An Einstein Model for Structural Change

To be more quantitative, we assign an internal energy Efee to the fee phase, Ebee to the bcc phase, and associated Einstein frequencies cofee and cobec- As a result, the free energies (neglecting anharmonicity and the electronic contribution) may be written as 

The resulting free energy difference AF = Fbee — Ffee may be written in turn as 

This simple model exhibits many of the qualitative features that govern the actual competition between different phases. First, by virtue of the fact that /cc Ebee, the low-temperature winner is the fee structure. On the other hand, because the bcc phonon density of states has more weight at lower frequencies (here we put this in by hand through the assumption cobee to fee), at high temperatures, the bcc phase has higher entropy and thereby wins the competition. These arguments are illustrated pictorially in fig. 6.14 where we see that at a critical temperature Tc, 

To make the elements of this model more concrete, we consider the way in which the transition temperature depends upon the difference in the Einstein frequencies of the two structural competitors. To do so, it is convenient to choose the variables a = (cob + coa)/2 and Aco = cob — coa) 2, where u a and cob are the Einstein frequencies of the two structural competitors. If we now further measure the difference in the two frequencies, Aco, in units of the mean frequency m according to the relation Aco = fco, then the difference in free energy between the two phases may be written as 

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