Chain of secular equations in the lattice vibration problem

The following theorem is proved. Any quadratic form I, which is composed of coordinates that transform according to the corepresentation D of the group Φ+KΦ of unitary and antiunitary operators and is invariant with respect to Φ+KΦ, can be written as U=∑IijAIjuIji*uIji, where uIji is the i-th coordinate transforming according to the irreducible representation (ICR) DIj of the sum D = ΣDIj (nonequivalent ICR's have different I’s while equivalent ICR's differ only as to their subscript j). It is a difficult case when D contains several identical type-b ICR's. This in fact is the case considered here. To ensure that the proof of the theorem would be of direct practical application we calculate the frequencies of crystal vibrations. In doing so we examine all versions of Brillouin-zone points and ICR types and we reduce the proof itself to the development of a method employing a chain of secular equations that would ensure transformation to the form of Eq. (1). The practical importance of the method is that only with it can the correct number of different frequencies be obtained in approximate numerical calculations. For the first time the theorem substantiates the assumption, made in the Landau-Lifshitz theory of second-order phase transitions, that a second-order transition can be related to the reversal of the sign of the coefficient of one second-order invariant. The situation, which is equivalent to the case of several identical type-b ICR's, has not been considered in the traditional scheme for analysis of lattice vibrations and hence, the traditional scheme can be regarded as being incomplete. The normal coordinates should be defined as coordinates that transform according to some ICR and correspond to a particular value of the frequency.

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