The note discusses the validity of the rotational invariance condition used in lattice dynamics. It is pointed out that rotations of non-central fields cannot be represented by displacements alone. Formulating the equations of lattice dynamics referred to Lagrangean co-ordinates imbedded in the crystal it is shown that the usual formulae do not strictly represent the physical requirement of rotational invariance. The result has considerable effect on several applications of the lattice theory. Β§ 1. Introduction
In order to develop the basic equations of the general lattice dynamics, where no assumptions regarding the interatomic potential functions are made, it is necessary to employ the various invariance relations obeyed by the potential energy of the 1a.ttice. It is well-known that in lattice dynamics, the crystal can be considered to be made up of atomic nuclei whose co-ordinates can be represented in terms of the cell indices m, p , q, . . . and the basis indices j , k, 1 . . . The rotational invariance condition1)2) can then be represented as The aim of the present note is to show that eq. ( ) has no basis in the general lattice dynamics, since the msthemat*ical formulation does not strictly represent the pyhsical condition of the invariance of the potential energy of the lattice under a rigid body rotation. Consider, as an example, a calculation of the properties of an arbitrary (triclinic) lattice restricting oneself to the first neighbour interactions. To an atom (rnj), let the first neighbour, which is one in number, be (pk). Remembering that Qap m -p is the force in the a direction on the (mj) atom due to the (pk) atom for unit relative displacement of the atoms in the p direcis symmetric (see eq. 9)) but otherwise completely arbitrary, since there is ( i k ) tion, the force constant scheme d j a p m -p in terms of a , p for fixed ( jk ) ( l) G.