Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 68

For the sake of simplicity, we study the ideal gas properties without taking into account the spin, which assumes that all particles are in the same spin state (the spin can easily be accounted for by adding the contributions of the different individual spin states). For a large physical system, the energy levels are very close and the discrete sum in (46) can be replaced by an integral. This leads to:

(47)

where the function is defined as (the subscript ig stands for ideal gas):

(48)

Figure 2 shows the variations of the function as a function of μ, for fixed values of β and the volume .

To deal with dimensionless quantities, one often introduces the “thermal wavelength” λ_T as:

(49)

We can then use in the integral of (48) the dimensionless variable:

(50)

and write:

(51)

Figure 2: Variations of the particle number for an ideal fermion gas, as a function of the chemical potential μ, and for different fixed temperatures T (β = 1/(kBT)). For T = 0 (lower dashed line curve), the particle number is zero for negative values of μ, and proportional to μ3/2 for positive values of μ. For a non-zero temperature T = T₁ (thick line curve), the curve is above the previous one, and never goes to zero. Also shown are the curves obtained for temperatures twice (T = 2T₁) and three times (T = 3T₁) as large. The units chosen for the axes are the thermal energy kBT₁ associated with the thick line curve, and the particle number , where λ_T1 is the thermal wavelength at temperature T₁.

Largely negative values of μ correspond to the classical region where the fermion gas is not degenerate; the classical ideal gas equations are then valid to a good approximation. In the region where μ ≫ kBT, the gas is largely degenerate and a Fermi sphere shows up clearly in the momentum space; the total number of particles has only a slight temperature dependence and varies approximately as μ3/2.

This figure was kindly contributed by Genevieve Tastevin.

with²:

(52)

where, in the second equality, we made the change of variable:

(53)

Note that the value of I_3/2 only depends on a dimensionless variable, the product βμ.

If the particles have a spin 1/2, both contributions and from the two spin states must be added to (46); in the absence of an external magnetic field, the individual particle energies do not depend on their spin direction, and the total particle number is simply doubled:

(54)