Gaussian Radial Dependence Research Articles

Localized ionospheric disturbance over the earthquake epicentre and modifications of Schumann resonance electromagnetic fields

The impact of seismogenic ionosphere non-uniformity is considered on the Schumann resonance (SR) signals observed both in the vertical electric and two orthogonal horizontal magnetic field components. The model is able to interpret the anomalous signals observed in Japan in possible association with the earthquakes (EQs) in Taiwan. Changes in the SR intensity arise from a localized reduction of the ionosphere over the EQ epicentre. Signals in the regular Earth–ionosphere cavity are described by using the knee model of vertical conductivity profile. It is suggested that pre-seismic and seismic activity lowers the knee altitude by 20 km at the centre of disturbance, and the perturbation has the Gaussian radial dependence in the 1 Mm zone. The diffraction problem in the cavity with a localized disturbance is resolved by using the Stratton–Chu integral equation. The data of Optical Transient Detector are used to model the source distribution. A pronounced modification is obtained in the intensity around the fourth mode frequency in a form of abrupt spectral changes. The model data are found to be similar to observations, and properties of the model seismic anomalies are discussed.

Model modifications in Schumann resonance intensity caused by a localized ionosphere disturbance over the earthquake epicenter

Abstract. This paper is a further extension of our latest observations and modeling by Hayakawa et al. (2005a), in which we discovered the anomalous behavior of Schumann resonance observed in Japan, in possible association with the Chi-chi earthquake in Taiwan. Schumann resonance intensity changes associated with a localized decrease in the lower ionospheric height over the earthquake epicenter are modeled. The knee model of the vertical conductivity profile of the ionosphere describes the regular Earth-ionosphere cavity, and the modified knee model is introduced for the disturbance. The localized ionosphere modification is of a Gaussian radial dependence; it has a 1-Mm radius, and the decrease reaches 20 km in the lower ionosphere height over the epicenter of the earthquake (Taiwan). The diffraction problem in the Earth-ionosphere cavity with a localized disturbance is resolved by using the Stratton-Chu integral equation. This solution is constructed for the case of natural resonance oscillations driven by independent random sources distributed worldwide. The data of the Optical Transient Detector (OTD) are used to introduce the source distribution. A pronounced increase in the intensity of the Schumann resonance is obtained around the fourth mode frequency (up to 20%) when thunderstorms are concentrated in Central America. The worldwide distribution of lightning strokes blurs and slightly reduces the effect (15% increase in intensity) for the observer in Japan and the localized nonuniformity positioned over Taiwan. A clear qualitative similarity is obtained in relation to the experimental data, indicating that records collected in Japan may be explained by the impact of a localized decrease in the lower ionosphere over the epicenter of the earthquake. It is admitted that the assumed conductivity decrease could only be caused by a severe change in the ionization in the middle atmosphere. It is not in the scope of this paper to discuss the possible mechanism, but rather to show that a closer and quantitative agreement with the experiment yields information about the form and size of the ionospheric modification and about the distribution of global thunderstorm activity during measurements.

Mean single-particle potentials and single-particle energies in a microscopic model of the nucleus

A method for calculating mean single-particle potentials and the corresponding single-particle energy levels in nuclei is presented. Specific formulas for these quantities are written for Slater determinant wave functions in the case of polarized orbitals and a central exchange nucleon-nucleon potential featuring a Gaussian radial dependence. The resulting theoretical estimates of single-particle properties of the nuclei considered in the present study are in satisfactory agreement with relevant experimental data.

SHELL MODEL CALCULATIONS OF 3T, 5He AND 6Li NUCLEI

The bases of the translation invariant shell model are used to construct the ground-state wave functions of 3 T , 5 He and 6 Li . For 3 T the bases used correspond to the number of quanta of excitation N up to ten. For 5 He and 6 Li the bases used correspond to the number of quanta of excitation N up to six. The model is applied to calculate the binding energy and the root mean square radius for 3 T , 5 He and 6 Li nuclei. The residual interactions used consist of central, tensor, spin-orbit and quadratic spin-orbit terms with Gaussian radial dependence. The parameters of these interactions are chosen in such away that they represent the long range attraction and the short range repulsion of nucleon interactions. It was found that this potential is more suitable for calculating the characteristics of these nuclei, and better than other potentials, such as our previous potentials which were represented by the parameters of long range attraction forces only. For 3T we obtained good agreement between calculated and experimental values of both the ground state binding energy and the root mean square radius. For 5 He and 6 Li nuclei we obtained an acceptable improvement with these calculations over other potentials.

Calculation of shell model energies for states in210Bi

A total of 34 experimental levels of {sup 210}Bi have been fitted with a generalized intermediate coupling model involving a n-p interaction with a Gaussian radial dependence with and without coupling to an octupole phonon. {chi}{sup 2} values are somewhat lower with the octupole parameter. Comparison of {chi}{sup 2} values with those obtained by fitting the same 34 experimental levels using the delta force radial dependence indicate the Gaussian radial dependence is 8{endash}10 times lower. The n-p interaction parameters are then used to calculate other {sup 210}Bi levels. {copyright} {ital 1997} {ital The American Physical Society}

The gamma-ray detection efficiency of a 118 cm 3 Ge(Li) detector for disc-sources of 7Be

The full energy peak detection efficiency of a 118 cm 3, closed end, coaxial Ge(Li) detector has been measured as a function of source to detector separation using disc-sources of 7Be with diameters ranging from 5 to 80 mm. The detector was characterized using a number of point-like sources of different energy placed on axis at different distances from the detector. The combined results were analysed in terms of the point-detector model and the Gaussian radial dependence model. The point-detector-model was found to provide a satisfactory parameterization of the axial variation of efficiency for point-sources. However the method did not predict the radius dependence of the efficiency for disc-sources. The Gaussian model provided much better fits in this case.

Charge density in pyroelectric lithium sulfate monohydrate at 80 and 298 K

Lithium sulfate monohydrate has been studied at 80 and 298 K by x-ray diffraction. The monoclinic crystal with space group P21 has lattice dimensions at 298 K of a=5.4553(1), b=4.8690(1), c=8.1761(2) Å, and β=107.337(2)°; lattice dimensions at 80 K were reported in our neutron study [J. Chem. Phys. 80, 423 (1984)]. Least-squares refinement based on 3486 (80 K) and 3390 (298 K) independent reflections, assuming a spherical atom model, results in final R( F 2) values of 0.024 (80 K) and 0.026 (298 K). Static deformation and charge density model refinement, based on Hirshfeld-type multipole functions, greatly reduces the residual electron density and gives R( F 2) values of 0.017 (80 K) and 0.016 (298 K). Refinement of the resulting multipole parameters within the Gaussian radial dependence model allows a qualitative estimation of the differences in electron densities between 80 and 298 K. A change of about 0.2 e Å−3 in the deformation density of the O–H bonds in the water molecule over this temperature range is related to the contraction in hydrogen bonding at 80 K. Changes in the sulfate oxygen atom lone-pair deformation densities, caused by contractions in the SO2−4–Li+ contacts between 298 and 80 K, are also found between the two temperatures. X-ray–neutron diffraction deformation density calculations indicate a substantial electron deficiency in the sulfur valence shell. Kappa refinement leads to a point charge model with positive atomic charge close to 1.8 on sulfur, about −1.0 on each sulfate oxygen atom, and −0.8 on the water oxygen atom. Interatomic distances at 80 and 298 K, particularly those based on a Gaussian radial dependence for the deformation density function, are in excellent agreement with the corresponding neutron diffraction values.

The ground states of the nucleide44Ti

Ground state binding energies and equilibrium harmonic oscillator energies are computed for the various ground state configurations of the doubly even nucleide44Ti. Employed for the study is a variational calculation carried out in a Cartesian harmonic oscillator basis of states employing a tripartite interaction potential composed of Wigner central forces coupled to Majorana exchange forces each having Gaussian radial dependence.

Self-focusing of a laser pulse in a transient plasma

Following Akhmanov's approach, self-focusing of a laser pulse in a transient plasma has been studied. The beamwidth parameter and hence the laser intensity and the frequency shift (time derivative of the phase) have been evaluated as a function of time and the distance of propagation. It is seen that the time dependence of the axial intensity changes appreciably as the pulse propagates. The present investigation is restricted to a pulse whose incident intensity has Gaussian radial dependence.

The residual n-p interaction in doubly odd deformed nuclei

Residual interaction calculations have been made for two-quasiparticle states in doubly odd rare earth nuclei. A comparison between theory and experiment has been made for 78 band head energy splittings and 20 odd-even shifts using a central two-body interaction with a Gaussian radial dependence that includes both short and long range components. Most of the splittings derived from definite configuration assignments were fitted well by the calculations. In addition, the interaction potential was reasonably well specified by the least squares fit to the splitting energies for a fixed pair of interaction radii. The odd-even shifts could not be fitted simultaneously with the splitting energies.

Studies of inelastic molecular collisions using impact parameter methods. I. Model calculations

In this paper we explore the behavior of an exponential approximation to the impact parameter amplitudes for model systems with potential matrices of simple form. In these matrices the magnitude of the coupling is governed by the difference in the quantum numbers of the coupled states. In the limit of large numbers of states the amplitude functions may be expressed as single integrals, which in many cases can be evaluated in terms of known mathematical functions. The simple structure of this result allows us to analyze the general features of the amplitude functions and the resulting cross section integrands versus impact parameter. Specific examples for potentials with Gaussian and exponential radial dependence are used for illustration. An exploration of the effect of varying the potential parameters for the Gaussian potential is made. In addition, the behavior of functions related to the generalized cross sections for spectral line broadening are considered. Finally, we explore the generalization of the simple model to more elaborate coupling schemes.

Triaxial states of light nucleides

The triaxial shapes of the first excited state of16O and of the ground states of24Mg and32S are studied in detail. Multipartite nucleon—nucleon interaction potentials, coupling Wigner central forces to Majorana exchange forces with both having Gaussian radial dependence, are employed in a variational calculation carried out in a Cartesian harmonic oscillator basis of states.

Shell-Model calculation of multipole transitions in 88Sr

A finite range force calculation using Gaussian radial dependence and employing Rosenfeld exchange mixture has been performed to calculate the various multipole transition rates in 88Sr. Fine agreement is obtained for the collective quadrupole and octupole states. In addition the dipole transition (3 − → 2 +) having a hindrance factor of 3.8 × 10 −3 is also found to be in very good agreement with the experiment.

Mathematical problem of 6 particles with Gaussian interactions

A differential-equation method is used to investigate the energy levels of6Li. The interaction assumed has a Gaussian radial dependence and contains central and tensor forces of equal range, the latter force being responsible for the splitting of theD-multiplet. The wave functions are expressed as linear combinations of the products of the multidimensional radial and surface harmonic parts. The radial parts (depending upon the nuclear radius) are allowed to take the best form by minimizing the total energy. The calculations for the splitting between the13S1 and13D3 levels, and between the13D2 and13D1 levels agree reasonably with the experimental data, the splitting between the13D2 and13D3 levels, apart from being only about 20 percent of the observed value, is also predicted to be in the wrong direction.

A study of light 4 n nucleides employing a three-part gaussian interaction

The properties of the light 4 n doubly even nucleides ranging from 4He to 40Ca are studied in a simple restricted Hartree-Fock calculation, employing an effective three-part two-body nucleon-nucleon interaction. This interaction consists of a trio of Wigner plus Majorana exchange forces each having Gaussian radial dependence. A rectangular harmonic oscillator basis of states is utilized. Two classes of parameter sets are studied: one generated from fitting the ground state energy and root-mean-square radius of 4He and ground state energy and density of nuclear matter, and one obtained from fitting the ground state energy and root-mean-square radius of both 4He and 40Ca. The resulting parameters are used to compute the ground state binding energies and harmonic oscillator energies of seven nucleides, 3Be, 12C, 20Ne, 14Mg, 28Si, 32S, and 36Ar, treated as being axially symmetric, with two possible configurations for all but 8Be and 12C considered. Moreover, 28Si is also analysed as a spherically symmetric nucleide. Additionally, the deformations of the axial nucleides are determined.

Symmetric-Core Collective Model for Odd-Odd Nuclei with Applications in the2s−1dShell

A model for odd-odd nuclei is described in which the odd neutron and proton are coupled to a symmetric, rotating core. The Hamiltonian consists of four parts: ${H}_{R}$, the Hamiltonian for a rotating core of fixed shape; ${H}_{p}$ and ${H}_{n}$, the Hamiltonians for a proton and a neutron moving in a symmetric oscillator potential with 1\ifmmode\cdot\else\textperiodcentered\fi{}s and ${1}^{2}$ terms; and ${V}_{\mathrm{pn}}$, the residual neutron-proton interaction. This latter was taken to have a Gaussian radial dependence with a Serber exchange mixture with parameters picked to reproduce the low-energy singlet and triplet scattering lengths. The energy eigenvalues were obtained by an exact diagonalization of the total Hamiltonian using a core-particle basis to which a truncation procedure is applied to account for the filling of the shells. The model has been applied to $^{28}\mathrm{Al}$, $^{30}\mathrm{P}$, $^{32}\mathrm{P}$, and $^{36}\mathrm{Cl}$, determining the core-strength parameter $P$, the deformation parameter $\ensuremath{\beta}$, and the well-depth parameter $\ensuremath{\lambda}$ by fitting to the known energy-level sequence. The state functions so obtained were used to calculate static magnetic dipole and electric quadrupole moments, and for $^{30}\mathrm{P}$ the mixing ratio ${\ensuremath{\delta}}^{2}$ for two ${2}^{+}$ to ground transitions. The fit to the measured energy levels is quite successful; however, only in $^{28}\mathrm{Al}$ and $^{30}\mathrm{P}$ is the residual neutron-proton interaction needed. The values obtained for the static moments are rather poor. The results are compared, where possible, to a shell-model calculation.

An improved S-state wave function for6Li

The lowest harmonic oscillator shell model wave function of 6Li, ψ0, is modified by applying the Gauss radial function Q = Σl<j6Jij (where Jij = exp(-rij2/rc2) and rc is the assumed range of nuclear forces) to it. It has been shown that Qψ0|H|Qψ0/Qψ0|Qψ0 gives a better approximation to the lowest eigenvalue than ψ0|H|ψ0/ψ0|ψ0. A typical short range two-body central interaction which is charge symmetric and has Gaussian radial dependence has been used in the Hamiltonian to investigate the improvement in the wave function. Considerable improvement has been found by this method.

Shell Model and the Levels ofZr91

A shell-model interpretation of low-lying positive-parity states of ${\mathrm{Zr}}^{91}$ has been attempted by taking into account the residual interactions between the two protons in the $2{p}_{\frac{1}{2}}$ and $1{g}_{\frac{9}{2}}$ configurations coupled to a neutron in the $2{d}_{\frac{5}{2}}$, $3{s}_{\frac{1}{2}}$, $1{g}_{\frac{7}{2}}$, and $2{d}_{\frac{3}{2}}$ single-particle states. A finite-range, central two-body force with a Gaussian radial dependence was assumed. Data on other nuclei in this mass region have been analyzed to provide a set of model parameters. Qualitative agreement has been obtained with the experimental data on ${\mathrm{Zr}}^{91}$.

Electronic Wave Functions for Polyatomic Molecules

A one-electron basis set made from functions with exponential and Gaussian radial dependence is proposed for constructing many-electron, many-center, electronic wave functions. Appropriate mixtures of these two types may greatly simplify the evaluation of many-center, two electron integrals and in addition make possible an improved description of the molecular bond. Plane waves may also be added to the basis set thus making an application to solids appear feasible.

The α, α interaction

The 8-body problem of the (α, α) scattering interaction is formulated by using the group resonance formalism, with two unpolarised α groups, and with the central nucleon-nucleon exchange force and α wave-function both having a gaussian radial dependence. Choosing parameters as far as possible the same as required by the low energy nucleon-nucleon data, the α binding energy, and the (α, n) scattering interaction, it was attempted to explain the S and D phases for low energy (α, α) scattering which was previously found to require a velocity-dependent interaction. The (α, α) scattering requires slightly different values of y = 4 w+2 b−2 h− m, both of which are lower than required by the (α, n) problem. It is shown that the S wave interaction essentially behaves as a potential at low energies. For low energies this potential effectively has a repulsive core with an attractive outside well—features which also emerged from a previous purely phenomenological analysis.