International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 5, Pages 301-312

Iterative solution of quadratic tensor equations for mutual polarisation

Wynand S. Verwoerd

Centre for Advanced Computational Solutions, AMAC Division, P.O. Box 84, Lincoln University, Canterbury, New Zealand

Received 16 October 2001

Copyright © 2002 Wynand S. Verwoerd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


To describe mutual polarisation in bulk materials containing high polarisability molecules, local fields beyond the linear approximation need to be included. A second order tensor equation is formulated, and it describes this in the case of crystalline or at least locally ordered materials such as an idealised polymer. It is shown that this equation is solved by a set of recursion equations that relate the induced dipole moment, linear polarisability, and first hyperpolarisability in the material to the intrinsic values of the same properties of isolated molecules. From these, macroscopic susceptibility tensors up to second order can be calculated for the material.