Brief Notes on the Fermi-Dirac and Bose-Einstein Distributions, Bose-Einstein Condensates and Degenerate Fermi Gases Last Update: 28 th December 2008

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1 Brif ots on th Frmi-Dirac and Bos-Einstin Distributions, Bos-Einstin Condnsats and Dgnrat Frmi Gass Last Updat: 8 th Dcmbr 8 (A)Basics of Statistical Thrmodynamics Th Gibbs Factor A systm is assumd to b abl to xchang both nrgy and particls with a rsrvoir. Th probability that th systm is in a spcifid quantum stat is, E P, () k = Boltzmann s constant (.8 x - Jouls. o K) T = absolut tmpratur ( o K). E = nrgy of th quantum stat of particls (Jouls) = chmical potntial (illustratd blow what this mans), Jouls = numbr of particls in th quantum stat of intrst. ot: Th probability is only proportional to th onntial rssion, not qual to it. Th actual probability rsults by normalising, that is, E P, () whr th Grand Sum,, is found by summing th probabilitis ovr all quantum stats and ovr all possibl numbrs of particls, Th Boltzmann Factor stats This is a spcial cas of th Gibb s factor whn th numbr of particls is fixd, i.. th systm cannot xchang particls frly but can xchang nrgy with th rsrvoir. Th probability that th systm is in a spcifid quantum stat is, E () E P (4) Th probability is only proportional to th onntial rssion, not qual to it. Th actual probability rsults by normalising, that is, P Z whr Z is known as th Partition Function and is found by summing th probabilitis ovr all quantum stats, stats E (5) E Z (6)

2 Dgnracy / Dnsity of Stats Th Gibbs and Boltzmann factors giv probabilitis for a spcific quantum stat. An nrgy lvl is dgnrat if thr is mor than on quantum stat with this nrgy. Suppos thr ar S(E) stats with nrgy E. Th probability of th systm bing in any stat with nrgy E is, (Diffusiv contact): (Fixd numbr of particls): E P, E S E (7) E P E S E (8) On important xampl of dgnracy is th nrgy stats for a Schrodingr particlin-a-box. Th numbr of stats with th sam wavnumbr, k (or, mor prcisly, th numbr of stats with wavnumbrs btwn k and k+dk) is, Vk ( k). dk dk (9) S whr V is th volum of th box. This also applis for light (photons). For non-rlativistic particls of mass m, th numbr of stats with th sam nrgy lvl, E (or, mor prcisly, th numbr of stats with nrgis btwn E and E+dE) is, V m E. de E de () S Exprssions (9) and () giv th sam numbr of stats, just in trms of diffrnt variabls. Equ.() follows from (9) by substituting k me /. Th function S is calld th dnsity of stats. (Th lttr usd for S varis btwn authors). If th particls also hav spin, thn (9) or () must b multiplid by th numbr of spin stats. E.g., for lctrons or photons ths rssions should b multiplid by. (B) Drivation of th Frmi-Dirac Distribution Frmions hav half-intgral spin and oby th Pauli Exclusion Principl, i.. thr cannot b mor than on idntical frmion in any singl quantum stat. If w considr a systm consisting of just on quantum stat, of nrgy, th Grand Sum bcoms simpl bcaus th only possibl numbrs of particls ar ithr or. Hnc () givs, From () th probability of th stat bing occupid is th Frmi-Dirac Distribution, () P ()

3 (C) Drivation of th Bos-Einstin Distibution Bosons hav intgral spin. Thr can b any intgral numbr of idntical bosons in th sam quantum stat. In Equs.(,,), th nrgy, E, is th nrgy of a quantum stat of particls. If th nrgy of a givn quantum stat is whn occupid by on particl, th total nrgy for particls in this stat is E. W may considr just this on quantum stat as th systm in qustion, so th Grand Sum bcoms, Whr, () (4) Th probability of th stat bing occupid by particls is thus, from (), P, (5) Th ctation valu for th numbr of particls in this stat of nrgy P, d d / d d is thus, This is th Bos-Einstin Distribution. It diffrs from th Frmi-Dirac distribution only as rgards th sign of th in th dnominator. But this diffrnc is crucial. For th Bos-Einstin distribution, th chmical potntial,, must b lss than th lowst nrgy lvl,, i.., so that - cannot b ngativ. This diffrs from th Frmi-Dirac distribution for which th chmical potntial may xcd th lowst nrgy stat and - may b ngativ. Th two distributions ar plottd against ( - )/ blow. Salint faturs ar, Th two distributions bcom ssntially th sam for ( - )/ > ; For bosons th occupation numbr bcoms infinit as ( - )/ tnds to zro; This mans that thr will b vry larg numbrs of bosons in any stat whos nrgy (strictly - ) is small compard with ; For frmions th man occupation numbr is ½ whn ( - )/ = ; This mans that about half th stats whos nrgy (strictly - ) is small compard with will b occupid by a frmion; For frmions th man occupation numbr tnds to whn ( - )/ bcoms larg and ngativ. This mans that as absolut zro is approachd, all thos stats with < occupid by a frmion. (6) will b

4 Occupancy umbr, <> Bos-Einstin Frmi-Dirac 5 4 ( - )/ (D) Bos-Einstin Condnsats As th tmpratur of a systm of bosons is rducd th valu of ( - )/ will bcom larg and positiv for any stats with >. Consquntly th man occupation numbr will bcom vry small for all such stats. If thr is a fixd numbr of bosons, it follows that thy must all nd up in stats with =. Consquntly, at absolut zro th valu of th chmical potntial for a systm of bosons must qual th ground stat nrgy, =. Morovr, at absolut zro w ct all th bosons to nd up in th ground stat (noting that for = th occupation numbr is divrgnt). In a sns, thrfor, thr is nothing surprising about Bos-Einstin condnsation. Th Bos-Einstin condnsat phas of a Bos gas consists simply of a larg proportion of th bosons in th ground stat. Howvr, th surprising thing is that th onst of Bos-Einstin condnsation is quit suddn (i.. it is a phas chang) and occurs wll abov absolut zro (.g. at a tmpratur of.7k for hlium-4 ). A simpl thortical rssion for th critical tmpratur for th onst of Bos- Einstin condnsation is drivd as follows:- Sinc = at absolut zro w approximat at sufficintly low tmpraturs, so that th ground stat is thus, from (6), for th ground stat. Th man occupancy of, whr th subscript dnots th Hlium-4 consists of protons, nutrons and lctrons, and hnc has an vn numbr of frmions. Ovrall it has spin zro, and hnc is a boson.

5 ground stat. From this w s that th tmpratur drivativ of th chmical potntial of a boson gas at low tmpratur, T k, dtrmins th occupancy of th ground stat. R-arranging givs up in th ground stat, so that. At absolut zro w ct all th particls to nd at low tmpraturs, whr is th total numbr of bosons. Hnc, if th numbr of particls is vry larg, must b xtrmly clos to. Using this approximation, (6) givs th man occupation numbr of any xcitd stat (i.. abov th ground stat) to b simply, (7) whr is th nrgy of th stat with rspct to th ground stat. Hnc, if th first xcitd stat has a (rlativ) nrgy which is much largr than, i.., if, thn th man occupancy of th xcitd stat will b small, <<. W want to know th total numbr of bosons in all xcitd stats. For this w nd to know how many stats thr ar, not just thir nrgis. This information is providd by th dnsity of stats, Equ.(). B: In using this form of dnsity of stats w ar rstricting attntion to massiv bosons, not photons. Th ctd numbr of particls with nrgy btwn and +d is th numbr of stats in this rang tims th man occupancy,, of ach, giving, V m d d ( ) (8) W hav changd th notation slightly to mphasis that th LHS (d ) mans th total numbr of particls in this nrgy rang, not just th occupancy of a singl quantum stat. To mak things a littl simplr, it is usual to assum that w ar daling with xcitd stat nrgis which ar quit a bit highr than th ground stat nrgy, so that w can ignor th distinction btwn and. Changing th variabl in (8) to x / thn givs, V m x dx d ( ) (9) x Th total numbr of particls in all xcitd stats (i.. not including th ground stat) is found by intgration to b, m x dx V () x Consistnt with prvious approximations th lowr limit of th intgral is st to zro, though rally it should b a small finit valu (i.. th gap btwn th ground stat and th first xcitd stat). Th last part of () is just a numrical constant which can

6 b valuatd numrically to b, x x dx.659 () Th total particl numbr dnsity is / V, whr is th total numbr of particls. Hnc, on dividing by, Equ.() givs th fraction of th particls in xcitd stats to b, m.659 () Clarly this only maks sns if th RHS valuats to a numbr lss than. Whn it dos not, thn th assumptions of th drivation hav brokn down. Whn it valuats to lss than, th rmaindr of th particls must b in th ground stat. Thus, th fraction of particls in th ground stat is, m.659 () Paus to rflct how xtraordinary this is. Equ.() mans that a substantial fraction of all th particls ar in th ground stat. In othr words, a substantial fraction of th boson gas is in a nw (suprfluid) phas. This is tru if th RHS of () is positiv, i.. if, m.659 Solving for th critical tmpratur for Bos-Einstin condnsation (BEC) givs,. (4) BEC (5) m Thus, th onst of Bos-Einstin condnsation will b at a highr tmpratur if, Th numbr dnsity of particls is gratr, or, Th particls ar lightr. Exampl: Hlium-4: umbr dnsity of liquid H 4 is.8 x 8 m -, and th mass of an H 4 atom is 6.68 x -7 kg. Hnc, x.8x BEC. 4.x J x Using k =.8x - J o K givs T BEC =. o K. This is a prtty rasonabl stimat givn that th onst of th suprfluid phas of H 4 is found to occur at.7 o K.

7 (E) Dgnrat Frmi Gass (on-rlativistic) Rcall th apparanc of th Frmi-Dirac distribution - s abov graph and Equ.(). R-itrating som of th salint faturs of th distribution, and making som conclusions For frmions th man occupation numbr tnds to whn ( - )/ bcoms larg and ngativ. This mans that as absolut zro is approachd, all thos stats with < occupid by a frmion. will b Th valu of th chmical potntial,, of a frmion at absolut zro is known as th Frmi Enrgy, F. At sufficintly low tmpraturs, spcifically whn << F, virtually all th stats with nrgis blow F will b occupid, and virtually no stats with nrgis abov F will b occupid. A frmion gas with << F, is said to b dgnrat. Th valu of th Frmi nrgy is asily valuatd. From Equ.() th total numbr of quantum stats with nrgis lss than or qual to F is, F F V m V m E de F whr w hav multiplid () by to account for th two spins stats of a spin ½ frmion. Th Frmi nrgy is, by dfinition, th nrgy for which th numbr of stats with nrgis not xcding this lvl quals th numbr of frmions prsnt. In trms of th numbr dnsity of frmions, / V, Equ.(6) r-arrangs to giv th Frmi nrgy as, F m ot that Equ.(7) for th Frmi nrgy is almost th sam as Equ.(5) for th nrgy at th onst of Bos-Einstin condnsation, apart from th constant factor (though only on can b rlvant for any givn particl typ, of cours). Exampl : Elctrons in Mtals For coppr, 8 8,.5x m and th lctron mass is m = 9. x - kg, giving, F J 7V 9. Hnc, using k =.8x - J o K givs th Frmi tmpratur, TF F / k, to b 8, o K. Consquntly th lctrons in mtals ar wll dgnrat, i.. tmpraturs blow th mlting point ar much smallr than T F. Givn that Equs.(5) and (7) for T BEC and T F ar so similar, why is th H 4 BEC tmpratur as low as ~ o K but T F for lctrons in mtals is so high? Th answr is simply bcaus of th factor of 7,4 mass diffrnc btwn th H 4 atom and an lctron. (6) (7)

8 Exampl : Hlium- H consists of two protons, on nutron and two lctrons. It thrfor has an odd numbr of frmion constitunts and must b a frmion. (It has spin ½ ). Hnc H is a vry diffrnt bast from H 4 sinc th lattr is a boson. Th similarity of Equs.(5) and (7), and th similarity of th masss of H and H 4, imply that w should ct a gas of H to b a dgnrat Frmi gas at tmpraturs blow, vry roughly, ~ o K or so. This is indd found to b th cas (for xampl by masuring its spcific hat). Exampl : Whit Dwarfs Whit dwarfs ar th nd point of th volution of stars in a crtain rang of initial masss. Thy hav a mass comparabl to th Sun, but with linar dimnsions about tims smallr, and hnc hav a man dnsity about a million tims gratr, with 6 ~ m. Th matrial of a whit dwarf is ntirly ionisd into nucli and fr lctrons. Th Frmi tmpratur for th lctron gas is thus, 9 T F ~ K ( F =.7 MV).8 9. Th tmpratur in th intrior of a whit dwarf is probably around 7 K, and crtainly not mor than ~ 8 K, so th lctron gas is dgnrat. In fact, it is th dgnracy prssur of th dgnrat lctron gas which prvnts th whit dwarf collapsing furthr undr th action of gravity. ot that th Frmi nrgy is comparabl with th rst mass nrgy of an lctron (.5 MV) so that ths lctrons ar bcoming rlativistic and th calculation is only just valid. It is this fact that limits th possibl siz of whit dwarfs. Rpating th calculation rlativistically rvals that lctron dgnracy prssur can support th star against gravitational collaps only if its mass is blow a crtain limit, which is about.46 tims th solar mass. Consquntly, stars with gratr mass than this rmaining at th nd of thir volution ar doomd to b crushd by gravity into an vn mor dns stat a nutron star, a quark star or a black hol. Finally, ar th protons in a whit dwarf dgnrat? Thir Frmi tmpratur diffrs from that of th lctrons only by th ratio of thir masss, 86, so T F ~ x 6 K. This is rathr lss than th likly tmpratur in th cntr of th whit dwarf, and so th protons ar probably not dgnrat. Consquntly th protons do not contribut significantly to supporting th star against gravitational collaps.

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