14. Ideal Quantum Gases II: Fermions

Transcription

1 University of Rhode Island Equilibrium Statistical Physics Physics Course Materials Ideal Quantum Gases II: Fermions Gerhard Müller University of Rhode Island, Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4. License. Follow this and additional works at: Abstract Part fourteen of course materials for Statistical Physics I: PHY525, taught by Gerhard Müller at the University of Rhode Island. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "4. Ideal Quantum Gases II: Fermions" (25). Equilibrium Statistical Physics. Paper. This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Equilibrium Statistical Physics by an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu.

2 Contents of this Document [ttc4] 4. Ideal Quantum Gases II: Fermions Fermi-Dirac functions. [tsl42] Ideal Fermi-Dirac gas: equation of state and internal energy. [tln69] Ideal Fermi-Dirac gas: chemical potentia.l [tsl43] FD gas in D dimensions: chemical potential I. [tex7] FD gas in D dimensions: chemical potential II. [tex8] Ideal Fermi-dirac gas: average level occupancy. [tsl44] Ideal Fermi-Dirac gas: isochores I. [tsl46] FD gas in D dimensions: statistical interaction pressure. [tex9] Ideal Fermi-Dirac gas: isotherms. [tln7] FD gas in D dimensions: isotherm and adiabate. [tex2] FD gas in D dimensions: ground-state energy. [tex2] Ideal Fermi-Dirac gas: heat capacity. [tsl45] FD gas in D dimensions: heat capacity at high temperature. [tex] FD gas in D dimensions: heat capacity at low temperature. [tex] Ideal Fermi-Dirac gas: isochores II. [tln73] Ideal Fermi-Dirac gas: phase diagram in infinite dimensions. [tln74] Stable white dwarf. [tex2] Unstable white dwarf. [tex22]

3 Fermi-Dirac functions [tsl42] f n (z) dx x n Γ(n) z e x +, z < Series expansion: f n (z) = l= ( ) l zl ln, z. Special cases: f (z) = z + z, f (z) = ln( + z), f (z) = z. Recurrence relation: zf n (z) = f n (z), n. Asymptotic expansion for z : [ (ln z)n f n (z) = + 2n(n ) (n k + ) Γ(n + ) = (ln z)n Γ(n + ) k=2,4,... [ + n(n ) π2 (ln z) n(n )(n 3) 7π4 36 (ln z) ( ) ] ζ(k) 2 k (ln z) k ] 2.5 f 5/2 2 f 2 f n (z).5 f 3/2 f f / z

4 Ideal Fermi-Dirac gas: equation of state and internal energy [tln69] Conversion of sums into integrals by means of density of energy levels: D(ǫ) = gv ( m ) D/2 ǫ D/2, V = L D. Γ(D/2) 2π 2 Fundamental thermodynamic relations for FD gas: pv k B T = k ln ( ) + ze βǫ k = dǫ D(ǫ) ln ( + ze βǫ) = gv f λ D D/2+ (z), T N = k z e βǫ k + = dǫ D(ǫ) z e βǫ + = gv f λ D D/2 (z), T U = k ǫ k z e βǫ k + = dǫ D(ǫ)ǫ z e βǫ + = D 2 k BT gv f λ D D/2+ (z). T Note: The range of fugacity has no upper limit: z. The chemical potential µ is unrestricted. The factor g is included to account for any existing level degeneracy due to internal degrees of freedom (e.g. spin) of the fermions. Equation of state (with fugacity z in the role of parameter): pv Nk B T = f D/2+(z) f D/2 (z) D = pv/nk B T Maxwell-Boltzmann z

5 Ideal Fermi-Dirac gas: chemical potential [tsl43] Fugacity z from x = f D/2 (z), where x = λd T v, v. = gv N, λ T = h 2 2πmk B T. Chemical potential [tex7]: µ k B T v = T T v ln z, T T v = [f D/2 (z)] 2/D. 2 D = µ/k B T F T/T F Reference temperature: k B T v = Λ v 2/D, Λ. = h2 2πm. For a complete list of reference values see [tln7]. Fermi energy: lim T µ = ǫ F = k B T F. Fermi temperature: T F T v = [Γ(D/2 + )] 2/D D D 2e.

6 [tex7] FD gas in D dimensions: chemical potential I (a) Start from the fundamental thermodynamic relation N = (gv/λ D T )f D/2(z) for the ideal Fermi- Dirac gas in D dimensions and use the reference temperature k B T v = Λ/v 2/D, v =. gv/n, Λ =. h 2 /2πm to derive the following parametric expression for the dependence on temperature T of the chemical potential µ: µ k B T v = T T v ln z, T T v = [f D/2 (z)] 2/D. (b) Derive the following expression for the Fermi energy ɛ F and the Fermi temperature T F : µ(t ) lim = ɛ F = T F = [Γ(D/2 + )] 2/D. T k B T v k B T v T v (c) Show that this result includes the familiar result, ɛ F = (h 2 /2m)(3N /4πgV ) 2/3 for D = 3. Solution:

7 [tex8] FD gas in D dimensions: chemical potential II Start from the results derived in [tex7] to infer the following expressions for the fugacity z and the chemical potential µ at T T F : ln z T F T, µ k B T F ( ) 2 π2 T (D 2). 2 T F Solution:

8 Ideal Fermi-Dirac gas: average level occupancy [tsl44] Average occupancy of -particle state at energy ǫ if system (with fixed N, V ) is at temperature T: n ǫ = e β(ǫ µ)+ with µ(t) from [tex7]. T/T F = T/T F =.8.6 <n ε >.4.2 T/T F = ε/ε F Note: n ǫ = 2 occurs at ǫ = µ(t). Limit T : µ(t) ǫ F, n ǫ Θ(ǫ F ǫ).

9 Ideal Fermi-Dirac gas: isochores I [tsl46] Reference values for temperature and pressure: k B T v = Λ v 2/D, T F T v = p F p v = p v = k BT v v ; [ Γ ( D 2 + Λ =. h2 2πm, v =. gv N. )] 2/D D D 2e. Isochore: p = T f D/2+ (z) p F T F f D/2 (z), [ ( ) 2/D T D = Γ T F 2 + f D/2 (z)]. Low-temperature limit [tex9]: lim T p p F = ( ) D 2 +. High-temperature asymptotic regime [tex9]: [ pv T [ ( )] ( ) ] D/2 D + 2 D/2+ Γ Nk B T F T F 2 + TF. T The excess pressure relative to the Maxwell-Boltzmann line may be called a manifestation of statistical interaction pressure D = p/p F Maxwell-Boltzmann T/T F

10 [tex9] FD gas in D dimensions: statistical interaction pressure Consider the isochore of an ideal Fermi-Dirac gas in D dimensions, as given by the parametric relation p = T f D/2+ (z) p v T v f D/2 (z), T = [ f D/2 (z) ] 2/D. T v where k B T v = Λ/v 2/D, p v = k B T/v, Λ. = h 2 /2πm, v. = gv/n. The upward deviation of this result from the Maxwell-Boltzmann result, p/p v = T/T v, is a manifestation of repulsive statistical interaction between fermions. (a) Calculate the high-t asymptotic dependence of p/p v on T/T v including the leading correction to MB behavior. (b) Calculate the low-t limit of p/p v. (c) Calculate the low-t limit of p/p F, where T F = T v [Γ(D/2 + )] 2/D is the Fermi temperature and p F = k B T F /v the associated reference pressure. (d) Compare the differently scaled statistical interaction pressures p/p v and p/p F at T = in the limit D. Solution:

11 Ideal Fermi-Dirac gas: isotherms [tln7] Reference values for reduced volume v. = gv/n and pressure p: v T = λ D T, p T = gk B T/λ D T. Parametric expression for isotherm: p p T = f D/2+ (z), v v T = [f D/2 (z)]. Isotherm at low density [tex2]: pv = const, v v T. Isotherm at high density [tex2]: pv (D+2)/D = const, v v T D = p/p T 2.5 Maxwell-Boltzmann v/v T

12 [tex2] FD gas in D dimensions: isotherm and adiabate (a) Show that the isotherm of an ideal Fermi-Dirac gas in D dimensions is described by the parametric relation p v = f D/2+ (z), = [f D/2 (z)], v T p T where v T = λ D T and p T = gk B T/λ D T are convenient reference values and v = gv/n is the reduced volume. (b) Show that the adiabate is described by the relation pv (D+2)/D = const for all values of v/v. (c) Show that the relation for the isotherm approaches Boyle s law, pv = const, for v v T and that it approaches the adiabate, pv (D+2)/D = const, for v v T. Solution:

13 [tex2] FD gas in D dimensions: ground-state energy Given are the following expressions for the average number of particles, the average energy, the average occupation number at T =, and the density of states for an ideal Fermi-Dirac gas in D dimensions: N = k n k, U = k n k ɛ k, n k = Θ(ɛ F ɛ k ), D(ɛ) = gv Γ(D/2) ( ) D/2 2πm ɛ D/2. h 2 Derive from these expressions the following results for the dependence of the ground-state energy per particle, U /N, on the Fermi energy ɛ F and for the dependence of the ground-state energy density U /V on the particle density N /V : U N = D D + 2 ɛ F, U V ( ) (D+2)/D N. V Solution:

14 Ideal Fermi-Dirac gas: heat capacity [tsl45] Internal energy: U = D 2 Nk BT f D/2+(z) f D/2 (z). Heat capacity [use zg n (z) = g n (z) for n ]: ( ) C V D = Nk B 2 + D2 fd/2+ (z) D2 f D/2+ (z) 4 f D/2 (z) 4 f D/2 (z). Low-temperature asymptotic behavior: C V D π2 Nk B 6 T T F. High-temperature asymptotic behavior: [ C V D D/2 Nk B 2 2 D/2 Γ(D/2) ( TF T ) D/2 ] D = 3 C V /Nk B.8.6 D = 2.4 D = T/T F

15 [tex] FD gas in D dimensions: heat capacity at high temperature The internal energy of the ideal Fermi-Dirac gas in D dimensions is given by the expression, U = N k B T D 2 f D/2+ (z) f D/2 (z). (a) Use this result to derive the following expression for the heat capacity C V = ( U/ T ) V N : ( ) C V D = N k B 2 + D2 fd/2+ (z) D2 4 f D/2 (z) 4 f D/2+ (z) f D/2 (z). Use the derivative / T of the result f D/2 (z) = N λ D T /gv with V = LD to calculate any occurrence of ( z/ T ) V N in the derivation. Use the recursion relation zf n(z) = f n (z) for n to further simplify the results pertaining to D 2. (b) Infer from this result the leading correction to the Maxwell-Boltzmann result, C V = (D/2)N k B, at high temperature. Solution:

16 [tex] FD gas in D dimensions: heat capacity at low temperature Use the results of [tex8] and [tex] to determine the low-temperature asymptotic behavior, C V D π2 N k B 6 T T F, of the heat capacity of the ideal Fermi-Dirac gas in D dimensions. Solution:

17 Ideal Fermi-Dirac gas: isochores II [tln73] Reference values for temperature and pressure: k B T v = Λ v 2/D, p v = k BT v v ; Λ. = h2 2πm, v. = gv N. Noncommuting limits z, D : z <, D : p = T f D/2+ (z) p v T v f D/2 (z) D T T v (ideal MB gas). D, z with D/2 = r ln z, r : p p v = f D/2+(z) D [f D/2 (z)] +2/D e + 2/D, T T v = [ f D/2 (z) ] 2/D D D 2 e ln z (pure Fermi sea)..2 D = p/p v T/T v

18 Ideal FD gas: phase diagram in D [tln74] Equation of state: { kb T, T > T pv = c (ideal MB gas) k B T c, T < T c (pure Fermi sea) Transition temperature: k B T c = Λ e, Λ. = h2 2πm. p v k B T 5 2

19 [tex2] Stable white dwarf Consider a burnt-out white dwarf star. For simplicity we assume that it consists of equal numbers N of electrons, protons, and neutrons. The electrons form a fully degenerate, nonrelativistic Fermi gas that prevents the star from collapsing into a neutron star or a black hole. (a) Under the assumption that the kinetic energy is predominantly due to the electrons and that the potential energy is predominantly gravitational in nature, show that the total energy of the star depends on N and R (radius) as follows: E = E kin + E pot = 3 2 m e ( ) 2/3 9π N 5/3 4 R m2 ng N 2 R, where m e, m n are the electron and neutron masses, and G is the universal gravitational constant. (b) Using a star of solar mass, m.99 3 kg, find the radius R wd in units of the solar radius, R m. Solution:

20 [tex22] Unstable white dwarf Consider a burnt-out white dwarf star of the same composition as described in [tex2] but with N so large that most of the electrons are ultrarelativistic, ɛ cp = kc, in the fully degenerate state. (a) Under similar assumptions as in [tex2] show that the expression of the total energy now reads E = E kin + E pot = c 3π ( ) 4/3 9π N 4/3 4 R 2 5 m2 ng N 2 R, where c is the speed of light. (b) Find the critical mass in units of the solar mass, m c /m, beyond which this star is unstable and thus prone to a gravitational collapse into a neutron star or a black hole. Solution: