- Einstein solid
The Einstein solid is a model of a solid based on three assumptions:
* Each atom in the lattice is a 3D
quantum harmonic oscillator
* Atoms do not interact with each other
* All atoms vibrate with the same frequency (contrast with the
While the first assumption is quite accurate, the second is not. If atoms did not interact with one another, sound waves would not propagate through solids.
The original theory proposed by
Einsteinin 1907 had a great historical relevance. The heat capacity of solids as predicted by the empirical Dulong-Petit lawwas known to be consistent with classical mechanics. However, experimental observations at low temperatures showed heat capacity vanished at absolute zero and grew monotonously towards the Dulong and Petit prediction at high temperature. By employing Planck's quantizationassumption Einstein was able to predict the observed experimental trend for the first time. Together with the photoelectric effect, this became one of the most important evidence for the need of quantization (remarkably Einstein was solving the problem of the quantum mechanical oscillator many years before the advent of modern quantum mechanics). Despite its success, the approach towards zero is predicted to be exponential, whereas the correct behavior is known to follow a power law. This defect was later remedied by the Debye Modelin 1912.
Heat capacity (
heat capacityof an object is defined as
, the temperature of the system, can be found from the
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 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.
Entropyof 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
We are now ready to compute the temperature
Inverting this formula to find "U":
Differentiating with respect to temperature to find :
Although Einstein model of the solid predicts the heat capacity accurately at high temperatures, it noticeably deviates from experimental values at low temperatures. See
Debye modelfor accurate low-temperature heat capacity calculation.
Heat capacity (
Heat capacity can be obtained through the use of the
canonical partition functionof an SHO.
substituting this into the partition function formula yields
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
Heat capacity of "one" oscillator is then
Heat capacity of the entire solid is given by :
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 "Einstein's 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
Debye versus Einstein).
* "Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme", A. Einstein, Annalen der Physik, volume 22, pp. 180-190, 1907.
* " [http://demonstrations.wolfram.com/EinsteinSolid/ Einstein Solid] " by Enrique Zeleny,
The Wolfram Demonstrations Project.
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