Certain systems in physics are commonly referred to as "Hamiltonian systems"; these are all closed systems: open systems, which may undergo damping or growth phenomena, are "non-Hamiltonian." The mathematical basis of this distinction is a little-known theorem of Helmholtz, stating necessary and sufficient conditions on the functions whose vanishing expresses the equations of motion, that a Lagrange function exist in a certain sense. When the Helmholtz conditions are satisfied, the equations of motion always can be written as $G_n(x,\dot{x},\ddot{x},t)\equiv \left(\frac{d}{\mathrm{dt}}\right)f_{2n}(x,\dot{x},t)-f_{1n}(x,\dot{x},t)=0\mathrm{}$, $n=1-N$, but many systems of physical importance having this form fail to satisfy those conditions. The present paper is the first of a planned series presenting a new classical Hamilton-Lagrange mechanics for systems having equations of the form just stated, restricted for convenience to the case det $\parallel \frac{\partial f_{2n}}{\partial {\dot{x}}_m}\parallel \ne 0$, but with otherwise arbitrary $f_{1n}$,$f_{2n}$. To get around the Helmholtz conditions in a way useful for application to modern theoretical physics (quantum mechanics and statistical mechanics), we have generalized the classical formalism. The new mechanics subsumes the old in a very reasonable and theoretically promising way. A formal quantization leads to a generalized Schrödinger equation of a state function whose natural physical interpretation is a density function (density "matrix"), rather than a "half density", or wave function. Another, for closed systems, gives the usual Schrödinger equation twice—once for $\psi$ and once for ${\psi }^{*}$, while a third gives a classical Hamilton-Jacobi partial differential equation.

The nuclear-spin dephasing of the noble gas ${\mathrm{Kr}}^{83}$ has been observed. A nonexponential decay signal, which is a function of the experimental cell in the apparatus, has been measured. The spin decay characteristics of ${\mathrm{Kr}}^{83}$ have been explained in terms of an interaction of the nuclear quadrupole moment of ${\mathrm{Kr}}^{83}$ with the electric field gradients present at the nucleus during a collision of the Kr atom with the walls of the experimental cell. Qualitative agreement only is shown, since a complete quantitative analysis of the experiment is not available at the present time.

A theory of saturation spectroscopy in three-level gas-vapor systems, including collisional effects, is presented. Using a model of collisions in which they are phase interrupting in their effect on level coherences and velocity changing in their effect on level population densities, the authors calculate the absorption profile of a weak probe field on a given transition when an arbitrarily strong pump field acts on a coupled transition. Line shapes are derived for an arbitrary collision kernel describing the velocity-changing collisions, and these line shapes are evaluated for the collision kernel proposed by Keilson and Storer. Various limiting forms for the line shapes are derived and representative line shapes are displayed. Several new features of the line shapes, including collisionally induced increased probe absorption are discussed. The line shapes are seen to reflect the various collisional processes that may occur in atomic or molecular systems.

That observed amplitudes of extended-x-ray-absorption fine structure (EXAFS) are less than those calculated theoretically has been attributed to the failure of the theory to take into account multielectron excitations during core ionization. To investigate this possibility, the authors have measured the satellite spectrum accompanying ionization of $3d$ electrons from ${\mathrm{Br}}_2$ by aluminum x rays. There is approximate agreement between the amplitude attenuation calculated from these experiments and that determined from EXAFS data, provided one assumes that shakeup occurs only if the outgoing photoelectron has a kinetic energy of several hundred eV, and makes allowance for a small contribution from high-energy satellites in the shakeup spectrum that would not be observed in the present experiments.

The authors introduce a procedure that combines measurements of absorption and emission by atoms to obtain relative oscillator strengths that are independent of temperature determination in the sources and of assumptions regarding local thermodynamic equilibrium. The experimental observations are formed into sets of transitions and required to satisfy defined ratios. The screened data are adjusted with a least-squares program to obtain optimized relative oscillator strengths and constants relating the observations to these values. With appropriate choices of input observations, the constants are proportional to upper-level lifetimes and lower-level populations. The procedure is illustrated by the published data of Whaling et al. and Smith and Kühne for 16 transitions in Tii. The relative oscillator strengths resulting from this procedure have calculated uncertainties between 5 and 17% (∼ 95% confidence level). Evidence is presented to suggest that these uncertainties have been overestimated. Calculated oscillator strengths are normalized to the atomic-beam absorption measurements of Bell et al. and to the experimental lifetimes of Roberts et al. and Whaling et al. The absolute oscillator strengths are determined with an uncertainty of 7-18%. The results indicate that the published lifetime for the level $y{}_{}{}^3{}_{}{}^{}D_2^0{}_{}{}^{}$ of Tii should be increased by 24%.

Exact solutions are presented for the equations describing the stationary population of a two-level atom in a nonmonochromatic ideal chaotic field. The dependence of the excited-state population on the Rabi frequency, bandwidth of the field, spontaneous decay, and detuning is studied and compared with the results obtained for phase-diffusing laser light. The validity of the decorrelation approximation is also investigated.

Formulas are derived for the mean values and variances of the energy distributions of the levels of an atomic configuration and of the radiative transitions between the levels of two configurations (in intermediate coupling). The variance ${\sigma }^2$ of the distribution of the eigenstate energies belonging to a given configuration is considered first: ${\sigma }^2$ is expressed as a linear combination of squares and cross products of the usual Slater electrostatic and spin-orbit radial integrals. It is shown how this expression can be used to check the numerical matrices of energy-integral coefficients. Then expressions are derived for the mean value and for the variance of the weighted distribution of the transition energies between two configurations (the weight of each transition being its strength) in the $n{l}^{N+1\mathrm{}}-n{l}^N{n}^{\prime }{l}^{\prime }$ and $n{l}^N{n}^{\prime }{l}^{\prime }-n{l}^N{n}^{\prime \prime }{l}^{\prime \prime }$ cases. This derivation is based on the second-quantization formalism. An extension is made to the case of complementary configurations. For transitions $n{l}^{N+1\mathrm{}}-n{l}^N{n}^{\prime }{l}^{\prime }$, an explicit formula is obtained for the shift between the mean energy of the transition array and the difference of the mean energies of the configurations. Numerical tables of the angular coefficients appearing in ${\sigma }^2$ are given for most cases where $l$, $l^{\prime }$, $l^{\prime \prime }\le 3$. The main application presented here concerns highly ionized spectra of molybdenum, with transitions between $3d^{N+1\mathrm{}}$ and $3d^{N}4p$, $3d^{N}4f$, $3d^{N}5p$, and $3d^{N}5f$. The agreement between experimental and theoretical (ab initio) mean wave numbers and variances is good. A discussion of the physical conditions of applicability of the results to experimental situations is given.

Using a high-resolution Bragg spectrometer equipped with a LiF crystal, the authors have measured the $K\alpha$ x rays of Fe produced by incident ${}_{}{}^{127}{}_{}{}^{}\mathrm{I}$ projectiles of energy 100 and 167 MeV. Distinct satellite structure corresponding to $K$ x-ray emission in the presence of multiple $L$ vacancies was resolved. The observed spectra consisted of six major components, from which it is concluded that initial $L_1$ vacancies transfer before x-ray emission via fast $2s\to 2p$ Coster-Kronig transitions. The authors have determined the energy, width, and intensity of the individual satellite peaks, and find that the number of $M$ vacancies increases with $L$ vacancies and that the degree of $M$-shell ionization is greater at the lower beam energy. Relative ionization intensities were determined by taking into account the effects of $L$-shell rearrangement prior to $K$-x-ray emission. It is found that these intensities are reasonably well represented by a binomial distribution giving ionization probabilities $p_L$ of 0.41 and 0.44 at 100 and 167 MeV, respectively.

A novel form of laser frequency switching is devised which extends coherent optical transient studies to a 100-psec time scale. The technique incorporates a traveling-wave electro-optic element which imposes with unit efficiency a uniform time-varying phase and thus a frequency shift on a cw laser beam. In contrast to earlier optical traveling-wave modulators which are driven by a microwave oscillator, here the optical phase change is induced rapidly and easily by a dc electric field pulse that propagates in a microwave guide either parallel or antiparallel to the light wave. This advance enables optical free-induction decay (FID) studies on a subnanosecond time scale and reveals such new features as a rapid first-order FID that dephases with the inhomogeneous dephasing time $T_2^{*}$. The well-known nonlinear FID can interfere with the first-order component at short times and decays over the much longer period $\frac{T_2}{[1+{(1+{\chi }^2{T}_1{T}_2)}^{\frac{1}{2}}]}$, where $\chi$ is the Rabi frequency. A complete analytical expression is derived for optical FID of a transition subject to homogeneous and inhomogeneous broadening and supports detailed observations of the sodium $D_1$ line.

The equations previously derived for the intensity fluctuations of a two-mode ring laser are applied to a homogeneously broadened laser. It is shown that the mean light intensity of the more lossy mode passes through a maximum and then tends to zero as the excitation is increased. The probability distribution of the intensity of each mode may exhibit two peaks, and spontaneous switching between the intensities associated with the peaks may occur. This is manifest in relative intensity fluctuations of the more lossy mode that can be much greater than unity. Estimates are given for the characteristic switching time.

The effect of cooperative atomic interactions on the photon statistics in a single-mode laser is studied on the basis of an equation of motion for the reduced-density operator of the field that was recently derived by Huang and Mandel. The corresponding antinormal ordering distribution function is shown to satisfy a Fokker-Planck equation. The steady-state solution of this equation is used to determine the photon-number distribution, the average intensity, and the intensity fluctuation.

The theory of third-harmonic generation is extended to the treatment of a generating electric field having a cylindrical spatial-mode structure. In the tight-focusing approximation it is proved that third-harmonic generation is only possible in negatively dispersive media ($\Delta k<\mathrm{}0\mathrm{}$). For a laser beam which consists of a pure $\mathrm{TE}{\mathrm{M}}_{\mathrm{pl}}$ mode, the generated field is a superposition of modes with a fixed angular mode number $3l$ and a radial mode number in the range from 0 to $3p$ with the exception of $3p-1$, which is always 0. The generated power has a functional dependence on $b\Delta k$ which is single peaked for $p=0\mathrm{}$ and multipeaked for $p\ne 0$. For a phase-matched situation, the conversion efficiency decreases as either $p$ or $l$ increases. However, for a system where the number density of the negatively dispersive medium is proportional to the mismatch wave vector $\Delta k$, the TE${\mathrm{M}}_{00}$ mode will not give the maximum conversion efficiency. It is noted that reliable determinations of refractive indices from phase-matching curves and third-order susceptibilities from absolute efficiencies of third-harmonic generation depend critically upon knowledge of the spatial structure of the generating field.

Parametric amplification may be obtained in a harmonic oscillator if its spring constant is replaced by a time-dependent function. It is shown that the quantum transition probabilities may be expressed in a simple closed analytic form which involves only the classical amplification factor of the oscillator. The authors also compute the time evolution of the density matrix which initially describes a thermodynamic ensemble at temperature $T$ and show that this matrix can similarly be expressed in such a closed form.

It is shown that the acceleration of an electron by the electrostatic fields of the ion-wave fluctuations can lead to enhanced electromagnetic waves in a plasma. The maximum emission arises from those electrons whose velocity is close to the phase velocity of the ion-wave oscillations. Application of our investigation to laboratory and astrophysical plasmas is discussed.

X-ray diffraction patterns of single-domain samples of the smectic $F$ phase of terephthalydene-bis-4-$n$-pentylaniline, obtained by melting single crystals of the crystalline phase, have enabled us to collect important information concerning the nature of this phase. The smectic layers are almost uncorrelated, and within the layers the order is characterized by a pseudohexagonal order analogous to that observed in the smectic $B$ phase with tilted molecules, but less well defined: the smectic $F$ phase is intermediate between the smectic $B$ and smectic $C$ phases. An intensity profile analysis performed on single-domain patterns is consistent with the two-dimensional structure of the smectic $F$ phase, and the investigation of the powder pattern indicates in addition the existence of another disorder type which remains to be determined.

The delay convention and the Maxwell convention, widely adopted in catastrophe theory, may be regarded as two extreme cases in a broad range of possibilities. A Fokker-Planck equation is introduced to interpolate between these two extremes. The two time scales associated with the Fokker-Planck equation are related to the critical curvature and the potential-barrier height. The canonical relaxation and diffusion time scale curves of the cusp catastrophe are presented. Quantitative conditions for the validity of one of these two conventions are given in terms of the time rate of change of the control parameters, the curvature at locally stable critical points, and a signal-to-noise ratio.

The microscopic dynamic behavior of some simple liquids has been investigated by observing the time evolution of the separation between members of pairs of atoms. The distribution function for this process was calculated in molecular-dynamics studies of several states of two model fluids: Lennard-Jones and simulated rubidium. The overall features of the results are described quite well by a simple model in which the distribution of pair separations satisfies Smoluchowski's equation with a potential of mean force. The model uses a nonlinear time scale determined from the single-particle motion.

A microscopic construction method of lattice defects which can be represented by dislocation configurations in simple-cubic lattices and for interaction by central forces is given. In contrast to standard theories, where dislocations are introduced via topological operations (Burgers circuit), the author starts from a microscopic ad hoc Hamiltonian which is suitable only for central-force interacting systems. The various sectors of this microscopic Hamiltonian are associated with dislocation configurations. A detailed discussion of the stability and symmetry properties of the microscopic Hamiltonian is given. Possible extensions of the theory to non-central-force interacting systems are pointed out.

The authors present a microscopic theory for the averages of any dynamical variable and in particular of fluctuations in nonequilibrium stationary states that are arbitrarily far from equilibrium, as long as the macroscopic gradients are sufficiently small. It is argued that the dynamics of the fluctuations are governed by the linearized macroscopic equations of motion (analogous to Onsager's hypotheses for equilibrium fluctuations). The fluctuation-dissipation theorem is examined and it is found that it does not hold in its equilibrium form. The authors find that the dissipation does not contain an important part of the information about the fluctuation, and attempt an interpretation of this fact.

The ratio of the diffusion coefficient $D$ to the mobility $\mu$ for electrons has been measured in liquid argon by using a modified Townsend apparatus. The measured ratio $\frac{\mathrm{eD}}{\mu }$ monotonically increases from ∼0.1 to ∼0.4 eV with an increase in the electric field from ∼2 to ∼10 kV/cm. The ratio $\frac{\mathrm{eD}}{\mu }$ is smaller than that in gaseous argon at the same value of $\frac{E}{N}$, where $N$ is the atomic density. The mean momentum-transfer cross section $〈\sigma 〉$ is estimated as a function of $〈\epsilon 〉$ on the assumption $〈\epsilon 〉=\frac{3eD}{2\mu }$, where $〈\epsilon 〉$ is the mean electron agitation energy. Thus the $〈\sigma 〉$ obtained is nearly constant (3∼4×${10}^{-17\mathrm{}}$ ${\mathrm{cm}}^2$) for $〈\epsilon 〉$ from 0.15 to 0.6 eV. This value of $〈\sigma 〉$ is almost the same as that estimated from the low-field mobility of electrons in liquid argon. This result is consistent with both Derenzo's experiment and Lekner's theory. It is concluded that there is no Ramsauer minimum at about 0.4 eV of $〈\epsilon 〉$, where the minimum exists in gaseous argon.