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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 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.

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.

Heat capacity (microcanonical ensemble)

The heat capacity of an object is defined as

C_V = left({partial Uoverpartial T}right)_V.

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

{1over T} = {partial Soverpartial 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).

N^{prime} = 3N

Possible energies of an SHO are given by

E_n = hbaromegaleft(n+{1over2}right)

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

varepsilon = hbaromega

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}-1right)!

. 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}-1right)!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 = lnOmega = ln{left(q+N^{prime}-1right)!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)lnleft(q+N^{prime}right)-N^{prime}ln N^{prime}-qln q.

Total energy of the solid is given by

U = {N^{prime}varepsilonover2} + qvarepsilon.

We are now ready to compute the temperature

{1over T} = {partial Soverpartial U} = {partial Soverpartial q}{dqover dU} = {1overvarepsilon}{partial Soverpartial q} = {kovervarepsilon} lnleft(1+N^{prime}/qright)

Inverting this formula to find U:

U = {N^{prime}varepsilonover2} + {N^{prime}varepsilonover e^{varepsilon/kT}-1}.

Differentiating with respect to temperature to find $C_V$ :

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

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

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 = varepsilonleft(n+{1over2}right)

substituting this into the partition function formula yields

begin{align} Z & {} = sum_{n=0}^{infty} e^{-varepsilonleft(n+1/2right)/kT} = e^{-varepsilon/2kT} sum_{n=0}^{infty} e^{-nvarepsilon/kT}=e^{-varepsilon/2kT} sum_{n=0}^{infty} left(e^{-varepsilon/kT}right)^n & {} = {e^{-varepsilon/2kT}over 1-e^{-varepsilon/kT}} = {1over e^{varepsilon/2kT}-e^{-varepsilon/2kT}} = {1over 2 sinhleft({varepsilonover 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 Erangle = u = -{1over Z}partial_{beta}Z

where

beta = {1over kT}.

Therefore

u = -2 sinhleft({varepsilonover 2kT}right){-coshleft({varepsilonover 2kT}right)over 2 sinh^2left({varepsilonover 2kT}right)}{varepsilonover2} = {varepsilonover2}cothleft({varepsilonover 2kT}right).

Heat capacity of one oscillator is then

C_V = {partial Uoverpartial T} = -{varepsilonover2} {1over sinh^2left({varepsilonover 2kT}right)}left(-{varepsilonover 2kT^2}right) = k left({varepsilonover 2 k T}right)^2 {1over sinh^2left({varepsilonover 2kT}right)}.

Heat capacity of the entire solid is given by $C_V = 3NC_V$ :

$C_V = 3Nkleft({varepsilonover 2 k T}right)^2 {1over sinh^2left({varepsilonover 2kT}right)}.$

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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Last updated on Monday July 21, 2008 at 10:47:08 PDT (GMT -0700)
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