Einstein Solid - Heat Capacity (canonical Ensemble)

Heat Capacity (canonical Ensemble)

Heat capacity can be obtained through the use of the canonical partition function of a single harmonic oscillator (SHO).

where

substituting this into the partition function formula yields

\begin{align} Z & {} = \sum_{n=0}^{\infty} e^{-\varepsilon\left(n+1/2\right)/kT} = e^{-\varepsilon/2kT} \sum_{n=0}^{\infty} e^{-n\varepsilon/kT}=e^{-\varepsilon/2kT} \sum_{n=0}^{\infty} \left(e^{-\varepsilon/kT}\right)^n \\ & {} = {e^{-\varepsilon/2kT}\over 1-e^{-\varepsilon/kT}} = {1\over e^{\varepsilon/2kT}-e^{-\varepsilon/2kT}} = {1\over 2 \sinh\left({\varepsilon\over 2kT}\right)}. \end{align}

This is the partition function of one SHO. Because, statistically, heat capacity, energy, and entropy of the solid are equally distributed among its atoms (SHOs), we can work with this partition function to obtain those quantities and then simply multiply them by to get the total. Next, let's compute the average energy of each oscillator

where

Therefore

Heat capacity of one oscillator is then

Up to now, we calculated the heat capacity of a unique degree of freedom, which has been modeled as an SHO. The heat capacity of the entire solid is then given by, where the total number of degree of freedom of the solid is three (for the three directional degree of freedom) times, the number of atoms in the solid. One thus obtains

which is algebraically identical to the formula derived in the previous section. The quantity has the dimensions of temperature and is a characteristic property of a crystal. It is known as the Einstein temperature. Hence, the Einstein Crystal model predicts that the energy and heat capacities of a crystal are universal functions of the dimensionless ratio . Similarly, the Debye model predicts a universal function of the ratio (see this article for further discussion).

Read more about this topic:  Einstein Solid

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