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11.2 Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by A 572 2 kT | ... 12

11.2 Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by A 572 2 kT | ... 12

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**11.2** Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by \[ \frac{A}{N} = \frac{3}{5} \epsilon_F \left[ 1 - \frac{5 \pi^2}{12} \left( \frac{kT}{\epsilon_F} \right)^2 + \cdots \right] \] Where: - \( A \) is the Helmholtz free energy. - \( N \) is the number of particles. - \( \epsilon_F \) is the Fermi energy. - \( k \) is the Boltzmann constant. - \( T \) is the temperature. This formula expresses the Helmholtz free energy per particle for an ideal Fermi gas at low temperatures, using an expansion in terms of a small parameter \(\left(\frac{kT}{\epsilon_F}\right)\). The terms following the ellipsis (\(\cdots\)) represent higher-order corrections that become significant at higher temperatures.
Transcribed Image Text:**11.2** Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by \[ \frac{A}{N} = \frac{3}{5} \epsilon_F \left[ 1 - \frac{5 \pi^2}{12} \left( \frac{kT}{\epsilon_F} \right)^2 + \cdots \right] \] Where: - \( A \) is the Helmholtz free energy. - \( N \) is the number of particles. - \( \epsilon_F \) is the Fermi energy. - \( k \) is the Boltzmann constant. - \( T \) is the temperature. This formula expresses the Helmholtz free energy per particle for an ideal Fermi gas at low temperatures, using an expansion in terms of a small parameter \(\left(\frac{kT}{\epsilon_F}\right)\). The terms following the ellipsis (\(\cdots\)) represent higher-order corrections that become significant at higher temperatures.
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