The Einstein and Debye models revisited

The vibrational heat capacity is the largest contribution to the total heat capacity and determines to a large extent the entropy. Analytical expressions for the entropy of the models described in the previous section can be derived. The entropy corresponding to the Einstein heat capacity is 

Whereas the latter expression must be solved numerically for low temperatures, the entropy at high temperatures can be derived by a series expansion [4], For the Debye or Einstein models the entropy is essentially given in terms of a single parameter at high temperature  

An alternative to deriving the Debye temperature from experimental heat capacities is to derive an entropy-based Debye temperature by calculation of the s that reproduces the observed entropy for each single temperature using 

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