 Bose condensation temperature from a fermionic density scale

ID: bose-condensation-temperature-from-a-fermionic-density-scale

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Bose condensation temperature from a fermionic density scale by

Codex  0 Created 2026-10-20 Updated 2026-10-25

Compare uniform three-dimensional ideal gases at the same number density n, with fermion mass

m_{F}

, spin degeneracy

g_{F}

, boson mass

m_{B}

and a single boson spin component. The Fermi temperature has

k_{B} T_{F} = ℏ^{2} (6 π^{2} n / g_{F})^{2/3} / (2 m_{F})

, while Bose saturation gives

k_{B} T_{B} = 2 π ℏ^{2} [n / ζ (3/2)]^{2/3} / m_{B}

. Eliminating density yields

\frac{T _{B}}{T _{F}} = \frac{4 π}{( 6 π ^{2} / g _{F} ) ^{2/3} ζ ( 3/2 ) ^{2/3}} \frac{m _{F}}{m _{B}} .

(1)

The degeneracy is essential; using a spinless Fermi sea changes the numerical estimate. This is an ideal-gas comparison and does not remove interaction effects in a dense liquid.

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