Coefficients of radial integral in the electrostatic interaction and their applications

2004 ◽  
Vol 82 (7) ◽  
pp. 517-522
Author(s):  
Z Chen ◽  
A Z Msezane
Keyword(s):  
Interaction Energy ◽  
Cross Sections ◽  
Electrostatic Interaction ◽  
Atomic State ◽  
Wave Functions ◽  
Radial Integral ◽  
Open Shells ◽  
Two Open Shells

We have derived the equations to calculate the electrostatic interaction energy and the coefficients of radial integrals between electrons ln or l1m and l′ of the atomic state |ln[S1L1]l1m[S2L2][ScLc]l′ SL >, where ln[S1L1], l1m[S2L2], and l′ are three open shells. The expressions have been checked against the formulas in the literature by reducing them to those for the case of atoms having two open shells. We demonstrate our formulas by evaluating the coefficients of the radial integrals in the interaction between the 2s or 2p4 and 3p electrons of the 2s2p4(2,4P)3p(3S,3P,3D) state of oxygen. Using these coefficients the wave functions and photoionization cross sections of oxygen 2s has been evaluated and compared with previous results. PACS Nos.: 31.15.Ne, 31.10.+z, 32.80.Fb

Molecular tapes in the structure of isoguaninium chloride

2017 ◽  
Vol 74 (1) ◽  
pp. 108-112 ◽  
Author(s):  
Urszula Anna Budniak ◽  
Paulina Maria Dominiak
Keyword(s):  
Crystal Structure ◽  
Single Crystal ◽  
Charge Density ◽  
Interaction Energy ◽  
Electrostatic Interaction ◽  
Electrostatic Interactions ◽  
The Other ◽  
X Ray Diffraction ◽  
X Ray ◽  
Diffraction Structure

Isoguanine, an analogue of guanine, is of intrinsic interest as a noncanonical nucleobase. The crystal structure of isoguaninium chloride (systematic name: 6-amino-2-oxo-1H,7H-purin-3-ium chloride), C5H6N5O+·Cl−, has been determined by single-crystal X-ray diffraction. Structure analysis was supported by electrostatic interaction energy (E es) calculations based on charge density reconstructed with the UBDB databank. In the structure, two kinds of molecular tapes are observed, one parallel to (010) and the other parallel to (50\overline{4}). The tapes are formed by dimers of isoguaninium cations interacting with chloride anions. E es analysis indicates that cations in one kind of tape are oriented so as to minimize repulsive electrostatic interactions.

A comparison of theory and experiment for photo-ionization cross-sections I. Neon and the elements from boron to neon

1951 ◽  
Vol 208 (1094) ◽  
pp. 408-418 ◽  
Keyword(s):  
Cross Sections ◽  
General Trend ◽  
Wave Functions ◽  
Ionization Cross ◽  
Hartree Fock ◽  
Quantal Theory ◽  
Ionization Cross Sections ◽  
Photo Ionization ◽  
Near Future ◽  
Dipole Length

The quantal theory of the continuous photo-electric absorption of radiation is briefly summarized, pàrticular attention being given to the alternative formulae available and to the accuracy to be expected in practical calculations. Detailed calculations are described for the photo-ionization cross-section of neon, a system for which it is understood that experimental data should be available in the near future. The calculation is made using Hartree-Fock wave functions and the two formulae of the dipole length and the dipole velocity. The corresponding cross-sections are found to be 5.8 and 4.4 x 10- 18 cm 2 . at the spectral head and to rise slowly with increasing frequency until a broad maximum is reached for an energy of the ejected electron of about 11 eV. A comparison is made with previous calculations on the elements from boron to neon ; the general trend of the results is discussed and improved estimates for boron and fluorine are given (10 x 10 -18 cm 2 . for boron and 4.3 x 10- 18 cm 2 . for fluorine at the spectral head).

scholarly journals Quantum electrodynamics with self-conjugated equations with spinor wave functions for fermion fields

2021 ◽  
pp. 2150086
Author(s):  
V. P. Neznamov ◽  
V. E. Shemarulin
Keyword(s):  
Cross Sections ◽  
Magnetic Moment ◽  
Anomalous Magnetic Moment ◽  
Quantum Electrodynamics ◽  
Lamb Shift ◽  
Wave Functions ◽  
Self Energy ◽  
Fermion Fields ◽  
Spinor Wave

Quantum electrodynamics (QED) with self-conjugated equations with spinor wave functions for fermion fields is considered. In the low order of the perturbation theory, matrix elements of some of QED physical processes are calculated. The final results coincide with cross-sections calculated in the standard QED. The self-energy of an electron and amplitudes of processes associated with determination of the anomalous magnetic moment of an electron and Lamb shift are calculated. These results agree with the results in the standard QED. Distinctive feature of the developed theory is the fact that only states with positive energies are present in the intermediate virtual states in the calculations of the electron self-energy, anomalous magnetic moment of an electron and Lamb shift. Besides, in equations, masses of particles and antiparticles have the opposite signs.

Direct Photodisintegration of 9Be in the Low and Sub-Giant Resonance Energy Region

1969 ◽  
Vol 24 (8) ◽  
pp. 1188-1195
Author(s):  
Terje Aurdal
Keyword(s):  
Cross Sections ◽  
Giant Resonance ◽  
Resonance Energy ◽  
Wave Functions ◽  
Nuclear Model ◽  
Final States ◽  
Core State ◽  
Radial Wave ◽  
Total Cross Sections ◽  
Reaction Channels

Abstract Photodisintegration cross sections for the reaction 9Be(γ,n) 8Be with photonenergies varied from threshold to about 17 MeV are calculated. As nuclear model is assumed a single particle shell model where the valence neutron outside the 8Be core is feeling a spherical field. The core state is assumed to be a mixture of the ground (0+) and the first excited (2+) state of the 8Be nucleus. The total cross sections are splitted up according to the different contributing reaction channels. The radial wave functions in initial as well as final states are of the Saxon-Woods type.

Gamow–Teller strength functions from scattering experiments

1987 ◽  
Vol 65 (6) ◽  
pp. 691-698 ◽  
Author(s):  
O. Häusser
Keyword(s):  
Shell Model ◽  
Cross Sections ◽  
Mean Field ◽  
Wave Functions ◽  
Structure Information ◽  
Model Wave ◽  
Spin Flip ◽  
Strength Functions ◽  
Gamow Teller ◽  
Good Agreement

We present here recent [Formula: see text] results from TRIUMF that are relevant to the determination of spin-flip isovector strength functions in nuclei. Distortion factors needed for the extraction of nuclear-structure information have been deduced from cross sections and analyzing powers in elastic scattering for several energies and targets. Nonrelativistic optical potentials obtained by folding effective nucleon (N)–nucleus interactions with nuclear densities are found to overpredict both elastic and reaction cross sections, whereas Dirac calculations that include Pauli blocking are in good agreement with the data. Spin observables (Snn and Ay) for the quasi-elastic region in 54Fe[Formula: see text] at 290 MeV provide some evidence for the reduction of the effective proton mass predicted in relativistic mean-field theories as a consequence of the attractive scalar field in the nuclear medium. The energy dependence of the effective N–nucleus interaction at small momentum transfers has been investigated using isoscalar and isovector 1+ states in 28Si as probe states. We find that the cross sections for the isovector transitions are in good agreement with predictions for the dominant Vστ part of the Franey–Love interaction. Gamow–Teller (GT) strength functions have been obtained in 24Mg and 54Fe from measurements of both cross sections and spin–flip probabilities Snn. The spin-flip cross sections σSnn are particularly useful in heavier nuclei to discriminate against a continuous background of ΔS = 0 excitations. In the (s, d) shell where full shell-model wave functions are available, the GT quenching factors [Formula: see text] are in good agreement with those from recent (p, n) and (n, p) experiments. We show that a state-by-state comparison of (p, p′) and (e, e′) results has the potential of identifying pionic current contributions in (e, e′). The GT quenching factors in 54Fe are smaller than in the (s, d) shell probably because of severely truncated shell-model wave functions, particularly those of the nuclear ground state.

scholarly journals The effect of a nuclear spin on the optical spectra.—III

1930 ◽  
Vol 127 (805) ◽  
pp. 407-416 ◽  
Keyword(s):  
Interaction Energy ◽  
Nuclear Spin ◽  
Energy Levels ◽  
Optical Spectra ◽  
Wave Functions ◽  
Orbital Momentum ◽  
Multiplet Structure ◽  
Multiple Wave ◽  
First Approximation ◽  
Intensity Ratios

In two recent papers the author has discussed the effect of a nuclear spin on the optical spectra by the method of multiple wave-functions. In these papers the interaction energy of the nuclear and electron spins was not taken into account, as has been pointed out by Hill. By its omission the equations were simplified considerably, without affecting the intensity ratios of the lines of the multiplet. The problem of finding the relative intensities is a purely kinematical one, depending as it does, to the first approximation, on the un­perturbed wave-functions. In the papers cited we used the interaction energy of the nuclear spin and orbital momentum to find the 4 i n + 2 wave-functions ( i n being the number of quanta of nuclear spin) which must replace the two wave-functions necessary to describe the electron spin fine structure. In order to describe the multiple energy levels correctly we must calculate the interaction energy of the two spins in addition to the energy increments already calculated in I and II. This is the first purpose of the present paper, and the work is carried out for the cases i n = ½, 1, 1½, 4½. It is found that in the case of the p ½ levels the interaction energy of the two spins is equal to that of the nuclear spin and orbital momentum, while for the p 3/2 levels the ratio is — ⅕. It is further found that the energy levels of the S terms are correctly given in I and II. As regards comparison with Jackson’s results in the case of cæsium, it would seen that, the separation of the p -levels being very small in comparison with that of the S-level, he has been able to observe the multiplet structure of the lines due to the separation of the S-level only. If we make this assumption it will be seen on reference to I that our results agree quite well with his observations.

scholarly journals On the quenching of Gamow–Teller strength

1987 ◽  
Vol 65 (6) ◽  
pp. 574-577 ◽  
Author(s):  
J. Rapaport
Keyword(s):  
Cross Sections ◽  
Wave Functions ◽  
Light Nuclei ◽  
Final States ◽  
Excitation Energies ◽  
Intermediate Energies ◽  
Quenching Factor ◽  
Configuration Mixing ◽  
Gamow Teller ◽  
Model Strength

The (p, n) reaction at intermediate energies has been used to measure differential cross sections in light nuclei to final states characterized with a ΔJπ = 1+ transfer (Gamow–Teller (GT) states). Experimental ft values for allowed beta-decay transitions in these nuclei are used to normalize the strength of the GT transitions in units of B(GT). This experimental GT strength is compared with predicted shell–model strength. For p-shell nuclei, the calculated excitation energies of the GT strength using Cohen and Kurath wave functions are in general agreement with the empirical GT distribution. Up to an excitation energy of about 20 MeV, the total experimental and calculated GT strengths are used to obtain the quenching factor, QF = Σ B(GT)exp/Σ B(GT)theor. It is found that QF decreases as the shell gets filled-up. The lowest value seems to occur for single-hole nuclei. This decrease may be explained by configuration mixing not specifically included in the calculations.

scholarly journals HADRON PRODUCTION IN NEUTRINO–NUCLEON INTERACTIONS AT HIGH ENERGIES

2003 ◽  
Vol 18 (04) ◽  
pp. 673-683
Author(s):  
M. T. HUSSEIN ◽  
N. M. HASSAN ◽  
W. ELHARBI
Keyword(s):  
Cross Sections ◽  
Particle Production ◽  
High Energy ◽  
Neutrino Energy ◽  
Wave Functions ◽  
Variation Method ◽  
Hypothetical Model ◽  
High Energies ◽  
Color Field

The multi-particle productions in neutrino–nucleon collisions at high energy are investigated through the analysis of the data of the experiment CERN-WA-025 at neutrino energy less than 260 GeV and the experiments FNAL-616 and FNAL-701 at energy range 120–250 GeV. The general features of these experiments are used as base to build a hypothetical model that views the reaction through a Feynman diagram of two vertices. The first of which concerns the weak interaction between the neutrino and the quark constituents of the nucleon. At the second vertex, a strong color field is assumed to play the role of particle production, which depend on the momentum transferred from the first vertex. The wave functions of the nucleon constituent quarks are determined using the variation method and relevant boundary conditions are applied to calculate the deep inelastic cross sections of the virtual diagram.

Fast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. III. Application to crystal structures via the Ewald and direct summation methods

2020 ◽  
Vol 76 (6) ◽  
pp. 630-651
Author(s):  
Daniel Nguyen ◽  
Piero Macchi ◽  
Anatoliy Volkov
Keyword(s):  
Interaction Energy ◽  
Crystal Structures ◽  
Electron Density ◽  
Small Molecule ◽  
Electrostatic Interaction ◽  
Multipole Moment ◽  
Interaction Energies ◽  
Acta Cryst ◽  
Löwdin Α Function ◽  
Direct Summation

The previously reported exact potential and multipole moment (EP/MM) method for fast and accurate evaluation of the intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density [Volkov, Koritsanszky & Coppens (2004). Chem. Phys. Lett. 391, 170–175; Nguyen, Kisiel & Volkov (2018). Acta Cryst. A74, 524–536; Nguyen & Volkov (2019). Acta Cryst. A75, 448–464] is extended to the calculation of electrostatic interaction energies in molecular crystals using two newly developed implementations: (i) the Ewald summation (ES), which includes interactions up to the hexadecapolar level and the EP correction to account for short-range electron-density penetration effects, and (ii) the enhanced EP/MM-based direct summation (DS), which at sufficiently large intermolecular separations replaces the atomic multipole moment approximation to the electrostatic energy with that based on the molecular multipole moments. As in the previous study [Nguyen, Kisiel & Volkov (2018). Acta Cryst. A74, 524–536], the EP electron repulsion integral is evaluated analytically using the Löwdin α-function approach. The resulting techniques, incorporated in the XDPROP module of the software package XD2016, have been tested on several small-molecule crystal systems (benzene, L-dopa, paracetamol, amino acids etc.) and the crystal structure of a 181-atom decapeptide molecule (Z = 4) using electron densities constructed via the University at Buffalo Aspherical Pseudoatom Databank [Volkov, Li, Koritsanszky & Coppens (2004). J. Phys. Chem. A, 108, 4283–4300]. Using a 2015 2.8 GHz Intel Xeon E3-1505M v5 computer processor, a 64-bit implementation of the Löwdin α-function and one of the higher optimization levels in the GNU Fortran compiler, the ES method evaluates the electrostatic interaction energy with a numerical precision of at least 10−5 kJ mol−1 in under 6 s for any of the tested small-molecule crystal structures, and in 48.5 s for the decapeptide structure. The DS approach is competitive in terms of precision and speed with the ES technique only for crystal structures of small molecules that do not carry a large molecular dipole moment. The electron-density penetration effects, correctly accounted for by the two described methods, contribute 28–64% to the total electrostatic interaction energy in the examined systems, and thus cannot be neglected.

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