Einstein solid state model exercise

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The discussion revolves around solving the Einstein solid state model using the canonical ensemble approach to find the partition function. A participant expresses uncertainty about the correctness of their calculations and seeks guidance on verification methods. Another contributor points out the need to clarify the addition of terms in the partition sum, emphasizing that the two sums over n_r and n_l should factorize. This highlights the importance of correctly applying the principles of statistical mechanics in the context of the problem. The conversation underscores the need for clear understanding of partition functions in equilibrium systems.
besebenomo
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Homework Statement
The system is composed of a set of N non-interacting harmonic oscillators which can be described as a two-dimensional Einstein solid model in the x, y plane. Suppose that each oscillator has mass m and charge q and that there is a magnetic field HH directed along the z direction.
Relevant Equations
Compute (when the system at thermal equilibrium at temperature T):
Internal Energy
Magnetization
Specific heat capacity\bigm
ass_1.png


I tried to solve it considering the canonical ensemble (since the system is at the equilibrium with temperature T) and started finding the partition function:
CodeCogsEqn(3).png
The problem is I am not sure if I have done it correctly and need help because I don't really know where to check.
 

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Why are you adding the two terms in your expression for the partition sum? You just have to use the expression and do the two sums over ##n_r## and ##n_l##. Obviously these two sums factorize!
 

Thread 'Help needed with Elliptic Integrals'

I want to find the solution to the integral ##\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}## I can see that ##\frac{d^2u}{d\theta^2} = A +Bu+Cu^2## is a Weierstrass elliptic function, which can be generated from ##\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3## (A = 0, B=-1, C=3) So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck. TerryW
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