Traceless cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media

Applequist, Jon

doi:10.1088/0305-4470/22/20/011

Cited by 79 publications

(113 citation statements)

References 9 publications

Supporting

Mentioning

112

Contrasting

Order By: Relevance

“…One can easily show that △R (n) vanishes with the use of r ·∇ n+1 r −1 = −(n + 1)∇ n r −1 [4]. Therefore, from (10) and (14), we finally obtain the following expression:…”

Section: Symmetric Traceless Tensor Casementioning

confidence: 94%

“…The nth (2 n -pole) Cartesian moment of volume/surface charge density with respect to the origin in V is defined as r n ρ v/s = r n−1 q 1v/s , where q 1v/s = rρ v/s is the volume/surface charge-induced dipole density, and r n = n rr · · · r is a Cartesian tensor of rank n (≥ 1) [4]. On the other hand, the nth Cartesian moment of volume/surface current density is defined as r n × j v/s = r n−1 q 2v/s [6], where q 2v/s = r × j v/s is termed the volume/surface current-induced dipole density.…”

Section: Definitionsmentioning

confidence: 99%

“…A symmetric tensor A (n) of rank n (≥ 1) is rendered traceless by the detracer operator D n defined in [4] as follows:…”

Section: Symmetric Traceless Tensor Casementioning

confidence: 99%

“…Hence, applying the detracer to symmetric tensors (10) and (14) simply changes the symmetric tensor r n into the symmetric traceless tensor R (n) , which is expressed as [3,4] …”

Section: Symmetric Traceless Tensor Casementioning

confidence: 99%

“…In the same article there was no mention of (i), probably because multipole moments were expressed by spherical harmonics. To understand the physical meaning of multipole moments, we need to represent them in terms of Cartesian coordinates [1,3,4,5,6].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Generalized integral formulation of electromagnetic Cartesian multipole moments

Niitsuma

2013

Journal of Electromagnetic Waves and Applications

View full text Add to dashboard Cite

Abstract.We study integral expressions of electromagnetic multipole moments of arbitrary order in Cartesian coordinates. The volume and surface integrals of charge-induced and current-induced multipole moment tensors are formulated and the relationship between them is discussed. Full surface integral expressions for the multipole moment are also obtained. We further extend the formulation to introduce another kind of dipole moment, which is similar to the charge-induced and current-induced multipole moments and is found in a vector decomposition formula. † This is an Author's Original Manuscript of an article whose final and definitive form, the Version of Record, has been published in the

show abstract

“…One can easily show that △R (n) vanishes with the use of r ·∇ n+1 r −1 = −(n + 1)∇ n r −1 [4]. Therefore, from (10) and (14), we finally obtain the following expression:…”

Section: Symmetric Traceless Tensor Casementioning

confidence: 94%

Section: Definitionsmentioning

confidence: 99%

“…A symmetric tensor A (n) of rank n (≥ 1) is rendered traceless by the detracer operator D n defined in [4] as follows:…”

Section: Symmetric Traceless Tensor Casementioning

confidence: 99%

“…Hence, applying the detracer to symmetric tensors (10) and (14) simply changes the symmetric tensor r n into the symmetric traceless tensor R (n) , which is expressed as [3,4] …”

Section: Symmetric Traceless Tensor Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalized integral formulation of electromagnetic Cartesian multipole moments

Niitsuma

2013

Journal of Electromagnetic Waves and Applications

View full text Add to dashboard Cite

show abstract

Torque and atomic forces for Cartesian tensor atomic multipoles with an application to crystal unit cell optimization

Elking¹

2016

J Comput Chem

View full text Add to dashboard Cite

New equations for torque and atomic force are derived for use in flexible molecule force fields with atomic multipoles. The expressions are based on Cartesian tensors with arbitrary multipole rank. The standard method for rotating Cartesian tensor multipoles and calculating torque is to first represent the tensor with n indexes and 3(n) redundant components. In this work, new expressions for directly rotating the unique (n + 1)(n + 2)/2 Cartesian tensor multipole components Θpqr are given by introducing Cartesian tensor rotation matrix elements X(R). A polynomial expression and a recursion relation for X(R) are derived. For comparison, the analogous rotation matrix for spherical tensor multipoles are the Wigner functions D(R). The expressions for X(R) are used to derive simple equations for torque and atomic force. The torque and atomic force equations are applied to the geometry optimization of small molecule crystal unit cells. In addition, a discussion of computational efficiency as a function of increasing multipole rank is given for Cartesian tensors. © 2016 Wiley Periodicals, Inc.

show abstract

The High‐Order Toroidal Moments and Anapole States in All‐Dielectric Photonics

Gurvitz

Ladutenko

Dergachev

et al. 2019

Laser & Photonics Reviews

190

165

View full text Add to dashboard Cite

All‐dielectric nanophotonics attracts ever increasing attention nowadays due to the possibility of controlling and configuring light scattering on high‐index semiconductor nanoparticles. It opens a room of opportunities for designing novel types of nanoscale elements and devices, and paves the way for advanced technologies of light energy manipulation. One of the exciting and promising prospects is associated with utilizing the so‐called toroidal moment, being the result of poloidal currents excitation, and anapole states, corresponding to the interference of dipole and toroidal electric moments. Here, higher‐order toroidal moments of both types (up to the electric octupole toroidal moment) are presented and investigated in detail via the direct Cartesian multipole decomposition allowing new near‐ and far‐field configurations to be obtained. Poloidal currents can be associated with vortex‐like distributions of the displacement currents inside nanoparticles, revealing the physical meaning of the high‐order toroidal moments and the convenience of the Cartesian multipoles as an auxiliary tool for analysis. High‐order nonradiating anapole states accompanied by the excitation of intense near‐fields are demonstrated. It is believed that the results are of high importance for both the fundamental understanding of light scattering by high‐index particles and a variety of nanophotonics applications and light governing on nanoscale.

show abstract

Traceless cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media

Cited by 79 publications

References 9 publications

Generalized integral formulation of electromagnetic Cartesian multipole moments

Generalized integral formulation of electromagnetic Cartesian multipole moments

Torque and atomic forces for Cartesian tensor atomic multipoles with an application to crystal unit cell optimization

The High‐Order Toroidal Moments and Anapole States in All‐Dielectric Photonics

Contact Info

Product

Resources

About