13. Ideal Quantum Gases I: Bosons

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1 University of Rhode Island Equilibrium Statistical Physics Physics Course Materials 5 3. Ideal Quantum Gases I: Bosons Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License his work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4. License. Follow this and additional works at: Abstract Part thirteen of course materials for Statistical Physics I: PHY55, taught by Gerhard Müller at the University of Rhode Island. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "3. Ideal Quantum Gases I: Bosons" (5). Equilibrium Statistical Physics. Paper. his 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 [ttc3] 3. Ideal Quantum Gases I: Bosons Bose-Einstein functions. [tsl36] Ideal Bose-Einstein gas: equation of state and internal energy. [tln67] BE gas in D dimensions I: fundamental relations. [tex3] Reference values for, V/N, and p. [tln7] Bose-Einstein condensation. [tsl38] Ideal Bose-Einstein gas: isochores. [tsl39] BE gas in D dimensions II: isochore. [tex4] BE gas in D dimensions III: isotherm and isobar. [tex5] Bose-Einstein gas: isotherms. [tsl4] Bose-Einstein gas: isobars. [tsl48] Bose-Einstein gas: phase diagram. [tln7] Bose-Einstein heat capacity. [tsl4] BE gas in D dimensions IV: heat capacity at high temperature. [tex97] BE gas in D dimensions V: heat capacity at low temperature. [tex6] BE gas in D dimensions VI: isothermal compressibility. [tex8] BE gas in D dimensions VII: isobaric expansivity. [tex9] BE gas in D dimensions VIII: speed of sound. [tex3] Ultrarelativistic Bose-Einstein gas. [tex98] Blackbody radiation. [tln68] Statistical mechanics of blackbody radiation. [tex5]

3 Bose Einstein functions [tsl36] Special cases: g n (z) Γ(n) g (z) dx x n z e x l z l ln, z. z z, g (z) ln( z), g (z) z. Riemann zeta function: Special values: g n () ζ(n). l l n. ζ(), Recurrence relation: ζ() π 6 π4 π6, ζ(4), ζ(6) zg n(z) g n (z), n. Singularity at z for non-integer n: g n (α) Γ( n)α n ( ) l + ζ(n l)α l, l! l α. lnz..5 g / g g 3/ g n (z).5 g g 5/ z

4 Ideal Bose-Einstein gas: equation of state and internal energy [tln67] Conversion of sums into integrals by means of density of energy levels [tex3]: V ( m ) D/ D(ǫ) ǫ D/, V L D. Γ(D/) π Fundamental thermodynamic relations for BE gas: pv k B k ln ( ) ze βǫ k dǫ D(ǫ) ln ( ze βǫ) V g λ D D/+ (z), N k U k dǫ z e βǫ k ǫ k z e βǫ k D(ǫ) z e βǫ V g λ D D/ (z), z <, D(ǫ)ǫ dǫ z e βǫ D k B V g λ D D/+ (z). Warning: he range of fugacity is limited to the interval z. At z, the expression for N must be amended by an additive term z/( z) to account for the possibility of a macroscopic population of the lowest energy level (at ǫ ). his amendment is only necessary for dimensionalities D >, i.e. for the cases with lim ǫ D(ǫ). Equation of state (with fugacity z in the role of parameter): pv Nk B g D/+(z) g D/ (z), z <. pv/nk B.8.6 D z

5 [tex3] BE gas in D dimensions I: fundamental relations From the expressions for the grand potential and the density of energy levels of an ideal Bose- Einstein gas in D dimensions and confined to a box of volume V L D with rigid walls, Ω(, V, µ) k B k ln( ze βɛ k ), D(ɛ) V ( m ) D/ ɛ D/ Γ(D/) π, derive the fundamental thermodynamic relations at fugacity z < in terms of the Bose-Einstein functions g n (z) and the thermal wavelength λ h /πmk B as follows: pv k B V λ D g D/+ (z), N V λ D g D/ (z), U D k B V λ D g D/+ (z). Solution:

6 Reference Values for, V/N, and p [tln7] he reference values introduced here are based on. h (i) thermal wavelength: λ Λ πmk B k B, (ii) MB equation of state: pv k B, v. V/N. Λ h πm. he reference values for k B, v, and p in isochoric, isothermal, and isobaric processes are k B v Λ v /D p v Λ v /D+ (v const.) v ( ) D/ ( ) D/+ Λ kb p Λ ( const.) k B Λ ( p ) /(D+) k B p Λ v p Λ ( ) D/(D+) Λ (p const.) p hese reference values are useful for bosons and fermions. Universal curves for isochores, isotherms, and isobars: p/p v versus / v at v const. p/p versus v/v at const. v/v p versus / p at p const. For fermions we will introduce alternative reference values based on the chemical potential (Fermi energy).

7 Bose-Einstein condensation [tsl38] Particles in the gas phase and in the Bose-Einstein condensate (BEC): N V g λ D D/ (z) + z z N gas + N BEC. Consider process at v const. Onset of macroscopic population of the lowest energy level begins when the fugacity locks in to the value z : z O(), z <, z O(N), z. c : c : N gas N, N BEC N. N gas N N BEC N [V/λD ]ζ(d/) ( ) D/ [V/λ D c ]ζ(d/), c N gas N ( ) D/. c.8 N BEC /N.6.4 D / v

8 Ideal Bose-Einstein gas: isochores [tsl39] Isochore at c [tex4]: p p v g D/+(z) [ gd/ (z) ] /D+, v [ g D/ (z) ] /D. Isochore at c (also valid asymptotically for v in D ): Critical temperature: ( ) D/+ p ζ(d/ + ). p v v c v [ζ(d/)] /D High-temperature asymptotic behavior: [ p p v v D/+ D D.57 D 3 D ( v ) D/ ]...8 p/p v.6.4. Maxwell-Boltzmann 3 C / v D / v

9 [tex4] BE gas in D dimensions II: isochore (a) From the fundamental thermodynamic relations for the Bose-Einstein gas in D dimensions (see [tln67]), derive the following parametric expression for the isochore at c : p g D/+ (z) p [ v gd/ (z) ], [ g /D+ D/ (z) ] /D, v where k B v Λv /D and p v Λv /D+ with Λ. h /πm are convenient reference values. (b) Calculate the leading correction to the Maxwell-Boltzmann result at high temperature. (c) Calculate the exact dependence of p/p v on / v at c in D >. Show that this result also holds asymptotically for v in dimensions D and D. Solution:

10 [tex5] BE gas in D dimensions III: isotherm and isobar (a) From the fundamental thermodynamic relations for the Bose-Einstein gas in D > dimensions (see [tln67]), derive the following expressions for the isotherm at v > v c and the isobar at c : p p g D/+ (z), v v [g D/ (z)] ; [ v gd/+ (z) ] D/(D+), v p g D/ (z) p [ g D/+ (z) ] /(D+). where v (Λ/k B ) D/, p Λ(k B /Λ) D/+, k B p Λ(p/Λ) /(D+), v p (Λ/p) D/(D+) with Λ. h /πm are convenient reference values for temperature and pressure and reduced volume. (b) Calculate the leading correction to the Maxwell-Boltzmann result for the isotherm at low density and for the isobar at high temperature. Solution:

11 Ideal Bose-Einstein gas: isotherms [tsl4] For D > we must again distinguish two regimes. At v > v c, all bosons are in the gas phase. At v < v c, a BEC is present. Only the bosons in the gas phase contribute to the pressure. Isotherm at v v c λ D /ζ(d/): p p g D/+ (z), v v [g D/ (z)]. Isotherm at v v c : p p p c p ζ(d/ + ).6 D.645 D.34 D 3 D Critical (reduced) volume: v c v [ζ(d/)] D D.383 D 3 D 3.5 D Maxwell-Boltzmann p/p v/v

12 Ideal Bose-Einstein gas: isobars [tsl48] A phase transition at c > takes place in all dimensions D. However, the existence of a BEC requires v c >, which is realized only for D >. Isobar at > c : v v p [ gd/+ (z) ] D/(D+), g D/ (z) p [ g D/+ (z) ] /(D+). Critical point: v c [ζ(d/ + )]D/(D+) v p ζ(d/) c p [ζ(d/ + )] /(D+) D D.383 D 3 D.57 D.779 D.884 D 3 D.5 v/v p.5 Maxwell-Boltzmann 3 D.5.5 / p

13 Ideal Bose-Einstein gas: phase diagram [tln7] D D 3 p p v k B v k B D p v k B 5 { kb, > pv c, < c k B c Λ. h πm. D : ransition at and v (transition line isochore). D 3: ransition at > and v >. D : ransition at > and v > (transition line isotherm).

14 Ideal Bose-Einstein gas: heat capacity [tsl4] Internal energy: U Nk B v D g D/+ (z) g D/ (z) D ζ(d/ + ), c, v ( ) D/+, c. v Heat capacity at c [use zg n(z) g n (z) for n ]: ( ) C V D Nk B + D gd/+ (z) D g D/+ (z) 4 g D/ (z) 4 g D/ (z). Heat capacity at c : ( ) ( )( ) D/ ( ) ( C V D Nk B + D D D ζ 4 + v + D ζ D + ) ( ) D/ 4 ζ ( ). D c High-temperature asymptotic behavior: [ C V D + D/ Nk B D/+ ( ) ] D/ v (/D)C v /Nk B D / v

15 [tex97] BE gas in D dimensions IV: heat capacity at high temperature he internal energy of the ideal Bose-Einstein gas in D dimensions and at c is given by the following expression: U N k B D g D/+ (z) g D/ (z). Use this result to derive the following expression for the heat capacity C V ( U/ ) V N : ( ) C V D N k B + D gd/+ (z) D 4 g D/ (z) 4 g D/+ (z) g D/ (z). Use the derivative / of the result g D/ (z) N λ D /V with V LD to calculate any occurrence of ( z/ ) V N in the derivation. Use the recursion relation zg n(z) g n (z) for n to further simplify the results pertaining to D. Solution:

16 [tex6] BE gas in D dimensions V: heat capacity at low temperature he internal energy of the ideal Bose-Einstein gas in D > dimensions and at c is given by the following expression: U D ( ) D/+ ζ(d/ + ) N k B v v (a) Use this result to derive the following expression for the heat capacity C V ( U/ ) V N : ( ) ( C V D N k B + D ζ D + ) ( ) D/ 4 ζ ( ), D c where c v [ζ(d/)] /D is the critical temperature and k B v Λ/v /D with v. V/N and Λ. h /πm a convenient reference temperature. (b) Show that the heat capacity is continuous at c if D 4 and discontinuous if D > 4. Find the discontinuity C V /N k B as a function of D for D > 4. (c) Infer from the result of [tex97] the leading singularity of C V /N k B at / v for D and D. hen show that these singularitues are consistent with the expression for C V /N k B obtained here in part (a) provided we substitute ( v / c ) D/ ζ(d/). Solution:

17 [tex8] BE gas in D dimensions VI: isothermal compressibility (a) Show that the isothermal compressibility, κ (/V )( V/ p) N, of the ideal BE gas in D dimensions at > c is g D/ p κ (z) g D/ (z)g D/+ (z), v v g D/ (z), where v.. V/N, v (Λ/kB ) D/., p kb /v, Λ. h /πm, and g n (z) are BE functions. Use zg n(z) g n (z) for n to simplify the results in D. (b) Sketch p κ versus v/v for v in D and for v v c in D 3, where v c /v [ζ(d/)] marks the onset of BEC. (c) Determine the nature of the singularity of κ as v/v in D,. Determine the critical compressibility p κ at v v c in D 3, 5. Solution:

18 [tex9] BE gas in D dimensions VII: isobaric expansivity o derive the parametric expression of the isobaric expansivity of the ideal BE gas at > c, [ (D p α p ) gd/+ p + (z)g D/ (z) ] g D/ (z)g D/+ (z) D p, [ g D/+ (z) ] D/+, where k B p Λ(p/Λ) /(D+), Λ. h /πm, and g n (z) are BE functions, establish first the general thermodynamic relation α p κ ( p/ ) v with v. V/N, the BE-specific relation C V N (D/)v( p/ ) v, and the results for C V and κ calculated in [tex97] and [tex8]. Solution:

19 [tex3] BE gas in D dimensions VIII: speed of sound (a) Start from the relation c (ρκ S ) / for the speed of sound as established in [tex8], where ρ m/v is the mass density and κ S the adiabatic compressibility. Use general thermodynamic relations between response functions to derive the following expression for c in terms of dimensionless quantities: mc k B (v/v ) (p κ ) [ + (/ p) (v/v )( p α p ) (p κ )(C V /N k B ) where v, p, p are defined in [tln7]. (b) Use the expressions derived in [tex9] for α p, in [tex8] for κ, and in [tex97] for C V to derive the result mc k B γ g D/+(z) g D/ (z), γ + D. (c) Relate the -dependence of mc to that of the isochore for v const and to that of the isobar for p const. ], Solution:

20 [tex98] Ultrarelativistic Bose Einstein gas Consider a Bose-Einstein gas with ultrarelativistic one-particle energy ɛ k c k cp in the grandcanonical ensemble at temperature and chemical potential µ. (a) Show that the one-particle density of states is D(ɛ) (4πV/h 3 c 3 )ɛ. (b) Calculate the pressure p( ), the internal energy U(, V ), and the average number of particles in excited states N ɛ (, V ). (c) Show that the heat capacity is C V /k B [6π 5 /5h 3 c 3 ]V (k B ) 3. Solution:

21 Blackbody radiation [tln68] Electromagnetic radiation inside cavity in thermal equilibrium at temperature. Grandcanonical ensemble of photons (ɛ ω cp, p k, spin s, bosonic, purely transverse). Density of states: D(ɛ) g 4πV h 3 c 3 ɛ with g independent polarizations. Average occupation number: n ɛ BE e βɛ. Number of photons with energies between ɛ and ɛ + dɛ: dn(ɛ) n ɛ BE D(ɛ)dɛ 8πV ɛ h 3 c 3 e βɛ dɛ. Spectral density inside cavity: [use dn(ɛ) V dn(ω) and ɛ ω]: dn(ω) dω V dn(ɛ) dɛ ω π c 3 e β ω. Spectral energy density inside cavity: du ωdn ρ(ω)dω. ρ(ω)dω ω π c 3 ω 8πν hν dω e β ω c 3 e βhν dν. Rate (per unit area) at which particles with (average) speed c escape from cavity through small opening [tex6]: dn/dt (N/V )c. 4 Spectral density of radiation: R(ω) c 4 Spectral energy density of radiation: dn(ω) dω ω 4π c e β ω. Q(ω) ωr(ω) ω 4π c ω e β ω (Planck radiation law). High frequencies: ultrarelativistic MB particles [use n ɛ MB e βɛ ]: Q(ω) ω3 4π c e βω (Wien radiation law). Low frequencies: equipartition law applied to electromagnetic modes: Q(ω) k B ω 4π c (Rayleigh Jeans radiation law).

22 [tex5] Statistical mechanics of blackbody radiation Electromagnetic radiation inside a cavity is in thermal equilibrium with the walls at temperature. his system can be described by a grandcanonical ensemble (with µ ) of photons (massless bosonic particles) with energy ɛ ω and density of states D(ω) (V/π c 3 )ω. (a) Show that the internal energy can be expressed in the form U(, V ) σv 4, σ π k 4 B 5 3 c 3 as postulated in a previous thermodydnamics problem [tex3]. (b) Show that the equation of state can be expressed in the form pv 3U(, V ) as was also postulated in [tex3]. Solution:

14. Ideal Quantum Gases II: Fermions

14. Ideal Quantum Gases II: Fermions University of Rhode Island DigitalCommons@URI Equilibrium Statistical Physics Physics Course Materials 25 4. Ideal Quantum Gases II: Fermions Gerhard Müller University of Rhode Island, gmuller@uri.edu