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In Wikipedia's article on Fermi Gases, they have the following equation for the chemical potential:

$$\mu = E_0 + E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 - \frac{\pi^4}{80} \left(\frac{kT}{E_F}\right)^4 + \cdots \right]$$ where $E_0$ is the potential energy per particle, $k$ is the Boltzmann constant and $T$ is the temperature.

I don't understand how they get the third term, particularly the 1/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1/12 factor (from a binomial expansion of $$\left( 1 + \frac{\pi^{2}}{8} \left(\frac{\tau}{E}\right)^{2}\right)^{-2/3}. $$

However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1/80.

In Wikipedia's article on Fermi Gases, they have the following equation for the chemical potential:

$$\mu = E_0 + E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 - \frac{\pi^4}{80} \left(\frac{kT}{E_F}\right)^4 + \cdots \right]$$ where $E_0$ is the potential energy per particle, $k$ is the Boltzmann constant and $T$ is the temperature.

I don't understand how they get the third term, particularly the 1/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1/12 factor from a binomial expansion of $$\left( 1 + \frac{\pi^{2}}{8} \left(\frac{\tau}{E}\right)^{2}\right)^{-2/3} $$

However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1/80.

In Wikipedia's article on Fermi Gases, they have an equation for the chemical potential; http://en.wikipedia.org/wiki/Fermi_gas .

I don't understand how they get the third term, particularly the 1/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1/12 factor (from a binomial expansion of $$\left( 1 + \frac{\pi^{2}}{8} \left(\frac{\tau}{E}\right)^{2}\right)^{-2/3}. $$

However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1/80.

In Wikipedia's article on Fermi Gases, they have the following equation for the chemical potential:

$$\mu = E_0 + E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 - \frac{\pi^4}{80} \left(\frac{kT}{E_F}\right)^4 + \cdots \right]$$ where $E_0$ is the potential energy per particle, $k$ is the Boltzmann constant and $T$ is the temperature.

I don't understand how they get the third term, particularly the 1/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1/12 factor (from a binomial expansion of $$\left( 1 + \frac{\pi^{2}}{8} \left(\frac{\tau}{E}\right)^{2}\right)^{-2/3}. $$

However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1/80.

In Wikipedia's article on Fermi Gases, they have an equation for the chemical potential; http://en.wikipedia.org/wiki/Fermi_gas .

I don't understand how they get the third term, particularly the 1/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1/12 factor (from a binomial expansion of $$ 1 + \frac{1}{8} \left(\tau\frac{\pi}{E}\right)^{-2/3}. $$

However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1/80.

In Wikipedia's article on Fermi Gases, they have an equation for the chemical potential; http://en.wikipedia.org/wiki/Fermi_gas .

I don't understand how they get the third term, particularly the 1/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1/12 factor (from a binomial expansion of $$\left( 1 + \frac{\pi^{2}}{8} \left(\frac{\tau}{E}\right)^{2}\right)^{-2/3}. $$

However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1/80.