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CRC Press eBooks

Ideal Fermi Gas

Michael J.R. Hoch

Published Apr 19, 2016• 0 citations

Abstract

To obtain an expression for the Fermi energy εF, dened in Section 12.3, consider a system of N fermions of spin 12 and mass m in a container of volume V at T = 0 K. In zero-applied magnetic eld, the mS = ± 12 spin states are degenerate, and it is convenient to introduce a degeneracy factor in expressions that involve the density of single particle states. e fermions ll the lowest energy states up to the energy εF = µ. By equating the number of particles to the number of states in the range 0 to εF, an expression for εF is obtained. In general, for fermions, N n er r r= ∑ = ∑ +−1 1/( ).( )b e m Conversion of the sum to an integral gives N f= ∞∫2 0 r e e e( ) ( ) ,d (13.1) where ρ(ε) is the density of states and the factor 2 is the spin degeneracy factor. f ee b e m( ) = +−1 1/( )( ) is the Fermi function, which at T = 0 K has the form f(ε) = 1 for e eʺ F and f(ε) = 0 for e e> F. is may be seen in Figure 12.1. At T = 0 K, Equation 13.1 may therefore be written as N = ∫2 0 r e ee ( ) .dF (13.2) From Equation 4.14, the density of single particle states for particles in a box is r e p e( ) ( / )( / ) / /= V m4 22 2 3 2 1 2 , and substitution in Equation 13.2 gives N V m V m = ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ e pF = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 m N V .

DOI: 10.1201/9781439850541-17

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To obtain an expression for the Fermi energy εF, dened in Section 12.3, consider a system of N fermions of spin 12 and mass m in a container of volume V at T = 0 K. In zero-applied magnetic eld, the mS = ± 12 spin states are degenerate, and it is convenient to introduce a degeneracy factor in expressions that involve the density of single particle states. e fermions ll the lowest energy states up to the energy εF = µ. By equating the number of particles to the number of states in the range 0 to εF, an expression for εF is obtained. In general, for fermions, N n er r r= ∑ = ∑ +−1 1/( ).( )b e m Conversion of the sum to an integral gives N f= ∞∫2 0 r e e e( ) ( ) ,d (13.1) where ρ(ε) is the density of states and the factor 2 is the spin degeneracy factor. f ee b e m( ) = +−1 1/( )( ) is the Fermi function, which at T = 0 K has the form f(ε) = 1 for e eʺ F and f(ε) = 0 for e e> F. is may be seen in Figure 12.1. At T = 0 K, Equation 13.1 may therefore be written as N = ∫2 0 r e ee ( ) .dF (13.2) From Equation 4.14, the density of single particle states for particles in a box is r e p e( ) ( / )( / ) / /= V m4 22 2 3 2 1 2 , and substitution in Equation 13.2 gives N V m V m = ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ e pF = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 m N V .

At low temperatures, the number of particles in the ground state, with ε = 0, becomes large, and care must be exercised in the evaluation of the integral. e density of states given by Equation 4.14 has the form ρ(ε) ∝ ε1/2, which gives ρ(ε) = 0 for ε = 0. e ground state is therefore excluded in the integral. For fermions, this introduces a negligible error because the exclusion principle prevents multiple occupancy of a particular state but for bosons this is not the case. At very low temperatures, the number of bosons in the ground state may be expected to become macroscopically large.

ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~

The laser cooling mechanisms described in Chapter 4 operate irrespective of the statistics of the atom, and they can therefore be used to cool Fermi species. The statistics of a neutral atom is determined by the number of neutrons in the nucleus, which must be odd for a fermionic atom. Since alkali atoms have odd atomic number Z , their fermionic isotopes have even mass number A . Such isotopes are relatively less abundant than those with odd A since they have both an unpaired neutron and an unpaired proton, which increases their energy by virtue of the odd–even effect. In early experiments, 40 K and 6 Li atoms were cooled to about one-quarter of the Fermi temperature. More recently, fermionic alkali atoms have been cooled to temperatures well below one-tenth of their Fermi temperature, and in addition a degenerate gas of the fermionic species 173 Yb (with I = 5/2) has been prepared. In the classical limit, at low densities and/or high temperatures, clouds of fermions and bosons behave alike. The factor governing the importance of quantum degeneracy is the phase-space density ϖ introduced in Eq. (2.24), and in the classical limit ϖ ≪ 1. When ϖ becomes comparable with unity, gases become degenerate: bosons condense in the lowest single-particle state, while fermions tend towards a state with a filled Fermi sea. As one would expect on dimensional grounds, the degeneracy temperature for fermions – the Fermi temperature T F – is given by the same expression as the Bose–Einstein transition temperature for bosons, apart from a numerical factor of order unity.