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Fermi sea energy and ideal degeneracy pressure

Physical scope

A three-dimensional ideal Fermi sea at fixed density, calculating its Fermi wave number, mean energy per particle, and grand thermodynamic pressure from the linked zero-temperature theorem. It is not a material equation of state or measured pressure.

Example

For degeneracy $g = 2$ and number density $ρ > 0$ in the declared homogeneous ideal model, the zero-temperature occupied region is the ball $∣ k ∣ < k_{F}$ , where $k_{F} = (3 π^{2} ρ)^{1 / 3}$ . The mean kinetic energy per particle is $3 E_{F} / 5$ and the ideal grand thermodynamic pressure is $2 ρ E_{F} / 5$ , with $E_{F} = ℏ^{2} k_{F}^{2} / (2 m)$ . These values are model predictions under the ideal-gas assumptions.

Verification

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), $g = 2$ , $ρ > 0$ , $α = ℏ^{2} / (2 m)$ , and the zero-temperature formulas of Ideal Fermi continuum and zero-temperature endpoint, with its SI convention: $m > 0$ is the mass parameter in kg, $ℏ > 0$ is the reduced Planck constant parameter in J s, $α$ has units J m $^{2}$ , $ρ$ is number per volume in m $^{- 3}$ , $k$ and $k_{F}$ have units m $^{- 1}$ , $E_{F}$ and $u / ρ$ have units J, $u$ has units J m $^{- 3}$ , and $p$ has units Pa $=$ J m $^{- 3}$ . The internal multiplicity $g = 2$ is dimensionless.

[F1] Ideal Fermi continuum and zero-temperature endpoint, Statement: $k_{F} = (6 π^{2} ρ / g)^{1 / 3}$ , $u = (3 / 5) ρ E_{F}$ , and $p = 2 u / 3$ .

1.1givenF1algebra

Setting $g = 2$ in [F1] gives $k_{F} = (3 π^{2} ρ)^{1 / 3}$ and $E_{F} = α k_{F}^{2} = ℏ^{2} k_{F}^{2} / (2 m)$ . Dividing the energy density $u$ by the number density $ρ$ gives mean energy per particle $3 E_{F} / 5$ .

2.1step 1.1F1algebra∎

The pressure identity in [F1] yields $p = (2 / 3) (3 / 5) ρ E_{F} = 2 ρ E_{F} / 5$ . The result applies to the ideal continuum model and supplies no material-specific or empirical pressure.

Depends on

Dependency tree · 2 levels

Sources

Mehran Kardar, MIT OpenCourseWare 8.333, Lecture 24, degenerate Fermi gas