Weakly degenerate ideal Fermi Dirac gas

Transcript

1 Quantum Statistics (Capter McQuarrie) alf integral spin Fermi Dirac electron, proton integral spin Bose Einstein deuteron, poton Composite particles odd # of fermions acts as fermion even # of fermions acts as a boson () ± ( ) ( ) VT e Ξ,, λ ± λ, () E () λε e ± βε ( λ ) βε βμ βε N ; n βε ± ± βε βε pv ± T n ± e βε βε upper FD lower BE solve () for λ, and substitute into () and ()

2 In general, can t solve analytically for λ N If λ small, λ classical statistics q (ig T and low densities) Even if particles are non interacting, quantum effects cause deviations from pv NT Wealy degenerate ideal Fermi Dirac gas βε N pv T n e βε + + βε ( λ ) ε / ( nx + ny + nz), nx, ny, nz,,... 8mV

3 Counting states for translational problem ε + + 8ma ( nx ny mz) 8ma degeneracy # ways can be written as ε nx + ny + nz Tae spere of radius 8ma ε n n n R x + y + z 8ma ε R for large R treat ε, R as continuous # states wit energy ε 4πR π 8ma ε Φ 8 6 /

4 π 6 # between ε+εδε, suc tat Δε/ε << ( ) ( ) ω Φ ε +Δε Φ ε / 8ma / / ( ε +Δε) ε / π 8ma ε ε... Δ + 4 T If ε, T K, m g, a cm, and Δε.ε 8 ω for N particle system, te degeneracy is muc larger: ω ~ / m ω π V εδε / m π V εdε

5 / βε m λ εe dε N π V βε + / m / βε ( ) pv π T V ε n + dε using te density of states from pages Expand in powers of λ and integrate A + ( ) λ λ λ N ρ +... V / / Λ Λ B p T + ( ) λ λ λ / 5/ Λ Λ ( ) solve λ λ ρ using A plug into B to get p as a function of ρ T

6 Write λ a + aρ+ a ρ +... ( a + aρ+ aρ +...) ( a + aρ+ aρ +...) ρ ( a + aρ+ aρ +...) + + Λ / / a a a a a ρ ρ+ + ρ + Λ a... a /... / / ( a...)... a a Λ a a / λ a ρ+ a ρ + a ρ + 6 Λ Λ ρ+ ρ + / ( ρ ) Λ λ ρλ + + / / ( ρλ )

7 p T λ λ λ / 5/ Λ p T p T p T ( a ) ( ) ρ+ aρ +... aρ+ aρ / Λ Λ ρ a Λ 6 + a ρ Λ + 5/... Λ 6 5/ 5/ ρ + ρ + Λ ρ... 8 a a / 5/ / 5/ Λ 6 6 Λ Λ p T ( a ρ +...) 5/ Λ Λ + Λ 6 ρ / ρ+ B ρ + ρ + nd virial coeff B... rd virial coeff Virial coefficients reflect deviations away from ideality B is +, tus increases pressure beyond tat for an ideal classical gas

8 Λ termal de Broglie wavelengt quantum effects < as de Broglie λ < Λ V Actually it is tat is a measure of quantum effects E + ( ) VT λ 5/ Λ Λ NT + ρ / get expansion for μ from λ e μ/t and S from G μne TS + pv Of course, above approac only valid if quantum corrections are small

9 Now consider te strongly degenerate Fermi Dirac gas A model for te electrons in a metal n f K βε βε β ( ε μ) + + e prob. a state is occupied + e ( ε ) βε ( μ) since ε is essentially continuous T μ μ states wit ε < μ are occupied states wit ε > μ are unoccupied f(ε) μ ε from 5 / m ω ( ε) dε 4π V εdε / m μ π ε ε N 4 V d # valence e (includes factor of for spin) 8π m / V μ / μ / / N m 8π V

10 ε at T, te levels are double occupied up to μ μ a finite T, te boundary is smeared out i.e., some electrons are excited leaving oles Even at room T f f ( ) ( ) ε ε < μ ε ε > μ is a good approximation μ / Fermi, TF T typically a few tousand degrees

11 / m μ / 4π ε ε μ E V d N T 5 K ZPE of FD gas only a very small fraction of te e are excited, so contribution to eat capacity is ~ equipartition teorem would lead us to expect for eac electron / m μ β( με) ( ) π ε + ε p 4 T n e d p m π ε ( μ ε) dε μ N 5 V μ / 4 ignore te zero point pressure on te order of ( 6 atm) S only one way to occupy levels at T K

12 It can be sow tat μ π η +..., η μ βμ μ ~ μ for temperatures for wic a metal is solid E 5 μ 5 E + πη +... μ 8 Note at T K, μ μ o C v π NT π T N μ / T f ~ 4 T cal/deg mol

13 wealy degenerate ideal Bose Einstein gas N βε βε βε ( λ ) pv T n e e βε λ <, oterwise get in denominator N βε + βε βε βε redefine ε to be zero ( λ) / / βε N λ m λε e ρ + π d, ( ) ε λ V V < ε > βε

14 / m ε> ε p βε n( λ ) π ε n( ) dε T V if λ << can ignore /V terms ρ g / ( λ ) Λ g p n λ n g 5/ ( λ ) T Λ p ρt Λ ρ / effective interaction between ideal bosons is attractive Λ E NT ρ /

15 Strongly degenerate ideal Bose Einstein gas T < T condensation into ground state Beavior seen for He 4 C V N From Wiepedia: A Bose Einstein condensate (BEC) is a state of matter of a dilute gas of wealy interacting bosons confined in an external potential and cooled to T near to absolute zero. Under suc conditions, a large fraction of te bosons occupy te lowest quantum state of te external potential, and quantum effects become apparent on a macroscopic scale. Tis state of matter was predicted by Bose and Einstein in Te first suc condensate was produced by Cornell and Wieman in 995 at te Univ. of Colorado NIST-JILA lab, using a gas of Rb atoms cooled to 7 (nk) []. Cornell, Wieman, and Ketterle (MIT) received te Nobel Prize in Pysics. Note: te fact tat Rb atoms act as bosons is due to interplay of electronic and nuclear spins. T T

16 From Wiepedia A Bose Einstein condensate (BEC) is a state of matter of a dilute gas of wealy interacting bosons confined in an external potential and cooled to T near to absolute zero. Under suc conditions, a large fraction of te bosons occupy te lowest quantum state of te external potential, and quantum effects become apparent on a macroscopic scale. Tis state of matter was predicted by Bose and Einstein in Te first suc condensate was produced by Cornell and Wieman in 995 at te Univ. of Colorado NIST-JILA lab, using a gas of Rb atoms cooled to 7 (nk) []. Cornell, Wieman, and Ketterle (MIT) received te Nobel Prize in Pysics. Note: te fact tat Rb atoms act as bosons is due to interplay of electronic and nuclear spins.

17 Ideal gas of potons potons mass ang mom ħ cavity emits/absorbs potons N not fixed cavity Assume armonic electromagnetic waves π Ext (, ) sin ( x ct) sin( xωt) λ ε υ ω; p / λ Consider blac-body radiation to be due to standing waves φ( x, t) sinxcosωt Fix at,l nπ/l

18 ε ħc (π/l) (n x + n y + n z ) E Q E βεn ( e ) n ε n π V( T) 5( c) 4 βε ( e ) Can be used to derive te Stepan-Boltzmann law Can also sow tat te cemical potential (follows from te fact tat te number of particles is not conserved) So could ave used te Bose-Einstein formulas wit λ

19 Density matrices All te expressions described above were derived assume tere are no interactions between particles ˆ,.. expectation value of operator ( ) E E E H E H H Q e M Me M qm M Q H E e e Q e e Q Tr e β β β β β β β ψ ψ ψ ψ ψ ψ Te trace is independent of te basis

20 ϕ a ψ n n n Q is te same wen evaluated over te ϕ M ( ˆ β H Tr Me ) Tr H ( e β ) β H e ρ β H Tr( e ) M Tr( Mˆ ρ) H l + U( r,, rn) m pir u( p,, r ) e e.f. of momentum operator ϕ N l i i pir ( r, rn) A( p,, pn) e dp,, dpn A p p r r e dr dr ( π ) i pir (,, N) ϕ (, N), N N Inverse Fourier transform

21 i Q *( r, rn) ( r', rn ') e e e N ϕ ϕ * dp,, dp dr, dr dr ', dr ' N N N i pir ' p r β H i, N i Q e e e p dr N Now adopt a strategy due to Kirwood i i i p ir ' p r β H i QM p r Cl βh i βh pir e e e e wp (, rn, β ) Fp (, rn, β ) Cl β H Q e w( p, rn, ) p, dr N N β F QM H F It is easier to wor wit β tis diff.eq. l w w w l l 4 β β p w { U [( U) + ( pi ) U] + ( pi U) } m m m Note we ave not included te symmetry of te wavefunction wic is wy te N! is missing. Contains te quantum corrections Tere is a w term but it does not contribute to Q