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A new renormalization group method is used to calculate the photon propagator in the high four-momentum regime. The assumption that this propagator is a self-similar object leads directly to a functional equation of evolution involving an invariant charge (effective coupling function) which, in turn, is a functional of the propagator. Numerical results produced by this theory depend on a single unknown, namely, the scale in which the four-momentum k is measured. Calculations are presented for several values of this scale. From these it is concluded that the transverse photon propagator behaves asymptotically as D⊥(k)~k2(λ-1) with the value of λ (the anomalous dimension) falling in the range (0.12, 1).

The task of computing the classical partition function for a bound aggregate of three particles is reduced to that of performing a 2-dimensional quadrature. A simple, analytical formula is obtained in the special case of three identical particles for which the mutual interactions are given by pair-additive square-well potentials.

The purpose of this investigation is to develop a computationally tractable procedure for obtaining theoretical estimates of the correlation functions characteristic of a chemically reactive small‐molecule fluid and for determining the spectral characteristics of the associated many‐body modes. Generalized Enskog‐type kinetic equations (appropriate both to low and high densities) are derived for fluid mixtures which include species capable of participating in bimolecular reactions such as AB+CD⇄AC+BD. The dynamics of the reactive events are treated according to the model of simple reactive spheres (SRS) and to a less restrictive, modified simple reactive sphere (MSRS) model, as well. The distribution functions of the various species are represented by bilinear expansions in terms of velocity basis functions and the associated expansion coefficients are identified with correlation functions of pairs of single‐particle dynamical variables. The kinetic equations are then transformed into coupled sets of line...

Numerical estimates are presented for the shear viscosity coefficient of a hard‐sphere suspension. Our calculations are based on a previously reported theory which uses a Fokker–Planck equation to describe the evolution of the solute many‐body distribution function. The hydrodynamic interactions of the solute particles are modeled by two‐body forces which include the short‐ranged, singular contributions due to lubrication. This produces estimates of the viscosity that agree significantly better with experiment than did our earlier calculations which neglected the effects of lubrication. Our predictions are compared with those of other theories, with results of computer simulations, and with experimental data.

Calculations are reported of the steady‐state shear and bulk viscosity coefficients of a suspension of freely rotating, hard spheres dispersed in a continuous Newtonian solvent. These calculations are based on a theory recently reported in this journal. Our theoretical predictions are compared with those of other theories, with the results of computer simulations, and with experimental data.

[ital Ab] [ital initio] calculations are performed on the [ital e][sup [minus]-]He[sub 2][sup +] complex using the [ital R]-matrix technique. A two-state approximation is utilized in which the [sup 2][Sigma][sub u][sup +] and the [ital A] [sup 2][Sigma][sub g][sup +] electronic states of He[sub 2][sup +], in the close-coupling expansion, are represented by truncated configuration-interaction (CI) wave functions obtained from multiconfiguration-self-consistent-field--complete-active-space-SCF (MCSCF-CASSCF) calculations. A (4[ital s],2[ital p],2[ital d]) Slater basis is used to perform bound-state calculations at SCF, full CI, and MCSCF-CASSCF levels, to gauge the quality of wave functions at various bond separations. Elastic cross sections are presented for selected bond lengths in the range 1.5--4.0[ital a][sub 0], for the [sup 3][Sigma][sub [ital u]][sup +] total symmetry. He[sub 2][sup *](1[sigma][sub [ital g]]1[sigma][sub [ital u]][sup 2][ital np][sigma] and [ital nf][sigma] [sup 3][Sigma][sub [ital u]][sup +]) resonances are detected at each separation and fitted to Breit-Wigner formulas in order to determine their energies and autoionization widths. In addition, low-lying He[sub 2][sup *](1[sigma][sub [ital g]][sup 2]1[sigma][sub [ital u]ns],[ital nd], and [ital ng] [sup 3][Sigma][sub [ital u]][sup +]) bound states are also calculated at several geometries.

A projection operator procedure is used to analyze the particle currents and the fluxes of momentum and energy in a multicomponent classical fluid. The projection is onto the fluctuations from equilibrium of the temperature, of the concentrations of all species present in the solution, and of the total density of momentum. Each of the flow rates is resolved into a sum of convective, dissipative, and random parts. Correlation function expressions are derived for the wave vector and frequency‐dependent Onsager kinetic coefficients appropriate to the laboratory frame of reference and to several flow reference frames as well. Special attention is devoted to the frame associated with the mass‐averaged, barycentric velocity and connections are considered between the results of the present investigation and those obtained in an earlier, closely related study which used a projection onto a subspace of that used here.

The authors have performed systematic quantal calculations of Ar-Ar elastic scattering events using accurate pair potentials. Specifically, they have calculated a large number of partial wave phaseshifts, total cross sections and resonance energies over a range covering most of the glory region. The cross sections from two accurate potentials are compared over one glory oscillation and over a region including several shape resonances. The resonance energies obtained from these two potentials also are compared. The authors discuss the inappropriateness of using Levinson's theorem to define the absolute scale of the phaseshifts for realistic atomic potentials. As an alternative, the authors define an 'absolute phaseshift' that explicitly accounts for the difference in phase between the true wavefunction and its free particle counterpart.

The Fokker–Planck formalism is used to construct a statistical‐mechanical theory of the linear viscoelastic response of a colloidal dispersion of freely rotating spheres. It is assumed that the Newtonian solvent exerts random, fluctuating forces upon the solute spheres and systematic, frictional forces as well. The direct interactions among the spheres are separated into short‐ranged, hard‐core repulsions and longer‐ranged, continuous forces. This model leads to a stress tensor for the suspension of the form σ=σSS+σPP+2σhy, with σPP denoting the contribution due to the translational motions of the suspended particles (P) and their mutual interactions. The term 2σhy arises directly from the frictional, hydrodynamic interactions between the spheres and the solvent and σSS is the stress tensor of the (continuum) solvent perturbed by the presence of the solute spheres. Formulas are obtained for the linear viscoelastic moduli associated with σPP and σhy and explicit procedures are presented for computing the n...

A new renormalization group procedure is used to resum the virial series for the viscosity of a hard-sphere suspension. Theoretical results are compared with experimental and numerical simulation data.

A new method of derivation of the moment conditions is presented and additional moment conditions are discovered.

A procedure is given for the analytic asymptotic evaluation of certain integrals which arise in the quantum‐mechanical theory of inelastic molecular collisions. The integrals are evaluated using WKB wavefunctions and the saddle‐point method. The method is equivalent to Landau's but is considerably more transparent and is easily applied to vibrational and rotational transitions for which there is no crossing of potential‐energy surfaces, as well as to electronic transitions near a crossing point. The special case of resonance collisions is discussed, and the method is illustrated with an example.

The projection operator methods of Mori and Zwanzig are used to construct kinetic equations for the single‐particle distribution functions of a fluid mixture. The particle interactions are pair‐additive sums of hard‐core and soft, continuous potentials. The mean field kinetic equations obtained by discarding the multicomponent memory functions are identical in form to van Beijeren and Ernst’s revised Enskog equations for a hard sphere mixture, but incorporate the radial and direct correlation functions characteristic of the composite interaction. The 13‐moments methods is used to construct an approximate solution of the mean field kinetic equation for a simple fluid and to relate the associated, zero‐frequency transport coefficients to those of the conventional Enskog theory. Our choice of interaction parameters optimally mimicks a Lennard‐Jones 12‐6 fluid and yields excellent estimates of viscosity.

Cross sections and polarizations of impact radiation have been calculated for 2s to 2p excitation of Li in collisions with He and Ne, and for 3s to 3p excitation of Na in collisions with He and Ne, for the (centre-of-mass) energy ranges ECM=0.2-100 keV (Li/Na-He), ECM=0.7-100 keV (Li-Ne) and ECM=1-100 keV (Na-Ne). The calculations have been based on three-state one-centre expansions of the electronic states, 2s, 2p0 and 2p+or-1 in the case of Li-He/Ne collisions, and 3s, 3p0 and 3p+or-1 in the case of Na-He/Ne collisions. For Li-He/Ne collisions a study of excitation cross sections and of polarizations of emissions calculated for two Baylis model potentials, two Bottcher-type model potentials and a Hartree-Fock frozen-core potential is reported. For Na-He/Ne collisions the results obtained by the Baylis model potential are reported.

Cross sections and polarization of impact radiation have been calculated for the excitation of Be/sup +/(2s) and Mg/sup +/(3s) ions in collisions with ground-state He and Ne atom targets in the energy range E/sub cm/ = 0.5--100 keV. Each ion-atom pair has been treated as a quasi-one-electron three-particle system consisting of the valence electron of the ion, the closed-shell ion core, and the closed-shell rare-gas atom. The interactions among these three have been represented by model potentials. Charge transfer and excitation of the rare-gas target atoms are neglected. We have obtained three- and seven-state (in the case of Mg/sup +/) close-coupling solutions to the impact-parameter equations, assuming a rectilinear trajectory for the motion of the two heavy particles. Comparisons are made of theoretical predictions based on two Bottcher-type model potential (V/sub B//sub 1/ and V/sub B//sub 2/) and the simple Hartree-Fock frozen-core potentials (V/sub HF/) for the electron--rare-gas interaction. The predictions based on V/sub HF/ agree better with experimental results than do those based on V/sub B//sub 1/ or V/sub B//sub 2/, especially in the case of Be/sup +/-Mg/sup +/--He collisions.

The results of a combined experimental and theoretical investigation of the excitation processes in Li-He collisions in the 1-100 keV energy range are presented. Experimentally the emission cross sections have been measured for the two lowest members of each of the 22S-n2P, 22P-n2S and 22P-n2D Li I Rydberg series, which are the strongest optical transitions observed. Due to the long lifetimes of the higher excited levels of the projectile the problem of radiation equilibrium in the collision cell requires attention. Theoretically, the collision has been treated as a quasi-one-electron system and close-coupling solutions to the impact-parameter equations have been obtained, using the (2s, 2p0, 2p+or-1, 3s, 3p0, 3p+or-1, 3d0, 3d+or-1, 3d+or-2) basis. A Hartree-Fock frozen-core potential has been used to describe the direct electrostatic interaction between the Li valence electron and the helium atom.

The classical-trajectory impact-parameter method is used as the basis for deriving an equation of motion for the electronic state of two colliding, composite particles, one a projectile atom or ion consisting of a single valence electron outside a closed inner shell and the other, a target species comprised of two electrons plus a closed inner shell and/or atomic nucleus. The three outer-shell electrons are treated as being indistinguishable from one another, whereas the inner-shell, core electrons serve as sources of static, local pseudopotentials. We adopt for the electronic basis, products of isolated target states and projectile states which are distorted by exclusion forces emanating from the target. The associated set of close-coupled equations are written in the same form as that specific to the quasi-one-electron model of a projectile, such as Li or Be/sup +/, colliding with a target such as an He or Mg atom, modeled as a structureless scattering center. We then extract from these equations an effective, one-electron Hamiltonian comprised of parts identifiable as Coulombic and exchange interactions, suitably modified by projection operators which take account of the exclusion forces stemming from the indistinguishability of the projectile and target electrons, i.e., from the Pauli principle. Although themore » formulas for these interactions may be too complicated for direct use in scattering calculations, they are essentially exact and so can serve as useful guides in the search for approximate, model interactions.« less

We extend the linear response-like derivation of the generalized Navier-Stokes equation to non-Newtonian flows with rate of strain-dependent transport coefficients. We derive a time correlation function expression for the viscosity tensor and point out possible ambiguities in the operational definitions of viscosity coefficients. Our analysis is specific to a suspension of polar, rod-like ferromagnetic particles. A commentary is included about the approximations that lead from the time correlation function and the molecular definition of the viscosity tensor to the standard, Brownian dynamics model used in the theoretical rheology of suspensions. Some theoretical difficulties and logical inconsistencies are pointed out. Preliminary results for the transport coefficients of dilute suspension of magnetic rod-like particles are presented.

For pt.I see ibid., vol.8, p.460 (1975). Cross sections for the transfer ionization process, He++Mg to He+Mg2++e, have been calculated using several different approximations for the wavefunction of the initial electronic state. The first was a frozen-orbital calculation using molecular orbitals associated with the neutral diatomic species HeMg. The second was a quasi-stabilization calculation involving configuration interaction among several frozen-orbital-type states. The final calculation used the hole-state model of Schwartz (1970) and of Bagus and Schaefer (1971). A critical analysis is made of these three methods and estimates provided for their conditions of applicability. Specifically, the results lead us to question whether the hole-state model is appropriate at all to this system.

Our recently proposed several-electron theory of collisional excitation in quasi-one-electron systems is applied to the resonance excitation process Be/sup +/(2s)+He..-->..Be/sup +/(2p)+He The general theory is reduced to that of a quasi-one-electron model by neglecting collisional excitation of the helium target atom. The velocity-dependent, nonlocal interactions of this theory give rise to strong short-ranged forces which account rather well for values of the excitation cross section, polarization of impact radiation, and coherence parameters observed experimentally for impact energies in the range 0.4

It is demonstrated that one can successfully generalize the moment method of dilute gas kinetic theory to a dense fluid composed of polyatomic molecules. The basic premise of the theory is Bogoliubov's assumption of the existence of a functional relationship between the pair and singlet distribution functions. The theory is rendered explicit by selecting for this functional the generalization to polyatomic fluids of a relationship first obtained by Severne and subsequently modified and generalized by Prigogine, Nicolis, and Misguich and by Davis. This leads to a closed set of approximate moment equations from which are extracted explicit, computationally tractable formulas for the characteristic relaxation times and frequency dependent transport coefficients associated with the transport of momentum, energy, and molecular internal angular momentum or spin.

A two-electron model of direct excitation in Li-Na collisions is developed, based upon one-electron eigenstates of the separated atoms. A valence-bond-type two-electron atomic basis, antisymmetrised and modified by the translational phase factors, is used in the expansion of the electronic scattering state. The time-dependent Schrodinger equation is solved by the impact parameter method assuming a rectilinear trajectory and constant relative velocity of the alkali cores. Solutions of the resulting close-coupled equations have been obtained in the high-energy limit which defines a model with complete neglect of exchange contributions. Various sets of basis functions, Li(2s/2p)Na(3s/3p), involving single and/or double (simultaneous) excitation of the alkali atoms have been used. The aim was to obtain first estimates of Li(2p) and Na(3p) excitation cross sections and polarisations as functions of collision energy and to probe the details of the direct-excitation mechanism. The single-excitation predictions match recent experimental cross sections at 50-60 keV centre-of-mass energy. The double-excitation contributions are significant but the present calculations tend to overemphasize their importance. Excitation out of the collision plane, an allowed two-electron excitation process, is important at centre-of-mass energies above 100 keV.

A kinetic equation is derved, at the mean field level, for a single‐component system composed of liquid and vapor phases separated by a planar interface. Three kinetic sum rules are derived and used to verify previously obtained configurational sum rules. The density gradient, which plays a key role in the interfacial dynamics, is discussed and used to construct a surface projection operator. The kinetic equation is then analyzed using this projection operator in conjunction with expansions in terms of momentum and spatial moments. The surface‐related terms of the kinetic equation lead to propagating surface modes which exhibit a dispersion identical to that of capillary waves.

For pt.I see ibid., vol.11, no.13, p.2377 (1978). Regional wavefunctions of the type proposed by Rudge have been used to construct estimates of the integral and differential cross sections for the parity-unfavoured 1Sg to 3Pg and 1Sg to 1,3Du transitions of helium. Both single- and multi-configurational ground-state wavefunctions are tested. Results obtained by using the Born-Oppenheimer approximation are found to be quite insensitive to the choice of ground-state wavefunction. Cross sections computed in this way are similar to previous Born-Oppenheimer results based on conventional continuous wavefunctions for the target states. However, calculations based on the Rudge variation principle are found to depend very sensitively on the choice of target wavefunction and to exhibit several other unexpected characteristics.

Integral and differential cross sections for the parity-unfavoured excitations of the 3Pg and 1,3Du states of helium have been calculated using the Born-Oppenheimer approximation for the scattering amplitude and a variety of estimates for the ground-state wavefunction of the helium atom. These cross sections are found to be much more sensitive to the choice of target wavefunction than single-excitation cross sections which previously have been investigated. Thus, the integral cross section for excitation of the 3Pg state is only half as large when calculated with a 35-term CI helium ground-state wavefunction as with a Hartree-Fock function.

For pt.I see ibid., vol.9, p/L403 (1976). In part I the authors incorrectly stated the asymptotic conditions on the wavefunctions of the classical trajectory theory and also erred in describing the consequences of time-reversal invariance of the choice of integration constants (and limits of integration) associated with the phases of these wavefunctions. These errors are corrected and it is shown that there are many phase factors which yield integral and differential cross sections that are identical with those calculated on the basis of the Bates and McCarroll (1958, 1962) 'travelling-wave phase factors'.

The continued fraction formalism of Mori is derived systematically and its application to nonequilibrium statistical mechanics is carefully studied. We choose applications from lattice dynamics and give general solutions for semi‐infinite and infinite harmonic oscillator chains in terms of finite continued fractions. Special cases of these general solutions are considered in detail.

The generalized moment method presented in the first paper of this series is applied here to three model systems. These models—rough spheres, rigid ellipsoids, and square‐well ellipsoids—have been selected not only because they are more tractable than others from a computational point of view but also because they mimic with varying degrees of success several characteristics of real polyatomic fluids. Formulas are derived for the transport coefficients and relaxation times descriptive of the establishment and maintenance of a steady state of transport of energy and of linear and angular momentum. To the approximation of Enskog these phenomenological coefficients can be expressed in terms of integrals of the same type as occur in the dilute gas theory. The integrands of these (which are trivial to evaluate in the rough sphere case) are complicated by the fact that they include the polyatomic generalization of the equilibrium pair distribution function. The formulas given here also include contributions (based upon the theories of Severne, of Prigogine, Nicolis and Misguich, and of Davis) from dynamical correlations that are not included in the Enskog approximations. These are more difficult to evaluate numerically.

Sum rules for inhomogeneous fluids are derived and then related to the surface tension of a liquid—vapor interface. It is found that this surface tension can be expressed in terms of the static correlation for the component of force normal to the interface. This formula is shown to be equivalent to two previous expressions for surface tension that generally are regarded as being exact. It is demonstrated that in the high-frequency or short-time limit this definition for surface tension leads directly to the standard hydrodynamic equation of motion for the normal component of the interfacial velocity.

The projection operator method of Zwanzig ant1 Fcshbach is uscd to construct the time dependent dcnsity opcrator associated with 21 binary scattering cvcnt. The formula tlcvclopcd to tlescribc this time tlcpcndcncc involves tirnc-ordcrcd cosine and sine projcctctl evolution (memory) supcroperntors. Both Schr(idinger and interaction picturc rcsults arc presented. The former is uscd to demonstrate the cquivalcncc of thc time rlependcnt solution of the von Ncumann equation and the more familiar, frequency tlcpcndcnt Laplacc transform solution. For two particular classcs of projection supcropcratoss projected dcnsity operators arc shown to bc equivalent to pro.jcctcd wavc fi~nctions. Except for thcsc two special cases, no projcctcd wavc function analogs of projcctcd dcnsity operators exist. Along with the dccouplcd-motions approximation. projected intcraction picturc tlcnsity operators arc applied to inelastic scattering cvcnts. Simple illustrations arc providctl of how this forn~alism is rclatcd to previously cstablishctl rcsults for two-state proccsscs, namely, the thcory of resonant transfer cvcnts, the first order Magnus approximation. ant1 thc Lantlau-Zcncr thcory. On utilisc la mdthodc d'opC~xtcurs dc prqjcction dc Zwanzig ct Fcshbach pour construirc I'opd~xtcur densitd ddpcndant du tcmps associd i~ un dvtncmcnt tlc tliffusion binnirc. La fi)l.rnulc dcvcloppdc pour ddcrirc ccttc ddpcndancc du temps fait intcrvcnir Ics pro.jcctions sinusoitlalcs ct cosinusoitlalcs tlc supcroperatcurs d'evolution (mcmoirc) ordonnds dans Ic tcmps. Lcs rdsult;~ts sont prdscntds sclon Ics dcux rcprdscntations tle Schriidingcr ct d'intcraction. La prcmitrc forrnc cst utilisdc pour dd~nontrcr I'dc~uivalcncc dc la solution tldpcnd~unt du tcmps clc 1'dqu;~tion dc von Ncumann avcc la solution dependant dc la frdqucncc, par t~xnsforrndcs dc Laplncc, qi~i cst plus f~unilitrc. On montrc clue pour tlcux classcs particulitrcs dc supcropdratcurs dc projcction Ics opdratcurs tlc tlcnsitds pro.jctds sont dquivalcnts anx hnctions d'ondc pro.jctCcs. Sauf dans ces dcux cas spdciaux. il n'cxistc aucunc klnction tl'ontlc projctdc analogue d'opdlxtcur Icur tlcnsitd projctd. Lcs opdratcurs dcnsitCs cn representation d'intcraction sont nppliquds. en memc temps quc 1'approxim;ltion tlc mouvcmcnts tldcouplds, h dcs dvtncmcnts tlc tliffusion indlasticluc. Dcs cxcrnplcs si~nplcs sont tlonnds pour illustrcr la liaison cntrc cc formalisme et Ics rdsultats dtablis pour dcs proccssus :I tlcux dtats. II savoir la thdoric dcs dv?ncmcnts tlc transfcrt resonant, I'approximation Magnus tlu premier ordrc ct In thdoric Lantlau-Zcncr. [Traduit par le journal]

The generalized Langevin equation formalism of Mori and Zwanzig and Akcasu and Dudenstadt is used to derive coupled kinetic equations for the distribution functions of a tagged test particle and of the untagged members of the surrounding background fluid. The interaction between any two of these particles is taken to be the sum of a long‐ranged continuous potential and a short‐ranged, hard‐core contribution. The distribution function of the background fluid is chosen to depend on the location of a solvent particle measured relative to the instantaneous position of the test particle. Because of this choice, the frequency matrix associated with the pair of distribution functions incorporates much of the coupling between the tagged and untagged particles which otherwise would be imbedded within their separate memory functions. Consequently, the mean‐field approximation obtained by neglecting the memory function of this two‐fluid theory is expected to be quite accurate. Explicit forms are derived for the coup...

Mori’s form of the generalized Langevin equation is applied to a system of particles whose pair interactions are sums of short‐ranged, rigid‐core repulsions and longer‐ranged continuous interactions. It is assumed that the memory function decays exponentially from its initial value at a rate characterized by a single, wave number dependent relaxation time. The test–particle kinetic equation resulting from this one key assumption is strongly reminiscent of that associated with the well‐known Rice–Allnatt theory. However, in addition to Fokker–Planck and Enskig operators similar to those of the Rice–Allnatt theory, it also includes coupling between rigid‐core scattering events and the diffusive, Brownian motions of the particles as well as contributions associated with three‐particle rigid‐core collisions. A systematic method of solution is developed and applied to a simplified form of this kinetic equation. Solutions generated by this procedure involve matrix elements of the linearized Enskog operator comp...

The two‐fluid mean‐field theory of the preceding paper is applied to the special case of a macroscopic test particle immersed in a quasicontinuous background fluid. Here, as in the preceding investigation, the particle interactions are taken to be pair‐additive sums of long‐ranged continuous and short‐ranged, rigid‐core contributions. The kinetic equation for the conditional singlet distribution function of untagged solvent particles is recast in terms of a related smoothly varying function which satisfies a boundary condition at the interface where this fluid comes into contact with the test particle. This formulation of the theory is then used to examine the case when the test particle is much larger and more massive than a solvent particle and when the free path length of the background medium is significantly smaller than the test particle diameter. In this limit the moments of the background fluid kinetic equation reduce to the linearized Navier–Stokes equations for pure solvent and the velocity moment of the boundary condition becomes identical with the macroscopic slip‐flow condition of hydrodynamics. From these, one can obtain the usual Stokes’ formula for the frictional force on the test particle. The theory also can be described by a kinetic equation for the background fluid which incorporates singular source terms arising from the rigid‐core interactions between the solvent and the test particle. In the case of a large, very massive test particle the moments of this equation are found to be generalizations to high density of the ‘‘source hydrodynamic equations’’ of Cukier, Kapral, Lebenhaft, and Mehaffey.

A theoretical investigation has been conducted to the exothermic transfer ionization reaction, He++Mg to He+Mg2++e-. The mechanisms assumed is autoionization of the diatomic ion HeMg+. To compute the potential energy curves and autoionization width descriptive of this process, Hartree-Fock single-configuration electronic wave-functions have been constructed for the initial discrete state, He+-Mg, and for the final continuum state, He-Mg+2-e-. As the internuclear separation varies from 3 to 5 bohr, the lifetime for autoionization grows from 8.75*10-14 to 3.2*10-13 s. The cross section for transfer ionization has been computed according to Miller's classical trajectory theory of Penning ionization. This cross section reaches a maximum value of 12.9 a02 for a collision energy equal to 1 hartree. Because of the inadequacy of our approximate potential energy function for the initial state, the theoretical cross section the authors have computed here is believed to be unreliable for energies less than this value.

The quantal theory of field-free associative ionization and Penning ionization, due to O'Malley, Bieniek, and others, is generalized to include the closely analogous laser-induced chemi-ionization processes A+B+h..omega -->..AB/sup +/+e/sup -/, A+B/sup +/+e/sup -/. The principal limitations of the theory are its reliance upon the two-state approximation and its neglect of Born-Oppenheimer terms. Formulas are derived for several scattering cross sections descriptive of the energy spectra and angular distributions of the reaction products. These cross sections vary with the direction of polarization of the laser radiation. The polarization dependences of the heavy-particle (A+B/sup +/) energy-angle double-differential cross section and of the associated heavy-particle (or photoelectron) energy distribution assume simple cosine-squared forms provided that there is a large number of contributing partial waves. Numerical calculations of these cross sections should be no more difficult to perform than for comparable field-free processes.

Excitation probabilities P(b,E) and the coherence parameters, lambda (b,E) and chi (b,E) have been calculated for 2S to 2P excitation of Be+(2s) and Mg+(3s) ions in collisions with ground-state He and Ne atom targets in the energy range ECM=0.5-100 keV and the impact parameter range b=0.1-5.0 au. Each ion-atom pair has been treated as a quasi-one-electron three-particle system consisting of the valence electron of the ion, the closed-shell ion core, and the closed-shell rare-gas atom. Close-coupled equations, based upon a rectilinear trajectory, have been solved for the amplitudes associated with a three-state (ns, np0, np1) one-centre expansion of the electronic wavefunction. A Hellmann-type pseudopotential has been used for the electron-ion core interaction; for the electron-rare-gas interaction an electrostatic model was chosen. The theory predicts that high energies and/or large impact parameters lambda will tend to zero and chi to tau /2. For finite energies (impact parameters) the function lambda (b) ( lambda (E)) exhibits pronounced oscillations which correlate with rapid changes (phase jump mod Delta chi mod approximately= pi ) of the phase angle chi (b) ( chi (E)).

The authors present an investigation of collisional excitations in systems characterised by the presence of two valence electrons localised outside closed shells in the energy range 1-300 keV. Two types of systems have been studied: (i) systems such as Mg-rare gases with both valence electrons localised on the projectile, and (ii) systems such as Li, K-Na with one valence electron localised on the projectile and one on the target. Experimentally, the total emission cross sections and polarisations have been measured for the Mg I 31S-31P, 33P-43S, s3P-33D, 33P-3p2 3P, Li I 22S-22P, Na I 32S-32P and K I 42S-42P transitions. The content metastable Mg(3s3p3P) atoms in the applied neutral Mg beams has been determined by means of an optical method based on the Wigner spin conservation rule in low-energy atomic collisions. The Mg(3s2)-He collision has been examined theoretically using a Hartree-Fock frozen-core potential to describe the interaction between the Mg valence electrons and the He atom.

A new derivation is presented of the properties of the travelling wave phase factors which are important in the semi-classical theory of high energy charge exchange processes. In the case of a rectilinear, constant velocity trajectory the phases obtained are those required by the Galilean invariance of the non-relativistic Schrodinger operator. For an arbitrary trajectory this invariance is not maintained and so there appear in the Schrodinger operator compensating terms which cannot be transformed away by a change of phase.

A response theory is constructed for systems with deterministic, nonrandom dynamics which are not governed by Hamiltonian mechanics. This theory is used to compute the response of a polymeric solution (modeled as elastic dumbbells suspended in a Newtonian solvent) to a flow field. This externally imposed flow displaces the system from its initial state of equilibrium and causes each function of state to respond in a characteristic way. The response function of greatest interest is the fluid stress tensor. We present a correlation function expression for this tensor and derive Kubo‐like formulas for a number of material functions, namely, the shear and extensional viscosities and the primary and secondary normal stress coefficients associated with laminar flows. Finally, an expression is obtained for the extra stress due to turbulent flow.

The memory function formulation of the time‐dependent Schrodinger equation is used to analyze the internal state problem associated with the decoupled motions or impact parameter approximation to binary collisional events. The exact, time‐orderd memory functions are then replaced with their time‐disordered counterparts. This generates a unitary approximation to the collision problem as well as simple, easily interpretable formulas for the transition probabilities. Two specific examples are considered in detail, namely, (i) the three‐state resonant transfer model and (ii) a ’’paired‐states’’ model for the dynamics of a many‐state problem.

Collision-induced coherent S to D excitation is investigated. A polarisation analysis of the subsequent D to P transition by the photon-particle coincidence method is not sufficient to completely determine the excitation amplitudes for the D state. Two distinct solutions are obtained. The electron distributions associated with these two are mirror images of one another in the plane determined by the normal to the scattering plane and the direction of maximum intensity of linearly polarised light, as measured from the direction of the normal. The Li(2s to 3d) excitation induced by collisions with helium is studied both theoretically and experimentally in the keV energy range. The measurements confirm the existence of a theoretically predicted low-energy structure in the circular polarisation. The results are interpreted by generalising concepts previously developed for S to P excitation in quasi-one-electron systems.