Theoretical studies in nuclear structure IV. Wave functions for the nuclear p -shell Part A. ‹ p n │ p n-1 p › fractional parentage coefficients

1951 ◽  
Vol 209 (1099) ◽  
pp. 502-524 ◽  
Keyword(s):  
Direct Method ◽  
Wave Functions ◽  
Irreducible Representations ◽  
Reciprocal Relation ◽  
Spin Coefficient ◽  
Linear Combinations ◽  
Orthogonal Representation ◽  
Fractional Parentage ◽  
Complete Set ◽  
A Charge

A complete set of wave functions is constructed for the whole of the nuclear p -shell (from p 3 to p 12 ). Following Racah, the wave functions for p n are expressed as linear combinations of totally antisymmetric wave functions for p n-1 , vector-coupled to the wave functions of the remaining particle. The coefficients in the linear combination are expressed as the product of an orbital coefficient, a charge-spin- coefficient and a weight factor equal to the square root of the ratio of the dimensions of two irreducible representations of permutation groups. Using the Young-Yamanouchi orthogonal representation of the permutation group, the orbital and charge-spin coefficients may be calculated independently. Specialization of the new method to the atomic p -shell and an alternative direct method of calculating the total parentage coefficients are described in the appendices. A reciprocal relation for the special unitary group, simplifying the calculation of both the orbital and the charge-spin coefficients, is described in an Addendum.

Studies in jj -coupling - II. Fractional parentage coefficients and the central force energy matrix for equivalent particles

1952 ◽  
Vol 214 (1119) ◽  
pp. 515-530 ◽  
Keyword(s):  
Invariant Theory ◽  
Short Range ◽  
Wave Functions ◽  
Central Force ◽  
Symplectic Groups ◽  
Energy Matrix ◽  
Linear Combinations ◽  
Special Transformation ◽  
Fractional Parentage

Jahn’s method of fractional parentage coefficients is adapted to obtain fully antisymmetric wave functions for the configurations j n of neutrons and protons, where j = 3/2, 5/2, 7/2, and n = 3 and 4. Invariant theory is used to obtain linear combinations of Slater integrals which have special transformation properties with respect to the unitary and symplectic groups. In this way it is shown how the ordering of levels in jj -coupling with short-range central inter­actions is determined by the eigenvalues of Casimir’s operator.

Theoretical studies in nuclear structure IV. wave functions for the nuclear p -shell. Part B. ( p n | p n-2< gt;

1953 ◽  
Vol 246 (912) ◽  
pp. 241-279 ◽  
Keyword(s):  
Nuclear Structure ◽  
Unitary Group ◽  
Wave Functions ◽  
Central Force ◽  
Theoretical Studies ◽  
Spin Orbit ◽  
Linear Combinations ◽  
Body Interaction ◽  
Fractional Parentage ◽  
Shell Part

The totally antisymmetric states for the p -shell nuclei previously given (part IV A, Jahn & van Wieringen 1951) arc transformed into the form of linear combinations of antisymmetric states for n - 2 particles vector-coupled to antisymmetric states of particles n and n - 1. The coefficients of the combinations, the ( n | n -2, 2) fractional parentage coefficients, are shown to be products of weight factors, orbital factors and charge-spin factors which are tabulated separately, the tabulation being simplified by use of the special unitary group. This form of the wave functions is the most suitable for the evaluation of any two-body interaction and has been used in a series of investigations (J. P. Elliott 1952, 1953; Robinson 1953) relating to the effect of tensor and spin-orbit forces on the levels of p -shell nuclei. The complete p -shell charge-symmetric central force matrix given by Racah (1950) with unspecified phases is recalculated with the standard (amended) phases of part IV A, with the help of the coefficients here tabulated.

scholarly journals SCATTERING, GREEN’S FUNCTIONS AND SYMMETRIES IN A TOTAL SPACE OF A DIRAC MONOPOLE FIBRE BUNDLE

1991 ◽  
Vol 06 (04) ◽  
pp. 577-598 ◽  
Author(s):  
A.G. SAVINKOV ◽  
A.B. RYZHOV
Keyword(s):  
Green's Functions ◽  
Principal Bundle ◽  
Fibre Bundle ◽  
Green’S Functions ◽  
Wave Functions ◽  
Total Space ◽  
Dirac Monopole ◽  
Hidden Symmetries ◽  
Scattering Wave ◽  
A Charge

The scattering wave functions and Green’s functions were found in a total space of a Dirac monopole principal bundle. Also, hidden symmetries of a charge-Dirac monopole system and those joining the states relating to different topological charges n=2eg were found.

MATRIX UNITS FOR THE GROUP ALGEBRA kGf = k((ℤ2 × ℤ2) ≀ Sf)

2009 ◽  
Vol 02 (02) ◽  
pp. 255-277
Author(s):  
B. Sivakumar
Keyword(s):  
Group Algebra ◽  
Irreducible Representations ◽  
Complete Set ◽  
Matrix Units

The irreducible representations of the group Gf := (ℤ2 × ℤ2) ≀ Sf are indexed by 4-partitions of f, i.e., by the set {[α]3[β]2[γ]1[δ]0|α ⊢ u3, β ⊢ u2, γ ⊢ u1, δ ⊢ u0, u0 + u1 + u2 + u3 = f}. This set is in 1 - 1 correspondence with partitions of 4f whose 4-core is empty. In this paper we construct the inequivalent irreducible representations of Gf. We also compute a complete set of seminormal matrix units for the group algebra kGf.

On particle-like representations of the complete homogeneous Lorentz group

1973 ◽  
Vol 74 (1) ◽  
pp. 149-160 ◽  
Author(s):  
J. A. de Wet
Keyword(s):  
Lorentz Group ◽  
Wave Functions ◽  
Unitary Groups ◽  
Irreducible Representations ◽  
Electron Wave ◽  
Precise Form ◽  
Homogeneous Lorentz Group ◽  
The Many ◽  
Matrix Contraction

In two previous papers (1, 2) representations of the unitary groups U4, U2 were found which described some of the properties of nucleons and electrons. In particular, the many electron wave functions were constructed from the irreducible representations of U2 restricted to the proper orthochronous Lorentz group Lp. In this paper the irreducible representations of U4 found in (1) will be shown to be also irreducible representations of the complete homogeneous Lorentz group L0 and the techniques of matrix contraction employed in (2) will be used to find the precise form of the matrices of the infinitesimal ring.

scholarly journals On the ionisation of gases by collision and the ionising potential for positive ions and negative corpuscles

1914 ◽  
Vol 90 (620) ◽  
pp. 398-411 ◽  
Keyword(s):  
Direct Method ◽  
Minimum Energy ◽  
Thin Foil ◽  
Experimental Conditions ◽  
Proper Position ◽  
Spherical Form ◽  
Positive Ions ◽  
Ionisation Chamber ◽  
Glass Tubes ◽  
A Charge

The object of the present research was to investigate the ionising properties of positive ions using a direct method, and to try experimentally to determine the minimum energy which is necessary for a positive ion to produce fresh ions by collision with a gas molecule. At the same time it appeared to be of interest to measure the ionising potential for negative corpuscles under experimental conditions somewhat different from those previously adopted. 1. Description of the Apparatus . The apparatus in the present investigation was used in two forms, represented in figs. 1 and 2. The part common to both forms consists of two pieces of glass tubing, A and B - A, 3·5 cm. in diameter, B, a little less the latter being blown out to a slightly spherical form, as shown in the diagram. The joint at C was made air-tight with sealing wax. By this means it was possible to take out the lower part very easily a process which was very often necessary in order to renew the platinum strip P, or to cover it with a fresh salt. In the narrow upper part of the tube A a brass rod was sealed with a round brass plate E at its end. This rod was insulated and surrounded by a brass tube connected to earth, to prevent it acquiring a charge from the glass. The plate E was used as an electrode to measure the ionisation, and was connected to a Wilson's tilted electroscope adjusted to a sensitiveness of 80 divisions per volt. The ionisation chamber was a wire-gauze cylinder D, through the open top of which passed the electrode E. The sides of the cylinder were made of brass gauze and the bottom of platinum gauze, which was fixed in clamps at the end of a brass rod which passed through a side tube and which held the cylinder in its proper position. The platinum strip P was of thin foil, 2 cm. long and 1 mm. wide. It could be heated electrically by means of the thick copper leads G and H, the current being supplied from a battery of six accumulator cells connected in parallel and regulated by wire resistances. The heating circuit was insulated. The copper leads G and H were covered with glass in order to make them more rigid and to keep them cool. These glass tubes and the leads themselves were sealed into two tubes joined to the spherical part of the lower tube. The tube B could be adjusted at the joint C, so as to fix the strip P at any distance from the bottom of the cylinder D.

Bestimmung der Kraftkonstanten der Hexafluoro-Komplexe des Technetium(IV) und Rhenium(IV) auf der Basis der D4h-Symmetrie

1970 ◽  
Vol 25 (2) ◽  
pp. 206-210 ◽  
Author(s):  
Wolfgang Krasser ◽  
Klaus Schwochau
Keyword(s):  
Force Field ◽  
Infrared Spectra ◽  
Force Constants ◽  
Irreducible Representations ◽  
Octahedral Complex ◽  
Site Symmetry ◽  
Complex Ions ◽  
Linear Combinations ◽  
Valence Force Field ◽  
Stretching Vibrations

Abstract The infrared spectra of potassium and cesium hexafluorotechnetate (IV) and -rhenate(IV) show a D4h-distortion of the octahedral complex ions. The splitting of degeneracies is not due to site symmetry. The linear combinations of the force constants of the irreducible representations A2u and Eu are determined in the valence force field on the basis of D4h-symmetry. As expected the force constants of stretching vibrations of [TcF6]-- are then smaller than those of [ReF6]--.

scholarly journals Induced vacuum bosonic current in a compactified cosmic string spacetime

2016 ◽  
Vol 41 ◽  
pp. 1660117 ◽  
Author(s):  
E. A. F. Bragança ◽  
H. F. Santana Mota ◽  
E. R. Bezerra de Mello
Keyword(s):  
Magnetic Flux ◽  
Cosmic String ◽  
Wave Functions ◽  
Complete Set ◽  
Current Densities

We analyze the bosonic current densities induced by a magnetic flux running along an idealized cosmic string considering that the coordinate along its axis is compactified. We also consider the presence of a magnetic flux enclosed by the compactificatified axis. To develop this analysis, we calculate the complete set of normalized bosonic wave functions obeying a quasiperiodicity condition along the compactified dimension. We show that in this context only the azimuthal and axial currents take place.

scholarly journals HANDLE OPERATORS OF COSET MODELS

1993 ◽  
Vol 08 (20) ◽  
pp. 1877-1889
Author(s):  
MICHAEL CRESCIMANNO
Keyword(s):  
Fixed Points ◽  
Irreducible Representations ◽  
Simons Theory ◽  
Simple Application ◽  
Linear Combinations ◽  
Explicit Formulae ◽  
Genus One ◽  
Coset Models ◽  
Chern Simons ◽  
Chern Simons Theory

Several interesting features of coset models “without fixed points” are easily understood via Chern-Simons theory. In this paper we derive explicit formulae for the handle-squashing operator in these cosets. These operators are fixed, linear combinations of the irreducible representations of the coset. As a simple application of these curious formulae, we compute the traces of all genus-one operators for several common cosets.

Determination of meloxicam using direct and indirect flow injection spectrophtometry

2020 ◽  
Vol 16 ◽  
Author(s):  
Ream Ismail Abed ◽  
Hind Hadi
Keyword(s):  
Flow Injection ◽  
Direct Method ◽  
Pharmaceutical Formulations ◽  
Physical Parameters ◽  
Official Method ◽  
Charge Transfer Reaction ◽  
Spectrophotometric Methods ◽  
Significant Difference ◽  
Key Enzyme ◽  
A Charge

Background: Meloxicam is a non-steroidal anti-inflammatory drug, known to inhibit cyclooxygenase, a key enzyme of prostaglandin biosynthesis at the site of inflammation. Objectives: Two sensitive and fast spectrophotometric methods (direct and indirect) using flow injection analysis (FIA) have been described for the assay of meloxicam in pure and in pharmaceutical formulations. Methods: The direct method (method A) depended on the coupling of meloxicam with diazotized procaine benzylpenicillin in alkaline medium, while the indirect method (method B) involved a charge transfer reaction between alkaline hydrolytic product of meloxicam as n-donor with metol (p-methylaminophenol sulfate) as a π- acceptor using sodium periodate as an oxidant. Results: The colored products exhibited absorption maxima at 492 and 656 nm for method A and B respectively. The methods obeyed Beer’s low over concentration ranges of 5-80 and 15-225 µg/mL, respectively. The detection limits were 2.73 and 5.26 µg/mL for direct and indirect methods, respectively. The chemical and physical parameters associated with FIA manifolds were optimized to construct the most sensitive and reproducible results. Conclusion: Methods described herein are cheap, environmental friendly and showed no significant difference with those of official method indicated no significant difference in accuracy and precision.

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