A one-parameter family of nonlinear integro-differential equations, which have been derived to model a wrinkled, premixed flame front, is investigated numerically. It is shown by computing positive Lyapunov exponents that these equations display a chaotic behavior. According to the value of the parameter, characterizing the equation, structures with a well-defined size may appear or not. We discuss this feature, and also the shape of the averaged energy spectrum.

From microscopic observations of droplets an intermediate state between $S_{Ã}$ and $S_{A_2}$ is located in a binary diagram of cyano mesogens. Considering the possible existence of a tetracritical point we suggest that this crenelated phase $S_{A_{\mathrm{cre}}}$ (because one antiphase domain grows at the expense of its neighbor) could be a necessary step between $S_{Ã}$ and $S_{A_2}$.

Many-body perturbation theory was applied to calculate the specific mass shift of the (4s4p) ${}_{}{}^1{}_{}{}^{}$,3P states in Ca relative to the 4s ground state of ${\mathrm{Ca}}^{+}$. The interaction between each valence electron and the core was treated in the same way as the level isotope shifts in the alkali metals in earlier work. The strong correlation between the two valence electrons in neutral Ca was treated self-consistently by iterative solution of the pair equation describing their interaction and its effect on the specific mass shift was evaluated in the same way as for helium. However, also the correlation between the pair of valence electrons and the core is important. All second-order and a large number of third- (and higher-) order diagrams of this type were included, but certain important third-order diagrams could not yet be evaluated. Our final results, -31 and -1533 MHz, respectively, for the 4 ${}_{}{}^1{}_{}{}^{}P$ and 4 ${}_{}{}^3{}_{}{}^{}P$ states between ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$, show only a qualitative agreement with the experimental values, -120(9) and -1116(8) MHz. This is not surprising, in view of the many competing effects with contributions larger than the final value. The remaining discrepancies are consistent with the expected size of the neglected diagrams and illustrate the importance of the core-valence correlation.

Total cross sections for single-electron capture from Li(2s) by ${\mathrm{H}}^{+}$, ${\mathrm{H}}_2$${\mathrm{}}^{+}$, ${\mathrm{He}}^{+}$, and ${\mathrm{Ne}}^{+}$ have been measured at impact energies between 2 and 20 keV. All data sets show maximum cross sections of about 5×${10}^{\mathrm{-}15}$ ${\mathrm{cm}}^2$ and are compared with other available measurements as well as calculations.

Electron-capture cross sections for low-velocity (${10}^6$–${10}^7$ cm/s) highly charged ${\mathrm{Ne}}^{\mathrm{q}+}$ (2≤q≤7) and ${\mathrm{Ar}}^{\mathrm{q}+}$ (2≤q≤10) projectiles incident on molecular- and atomic-hydrogen targets have been measured. A recoil-ion source that used the collisions of fast heavy ions (1 MeV/amu) with target gas atoms was utilized to produce slow highly charged ions. Atomic hydrogen was produced by dissociating hydrogen molecules in a high-temperature oven. Measurements and analysis of the data for molecular- and atomic-hydrogen targets are discussed in detail. The measured absolute cross sections are compared with published data and predictions of theoretical models.

We present measurements of the Stark mixing of the n=2 excited states of hydrogen atoms resulting from the interaction of 800-keV/amu hydrogen projectiles with thin carbon foils. Various excitation mechanisms have been studied in which the projectiles are atomic or molecular species and the electron associated with the excited state originates from the incident projectiles or results from a capture in the target. From these measurements we deduce the relative fractions of the 2p and 2s sublevels in the various cases. We show in particular that the 2s sublevel is rather insensitive to the excitation process.

A detailed study of electronic energy transfers in Kr-Xe mixtures has been made using a 250-keV electron accelerator to excite Kr-Xe mixtures over a wide range of Kr-host-gas pressure and Xe-impurity concentrations. Kr pressure ranged from 25 to 900 Torr with Xe concentrations varying from 0.1% to 10% of Kr partial pressure. Emission spectra taken in the vacuum-ultraviolet (vuv) region indicate that energy is transferred efficiently from Kr to Xe, with radiation from the Xe${(}^3$$P_1$) state becoming greatly enhanced as the Xe-impurity concentration is increased. Emission spectra from the vuv also show that the Xe${(}^1$$P_1$) state is populated by absorption of photons emitted from the ${\mathrm{Kr}}_2$${\mathrm{}}^{\mathrm{*}}$ first continuum. Two-body quenching rates with Xe ground-state atoms have been found for the Kr${(}^1$$P_1$) [$k_q$(1165 Å)=1.2×${10}^5$ ${\sec }^{\mathrm{-}1}$/Torr] and for the Kr${(}^3$$P_1$) [$k_q$(1236 Å)=9.5×${10}^5$ ${\sec }^{\mathrm{-}1}$/Torr] resonance states. Time-resolved measurements have also been made on the ${\mathrm{Kr}}_2$${\mathrm{}}^{\mathrm{*}}$ first and second continua. These measurements show the Kr${(}^3$$P_1$) state is the atomic precursor of the ${\mathrm{Kr}}_2$${\mathrm{}}^{\mathrm{*}}$ first continuum. The results obtained in the ${\mathrm{Kr}}_2$${\mathrm{}}^{\mathrm{*}}$ second continuum are rate limited by the decay of Kr${(}^3$$P_1$) at large Xe-impurity concentrations.

The cross sections for the double ionization of ${\mathrm{H}}_2$ by 0.08–1.0-MeV/amu ${\mathrm{D}}^{+}$ and 0.40–3.5-MeV/amu ${\mathrm{H}}^{+}$ are reported for the ${\mathrm{H}}_2$ internuclear axis oriented at 90° and 30° relative to the ion-beam direction. There is no apparent difference between the cross sections at the 90° orientation and at the 30° orientation. Bethe theory does not give qualitative agreement with the experimental results.

An iterative method is presented for the evaluation of the matrix representation of the group generators in the unitary-group approach to many-electron systems. The method yields, in addition, closed-form expressions and selection rules for nonvanishing matrix elements. Generator products (two-body operators) are treated by a scalar-product method.

Matrix elements of the two-electron operators ${|{\overrightarrow{\mathrm{r}}}_i-{\overrightarrow{\mathrm{r}}}_j|}^{\pm 1}$ decompose into a finite sum of products of angular integrals and radial integrals when a one-electron basis of the form ${\psi }_{l,m,n}\left(\overrightarrow{\mathrm{r}}\right)=R_{l,n}\mathrm{}\left(r\right)\mathrm{}{Y}_{l,m}(\theta ,\phi )$ is used. New methods for computing these matrix elements are given for the choice $R_{l,n}\mathrm{}\left(r\right)=N_{l,n}{\mathrm{}\left(\mathrm{br}\right)\mathrm{}}^1\exp (-\beta r)L_n^{\left(\alpha \right)}\mathrm{}\left(\mathrm{br}\right)\mathrm{}$ where $L_n^{\left(\alpha \right)}$ is a generalized Laguerre polynomial and $N_{l,n}$ is a normalizing factor. Recursive algorithms are used for efficiency. These algorithms are arranged so that those parts of the calculation which are susceptible to rapid accumulation of roundoff error can be done in exact integer arithmetic.

We report a comprehensive set of calculations of the lifetimes of Rydberg states in alkali-metal atoms, using realistic potentials to represent the atomic core. The effects due to core polarizability, spin-orbit interaction, and blackbody radiation are explicitly included. A complete cross reference listing of all available experimental lifetimes is also attempted. The results compare well with experiment for all principal quantum numbers $n$ examined and should provide guidance for future experimental investigations. The relation of the present approach to other works is discussed.

We report a comprehensive set of calculations of the Rydberg-state lifetimes in the helium atom, using a realistic potential to represent the atomic core. The effects due to blackbody radiation are explicitly included. The overall results compare well with experiment for all principal quantum numbers examined and should provide guidance for future experimental investigations. They are used here to single out incorrect experimental determinations of lifetimes. Our calculated oscillator strengths are in excellent agreement with predictions of previous and very sophisticated calculations.

The notion of reduced density operators is generalized and placed in the context of abstract operator algebras. In this setting it is shown that the representability problem is one of finding positive extensions to positive linear functionals, a problem that has been addressed by M. G. Krein in the 1930s. An extension of Krein's theorem is given and used in the analysis of the fermion representability problem.

The energies of the $\left(2s3p\right){}_{}{}^{1,3}{}_{}{}^{}P$ states of the Be sequence ions N iv-Ne vii are calculated, with use of a general-model-space many-body perturbation theory (MBPT) to third order. The triplet states are described accurately by a one-configuration model ($P$) space including only $2s3p$, but the singlets are not; the singlet-triplet separation has the right sign (singlet below triplet) but is only 30% of experiment. Incorporation of the $2p3s$ configuration in the model space greatly improves the results. This incomplete, two-configuration model space yields virtually the same energies as the larger, complete model space required by other quasidegenerate MBPT methods, at one-fifth the computational cost.

For electrons moving independently in a $d$-dimensional bare point-charge potential, the Euler equation of density-functional theory in the limit of large numbers of electrons, $N$, is combined with the virial theorem to relate the sum of eigenvalues $E$ to the chemical potential $\mu$. The result is $\frac{E}{N\mu }=\frac{d(4-d)\mathrm{}}{(4+2d-d^2)}\equiv {\alpha }^{-1\mathrm{}}$. This result (I) is verified by direct calculation of both $E$ and $\mu$ for $d=1,2, \mathrm{and} 3$ in the asymptotic limit of large $N$. Neither does there seem to be any difficulty in applying (I) for $d\ge 5$. But evidently $\frac{E}{N\mu }=0\mathrm{}$ for $d=4\mathrm{}$; an immediate consequence of the virial theorem for this dimensionality. Viewed as a continuous function of $d$, $\frac{E}{N\mu }$ is singular at $d=1+\sqrt{5}$ and is negative for $1+\sqrt{5}<d<\mathrm{}4\mathrm{}$. Excluding this pathological regime of $d$, it can be shown from (I) that, for a point charge $\mathrm{Ze}$, $E=F(Z,d)\mathrm{}{N}^{\alpha }$, where the exponent $\alpha$ is given by (I).

Cross sections for the production of fast ${\mathrm{I}}^0$ and ${\mathrm{I}}^{+}$ particles from ${\mathrm{I}}^{-}$ negative ions in single collisions with He, Ne, Ar, Kr, and Xe targets are reported. The single-electron-detachment cross sections, which previously have been found to reach roughly constant values in other targets at about 100-eV center-of-mass energy, continue to rise until about 8 keV in neon, supporting the suggestion that the (${\mathrm{I}}^{-}$Ne) molecular state does not cross into the continuum. The double-electron-detachment cross sections do not show the inverse target-mass dependence which has been found for ${\mathrm{F}}^{-}$—rare-gas double-electron-detachment collisions.

Simple analytical formulas are given that allow the rapid estimation of cross sections for $l$ mixing, $n$ changing, and ionization of Rydberg atoms with atomic and molecular targets. These formulas are derived by using the quasifree-electron model and the impulse approximation. The $l$ dependence of the cross section is accounted for only via the atomic energy. The quasielastic process of angular-momentum mixing caused by either the short- or long-range interaction between the Rydberg electron and the neutral perturber is studied and the corresponding cross sections are evaluated. The Rydberg-atom—molecule inelastic transitions ($n$ changing and ionization) caused by the $e^{-}$-dipole or $e^{-}$-quadrupole interaction are investigated. It is shown that the $n$-changing reactions occur via quasiresonant internal-energy transfer between the two colliding partners. The influence of the relevant physical parameters is clearly stated for the three types of interaction (short-range, $e^{-}$-dipole, $e^{-}$-quadrupole). Comparison with more sophisticated approaches and experimental data shows that the accuracy of the formulas is about a factor of 2.

For the purpose of astrophysical, aeronomical, and laboratory applications, we seek a precise independent-particle model for electrons in atomic negative ions from $Z=3\mathrm{}$ up to $Z=85\mathrm{}$. We obtain analytic potentials whose inner energy levels are close to the experimental levels of neutral atoms but whose last electron eigenvalue is adjusted so as to be in precise agreement with the electron affinity of negative ions. Systematic trends in the potential parameters are discussed.

Electron-hydrogen binding energy and scattering observables below the first excitation threshold are calculated with three-body integral equations reduced to effective two-body Lippmann-Schwinger-type equations by means of the quasiparticle method. The effective potentials occurring in this formulation are determined in lowest-order and in first-order quasi-Born approximation (QBA). In first-order QBA, our results are closely related to those obtained in the static exchange approximation. Polarization effects are only partly contained in the low-order quasi-Born-approximation terms considered, a fact discussed in the detailed analysis of our results.

The rearrangement collision problem of positronium formation into arbitrary $n$,$l$,$m$ states by positrons from hydrogen atoms in the ground state has been investigated employing the first Born approximation and the first-order exchange approximation. It is shown that for fixed $l$,$m$, $n^3$ times the asymptotic cross section, if multiplied by the factor ${\mathbf{\Pi }}_{\alpha =0\mathrm{}}^l(1-{\alpha }^2{n}^{-2\mathrm{}})$, gives a good estimate of the $n^3$ cross section of the corresponding $n$,$l$,$m$ state when $n$ is not too low. For $s$ state the first-order-exchange-approximation results are greater than those of the first Born approximation. The momentum-transfer cross sections in the first Born approximation overestimate the first-order-exchange-approximation results throughout the energy range considered.