Thermal scattering of X-rays by a close-packed hexagonal lattice

The dynamical properties of a crystal for small vibrations can be described by the set of coefficients of the potential energy forming the dynamical matrix. The elastic constants and many other observable quantities can be calculated in terms of the elements of the dynamical matrix, but, in general, the reverse does not hold. On the assumptions that only central forces need to be considered, and that only next-neighbour atoms act on one another, the dynamical matrix for a close-packed hexagonal lattice is expressed in terms of one atomic constant, which can be determined by comparing the expressions for the elastic constants with experiment. The Fourier transform of the dynamical matrix and its reciprocal, which in first approximation is proportional to the scattering matrix, are then calculated. A diagram of the equidiffusion lines, which covers a part of reciprocal space containing sixteen lattice points, is drawn. The diagram shows that the `extra spots' are surrounded by a weak background which exhibits considerable fine structure. The equidiffusion lines constructed for the vicinity of the selective reflections (Jahn case) agree with those calculated by Begbie for beryl. No trace is found of the intense star pattern observed by Lonsdale for ice and ammonium fluoride.