6 Pages • 1,286 Words • PDF • 426.5 KB
Uploaded at 2021-09-24 09:50
This document was submitted by our user and they confirm that they have the consent to share it. Assuming that you are writer or own the copyright of this document, report to us by using this DMCA report button.
Einstein solid - Wikipedia
1 de 6
https://en.wikipedia.org/wiki/Einstein_solid
Einstein solid The Einstein solid is a model of a solid based on two assumptions: Each atom in the lattice is an independent 3D quantum harmonic oscillator All atoms oscillate with the same frequency (contrast with the Debye model) While the assumption that a solid has independent oscillations is very accurate, these oscillations are sound waves or phonons, collective modes involving many atoms. In the Einstein model, however, each atom oscillates independently. Einstein was aware that getting the frequency of the actual oscillations would be difficult, 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]
Contents Historical impact Einstein's theory of specific heats Heat capacity (microcanonical ensemble) Heat capacity (canonical ensemble) See also References External links
Historical impact The original theory proposed by Einstein in 1907 has 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 pieces of evidence for the need of quantization. Einstein used the levels of the quantum mechanical oscillator many years before the advent of modern quantum mechanics.
Einstein's theory of specific heats 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
08/04/2018 22:14
Einstein solid - Wikipedia
2 de 6
to zero as a
https://en.wikipedia.org/wiki/Einstein_solid
power law, which matches experiment. This modification is called the Debye model, which appeared in
1912. When Walther Nernst learned of Einstein's 1906 paper on specific heat,[2] he was so excited that he traveled all the way from Berlin to Zürich to meet with him.[3]
Heat capacity (microcanonical ensemble) The heat capacity of an object at constant volume V is defined through the internal energy U as
, the temperature of the system, can be found from the entropy
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 for "Simple Harmonic Oscillators").
Heat capacity of an Einstein solid as a function of temperature. Experimental value of 3Nk is recovered at high temperatures.
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 an SHO is increased. Next, we must compute the multiplicity of the system. That is, compute the number of ways to distribute task becomes simpler if one thinks of distributing
or separating stacks of pebbles with
pebbles over
quanta of energy among
SHOs. This
boxes
partitions
08/04/2018 22:14
https://en.wikipedia.org/wiki/Einstein_solid
Einstein solid - Wikipedia
3 de 6
or arranging pebbles and
partitions
The last picture is the most telling. The number of arrangements of arrangements of
pebbles and
partitions is
objects is
. So the number of possible
. However, if partition #3 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. Entropy of 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
since there are q energy quanta in total in the system in addition to the ground state energy of each oscillator. Some authors, such as Schroeder, omit this ground state energy in their definition of the total energy of an Einstein solid. We are now ready to compute the temperature
08/04/2018 22:14
Einstein solid - Wikipedia
4 de 6
https://en.wikipedia.org/wiki/Einstein_solid
Elimination of q between the two preceding formulas gives for U:
The first term is associated with zero point energy and does not contribute to specific heat. It will therefore be lost in the next step. Differentiating with respect to temperature to find
we obtain:
or
Although the 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 how to calculate accurate low-temperature heat capacities.
Heat capacity (canonical ensemble) Heat capacity is obtained through the use of the canonical partition function of a simple harmonic oscillator (SHO).
where
substituting this into the partition function formula yields
08/04/2018 22:14
Einstein solid - Wikipedia
5 de 6
https://en.wikipedia.org/wiki/Einstein_solid
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
where
Therefore,
Heat capacity of one oscillator is then
Up to now, we calculated the heat capacity of a unique degree of freedom, which has been modeled as an SHO. The heat capacity of the entire solid is then given by three (for the three directional degree of freedom) times
, where the total number of degree of freedom of the solid is , the number of atoms in the solid. One thus obtains
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
the Einstein temperature.[4] Hence, the Einstein Crystal model predicts that the energy and heat capacities of a crystal are universal functions of the dimensionless ratio function of the ratio
. Similarly, the Debye model predicts a universal
.
08/04/2018 22:14
Einstein solid - Wikipedia
6 de 6
https://en.wikipedia.org/wiki/Einstein_solid
See also Kinetic theory of solids
References 1. Mandl, F. (1988) [1971]. Statistical Physics (2nd ed.). Chichester·New York·Brisbane·Toronto·Singapore: John Wiley & sons. ISBN 978-0471915331. 2. Einstein, Albert (1906). "Theorie der Strahlung und die Theorie der Spezifischen Wärme" (http://einsteinpapers.press.princeton.edu/vol2-trans/228) [Planck's theory of radiation and the theory of specific heat]. Annalen der Physik. 4. 22: 180–190, 800. Bibcode:1906AnP...327..180E (http://adsabs.harvard.edu /abs/1906AnP...327..180E). doi:10.1002/andp.19063270110 (https://doi.org/10.1002%2Fandp.19063270110). Retrieved 2016-03-18. 3. Stone, A. D. (2013). Einstein and the Quantum: The Quest of the Valiant Swabian (http://press.princeton.edu/titles /10068.html). Princeton University Press. p. 146. ISBN 978-0-691-13968-5. 4. Rogers, Donald (2005). Einstein's other theory: the Planck-Bose-Einstein theory of heat capacity. Princeton University Press. p. 73. ISBN 0-691-11826-4.
External links Zeleny, Enrique. "The Wolfram Demonstrations Project - Einstein Solid" (http://demonstrations.wolfram.com /EinsteinSolid/). Retrieved 2016-03-18.. Retrieved from "https://en.wikipedia.org/w/index.php?title=Einstein_solid&oldid=821008041"
This page was last edited on 17 January 2018, at 22:04. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
08/04/2018 22:14