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H. M. Krutter

Bio: H. M. Krutter is an academic researcher from Massachusetts Institute of Technology. The author has contributed to research in topics: Elementary charge & Kinetic energy. The author has an hindex of 1, co-authored 1 publications receiving 127 citations.

Papers
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TL;DR: In this paper, the Thomas-Fermi method is applied to metals, by replacing each atom by a sphere, assuming the potential to be spherically symmetrical within it, and solving the Thomas Fermi equation subject to the boundary condition that the electronic charge within the sphere shall balance the nuclear charge, rendering it electrically neutral.
Abstract: The Thomas-Fermi method is applied to metals, by replacing each atom by a sphere, assuming the potential to be spherically symmetrical within it, and solving the Thomas-Fermi equation subject to the boundary condition that the electronic charge within the sphere shall balance the nuclear charge, rendering it electrically neutral. Calculations are presented giving potential field, charge density, and kinetic, potential, and total energy of the metal, as function of lattice spacing. The virial theorem is verified for the energy. The total energy shows no minimum, the pressure being always positive. Calculations are also made using the Dirac method of correcting for exchange, for three atoms, Li, Na and Cu. The exchange lowers the energy, but still not quite enough to produce a minimum of energy and an equilibrium at zero pressure. The result should be useful as a first approximation in self-consistent field approximations for the structure of metals, and could be adapted to give approximate treatment for matter under very high pressure, as in stars.

128 citations


Cited by
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Francis Birch1
TL;DR: In this paper, the authors derived a general equation for the variation of the quantity,, in a homogeneous gravitating layer with an arbitrary gradient of temperature, and discussed the parameters of this equation in terms of the experimental and theoretical relations for solids.
Abstract: The observed variation of the seismic velocities with depth, below the crust, is examined with reference to the variation to be expected in a homogeneous medium. A general equation is derived for the variation of the quantity, , in a homogeneous gravitating layer with an arbitrary gradient of temperature. The parameters of this equation are then discussed in terms of the experimental and theoretical relations for solids. The principal parameter is (∂KT/∂P)T, the rate of change of isothermal incompressibility with pressure, which can be found for large compressions from Bridgman's measurements. Comparison of observed and expected rates of variation of ϕ throughout the Earth's interior leads to conclusions regarding homogeneity and, with a larger uncertainty, to estimates of temperature. A shadow zone at a depth of about 100 km, as suggested by Gutenberg, may be accounted for by a gradient of temperature of about 6°/km in a homogeneous layer of ultrabasic rock. Between depths of about 900 and 2,900 km, the mantle appears to be substantially uniform, and at a relatively uniform temperature of the order of several thousand degrees. Between about 200 and 900 km, the rate of rise of velocity is too great for a homogeneous layer, and indicates a gradual change of composition, or of phase, or both. New phases are required to account for the high elasticity of the deeper part of the mantle (below 900 km), and it is suggested that, beginning at about 200 to 300 km, there is a gradual shift toward high-pressure modifications of the ferro-magnesian silicates, probably close-packed oxides, with the transition complete at about 800 to 900 km. There may also be a concentration of alumina, lime, and alkalis toward the upper part of the mantle, in and above the transitional layer but below the crust, existing in minerals of high elasticity such as garnets and jadeites. The transitional layer appears to hold the key to a number of major geophysical problems. The velocities in the core and inner core are also reviewed. The inner core is most simply interpreted as crystalline iron, the outer part as liquid iron, perhaps alloyed with a small fraction of lighter elements. The density and compressibility of iron at high pressures are estimated with the aid of the experimental compressions of the alkali metals; the central density is found to be about 15. Several other recent proposals regarding the crust are discussed.

2,142 citations

Journal ArticleDOI
TL;DR: The Thomas-Fermi approximation in quantum mechanics has been studied extensively in the literature as mentioned in this paper, including in the context of quantum mechanics, but without the application of quantum computers.
Abstract: (1957). The Thomas-Fermi approximation in quantum mechanics. Advances in Physics: Vol. 6, No. 21, pp. 1-101.

346 citations

Book ChapterDOI
TL;DR: Theoreme du viriel classique and theoreme de viriel en mecanique quantique as mentioned in this paper were derived from Hartree-Fock-Slater and Thomas-Fermi-Dirac.
Abstract: Mise au point. Theoreme du viriel classique et theoreme du viriel en mecanique quantique. Applications aux equations d'etat, aux modeles de Hartree-Fock-Slater et Thomas-Fermi-Dirac, a la liaison chimique. Theoreme de l'hyperviriel. Moments de l'equation de Fokker-Planck

125 citations

Journal ArticleDOI
TL;DR: In this article, a self-consistent thermodynamic description of silicate liquids applicable across the entire mantle pressure and temperature regime was developed, which combines the finite strain free energy expansion with an account of the temperature dependence of liquid properties into a single fundamental relation.
Abstract: SUMMARY We develop a self-consistent thermodynamic description of silicate liquids applicable across the entire mantle pressure and temperature regime. The description combines the finite strain free energy expansion with an account of the temperature dependence of liquid properties into a single fundamental relation, while honouring the expected limiting behaviour at large volume and high temperature. We find that the fundamental relation describes well previous experimental and theoretical results for liquid MgO, MgSiO3 ,M g2SiO4 and SiO2. We apply the description to calculate melting curves and Hugoniots of solid and liquid MgO and MgSiO3. For periclase, we find a melting temperature at the core–mantle boundary (CMB) of 7810 ± 160 K, with the solid Hugoniot crossing the melting curve at 375 GPa, 9580 K, and the liquid Hugoniot crossing at 470 GPa, 9870 K. For complete shock melting of periclase we predict a density increase of 0.14 g cm −3 and a sound speed decrease of 2.2 km s −1 . For perovskite, we find a melting temperature at the CMB of 5100 ± 100 K with the perovskite section of the enstatite Hugoniot crossing the melting curve at 150 GPa, 5190 K, and the liquid Hugoniot crossing at 220 GPa, 5520 K. For complete shock melting of perovskite along the enstatite principal Hugoniot, we predict a density increase of 0.10 g cm −3 , with a sound speed decrease of 2.6 km s −1 .

116 citations