studied. The results are analyzed in detail by use of the best eigenvectors available and the effective-operator formalism. After making suitable corrections for hfs and Zeeman interactions with neighboring atomic states, values for a number of radial integrals are deduced by treating them as adjustable parameters. The value obtained for the electric-quadrupole moment of the ${\mathrm{Tb}}^{159}$ nuclear ground state is $Q\left({\mathrm{Tb}}^{159}\right)=+1.34\mathrm{}\pm 0.11$ b, in good agreement with earlier results. This value of $Q$ has been corrected for Sternheimer shielding effects in both the $4f$ and $5d$ electron shells; it results from least-squares fits to the quadrupole hfs measurements in all 17 atomic states studied in both configurations.

Coherent spontaneous emission from a system of $N$ two-level atoms interacting with a quantized radiation field is treated for the case in which Dicke's "cooperation number" $r$ is macroscopically large. It is shown how to modify the quasiclassical approach to this problem to incorporate quantum effects that are lost in the self-consistent-field approximation. The statistics of the emitted radiation are found to vary markedly with the initial state of the system of atoms. The photon statistics tend to that typical of blackbody radiation when the initial state of the atomic system is that which would result from incoherent pumping ($m\sim r$). When, on the other hand, the atoms are initially in a superradiant state ($m\ll r$), the emitted radiation may be represented approximately by a coherent state.

The radiative decay of an atom with two excited states coupled by an external perturbation is investigated. The differential equations of motion are Fourier transformed and the probability amplitudes are obtained by contour integration. The real parts of the poles in the complex plane are the perturbed energies of the excited states, and the imaginary parts yield the decay characteristics. The decay probabilities of the excited states contain three different decay terms; two exponential decays and one modulated exponential decay. The probabilities of the final states give the frequency distribution of the emitted photons as a function of time. In an Appendix, the Heitler-Ma formalism is used to eliminate the final states of the system, and the resulting equations which contain damping terms are compared with the phenomenological method.

Formulas are developed which account for the details of the high-resolution photoabsorption spectrum of ${\mathrm{H}}_2$ near threshold observed by Herzberg. The treatment illustrates the application of Seaton's multichannel quantum defect method under schematic conditions, and might serve as a basis for further developments.

The extended-polarization-potential method has been used to calculate the differential cross sections for elastic electron-helium scattering. The results are presented in tabular form for collision energies between 1 and 95 eV and a selection of scattering angles from 0 to 180 deg. Total scattering cross sections are also given for all energies considered.

The reaction rate constants or the cross sections for charge-exchange reactions of the ions ${\mathrm{O}}^{-}$, ${\mathrm{O}}^{+}$, $\mathrm{O}_2^{}{}_{}{}^{-}$, and $\mathrm{O}_2^{}{}_{}{}^{+}$ with ${\mathrm{O}}_2$ have been measured over a range of values of $\frac{E}{p_0}$. The rate constant for the ${\mathrm{O}}^{-}$ exchange rises from 2 × ${10}^{-13\mathrm{}}$ ${\mathrm{cm}}^3$/sec at $\frac{E}{p_0}$ of 25 to 2.7 × ${10}^{-10\mathrm{}}$ ${\mathrm{cm}}^3$/sec at $\frac{E}{p_0}$ of 300 V ${\mathrm{}\left(\mathrm{cm}\mathrm{Torr}\right)\mathrm{}}^{-1\mathrm{}}$. For ${\mathrm{O}}^{+}$, the rate constant rises from 1.4 × ${10}^{-11\mathrm{}}$ ${\mathrm{cm}}^3$/sec at $\frac{E}{p_0}$ of 20 to 2.9 × ${10}^{-11\mathrm{}}$ at $\frac{E}{p_0}$ of 55. The resonant charge-exchange cross section for $\mathrm{O}_2^{}{}_{}{}^{+}$ in ${\mathrm{O}}_2$ is 40 ${\mathrm{\AA }}^2$, and for $\mathrm{O}_2^{}{}_{}{}^{-}$ is 25 ${\mathrm{\AA }}^2$ at energies of about 0.2 eV. The drift velocities of ${\mathrm{O}}^{-}$, $\mathrm{O}_2^{}{}_{}{}^{-}$, and $\mathrm{O}_2^{}{}_{}{}^{+}$ in pure ${\mathrm{O}}_2$ are also presented. The mobilities for the molecular ions emerge as 2.4 ± 0.1 ${\mathrm{cm}}^2$ ${\mathrm{}\left(\mathrm{V}\sec \right)\mathrm{}}^{-1\mathrm{}}$ for ions of both signs when extrapolated to the limit of $\frac{E}{p_0}$ of zero, and to $p_0$ of 760 Torr. The values for ${\mathrm{O}}^{-}$ lie appreciably above those of Eiber and of Beaty. A method of producing relatively large currents of ${\mathrm{O}}^{-}$ has been found. It depends on using higher values of $\frac{E}{p_0}$ then might be expected from attachment-coefficient data.

Differential charge-transfer cross sections $\sigma$ are measured for ${\mathrm{He}}^{+}$ in Ar at energies $E$ from 65 to 300 eV and angles $\theta$ from 1° to 30°. The relative measurements are calibrated by comparison with total charge-transfer measurements of Koopman. From reduced plots of $\rho =\theta sin\theta \sigma \left(\theta \right)$ versus $\tau =E\theta$, two, and perhaps three families of oscillations are identified, with reduced angular thresholds at $E\theta \cong 300,970, \mathrm{and} 1250$ eV deg and with spacing $\Delta \theta$ between peaks near threshold related to the wave number $k$ by the rule $\Delta b=\left(\frac{2\pi }{k\Delta \theta }\right)\cong 0.51,0.25, \mathrm{and} 0.09$ a. u., respectively. Using potential curves constructed from earlier information on elastic scattering, the first family is identified as arising from a crossing at 2.9 a. u. to a state dissociating to $\mathrm{He} \left(1\mathrm{}{s}^2\right)+{\mathrm{Ar}}^{+}\left(3s3\mathrm{}{p}^6\right)$, and the second from a crossing at 2.3 a. u. to a state producing $\mathrm{He} \mathrm{}\left(1s2s\right)+{\mathrm{Ar}}^{+}\left(3\mathrm{}{s}^{2}3p^5\right)$. Further crossings at distances between 2.25 and 1.95 a. u. leading to other excited-state channels are apparently responsible for more than half of all the charge transfer at these energies.

Decay lifetimes of seven vacuum-ultraviolet O I and O II emission multiplets have been measured using a pulsed electron beam in low-pressure samples of ${\mathrm{O}}_2$, CO, NO, and ${\mathrm{H}}_2$O. The lifetimes and cascade patterns of the lines are consistent with known autoionization processes. In two cases, spin-orbit-induced autoionization rates are determined. Absolute electron excitation functions for 8447—Å O I have been measured in the above gases. Relative excitation functions for 1304-Å O I are normalized to the 8447-Å absolute cross section by means of a time-domain cascade analysis. A method for computing relative cross sections from $N$-exponential decays is outlined.

The Born approximation is used to calculate the cross sections when a hydrogen atom in an excited state with the principal quantum number $n$ collides with a hydrogen atom in the ground state. Calculation has been carried out when the first atom is ionized while the second atom has any final state. The cases considered are $n=1-4\mathrm{}$. A closure approximation originally used by Lodge is also used here for high impact energy calculations. We find disagreement with some of Lodge's numerical results. At sufficiently high impact energies, the cross sections are inversely proportional to the energies. Coefficients of proportionalities for all cases are given. When the incident atom is in a state with a high principal quantum number such that the relation $\left(\frac{m}{M}\right) K^2{a}_0^2>n^2\gg 1$ is satisfied, with the impact energy $K^2$ in Ry, the electronic and the atomic masses $m$ and $M$, and the principal quantum number $n$ considered, the cross section is given as $Q_0=4\left(\frac{M}{m}\right) (\ln 4n^2-\frac{1}{2}) \left(\pi a_0^2\right) \frac{\mathrm{Ry}}{K^2}$.

Exact and approximate methods for determining the momentum distribution of electronic systems from Compton scattering measurements are presented. The method used previously to analyze Compton scattering measurements, the impulse approximation (IA), is derived from first principles, and its accuracy is compared with the exact calculations for Compton scattering from a hydrogenic system. It is shown that the IA gives very accurate results for weakly bound electrons and that exact calculation may only be necessary to substract out the contributions to Compton scattering from deeply bound core electrons. Experimental results for Compton scattering from helium are presented as a test of the above ideas. Analyzing the results of the experiment in the IA gives a momentum distribution for the weakly bound helium electrons which is in excellent agreement with the momentum distribution obtained from Clementi Hartree-Fock wave functions.

Within the limits set by our experimental apparatus, large-amplitude zero-field oscillations of H fine-structure amplitudes treated in a recent letter of Macek were not observed in polarization—versus—flight-path measurements on ${\mathrm{H}}_{\beta }$, ${\mathrm{H}}_{\gamma }$, and ${\mathrm{Ly}}_{\alpha }$ emissions from suddenly excited H atoms (50- and 150-keV incident protons). Random initial phases for orbital and magnetic substates and approximately equal magnetic substate populations are indicated, in contrast to binary electron capture in gases. A successful test of the theoretical polarization of $2s$ Stark quench radiation was also made.

Br ions with energies of 13.9 and 25 MeV from the Oak Ridge tandem accelerator have been used to measure ionic charge-state distributions in dilute gas targets of ${\mathrm{H}}_2$, He, and Ar. A least-squares technique has been developed to obtain complete sets of electron charge-exchange cross sections from any number and range of measured nonequilibrium charge distributions. Characteristic irregularities, which originate in the atomic shell structure of the Br ions, have been found in the cross sections for capture of a single electron. Multiple electron capture and loss, as well as equilibrium charge distributions, are discussed.

Average electron number density and electron-temperature measurements are presented for the mercury vapor afterglow in a hot-cathode cylindrical discharge tube at a pressure range where ambipolar diffusion predominates. The average electron number density is measured using cylindrical microwave cavities in the T${\mathrm{M}}_{010}$ and dipole resonance modes. The electron temperature is measured by means of a time-resolving 3.0-GHz radiometer. Three pressure regions are identified: (1) $p_0<0.08\mathrm{}$ Torr, where $\frac{p_0}{{\tau }_n}$ increases with $p_0$, (2) $0.08<\mathrm{}{p}_0<0.18\mathrm{}$ Torr, where $\frac{p_0}{{\tau }_n}$ is roughly constant, and (3) $0.18 \mathrm{Torr}<p_0$, where $\frac{p_0}{{\tau }_n}$ increases. Here $p_0$ and ${\tau }_n$ are the pressure reduced to 0 °C and the density decay time constant, respectively. The work presented here is concerned mainly with the first two pressure regions. In the lowest pressure region, electron-temperature measurements in the afterglow under experimental conditions identical to those used for density measurements are used to determine after which point in time the electrons reach the neutral-atom temperature. This information, along with the density decay time constant, is used to predict a mobility of 0.21 ${\mathrm{cm}}^2$ ${\mathrm{V}}^{-1\mathrm{}}$ ${\sec }^{-1\mathrm{}}$ at 760 Torr for ${\mathrm{Hg}}^{+}$ in mercury vapor at 443 °K. This value is in good agreement with previous theoretical and experimental work. In the second pressure range, the electron-temperature measurements indicate that, after an initially rapid drop, a slow decay of electron temperature occurs from roughly 1050 to 600 °K between 0.5 and 5 msec. In spite of this temperature change, the density decay curves were exponential over two decades. This curious observation, which has led previous workers to predict a constant electron temperature around 2000 °K for periods of several milliseconds in the afterglow, has been explained by postulating that the ion mobility could vary over this period. The reasons for this assumption are given in the paper and involve Coulomb collisions which are very important in these experiments. The initially rapid electron-temperature decay occurring during the first 0.5 msec of the afterglow is pressure independent and is well explained by electron-ion elastic collisions. The slow electron temperature decay after 0.5 msec which occurs in the second pressure region is pressure dependent and is explained by a balance between energy gain from collisions of the second kind in the reaction $e+{}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}\to {}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}+e$ and losses due to electron-ion elastic collisions. A complete discussion of these processes is given and the neglect of other mechanisms justified.

The statistical distribution of the number of photoelectrons emitted in a short time interval from a phototube illuminated by laser light scattered from a fluid is examined. Moments of this distribution measure correlated scattering of photons and are related to density correlations of various orders in the fluid. It is pointed out that a measurement of the second moment will indicate whether corrections are needed to the usual interpretation of the scattered-light spectrum. Deviations of the higher moments from the value for uncorrelated scattering would allow measurement of the spatial dependence of multiparticle correlations. The strength of the correlated scattering contribution to the various moments is estimated near a critical point, where it is expected to be largest, using a scaling-law equation of state and available data for simple fluids. According to this estimate, measurement of these multi-particle correlations will be very difficult in a simple fluid. It is pointed out that it may be easier near the critical mixing point of a binary fluid.

An analysis of cavity-mode-mixing effects is given for a variety of cavity geometries and coupling parameters in the presence of a continuous transverse aperture velocity. The paper represents an extension of the previous work by Fox and Li, and Boyd and Gordon to include the effects of time-dependent changes in the transverse aperture location. The average beam profile in the scanning reference frame is calculated through a continuous reexpansion of the optical field in terms of the set of self-reproducing modes characteristic of the same cavity under static transverse boundary conditions. Results are specifically given for square aperture, plane-parallel, concentric, and confocal geometry for light, plane polarized orthogonally to the scan velocity. The effects of a random variation in the mirror-transmission coefficient are calculated, and it is shown that these effects are largely averaged out in the scanning process. Cavities of the type considered earlier by Pole and Myer and by Beiser are specifically considered. A method of evaluating different real systems with the present calculations is discussed; it is shown that a "push-pull"-type scanning laser should have an inherent scan velocity roughly twice that for the single-ended scanning laser geometry previously considered for the same degree of beam degradation and other cavity parameters. A method is developed for the derivation of self-reproducing confocal modes which arise with stationary, continuously varying aperture transmission functions. The method is specifically applied to a confocal Gaussian gain aperture and to a nonsymmetric confocal gain aperture. It is shown that the odd-symmetry mode of the Gaussian gain aperture dominates over the even-symmetry mode at low-gain coefficients, in opposition to the normal diffraction-limited rectangular-transmission-aperture case. An extension of the scanning problem is given to include the transverse displacement of a continuous gain aperture; it is shown formally that the solution of the continuous-aperture problem goes continuously over to the rectangular-aperture results at low values of the gain. The method is specifically applied to a hypothetical scanning laser employing swept Gaussian and nonsymmetric gain apertures.

It is shown that the quantization of circulation in superfluid helium can be understood in a fundamental way by an argument analogous to that used by Byers and Yang for flux quantization in superconductors. The ac Josephson effect in helium also follows simply from this argument.

We calculate the dielectric response function for a strongly correlated electron liquid from a solution of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy; the present result improves the earlier formulas derived by Hubbard and by Singwi et al.

The quantum hydrodynamics of Landau is here regarded as a means of calculating phonon-phonon interactions in liquid helium II. This theory is tested by applying it to the low-density Bose-Einstein gas, at low temperatures and long wavelengths, it agrees with the microscopic theory of the Bose-Einstein gas. This agreement justifies the past use of the hydrodynamic theory to calculate the Landau-Rumer sound absorption in liquid helium; however, the calculated absorption is roughly half that measured by Abraham, Eckstein, Ketterson, Kuchnir, and Vignos. Hence the higher-order terms in the hydrodynamic theory are considered; they probably cannot account for the discrepancy.

In this paper we have calculated the emission of bremsstrahlung radiation of nonrelativistic electrons in Landau levels in a dense plasma. We have found that the radiation rate is inversely proportional to the electron momentum, which is characteristic of one-dimensional gases.