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Fermi energy:

[tex]E_f =\frac{\hbar^2 \pi^2}{2m L^2} n_f^2[/tex]

The number of states with energy less than [tex]E_f[/tex] is equal to the number of states that lie within a sphere of radius [tex]|\vec{n}_f|[/tex] in the region of n-space where nx, ny, nz are positive. In the ground state this number equals the number of fermions in the system:

[tex]N = 2\times\frac{1}{8}\times\frac{4}{3} \pi n_f^3[/tex]

The factor of two is because there are two spin states, the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive, the octant of positive quantum numbers.

[tex]n_f=\left(\frac{3 N}{\pi}\right)^{1/3}[/tex]

According to Wikipedia (ref. 1), the degenerate matter total energy is obtained by integrating the Fermi energy of each state over the allowed states:

[tex]E_t = {\int_0}^{N_0} E_f(N) dN = {3\over 5} N_0 E_f[/tex]

However, according to ref. 2, total energy is:

[tex]E_t = \frac{1}{8} \int_0 ^{N_0} E_f(N) dN[/tex]

1/8 is from the octant of positive quantum numbers in a sphere region where all n are positive.

My question is, why is the octant of positive quantum numbers included in the ref. 2 integration, but not included in the Wikipedia integration?

If the octant of positive quantum numbers is included in the integration, then why is the number of spin states also not included?

Also, I noticed that the Fermi energy equation in ref. 1 does not match the equation in ref. 2. Does this mean that the ref. 2 solution is incorrect?

Reference:

http://en.wikipedia.org/wiki/Fermi_energy" [Broken]

http://www.sfu.ca/~boal/385lecs/385lec18.pdf" [Broken]

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# Degenerate matter total energy

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