Calculated Coulomb Energy Research Articles

The accuracy of the Gaussian-and-finite-element-Coulomb (GFC) method for the calculation of Coulomb integrals

We analyze the accuracy of the Coulomb energy calculated using the Gaussian-and-finite-element-Coulomb (GFC) method. In this approach, the electrostatic potential associated with the molecular electronic density is obtained by solving the Poisson equation and then used to calculate matrix elements of the Coulomb operator. The molecular electrostatic potential is expanded in a mixed Gaussian-finite-element (GF) basis set consisting of Gaussian functions of s symmetry centered on the nuclei (with exponents obtained from a full optimization of the atomic potentials generated by the atomic densities from symmetry-averaged restricted open-shell Hartree-Fock theory) and shape functions defined on uniform finite elements. The quality of the GF basis is controlled by means of a small set of parameters; for a given width of the finite elements d, the highest accuracy is achieved at smallest computational cost when tricubic (n = 3) elements are used in combination with two (γ(H) = 2) and eight (γ(1st) = 8) Gaussians on hydrogen and first-row atoms, respectively, with exponents greater than a given threshold (αmin (G)=0.5). The error in the calculated Coulomb energy divided by the number of atoms in the system depends on the system type but is independent of the system size or the orbital basis set, vanishing approximately like d(4) with decreasing d. If the boundary conditions for the Poisson equation are calculated in an approximate way, the GFC method may lose its variational character when the finite elements are too small; with larger elements, it is less sensitive to inaccuracies in the boundary values. As it is possible to obtain accurate boundary conditions in linear time, the overall scaling of the GFC method for large systems is governed by another computational step-namely, the generation of the three-center overlap integrals with three Gaussian orbitals. The most unfavorable (nearly quadratic) scaling is observed for compact, truly three-dimensional systems; however, this scaling can be reduced to linear by introducing more effective techniques for recognizing significant three-center overlap distributions.

Coulomb Energy of Traps in Semiconductor Space Charge Regions

Large Coulomb barriers exceeding ΔE≊250 meV are estimated for capture and emission rates of trap centers in semiconductor space‐charge regions. Depending on the charge state of the trap, the capture rate or both the capture and emission rates are activated or deactivated, respectively. The Coulomb energy raises the equilibrium energy state of a trap center that is repulsively charged when occupied. Quantitative agreement of the calculated Coulomb energy is obtained with trapping rates for single individual interface traps in metal‐oxide‐semiconductor (MOS) structures measured by random telegraph signals. The Coulomb barrier is reduced in MOS capacitors by partial screening due to mobile charge carriers in the inversion channel. The Coulomb energy can be externally controlled in MOS structures by the gate bias voltage.

Coulomb energy of traps in semiconductor space-charge regions

Large Coulomb barriers exceeding ΔE≊250 meV are estimated for capture and emission rates of trap centers in semiconductor space-charge regions. Depending on the charge state of the trap, the capture rate or both the capture and emission rates are activated or deactivated, respectively. The Coulomb energy raises the equilibrium energy state of a trap center that is repulsively charged when occupied. Quantitative agreement of the calculated Coulomb energy is obtained with trapping rates for single individual interface traps in metal-oxide-semiconductor (MOS) structures measured by random telegraph signals. The Coulomb barrier is reduced in MOS capacitors by partial screening due to mobile charge carriers in the inversion channel. The Coulomb energy can be externally controlled in MOS structures by the gate bias voltage.

Coulomb energy of trapping in MOS interface states

Large Coulomb barriers exceeding several hundred meV are estimated for capture and emission rates into trap centers at the SiO 2-Si interface of MOS structures. The Coulomb energy is added to the equilibrium level energy for a trap center that is repulsively charged when occupied. Quantitative agreement of the calculated Coulomb energy is obtained with trapping rates measured for single individual traps in sub- μm MOSFETs by random telegraph signals.

Nucleon-nucleon exchange density using the Wigner transform

An expression for nucleon-nucleon exchange density has been derived for large closed-shell nuclei using the Wigner transform. It is shown that it is a highly peaked function of the distance mod r1-r2 mod of the two nucleons. Using this density, an expression is derived for the Coulomb energy of a heavy nucleus. The exchange part turns out to be the same as the one obtained using the plane-wave approximation. It is shown that the calculated Coulomb energy difference between the isobars 41Sc and 41Ca is in fair agreement with its experimental value.

X-ray photoemission study of Hg clusters on Hg1−xCdxTe surfaces

Hg1−xCdxTe (111)B surfaces have been studied with x-ray photoelectron spectroscopy (XPS). A surface shift is deduced from a careful analysis of Te 4d core-level spectra. The presence of small Hg clusters on these surfaces is observed, and the size is estimated from the XPS data (R=5–20 Å). The positive binding energy shift for these clusters agrees very well with the calculated Coulomb energy due to the positive charge which appears on the clusters during the photoemission process. The origin of these clusters is briefly discussed.

Trinucleon Coulomb energy with inclusion of a three-nucleon force

We calculate binding energy difference between $^{3}\mathrm{H}$ and $^{3}\mathrm{He}$ using the exact harmonics method where Coulomb force is treated nonperturbatively. A variety of two-nucleon potentials with or without a two-pion-exchange three-body force (Fujita-Miyazawa type) has been used properly. We compare our results with the hyperspherical formula and previous calculations. Calculated Coulomb energy agrees roughly with the formula; but the significant difference indicates that the perturbative treatment is not sufficient.NUCLEAR STRUCTURE $^{3}\mathrm{He}$, Coulomb energy, harmonics calculation.

Phenomenological Coulomb interaction for microscopic nuclear structure calculations

The proton charge distribution, deduced from electron-proton scattering data, was used to calculate the Coulomb interaction between protons in nuclei in the $f\ensuremath{-}p\ensuremath{-}g$ shell. The Coulomb energies and shifts between members of the $A=42$ isotopic spin triplet were calculated.NUCLEAR STRUCTURE Derived the Coulomb potential between protons in $f\ensuremath{-}p\ensuremath{-}g$ shell. Calculated Coulomb energy in $^{42}\mathrm{Ti}$ and shifts ($^{42}\mathrm{Ti}$-$^{42}\mathrm{Ca}$) and ($^{42}\mathrm{Sc}$-$^{42}\mathrm{Ca}$). Compared with experiment.

Influence of 2p-2h excitations in the41Sc-41Ca Coulomb energy shift

The influence of 2p-2h core excitations on the41Sc-41Ca Coulomb energy difference is studied. A simple parametrization of the main wave function components shows that the calculated energy shift could only agree with experiment for rather unrealistic values of the amplitudes. Using an effective interaction appropriate for this region, we find that when this kind of excitations is included, the calculated Coulomb energy shift is increased by 105keV.

Shell-model calculations in the sd shell. IX. Masses, spectra and beta decays of the mass-24 nuclei

For pt.VIII see ibid., vol.3, no.7, p.919 (1977). The authors present the results of a shell-model investigation of the masses, spectra, Coulomb energies and beta decays of the mass-24 nuclei. They have used the effective interaction of Chung and Wildenthal together with an empirically-determined Coulomb interaction. The ab initio use of a reasonably good Coulomb interaction reveals the presence of quite large errors in the calculated energies of certain states and hence of serious imperfections in the nuclear effective interaction. The calculated Coulomb energy differences in the mass-24 system have been used to improve the empirical Coulomb matrix elements.

Evidence for diffusive relaxation along the mass asymmetry coordinate in the reaction 197Au + 620 MeV 86Kr

Nuclei have been identified by atomic number up to Z = 50 using ΔE- E telescopes. Kinetic energy spectra, charge and angular distributions have been measured from θ lab = 10–80° . At angles removed from the grazing, a single peak with a mean energy somewhat below the calculated Coulomb energy is observed for all elements. Near the grazing angle, a much broader peak appears, which extends from near elastic energies down to the Coulomb barrier. The charge distributions are peaked near the projectile Z and demonstrate a strong shape dependence on the angle of observation. The angular distributions for elements near the projectile are strongly side peaked; however, as Z is increased or decreased from Z = 36, they gradually become forward peaked. The dependence of the charge and angular distributions on energy dissipation is discussed as well as the patterns observed in the two-dimensional Wilczynski plots. Diffusion model calculations reproduce the experimental data both qualitatively and quantitatively. This successful application of the diffusion model, which was originally developed to explain N, Ne and Ar induced reactions, to the present system is strong evidence that no essential differences exist between the “quasi-fission” process observed in Kr bombardments of heavy targets and the deep-inelastic phenomena seen with lighter ions.

Charge Asymmetry Effects and the Trinucleon Binding-Energy Difference

We demonstrate, using a separable-potential model, that the discrepancy between the observed $^{3}\mathrm{H}$-$^{3}\mathrm{He}$ binding-energy difference and the calculated Coulomb energy does not necessarily imply that $|{a}_{nn}|>|{a}_{pp}|$. This possibility arises from the relative insensitivity of the trinucleon binding energy to the scattering length as compared to the sensitivity to the effective range.

Coulomb energies and charge asymmetry of nuclear forces

Charge-symmetry-breaking potentials suggested in the literature to resolve the discrepancy between calculated Coulomb energy differences of analog states and the experimental values, are considered in detail. We calculate the contributions of these potentials to the ground state energy differences of the mirror nuclei 3He 3H, 15O 15N, 17F 17O, 39Ca 39K and 41Sc 41Ca. It turns out, due to the short range character of these symmetry-breaking potentials, that their inclusion may resolve the 3He 3H difficulty but not the 41Sc 41Ca discrepancy.

Coulomb energy and the mass difference of 3H and 3He

The calculated Coulomb energy is large enough to account for the mass difference of 3H and 3He.

The density distribution of He 6

A two-parameter density distribution has been obtained for He 6 using an oscillator shell model distribution with different oscillator length parameters a s and a p for the s and p-nucleons respectively. One relation between a s and a p is obtained by equating the calculated Coulomb energy difference between He 6 and the lowest T = 1 state of Li 6 with the empirical value while a second relation has been obtained using a method due to Überall which gives good agreement for the form factor of Li 6 and which assumes that the s-nucleon distribution is due to a recoiling α-particle. One then obtains a s = 1.79±0.04 fm, a p = 2.60±0.06 fm and 〉r 2〉 1 2 = 2.78±0.05 fm for the r.m.s. radius of the point nucleon distribution. Compared with Li 6 the extension of the s-nucleons is only slightly greater. However, 〈r 2〉 1 2 and the extension of the p-nucleons is significantly greater, consistent with the smaller binding energy of the p-nucleons in He 6. The effect of the Coulomb interaction in extending the p-nucleons in the T = 1 state of Li 6 beyond those in He 6 is shown to be negligible.

Coulomb Energy ofHe3and Charge Distribution of Nucleon

The effect of finite nucleon size on the Coulomb energy of ${\mathrm{He}}^{3}$ has been investigated. The experimental value of the difference between the binding energies of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{3}$ is 0.764 MeV, while the calculated Coulomb energy is approximately equal to or greater than 1.0 MeV if the nuclear force has no repulsive core. We show that, if the finite size of nucleon is taken into consideration, the Coulomb energy of ${\mathrm{He}}^{3}$ is reduced by about 15-20%. The effect of finite charge distribution is determined mainly by the mean square radius. If there is a hard core (with radius $D$), the calculated Coulomb energy (assuming point nucleons) is already smaller, with the values 0.8-0.9 MeV for $D=0.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm, \ensuremath{\sim}0.7 MeV for $D=0.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm. The reduction of Coulomb energy due to the finite size is about 8% and 3%, respectively, for two different models. The Coulomb potential between extended unpolarized nucleons is given in closed form for exponential and Yukawa charge distributions.