Einstein solid: Information from Answers.com

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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
Each atom is attached to a preferred position in space by a spring
All springs have same frequency (contrast with the Debye model)

While the assumption that a solid has independent oscillations is very accurate, these oscillations are sound waves, collective modes involving many atoms. In the Einstein model, each atom oscillates independently. Einstein was aware that the frequency of the actual oscillations would be different, 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 Historical impact
2 Heat capacity (microcanonical ensemble)
3 Heat capacity (canonical ensemble)
4 See also
5 References
6 External links

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 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. As the temperature goes up, the specific heat goes up until it approaches the Dulong and Petit prediction at high temperature.

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 evidence for the need of quantization. Einstein used the levels of the quantum mechanical oscillator many years before the advent of modern quantum mechanics.

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 $T 3$ power law, which matches experiment. This modification is called the Debye Model, which appeared in 1912.

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 at constant volume V is defined through the internal energy U 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.

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 $T / T E$ . Similarly, the Debye model predicts a universal function of the ratio $T / T D$ (see Debye versus Einstein).

References

"Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme", A. Einstein, Annalen der Physik, volume 22, pp. 180-190, 1907.

External links

"Einstein Solid" by Enrique Zeleny, The Wolfram Demonstrations Project.

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Einstein solid

Contents

Historical impact

Heat capacity (microcanonical ensemble)

Heat capacity (canonical ensemble)

See also

References

External links