Equilibrium and monoenergetic distribution functions in the laboratory frame are derived from first principles. Thermodynamically consistent macroscopic properties of relativistic beam systems are defined and discussed as a foundation for specific applications. Characteristics of superpinched relativistic electron beams such as isotropy in the lab frame, rms cone angle of the electrons with respect to the beam axis, and the energy flux across a unit surface at an arbitrary angle to the beam axis are obtained. These properties provide important information on the expected symmetry of irradiation of fusion targets by focused electron beams. A linearized kinetic theory is outlined which leads to an integral expression for the general form of the relativistic plasma dispersion function.

Baye's theorem, when applied to the interpretation of a set of measurements with assumed multivariate Gaussian a priori distributions of the physical quantities to be determined, leads to explicit formulas for the most likely interpretation and the corresponding reduction in entropy. We point out an interesting complementarity in two equivalent expressions for the entropy decrease, one of which involves the determinant of a matrix connecting points in physical space, while the other involves the determinant of a matrix connecting the individual measurements. Off-diagonal elements of the latter matrix indicate the degree of redundancy of the information obtained from the corresponding pairs of measurements. Techniques for evaluating the Bayesian expressions for the most likely interpretation are briefly outlined.

Recently, Liefeld, Burr, and Chamberlain observed a structure just below the cutoff frequency of the continuous x-ray spectrum of lanthanum. In an attempt to understand the origin of this feature, we have numerically calculated the tip-region electron-lanthanum atomic bremsstrahlung spectrum, obtained in an independent-electron approximation with the Hartree-Slater potential, for incident-electron kinetic energies from 400 to 1500 eV. The calculations dogive a structure qualitatively similar to the observation. The enhancement occurs in the $f$-wave channel for the final scattered electron, which exhibits a resonance. Such a resonance may be expected because at this point in the periodic table the $4f$ subshell is about to start filling. A more quantitative description would require inclusion of many-electron and solid-state effects.

Argon $L_1$ fluorescence yields have been calculated for the individual multiplet states of ions containing from three to six $M$-shell vacancies. Effective fluorescence yields for these charge states are computed for statistically distributed states and for distributions that would arise in sequential stripping. For Ar $L_1{M}^6$, the latter result is ∼ 430 times smaller than the former.

The Hartree-Fock-Slater (HFS) independent-particle model in the form developed by Hermann and Skillman (HS) and the Green, Sellin, and Zachor (GSZ) analytic independent-particle model are being used for many types of applications of atomic theory to avoid cumbersome, albeit more rigorous, many-body calculations. We here examine the single-electron eigenvalues obtained with these models and find that the GSZ model is capable of yielding energy eigenvalues for valence electrons which are substantially closer to experimental values than are the results of HS-HFS calculations. With the aid of an analytic representation of the equivalent HS-HFS screening function, we identify the difficulty with this model as a weakness of the potential in the neighborhood of the valence shell. Accurate representations of valence states are important in most atomic applications of the independent-particle model.

It is argued that a straightforward physical interpretation of the intensity-dependent ionization potentials computed by Choi, Henneberger, and Sanders is not possible.

The argument of Vermani and Beers is centered around two examples. The smallness of the discrepancy noted in the case of the harmonic oscillation reinforces our confidence in the validity of the Kramers transformation method for the potential case. The generalization of the method to the two-level system is fundamentally different from the potential case. We are thus unwilling to accept conclusions drawn from this particular model as having general validity.

Four new resonances, located at 10.73, 12.55, 14.40, and 15.65 eV, are observed in electron scattering by atomic oxygen. Two of these resonances, located at 10.73 and 15.65 eV, have been predicted theoretically to occur at 10.63 [$\left({}_{}{}^4{}_{}{}^{}S\right)3\mathrm{}{p}^2{}_{}{}^6{}_{}{}^{}P$] and 15.75 eV [$\left({}_{}{}^2{}_{}{}^{}P\right)3\mathrm{}{p}^2{}_{}{}^4{}_{}{}^{}D$] respectively. We attribute the two other resonances at 12.55 eV and 14.40 eV, to the configurations $\left({}_{}{}^2{}_{}{}^{}D\right)3s3p$ and $\left({}_{}{}^2{}_{}{}^{}P\right)3s3p$. With the addition of these new identifications, a total of ten resonances are now known to occur in atomic oxygen, completing the triad of resonance configurations, $3s^2$, $3s3p$, and $3p^2$, associated with each of the three lowest (${}_{}{}^4{}_{}{}^{}S, {}_{}{}^2{}_{}{}^{}D, {}_{}{}^2{}_{}{}^{}P$) positive ion states.

A minimum-variance principle for quasibound states is defined and utilized to investigate the corresponding energy resonances. The variational principle yields simultaneously a resonance energy $E$ and a square-integrable wave function $\varphi$, such that minimum variance is obtained for arbitrary variations of a restricted class of square-integrable functions as well as with respect to variations of $E$. It is further shown that the optimum energy $E_0$ obtained from this method can simultaneously be written as an expectation value of the actual Hamiltonian with respect to $\varphi =\varphi \left(E_0\right)$. The example of the Stark effect in the hydrogen atom is studied. It is shown that the variationally obtained resonance energy coincides with the real part of the complex pole of the $m$ function of Weyl, related to the Green's function of the system under consideration. It is also shown that the corresponding numerical application of the Rayleigh-Ritz variational method only gives meaningful results for field intensities below 0.06 a.u., as compared with the "exact" results of Hehenberger, McIntosh, and Brändas.

The moments $S\left(\mu \right)$ and $L\left(\mu \right)=\frac{dS\left(\mu \right)}{d\mu }$ for $-6\mathrm{}\le \mu \le 1$ are derived from comprehensive Hartree-Slater oscillator-stength distributions for He through Ar. For $\mu \le -2\mathrm{}$, these moments are governed by valence excitations only, and therefore exhibit a pronounced periodic variation that repeats in each row. Inner shells begin to contribute appreciably to $S(-1\mathrm{})\mathrm{}$, which retains a periodic variation superimposed upon an over - all increase with increasing atomic number $Z$. For $\mu \ge 0$, the $Z$ dependence of the moments becomes dominated by inner-shell contributions; as $\mu$ increases, the over-all increase with $Z$ becomes more rapid. Another perspective of the voluminous data is gained by plotting $logS\left(\mu \right)$ vs $\mu$. The plot reveals three classes of behavior—"tight," "intermediate," and "loose" atoms. Comparisons with experiment and more detailed calculations are made where possible.

Theoretical expressions for the Auger rates of various spectroscopic terms of electronic configurations $1s^{1}2s^{2}2p^n$, $1s^{1}2s^{0}2p^n$, and $1s^{1}2s^{1}2p^n$ ($n=6\mathrm{}\to 1$) are presented. Numerical values of the x-ray transition energies and the multiplet partial fluorescence yields are given for variously ionized neon.

Emission spectra observed from electron-beam—excited atmospheric-pressure gas mixtures of Xe and ${\mathrm{I}}_2$ are reported and analyzed in terms of an ionic alkali-halide-like excited state ${\mathrm{Xe}}^{+}$${\mathrm{I}}^{-}$. A simple theory is given to analyze the observed spectra, and predictions of the uv spectra of other inert-gas monohalides are made. This class of molecules appears to offer interesting possibilities for production of new uv lasers.

The normalized fluorescence spectrum of the Li 6708-Å resonance line perturbed by the noble gases has been measured for 100 Å either side of the resonance line, at a temperature of 400 ± 10°C. The noble-gas pressure range of 100-1800 Torr was investigated. It was observed that the line wings scale linearly with this pressure for $2<\left|\Delta \lambda \right|<\mathrm{}100\mathrm{}$ Å, and it was was demonstrated that a Lorentzianbroadened line core can be observed for this pressure range. The Li density was ∼ ${10}^{10}$/${\mathrm{cm}}^3$ and was varied to establish the fluorescence spectrum in the limit of no radiation entrapment. The central region of the line has been measured with high resolution and compared to calculated profiles due to the various fine structure and isotopic line components, each broadened by a convolution of Lorentz, Doppler, and instrumental profiles. The shift and Lorentz broadening rates, as well as the wavelengths for transition to non-Lorentzian behavior, are reported. In contrast to the heavier alkali-metal-noble-gas cases, no distinctive far-wing features such as satellites or sharp inflections are observed. This is consistent with the much larger collision velocities for the Li case.

The Fresnel reflectivity and transmissivity coefficients for a semi-infinite medium are derived for normal and oblique incidence using a particle approach and a probabilistic condition at the boundary. The method depends in part on the behavior of the momentum of light in ordinary refractive media, and this subject is discussed. The method clarifies why the Schrödinger equation for the comparable problem (potential-step) and Maxwell's equations yield the same coefficients for normal incidence. The use of the phase velocity rather than the group velocity in formulating the probabilistic condition is interpreted from a quantum viewpoint, although the absence of Planck's constant in the analysis suggests one is dealing with a kind of nonclassical but prequantal indeterminacy. An analysis and review of the momentum, inertia, and energy-momentum tensor of light in ordinary refractive media is also given.

The quantum theory of a one-dimensional optical cavity is developed with emphasis on the presence of output coupling. First the resonant mode is defined and calculated classically. Then the field is decomposed into a sequence of modes by introducing an imaginary boundary at a large distance. The mode functions are proved to be orthogonal with respect to an integration with the dielectric constant as a weighting factor. The Hamiltonian of the field is shown to be equivalent to that of a collection of independent harmonic oscillators, the mass of which is a function of the frequency of oscillation. This equivalent of mass appears on normalizing the mode functions and proves to carry information on the structure of the cavity. Quantization of the field is carried out, and the commutation relation for the electric fields inside and outside the cavity is derived. The commutator is composed of an infinite set of derivatives of $\delta$ functions, which discloses the effect of the presence of output coupling and that of the small size of the cavity on the radiation field. Also, the commutator is shown to have some properties common with the classical resonant mode, i.e., the property of exponential decay and its rate.

In this paper we develop a theoretical analysis which describes a coherent transient effect of two entering pulses—one laser and one Stokes—in a Raman-active medium. The field (Maxwell) and atomic (Schrödinger, density-matrix representation) equations are coupled in a self-consistent manner in time and space by virtue of the resultant second-order nonlinear macroscopic polarizations at the two frequencies. This treatment is accomplished under the circumstances where atomic coherence plays a predominant role. The equations derived here may be referred to as "Raman Bloch-Maxwell equations." The resultant equations yield a new aspect of transient stimulated Raman scattering and thus coherent Raman propagation. The transient-pulse behavior may be understood as self-induced modulation of coherent amplification and absorption. The self-induced modulation for the step-function input pulse is described by a combination of Jacobian elliptic functions depending on the input intensity. Computer calculations of transient-pulse evolution reveal interesting pulse breakup phenomena with peak amplification, energy transmission, pulse advance or delay, and pulse-narrowing effects.

Stark-broadened profiles of the Balmer lines $H_{\alpha }$ and $H_{\beta }$ have been measured by means of a high-pressure electromagnetically driven shock tube, at electron densities $N_e\sim {10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and temperatures $\kappa T\sim 1.5$ eV. The measured profiles of $H_{\alpha }$, down to ∼5% of peak intensity, are in much better agreement with the theoretical profiles of Kepple and Griem than with the results of more recent computations by Vidal, Cooper, and Smith. This suggests that for hydrogen lines with significant upper- and lower-state broadening, only elastic scattering contributions to the upper-lower state interference term should be included in the line-broadening operator. For $H_{\beta }$, agreement is obtained with both theories except within the central dip, the discrepancies between measured and predicted modulations being somewhat larger than those obtained in recent arc experiments. Our results for $H_{\beta }$ can be interpreted as indicating the presence of a transition layer of much lower electron density near the walls rather than the necessity for including the effects of ion dynamics in the calculations. The first-order dynamical correction to the Holtsmark profile for a single Stark component is considered in analogy with stellar dynamics, and found to be of negligible importance under our conditions. Additional measurements of the central structure of $D_{\beta }$ and the He i 4471-Å line are consistent with these conclusions.

Generalized oscillator strengths $f\left(K\right)\mathrm{}$ and total cross sections $\sigma$ are calculated within the first Born approximation for elastic scattering and for spin-allowed electronic transitions from the $3s^{2}3p$ ground state to the $3s^2\mathrm{nd}$ ($n=3, \dots , 7$), $3s^{2}4s$, and $3s^{2}4p$ Rydberg states. Exchange effects are included in the cross sections via the Bonham-Ochkur-Rudge approximations. Hartree-Fock and superposition-of-configurations (SOC) wave functions are utilized to examine their effect on $f\left(K\right)\mathrm{}$ and $\sigma$. $f\left(K\right)\text{'}s$ obtained from SOC wave functions to varying accuracy are plotted and compared, and $\sigma \text{'}\mathrm{s}$ calculated with HF and SOC wave functions are presented. Results calculated with a one-electron model using scaled hydrogenic orbitals to generate the $f\left(K\right)\mathrm{}$ are compared with those obtained with Hartree-Fock and SOC wave functions.

A mathematical picture of the transition to turbulence in statically stressed fluid systems is proposed. Systems are classified on the basis of the Hopf bifurcation theorem. One category, "inverted bifurcation," contains flows that exhibit hysteresis, finite-amplitude instabilities, and an immediate transition to turbulent behavior. A mathematical model due to Lorenz which manifests this kind of behavior is discussed. The other category, "normal bifurcation," includes flows in which, as the stress increases, a time-periodic regime precedes turbulence. A model of low-Prandtl-number convection falls into this category. The transition to nonperiodic behavior in this model is studied and found to proceed in accord with abstract mathematical proposals of Ruelle and Takens. Semiquantitative agreement with the experimental fluctuation spectrum of Ahlers is also obtained.