Yasuo Tomishima

Solution of the Thomas-Fermi-Dirac Equation with a Modified Weizsäcker Correction

According to the result obtained in a previous paper (J. Phys. Soc. Japan 20 (1965) 1051), the Weizsacker correction is introduced into the TFD theory with a constant weighting factor λ(=0.2) and this extended statistical equation is solved for the electron distribution of the inert gas atoms Ne, Ar, Kr and Xe. The values of the total energy calculated on this basis agree very well to the Hartree-Fock values. The diamagnetic susceptibilities, however, still have a poor coincidence although the present model gives definitely better results compared to the simple TF or TFD theory. One electron energy values or the occupied states in Kr atom are also calculated by the Bohr-Sommerfeld quantization condition.

Thomas-Fermi Theory for Atoms in a Strong Magnetic Field

Along the general scheme of Sondheimer and Wilson, the kinetic energy density o£ an electron gas under constant magnetic field is expressed as a functional o£ the electron density at absolute zero of temperature. On this basis, the statistical theory for atoms in a magnetic field is formulated, which includes the theory developed by Banerjee et al. as an extreme of high magnetic field. Some numerical results on the atomic radius, the total energy etc. are also shown for free neutral Ne atom.

Calculation of the Cohesive Energy of Zincblende

The cohesive energy of zincblende has been calculated quantum mechanically by introducing a covalency correction into the usual pure ionic binding. The cohesive energy per ion-pair thus obtained is 1.46 in atomic unit which is very near the observed value 1.36 derived by means of Born-Harbers energy cycle, and is certainly greater than the value calculated for purely ionic binding by about 60%. This method of calculation would be generally applicable not only for sphalerite-type crystals but also for wurzite-type ones.

A Relativistic Thomas-Fermi Theory

By introducing a modified Weizsacker correction, a relativistic Thomas-Fermi theory is developed without any divergence difficulties. The electron density distribution and the total energy are obtained for several free neutral atoms and positive ions.

On the Influence of the Packing on the Atomic Scattering Factor Based on the Thomas-Fermi Theory

The influence of the juxtaposition in the crystal on the atomic scattering factor has been investigated on the basis of the Thomas-Fermi(abbreviated as TF) theory, using the TF functions appropriate to the packed neutral atoms. It is pronounced especially for the smaller values of sin θ/λ· Z -1/3 . The values of the atomic scattering factor for free neutral atom have been recalculated anew, using the most accurate Miranda values of the free neutral TF function.

TFD Functions for Non-zero Temperatures and Equations of State Based on Them

By means of the Lidiard distribution function for free electron gas, the Thomas-Fermi-Dirac (TFD) equations for non-zero temperatures are derived and integrated numerically for Fe. On the basis of the TFD functions there are given also the equations of state of metals under extremely high pressures.

Monte Carlo Solutions of Schrödinger's Equation for H+2 Ion in Strong Magnetic Fields

The analytical expressions suitable for the Monte Carlo calculation to obtain the solution of Schrodingers equation of hydrogen molecular ion in a strong magnetic field are derived. The wave functions, the energy values and the equilibrium internuclear distances of 1σ g state of H + 2 are obtained numericallly through the Monte Carlo simulation and compared with other results based on the variational method. The agreement between them is fairly good over a wide range of magnetic field. The calculation of the energy values of 1π g state of H + 2 for various internuclear distances taking a constant magnetic field as a parameter, shows that the antibonding 1π g state in the absence of the external magnetic field changes to a bonding state with an increasing magnetic field. The lowest energy values and the equilibrium internuclear distances of 1π g state are also calculated for various magnetic field.

Lattice Defects in Zincblende : Part II. Formation Energy of Lattice Defects

The formation energies of various types of lattice defects in zincblende are calculated by means of Mott-Littletons method. A phenomenological expression of interionic potential which contains the covalency binding energy is used in the calculation. In addition, the total energy when the lattice around a defect is not in equilibrium is computed at several positions of the surrounding ions of the defect near the equilibrium point. The results of calculation show that the formation energies of Schottky defect, double hole and Frenkel defect by zinc ion are estimated to be 4∼6 eV. The result that the formation energy of Schottky defect is nearly equal to that of double hole is qualitatively in agreement with the experimental results which have been obtained with pure host crystal of ZnS phosphor.

Lattice Defects in Zincblende Part I. Phenomenological Expressions of Interionic Potentials

As the basis of the calculation on formation energies of lattice defects, phenomenological expressions of interionic potentials are derived. In zincblende crystal which is partly ionic and partly covalent, the effective charge on the constituent ions is reduced to a certain amount Z e ( Z =1.66, e : magnitude of the electronic charge), whereas covalent attraction between two ions adjacent to each other is generated, which is assumed to be exponential form - b c exp (- r /ρ c ) such as the usual expression of exchange repulsion. As a preliminary calculation to Part II, the lattice energy variation when one anion or cation is displaced from its regular position in a perfect crystal is calculated on taking account of the Madelung, Van der Waals, exchange repulsion, covalency attraction energies and the polarization energies of the displaced ion and all other surrounding ions. All the physical constants already known in references and derived in our calculation are tabulated in Table III.

Inhomogeneity Correction to the Thomas-Fermi Atom in a Strong Magnetic Field

The inhomogeneity correction to the statistical theory of the atom m a uniform strong magnetic field is derived by using the March-Murray perturbation expansion theorem of the canonical density matrix modified to give the Thomas-Fermi electron density as the first term of the expansion. The inhomogeneity correction makes the atom non-spherical and it is concluded that as the magnetic field strength increases the electron cloud of the atom contracts in the direction perpendicular to the applied magnetic field as will be expected.