Einstein solid

Heat capacity

For a thermodynamic approach, the heat capacity can be derived using different statistical ensembles. All solutions are equivalent at the thermodynamic limit.

Microcanonical ensemble

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

$${\displaystyle C_V={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_V.}$$

${\displaystyle T}$, the temperature of the system, can be found from the entropy

$${\displaystyle \frac{1}{T}=\frac{\mathrm{\partial }S}{\mathrm{\partial }U}.}$$

To find the entropy consider a solid made of ${\displaystyle N}$ atoms, each of which has 3 degrees of freedom. So there are ${\displaystyle 3N}$ quantum harmonic oscillators (hereafter SHOs for "Simple Harmonic Oscillators").

$${\displaystyle N^{\mathrm{\prime }}=3N}$$

Possible energies of an SHO are given by

$${\displaystyle E_n=\hslash \omega \left(n+\frac{1}{2}\right)}$$

where the n of SHO is usually interpreted as the excitation state of the oscillating mass but here n is usually interpreted as the number of phonons (bosons) occupying that vibrational mode (frequency). The net effect is that the energy levels are evenly spaced, and one can define a quantum of energy due to a phonon as

$${\displaystyle \epsilon =\hslash \omega }$$

which is the smallest and only amount by which the energy of an SHO is increased. Next, we must compute the multiplicity of the system. That is, compute the number of ways to distribute ${\displaystyle q}$ quanta of energy among ${\displaystyle N^{\mathrm{\prime }}}$ SHOs. This task becomes simpler if one thinks of distributing ${\displaystyle q}$ pebbles over ${\displaystyle N^{\mathrm{\prime }}}$ boxes

or separating stacks of pebbles with ${\displaystyle N^{\prime }-1}$ partitions

or arranging ${\displaystyle q}$ pebbles and ${\displaystyle N^{\prime }-1}$ partitions

The last picture is the most telling. The number of arrangements of ${\displaystyle n}$ objects is ${\displaystyle n!}$. So the number of possible arrangements of ${\displaystyle q}$ pebbles and ${\displaystyle N^{\prime }-1}$ partitions is ${\displaystyle \left(q+N^{\prime }-1\right)!}$. However, if partition #3 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 ${\displaystyle q!}$ identical quanta arrangements, and ${\displaystyle (N^{\prime }-1)!}$ identical partition arrangements. Therefore, multiplicity of the system is given by

$${\displaystyle \mathrm{\Omega }=\frac{\left(q+N^{\mathrm{\prime }}-1\right)!}{q!(N^{\mathrm{\prime }}-1)!}}$$

which, as mentioned before, is the number of ways to deposit ${\displaystyle q}$ quanta of energy into ${\displaystyle N^{\mathrm{\prime }}}$ oscillators. Entropy of the system has the form

$${\displaystyle S/k=\ln \mathrm{\Omega }=\ln \frac{\left(q+N^{\mathrm{\prime }}-1\right)!}{q!(N^{\mathrm{\prime }}-1)!}.}$$

${\displaystyle N^{\mathrm{\prime }}}$ is a huge number—subtracting one from it has no overall effect whatsoever:

$${\displaystyle S/k\approx \ln \frac{\left(q+N^{\mathrm{\prime }}\right)!}{q!N^{\mathrm{\prime }}!}}$$

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

$${\displaystyle S/k\approx \left(q+N^{\mathrm{\prime }}\right)\ln \left(q+N^{\mathrm{\prime }}\right)-N^{\mathrm{\prime }}\ln N^{\mathrm{\prime }}-q\ln q.}$$

Total energy of the solid is given by

$${\displaystyle U=\frac{N^{\mathrm{\prime }}\epsilon }{2}+q\epsilon ,}$$

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

$${\displaystyle \frac{1}{T}=\frac{\mathrm{\partial }S}{\mathrm{\partial }U}=\frac{\mathrm{\partial }S}{\mathrm{\partial }q}\frac{dq}{dU}=\frac{1}{\epsilon }\frac{\mathrm{\partial }S}{\mathrm{\partial }q}=\frac{k}{\epsilon }\ln \left(1+N^{\prime }/q\right)}$$

Elimination of q between the two preceding formulas gives for U:

$${\displaystyle U=\frac{N^{\mathrm{\prime }}\epsilon }{2}+\frac{N^{\mathrm{\prime }}\epsilon }{e^{\epsilon /kT}-1}.}$$

The first term is associated with zero point energy and does not contribute to specific heat. It will therefore be lost in the next step.

Differentiating with respect to temperature to find ${\displaystyle C_V}$ we obtain:

$${\displaystyle C_V=\frac{\mathrm{\partial }U}{\mathrm{\partial }T}=\frac{N^{\mathrm{\prime }}{\epsilon }^2}{k{T}^2}\frac{e^{\epsilon /kT}}{{\left(e^{\epsilon /kT}-1\right)}^2}}$$

or

$${\displaystyle C_V=3Nk{\left(\frac{\epsilon }{kT}\right)}^2\frac{e^{\epsilon /kT}}{{\left(e^{\epsilon /kT}-1\right)}^2}.}$$

Although the Einstein model of the solid predicts the heat capacity accurately at high temperatures, and in this limit

${\displaystyle \underset{T\to \mathrm{\infty }}{lim}{C}_V=3Nk}$, which is equivalent to Dulong–Petit law, the heat capacity noticeably deviates from experimental values at low temperatures. See Debye model for how to calculate accurate low-temperature heat capacities.

Canonical ensemble

Heat capacity is obtained through the use of the canonical partition function of a simple quantum harmonic oscillator.

$${\displaystyle Z=\sum _{n=0}^{\mathrm{\infty }}{e}^{-E_n/kT}}$$

where

$${\displaystyle E_n=\epsilon \left(n+\frac{1}{2}\right)}$$

substituting this into the partition function formula yields

$${\displaystyle \begin{array}{cc}Z& =\sum _{n=0}^{\mathrm{\infty }}{e}^{-\epsilon \left(n+1/2\right)/kT}=e^{-\epsilon /2kT}\sum _{n=0}^{\mathrm{\infty }}{e}^{-n\epsilon /kT}\\ & =e^{-\epsilon /2kT}\sum _{n=0}^{\mathrm{\infty }}{\left(e^{-\epsilon /kT}\right)}^n=\frac{e^{-\epsilon /2kT}}{1-e^{-\epsilon /kT}}\\ & =\frac{1}{e^{\epsilon /2kT}-e^{-\epsilon /2kT}}=\frac{1}{2\sinh \left(\frac{\epsilon }{2kT}\right)}.\end{array}}$$

This is the partition function of one harmonic oscillator. Because, statistically, heat capacity, energy, and entropy of the solid are equally distributed among its atoms, we can work with this partition function to obtain those quantities and then simply multiply them by ${\displaystyle N^{\mathrm{\prime }}}$ to get the total. Next, let's compute the average energy of each oscillator

$${\displaystyle ⟨E⟩=U=-\frac{1}{Z}{\mathrm{\partial }}_{\beta }Z}$$

where

$${\displaystyle \beta =\frac{1}{kT}.}$$

Therefore,

$${\displaystyle U=-2\sinh \left(\frac{\epsilon }{2kT}\right)\frac{-\cosh \left(\frac{\epsilon }{2kT}\right)}{2{\sinh }^2\left(\frac{\epsilon }{2kT}\right)}\frac{\epsilon }{2}=\frac{\epsilon }{2}\coth \left(\frac{\epsilon }{2kT}\right).}$$

Heat capacity of one oscillator is then

$${\displaystyle c_V=\frac{\mathrm{\partial }U}{\mathrm{\partial }T}=-\frac{\epsilon }{2}\frac{1}{{\sinh }^2\left(\frac{\epsilon }{2kT}\right)}\left(-\frac{\epsilon }{2k{T}^2}\right)=k{\left(\frac{\epsilon }{2kT}\right)}^2\frac{1}{{\sinh }^2\left(\frac{\epsilon }{2kT}\right)}.}$$

Up to now, we calculated the heat capacity of a unique degree of freedom, which has been modeled as a quantum harmonic. The heat capacity of the entire solid is then given by ${\displaystyle C_V=3N{c}_V}$, where the total number of degree of freedom of the solid is three (for the three directional degree of freedom) times ${\displaystyle N}$, the number of atoms in the solid. One thus obtains

$${\displaystyle C_V=3Nk{\left(\frac{\epsilon }{2kT}\right)}^2\frac{1}{{\sinh }^2\left(\frac{\epsilon }{2kT}\right)}.}$$

which is algebraically identical to the formula derived in the previous section.

The quantity ${\displaystyle T_{\mathrm{E}}=\epsilon /k}$ 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 ${\displaystyle T/T_{\mathrm{E}}}$. Similarly, the Debye model predicts a universal function of the ratio ${\displaystyle T/T_{\mathrm{D}}}$, where ${\displaystyle T_{\mathrm{D}}}$ is the Debye temperature.

Limitations and succeeding model

In Einstein's model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a ${\displaystyle T^3}$ power law, which matches experiment. This modification is called the Debye model, which appeared in 1912.