Repulsive Region Research Articles

Density profile of a quantized vortex line in superfluidHe4

The relative-density amplitude profile of a quantized vortex line in He II is calculated from a generalized Gross-Pitaevskii (GP) equation. The ordinary GP equation is a Hartree nonlinear integrodifferential equation for the order parameter of an interacting system of bosons. The generalization is made by replacing the two-body potential in the average field by a local $T$ matrix, which is calculated from a realistic interatomic potential for helium. In configuration space the $T$ matrix is similar to the potential, except in the repulsive region where it is much softer. The only adjustable parameter in the generalized GP equation is the condensate fraction for which a theoretical value of 0.10 and an experimental value of 0.024 are used. In all cases the density profile is a monotonically increasing function with an effective core radius of from 3.7 to 4.7 \AA{}.

Resonance vibrational energy transfer in liquids in the repulsive potential region

It was found that the resonance vibrational energy transfer caused by the short−range repulsive part of the intermolecular (perturbing) potential may play an important role as mechanism to produce broadening of vibrational band contours of molecules in the liquid state. The value of the full bandwidth at the half peak height (Δν1/2) for the ν4 (C−H deformation) fundamental of neat chloroform (8.2 cm−1) was reduced to 6.2 cm−1 at infinite dilution with deuterochloroform. This difference in Δν1/2 can be reproduced by either the above repulsive mechanism or by resonance energy transfer caused by the long−range transition dipole−transition dipole interactions. Physical aspects of these mechanisms are discussed and compared with those of NMR relaxation. The significance of the repulsive resonance transfer process with respect to transition state theory is indicated.

Rigid cage effect on ICl photodissociation and B O+ fluorescence in rare gas matrices

The relaxation dynamics of matrix isolated ICl have been studied via time and wavelength resolved emission. Vibrationally relaxed B O+ fluorescence occurs following irradiation both above and below the gas phase dissociation barrier. The fluorescence quantum yield is near unity independent of excitation wavelength, and I35Cl and I37Cl progressions clearly separate in the emission spectrum. The excitation spectrum shows discrete thresholds for v′ = 0−3 and only smooth structure above the dissociation barrier, indicating that constrictive cage forces cause no significant vibrational quantization in the B O+ repulsive region. Irradiation below the v′ = 0 threshold also yields B O+ fluorescence by a weak sequential two photon process. Vibrational line shapes show that excitation below the barrier produces relaxed phonon emission, while irradiation above the barrier produces unrelaxed phonon structure. It is proposed that unrelaxed emission occurs from metastable orientational (librational) local minima in the v′ = 0 B O+ state. The photodissociation−recombination sequence appears to be a general method for populating phonon states not Franck−Condon accessible in bound−bound absorption.

Distorted atoms in molecules (DAM) model for ordinary covalent molecules. II. Force constants for H2, LiH, and BeH+ as well as medium range repulsion of H + H(3Σu) and He + He

The distorted atoms in molecules valence-bond model is used to determine potential energy curves near equilibrium for H2, LiH, and BeH+ as well as in the medium range repulsive region for H+H(3Σu) and He+He. It is shown that the force constants that we obtain by numerical differentiation are, in principle, equal to those that would be found by double perturbation theory. The usual (single) perturbation method, however, contains a first-order error. Our values for all force constants including the anharmonic ones lie within 10% of experiment except for k2 of BeH+. The somewhat larger (20%) error in the latter case is probably due to omission of ionic-covalent mixing.

Study of the electronic structure of molecules. XXII. Correlation energy corrections as a functional of the Hartree-Fock type density and its application to the homonuclear diatomic molecules of the second row atoms

A semiempirical functional of the Hartree-Fock type density, previously obtained for the Hartree-Fock atomic functions and applied with good success to the second row hydrides (in paper XXI of this series), is now applied to the molecules H2(X 1Σg+), Li2(X 1Σg+), Be2(1Σg+), B2(X 3Σg−), C2 (x1 Σg+), N2(X 1Σg+), O2(X 3Σg−), and F2(X 1Σg+). For each molecule the potential energy curve is computed from the repulsive region to large internuclear distances. In order to follow the dissociation to the correct limits, we have added to the Hartree-Fock configuration of each molecule those configurations which are needed for describing proper dissociation (HFPD functions). It is found that for certain molecules (Li2, B2, C2, and O2) there are configurations which contribute quite significantly in the bonding regions and must be added to the HFPD function to reach a proper reference state function as previously defined (see paper XXI). With our functional of the density applied to the HFPD functions, the calculated dissociation energies are, with the experimental values given in parentheses, 4.69 eV (4.75 eV), 0.65 eV (1.14 eV), 1.64 eV (3.0 eV), 3.48 eV (6.36 eV), 9.34 eV (9.91 eV), 3.75 eV (5.21 eV), and 1.37 eV (1.66 eV) for H2, Li2, B2, C2, N2, O2, and F2, respectively. More realistic dissociation energies are obtained for Li2 (1.03 eV), B2 (3.06 eV), C2 (5.21 eV), and O2 (4.64 eV) if we use the proper reference state functions. The Σg+1 state of Be2 arising from the ground state atoms is repulsive. The agreement with the experimental dissociation energy is satisfactory but not as good as what we have previously obtained for the second row hydrides. However, owing to the extreme simplicity of the method it appears in a preliminary way that the combination of a Hartree-Fock type function and the functional of its density (of the form that we have proposed) is capable of yielding sufficiently reliable results for very little computational and human effort. Since the parametrization of the density functional was made for the atomic systems and not changed in the molecular formation, our method does not suffer from the need of variation of parameters introduced ad hoc and differing from molecule to molecule.

Study of the electronic structure of molecules. XXI. Correlation energy corrections as a functional of the Hartree-Fock density and its application to the hydrides of the second row atoms

A semiempirical functional of the Hartree-Fock density is presented for estimates of the correlation energy correction. The functional is similar to the one proposed by Gombas and is (a) parametrized with reference to few atomic systems, and (b) is modified as to reproduce the atomic correlation correction not only for closed but also for open shell systems. The functional is then applied to the ground state function of the hydrides LiH(1Σ+), BeH(2Σ+), BH(1Σ+), CH(2Π), NH(3Σ−), OH(2Π), and HF(1Σ+). Several internuclear distances have been considered for each hydride, scanning the potential energy curve from the repulsive region to the dissociation products (∼ 10 a.u.). For these points a simple multiconfigurational function (consisting of no more than three configurations) was computed to obtain Hartree-Fock functions with proper dissociation behavior (H-F-P-D functions). The semiempirical functional was applied both to the traditional H-F functions and the H-F-P-D functions in order to study how to select a proper reference wavefunction. The computed binding and a number of spectroscopic constants from vibrational and rotational analysis are discussed. The new semiempirical functional of the electronic density appears to be a rather promising one for reliable estimates of the correlation energy correction.

Interaction of vortices in type II superconductors and the behavior nearH c1 at arbitrary temperature

The interaction between well-separated Abrikosov vortices is calculated using Eilenberger's version of the Gorkov theory. Let λ and ξ be the decay lengths determining the asymptotic behavior of the magnetic field and the order parameter for an isolated vortex. Then the onset of attraction in the limit of large vortex separation is either given by λ=ξ or by λ becoming complex, whichever occurs first. It is shown that in general a first order transition atHc1 takes place already in the region of asymptotic repulsion because of the appearance of a minimum in the interaction energy at finite vortex separation.

The direct determination of potential energy functions from second virial coefficients

Methods which enable the intermolecular pair potential energy curve to be determined from second virial coefficient data are critically examined. Those techniques which involve the inverse Laplace transform, though theoretically attractive, are found to be ineffective with experimental data. New approximate methods are proposed and found to be successful in characterizing both the repulsive and attractive regions of the potential energy function. The procedures were tested on simulated data calculated from a number of known potential energy functions and were shown to reproduce the original potentials with high accuracy. The techniques were then applied to experimental second virial coefficient data for argon to give the repulsive potential energy curve above 300 K and the width of the upper portion of the potential well as a function of its depth. When used in conjunction with similar techniques for the inversion of gas viscosity data, these methods enabled the potential energy curve for argon to be established over a wide range of intermolecular separation.

Electrostatics of Diffuse Anisotropic Interfaces. III. Point Charge and Dipole Image Potentials for Air-Water and Metal-Water Interfaces

Within the framework of macroscopic continuum theory, the point charge and point dipole image potentials in the presence of a diffuse dielectric interface are calculated. We consider the cases of both short and long range diffuseness for air-water interfaces. In the first category we propose a model profile for which the charge image potential is devoid of singularities and is an integral over elementary functions. The calculated image potential goes through a maximum and minimum as the charge is taken from the high to the low dielectric constant medium. The heights of the maximum and minimum may be estimated in terms of a physical cutoff, characteristic of the gradient of the dielectric constant, D, in the diffuse region, using the crude model of an abrupt interface. For a diffuse layer, the very large but finite electrostatic barrier for an ion is determined by the ratio of a distance characteristic of the bulk electrolyte and a distance characteristic of the maximum gradient of D. The image potential for a point dipole, normal or tangential to the diffuse interface, is exhibited in the repulsive region, for the long range diffuse interface. For metal-water interfaces, encountered in electrochemistry, the effects of diffuseness of the inner region, adjacent to the metal, are examined. One effect of diffuseness is to remove singularities in the image potential and to increase it in the relevant part of the inner region. The other effect is to decrease the effective attractive interaction between charges. For both the air-water and metal-water interfaces, the limitations of the commonly used constant dielectric profile are discussed.

Simple numerical solution of the partial-wave Lippmann-Schwinger equation

The partial-wave Lippmann-Schwinger integral equation is reduced to a system of simultaneous linear equations at negative energies and the negative energy solution is analytically continued to positive energies by means of Pade approximants. This method of solution is suitable for application to interactions which contain both attractive and repulsive regions as well as to purely attractive or purely repulsive potentials.

Three-Body Quantum Correction to the Free Energy of a Simple Liquid and the Role of Pair and Triplet Potentials in Isotopic Distillation

The lowest-order quantum correction to the partition function and free energy of a simple dense fluid is generalized to take account of many-body forces. The three-body quantum correction is calculated using the triple-dipole potential for the nonadditive interaction, assuming the superposition approximation for the triplet correlation function, and taking the pair correlation functions from x ray and neutron diffraction experiments. In the case of liquid Ar near the triple point, the three-body correction is small enough to be neglected in connection with isotopic distillation experiments. These experiments provide values for 〈 ∇2U 〉, where U is the total interaction energy of the molecules in the liquid, and the results can be used in conjunction with experimental radial distribution functions to test empirical pair potentials u(r) for the Ar—Ar interaction. Several forms of u(r) are investigated and the Barker—Pompe, Kingston, and (exp-6) potentials are found to give the best results for 〈 ∇2U〉. The values of 〈 ∇2U 〉 test the repulsive region of the potential and are much less sensitive to the form of u(r) than is the vapor pressure of the liquid near the triple point. The vapor-pressure and isotopic-distillation tests are compared.

Corresponding States: A Universal Reduced Potential Energy Function for Spherical Molecules

Second virial coefficients and dilute-gas transport coefficient data for the inert gases neon to xenon are satisfactorily interpreted on the basis of a universal reduced pair potential energy function. However, molecular beam data and high-temperature viscosity data, considered together, provide evidence that the repulsive region of this curve is much harder for neon than for the other rare gas molecules.

Perturbation theory for the radial distribution function

Perturbation theory is used to consider expansions for the radial distribution function, g 2(r), of a fluid with a soft core. We consider the Lennard-Jones (12, 6) potential and divide it into repulsive and attractive regions. In the repulsive region we expand the function exp (β u(r))g 2(r) about a hard sphere value. For the first-order contribution of the attractive region we consider a simple approximation to the exact analytical expression. The resulting g 2(r) is accurate at densities below about ρσ 3=0·5.

Evaluation of Transport Properties of Gases: The Bruch–McGee Potential for Helium

The transport properties and second virial coefficient of helium have been calculated for two hybrid intermolecular potentials recently proposed by Bruch and McGee, and the results are compared with experimental values. Tables of transport collision integrals for the helium isotopes which are accurate to several parts per thousand were prepared for the temperature range 0.8–2500°K in order to adequately span the existing data. The collision integrals were computed by an exact quantum phase-shift calculation at low temperatures followed by a JWKB approximation and classical calculation at intermediate and high temperatures, respectively. The bulk properties were then evaluated using the prepared tables. The agreement with experimental measurements for the various properties at low temperatures is rather poor when compared with the results calculated using the Lennard-Jones (12–6) potential. We conclude that this is due to excessive depth of the potential well. On the other hand, the calculations above approximately 150°K agree quite well with experimental data which leads us to the conclusion that the low-energy repulsive region of either hybrid potential is a good representation of the helium–helium interaction. Since the potentials are essentially identical in this region and independent of the long-range attractive terms, it is suggested that for practical applications the Morse-VDD potential be truncated to the much simpler Morse potential with the given parameters and its region of validity restricted to temperatures above 150°K.

Perturbation Theory and Equation of State for Fluids. II. A Successful Theory of Liquids

The perturbation theory previously shown to give good results for the equation of state of a square-well fluid at liquid densities and temperatures is applied to more realistic potentials with soft repulsion, in particular the 6:12 potential. For an arbitrary potential function a modified potential is defined involving three parameters, namely a hard-sphere diameter, an inverse-steepness parameter for the repulsive region, and a depth parameter for the attractive region. When the latter parameters are zero, the modified potential becomes the hard-sphere potential; when they are one, it becomes the original potential. The configuration integral is expanded in a double-power series in the inverse-steepness and depth parameters, the hard-sphere diameter being chosen so that the first-order term in the inverse-steepness parameter is zero. The first-order term in the depth parameter is evaluated essentially exactly and the second-order term approximately: other second-order terms and all higher-order terms are neglected. The resulting equation of state is in good agreement with molecular dynamics, Monte Carlo results, and experimental data for argon at all temperatures and densities relevant for fluids.

Kinetics of the Two-Body Radiative Association of Atoms

A treatment of radiative association of atoms based on equilibrium statistical mechanics is presented for the case where colliding atoms enter the repulsive region of an excited state and radiatively stabilize by undergoing electronic transition to the ground state. The final rate expression is the same as that derived from scattering theory when the potential is pure repulsive. The equilibrium treatment seems to have some advantages, for this special interaction, and may actually be somewhat more general than at first appears to be the case.

Properties of Dilute Mixtures ofHe3in LiquidHe4at Low Temperatures

Properties of dilute mixtures of ${\mathrm{He}}^{3}$ in liquid ${\mathrm{He}}^{4}$ are calculated on the assumption that, for the most part, they behave as low-density fermion systems with an effective ${\mathrm{He}}^{3}$ Hamiltonian. Expressions for ${\mathrm{He}}^{3}$ scattering amplitudes in terms of phase shifts are obtained, and the measured spin diffusion coefficient $D$ for temperature $T$ below 1\ifmmode^\circ\else\textdegree\fi{}K is used to determine the effective interaction between ${\mathrm{He}}^{3}$ atoms. It is found to have a strong repulsive region with a longer-range attraction which is deeper but of shorter range than the corresponding part of the Van der Waals force. At low momenta, the phase shifts are small, and at low temperatures, the effects are weak. At higher temperatures, larger momenta are probed, and the scattering is more significant. The phase-separation curve and the low-temperature viscosity, thermal conductivity, spin susceptibility, compressibility, and specific heat are calculated and agree with existing experimental evidence. It is found that the mixtures should undergo a fermion superfluid phase transition, but the maximum transition temperature is ${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{}K. Evidence for the transition should be found in a 4% decrease of D${\mathrm{T}}^{2}$ between ${10}^{\ensuremath{-}2\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ and 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$ \ifmmode^\circ\else\textdegree\fi{}K.

NUCLEAR MATTER AND THE LONG-RANGE PART OF THE NUCLEON–NUCLEON POTENTIAL

Elementary nuclear-matter calculations are performed with five different central nucleon–nucleon potentials. These are all static with a hard core of radius 0.4 fm and an OPEP tail, but are characterized by vastly different forms in the intermediate region. It is concluded that nuclear matter is insensitive to the precise form of the central part of the nucleon–nucleon potential everywhere beyond the short-range repulsive region, provided the nucleon–nucleon data are well fitted.

Shock compression of liquid hydrogen

High explosives have been used to shock liquid hydrogen to 39 500 bars, 5·2 cm3/g and an estimated temperature of 1100°k from an initial state near the normal boiling point, 20·5°k and 14·1 cm3/g. How far the melting curve and the equation of state can be pursued by shock wave experiments is discussed. The derived intermolecular potential between two hydrogen molecules is shown to be inadequately represented by the previously postulated Lennard-Jones 6–12 potential in the repulsive region between 2·0 and 2·7 A. The power of the inverse intermolecular distance is shown instead to be 8·5 in that region if a purely repulsive potential is used.

Effective nuclear interactions

The energy level spectra of O 17, O 18, O 19 and O 20 are analysed in the framework of the nuclear shell model. The matrix elements of the effective interaction in various two-particle states are deduced. It is shown that the singlet even interaction is not only short-ranged and attractive, but also contains a strongly repulsive central region. This demonstration of a repulsive non-singular core is the essential conclusion of this paper. It is also suggested that it may not be possible to represent the effective triplet odd interaction by a simple central potential. An experimental determination of the 3 + state in O 18 would throw further light on this question. The excitation energy of this state is predicted.