Irreducible spherical representation of some fourth-rank tensors

Issue title: Atomic and molecular nonlinear optics: Theory, Experiment and Computation A homage to the pioneering work of Stanisław Kielich (1925-1993)

Guest editors: G. Maroulis, T. Bancewicz, B. Champagne and A.D. Buckingham

Article type: Research Article

Authors: Bancewicz, Tadeusz^{a; *} | Ożgo, Zdzisław^b

Affiliations: [a] Nonlinear Optics Division, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland | [b] Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland

Correspondence: [*] Corresponding author. E-mail: [email protected], http://zon8.physd.amu.edu.pl/~tbancewi.

Abstract: A process of decomposition of fourth-rank tensors into parts irreducible with reference to the continuous group of rotation is presented. The transformation matrices between the Cartesian and spherical reducible and irreducible fourth-rank tensors are given and discussed. We have focused our attention to the purely dipolar fourth-rank tensor Ci;jkl symmetric to its last three indices and the dipole-octopole fourth-rank multipolar polarizability tensor E(1,3). The fourth-rank tensors intervene in a number of very important nonlinear optics processes like Kerr effect, intensity-dependent refractive index phenomena, four wave mixing, third harmonic generation and several other effects. Tensors of this kind are also very important in piezo-electric phenomena and in elasticity studies. Cartesian tensor index permutation is discussed as well as its influence on the tensor irreducible spectrum is studied. Several examples concerning fourth-rank tensors are given.

DOI: 10.3233/JCM-2010-0314

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 10, no. 3-6, pp. 129-138, 2010