Einstein Solid - Heat Capacity (microcanonical Ensemble)

Heat Capacity (microcanonical Ensemble)

The heat capacity of an object at constant volume V is defined through the internal energy U as

, the temperature of the system, can be found from the entropy

To find the entropy consider a solid made of atoms, each of which has 3 degrees of freedom. So there are quantum harmonic oscillators (hereafter SHOs).

Possible energies of an SHO are given by

or, in other words, the energy levels are evenly spaced and one can define a quantum of energy

which is the smallest and only amount by which the energy of an SHO can be incremented. Next, we must compute the multiplicity of the system. That is, compute the number of ways to distribute quanta of energy among SHOs. This task becomes simpler if one thinks of distributing pebbles over boxes

or separating stacks of pebbles with partitions

or arranging pebbles and partitions

The last picture is the most telling. The number of arrangements of objects is . So the number of possible arrangements of pebbles and partitions is . However, if partition #2 and partition #5 trade places, no one would notice. The same argument goes for quanta. To obtain the number of possible distinguishable arrangements one has to divide the total number of arrangements by the number of indistinguishable arrangements. There are identical quanta arrangements, and identical partition arrangements. Therefore, multiplicity of the system is given by

which, as mentioned before, is the number of ways to deposit quanta of energy into oscillators. Entropy of the system has the form

is a huge number—subtracting one from it has no overall effect whatsoever:

With the help of Stirling's approximation, entropy can be simplified:

Total energy of the solid is given by

since there are q energy quanta in total in the system in addition to the ground state energy of each oscillator. Some authors, such as Schroeder, omit this ground state energy in their definition of the total energy of an Einstein solid.

We are now ready to compute the temperature

Inverting this formula to find U:

Differentiating with respect to temperature to find :

or

Although the Einstein model of the solid predicts the heat capacity accurately at high temperatures, it noticeably deviates from experimental values at low temperatures. See Debye model for how to calculate accurate low-temperature heat capacities.