Fermi-Dirac and Bose-Einstein

Transcription

1 Fermi-Dirac and Bose-Einsein Beg. chap. 6 and chap. 7 of K &K Quanum Gases Fermions, Bosons Pariion funcions and disribuions Densiy of saes Non relaivisic Relaivisic Classical Limi Fermi Dirac Disribuion Fermi Energy Elecrons in solids Nuclear maer Whie Dwarf Bose Einsein Disribuion Bose-Einsein Condensaion Liquid Helium

2 Quanum Gases Bosons, Fermions Consider quanum sysem and is quanum saes Quanum Field Theory => Ineger spin= Boson = number of paricles in a given sae is arbirary Half Ineger spin=fermion= a mos one on each orbial: Pauli exclusion principle Pariion Funcions and mean occupaion numbers Thermodynamical sysem=sae of energy ε Use Gibbs mehod and grand pariion funcion Fermion A mos one Z = + exp µ ε s( ε) =τ logz µ = Boson Sum on all inegers Z = s( ε) = τ logz µ = exp µ ε s( ε) = + exp µ ε exp s µ ε = s = exp µ ε exp µ ε τ s( ε) = exp µ ε τ 2 exp ε µ + exp ε µ

3 Ideal quanum gases Ideal gas approximaion Saes are no modified by presence of oher paricles Densiy of saes g mulipliciy x densiy in phase space i d 3 x d 3 p h 3 p ε change of variable o energy p 2 dp dω g i Ω h 3 = D(ε)dε where D(ε) is he densiy of saes Non relaivisic dp = mdε p = m 2 ε = p2 2m p2 = 2mε dε ε D ε ( )dε = 4πg i p 2 dp h 3 = g i 4π 2 2m εdε Ulra-Relaivisic ε = pc p D( 2 dp ε)dε = 4πg i h 3 = g i 2π 2 3 c 3 ε2 dε 3

4 Behavior Sign is criical for (ε-µ)/τ small Fermi-Dirac s( ε ) = Bose-Einsein Bose condensaion s( ε ) = exp ε µ = + 2 for ε = µ = for ε << µ exp ε µ for ε µ For (ε-µ)/τ large, classical limi Occupaion number << s( ε) exp µ ε = λ exp ε independen of F.D. or B.E. Same old resuls of Bolzmann /Gibbs! Prob( ε) = s( ε) < N >= V s( ε) N D ( ε )dε = V exp µ exp ε d 3 p h 3 < N >= V exp µ n n Q µ = log 4 <s(ε)> n Q µ ε

5 Thermo funcions for ideal quanum Number of Paricles N = V s( ε ) D( ε )dε = V D( ε )dε exp ε µ ± µ(τ) se by requiremen ha N=oal number of paricles Energy U = V case of Black Body N γ α τ 3 εd( ε)dε exp ε µ ± Enropy σ ( ε) = ( τ log Z) - Bose- Einsein + Fermi-Dirac τ = ε µ s + log Z log prob s =< s > τ = ε µ s ± log( ± s ) τ = ± ( ± s )log( ± s ) s log s 5 + Bose- Einsein - Fermi-Dirac ( )

6 Fermi Gas Ground Sae (Non relaivisic) s( ε) Fermi Energy Calculaion : d 3 xd 3 p densiy of saes g i and ε = p 2 h 3 2m N = V 2m 2 ε F 2π 2 2 εdε Wach ou:! here n=densiy Ex: Elecrons in meal Energy Free Energy Pressure ε F Spin 2 g i = 2 ε τ << µ ε F = 5 ev v F = 8 cm/s = 3 3 c U = V s( ε ) = exp ε µ for ε < µ + for ε > µ ε F = µ ( τ = ) N = V s ( ε ) D( ε)dε = V D( ε)dε ε F εd( ε)dε = 3 5 Nε F F = V ε F F( ε)d( ε)dε F = U = 3 5 Nε F p = F = 2 V τ 5 nε F V 3 ε F Kiel+Kroemer do his calculaion wih ineger n densiy Spin /2 ε F = 2 ( 2m 3π2 n) 2 3 wih n = N V F( ε ) =U a τ =! σ = for s( ε ) = or Repulsive!

7 Fermi Gas Ground Sae (Relaivisic) Noe ha even a zero emperaure very large kineic energies: in some case ulra-relaivisic τ << µ Fermi Energy ε F = µ ( τ = ) ε F N = V D ( ε )dε = V 3π 2 3 c 3 ε F 3 ε F = ( 3π 2 n) 3 c Energy Pressure U = F = V ε F εd( ε)dε = 3 4 Nε F p = F = V τ 4 nε F V 4 3 Noe: same general relaion P = /3u (ulra relaivisic) 7

8 Pauli exclusion principle Inerpreaion 2 fermions canno be in he same quanum sae! In paricular no a he same posiion=> fermion does no have he full volume available bu only Heisenberg uncerainy principle Δp x Δx => large random momena V N = n Δp x n / 3 Fermi energy: Relaivisic Non relaivisic ε F E Δp2 2m 2 2m ε F E cδp cn /3 n2 /3 =>pressure 8

9 Hole and Elecron Exciaions Symmery of Fermi Dirac disribuion f ( ε,τ ) = exp ( ε µ ) + Wriing ε = µ +δ f ( ε,τ ) = exp δ + f ( ε,τ ) = describes Fermions of energy -ε and chem.po. - µ exp δ + Basic symmery (excep for lower bound a δ =-µ ). The disribuion can be described eiher as he presence of elecrons or he absence of elecrons (=holes). Decomposiion ino hole-like and elecron-like exciaions f ( ε,τ ) N V = D ( ε ) f ( ε,τ ) dε = ε F D( ε ) dε D( ε ) f ( ε,τ ) dε = ε F D( ε ) f ( ε,τ ) dε + D( ε ) f ( ε,τ ε ) dε ε F D( ε ) f ( ε,τ ε ) dε = ε F D( ε )( f ( ε,τ )) dε F B ="elecrons" A ="holes" 9

10 Elecron and hole energy Change of energy wih emperaure f ( ε,τ ) ε F u( τ ) u dε ε F εd( ε ) dε dε + εd ε dε ε F εd ε dε ( ) = εd( ε ) f ( ε,τ ) = ε F εd( ε ) f ( ε,τ ) ( ) f ( ε,τ ) Noe ha as referenced o he Fermi energy, he energy of holes are opposie o ha of he corresponding missing elecrons and are posiive ε ε F ε ε F +ε ε F F D( ε ) f ( ε,τ ) dε + D( ε ) f ( ε,τ ε ) dε ε F D( ε ) dε F = = ( ε ε F )D( ε ) f ( ε,τ ) dε + ε F ( ε F ε )D( ε ) f ( ε,τ ) dε ε F "elecrons" Elecron like exciaion: increasing energy k Hole like exciaion: increasing energy "holes" ( ) ( )

11 Elecrons in crysals: Quanum Saes Energy Band srucure ε v ε c E Gap energy Chap. 7)) conducion band valence band k Periodiciy of he laice (e.g., spacing a) Periodic zones in momenum space k k + G wih G = 2π a Resonan unelling: free propagaion of specific modes (Bloch Waves) Discree E( k) These relaionships do no necessarily overlap in => gap (cf. Kiel, inroducion o Solid Saes Physics ev Seen in projecion on he energy axis: energy bands Valence band Conducion band Meal: Fermi level = chemical poenial in conducion band => conducion can be described by free Fermi gas Insulaor: Fermi level in gap

12 Order of magniude Phys 2 (F26) 7 Fermi Dirac/Bose τeinsein Elecrons in meals bu conservaion of number of elecrons Then use: df /d only large close o µ does no vary fas wih τ ε F large negaive τ Change variable ( )τ ε F 5 ev T F 5 4 K >> T lab τ=o : very good approximaion a room emperaure and below Common noaion: f ( ε,τ) = s( ε) τ du Hea Capaciy C el = k B dτ = k d B VεD ( ε )f ( ε,τ ) dε = k B V εd ε dτ C el k B VD ε F C el k B VD ε F ε F ε = ε F µ ε F 2 ( ) d ( ) dτ f ε,τ d D( ε) f ε,τ dτ ( ) dε = ε F D( ε) d dτ f ( ε,τ ) dε = C el = k B V ε ε df F dε k dτ B VD ε F ε x = ε ε F τ x 2 e x dx k ( e x +) 2 B VD( ε F )τ df dτ ( ε ε F )D ε ( ε ε F ) τ 2 ( ) 2 ε ε F x 2 e x τ 2 ( ) 2 e x + ( ) df dε dτ ( ) exp ε ε F τ ( ( ) + ) 2 exp ε ε F τ exp ε ε F τ exp ε ε F τ + 2 dε dx = π2 3 k BVD( ε F )τ dε

13 Finally Hea capaciy of meals Hence aking ino accoun phonons C o = C el + C ϕ = γt + AT 3 f ( ε,τ ) Imporan puzzle hisorically Very few elecrons affeced C el << 3 2 Nk B ε F ε 3

14 Example Ge,Si,GaAs from Sze Physics of Semiconducor devices p3 Wiley-InerScience 98 4

15 Insulaors: Densiy of saes cf. K&K chap 3 In paricular inrinsic semiconducors (no role of impuriies) Saisical disribuion Sill good approximaion o consider free elecrons as quanum ideal gas => occupaion number Densiy of saes ε conducion band valence band f ( ε) = exp ( ε µ )/τ We can hen ge he oal number of elecrons n et = ε c Gap ε v ( ) + f ( ε )D( ε)dε = f ( ε)d h ( ε )dε + f ( ε)d e ( ε)dε ε v ε v = D exp ( µ ε h ε )/ ( )dε + ε D (ε) ( ) + D e (ε)dε = 2 4π 2 D h (ε)dε = 2 4π 2 ε c ε c 2m* e 2 2m* h 2 exp ( ε µ )/τ 2 spin saes 3 2 (ε εc )dε 3 2 Elecron sae densiy below he gap (ε v ε)dε ( ) + D e ε Parabolic a gap edge ( )dε 5

16 Elecrons and Holes This can be rewrien as ε n et = n v vt exp (( µ ε )/τ ) + D h ( ε )dε + exp( ( ε µ )/τ ) + D e ( ε )dε ε c ε v oal in valence band a zero emperaure =oal number of elcrons exp (( µ ε )/τ ) + D h ( ε )dε = exp( ( ε µ )/τ ) + D e ( ε )dε ε c holes holes elecrons elecrons Therefore we can describe he elecron populaion by wo non relaivisic gases : holes and elecrons (cf. wha we did wih meals). The equaliy of he number of holes and elecrons fixes he chemical poenial Fermi level: in he middle of he gap if m h *=m e * Measuring from he edge of he valence and conducion band respecively n e = 2 4π 2 for τ << ε v = n h = 2 4π 2 * 2m e 2 * 2m h 2 3/2 3/2 exp( ( ε ' ( µ ε c )) / τ ) + ε 'dε ' n Qe exp ε c µ wih n Qe = 2 n Qh exp µ ε v exp ε ' ε v µ wih n Qh = 2 (( ( )) / τ ) + ε 'dε ' m * eτ 2π 2 m * hτ 2π

17 from Sze Physics of Semiconducor devices p85 Wiley-InerScience 98 Example Ge,Si,GaAs 7

18 Deermining he chemical poenial f No impuriies: inrinsic semiconducors log n e (µ) log n h µ ( ) ε v µ ε ε c ε v µ µ ε c 8

19 Semiconducors ε v ε c Large role of impuriies: localized saes (No band!) in gap If hey are shallow ( 4meV (Si) mev (Ge)) can be excied a room emperaure. This modifies oally he behavior! k ε D ε A Donors Accepors d o d + + e a a o + e noe: 2 A sae because a bond is missing and he missing elecron can be spin up or down, A- bond esablished (pair of elecrons of aniparallel spins) : sae The number of free elecrons(holes) is no more consan Can be increased by donors and decreased by accepors Bu we need o keep charge neuraliy = mehod o compue he Fermi level For large enough impuriies concenraion, he Fermi level can move close o he edge of he gap (Thermally generaed) conduciviy eiher dominaed by elecron like exciaion: negaive carriers (n ype) hole like exciaion: posiive carriers (p ype) 9 n d = n d + + n d o n a = n a + n a o

20 Donors Semiconducors cf. K&K fig 3.6 d d + + e exp ε + n d + = n d exp ε + + 2exp ε µ = n d + 2exp µ ε d.4 ev Si wih ε d ε ε + ε c. ev Ge negaive carriers (n ype) Accepors a + e d + exp ε µ τ n a = n a 2exp ε + exp ε µ = n d + 2exp ε µ a.4 ev Si wih ε a ε ε ε v +. ev Ge posiive carriers (p ype) log n e (µ) log n h µ ( ) ( ) log n d + µ log n e (µ) log n h µ 2 ( ) ( ) log n d + µ Elecric neuraliy n e = n d + + n h n d + + µ ε v ε c µ µ ε d Elecric neuraliy n e + n a = n h n a ε v ε c µ ε a -

21 Oher examples of degenerae Fermi gas 3 He Spin /2 cf. Problem se Very differen behavior from 4 He: phase separaion Basis for diluion refrigeraors (K&K Chaper 2: evaporaion of 3 He ino superfluid 4 He which acs as an excellen pump: Works down down o mk) Nuclear Maer n p n n 5 37 N p N n A 2 R.3 3 A3 cm cm -3 ε F = 4 2 J 3 MeV T F 3 K Fermi momenum 948 sub hreshold producion LBL α( 38 MeV)+ 2 C π

22 Whie Dwarfs and Neuron Sars Whie dwarf sars (and core of massive sars) ρ 6 g/cm 3 n 3 cm -3 ε F.5 3 J 3 5 ev T F 3 9 K >> T sar => Degenerae Fermi gas Fermi pressure balances graviy => Condiion for equilibrium Look a oal energy of sysem: Assume consan densiy ρ = Graviaional poenial energy U G GM2 Kineic energy (neglec angular momenum) R U FD nvε F Mε F Non relaivisic ε F n 2 / 3 M2 / 3 R 2 5/3 am bm 2 U T = R 2 R minimum of oal energy: sable! Ulra Relaivisic ε F n / 3 M/3 same R dependence as Rgraviaional energy ( U T = am 4 /3 bm 2 ) R Degeneracy pressure canno balance graviy if M oo big! Chandrasekhar limi.4 M Neuron sars ε F 3 MeV Same sory for neurons (uncerainy of equaion of sae) Similar Chandrasekhar limi if larger => black hole 22 3 M U T Rel M 4 /3πR 3 n M R 3 U U FD Non Relaivisic U T NR U U G U FD Relaivisic U G R R

23 Whie Dwarf Explosions: SN Ia An acceleraing universe? Supernovae Type Ia a high redshif (2 groups) Ω m -Ω Λ Disan supernovae appear dimmer han expeced in a fla universe Poenial problems Are supernova properies really consan? Dus? The Uninvied Gues: Dark Energy Large negaive energy a = G u acceleraion r 2 c p V energy densiy pressure GR graviaional mass Luminosi y Fainer 23 Ω m = Ω Λ = ime Graviy becomes repulsive! Disance

24 Bose-Einsein Condensaion Calculaion of chemical poenial Le us ake he origin of energy a he ground sae As usual he chemical poenial can be compued by imposing he average number of paricles in he sysem Separaing beween he ground sae s= and he oher saes exp µ + s > exp ε = N s µ A low enough emperaure, we can solve he equaion by having exp µ and pu nearly all he paricles in he ground sae, wih he occupaion of he higher saes given by exp ε µ s exp ε << N s or exp ε s >> + 24 N τ <<ε sn Bose Einsein condensaion For large densiies no a very low emp. phenomenon

25 B-E Condensaion: Quaniaive Approach Chemical poenial a low emperaure Wih he energy origin a ground orbial. The occupaion number of he ground sae is f (,τ ) = s( ) = exp µ τ for µ << τ µ for 4 He (µ< because <ε o =) Comparison wih 2nd sae Canno use coninuous approximaion For 4 He bu f,τ ( ) = N µ τ N N N = 22 cm -3 µ =.4 45 J a T =K and V = cm 3 ε( n x,n y,n z ) = 2 ( π ) 2 2m L n x + ny + nz ε(,, ) = 3 2 ( π ) 2 2m L ε( 2,,) = 6 2 π 2m L Δε = ε( 2,,) ε(,, ) = 3 2 2m ( ) 2 π L ( ) ( ) 2 = J for L = cm>> µ => NΔε = J >> τ = f ([ 2,, ],τ ) = ( ) exp Δε µ τ << ( exp µ ) τ = N 25

26 Bose-Einsein Condensaion(2) Temperaure dependence Calculae separaely condensed phase and normal phase N = f (,τ ) + f ( ε,τ ) condensed ε( 2,,) excied Replace sum by inegral N exp µ + V 3/2 2m 4π 2 2 ε /2 dε exp µ exp ε exp µ,.36 π N exc = 2.62 mτ x /2 dx e x 2π 2 3 /2 V = 2.62n Q V Defining he Einsein condensaion emperaure τ E = 2π 2 N m 2.62 V 2/3 N exc = N τ τ E 3/2 For large densiies, τ E is no very small 26

27 Properies of 4 He Liquid Helium 4 Loose coupling =>liquid (4.2K a Am) ideal gas 4 He has spin => boson 3 He spin /2 => expec condensaion a 3.K Experimenally lambda poin 2.7K (Landau emp.) Phase ransiion => peculiar properies Macroscopic quanum sae iϕ(, x ) Ψ = n /2 e C 4 He 2.7K T cf energy of elecromagneic field E n ω ˆ E e ( ) i ω k x => Quanizaion phenomena =>Superfluidiy e.g. Vorex Equivalen of Josephson effec Δϕ = 2nπ where n is ineger 27

28 Liquid Helium Superfluidiy Consider an objec of mass M going hrough helium a velociy V Because coheren quanum sae, canno ransfer energy o single aom: only possibiliy is ransfer o exciaions phonons Single aom (no superfluidiy) Conservaion energy and momenum in collision phonons ω k => subracing / 2MV 2 = / 2 MV' 2 + / 2mv 2 / 2MV 2 = / 2 MV' 2 + ω k MV = MV ' +mv MV = MV ' + k ( MV mv ) 2 = ( MV ') 2 MV k ( ) 2 = ( M ) 2 M 2 V 2 mmv 2 = M 2 V' 2 M 2 V 2 2 ω k M = M 2 V' 2 2mMv V + m( m + M)v 2 = v = M m + M V cosθ only possible if V is large enough (>velociy of sound) V ' 2 M k V + 2 k ω k M = 2 ω k 2 ω k = 2Mc s ( ) > V cosθ c s c s 2 V cosθ 2 M ω k c s V cosθ + 2 ω k M = k 28

29 Much cleaner sysem: Alcali Vapors BE condensaion for aoms demonsraed in 995 => 2 Nobel Prize in Physics awarded joinly o Eric A. Cornell of NIST / JILA; Wolfgang Keerle of MIT; and Carl E. Wieman of CU / JILA. Time sequence of images showing one cycle of he ringing of a Bose- Einsein condensae (BEC) in he JILA TOP (ime-averaged orbiing poenial) rap afer being driven by srong oscillaions of rap poenial. 29

30 Cooling in a rap See:hp://bec.nis.gov/ BEC Aoms Images of he velociy disribuions of he rapped aoms Lef: jus before he appearance of he Bose-Einsein condensae Cener: jus afer he appearance of he condensae Righ: afer furher evaporaion. 3

31 Pairing of Fermions Superconduciviy Pairing of elecrons s= (Cooper pairs) <= phonon ineracion Bu condensaion heory bad approximaion (no free) τ superconduciviy << τ condensaion 3 He Similar effecs Zero resisance Quanizaion of flux : Vorices Spin /2 2 phases of pairing s= similar o superconduciviy bu magneic properies τ condensaion =.95mK and 2.5mK 3

32 Energy Densiy of Ulra Relaivisic Gases Generalizaion Imporan for behavior of early universe (energy densiy =>expansion) Densiy of energy x 3 dx e x = π 4 5 Bosons g π c 3 k BT Suppose ha paricles are non degenerae (µ<<τ) f ( ε ) exp ε ± u = εd ( ε ) f( ε)dε wih D( ε)dε = g ε 2 dε 2 π 2 c 3 3 ( ) 4 = g 2 a BT 4 u = where g is he spin mulipliciy u = g τ 4 x 3 x 3 dx dx e 2 π 2 c 3 3 x = e x ± g bosons g fermions a B 2 T 4 Fermions 7 g 8 2 Effecive number of degrees of freedom for a relaivisic π 4 5 π 2 ( 5 3 c 3 k BT) 4 = 7 g 8 2 a BT 4 32