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.
[1] The original theory proposed by Einstein in 1907 has great historical relevance.
The heat capacity of solids as predicted by the empirical Dulong–Petit law was required by classical mechanics, the specific heat of solids should be independent of temperature.
But experiments at low temperatures showed that the heat capacity changes, going to zero at absolute zero.
By employing Planck's quantization assumption, Einstein's theory accounted for the observed experimental trend for the first time.
Together with the photoelectric effect, this became one of the most important pieces of evidence for the need of quantization.
For a thermodynamic approach, the heat capacity can be derived using different statistical ensembles.
The heat capacity of an object at constant volume V is defined through the internal energy U as
Possible energies of an SHO are given by 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).
Therefore, multiplicity of the system is given by which, as mentioned before, is the number of ways to deposit
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 Elimination of q between the two preceding formulas gives for U: The first term is associated with zero point energy and does not contribute to specific heat.
we obtain: or Although the Einstein model of the solid predicts the heat capacity accurately at high temperatures, and in this limit
, 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.
Heat capacity is obtained through the use of the canonical partition function of a simple quantum 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
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 a quantum harmonic.
One thus obtains which is algebraically identical to the formula derived in the previous section.
has the dimensions of temperature and is a characteristic property of a crystal.
[2] 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
In Einstein's model, the specific heat approaches zero exponentially fast at low temperatures.
The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested.
This modification is called the Debye model, which appeared in 1912.
