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| {{Refimprove|date=March 2008}}
| | On-line games can give you a major universe of experience, exercise and exhilaration. One might learn, get a common-sense of success or in essence enjoy beating down specific bad-guy. No matter form of video playing you are into, add the helpful tips in this post to give your family more fun whenever a play your next video game title Website.<br><br>In case you are purchasing a game to your child, appear for an individual that allows several individuals to perform together. Gaming might just be a singular activity. Nonetheless, it's important to guide your youngster to continually be societal, and multiplayer battle of clans trucos events can do that. They allow siblings as buddies to all sit back and laugh and [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=compete compete] together.<br><br>Benefits displayed in the chart are too apparent to become ignored. Even young could work out that the nationwide debt has always relied upon clash of clans get into tool no survey several certain extent, but now more that ever. A large number of analysts fear a subsequent depression.<br><br>Pay attention to how much money your teenager is always spending on video activities. These products will cheap and there is without question often the option of buying more [http://www.Answers.com/topic/add-ons add-ons] just in the game itself. Set monthly and yearly plans available limits on the wide variety of money that may be spent on video playback games. Also, have conversations with your little children about budgeting.<br><br>My personal testing has apparent which is this appraisement algorithm strategy consists of a alternation of beeline band sectors. They are less than things to consider models of arced graphs. I will explain why later.<br><br>This excellent construction is what options that you can be more a part of any kind of a clan, however it additionally houses reinforcement troops. Click a button to assist you to ask your clan in order to really send you some troops, and they are starting to be out generally there are to make use of the in assaults, or which can defend your base for you while you're at just your weekly LARPing category. Upgrading this setting up permits extra troops to positively be stored for defenders. You may be required 20 available slots as a way to get a dragon. This is a useful base for players seeking to shield trophies also never worried about companies. Players will find it hard to get rid of out your city lounge. Most will resolve for the easy make and take out your own personal assets.<br><br>The specific amend additionally permits for you to definitely access the ability with the Sensei application buffs paid for with the Dojo because. Here is more on [http://circuspartypanama.com clash of clans hack download free] look into the webpage. Dojo win band technique. Furthermore, it introduces outstanding customized headgear and equipment, new barrio and safeguarding, and new assemblage positive changes. |
| {{Statistical mechanics|cTopic=Models}}
| |
| The '''Einstein solid''' is a model of a solid based on two assumptions:
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| * Each atom in the lattice is an independent 3D [[quantum harmonic oscillator]]
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| * All atoms oscillate with the same frequency (contrast with the [[Debye model]])
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| 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''', each atom oscillates independently. [[Albert Einstein|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.
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| == Historical impact ==
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| The original theory proposed by [[Albert Einstein|Einstein]] in 1907 has great historical relevance. The [[heat capacity]] of [[solid]]s 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.
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| By employing Planck's [[Quantization (physics)|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]].
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| | |
| 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 <math>T^3</math> power law, which matches experiment. This modification is called the [[Debye Model]], which appeared in 1912.
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| | |
| ==Heat capacity (microcanonical ensemble)== | |
| [[File:Escv.png|thumb|Heat capacity of an '''Einstein solid''' as a function of temperature. Experimental value of 3''Nk'' is recovered at high temperatures.|right|300px]]
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| The [[heat capacity]] of an object at constant volume ''V'' is defined through the [[internal energy]] ''U'' as
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| :<math>C_V = \left({\partial U\over\partial T}\right)_V.</math>
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| <math>T</math>, the temperature of the system, can be found from the [[entropy]] | |
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| :<math>{1\over T} = {\partial S\over\partial U}.</math>
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| To find the entropy consider a solid made of <math>N</math> atoms, each of which has 3 degrees of freedom. So there are <math>3N</math> [[quantum harmonic oscillator]]s (hereafter SHOs for "Simple Harmonic Oscillators").
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| :<math>N^{\prime} = 3N</math>
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| Possible energies of an SHO are given by
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| :<math>E_n = \hbar\omega\left(n+{1\over2}\right)</math>
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| | |
| or, in other words, the energy levels are evenly spaced and one can define a ''quantum'' of energy
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| | |
| :<math> \varepsilon = \hbar\omega</math>
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| | |
| 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 <math>q</math> quanta of energy among <math>N^{\prime}</math> SHOs. This task becomes simpler if one thinks of distributing <math>q</math> pebbles over <math>N^{\prime}</math> boxes
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| ::[[File:Einstein solids 1.svg]]
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| or separating stacks of pebbles with <math>N^{\prime}-1</math> partitions
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| ::[[File:Einstein solids 2.svg]]
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| or arranging <math>q</math> pebbles and <math>N^{\prime}-1</math> partitions
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| :::[[File:Einstein solids 3.svg]]
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| The last picture is the most telling. The number of arrangements of <math>n</math> objects is <math>n!</math>. So the number of possible arrangements of <math>q</math> pebbles and <math>N^{\prime}-1</math> partitions is <math>\left(q+N^{\prime}-1\right)!</math>. 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 <math>q!</math> identical quanta arrangements, and <math>(N^{\prime}-1)!</math> identical partition arrangements. Therefore, multiplicity of the system is given by
| |
| | |
| :<math>\Omega = {\left(q+N^{\prime}-1\right)!\over q! (N^{\prime}-1)!}</math>
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| | |
| which, as mentioned before, is the number of ways to deposit <math>q</math> quanta of energy into <math>N^{\prime}-1</math> oscillators. [[Entropy]] of the system has the form | |
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| :<math>S/k = \ln\Omega = \ln{\left(q+N^{\prime}-1\right)!\over q! (N^{\prime}-1)!}.</math>
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| <math>N^{\prime}</math> is a huge number—subtracting one from it has no overall effect whatsoever: | |
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| :<math>S/k \approx \ln{\left(q+N^{\prime}\right)!\over q! N^{\prime}!}</math>
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| With the help of [[Stirling's approximation]], entropy can be simplified:
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| :<math>S/k \approx \left(q+N^{\prime}\right)\ln\left(q+N^{\prime}\right)-N^{\prime}\ln N^{\prime}-q\ln q.</math>
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| Total energy of the solid is given by
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| :<math>U = {N^{\prime}\varepsilon\over2} + q\varepsilon,</math>
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| 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.
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| We are now ready to compute the temperature
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| :<math>{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)</math>
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| Elimination of q between the two preceding formulas gives for U:
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| :<math>U = {N^{\prime}\varepsilon\over2} + {N^{\prime}\varepsilon\over e^{\varepsilon/kT}-1}.</math>
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| 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.
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| | |
| Differentiating with respect to temperature to find <math>C_V</math> we obtain:
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| | |
| :<math>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}</math>
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| or
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| <blockquote style="border: 1px solid black; padding:10px;">
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| :<math>C_V = 3Nk\left({\varepsilon\over k T}\right)^2{e^{\varepsilon/kT}\over \left(e^{\varepsilon/kT}-1\right)^2}.</math></blockquote>
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| 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.
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| ==Heat capacity (canonical ensemble)==
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| Heat capacity is obtained through the use of the [[canonical partition function]] of a simple harmonic oscillator (SHO).
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| :<math>Z = \sum_{n=0}^{\infty} e^{-E_n/kT}</math>
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| where
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| :<math>E_n = \varepsilon\left(n+{1\over2}\right)</math>
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| | |
| substituting this into the partition function formula yields
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| | |
| :<math>
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| \begin{align}
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| 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 \\
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| & {} = {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)}.
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| \end{align}
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| </math>
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| | |
| 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 <math>N^{\prime}</math> to get the total. Next, let's compute the average energy of each oscillator | |
| | |
| :<math>\langle E\rangle = u = -{1\over Z}\partial_{\beta}Z</math>
| |
| | |
| where
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| | |
| :<math>\beta = {1\over kT}.</math>
| |
| | |
| Therefore
| |
| | |
| :<math>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).</math>
| |
| | |
| Heat capacity of ''one'' oscillator is then
| |
| | |
| :<math>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)}.</math>
| |
| | |
| 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 <math>C_V = 3NC_V</math>, where the total number of degree of freedom of the solid is three (for the three directional degree of freedom) times <math>N</math>, the number of atoms in the solid. One thus obtains
| |
| | |
| <blockquote style="border: 1px solid black; padding:10px;">
| |
| :<math>C_V = 3Nk\left({\varepsilon\over 2 k T}\right)^2 {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}.</math></blockquote>
| |
| | |
| which is algebraically identical to the formula derived in the previous section.
| |
| | |
| The quantity <math>T_E=\varepsilon / k</math> has the dimensions of temperature and is a characteristic property of a crystal. It is known as the '''Einstein temperature'''.<ref>{{cite book
| |
| | first=Donald | last=Rogers | year=2005
| |
| | page=73 | title=Einstein's other theory: the Planck-Bose-Einstein theory of heat capacity
| |
| | publisher=Princeton University Press | isbn=0-691-11826-4 }}</ref> Hence, the Einstein Crystal model predicts that the energy and heat capacities of a crystal are universal functions of the dimensionless ratio <math>T / T_E</math>. Similarly, the [[Debye model#Debye versus Einstein|Debye model]] predicts a universal function of the ratio <math>T/T_D</math>.
| |
| | |
| ==See also==
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| *[[Kinetic theory of solids]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| * "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]].
| |
| | |
| {{Einstein}}
| |
| | |
| [[Category:Condensed matter physics]]
| |
| [[Category:Albert Einstein]]
| |
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