Vector harmonic functions decaying harmonics

Now, the solution for p must be constructed solely from Cl and the general position vector x in the form of the vector harmonic functions. Now, there is only a single scalar function that can be constructed from Cl and the decaying harmonics, (8-5), that is linear in Cl, namely,... 

The function u(H> is harmonic, decaying, linear in U and a true vector. Again, examining the decaying vector harmonic functions (8-5), we find that there are only two products of U and the decaying harmonics that are true vectors. Thus the most general form for u(//) is... 

It follows, therefore, from (8-3) that u = u(ff) must be a decaying harmonic function, linear in Cl and a true vector. The only combination of Cl and the vector harmonic functions that satisfies these conditions is... 

So Sap(q, co) describes the spectrum of density fluctuations at wave vector q. At low temperatures the crystal dynamics consist of phonon vibrations and Fap(q, t) is a superposition of harmonic oscillations so 5a (q, phonon frequencies corresponding to wave vector q. At higher temperatures translational motion occurs and the associated correlations should simply decay in time, giving rise to a peak in Sa/ (q, co) that is centred on co = 0, and therefore called the quasi-elastic peak, with subsidiary phonon peaks at the appropriate values of co. A nice example is shown in a paper by Gillan (1986). 

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