NOTES FOR CHAPTER 17. THE BOLTZMANN FACTOR AND PARTITION FUNCTIONS. Equilibrium statistical mechanics (aka statistical thermodynamics) deals with the

Transcription

1 OTE FOR CHAPTER 7. THE BOLTZMA FACTOR AD PARTITIO FUCTIO Equlbrum statstcal mchancs (aa statstcal thrmodynamcs) dals wth th prdcton of qulbrum thrmodynamc functon, (.g. nrgs, fr nrgs and thr changs) from th bass of atomc- molcular-lvl dscrptons of th sampl of mattr bng consdrd. uppos you wantd to prdct th avrag total nrgy.. (KE + POT) of som systm. To gt t, you would nd two quantts: () th nrgy of vry stat that th systm can b n - th E s (2) th probablty that th systm can b found n ach of thos stats - p. Thn th avrag nrgy <E> can b calculatd from <E> p E sum ovr all possbl stats Th E s ar th nrgs of all th stats th systm can hav. Ths ar gvn at last n prncpl by th soluton of th chrödngr quaton of th systm. Th p s ar th probablts f obsrvng ( fndng) th systm n ach of ths stats. Th Boltamann fact says that f a systm has stats wth nrgs E, E 2, E 3, thn p E / T B () whr B s th Boltzmann fact, T s th absolut tmpratur, β s th symbol f / B T. Th systm partton functon Q s ust th sum of th Boltzmann facts ovr all possbl stats.. Wrtng p β E Q (2) -β E s quvalnt to wrtng p K, and K can b dtrmnd by th -β E fact that th p s hav to b nmalzd.. that: p (3)

2 that E K β K -β E K (4) β E o th nmalzd p s ar gvn by: - β E p K (5) [ub (4) nto (5)] by p (6) β E -β E -β E p (7) Q o th systm partton functon Q has th proprty that (/Q) s th nmalzaton constant that convrts th Boltzmann fact β E to th nmalzd probablty wrttn n quaton (7). F many systms of non-ntractng partcls.. dal gass (whch ral gass bcom at hgh dluton) th chrödngr quaton f th total systm nrgy of all th allowd stats th E s - can actually b obtand numrcally. Ths s bcaus, f th partcls do not ntract wth ach othr, th systm nrgy E s ust th sum of ndvdual partcl nrgs,.. E (, V) ε + ε ε (8) 2 Th ε s ar th ndvdual partcl nrgs (gnvalus). F atoms ths ar gvn by: 2

3 2 h ε n,n n (n x + n y + n z ) (9) x y z 2 8ma whr a V /3, m mass, h Planc s constant, and th n x, n y, n z ar ntgrs:,2,3. E s wrttn as a functon of and V th dpndnc of s sn through quaton (8), th dpnd on V through quaton (9) (a 2 n quaton (9) s th sam as V 2/3 ). Orgn of th Fm of th Boltzmann Fact To do ths.. to gt som undrstandng of why th Boltzmann fact s of th fm β E p (0) w ntroduc th concpt of an nsmbl. An nsmbl s a hug collcton of macroscopc systms, all wth th sam valus of, V, T. Th rqurmnt of th sam constant T s mt by th noton of a hat bath Fgur 7. Each systm has th sam, V, T but can b n any of th quantum stats consstnt wth th spcfc valus of and V. cton 7.5 Th rlaton of th avrag prssur to th partton functon. To gt ths, start wth th thrmodynamcs rsult. E P and ta th nsmbl avrag < > of both sds to gt P E - E E P - - p (, V,β) () V Also P p (, V,β)P (, V) (2) Comparng th RH s of () and (2); w gt 3

4 E P (,V) - V (3) ub (3) nto (2): E P p (, V,β) - otc also that from P -β E (,V) E - Q V (4) Q(, V,β) β E (,V) w gt Q,β V β E (,V),β V - β E (,V),β (-β E ) -β E (,V) (-β E ) V,β -β E (,V) β E,β Q E β E (,V) - β, β V,β (5) Dvd both sds b βq: 4

5 β Q Q E (6) -β E -, β Q V,β nc RH (4) RH (6), P β Q Q,β P lnq (7) β,β A basc postulat of statstcal mchancs s that th nsmbl avrag of a mchancal varabl (varabls E and P ths ar assocatd wth th st law of thrmodynamcs xprssd n trms of hat and w but not T) s qual to th obsrvd valu of th varabl. E E (8) P P (9) obsrvd nsmbl avrag Thus, f w can calculat Q f som systm w can calculat ts prssur fm quatons (7), and ts nrgy from (8), wth E calculatd from (7.20) (7.2) of th txt. Dfntons Indpndnt Partcls: ar thos that bhav as though th othr partcls n th systm do not xst. F a systm of ndpndnt partcls Q,,... β(ε + ε + ε +.) β ε β ε β ε... 5

6 q ontractng Partcls: Ths s a war condton than ndpndnc. In a systm of nonntractng partcls, th PE of ntracton btwn all th partcls s zro..g. A dlut gas of frmons conssts of nonntractng partcls, but th frmons ar not ndpndnt bcaus thy oby th Paul Excluson Prncpl f nonntractng frmons: Q (...) β(ε + ε + ε +...) q 7-6 ystm Partton Functon f a ystm Comprsd of Indpndnt Dstngushabl Molculs Atoms. Ths s a hypothtcal modl. Thr rally s no such systm.. a systm of ndpndnt, dstngushabl atoms molculs dos not xst. Howvr f som systms t s a farly good approxmaton. An xampl of such a systm s a modl calld a prfct crystal. In a prfct crystal, th atoms molculs do not mgrat and ach s dntfd wth a spcfc st of co-dnats. Whl thy ntract wth thr ngbours (ths ntracton s rsponsbl f thr vbraton about thr lattc ponts), th vbraton of ach partcl s somtms only waly crlatd wth th vbraton of ts nghbours and thn w can, as an approxmaton, trat th vbratons as though thy occur ndpndntly of th vbratons of all th nghbours. Th pctur s that ach atom vbrats about ts lattc pont, surroundd by nghbs whch, as far as th vbrat n quston s concrnd, ar frozn n thr rspctv lattc locatons. Each atom molcul of th prfct crystal s thought of n ths way. F such as systm, th total nrgy E Ρ (,V) s gvn by th sum of th nrgs of all th vbratons: 6

7 ndx dntfyng th partcl a b c E l (, V) ε (V) + ε (V) + ε (V) +... LOOOOMOOOO nrgy stat of partcl trms - on f ach vbrat Hr, th systm partton functon bcoms: Q(, V, T) β E sum ovr (systm PF) systm stats a b c -β ( ε ( V ) + ε ( ) V ε LOOMOO trms L OMO O summatons a β ε ( V ) b β ε ( V ) c β ε ( V ) q a (V, T) q b (V, T) q c (V, T)... LOOOOOO MOOOOOO product trms nc all th vbrats ar dntcal (.g. a lattc of Ar atoms). q a (V,T) q b (V,T) q c (V,T). Thn Q q(v,t) (systm PF) (f ndpndnt, dstngushabl partcls) whr q(v,t) β ε (V) 7.7 Th Partton Functon f onntractng Indstngushabl Partcls In systms whrn th partcls ar all of th sam typ (.g. gasous CO 2 ) thy ar of cours ndstngushabl n that ach partcl s proprts ar dntcal to thos of th othr partcls. Also, th partcls ar ndstngushabl bcaus thy cannot b dntfd wth any 7

8 dfnt locaton (st of co-dnats) n th systm. W wll vntually show that f such a systm at hgh dluton, th systm partton functon Q s approxmatly gvn by: [q(v,t)] Q! whr th! crcts f partcl ndstngushablty and, as bf, q( V, T ) β ε ( V ) partcl nrgy stats But frst w hav to dal wth th fact that all partcls ar thr bosons frmons. Frmons: ar partcls whos wav functons ar antsymmtrc whn two partcls xchang locaton. Also frmons hav ½-ntgral spns. Exampls of frmons: lctrons, protons, nutrons. Bosons: ar partcls whos wav functons ar symmtrc whn two partcls xchang locatons. Bosons hav ntgr spns..g. photons (hav spn ) dutrons (hav spn 0). Also whl frmons oby a Paul Excluson Prncpal bosons do not. Thus whl no two dntcal frmons n a systm can occupy th sam nrgy stat, thr s no such rstrcton on bosons. F a systm comprsd of nonntractng ndstngshabl partcls, th total nrgy s E ε + ε + ε +... (not: no suprscrpts lablng th partcl) LOO MOO trms and th systm partton functon s: Q(, V, T)... β (ε + ε + ε +...) β (ε + ε + ε +...) Q(, V, T) (20),,... 8

9 Howvr Q(,V,T) of th systm cannot b smply rlatd to th sngl partcl q(v,t), as happns f a systm of ndpndnt dstngushabl partcls, bcaus of rstrctons mposd by ndstngushablty whch crats rstrctons both f frmons and bosons, and by th Paul Excluson Prncpl whch crats rstrctons f frmons..g.. Frmons (xampl 7.5). F a systm comprsd of two dntcal nonntractng frmons ach wth nrgy stats: ε, ε 2, ε 3 and ε 4, lst th systm nrgs that would occur n th systm partton functon sum n quaton (20) abov. oluton Q(2,V,T) 4 4 β(ε + ε ) Th only systm stats that occur ar: ε + ε 2 ε 2 + ε 3 ε + ε 3 ε 2 + ε 4 ε + ε 4 ε 3 + ε 4 Thus thr ar sx, not sxtn (4 4), trms n th sum f Q(2,V,T) f ths systm. Th sx trms you gt by wrtng th abov sx trms n rvrs dr (.g. ε 2 + ε, ε 3 + ε, tc.) do not occur bcaus partcl ndstngushablty causs ths sx trms to rfr to th sam systm stats as thr (rvrs dr) countrpart lstd abov... " " " 2" " " " 2" 9

10 ar th sam systm stat bcaus partcls on and two ar ndstngushabl. Thus ndstngushablty lmnats sx of th sxtn trms that would ars n an unrstrctd sum. Also, th Paul Excluson Prncpl lmnats th four trms: (ε + ε ),2,3,4. o th complcaton n valuatng Q(,V,T) n trms of q(v,t) f systm of frmons s that th sums ovr and cannot b don ndpndntly of ach othr bcaus of th abov rstrctons. ow consdr a systm of two bosons, ach wth four allowd nrgy stats, and lt us numrat th allowd trms. Thr ar tn. ε + ε 2 ε 2 + ε 3 ε + ε ε + ε 3 ε 2 + ε 4 ε 2 + ε 2 ε + ε 4 ε 3 + ε 4 ε 3 + ε 3 as wth frmons ε 4 + ε 4 no Paul Excluson Prncpl rstrcton But th rstrcton du ndstngushabllty, causs sx trms (th sam ons as bf wth frmons) to b xcludd.g. (ε 2 + ε ) rfrs to th sam systm stat as (ε + ε 2 ), and so s not countd as a sparat trm. Ths rstrctons also of cours apply to partcl boson systms..g. f a systm of bosons and 0 partcl stats, th trms ε rows ε ε 0 + ε0 + ε ε0 2 trms 0 + ε + ε ε 0 + ε 2 tc ε ε All rprsnt on systm stat bcaus of partcl ndstngushablty, and not th systm stat that ar ndcatd by th rows shown abov

11 nc all partcls ar thr frmons bosons, ths complcatons man that an xact quaton rlatng Q(,V,T) f a systm, to q(v,t) f ach partcl n th systm dos not xst. (xcpt f th artfcal cas of ndpndnt dstngushabl partcls whr Q q ). Consdr howvr th stuaton whr th numbr of partcl stats that can b populatd by ach partcl gratly xcds th numbr of partcls. W wll fnd that f a numbr of systms of ntrst to us, ths wll b th cas. Clarly, whn ths occurs, t wll b vry unlly to fnd two m partcls n th sam partcl stat. Whl most quantum-mchancal systms hav an nfnt numbr of quantum stats (that ars from an nfnt numbr of solutons of th chrödngr quaton), what s rlvant hr s th numbr of stats lly to b populatd to a non-nglgbl dgr by ach partcl. Roughly spang, th stats populatd by a partcl ar thos whos nrgs ar not too much gratr than th avrag nrgy of th partcl. Ths avrag nrgy s of th dr of T (.g. th avrag nrgy of an atomc dal gas partcl s 3/2 T). so f w hav a systm f whch th numbr of quantum stats whos nrgy ar T, gratly xcds that numbr of partcls thn f that systm t wll b xtrmly unlly to vr hav two m partcls n th sam stat. In ths cas w can drv an xprsson rlatng Q(,V,T) to q(v,t) that s an xcllnt approxmaton. It s Q(, V, T) [q(v,t)]! (2) (f ndpndnt, ndstngushabl partcls - thr bosons frmons) W now show why ths occurs. Frst, quaton (2) can only b drvd f th total nrgy of th systm E(,V) can b wrttn a th sum of ndvdual partcl nrgs. Ths happns only f dal gas partcls whr th ntrmolcular PE of ntracton s zro. o quaton (2) wll only b vald f systms of nonntractng partcls.

12 onndpndnt, onntractng Frmons F dfntnss, suppos w hav a systm of 3 nonntractng frmons ach of whch has xactly 4 allowd nrgy stats. Th fact that thy collctvly comprs on systm mpls that thy nflunc on anothr through th Paul Excluson Prncpl. Th xact systm Q hr would b Q α -β E α (Ρ # systm stats) Q 4 ( ) β(ε + ε + ε partcl stats ) (xact) Th numbr of trms n ths sum (Ρ) s: (ε + ε + ε ) f th frmons ar dstngushabl. Howvr f ndstngushabl frmons (any pur systm othr than a prfct crystal) thr wll b far fwr trms snc th trm (ε 2 + ε + ε 3 ) s not dffrnt from th trm (ε + ε 2 + ε 3 ) tc. f ndstngushabl partcls. o th numbr of dstnct contrbutng stats to th sum wll b only 4: (ε + ε 2 + ε 3 ) (ε + ε 2 + ε 4 ) (ε + ε 3 + ε 4 ) (ε 2 + ε 3 + ε 4 ) Ths coms from th followng. Lt th numbr of dstnct contrbutng trms b t 2

13 total numbr of trms wth all subscrpts dffrnt t numbr of ways ths subscrpts can b prmutd t o Q f ths systm of nonntractng, nonndpndnt, ndstngushabl frmons s Q β(ε + ε 2 + ε3 ) β(ε + ε 2 + ε 4 ) β(ε + ε3 + ε 4 ) β(ε 2 + ε3 + ε 4 ) Gnrally t trms n th product QOOOOOO UOOOOOO ` ( )( 2)...( [ ])!! whr t numbr of dstnct contrbutng trms n sum f Q numbr frmons; Ρ numbr systm stats f dstngushabl frmons. numbr of nrgy stats avalabl to ach frmon. o th way to convrt a Q f dstngushabl frmons to a Q f ndstngushabl frmons s to dvd th [ ( -)( -2) ( -(-)] trms that ars f th dstngushabl frmons by!. In ths xampl w hav Q(DIT) 4 ( ) -β ( ε + ε + ε ) (24 trms) (22) To gt Q(IDIT), w hav to dvd Q(DIT) by! to rmov th trms, whch f ndstngushabl frmons ar ovr-countd n (22), o: Q(IDIT)! ( ) trms QOO UOO ` -β(ε + ε + ε +...) (23) -β E Q(IDIT) (24)! 3

14 whr ( )( 2)... ( [ ]) LOOOOOO MOOOOOOO trms n product To summarz, th xact xprssons gvn n quatons (23) (24) f Q(IDIT) f nonndpndnt nonntractng frmons, cannot b xactly crctd to th fm q(v, T) Q bcaus th q trm coms from an unrstrctd multpl sum and not th! proprly rstrctd multpl sum such as th on shown n quaton (23). onntractng Bosons Agan, f dfntnss, consdr a systm of 3 nonntractng bosons, ach of whch has xactly 4 allowd nrgy sats. Th xact systm Q s: Q(DIT) m -β E (m # systm stats) Hr, f dstngushabl bosons, m snc ach trm f E loos l (ε + ε + ε ) whr ach ε, ε, ε can b any of ε, ε 2, ε 3, ε 4. o n gnral, f dstngushabl bosons, th systm Q s gvn by: whr numbr systm stats m ( ), Q m β E numbr nrgy stats avalabl to ach boson, and E ε + ε + ε +... LOO MOO trms Wrttn n trms of partcl nrgy stats, ths bcoms, Q (,,,...) β(ε + ε + ε +...) LOO MO O trms 4

15 β ε β ε β ε Q... LOOOO MOOOO trms Q q q q... LOMO trms Q q Crctng th xprsson Q q so as to ma t applcabl to ndstngushabl bosons s complcatd bcaus th! crcton fact usd f frmons dos not apply to bosons. Ths s bcaus! proprly crcts f ovr - countng only whn all th partcl subscrpts ar dffrnt. Ths condton dos not apply to bosons whr m than on boson can b n th sam stat. Thr s no sngl gnral crcton fact that crcts q f dstngushabl bosons so as to fm a crct xprsson, f Q n trms of q, f ndstngushabl bosons. o, th concluson s that both f frmons and bosons, no xact xprsson xsts that rlats th partcl q(v,t) to th systm Q(,V,T) (othr than th artfcal cas of ndpndnt dstngushng partcls, f whch Q(,V,T) q(v,t ). Howvr, th rlaton Q(,V,T) q(v,t) /! wll b a good approxmaton f systms of ral nonntractng frmons and ral nonntractng bosons provdd th numbr of partcl stats whos nrgy < T gratly xcds th numbr of partcls. Ths s tru f bosons bcaus th sum: 5

16 Q β(ε + ε + ε +...) (,,,...) β ε q wll contan only a vry small propton of trms f whch two m subscrpts ar th sam. As long as ths s so, th fact /! farly accuratly crcts f th ovr - count that occurs n tryng to us th dstngushabl partcl xprsson (.. q ) so as to apply to ndstngushabl partcls. Ths s tru f frmons, bcaus as long as th contrbuton of trms contanng 2 m dntcal subscrpts (.g. ε + ε 2 + ε 2 + ε 3 + ) s small, thn th rstrctd sum approprat to frmons bcoms almost dntcal to th unrstrctd sum applcabl to ndpndnt partcls.. (...) β ( ε ε ε...) β ( ε ε ε...) ( ) (...) β(ε + ε + ε +...) LOO MO O trms q provdd th propton of subscrpts n (ε + ε + ε + ) wth th sam valu s small. Concluson F ral, nonntractng, ndstngushabl frmons and bosons Q(, V, T) q(v, T)! (25) provdd th numbr of partcl stats of nrgy < T ( *) gratly xcds th numbr of partcls (). ystms of nonntractng, ndstngushabl partcls, (.g. dal gass, both atomc and molcular) that oby quaton (25) ar sad to oby Boltzmann tatstcs. Crtron: It turns out that th condton ndd f quaton (25) to b vald.. * >> s quvalnt to th opratonally usful condton 6

17 2 h V 8mT 3/2 << Clarly th condton s favourd by systms of low dnsty, larg mass(m), and hgh V tmpratur (T). 7