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.

Matrix Elements of the Spin–Orbit Coupling for an ln Configuration in a Crystalline Electric Field

1972 ◽  
Vol 50 (2) ◽  
pp. 78-83 ◽  
Author(s):  
H. A. Buckmaster ◽  
R. Chatterjee ◽  
Y. H. Shing
Keyword(s):  
Electric Field ◽  
General Expression ◽  
Ground State ◽  
Orbit Coupling ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Crystalline Electric Field ◽  
State Splitting ◽  
The Matrix

A general expression for the matrix elements of the spin–orbit coupling for an ln configuration in a crystalline electric field of arbitrary symmetry is derived using Racah formalism. This calculation is an extension of Lulek's treatment of this problem for an l1 configuration. This general expression is used to calculate the contribution to the ground-state splitting for the S-state lanthanide ion Gd3+ (4f7; 8S7/2)in an axial crystalline electric field of a second-order perturbation mechanism involving the matrix element of the spin–orbit coupling. It is shown that this mechanism, which was proposed by Lulek is incapable of explaining the observed ground-state splitting.

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.

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.

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.

scholarly journals Dirac equations with confining potentials

2015 ◽  
Vol 30 (01) ◽  
pp. 1550002 ◽  
Author(s):  
J. H. Noble ◽  
U. D. Jentschura
Keyword(s):  
Bound State ◽  
Negative Energy ◽  
Low Energy ◽  
Spin Orbit Coupling ◽  
Matrix Elements ◽  
Dirac Equations ◽  
Dirac Particles ◽  
Negative Energy States ◽  
The Matrix ◽  
Effective Operators

This paper is devoted to a study of relativistic eigenstates of Dirac particles which are simultaneously bound by a static Coulomb potential and added linear confining potentials. Under certain conditions, despite the addition of radially symmetric, linear confining potentials, specific bound-state energies surprisingly preserve their exact Dirac–Coulomb values. The generality of the "preservation mechanism" is investigated. To this end, a Foldy–Wouthuysen transformation is used to calculate the corrections to the spin-orbit coupling induced by the linear confining potentials. We find that the matrix elements of the effective operators obtained from the scalar, and time-like confining potentials mutually cancel for specific ratios of the prefactors of the effective operators, which must be tailored to the preservation mechanism. The result of the Foldy–Wouthuysen transformation is used to verify that the preservation is restricted (for a given Hamiltonian) to only one reference state, rather than traceable to a more general relationship among the obtained effective low-energy operators. The results derived from the nonrelativistic effective operators are compared to the fully relativistic radial Dirac equations. Furthermore, we show that the preservation mechanism does not affect antiparticle (negative-energy) states.

scholarly journals Antiferromagnetism and crystalline electric field excitations in tetragonal NaCeO2

2021 ◽  
Vol 103 (2) ◽  
Author(s):  
Mitchell M. Bordelon ◽  
Joshua D. Bocarsly ◽  
Lorenzo Posthuma ◽  
Arnab Banerjee ◽  
Qiang Zhang ◽  
...  
Keyword(s):  
Electric Field ◽  
Crystalline Electric Field

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

Antiferromagnetism and Crystalline Electric Field Effect of Ce2Pt3Si5

2008 ◽  
Vol 77 (12) ◽  
pp. 124707 ◽  
Author(s):  
Yuji Muro ◽  
Masayuki Nakano ◽  
Kiyoichiro Motoya
Keyword(s):  
Electric Field ◽  
Field Effect ◽  
Electric Field Effect ◽  
Crystalline Electric Field
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