Einstein solid

The Einstein solid is a model of a crystalline solid that contains a large number of independent three-dimensional quantum harmonic oscillators of the same frequency. The independence assumption is relaxed in the Debye model. While the model provides qualitative agreement with experimental data, especially for the high-temperature limit, these oscillations are in fact phonons, or collective modes involving many atoms. Albert Einstein was aware that getting the frequency of the actual oscillations would be difficult, but he nevertheless proposed this theory because it was a particularly clear demonstration that quantum mechanics could solve the specific heat problem in classical mechanics.

The Einstein solid is a model of a crystalline solid that contains a large number of independent three-dimensional quantum harmonic oscillators of the same frequency. The independence assumption is relaxed in the Debye model. While the model provides qualitative agreement with experimental data, especially for the high-temperature limit, these oscillations are in fact phonons, or collective modes involving many atoms. Albert Einstein was aware that getting the frequency of the actual oscillations would be difficult, but he nevertheless proposed this theory because it was a particularly clear demonstration that quantum mechanics could solve the specific heat problem in classical mechanics.

Heat capacity

For a thermodynamic approach, the heat capacity can be derived using different statistical ensembles. All solutions are equivalent at the thermodynamic limit. C_V = \left(\frac{\partial U}{\partial T}\right)_V. T , the temperature of the system, can be found from the entropy \frac{1}{T} = \frac{\partial S}{\partial U}. To find the entropy consider a solid made of N atoms, each of which has 3 degrees of freedom. So there are 3N quantum harmonic oscillators (hereafter SHOs for "Simple Harmonic Oscillators"). N^{\prime} = 3N Possible energies of an SHO are given by E_n = \hbar\omega\left(n+{1\over2}\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 \varepsilon = \hbar\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 q quanta of energy among N^{\prime} SHOs. This task becomes simpler if one thinks of distributing q pebbles over N^{\prime} boxes or separating stacks of pebbles with N' - 1 partitions or arranging q pebbles and N' - 1 partitions The last picture is the most telling. The number of arrangements of n  objects is n! . So the number of possible arrangements of q pebbles and N'-1 partitions is \left(q + N' - 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 q! identical quanta arrangements, and (N'-1)! identical partition arrangements. Therefore, multiplicity of the system is given by \Omega = {\left(q+N^{\prime}-1\right)!\over q! (N^{\prime}-1)!} which, as mentioned before, is the number of ways to deposit q quanta of energy into N^{\prime} oscillators. Entropy of the system has the form S/k = \ln\Omega = \ln{\left(q+N^{\prime}-1\right)!\over q! (N^{\prime}-1)!}. N^{\prime} is a huge number—subtracting one from it has no overall effect whatsoever: S/k \approx \ln{\left(q+N^{\prime}\right)!\over q! N^{\prime}!} With the help of Stirling's approximation, entropy can be simplified: S/k \approx \left(q+N^{\prime}\right)\ln\left(q+N^{\prime}\right)-N^{\prime}\ln N^{\prime}-q\ln q. Total energy of the solid is given by U = {N^{\prime}\varepsilon\over2} + q\varepsilon, 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 \frac{1}{T} = \frac{\partial S}{\partial U} = \frac{\partial S}{\partial q} \frac{dq}{dU} = \frac{1}{\varepsilon} \frac{\partial S}{\partial q} = \frac{k}{\varepsilon} \ln\left(1 + N'/q\right) Elimination of q between the two preceding formulas gives for U: U = {N^{\prime}\varepsilon\over2} + {N^{\prime}\varepsilon\over e^{\varepsilon/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 C_V we obtain: C_V = {\partial U\over\partial T} = {N^{\prime}\varepsilon^2\over k T^2}{e^{\varepsilon/kT}\over \left(e^{\varepsilon/kT}-1\right)^2} or C_V = 3Nk\left({\varepsilon\over k T}\right)^2{e^{\varepsilon/kT}\over \left(e^{\varepsilon/kT}-1\right)^2}. Although the Einstein model of the solid predicts the heat capacity accurately at high temperatures, and in this limit \lim_{T\to\infty} 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. Z = \sum_{n = 0}^{\infty} e^{-E_n/kT} where E_n = \varepsilon\left(n+{1\over2}\right) 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} \\[1ex] &= e^{-\varepsilon/2kT} \sum_{n=0}^{\infty} \left(e^{-\varepsilon/kT}\right)^n = {e^{-\varepsilon/2kT}\over 1-e^{-\varepsilon/kT}} \\[1ex] &= {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 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 N^{\prime} to get the total. Next, let's compute the average energy of each oscillator \langle E\rangle = U = -{1\over Z}\partial_{\beta}Z where \beta = \frac{1}{kT}. Therefore, U = -2 \sinh\left({\varepsilon\over 2kT}\right){-\cosh\left({\varepsilon\over 2kT}\right)\over 2 \sinh^2\left({\varepsilon\over 2kT}\right)}{\varepsilon\over2} = {\varepsilon\over2}\coth\left({\varepsilon\over 2kT}\right). Heat capacity of one oscillator is then c_V = {\partial U\over\partial T} = -{\varepsilon\over2} {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}\left(-{\varepsilon\over 2kT^2}\right) = k \left({\varepsilon\over 2 k T}\right)^2 {1\over \sinh^2\left({\varepsilon\over 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 C_V = 3Nc_V , where the total number of degree of freedom of the solid is three (for the three directional degree of freedom) times N , the number of atoms in the solid. One thus obtains C_V = 3Nk\left({\varepsilon\over 2 k T}\right)^2 {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}. which is algebraically identical to the formula derived in the previous section. The quantity T_{\rm E}=\varepsilon / 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 T / T_{\rm E} . Similarly, the Debye model predicts a universal function of the ratio T/T_{\rm D} , where T_{\rm D} is the Debye temperature.

See also

Kinetic theory of solids

References

External links

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