Quantum GasQuantum concentration:Classical gas:Quantum gas:If concentration is fixed, then temperature is important variable:Classical gas:Quantum gas:Degenerate gas:Classical entropy diverges at low temperatures:Quantum entropy will be zero since both bosons and fermions will be in the ground state…()2322 hπτMnQ≡QQnnnn≥<<3220)2( nMnnQhπττ≡=⇔=000ττττττ<<≤>>()τσττlog2325)log(023NnnNnQQ⎯⎯→⎯+=⇒∝→5 bosons or fermions in a box at zero temperatureEnergy spectrum in 3D:Bosons: Fermions:lowest level 3 lowest levels22222222 ,2zyxnnnnnmL++==hπεnAbout metalsValence electrons of atoms in a crystal lattice are free to move:Simplest : 1stgroup of the periodic table: Li, Na, K, CsN atom cores (15% of the space for Na) + N conduction electronsClassical theory (Maxwell-Boltzmann d. f.): Can explain: Ohm’s law, relation between thermal and electrical conductivities (Widemann-Franz law) Cannot explain: heat capacity, magnetic susceptibility, large mean free path (up to 108 lattice constants)•Electron mean free path is large because:(1) strictly periodic lattice does not scatter electrons(2) e-e scattering is weak because of the Pauli principleFree electron Fermi Gas in three dimensionsEnergy spectrum:Occupancy:Total number of particles/occupied orbitals: integration rule (including spin):Radius of a Fermi sphere in n-space:22222222 ,2zyxnnnnnmL++==hπεnFFFfεεεετεεετ><⎩⎨⎧==→+−=⇒=−nnnn,0,11))(exp(1)(030234812)(FnndnnfNFππε=⋅==∫∑nn31)3(πNnF=Free electron Fermi Gas in three dimensionsFermi energy/Fermi temperature:Fermi wavevector:Fermi velocity:All are determined by density!⇒⎟⎠⎞⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛===≡3232222222322πππετεNVmnmLFnFFFhh312322222)3()3(22nknmmkFFFππε=⇒=≡hhmnmkvFF312)3(πhh==3222)3(2nmFπεh=Density of states in 3DAverage values:D: Density of one-particle states (orbitals)# of orbitals per unit energy:dN – number of orbitals with energies between εand ε+ d εεεddDn=)(⇒=⋅=⇒=⇒=⇒=⇒==⋅⋅=⋅⋅===3221222222222221)(212...4)81(2)(αεπεααεπεεαεαεαεαπεεπεπεεεDddnnnnmLddnnddnnddndndnddDzyx321hnεπε232222)(⎟⎠⎞⎜⎝⎛=hmVD}∫∫∑===εεεεεεεdDddXfdXfXfX)()()()())(()(nnnnnnnDensity of statesAnother expression for DOS:Another way to get DOS:# of orbitals below ε:Average values:Density of occupied orbitals:Zero-temperature gas (ground state):Check:∫=εεεεdXfDX )()()(∫∫∑=⇒==FFdDNdXDXXεεεεεεε00)()()(nn)()(εεfDNNdNNFFFF===∫232302123322323εεεεεε⇒=⇒=⇒==εαπεεπαπεεα323332222)(3)3(DNNnFFFFεεε2323)(FND =εαπεεεαππεα32333222)(3)3(==⇒=⇒=ddNDNNTotal energy of degenerate Fermi gasTotal ground state (kinetic) energy:Direct way:Using DOS:Heat capacity (estimate):Only electrons close to the Fermi level can be thermally excited:...10...242812252230422302222210====⎟⎟⎠⎞⎜⎜⎝⎛⋅==∫∫∑=FnnnnnmLdnnmLdnnnmLUFFFhhhππππεnFNUε530=FFFFNNdNdDUFFεεεεεεεεεεε53522323)(25230232300====∫∫FvFNUCNUTUετττετ~~)(0∂∂=⇒⋅−Heat capacity of cold Fermi gasEnergy at low, but finite temperature:Heat capacity at constant volume:Approximations:Calculate derivative:NdfDdfDUFFεετμεεεεετμεεετ+−==∫∫∞∞00),,()()(),,()()(∫∫−≅−=⎟⎠⎞⎜⎝⎛∂∂≡ετεεεετεεεττdddfDdddfDUCFFFVV)()()()()()()()()(0FFFDDddfεεεεδτετμτ≈⇒−∝⇒≈⇒≈222)1()1( 1exp1+=⎟⎠⎞⎜⎝⎛−−⋅+−==⇒+=⇒−=xxFxxFeexeeddxdxdfddfxfxττεετττεεHeat capacity of cold Fermi gasChange variables:Heat capacity:# of excited electrons:Energy of each excitation:44344213213)1(23)()1()(2222πτεετττετε∫∫∞∞−∞−+≅+⋅= dxeexNDdxeexxDCxxFFxxFVF⇒−∞→−===−τετετεεFFxdxdx0 , ,τεπ)(32FVDC =FVNCετπ22=...~~~~2FVFFNCNUNετεττετ⇒⇒⎭⎬⎫Finite temperature energyGround state energy:FFNNUετπε22453+=⇒⎟⎠⎞⎜⎝⎛∂∂=⎟⎠⎞⎜⎝⎛∂∂==VVFVUUNCττδτετπ))((22⇒===∫∫FFVNdNdCUετπττεπττδ222412)()(0,53000τδτετUUUNUUF+=⇒≠==⇒=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+=22453FFNUετπεEntropy and pressure: equation of stateEnergy:Entropy:Free energy:Pressure:FFNNUετπε22453+=ττττστστετπ∝==⇒⎟⎠⎞⎜⎝⎛∂∂≡=∫VVNVVFVCdCCNC)( ;2,2FFFVNUNNCUUFεετπεττσ53453022=≈−=−=−=3232225332−∝=⇒⎟⎟⎠⎞⎜⎜⎝⎛= VNFVNmFFεπεh35052533232nnVNVUVFpFF∝===⎟⎠⎞⎜⎝⎛∂∂−=εετconst35=⎟⎠⎞⎜⎝⎛NVpFinite temperature chemical potentialCorrections:Symmetry:This is the result of growing DOS in 3D!)(0,0τδμεμτεμτ+=⇒≠=⇒=FFNdDdfDNδεεετμεεττμ+==∫∫∞43421on depends)(00)(),,()()(1)(1))(exp(1)(δμδμτμεε−−=+⇒+−=−−−fff[]444443444442144443444421μμμδδμδμδδμδμδ below holes0 above electrons0)(1)()()(∫∫∞→∞−−−−++= dfDdfDN[]0)()(2)()()(00>+≅+−−+=∫∫∞=∞δδδμεεδδμδμδμδμεdfddDdfDDN⇒=+=+=+∫∫∫∞∞∞220200121exp1)exp()(τπττδδδδδδμxxdxddfFinite temperature chemical potentialSolve directly:Solution:⇒+==∫μεμεετπεεddDdDN)(6)(220⇒⎟⎟⎠⎞⎜⎜⎝⎛−≈⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−≈⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡≈⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−=⇒⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=⇒⋅+⋅=⇒⋅=⋅=⇒=−−=∫22322232212221222321232223232123232302383218118181212362123)( ;3223)(23)(FFFFFFFFFFFFFNNNNddDNdDNDετπετπεμετπεμεμετπεμμετπμεμεεεμεεεεεεμεμ321⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−=22121FFετπεμExamples of Fermi gases1). Metals 2). White dwarf stars free electrons3). Nuclear matter3phelτγτACCCVVV+=+=K 10~4BFFkTε≡K 10~eV 103cm 1095-330FFTn ⇒×≈⇒=ε)(H A74.0A 0.01~ separationar internucleatom/cm 102g/cm 10g/cm 10-10cm 10701.0~ ,g 102~2oo3306741033<⇒×≈⇒==⇒×=×=−ASSVRRMMρρK 10~K 10~97FTT <<K 10~V 103~cm 10)34(cm) 103.1( 1173-3833113FFAATeVAnRVAR⇒×⇒≅≅⇒=⇒××≅−επSummaryconst)3(223)(21214533532222322222=⎟⎠⎞⎜⎝⎛≡===⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−=⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+=NVpnmNDNCNUFFFFVFFFFτπεεεεετπετπεμετπεhBose gas in three dimensions: chemical potentialTo