Calculation of the scattering law

The INS intensity, 5 (g,fo), as calculated from the Scattering Law, Eq. (2.41), is related to the mean square atomic displacements, weighted by the incoherent scattering cross sections. What is required to calculate this quantity is the mean square atomic displacement tensor, Bi, and this can be obtained from the crystalline equivalent of L/ ( A2.3), the normalised atomic displacements in a single molecule Eq. (4.20). This is and was introduced above, in Eq. (4.55). We have seen how 

This is entirely equivalent to integrating over both k and ox, but expressing the distribution in ox, by the local density of states g o, ) ( 2.6.2). If  

However, in our convention, see Eq. (2.55), g(fiv) integrates to unity across its local spectral range, we thus avoid the extra step of Eq. (4.58) and work directly with Li Xaxv). First we simphfy Eq. (4.56), replacing the integral by a sura limited to a selection of discrete k points. For example in the case of Na[FHF], as we shall see below, we construct a three-dimensional grid of 16x16x16 points evenly distributed across the k space of the first Brillouin zone. Individual values (at each 

The B oXv) are manipulated appropriately ( A2.5) to generate the required S (Q,co) values (again g(o)v) remains absent). 

Although the preferred method of calculating a spectrum is to perform an ab initio calculation on an extended solid, extracting frequencies and displacements across the Brillouin zone, on a fine A-grid, this approach can be computationally very expensive. In plane wave codes like CASTEP [18], CPMD [19], TWSCF [20], VASP [21], ABINIT [22], and some others, the number of plane waves that are taken into consideration, the selected correlation fimctional and the choice of pseudopotential will all have an impact on the quality of the calculations. Some codes (e.g. ABINIT) alleviate the problem by permitting frozen phonon calculations at the symmetry zone boundary, i.e. (0,0,0), (l/2,0,0), (l/2,l/2,0) and (l/2,l/2,l/2) and so determine the dynamical matrix at these points. The code then interpolates values of the d3mamical matrix for all the points within the Brillouin zone and uses these to calculate the solution to the vibrational problem inside the zone. 

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