1549380323-Statistical Mechanics Theory and Molecular Simulation

424 Quantum ideal gases an electron from one of the occupied energy levels—is closely relatedto the Fermi energy. 11.5.3 Zero-te ...

The ideal fermion gas 425 〈N〉= 4 πg 3 ( m 2 π ̄h^2 ) 3 / 2 V εF^3 /^2. (11.5.39) By a similar procedure, we can obtain an expres ...

426 Quantum ideal gases E= ∫ D(V) dr E V =CK ∫ D(V) drρ^5 /^3 =V CKρ^5 /^3. (11.5.46) In one of the early theories of the electr ...

The ideal fermion gas 427 E=− ( ∂ ∂β lnZ(ζ,V,T) ) ζ,V , (11.5.49) from which it follows that E= 3 2 β V g λ^3 ∑∞ l=1 (−1)l+1ζl l ...

428 Quantum ideal gases where Cx=− 3 4 ( 3 π ) 1 / 3 e^2. (11.5.56) As we did for the kinetic energy, the volume factor in eqn. ...

The ideal fermion gas 429 Performing the angular integrals, we obtain ρ 1 (s) = 4 π V ∫∞ 0 dn n^2 θ(εF−εn) L 2 πins ( e^2 πins/L ...

430 Quantum ideal gases = V π^3 ∫∞ 0 1 s^5 [ sin(kFs)− s lF cos(kFs) ] 2 , (11.5.68) wherekF= 2πnF/L. If we now introduce the ch ...

The ideal fermion gas 431 due to thermal fluctuations. These excitations will give rise to smallfinite-temperature corrections t ...

432 Quantum ideal gases CV 〈N〉k = π^2 kT 2 εF . (11.5.78) From eqn. (11.5.77), the pressure can be obtained immediately as P= 2 ...

The ideal boson gas 433 =− 4 V g √ πλ^3 ∫∞ 0 dx x^2 ln(1−ζe−x 2 )−gln(1−ζ), (11.6.4) where the change of variables in eqn. (11.5 ...

434 Quantum ideal gases ρλ^3 g = (a 1 ρ+a 2 ρ^2 +a 3 ρ^3 +···) + 1 23 /^2 (a 1 ρ+a 2 ρ^2 +a 3 ρ^3 +···)^2 + 1 33 /^2 (a 1 ρ+a 2 ...

The ideal boson gas 435 Pλ^3 gkT = ∑∞ l=1 ζl l^5 /^2 − λ^3 V ln(1−ζ) ρλ^3 g = ∑∞ l=1 ζl l^3 /^2 + λ^3 V ζ 1 −ζ . (11.6.16) We wi ...

436 Quantum ideal gases 0 0.2 0.4 0.6 0.8 1 z 0 0.5 1 1.5 2 2.5 3 g3/2 (z) Fig. 11.2 The functiong 3 / 2 (ζ). prefactor. It is, ...

The ideal boson gas 437 0 1 2 3 vg/l 3 z 1 1/R(3/2) Fig. 11.3Plot of eqn. (11.6.28). Forρλ^3 /g < R(3/2),ζwill be different f ...

438 Quantum ideal gases from which it can be seen that the average occupation of each energy level is given by 〈fnm〉= ζe−βεn 1 − ...

The ideal boson gas 439 〈f 0 m〉= 〈N〉 g . (11.6.35) Summing both sides of eqn. (11.6.35) overmcancels the degeneracy factorgon th ...

440 Quantum ideal gases The occupation of the ground state undergoes a transition from afinite value at T = 0 to zero atT =T 0 , ...

The ideal boson gas 441 V P T 1 T 2 T 3 Fig. 11.5Plot of the isotherms of the equation of state in eqn. (11.6.43). HereT 1 > ...

442 Quantum ideal gases 3/2 T 0 T CV <N>k Fig. 11.6CVas a function ofTfrom eqn. (11.6.45). ForT < T 0 , the curve incre ...

Problems 443 11.3. Determine how the average energy of an ideal gas of identicalfermions in one dimension at zero temperature de ...