Pure crystals wave vector

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

LDA methods have been employed to investigate solids in two types of approaches. If the solid has translational symmetry, as in a pure crystal, Bloch s theorem applies, which states that the one-electron wave function n at point (r + ft), where ft is a Bravais lattice vector, is equal to the wave function at point r times a phase factor ... 

A remarkable effect of strain modulation of the exchange interactions on the elastic properties of the LiTbF4 ferromagnet (see eq. 105) was observed by Aukhadeev et al. (1983). For example, below Tq the velocity v of purely transverse ultrasonic waves of frequency 13 MHz with the wave vector along the [001] direction of the crystal increases sharply, as is evident in fig. 29a which shows the relative variations... 

We restrict the attention to periodic solids, molecular crystals. The excitations are characterized by wave vectors q, that lie in the first Brillouin zone of the lattice considered. These excitations are not necessarily pure translational phonons, librons or vibrons, in general they will be mixed. Much experimental information has been collected about such excitations, by infrared and Raman spectroscopy and, in particular, by inelastic neutron scattering. Due to the optical selection rules infrared and Raman spectra can only probe the = 0 excitations. By neutron scattering one can excite states of any given q and thus measure the complete dispersion (wave vector dependence) of the phonon and vibron frequencies. 

It has been shown by computer simulation [109-111] and density functional theory [106, 108] that the soft, purely repulsive, radially symmetric potential, V(r), will form cluster crystals at sufficiently high density if its Fourier transform, V(fe), becomes negative for a range of wave vectors. Within mean field approximation, the stability limit of the homogeneous liquid is given by the X-line [108]... 

Up to now the reasoning is purely geometrical and the vector s is not yet connected to the resulting diffraction pattern. From FT considerations, it is clear that the interaction between the object in real space and the incident wave X is represented in the reciprocal space by interaction between the reciprocal lattice and the reciprocal of the incident wave (its FT). The FT of a planar wave A, is a sphere of radius 1/A (Ewald sphere). To obtain a scattered beam it is thus necessary and sufficient that a node of the reciprocal lattice be on the sphere. Figure 1.1c shows that this is generally not the case (i.e., the chance for a crystal to produce a scattered beam is not always realized). 

---