Vibrational Contribution to Structure

Polymorphism, that is, the ability of atoms of a given element to imite into more than one crystal structure, is very common among the elements [55]. More than one-third of them show polymorphism at ordinary pressure. The free energy curves of two polymorphic structures intersect each other. There must be a difference in heat capacities of these structures. For metals, important contributions to the heat capacity come from the lattice vibration, the conduction electrons and in some cases magnetic excitation or localized electron states. 

The free energy due to harmonic lattice vibrations (or equivalently the Debye temperature) is approximately the same for bcc, fee, and hep structures but with a significant tendency for the bcc value to be a few percent lower. The more open bcc structure has a transverse phonon mode with a particularly low frequency which causes a more rapid decrease in the free energy with temperature. On cooling, sodium and lithium transform partially from bcc to hep at very low temperatures (0.1-0.2 Tm). Calcium, strontium, beryllium, and thallium transform to a bcc phase at high temperatures (0.66-0.98 Tm) when there is a considerable anharmonic contribution to the free energy. 

The electrons in simple metals are well described by a free electron gas. Since the atomic volume is changed by only a few percent as a result of polymorphic transformations, the electronic contribution to a free energy does not show any significant structure dependence for these metals. For transition metals, the elec- 

The high-temperature stabilization of the bcc lattice is the prototypical example of the importance of vibrational contributions. Friedel proposed that the larger vi- 

For a temperature of order 1000 K, the contribution T A S to the free energy is about 0.052 eV/atom which is significant on the scale of the energy differences between the bcc and fee and hep phases. 

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