Lifshitz invariants

Inserting this in Eq. (216) we find that the Lifshitz invariant (which has a composition rule slightly reminiscent of angular momentum, cf. Lj( = xpy-ypx) in the cholesteric case has the value equal to q, the wave vector. In fact we can gain some familiarity with this invariant by starting from an expression we know quite well, the Oseen-Frank expression for the elastic free energy Eq. (96). Because of its symmetry, this expression cannot describe the cholesteric state of a nematic which lacks reflection symmetry and where the twisted state represents the lowest energy. Now, if there is a constant twist with wave vector q, the value of n Fxn in the K22 term equals-q. The expression Eq. (96) therefore has to be renormalized to... 

Inserting this in Eq. (216) we find that the Lifshitz invariant equals... 

Before we add the Lifshitz invariant to our Landau expansion Eq. (157) another consequence of the helix may be pointed out. It is the fact that, as we have already... 

It is remarkable how many people think it difficult to compute van der Waals forces directly by using the Lifshitz theory. Invariably, after a few minutes instruction, there is the reaction "I didn t know how easy it was." Essentially it is a matter of introducing tabulated experimental information for e s and numerically summing or integrating for the interaction energy. 

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