- Einstein solid
The

**Einstein solid**is a model of a solid based on three assumptions:

* Each atom in the lattice is a 3Dquantum harmonic oscillator

* Atoms do not interact with each other

* All atoms vibrate with the same frequency (contrast with theDebye 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 empiricalDulong-Petit law was known to be consistent withclassical 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'squantization assumption Einstein was able to predict the observed experimental trend for the first time. Together with thephotoelectric 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 modernquantum 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 theDebye Model in 1912.**Heat capacity (**microcanonical ensemble )The

heat capacity of an object is defined as:$C\_V\; =\; left(\{partial\; Uoverpartial\; T\}\; ight)\_V.$

$T$, 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 oscillator s (hereafter SHOs).:$N^\{prime\}\; =\; 3N$

Possible energies of an SHO are given by

:$E\_n\; =\; hbaromegaleft(n+\{1over2\}\; ight)$

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 $left(q+N^\{prime\}-1\; ight)!$. 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

:$Omega\; =\; \{left(q+N^\{prime\}-1\; ight)!over\; q!\; (N^\{prime\}-1)!\}$

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:$S/k\; =\; lnOmega\; =\; ln\{left(q+N^\{prime\}-1\; ight)!over\; q!\; (N^\{prime\}-1)!\}.$

$N^\{prime\}$ is a huge number—subtracting one from it has no overall effect whatsoever:

:$S/k\; approx\; ln\{left(q+N^\{prime\}\; ight)!over\; q!\; N^\{prime\}!\}$

With the help of

Stirling's approximation , entropy can be simplified::$S/k\; approx\; left(q+N^\{prime\}\; ight)lnleft(q+N^\{prime\}\; ight)-N^\{prime\}ln\; N^\{prime\}-qln\; q.$

Total energy of the solid is given by

:$U\; =\; \{N^\{prime\}varepsilonover2\}\; +\; qvarepsilon.$

We are now ready to compute the temperature

:$\{1over\; T\}\; =\; \{partial\; Soverpartial\; U\}\; =\; \{partial\; Soverpartial\; q\}\{dqover\; dU\}\; =\; \{1overvarepsilon\}\{partial\; Soverpartial\; q\}\; =\; \{kovervarepsilon\}\; lnleft(1+N^\{prime\}/q\; ight)$

Inverting this formula to find "U":

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

Differentiating with respect to temperature to find $C\_V$:

:$C\_V\; =\; \{partial\; Uoverpartial\; T\}\; =\; \{N^\{prime\}varepsilon^2over\; k\; T^2\}\{e^\{varepsilon/kT\}over\; left(e^\{varepsilon/kT\}-1\; ight)^2\}$

or

:$C\_V\; =\; 3Nkleft(\{varepsilonover\; k\; T\}\; ight)^2\{e^\{varepsilon/kT\}over\; left(e^\{varepsilon/kT\}-1\; ight)^2\}.$

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

:$E\_n\; =\; varepsilonleft(n+\{1over2\}\; ight)$

substituting this into the partition function formula yields

:$egin\{align\}Z\; \{\}\; =\; sum\_\{n=0\}^\{infty\}\; e^\{-varepsilonleft(n+1/2\; ight)/kT\}\; =\; e^\{-varepsilon/2kT\}\; sum\_\{n=0\}^\{infty\}\; e^\{-nvarepsilon/kT\}=e^\{-varepsilon/2kT\}\; sum\_\{n=0\}^\{infty\}\; left(e^\{-varepsilon/kT\}\; ight)^n\; \backslash \; \{\}\; =\; \{e^\{-varepsilon/2kT\}over\; 1-e^\{-varepsilon/kT\; =\; \{1over\; e^\{varepsilon/2kT\}-e^\{-varepsilon/2kT\; =\; \{1over\; 2\; sinhleft(\{varepsilonover\; 2kT\}\; ight)\}.end\{align\}$

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

:$langle\; E\; angle\; =\; u\; =\; -\{1over\; Z\}partial\_\{eta\}Z$

where

:$eta\; =\; \{1over\; kT\}.$

Therefore

:$u\; =\; -2\; sinhleft(\{varepsilonover\; 2kT\}\; ight)\{-coshleft(\{varepsilonover\; 2kT\}\; ight)over\; 2\; sinh^2left(\{varepsilonover\; 2kT\}\; ight)\}\{varepsilonover2\}\; =\; \{varepsilonover2\}cothleft(\{varepsilonover\; 2kT\}\; ight).$

Heat capacity of "one" oscillator is then

:$C\_V\; =\; \{partial\; Uoverpartial\; T\}\; =\; -\{varepsilonover2\}\; \{1over\; sinh^2left(\{varepsilonover\; 2kT\}\; ight)\}left(-\{varepsilonover\; 2kT^2\}\; ight)\; =\; k\; left(\{varepsilonover\; 2\; k\; T\}\; ight)^2\; \{1over\; sinh^2left(\{varepsilonover\; 2kT\}\; ight)\}.$

Heat capacity of the entire solid is given by $C\_V\; =\; 3NC\_V$:

:$C\_V\; =\; 3Nkleft(\{varepsilonover\; 2\; k\; T\}\; ight)^2\; \{1over\; sinh^2left(\{varepsilonover\; 2kT\}\; ight)\}.$

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*** " [

*http://demonstrations.wolfram.com/EinsteinSolid/ Einstein Solid*] " by Enrique Zeleny,The Wolfram Demonstrations Project .

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Einstein–Cartan theory**— in theoretical physics extends general relativity to correctly handle spin angular momentum. As the master theory of classical physics general relativity has one known flaw: it cannot describe spin orbit coupling , i.e., exchange of intrinsic… … Wikipedia**Einstein Observatory**— (HEAO 2) was the first fully imaging X ray telescope put into space and the second of NASA s three High Energy Astrophysical Observatories. Named HEAO B before launch, the observatory s name was changed to honor Albert Einstein upon its… … Wikipedia**Solid-state physics**— Solid state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. The bulk of solid state physics theory and research is focused on crystals, largely because the periodicity of atoms in a crystal mdash; … Wikipedia**Einstein , Albert**— (1879–1955) German–Swiss–American theoretical physicist Einstein was born at Ulm in Germany where his father was a manufacturer of electrical equipment. Business failure led his father to move the family first to Munich, where Einstein entered… … Scientists**Albert Einstein**— Einstein redirects here. For other uses, see Einstein (disambiguation) … Wikipedia**List of things named after Albert Einstein**— This is a list of things named after Albert Einstein. Scientific and mathematical concepts * Higher dimensional Einstein gravity * Einstein solid * Einstein force * Einstein s constant * Einstein relation (kinetic theory) * Stark Einstein law *… … Wikipedia**Bose–Einstein condensate**— A Bose–Einstein condensate (BEC) is a state of matter of bosons confined in an external potential and cooled to temperatures very near to absolute zero (val|0|u=K, val| 273.15|u=°C, or val| 459.67|u=°F ). Under such supercooled conditions, a… … Wikipedia**Celebrating the Centennial of Einstein's Miraculous Year**— ▪ 2006 by John J. Dykla By his 26th birthday, in 1905, Albert Einstein had not yet obtained his doctorate in physics or obtained an academic teaching position. He had published five papers in the premier German physics journal, Annalen der… … Universalium**Modele d'Einstein**— Modèle d Einstein En physique statistique et en physique du solide, le modèle d’Einstein est un modèle permettant de décrire la contribution des vibrations du réseau à la capacité calorifique d’un solide cristallin. Il est basé sur les hypothèses … Wikipédia en Français**Modèle d’Einstein**— Modèle d Einstein En physique statistique et en physique du solide, le modèle d’Einstein est un modèle permettant de décrire la contribution des vibrations du réseau à la capacité calorifique d’un solide cristallin. Il est basé sur les hypothèses … Wikipédia en Français