Digital Commons Network^™

Fourier Transform Of The Product Of N One-Center Hydrogenic Orbitals, Jack C. Straton

Physics Faculty Publications and Presentations

Integrating the radial part of the Fourier transform of the product of N hydrogenic orbitals results in an associated Legendre function that can be reduced to a finite series of elementary functions. This transform is found to depend on a polynomial in the wave vector k divided by a binomial in k² raised to a power that is the sum of principle quantum numbers. This form facilitates the analytical reduction of integrals arising from orthogonalization corrections in atomic processes. Transforms for the product of orbital pairs (1s,1s) through (1s,3d) are given …

Shell Correction For The Stopping Power Of K Electrons, P.T. Leung, M. L. Rustgi, S. A. Long

Physics Faculty Publications and Presentations

In view of the inapplicability of the asymptotic expressions for the stopping number available in the literature at high energies, an alternative approach is taken to compute the shell correction to the stopping number of K electrons. Anholt's formula (1979) for the K-shell ionization has been used to calculate the excitation function for longitudinal interaction and numerical integration over energy has been carried out to evaluate the shell correction. Comparison with other theoretical calculations is made. It is proposed that, with the inclusion of relativistic effects, an asymptotic expansion of the stopping number with a leading-term logarithmic in the energy …

Relativistic Corrections To The Bethe Sum Rule, P.T. Leung, M. L. Rustgi, S. A. Long

Physics Faculty Publications and Presentations

Relativistic corrections to order alpha² to the Bethe sum rule have been obtained for a one-electron system employing the Foldy-Wouthuysen transformation. The results have wide applications in high-Z systems at large momentum transfers.

Fock-Tani Transformation And A First-Order Theory Of Charge Transfer, P. C. Ojha, M. D. Girardeau, J. D. Gilbert, Jack C. Straton

Physics Faculty Publications and Presentations

A unitary (Fock-Tani) transformation of the second-quantized Hamiltonian breaks the interaction into its component parts, e.g., elastic scattering, inelastic scattering, rearrangement interaction, etc. The interaction for a particular process is ‘‘weaker’’ than the overall interaction; this is reflected in certain orthogonality corrections which appear in a perturbation expansion of the T-matrix element. As a result, the internuclear potential makes a negligible contribution of order m_e/m_p to the first-order amplitude for charge transfer. We find very good agreement with experimental and the best available theoretical results for the total cross section for the reaction p+H(1s)→H(1s)+p for …

An Approximate Variational Method For Improved Thermodynamics Of Molecular Fluids, M. S. Shaw, J. D. Johnson, John D. Ramshaw

Physics Faculty Publications and Presentations

For a certain class of thermodynamic perturbation theories, a generalization of the Gibbs-Bogoliubov inequality holds through second order of perturbation theory and for a subset of terms the inequality is true to infinite order. Using this approximate variational principle, a perturbation theory is chosen for which the Helmholtz free energy of the reference system is minimized under the constraint that the first order term is identically zero. We apply these ideas to the determination of effective spherical potentials that accurately reproduce the thermodynamics of nonspherical molecular potentials. For a diatomic-Lennard-Jones (DLJ) potential with l ∕σ = 0.793, the resulting spherical …

General Formula For The Inelastic Scattering Of Relativistic Spin-½ Particles By Heavy Atoms, P.T. Leung, M. L. Rustgi, S. A. Long

Physics Faculty Publications and Presentations

A general differential-cross-section formula for the inelastic scattering of relativistic spin-½ particles by heavy atoms is derived in the Born approximation and the structure and polarization of the incident particle is incorporated. The results reduce to those of Fano (1963) and of Turner et al. (1973) in well-known limiting cases.

Energy Losses Of Solar Neutrinos And The Oscillation Hypothesis, P.T. Leung, S. Boedo, M. L. Rustgi

Physics Faculty Publications and Presentations

A formula for the stopping power of neutrinos interacting via the standard weak-interaction model, but incorporating the possibility of neutrino oscillations among the three flavors, is derived. The results are applied to study the solar-neutrino anomaly and it is found that the anomaly cannot be accounted for by many orders of magnitude from consideration of the energy losses of the neutrinos interacting with the solar matter, even if the oscillation hypothesis is found to be valid.

Coexistence And Spinodal Curves In Directionally Bonded Liquids Using The Four-Cluster Approximation, Erik Bodegom, Paul H. Meijer

Physics Faculty Publications and Presentations

We derive the phase diagrams and spinodals of binary liquid systems with anisotropic interactions, such as hydrogen-bonded molecules. The work is based on the four-particle cluster variation method, using a different potential for different contact points. It is shown that the introduction of a cluster larger than previously used by Barker and Fock, leads to a considerable improvement in the shape of the phase diagram and avoids some of the difficulties encountered in their calculation. Phase diagrams are displayed for various choices of the parameters: the number of contact points, the interaction potential, and the order of the approximation.

Probability Densities And The Random Variable Transformation Theorem, John D. Ramshaw

Physics Faculty Publications and Presentations

D. T. Gillespie has recently derived and discussed a random variable transformation (RVT) theorem relating the joint probability densities of functionally dependent sets of random variables.....

It is hoped that the present discussion will help to disseminate this basic relation among a wider circle of nonspecialists.

Ionization Of Heavy Atoms By Polarized Relativistic Protons, P.T. Leung, M. L. Rustgi

Physics Faculty Publications and Presentations

The effect due to the polarization of the incident high-energy (~GeV) proton beam on the ionization cross section of heavy atoms is investigated. In particular, with the use of Darwin wave functions for the atomic electron, the effect for hydrogenlike atoms is worked out. A numerical illustration shows that the polarization effect is completely insignificant. The authors also conclude that the K-shell ionization process is completely insensitive to the polarization of the incident beam.

Stopping Power Of K Electrons At Extreme Relativistic Energies, P.T. Leung, M. L. Rustgi

Physics Faculty Publications and Presentations

The recent work of Anholt (1979) on K-vacancy production by relativistic projectiles has been applied to calculate the stopping power of the K electrons. The results show that for protons of energy ~10³ GeV and heavy target elements, the relativistic contributions to the stopping power amount to several times the results due to the longitudinal terms obtained from Walske's (1956) work.

Structural Effects On K-Vacancy Production By Protons Of Extreme Relativistic Energies, P.T. Leung, M. L. Rustgi

Physics Faculty Publications and Presentations

The effect on the K-vacancy-production cross section due to the finite distribution of charge and magnetic moment of the incident proton is examined. Numerical results are given for two target elements, namely, Ni and U. The results show that even at energy as high as 10³ GeV the incident proton can still be regarded as a point charge. The effect on the stopping power of matter is discussed.

Massless Neutrinos And The Stellar Stopping Power Via The V-A And Weinberg Interactions, P.T. Leung, M. L. Rustgi

Physics Faculty Publications and Presentations

A formula for the stopping power of matter for neutrinos and antineutrinos, interacting with electrons via the weak interaction in the context of both the V-A theory and the Weinberg theory is derived in the high-Q approximation and applied to study the solar-neutrino anomaly. The result shows that neutrinos extending in energy from 1 to 10 MeV produced inside the Sun escape it essentially without any loss of energy. In agreement with previous investigators, Rustgi, Leung, Turner, and Brandt (1981). It is found that the neutrinos should retain the energy given them at their creation essentially forever and permeate the …

Comparison Of Bethe's Formula For Magnetic-Moment Transitions In Atoms With The Generalized Rosenbluth's Equation, P.T. Leung, M. L. Rustgi, J. E. Turner

Physics Faculty Publications and Presentations

An error in an old but often quoted paper by Bethe (1935) on the cross section for an atomic transition caused by the magnetic moment of the projectile is corrected by comparing it with the generalized Rosenbluth's formula (1950).

Dielectric Constant In Fluids Of Classical Deformable Molecules, John D. Ramshaw

Physics Faculty Publications and Presentations

Classical statistical mechanical description of dielectric fluids is further discussed. Dipole moment correlations of polarizable molecules are discussed.

Existence Of The Dielectric Constant In Fluids Of Classical Deformable Molecules, John D. Ramshaw

Physics Faculty Publications and Presentations

The existence of the dielectric constant epsilon is investigated for fluids composed of classical deformable (polarizable) molecules. The development is based upon generalized functional-derivative relations which involve joint distributions in molecular positions r/sub k/ and dipole moments ..mu../sub k/. Sufficient conditions for the existence of epsilon are expressed in terms of the generalized direct correlation function c(12) = c(r/sub 1/, ..mu../sub 1/; r/sub 2/, ..mu../sub 2/). It is found that epsilon exists if -kTc(12) depends only on relative positions and dipole moment directions (in addition to Vertical Bar..mu../sub 1/Vertical Bar and Vertical Bar..mu../sub 2/Vertical Bar), and becomes asymptotic to the …

Massive Neutrinos And The Stellar Stopping Power Via The Neutrino Magnetic Moment, P.T. Leung, M. L. Rustgi, J. E. Turner, Werner Brandt

Physics Faculty Publications and Presentations

An exact expression is derived within the high-Q approximation for the spectrum of energy losses of neutrinos interacting via their magnetic moment with electrons. A formula for the stopping power of matter for neutrinos is derived and applied to the study of the solar neutrino anomaly. The total inelastic cross section is given. An upper limit is deduced for the neutrino magnetic moment from the antineutrino spectrum as recently parameterized by Lee (1979). It confirms the bound reported by Cowan and Reines. The distance for stellar neutrinos to thermalize by the interactions studied here, through encounters with stars is found …

Existence Of The Dielectric Constant In Dipolar Fluid Mixtures, John D. Ramshaw, Norman D. Hamer

Physics Faculty Publications and Presentations

The existence of the dielectric constant epsilon is investigated for fluid mixtures of rigid polar molecules. The investigation is performed using the functional-derivative formalism for mixtures, and is closely analogous to that previously carried out for pure dipolar fluids (J. Chem. Phys. 68, 5199 (1978)). Sufficient conditions for the existence of epsilon are obtained in terms of the direct correlation function matrix c/sub alphabeta/(12). It is found that epsilon exists if c/sub alphabeta/(12) depends only on relative positions and orientations, and becomes asymptotic to -theta/sub alphabeta/(12)/kT at long range, where theta/sub alphabeta/(12) is the dipole--dipole potential between a molecule of …

Time-Dependent Direct Correlation Function, John D. Ramshaw

Physics Faculty Publications and Presentations

A time-dependent direct correlation function (TDDCF) is defined and discussed. The definition (which differs from that proposed by Percus and Yevick) is motivated by linear response theory. The TDDCF consequently has a functional-derivative or stimulus-response interpretation closely analogous to that of the static direct correlation function. Mean-field considerations suggest that the TDDCF separates into a term which is simply related to the pair potential and a spatially shorter-ranged remainder. This behavior is confirmed in the Vlasov-Zwanzig approximation.

Brownian Motion In A Flowing Fluid Revisited, John D. Ramshaw

Physics Faculty Publications and Presentations

It is shown how the phenomenon of osmosis may be treated using the phenomenological theory of Brownian motion in a flowing fluid. The theory is also generalized to include viscous stresses in the particle and mixture momentum equations.

Stopping Power Of Matter For Alpha Particles At Extreme Relativistic Energies, P.T. Leung, M. L. Rustgi, J. E. Turner

Physics Faculty Publications and Presentations

The stopping power of matter for alpha particles at extreme relativistic energies has been calculated by incorporating the charge form factor. A table is presented for aluminum, copper, and lead. It is found that at the highest energies considered, inclusion of form factor reduces the mass stopping power by about 6-8% in comparison with that predicted by Bethe's relativistic formula.

Dielectric Saturation In Dipolar Fluids. I. The Single-Molecule Distribution Function, John D. Ramshaw

Physics Faculty Publications and Presentations

The functional-derivative formalism is used to investigate the form of the equilibrium single-molecule distribution function n(1) in a finite fluid system of rigid polar molecules subjected to a strong external electric field that varies slowly with position. The investigation is based on the assumption that the long-range asymptotic behavior of the direct correlation function is independent of the external field, and is hence the same as in the unperturbed fluid. This assumption implies that n(1) has the form of a single-molecule Boltzmann factor in which the interaction energy is that of a deformable quasidipole with the local Lorentz electric field …

Debye–Hückel Theory For Particles Of Arbitrary Electrical Structure, John D. Ramshaw

Physics Faculty Publications and Presentations

Classical linearized Debye–Hückel theory is formulated for a finite fluid system, of arbitrary shape, composed of rigid particles with arbitrary internal electrical structure. The multipole description is eschewed in favor of the more basic description of a particle in terms of its charge density function. This function is left arbitrary, so the particles may be charged or neutral, polar or nonpolar, etc. The theory implies that the direct correlation function c(12)=−v(12)/k T, where v(12) is the Coulomb interaction energy between the charge densities of particles 1 and 2. In the case of uncharged polar …

Partial Chemical Equilibrium In Fluid Dynamics, John D. Ramshaw

Physics Faculty Publications and Presentations

An analysis is given for the flow of a multicomponent fluid in which an arbitrary number of chemical reactions may occur, some of which are in equilibrium while the others proceed kinetically. The primitive equations describing this situation are inconvenient to use because the progress rates ω [subscript s] for the equilibrium reactions are determined implicitly by the associated equilibrium constraint conditions. Two alternative equivalent equation systems that are more pleasant to deal with are derived. In the first system, the ω [subscript s] are eliminated by replacing the transport equations for the chemical species involved in the equilibrium reactions …

Brownian Motion In A Flowing Fluid, John D. Ramshaw

Physics Faculty Publications and Presentations

A phenomenological theory is developed for Brownian motion in a flowing incompressible fluid. The Brownian particles are regarded as an ideal gas subject to a position- and time-dependent force field that represents interactions with the host fluid. This approach immediately leads to deterministic partial differential equations of motion for the Brownian particles. These equations are then examined in the limit of large friction, in which they imply an expression for the diffusional mass flux of Brownian particles. This expression is a sum of terms representing concentration, forced, thermal, and pressure diffusion. Comparisons are made with earlier work, and with the …

Comments On The Theory Of Dipolar Fluids, John D. Ramshaw

Physics Faculty Publications and Presentations

The theory of dipolar fluids posbulated by Chan, Mitchell and Ninham is commented upon.

Existence Of The Dielectric Constant In Fluids Of Nonlinear Rigid Polar Molecules, John D. Ramshaw

Physics Faculty Publications and Presentations

The existence of the dielectric constant epsilon is investigated for fluids composed of nonlinear rigid polar molecules. The investigation is performed using the functional-derivative approach previously employed to establish sufficient conditions for the existence of epsilon in fluids of linear (axially symmetric) molecules. It is shown that these same conditions are sufficient for nonlinear molecules of arbitrary symmetry. An expression for epsilon in terms of the direct correlation function emerges automatically from the development. This expression, which involves the inversion of a 3 x 3 matrix, is a slight generalization of one obtained earlier by Hoye and Stell using an …

Existence Of The Dielectric Constant In Rigid-Dipole Fluids: The Functional-Derivative Approach, John D. Ramshaw

Physics Faculty Publications and Presentations

In a previous article [J. Chem. Phys. 57, 2684 (1972)] sufficient conditions were established for the existence of the dielectric constant ɛ in rigid-dipole fluids. One of these conditions was an unrealistic restriction on the angular dependence of the direct correlation function c (12) at short range. Here it is shown that this restriction can be removed without altering the previous conclusions. Consequently, ɛ rigorously exists if c (12) depends only upon relative positions and orientations of molecules 1 and 2, and becomes asymptotic to -φd(12)/kT at long range, where φd(12) is the dipole-dipole potential. The development is based upon …

Debye–Hückel Theory For Rigid‐Dipole Fluids, John D. Ramshaw

Physics Faculty Publications and Presentations

The dipolar analog of classical linearized Debye–Hückel theory is formulated for a finite fluid system of arbitrary shape composed of rigid polar molecules. In contrast to the ionic case, the dipolar Debye–Hückel (DDH) theory is nonunique due to an inherent arbitrariness in the choice of a local field E*. This nonuniqueness is expressed in terms of a parameter ϑ related to the ellipticity of the spheroidal cavity used to define E*. The theory then leads to an expression for the direct correlation function c (ϑ) as a function of ϑ. Only the short‐range part of c (ϑ) depends upon ϑ; …

On The Reduction Of Many-Body Dielectric Theories To The Onsager Equation, John D. Ramshaw

Physics Faculty Publications and Presentations

An approximate theory for the dielectric constant ℇ of a dense polar fluid was derived by Ramshaw, Schaefer, Waugh, and Deutch (RSWD). In the present article, the RSWD theory is generalized and made rigorous by another method of derivation. The result is a rigorous expression for ℇ which differs from the RSWD expression by the presence of a fluctuation term. Both the rigorous expression and the RSWD expression are then specialized to the Onsager model. It is found that the rigorous expression for ℇ reduces to the Onsager equation, but that the RSWD expression does not because the fluctuation term …