The cyclic boundary conditions and crystal vibrations

Abstract

In considering the vibrational properties of a crystal, a rigorous finite transformation of the particle displacements from their reference configuration is introduced. This transformation shows that an arbitrary set of such displacements may be regarded as made up of a rotation, a translation, a homogeneous deformation of the reference configuration, and a set of inhomogeneous deformational orthogonal modes. For a three-dimensional crystal, there are 3 N – 12 such inhomogeneous modes, which, in the limit of a large crystal can be considered wave-like. In the usual treatment beginning with the cyclic boundary conditions, 3 N wave-like modes are assumed and rotational displacements, for example, must be ignored. The present treatment accounts satisfactorily for all degrees of freedom, including rotational. Because of the non-singular nature of the above transformation, the transformation of the above modes to the normal modes proves that some normal modes are admixtures of inhomogeneous and homogeneous modes and therefore cannot possibly satisfy the Born cyclic boundary conditions. The vibrational hamiltonian is shown to contain the elastic energy and the elastic–phonon interaction terms as well as the usual wave energies. In the limit of a large crystal, it is shown that, for all processes involving phonons, the homogeneous coordinates may be regarded as effectively static, in much the same way as, in a simple theory of the Earth–Sun motion, the Sun, because of its large inertial mass, is considered stationary and its position coordinates static. The above transformation enables the case of a crystal, free or confined in a container, to be satisfactorily discussed. It is proved that the quantum mean value of the tensor whose independent elements define the homogeneous coordinates is, in the limit of a large crystal, equal to the strain tensor of the container, when it is being used to deform the crystal by being itself homogeneously deformed. A rigorous quantum treatment of crystal elastic constants may then be developed. For practical use, the 3 N – 12 inhomogeneous modes may be assumed to obey the cyclic boundary conditions. Thus a satisfactory complete basic treatment of lattice dynamics may be given which accounts for all degrees of freedom including rotation.