Taro Kihara

Crystal Structures and Intermolecular Forces of Rare Gases

By means of models of intermolecular potential U ( r ), the total energy of molecular crystals is calculated provided the additivity of intermolecular forces is valid. If the Lennard-Jones model, U ( r )= U 0 ( s -6) -1 [6( r 0 / r ) s - s ( r 0 / r ) 6 ], is used, the lattice of hexagonal closest packing has always a lower energy than that of cubic closet packing. If the model, U ( r )= U 0 (σ-6) -1 [6 exp (σ-σ r / r 0 )-σ( r 0 / r ) 6 ] is used, there is a critical value of σ, σ 0 =8.675, above which the hexagonal is stable and below which the cubic is stable. In general, in order that the cubic structure has a lower energy as in the case of rare gases, it is necessary (although not sufficient) that the hollow part of the intermolecular potential is wide enough, much wider than that of U ( r ) = U 0 [ r 0 / r ) 12 -2( r 0 / r ) 6 ] for instance. These results are in agreement with the conclusions which Kihara has obtained by investigating the third virial coefficients of the equation of state for rare g...

The Second Virial Coefficient of Non-Spherical Molecules

The Lennard-Jones model of molecules of spherical symmetry has been generalized to non-spherical models without sacrificing analytical integrability of the second virial coefficient. The essential generalization consists in the definition of the intermolecular distance, ρ, the intermolecular potential, U , being supposed to be a function of ρ only and \begin{aligned} U(\rho){=}U_{0}\left[\frac{m}{n-m}\left(\frac{\rho_{0}}{\rho}\right)^{n}-\frac{n}{n-m}\left(\frac{\rho_{0}}{\rho}\right)^{m}\right],\quad n{>}m{>}3. \end{aligned} As the simplest extension ρ is defined by the shortest distance between molecule cores, as which thin rods (disks) are adopted in case of prolonged (flat) molecules. Next, ρ is so defined that the model becomes an attracting spheroid. For these models the second virial coefficient has been integrated analytically and tabulated. The model Constants have been determined for H 2 , N 2 , C 2 H 4 and CO 2 .

Statistics of Two-Dimensional Lattices with Many Components

The statistics of the two-dimensional Ising lattice, for which each lattice site may take one of two states, has been partly generalized to the following model: each lattice site may take one of s states and each pair of neighboring sites in different states has a common excess energy, J , compared with a pair of neighboring sites in a same state. The partition function for the square lattice has a temperature symmetry and the Curie temperature, T c , is given by \begin{aligned} (s-1)\exp(-2J/kT_{c})+2\exp(-J/kT_{c})-1{=}0, \end{aligned} ( k is the Boltzmann constant). Although the partition function itself has only been obtained in the form of power series valid at low and high temperature regions, it is highly probable that for any finite s the specific heat is infinite at the Curie temperature. The transition is not the normal phase transition of the first order which the Bragg-Williams approximation predicts for our lattice with s ≧3.

Non-additive Intermolecular Potential in Gases I. van der Waals Interactions

The potential energy of van der Waals interactions between three spherically symmetric atoms at large separations has been investigated both by perturbation theory and by variation method. Without using any particular atomic models the van der Waals interaction is derived, from second and third-order perturbations, in the form \begin{aligned} -\mu_{12}{r_{12}}^{-6}-\mu_{23}{r_{23}}^{-6}-\mu_{31}{r_{31}}^{-6}+\nu(r_{12}r_{23}r_{31})^{-3}(3\cos\theta_{1}\cos\theta_{2}\cos\theta_{3}+1), \end{aligned} where r i j is the distance between i th and j th atoms, θ i s are inner angles of a triangle formed by the three atoms. An approximate relation between µ and ν is found by variation method: \begin{aligned} \nu{=}\frac{2R_{1}R_{2}R_{3}(R_{1}+R_{2}+R_{3})}{(R_{1}+R_{2})(R_{2}+R_{3})(R_{3}+R_{1})}, \end{aligned} in which \begin{aligned} \frac{1}{R_{1}}{=}\frac{1}{\mu_{12}\alpha_{3}}+\frac{1}{\mu_{31}\alpha_{2}}\frac{1}{\mu_{12}\alpha_{3}}-\frac{1}{\mu_{12}\alpha_{3}}\quad\text{etc}, \end{aligned} α i being the po...

On the Coefficients of Irreversible Processes in a Highly Ionized Gas

An effective interparticle potential, which is a modified Debye-Huckel potential, is introduced and discussed in order to eliminate simultaneous interactions between three particles or more in an ionized gas. For this potential weak-interaction asymptotes of collision cross sections necessary for the theory of irreversible processes are calculated without making further approximations. These cross sections are applied to relaxation between ion and electron temperatures and to the coefficient of viscosity.

Virial Coefficients and Intermolecular Potential of Helium

The intermolecular potential of helium has been determined from an analysis of the second and the third virial coefficients above the Boyle temperature, where the virial coefficients can be expanded into power series on h 2 , the square of Plancks constant. The series have been calculated up to the term proportional to the third power of h 2 in case of the second virial coefficient and up to the term proportional to the first power of h 2 in case of the third virial coefficient. The following conclusion has been drawn. The two-body system of He 4 atoms has a discrete energy level while the two-body system of He 3 atoms has no discrete level.

Non-additive Intermolecular Potential in Gases : II. Cluster Integrals

The van der Waals interaction discussed in Part I are applied to the equation of state of rare gases. The potential energy between three identical atoms is assumed to be \begin{aligned} U(r_{12},r_{23},r_{31}){=}U(r_{12})+U(r_{23})+U(r_{31})+\nu(r_{12}r_{23}r_{31})^{-3}(3\cos\theta_{1}\cos\theta_{2}\cos\theta_{3}+1). \end{aligned} Here r i j is the distance between i th and j th atoms, θ i s are inner angles of the triangle formed by the three atoms, and \begin{aligned} U(r){=}\lambda r^{-12}-\mu r^{-6}. \end{aligned} Use is made of a relation proved in Part I, 4ν=3αµ, in which α is the polarizability. For this model of the intermolecular potential we have calculated the third cluster integral including quantum effect. By virtue of the non-additive part of the intermolecular potential agreement between observed and calculated values is very good.

MACROSCOPIC FOUNDATION OF PLASMA DYNAMICS

A fluid which comprises several species of charged and neutral particles is treated macroscopically. Basic equations, energy and momentum theorems, and law of similarity are discussed. The condition under which current equation reduces to the usual form is considered; and one-dimensional transverse waves are treated as an example. Quasi-stationary phenomena are examined as a particular case; and it is pointed out that one of the usual basic equations of hydromagnetics, the Ohms law J =σ( E + v × B ), should be replaced by rot( E + v × B -σ -1 J )=0, v being the velocity of the mean mass flow. An axially symmetric solution is obtained and applied to a self-pinched column, whose stability is explained in an elementary manner.

Second Virial Coefficient between Unlike Molecules

By means of the core model of molecules introduced by Kihara in Revs. Modern Phys. 25 (1953) potentials between unlike molecules in nonpolar gases have been determined from the second virial coefficients. The intermolecular potential U A B for the pair of molecules ( A , B ) is assumed to be a function of the shortest distance ρ between the cores of the molecules, and furthermore \begin{aligned} U_{AB}(\rho){=}U_{0AB}\left[\left(\frac{\rho_{0AB}}{\rho}\right)\right]^{12}-2\left(\frac{\rho_{0AB}}{\rho}\right)^{6}\bigg]. \end{aligned} Here U 0 A B and ρ 0 A B are constants which have been determined. Regarding these model constants the following relations have been verified:

Multipole Interaction Stabilizing Cubic Molecular Crystals

The stability of several cubic molecular crystals is investigated on the basis of multipole interaction between the molecules. The stability of the cubic close-packed structure of heavy rare-gas crystals is due to the repulsion between electrical octapoles induced in atoms in the hexagonal close-packed structure. The face-centered cubic structures of carbon dioxide and cyclohexane-hexahalides and the body-centered cubic structures of silicon-tetrafluoride and hexamethylene-tetramine are configurations with the maximum electrostatic attraction between permanent charge distributions in the molecules. Pingpong balls containing two or four pieces of magnet represent these molecules with permanent multipoles; and it is demonstrated that a proper “crystal” of such balls (suspended in water) is stable.