Parity, time, charge and Hermitian conjugations of Racah tensor operators

1983 ◽  
Vol 61 (12) ◽  
pp. 1613-1617 ◽  
Author(s):  
R. Chatterjee ◽  
J. A. Tuszyński ◽  
H. A. Buckmaster
Keyword(s):  
Magnetic Properties ◽  
Electric Properties ◽  
Tensor Operator ◽  
Matrix Elements ◽  
Selection Rules ◽  
Angular Momenta ◽  
The Matrix ◽  
Polar Tensor ◽  
The Relationship ◽  
Tensor Operators

The relationship between the parity P, time θ, charge C, and Hermitian h conjugation operators and the irreducible Racah tensor operators is reexamined. Polar tensor operators (describing electric properties) are distinguished from axial tensor operators (describing magnetic properties and angular momenta) on the basis of their individual parity and time conjugation properties. However, the effect of the Pθ product conjugation is identical for both classes and for even rank is equivalent to the Racah definition for the Hermitian conjugation of a tensor operator. It is shown that this property separates the Racah tensor operators from other vector quantities like linear momentum which cannot be represented by such operators. The selection rules due to parity and time conjugation and Hermitian conjugation that arise in the calculation of the matrix elements of the tensor operators and their products are then obtained self-consistently using the Wigner–Eckart theorem.

Zero-field splitting calculations. Part II. Tabulation of the reduced matrix elements of the irreducible double tensor operators

1976 ◽  
Vol 54 (12) ◽  
pp. 1228-1233 ◽  
Author(s):  
R. Chatterjee ◽  
M. R. Smith ◽  
H. A. Buckmaster
Keyword(s):  
The Other ◽  
Tensor Operator ◽  
Zero Field ◽  
Matrix Elements ◽  
Zero Field Splitting ◽  
Field Splitting ◽  
The Matrix ◽  
The Many ◽  
Relationship Of ◽  
The Relationship

The many-electron reduced matrix elements of the double tensor operator [Formula: see text] are tabulated for the pn, dn and fn configurations. These tables have been calculated from the Racah formalism and form an extension of the tabulations given by Nielson and Koster for K2 = 1. The relationship of this tensor with the other double tensors and the calculation of the matrix elements is summarized. See Chatterjee et al for an application.

New relativistic crystalline electric field effective operators

1989 ◽  
Vol 67 (6) ◽  
pp. 632-633 ◽  
Author(s):  
R. Chatterjee ◽  
H. A. Buckmaster
Keyword(s):  
Electric Field ◽  
Tensor Operator ◽  
Operator Technique ◽  
Matrix Elements ◽  
Effective Operator ◽  
Crystalline Electric Field ◽  
The Matrix ◽  
Definition Of ◽  
Effective Operators

This paper shows that the use of the Landau and Lifshitz definition of the Hermitian adjoint of a tensor operator combined with the triangular rule leads to new terms in the relativistic "effective operator technique" formulation of a crystalline electric field, and that the matrix elements of these terms are significant for S-state ions.

Magnetic hyperfine interactions in the electronic spectra of diatomic molecules II. Magnetic interactions in Hund’s case ( c ) and the spectram of BiO

1967 ◽  
Vol 300 (1463) ◽  
pp. 487-495 ◽  
Keyword(s):  
Hyperfine Interaction ◽  
Diatomic Molecules ◽  
Hyperfine Interactions ◽  
Electric Quadrupole ◽  
Magnetic Interactions ◽  
Magnetic Hyperfine ◽  
Matrix Elements ◽  
Magnetic Hyperfine Interaction ◽  
Angular Momenta ◽  
The Matrix

The magnetic hyperfine interaction in Hund’s case ( c ) diatomic molecules is investigated by the method of spherical tensors and time-reversed angular momenta. Explicit expressions for the matrix elements are given, so too are expressions for the first and second order energies for an |Ω| = 1/2 state. The equations are applied to the electronic spectrum of BiO (Barrow, Gissane & Richards 1967) and the observed line widths are accounted for in terms of a magnetic hyperfine interaction with the 209 Bi nucleus. The electric quadrupole hyperfine interaction in BiO is also investigated and shown to be incapable of accounting for the observed effects.

MATRIX ELEMENT AND COMPLEX l PLANE EVALUATION OF TWO-LOOP VACUUM AMPLITUDES IN QED ON S4

1995 ◽  
Vol 10 (09) ◽  
pp. 1281-1327 ◽  
Author(s):  
B.A. HARRIS ◽  
G.C. JOSHI
Keyword(s):  
Field Theory ◽  
Angular Momentum ◽  
Matrix Element ◽  
Free Field ◽  
Dimensional Regularization ◽  
Analytic Form ◽  
Selection Rules ◽  
Angular Momenta ◽  
Abelian Field ◽  
The Matrix

In this paper we further develop our matrix element and complex angular momentum summation techniques, in order to calculate both the one-loop free field and two-loop interacting vacuum diagrams in field theory on a four-sphere. In the case of the free field diagrams, we show how the sums may be evaluated by integrating over an analytic function with both poles and branch cuts where the discontinuity across the cuts determines the result. We then extend the matrix element formalism to multiple angular momenta involving the addition of angular momenta and the associated Clebsch-Gordon type selection rules. This then allows us to evaluate the matrix elements of two-loop diagrams in spherical QED as a function of the three angular momenta in the diagram. The selection rules allow us to cast the triple angular momentum sum into a form which enables evaluation again by contour integration. The result is obtained in analytic form using dimensional regularization for the previously obtained spinor case, and also for scalar QED, which we believe is a new result. Finally, we discuss the applicability of this method for calculations in non-Abelian field theory which we believe cannot be performed using earlier methods.

The matrix elements of tensor operators for configurations of three equivalent electrons

1959 ◽  
Vol 250 (1263) ◽  
pp. 562-574 ◽  
Keyword(s):  
Irreducible Representations ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Quantum Numbers ◽  
Fractional Parentage ◽  
The Matrix ◽  
Tensor Operators

The formulae of Redmond are used to construct expressions for the fractional parentage coefficients relating the configurations l 3 and l 2 . The explicit occurrence of godparent states is avoided for the quartet states of f 3 and also for a sequence of doublet states. The latter are defined by the set of quantum numbers f 3 WUSLJJ 2 , where W and U are irreducible representations of the groups R 7 and G 2 . Matrix elements of the type ( f 3 WUSL || U k || f 3 W'U'SL' ), where U k is the sum of the three irreducible tensor operators u k corresponding to the three f electrons, are tabulated for k = 2, 4 and 6 and for all values of W, U, S and L .

scholarly journals Electroelastic Field Concentration on Elliptical Pores in Textured Media

1996 ◽  
Vol 25 (2-4) ◽  
pp. 237-244 ◽  
Author(s):  
T. D. Shermergor ◽  
V. B. Yakovlev
Keyword(s):  
Random Fields ◽  
Piezoelectric Properties ◽  
Tensor Operator ◽  
Pore Shape ◽  
Electroelastic Field ◽  
Crystalline Inclusions ◽  
The Matrix ◽  
Tensor Operators

A method of calculating the tensor operator of the electroelastic field concentration in the vicinity of two kinds of inhomogeneities (pores and crystalline inclusions) is developed. It is assumed that the shape of inhomogeneities is an ellipsoid. The materials of the matrix and/or inclusions have piezoelectric properties. Effective piezoelectric properties of textured polycrystalline media are calculated by means of the general singularity approximation of the theory of random fields. Methods for the analysis of the operators of electroelastic field concentration on the surface of the inclusion are used. The influence of the ellipsoidal pore shape on the tensor operators of electroelastic field concentration is considered. Data are shown in figures.

CHIRALITY-ASYMMETRY MACROSCOPIC FORCE INDUCED BY AXION BETWEEN THE α-QUARTZ AND COPPER BLOCK

2012 ◽  
Vol 26 (06) ◽  
pp. 1150035
Author(s):  
YONGHONG HU ◽  
YUNYI WU
Keyword(s):  
Effective Potential ◽  
Quartz Crystal ◽  
Coupling Constants ◽  
Matrix Elements ◽  
Copper Block ◽  
Compton Wavelength ◽  
Selection Rules ◽  
Space Group ◽  
Valence Electrons ◽  
The Matrix

A macroscopic force induced by the effective potential suggested by Moody and Wilczek between α-quartz crystal and copper block is studied in detail. The matrix elements in the force formula are analyzed according to representations of the non-symmorphic space group P 3121 of α-quartz crystal. The asymmetry distribution of valence electrons and the selection rules do not cancel the force. We conclude that there exists a CP-violating macroscopic force. According to the limit of the production of the axion coupling constants gs gp/ℏc ≤ 10-26 at the Compton wavelength λ = 10-3 m, the macroscopic force between a 0.08 × 0.08 × 0.02 m3 block of α-quartz and a 0.08 × 0.08× 0.01 m3 copper block with a separation being 0.5 × 10-3 m in between, is estimated at less than 3.09 × 10-26 N.

scholarly journals Colour?Spin Matrix Elements for Multiquark Systems

1980 ◽  
Vol 33 (6) ◽  
pp. 951 ◽  
Author(s):  
RP Bickerstaff ◽  
BG Wybourne
Keyword(s):  
Classification Scheme ◽  
Spin Operator ◽  
Tensor Operator ◽  
Matrix Elements ◽  
S Wave ◽  
Strange Quarks ◽  
Colour Singlet ◽  
Singlet States ◽  
The Matrix ◽  
Relevant Interaction

Tensor operator methods have been developed for calculating the matrix elements of the two-particle colour-spin operator that arises in the calculation of the quark-gluon interaction in the MIT bagmodel treatment of the S-wave colour singlet states of multiquark hadrons. A group classification scheme for multi quark states which distinguishes the nonstrange and strange quarks, and thus avoids the occurrence of hidden strangeness ss pairs, is constructed. This scheme has the added advantage of avoiding any need to approximate the strangeness dependence of the relevant interaction integrals. The colour-spin matrix elements for all the q4 if colour singlet states and for the strangeness - 2 states of q6 are given by way of examples. A number of checking procedures have been developed to ensure the correctness of the calculated matrix elements.

A Study of Normalized Population Diversity in Particle Swarm Optimization

2013 ◽  
Vol 4 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Shi Cheng ◽  
Yuhui Shi ◽  
Quande Qin
Keyword(s):  
Particle Swarm Optimization ◽  
Particle Swarm ◽  
Pso Algorithm ◽  
Population Diversity ◽  
Matrix Elements ◽  
Swarm Optimization ◽  
Multimodal Function ◽  
The Matrix ◽  
The Relationship ◽  
Solution Matrix

The values and velocities of a Particle swarm optimization (PSO) algorithm can be recorded as series of matrix and its population diversity can be considered as an observation of the distribution of matrix elements. Each dimension is measured separately in the dimension-wise diversity, on the contrary, the element-wise diversity measures all dimension together. In this paper, PSO algorithm is first represented in the matrix format, then based on the analysis of the relationship between pairs of vectors in PSO solution matrix, different normalization strategies are utilized for dimension-wise and element-wise population diversity, respectively. Experiments on benchmark functions are conducted. Based on the simulation results of ten benchmark functions (include unimodal/multimodal function, separable/non-separable function), the properties of normalized population diversities are analyzed and discussed.

scholarly journals Crystal Field Effects in S-state Ions Due to Exchange Polarization

1976 ◽  
Vol 29 (3) ◽  
pp. 177 ◽  
Author(s):  
DJ Newman
Keyword(s):  
Crystal Field ◽  
Electron Correlation ◽  
Ground State ◽  
Matrix Elements ◽  
Open Shells ◽  
The Matrix ◽  
New Type ◽  
Relationship Of ◽  
The Relationship ◽  
Exchange Polarization

It is shown that the matrix elements of the tensor operators describing a new type of crystal field can be calculated using the properties of the spin ~ quasi-spin transformation. The relationship of this field to electron correlation in open shells is clarified, and its contribution to the ground state splittings of S-state ions is discussed

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