Einstein solid

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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 Debye model)

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.

1 Historical impact
2 Heat capacity (microcanonical ensemble)
3 Heat capacity (canonical ensemble)
4 References
5 External links

[edit] Historical impact

The original theory proposed by Einstein in 1907 had a great historical relevance. The heat capacity of solids as predicted by the empirical Dulong-Petit law was 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 quantization assumption 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 $T 3$ power law. This defect was later remedied by the Debye Model in 1912.

[edit] Heat capacity (microcanonical ensemble)

Heat capacity of an Einstein solid as a function of temperature. Experimental value of 3Nk is recovered at high temperatures.

The heat capacity of an object is defined as

$C_V = \left({\partial U\over\partial T}\right)_V.$

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

${1\over T} = {\partial S\over\partial U}.$

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

$N^{\prime} = 3N$

Possible energies of an SHO are given by

$E_n = \hbar\omega\left(n+{1\over2}\right)$

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

$\varepsilon = \hbar\omega$

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 $q$ 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 $N^{\prime}-1$ partitions

or arranging $q$ pebbles and $N^{\prime}-1$ 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 $N^{\prime}-1$ partitions is $\left(q+N^{\prime}-1\right)!$ . 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 $q!$ identical quanta arrangements, and $(N^{\prime}-1)!$ 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 $N^{\prime}-1$ 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

$U = {N^{\prime}\varepsilon\over2} + q\varepsilon.$

We are now ready to compute the temperature

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

Inverting this formula to find U:

$U = {N^{\prime}\varepsilon\over2} + {N^{\prime}\varepsilon\over e^{\varepsilon/kT}-1}.$

Differentiating with respect to temperature to find $C V$ :

$C_V = {\partial U\over\partial T} = {N^{\prime}\varepsilon^2\over k T^2}{e^{\varepsilon/kT}\over \left(e^{\varepsilon/kT}-1\right)^2}$

$C_V = 3Nk\left({\varepsilon\over k T}\right)^2{e^{\varepsilon/kT}\over \left(e^{\varepsilon/kT}-1\right)^2}.$

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 model for accurate low-temperature heat capacity calculation.

[edit] Heat capacity (canonical ensemble)

Heat capacity can be obtained through the use of the canonical partition function of an SHO.

$Z = \sum_{n=0}^{\infty} e^{-E_n/kT}$

where

$E_n = \varepsilon\left(n+{1\over2}\right)$

substituting this into the partition function formula yields

$\begin{align} Z & {} = \sum_{n=0}^{\infty} e^{-\varepsilon\left(n+1/2\right)/kT} = e^{-\varepsilon/2kT} \sum_{n=0}^{\infty} e^{-n\varepsilon/kT}=e^{-\varepsilon/2kT} \sum_{n=0}^{\infty} \left(e^{-\varepsilon/kT}\right)^n \\ & {} = {e^{-\varepsilon/2kT}\over 1-e^{-\varepsilon/kT}} = {1\over e^{\varepsilon/2kT}-e^{-\varepsilon/2kT}} = {1\over 2 \sinh\left({\varepsilon\over 2kT}\right)}. \end{align}$

This is the partition function of one 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 $N^{\prime}$ 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

$\beta = {1\over kT}.$

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

$C_V = {\partial U\over\partial T} = -{\varepsilon\over2} {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}\left(-{\varepsilon\over 2kT^2}\right) = k \left({\varepsilon\over 2 k T}\right)^2 {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}.$

Heat capacity of the entire solid is given by $C V = 3 N C V$ :

$C_V = 3Nk\left({\varepsilon\over 2 k T}\right)^2 {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}.$

which is algebraically identical to the formula derived in the previous section.
The quantity $T_E=\varepsilon / k$ 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 $T / T E$ . Similary, the Debye model predicts a universal function of the ratio $T / T D$ (see Debye versus Einstein).