On the symmetries of spherical harmonics

1957 ◽  
Vol 53 (2) ◽  
pp. 343-367 ◽  
Author(s):  
S. L. Altmann
Keyword(s):  
Irreducible Representation ◽  
Band Structure ◽  
Symmetry Group ◽  
Spherical Harmonics ◽  
Lower Order ◽  
Central Atom ◽  
Real Form ◽  
Cellular Method ◽  
Hybrid Orbitals

It is often necessary to obtain expansions in spherical harmonics that belong to a given irreducible representation of a symmetry group. This is the case, for instance, when the cellular method is applied to investigate the band structure of a metal, where expansions are required that reproduce the symmetry of the group of the k vector (see Bouckaert, Smoluchowski and Wigner(4); von der Lage and Bethe (9)). Another instance where such expansions are necessary appears when hybrid orbitals are obtained for a central atom in a molecule of given symmetry (Kimball (8)). In this case lower order spherical harmonics are considered and tables for them up to l = 2 (functions s, p and d in real form) are given in the literature (cf. for example Eyring, Walter and Kimball (5)). However, interest has recently arisen in hybrids that include f functions (Shirmazan and Dyatkina (12)) and an extension of these tables appears to be desirable.

Band-structure calculations of BN by the self-consistent variational cellular method

1990 ◽  
Vol 41 (3) ◽  
pp. 1691-1694 ◽  
Author(s):  
E. K. Takahashi ◽  
A. T. Lino ◽  
A. C. Ferraz ◽  
J. R. Leite
Keyword(s):  
Band Structure ◽  
The Self ◽  
Band Structure Calculations ◽  
Self Consistent ◽  
Cellular Method ◽  
Structure Calculations

The cellular method for a close-packed hexagonal lattice I. Theory

1958 ◽  
Vol 244 (1237) ◽  
pp. 141-152 ◽  
Keyword(s):  
General Theory ◽  
Spherical Harmonics ◽  
Hexagonal Lattice ◽  
Unit Cell ◽  
Irreducible Representations ◽  
Space Group ◽  
Different Types ◽  
Group A ◽  
Cellular Method

The general theory of the irreducible representations of a space group with two atoms per unit cell is discussed. A particular application of it is made for the group of k = 0 for the close-packed hexagonal lattice. This leads to the determination of the spherical harmonics with the symmetry of this group. A technique is described to determine the boundary and continuity conditions on the surface of the Wigner-Seitz polyhedron. It is pointed out that these vary for different types of points on this surface and complete tables for them are given.

scholarly journals Spherical HarmonicsYlm(θ,ϕ): Positive and Negative Integer Representations ofsu(1,1)forl-mandl+m

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
H. Fakhri
Keyword(s):  
Angular Momentum ◽  
Irreducible Representation ◽  
Lie Algebra ◽  
Spherical Harmonics ◽  
Compact Manifold ◽  
Unitary Irreducible Representation ◽  
Irreducible Representations ◽  
Angular Momentum Operator ◽  
Quantum Numbers ◽  
Azimuthal Quantum Number

The azimuthal and magnetic quantum numbers of spherical harmonicsYlm(θ,ϕ)describe quantization corresponding to the magnitude andz-component of angular momentum operator in the framework of realization ofsu(2)Lie algebra symmetry. The azimuthal quantum numberlallocates to itself an additional ladder symmetry by the operators which are written in terms ofl. Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative(l-m)- and(l+m)-integer discrete irreducible representations forsu(1,1)Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation ofsu(2)compact Lie algebra via theYlm(θ,ϕ)’s for a givenl, we can also representsu(1,1)noncompact Lie algebra by spherical harmonics for given values ofl-mandl+m.

On the symmetries of spherical harmonics

1963 ◽  
Vol 255 (1054) ◽  
pp. 199-215 ◽  
Keyword(s):  
Irreducible Representation ◽  
Spherical Harmonics ◽  
Irreducible Representations ◽  
Linear Combinations ◽  
Point Groups

This paper extends the work described in a previous paper by one of the authors (Altmann 1957). The spherical harmonics that belong to the irreducible representations of the cubic groups are now given up to and including l = 12. Also, for all point groups the expansions in spherical harmonics that are given belong to the separate columns of the irreducible representations (whereas before they were linear combinations of such functions). Accordingly, full tables for the irreducible representations for all crystallographic point groups are required and are given in the paper. Finally, a technique is described, and used throughout in the tables, to orthogonalize several expansions that belong to the same column of the same irreducible representation. Therefore, the different expansions listed in the tables are always fully orthogonal.

scholarly journals A Hot-Hole Transport Model Based on Spherical Harmonics Expansion of the Anisotropic Bandstructure

VLSI Design ◽  
1998 ◽  
Vol 6 (1-4) ◽  
pp. 205-208 ◽  
Author(s):  
H. Kosina ◽  
M. Harrer
Keyword(s):  
Monte Carlo ◽  
Band Structure ◽  
Spherical Harmonics ◽  
Transport Model ◽  
Hole Transport ◽  
Brillouin Zone Boundary ◽  
Hole Energy ◽  
System A ◽  
Expansion Technique ◽  
Valence Bands

To represent the valence bands of cubic semiconductors a coordinate transformation is proposed such that the hole energy becomes an independent variable. This choice considerably simplifies the evaluation of the integrated scattering probability and the choice of the state after scattering in a Monte Carlo procedure. In the new coordinate system, a numerically given band structure is expanded into a series of spherical harmonics. This expansion technique is capable of resolving details of the band structure at the Brillouin zone boundary and hence can span an energy range of several electron-volts. Results of a Monte Carlo simulation employing the new band representation are shown.

scholarly journals Seeking SUSY fixed points in the 4 − ϵ expansion

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Pedro Liendo ◽  
Junchen Rong
Keyword(s):  
Fixed Point ◽  
Irreducible Representation ◽  
Fixed Points ◽  
Symmetry Group ◽  
Potts Model ◽  
Arbitrary Integer ◽  
New Family ◽  
Finite Subgroups

Abstract We use the 4 − ϵ expansion to search for fixed points corresponding to 2 + 1 dimensional $$ \mathcal{N} $$ N =1 Wess-Zumino models of NΦ scalar superfields interacting through a cubic superpotential. In the NΦ = 3 case we classify all SUSY fixed points that are perturbatively unitary. In the NΦ = 4 and NΦ = 5 cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group (irreducible fixed points). For NΦ = 4 we show that the S5 invariant super Potts model is the only irreducible fixed point where the four scalar superfields are fully interacting. For NΦ = 5, we go through all Lie subgroups of O(5) and use the GAP system for computational discrete algebra to study finite subgroups of O(5) up to order 800. This analysis gives us three fully interacting irreducible fixed points. Of particular interest is a subgroup of O(5) that exhibits O(3)/Z2 symmetry. It turns out this fixed point can be generalized to a new family of models, with NΦ = $$ \frac{\mathrm{N}\left(\mathrm{N}-1\right)}{2} $$ N N − 1 2 − 1 and O(N)/Z2 symmetry, that exists for arbitrary integer N≥3.

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