Multipolar exchange interaction: Difference between revisions

Revision as of 07:30, 24 August 2015

Revision as of 17:55, 24 August 2015

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==== Multi-exchange Description ====

SinceUsing the addition rules of tensor operators, a product of twoa rank n tensor operatorsand a rank m tensor can begenerate ranka <math>tensor 0with \simrank 2nn+m ~ |n-m|. </math>Therefore, a high rank tensor can be expressed as the product of low rank tensors. This convention is useful to interpret the high rank multipolar exchange terms as a "multi-exchange" process of dipoles (or pseudospins). For example, for the spherical cubic tensors of <math> J=1 </math> case, we have

:<math> Y_{2}^{-2}=2Y_{1}^{-1}Y_{1}^{-1} </math>

:<math> Y_{2}^{-1}=\sqrt{2}(Y_{1}^{-1}Y_{1}^{0}+Y_{1}^{0}Y_{1}^{-1}) </math>

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:<math> Y_{2}^{+1}=\sqrt{2}(Y_{1}^{0}Y_{1}^{+1}+Y_{1}^{+1}Y_{1}^{0}) </math>

:<math> Y_{2}^{+2}=2Y_{1}^{+1}Y_{1}^{+1} </math>

If so, a quadrupole-quadrupole interaction can be considered as a two step dipole-dipole interaction (see next section), e.g <math> Y_{2_{i}}^{+2_{i}}Y_{2_{j}}^{-2_{j}}=4Y_{1_{i}}^{+1_{i}}Y_{1_{i}}^{+1_{i}}Y_{1_{j}}^{-1_{j}}Y_{1_{j}}^{-1_{j}} </math>. However, it is not a perturbation expansion but just a mathematical technique. The high rank multi-exchange terms are necessary smaller than low rank terms. In many cases, high rank terms are often more important than low rank terms<ref name="Review"/>.

== Multipolar Exchange Interactions ==