PH 605. Thermal and Statistical Physics. Part II: Semi-Classical Physics Quantum Statistics. course-webpage:

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1 Scool of Pyscal Sceces PH 65 emal a Statstcal Pyscs Pat II: Sem-Classcal Pyscs Quatum Statstcs couse-webpage: ttp://wwwm.uc.ac.u/m/staff/pb/teac/ph65/ph65.tml. Pete lümle pb@uc.ac.u oom Poe: 8

2 Syllabus. Sem-Classcal Pyscs.... e oltzma stbuto eve!..... A smple eample..... Geealsato.... e Sem-Classcal Pefect Gas efto of te sem-classcal mo-atomc pefect gas stgusable / stgusable patcles? Cotbutos of ffeet types of moto to..... e esty of states Patto fucto fo taslatoal moto t Patto fucto of teal moto t Patto fucto of molecula otato ot Patto fucto of molecula vbato vb Patto fuctos a compaso to emetal ata topy a egy of te Sem-Classcal Gas topy of a mo-atomc gas te Sacu-etoe equato e etopy of mg-te Gbbs paao..... e pcple of te equpatto of eegy.... alty a Lmt of te Sem-Classcal escpto e classcal lmt Mawell velocty stbuto a classcal gas otatoal specfc eat of atomc molecules- oto/paa H.... Quatum Statstcs Ieal Sols ste's teoy of a eal cystal ebye's teoy of a eal cystal Quatum Statstcs ose-ste statstcs Fem-ac statstcs Compaso of oltzma a F statstcs etemato of α Systems wt vaable patcle umbe e Ga patto fucto Applcato to Femo/oso-Systems Fee electos metals Paul-paamagetsm e pefect poto gas - blac-boy aato ose-ste coesato Supecouctvty a supefluty C emoyamcs of stas...

3 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: ecommee oos / acgou eag fo ts seco pat: F. Mal "Statstcal Pyscs" Wley 988 [QC75 7 copes]. aele: "emal Pyscs" Cambge Uvesty Pess 999 [IS: ] Sem-Classcal Pyscs. e oltzma stbuto eve! e oltzma stbuto was touce te last secto of pat I see. Mallett s scpt. ecap: mco-state: ceta assgmet of patcles to ceta eegy state maco-state: ealse by may mco-states sum of mco-states Luwg oltzma coclue fom te law of emoyamcs tat te maco-state wt te most mco-states s te most stable equlbum emembe S l Ω... A smple eample ee ae stgusable epeet a etcal patcles A a C. ey ae allowe to occupy ffeet eegy states: a e.g. amoc oscllato. e total eegy of te system amouts to. e occupato umbes ae a. ow we ae gog to ty to f te umbe of maco-states by wc te system ca be ealse. maco-state I II III We see tat tee ae oly possble maco-states fo te system. e et questo s te: How may mco-states ae possble to ealse eac maco-state? ote: We ecall/ealse tat ecage of patcles te same mco-state oes t geeate a ew mco-state! eegy state maco-state I maco-state II maco-state III A C A A C C C A C A AC C AC A C C A A - o. of mco-states 6. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

4 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: Hece maco-state II as te gest statstcal wegt Ω o temoyamc pobablty o umbe of aagemets; ofte wtte as W fo Gema Wascelcet pobablty Ω II 6 ts eample... Geealsato Ispe fom ts eample we wat to geealse ts fo patcles. Fo sgle occupato t ca be ectly coclue tat te gest statstcal wegt s gve by all pemutatos o Ω! Howeve f we cose cases wc te occupato umbe ca evetually become lage ta > we ae oveestmatg by ts meto. s s because te pemutato of patcles eac vual mco-state oes t geeate a ew mco-state. Hece we ee te followg coecto:! Ω [..]!!!... ote: e meag of ts equato ca easly be cece o te pevous eample. maco-state I:! 6 Ω I!!!! maco-state II:! 6 Ω II 6!!!! maco-state III:! 6 Ω III!!!! 6 We also ow tat equlbum te oltzma etopy equato tells us tat te most pobable s ealse fo mamum etopy o S l Ω ma [..] Statg fom tese facts we ow wat to eve te equato fo te oltzma-stbuto: Gve: patcles stgusable epeet a etcal eegy states:... wt occupato umbes... atoally we ca establs te followg bouay cotos: a total umbe of patcles b total eegy costat [..] costat [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

5 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: ou goal s summase eq. [..]: We ave to f te mamum statstcal wegt o Ω!! mamum [..5] o smplfy ts tas we ealse tat we Ω as a mamum lω also must ave a mamum because te logatm s a mootoc fucto. s eables us to use Hece eq. [..5] becomes: l Ω l Stlg s appomato: l! l fo lage : [..6] [..6]!! l! l l l! l! l! ote: we wll late see tat fo ealstc cotos te last step applyg Stlg s fomula to ece lage s satsfe. l Ω l l l l [..] l Ω l l [..7] e mamum statstcal wegt eq. [..5] s te gve fo: l Ω [..8] ate ta ffeetatg wt espect to te occupato umbes t s stuctve to cose small cages symbol δ of te occupato umbe. Hece eq. [..8] becomes. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

6 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:5 δ l Ω δ l pouct ule δ l δ l δ δ l Ω δ l δ [..9] quato [..9] ca be combe wt te costat bouay cotos eqs. [..] a [..]. δ l Ω δ l δ δ δ δ δ mamum cost. must ot cage cost. must ot cage e easest way to solve suc a equato system o to combe te cotos s te meto of ueteme Lagage multples ecap: M. oas: Matematcal Metos te Pyscal Sceces e. Wley 98 page 7ff.. s gves: δ l δ λ δ β δ [ l λ β ] δ [..] wee λ a β ae te yet ueteme multples. e fst tem δ [..] ca be cose abtaly to be ay umbe as log as te last two ae cose to fulfl a te cotos above. ut geeally te followg coto must ol: l λ β l wt [ λ ] α. λ β [ λ ] [ β ] α [ β ] [..] Wat s left to o? We ave to f essos fo α a β. etemato of α: Fom eq. [..] α [ β ] α [ β ] α β e [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

7 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:6 etemato of β: Fom eq. [..] ow a tc: α e [..] β e e β β β β e β e β l β e β β e e oes t epe o β but α oes ece [.. ] α [..] l α α l [..] β β Fom eq. [..] we also ow: S l Ω [..7] β β αe β β β [ l α e l α β ] [ α α e αβ l l e ] l l [..] l [ l α β] α e [.. ] α l [..] α S l [..5] fom te pevous pat Mawell elatos: we ecall we use ee fo te total teal eegy -ate ta U- because t typcally use QM otato U S S S β [..5 ] β [..6] S S β [ l α β ] β [..] β β β. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

8 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:7. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy a wt eq. [..6] we get: β β β β β β S β [..7] f we ow set ts equato a eq. [..] to eq. [..]: [ ] β α we get te oltzma-stbuto: [..7] wee s calle patto fucto fom Gema: ustassumme sum of states of estece We see ow a statstcal stbuto ca be eve fom vey smple assumptos. e oly assumpto was eq. [..]: l Ω ma S wc s te statstcal tepetato of te seco law of emoyamcs. We wll apply te same fomalsm to obta ote quatum statstcal stbutos late.

9 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:8. e Sem-Classcal Pefect Gas ow we wat to apply oltzma statstcs to smple systems. e most smple system s te pefect o eal gas... efto of te sem-classcal mo-atomc pefect gas sem classcal QM eegy to eteme levels mo atomc oteal eegy o stuctue pefect gas umbe of patcles s muc smalle ta avalable eegy levels o egeeacy s s some yb betwee quatum-mecacal a classcal beavou oweve we assume to be able to la moe effects moe pecsely ta te pue classcal pctue. efto: etcal gas patcles molecules o atoms oly vey wea o teacto betwee te patcles patcles ae sepaate low esty eegetc tems ts meas tat te potetal eegy of teacto s eglgble compae to te etc eegy <<K tee ae muc moe avalable eegy levels ta patcles classcal cotuum of QMescbable scete eegy levels. e gas sall ave te followg popetes: cossts of patcles [..] te vual eeges -eac patcle ca est - ae: fomg a complete set of scete quatum states wt o ece te state of te system s: patcles quatum state wt eegy patcles quatum state wt eegy patcles quatum state wt eegy patcles quatum state wt eegy t as a total teal eegy of [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

10 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:9.. stgusable / Istgusable Patcles? ecause te patcles ae cose to be o-teactg we ca select a sgle patcle epeet of te otes to epeset a typcal patcle. Hece mco-state: [..] maco-state:* [..] *oweve te latte we o ot ow ow to calculate! Fo te calculato of te patto fucto we ave to stgus two vey ffeet cases: a te system cossts of stgusable etcal o-teactg patcles.e. te ecage of patcles esults a ew state e.g. eal cystal state a state b b te system cossts of stgusable etcal o-teactg patcles.e. te ecage of patcles oes O esult a ew state e.g. eal gas state a state b. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

11 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy case a stgusable patcles cystal of patcles: Hece fom eq. [..]: states all ove s sum of poucts us to fty so we ca be sue to f all possble pemutatos of eegy levels a patcle catos. Hece we ca eplace t by a pouct of sums: eegy level patcle cato

12 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy o j j j j [..] [..5] f all te patcles ae etcal j fo stgusable patcles j [..6] case b stgusable patcles gas of patcles: Wat s gog to appe f we woul use te same fomula to calculate te patto fucto of te patcles a gas???? o llustate tat ts s WOG we cose two stgusable patcles : ffeet states patcles te same state bot patcles? Howeve te patcles te seco tem ae coute twce but sce te patcles ae stgusable ts oes't geeate a ew state because: tee s o emetal meto to stgus tese two states a gas because te patcles ae ot fe space Hece: couts te patcles ffeet states falsely twce! We ca coect fo tat by:

13 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: So ow about a geealsato fo patcles? e fomula [..6] woul cout all possble pemutato of eegy levels wogly. Hece we ave to coect by! all patcles te same state some patcles same states! [..7] all patcles ffeet states s woul epeset te coect fomula wee te cetal tem we woul ave sums ove stuatos wee e.g. just two patcles ae te same state ece to be coecte by / a suc wee ae te same state ece coecto by /6 etc...s s vey messy! ow we ca use te sem-classcal agumets fom te efto secto... e pobablty s vey small fo suc a gas tat ay sgle patcle state s occupe by moe ta oe patcle! We ae someow scussg a cacatue of te classcal cotuum of eeges wc oweve as to be uestoo as beg splt up to scete coutable -QM-le- levels of fte umbe sce tee ae fte levels but a fte umbe of patcles most states ae empty o vey few states ae occupe by a sgle patcle a a sgfcatly small umbe of states cotas moe ta oe patcle egeeacy sem-classcal gas: umbe of patcles << umbe of eegy-states s agumet geatly smplfes eq. [..7] because oly te last tem couts! fo stgusable patcles:! [..8]! ow we ae eay to calculate te ffeet cotbutos to te sgle patcle patto fucto. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

14 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:.. Cotbutos of ffeet types of moto to? emo-yamcs: eteme by te temally actvate yamcs of patcles! Hece we ave to eal wt moto! ffeet types of patcle moto: taslatoal moto teal motos a otato b vbato [ c electoc ectato] taslato otato vbato e eegy of a state s te: t t a b [..9] taslato stateof al moto stateof teal moto a a b t a t b [..a] but te types of moto ae epeet! ece t t a b [..b] a b t t t wee s te sum ove all eegy states of taslatoal moto of te patcle epeet of stuctue s te sum ove all eegy states of teal moto of a molecule sce tey ae all a t etcal but epes o ts stuctue... atomc vbato taslato. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

15 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: Summase: te molecula patto fucto: t t [..] t : wll apply to ay pefect gas epeet of stuctue e.g. He e O CO CO H O C H 5 OH etc. t p mv wt p p m classcally ay value of p s possble sem-classcally: tee mgt be estctos ue to QM t : epes o te teal stuctue wc s te same fo all etcal patcles. ecause te patcles o ot teact t oes t epe o te macoscopc state P of te system. ot oweve epe o te tempeatue wc s esposble fo wc eegy-levels ae occupe/actvate O te ote a tempeatue s efe by tese motos so we mgt loo at ts fom just te ote se of te equato! z.. e esty of states Fo ts we ecap te QM-teatmet of a patcle a -bo: y eefoe we ave to solve te tme-epeet Scöge equato ħ ˆ m ψ y z y z ψ y z ψ y z fo a patcle a bo wt te followg bouay cotos see fgue: y z fo a a y a a z a fo all ote equg ψ y z o all faces of te bo. [you mgt efe to ttp://wwwm.uc.ac.u/m/staff/pb/teac/ph5/ph5.tml]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

16 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:5. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy s as te soluto: wt 8 s s s 8 ψ a a a m a z a y a a a a z y wc smplfes by substtuto wt te wave-vecto : a a a a a a a [..] we ca fute smplfy by assumg a a a a cubc bo volume: bo a [..] a fom e ogle s equato we get: λ p wt λ [..] ece p o ħ p [..] We wat to escbe a calculate te poblem of ow may states ae a ceta volume - space! s s because -space te ffeet eeges/states efe by te quatum umbes ae equally space see fgue wc maes te calculato muc ease.

17 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:6 ote tat te spees ts fgue o epeset states ot patcles!. e volume of eac quatum state -space s te: e esty of lattce pots o allowe -vectos s: o calculate te esty of te states we wat to ow te umbe of suc omal moes of stag waves wt a -vecto wose magtue les te teval to. s umbe ca be obtae appomate by calculatg te umbe of -lattce pots wc le betwee two specal sells aus a aus ollow spee of tcess. s wll gve appomately te coect aswe fo a ese lattce. ecause a macoscopc bo s bg te assumpto s val. [..5] a a ρ [..6] Caeful: > oly oe postve octat of te sells must be coute y z > s octat as a volume: octat 8 postve octat suface tcess eefoe te umbe of moes of stag waves wt betwee a s [..6 ] [.. ] a bo ρ a 8 8 esty of states -space: [..7] usg eqs. [..] a [..] ca be substtute fo v λ. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

18 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:7 ow we ae eay to eteme te sgle-patcle patto fucto fo te ffeet type of motos...we stat wt te taslatoal...5 Patto fucto fo taslatoal moto t Fo taslatoal moto we ave tee egees of feeom cf. eq. [..]. t y z [..8] ece te eegy ca be sepaate by compoets. e eegy- egevalues of te Scöge-equato te pevous pat ae gve by: o fo a smple cubc bo a a wt [..9] 8ma 8ma [..] ħ m wt [..] a wt eq. [..]: t y z wt a... t y z because tee s o pefee ecto of moto te tee sums ae equvalet a we ave to solve: t wt ħ m 8ma [..] ow we ave to cose te esty of te -values egeeacy of te state. Fo lage macoscopc mesos of te bo a >> te ae vey close sem-classcal cotuum so tat goo appomato te sum eq. [..] ca be eplace by a tegal. ut caeful: quato [..7] was aleay obtae fo coutg te esty of -values y a z a tee-mesoal faso. Hece we ave to op te powe of we usg te esty of states --space otewse we woul cout te same twce!. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

19 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:8. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy t [..] m a a m a a [..] [..7 ] wt ħ ħ Patto fucto fo taslatoal moto sgle patcle: t m m ħ [..] Alteatve: A moe ect appoac uses te soluto of te Scöge-equato. s aleay clues te possble egeeacy of states so we ca op cluso by te esty of states -space t solve te tegal: ma 8 [..] we substtute: m a ma 8 8 m a m a 8 qe! [.. ] t m m a m a s ect appoac avos te stocal agumet about esty of states wc oweve s qute useful fo a vaety of poblems!

20 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:9 ow we ae eay to apply ou owlege!!! efoe we o ts we ecap ow temoyamc popetes ca be eve fom te patto fucto. cf. pat I of lectue teal aveage eegy: etopy: pessue: l l U [..] l l S l [..5] l l P [..6] Helmoltz fee eegy: F S l [..7] etalpy: H l l P [..8] l l Gbbs fee etalpy: l G F P l [..9] l A pefect gas of stgusable patcles o states as a patto fucto eq. [..8]. s combato wt eq. [..] allows us te etemato of te patto! fucto ue to te taslatoal moto of te gas patcles. Hece te teal eegy of te gas ca be calculate usg eq. [..]: t [..] l l m! oes' t epeo C l C l l! l / aslatoal eegy fo patcles of a pefect gas: t [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

21 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: O fo mole A of pefect gas: ~ t eal gas costat 8.5 J/mol K A s s -fo stace- total ageemet wt etc gas teoy see QC. Specfc eat ue to taslatoal moto fo patcles of a pefect gas: ~ ~ t t C cost. [..]..6 Patto fucto of teal moto t Wat o we ow? Fom secto.. we ecap: eq. [..9] t t t ot vb elc otatoal cotbuto cotbut vbatoa o l cotbuto of electoc ectato a we a sow eq. [..] tat t t we ow: I te cose sem-classcal lmt we ca aga assume o cose a lage umbe of eegy egevalues states. e ae combatos of tese egevalues we sot tem wc ae epeet see also sepaato of vaables QM-poblems t t t t t t ot etc. ot vb vb elc elc. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

22 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: a we ca use te same agumets as gettg fom eq. [..9] to [..] o t t t t ot vb elc t t ot vb elc Patto fucto ue to teal motos: t [..] ot vb elc wee te patto fucto of otatoal eegy electoc ectato* elc *ot calculate ts lectue ot vbatoal eegy vb a ue to ae all epeet of te macoscopc state of te system!..7 Patto fucto of molecula otato ot We ecap te QM-escpto a solutos fo etals see ttp://wwwm.uc.ac.u/m/staff/pb/teac/ph5/ph5.tml ˆ -Scöge-equato: Λ Y θ φ l l Y θ φ l m l m Hamltoa specal pola co-oates Legea wavefucto: specal amocs Legea: Λˆ s θ θ s θ θ s θ φ egevalues: l ħ l l I wt momet of eta: I µ wee µ s te euce mass. Quatzato cotos: l a l m l. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

23 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy Hece I l l l ot ħ [..] ecause m s ot peset eq. [..] but spas l m l -we emembe- tat eac eegy level s l-fol egeeate. Hece ot ot l l l I l l l l ħ If te momet of eta o mass s bg* o te tempeatue g te eegy levels ae vey close aga sem-classcal agumet a we ae allowe to eplace te sum by a tegal. ot l I l l l ħ we substtute: l l I l l I I l l ħ ħ ħ [ ] ot ħ ħ ħ ħ I e I e I l I l Patto fucto fo molecula otato: ot 8 I I ħ [..] Commet o symmety coseatos: fo atoms e.g. O tee ae atoal egeeate states oly two egees of otatoal feeom I oe to coect eq. [..] a symmety facto σ s clue so tat ot σ ħ I we wll scuss ts late * ot tue fo stace fo H a H we wll see late!

24 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: At te bottom of page we a state tat t s te sum ove all eegy states of teal moto of a molecule. We also see tat eq. [..] oes t epe o macoscopc paametes le cf. eq. [..] wc s goo because otewse ou agumets secto.. woul ave bee wog! We ca cose tese molecules as vuals. ac s otatg epeetly of te otes! It oes t matte f tee s o of tese molecules te otato of molecule A oes ot tefee wt tat of a vce vesa. So f we woul someow magcally ecage te eeges o postos we woul geeate a ew mco-state e.g. state : Molecule A at posto as otatoal eegy A a at posto as eegy ; state : Molecule at posto as otatoal eegy a A at posto as eegy A. [s s ffeet fo te taslatoal moto! Wy?] Hece ot ot [..] How about te temoyamc popetes of otatoal moto? teal eegy ue to otatoal moto: ot l ot [..] I l ħ l ot l [ C] l ot otatoal eegy fo patcles of a pefect gas: ot [..5] o fo mole mola: ~ ot [..6] a ~ ~ ot ot C cost. [..7]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

25 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:..8 Patto fucto of molecula vbato vb Aga we ecap te QM-escpto a solutos fo etals see ttp://wwwm.uc.ac.u/m/staff/pb/teac/ph5/ph5.tml -Scöge equato fo amoc oscllato: ħ ˆ m ψ mω ψ ψ as te solutos: ψ s ξ s H s ξ e ξ wt ξ m mω ħ a egevalues s s ħ ω wt s We ow apply te same poceues as befoe: a s s ħ ω wt s s vb vb s s vb s s [..8] [..9] Howeve ts case we caot smply eplace te sum by a tegal! Wy ot? We wll see we we calculate te sepaato of te eegy levels tat eve fo a sem-classcal gas tey ae too fa apat to justfy suc a step. I te et capte we wll calculate tese sepaatos a cec ou poceues te tegato woul eque tempeatues wee - oay molecules- te cemcal bos woul stegate. I oe to get a pyscally meagful esult we -ufotuately- ave to solve te sum! Fst we smplfy te agumet by substtutg ece eq. [..9] smplfes to s te we multply eq. [..] wt vb s [..] e : vb e s s s s [..] ow we subtact eq. [..] fom [..] a get:. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

26 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:5. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy vb e e e e e e s s e s we see tat we we calculate te sum all tems cacel ecept fo s -/. Hece vb vb e e e-substtute: Patto fucto fo molecula vbato: vb [..] a* vb vb [..] *ue to te same agumets wc lea to eq. [..]! Alteatvely: fo wt vb > < s s s s s s s e sum ca be esse as a geometc sees α - ece α < fo wc fo... < α α α α α α s s Hece vb / e e

27 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:6. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy How about te temoyamc popetes? teal eegy: l l l l l vb vb vb vb batoal eegy fo patcles of a pefect gas: vb [..] fo te mola eegy ~ vb we smply ave to eplace eq. [..] by A. So wat woul ave appee f we woul ave cose a tegato ate ta a summato? Let s cec o a evg a ypotetcal patto fucto fom eq. [..]: e e e s s e s s s yp vb A A A A yp vb A yp vb l l l ~ a cost. ~ ~ yp vb vbyp C wc also coespos to te equpatto teoem wc wll be touce late! s s just fo evso! W O G!!

28 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:7 pemetal ata fo valato of te cose appoac: Specfc eat of oyge gas at oom-tempeatue: Fom I-spectoscopy: Hece O.67 ~ C vbyp s cetaly wog! Hz ~ K C O. ow about eq. [..] ~ C vb ~ vb A plug te emetal values : ~ vb C A [..5] ~ C vb 6.6 Js.67 Hz J/K K ~ vb 75 C ecellet ageemet! 75 Close specto of equato [..] eveals tat beses te zeo-pot eegy tem of te amoc oscllato oly a facto of te ge vbatoal states s ecte at omal tempeatues. So te sem-classcal agumet s ot val ts case! Howeve we ca ypotetcally cec f eq. [..] a te euce eq. [..5] gve te coect sem-classcal agumet by sg te tempeatue eefoe we o sometg typcal we touce a caactestc tempeatue fo te oyge-eample above: K ~ C vb. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

29 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:8 lm ~ C vb lm lm e L' Hosptal lm lm ctcal tems oly lm..9 Patto fuctos of te pefect gas a compaso to emetal ata moe ~ C ~ taslatoal m otatoal vbatoal wt 8 e I e A A e e we ca touce caactestc tempeatues geeally by [..6] ħ ece: vb a ot I ote: e caactestc tempeatue ca be assocate wt a facto of ecte states at a tempeatue fo e - 7% o about a t emetal values at K moe wavelegt spectoscopy [K] otatoal.5 - cm mco-wave.5- vbatoal - µm I 8-5 electoc ectato. -7 m U/IS - 5 fo omal tempeatues we ca cetaly eglect te electoc pat. otatoal levels ae completely ecte sem-classcal agumet val vbatoal levels ae patally ecte sem-classcal agumet ot val. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

30 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:9 ect compaso fo ffeet molecules vb [K] ot [K] O * HCl 5 H * 6 85 mass * fo omouclea atomc molecules we wll lea at te e of ts fst pat tat eq. [..] s ot etely coect especally at low tempeatues.. topy a egy of te sem-classcal Gas.. topy of a Mo-atomc Gas te Sacu-etoe-equato Fo a mo-atomc pefect gas e.g. oble gases: He e A... wtout electoc ectato all we ave to cose s te eegy ue to taslatoal motos. batos o otatos of a sgle atom o ot geeate ew mco-states. t I secto..5 we ave see ow fom eq. [..]: m t a eq. [..8]! te teal eegy of taslatoal moto was calculate to be t [..] ow we wat to calculate te etopy. Fo ts pupose we stat wt eq. [..7] fo te Helmoltz fee eegy: F S l t l m! l l! l! m usg Stlg s appomato eq. [..6] F l l m l. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

31 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: t t F m S l l l s s te Sacu-etoe* equato S 5 l l m l t [..] *Otto Sacu a Hugo M. etoe ca. 9 t s mpotat because t as o abtay substace specfc costats. Hece t ca be apple to ay atomc gas! eample: etopy fo vaposato of e: fst we calculate te cage of etopy by gog toug all pases of e: emetal ata a sol: assumg SK K melt.55k: m - - S C P l.9 Jmol K usg ebye s teoy fo C P - see late b meltg at m : S m H m c lqu: m bol 7.K bolg vapose at b : S m b S J/mol.55 K b m H b b.6 Jmol - K -.85 J mol - C Pl K 758 J/mol 7. K e total cage of etopy fom a- s te: S7.K e Sacu-etoe-equato gves: S7.K s a pefect gas wt S s s a amazg vefcato of ou statstcal teatmet! Jmol - K - - K J mol 96.5 J mol - K - aga assumg tat at K e. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

32 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:.. e etopy of mg-e Gbbs Paao Fom te Sacu-etoe equato te etopy s coectly escbe as a etesve popety of a gas S. I te ealy ays of Statstcal emoyamcs log scussos esulte fom te fact tat te stgusablty of te gas molecules was ot uestoo a ot tae to accout see scusso o page. ese scussos a poblems ae summase te Gbbs-paao of stocal mpotace oly! Cose a bo ve to two equal volumes. Oe pat s empty vacuum a te ote cotas gas patcles. Wat s te cage of etopy we te sepaato s emove as llustate? F F t b a b sepaato emove m ece F a t t l l l t t sce oes t epe o a te tempeatue s costat: S l as ecte ow cose te same aagemet but te two volumes tally beg flle wt two ffeet gases. a b sepaato emove ow eac gas eeces a cease etopy of S l. Hece to total cage etopy s S l Fally cose te two volumes tally beg flle wt te same gas. a b sepaato emove t. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

33 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: e ogal agumetato cosee te gas patcles to be stgusable. e case s etcal wt a oe woul get a cage of etopy of S l altoug otg as cage! s s te paao! e coect agumetato ealses tat te patcles ae stgusable! efoe emovg te sepaato: Afte emovg te sepaato: a b!!! o usg eq. [..7]: a l!! Afte emovg te sepaato: F l l l! a F [ l l l l l l ] efoe emovg te sepaato: F [ l l ] ece S wc s coect! [ ] b.. e Pcple of te qupatto of egy eoem: A classcal temoyamc system temal equlbum at a tempeatue g wc te eegy of te wole system epes o te squae of oe pase-space cooate* e.g. mometum posto as a eegy assocate wt t. * alteatvely: ts co-oate appeas squae te assocate Hamltoa eac egee of feeom cotbutes to te total eegy. Questo: If ts s tue wy te eq. [..6] compaso to te tee egees of otato te fgue o page? evato of te teoem: Co-oates pase-space fo patcles: p p p ; y y y ; p y p y p ; z z y z ; p z z p all togete: 6 co-oates. ; p z. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

34 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy We wat to smplfy ts by toucg a geeal co-oate η < < η η z z y y p p p p p p z z y y a 6 wt ; ; ; ; ; Fom te teoem we ow te coto fo te eegy of te t state: ote A η [..] wee ote epesets ote eegy cotbutos wc ae epeet o η. e umbe of molecules populato wt a ceta eegy s te gve by te oltzmastbuto: η η A A 6 ote ote [..] [..] We ae ot teeste te cotbutos of ote because we oly wat to calculate te pobablty o cotbutos assocate wt η. We ca get of ote by summg ove all just fo tese eeges ote o ealsg assumg tat t oes t epe o te state. s wll gve us te ese pobabltes P η assocate wt η : A A P η η η η η [..] I ts equato we ave also eplace te sum ove states by a tegal ove co-oates. We ae allowe to use te tegal because te teoem s statg a classcal system. Sce te eegy epes o te co-oate η tegato ove all co-oate-space also couts all eeges. e te aveage eegy of te system s gve by: A A A P A η η η η η η η η

35 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy wt te gve eteme tegals: a a e a a a e a fo a > we get A A A A qe! alteatvely we ca also calculate te aveage eegy by calculatg a esemble aveage of te vual eeges: above see ote ote A A A ote A ote A A ote ote A A η η η η η η η η η η η η η η Hece te mola aveage eegy fo molecules ca be estmate usg te equpatto teoem: vb ot ~ f f a te mola specfc eat: vb ot ~ f f C f ot fo lea molecules f ot fo o-lea molecules f vb epes o umbe of cemcal bos a agles te molecule but s selom actve at omal tempeatues taslato

36 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:5 e followg gap sows C ~ ~ C [] 7 5 vesus fo a atomc molecule tas.ot.vbato taslato of atoms taslatootato taslato atomc molecule ot vb ssoc e temoyamc popetes of polyatomc molecules s eteme by te stuctue pectos ae possble but moe ffcult. Howeve te same basc eas ca be apple but specal cae as to be tae we a egee of feeom s accessble ecte. ee eve est stuatos wee a vbatoal egee of feeom ca mutate to a otatoal egee of feeom..g. Hee otato ue to volumous pats of te molecule at low tempeatue geeates vbato aou te bo. At suffcetly ge tempeatue tee s te eoug eegy to o pefom te otato C te eceases wt ceasg. alty a Lmt of te Sem-classcal escpto.. e Classcal Lmt We wat to eve te lmts fo a sem-classcal teatmet. Have a loo at secto.. o page 8 to ecap te efto. Fo ts pupose we wll aalyse taslatoal cotbutos to te eegy of a system oly.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

37 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:6 quato [..] gves te patto fucto fo taslatoal moto: t m e pobablty P s of a patcle beg a patcula state s wt eegy s gve by te oltzma stbuto t s of taslatoal moto P s t t s [..] We tee ae patcles te mea occupato umbe s of eegy levels aveage umbe of patcles state s ca be esse as s P s [..] ow we use te efto of te sem-classcal system. I patcula te fact tat te umbe of patcles s muc smalle ta te umbe of avalable eegy levels see page 8 a ece a s << fo all s [..] s P s t s t [..] m s << t [..] If tat s te coto fo all s te t must be somewat epeet of te eegy of eac state ove a boa tempeatue age. Hece we equest te followg to be tue fo all states s: Coto fo te sem-classcal egme: m << [..5] Hstocally ts was tue because te ogal wo of oltzma a Mawell befoe Plac s scovey of. We also ecogse satsfactoly tat g tempeatues a low estes / ae ecessay to fulfl eq. [..5] complete ageemet wt te eftos a scussos pat... Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

38 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:7 ow we wat to compae tese esults wt some QM-coseatos. eefoe we stat wt e ogle s law: λ p m t a fom eq. [..] fo t ece λ [..6] m s we substtute to eq. [..5]: λ << m m It s useful to touce a aveage stace of te patcles λ <<. Hece we ca appomate tat te sem-classcal lmt: te [..7] [..8] λ << [..9] I wos: e e ogle wavelegt as to be small compae to aveage stace of te patcles. e te wave-popetes of te patcles ca be eglecte a quatummecacal tefeece of te wavefuctos ca be eglecte. We ae allowe to teat te costtuets as patcles obeyg classcal ewtoa mecacs. eample: He gas at oom tempeatue. esty ca. atoms/cm m g fom eq. [..6] K -7 cm λ 8-9 cm We see tat fo oay cotos te classcal lmt fom eq. [..9] s fulflle! e lmt s volate e.g. coese He lqu electos metals etc see pat. Geeally we ave to cose to abols te classcal lmt fo low mass low tempeatue g estes. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

39 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:8.. Mawell velocty stbuto a classcal gas As a last eample of classcal teatmet we wat to eve Mawell s velocty stbuto fo a pefect classcal gas wt ou ew statstcal stumets. ecap st yea mateal Fom eq. [..] we ectly get te mea umbe of patcles wt a mometum of magtue p: s p m m t s wt p m t s p m Hece te pobablty Pp of fg a sgle patcle avg a mometum betwee p a pp s: P p p p p [..] wee p s te esty of states. fom eq. [..7]: a fom eq. [..]: p ece: pp p p p p [..] a eq. [..] becomes: P p p p p m p m P p p p m m p p [..] s s Mawell s mometum stbuto! efoe we cotue we cec f eq. [..] ees a atoal omalsato costat! we oly cosee te magtue of te mometum ece tegato fom to P p p m p m p m m aleay omalse sgle patcle p. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

40 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page:9. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy I oe to obta a velocty stbuto we substtute p mv: to eq. [..]: v m m v m m v m v v P Mawell s velocty stbuto v mv v m v v P [..] see also gaps et page Cec you abltes evg te followg fom eq. [..]: most pobable spee ma. of Pv m v m.p. [..] mea spee: m v v vp v 8 [..5] seco momet: m v v P v v [..6] ote: ts s equvalet to v m a te ms oot-mea-squae spee: m v v ms [..7] As a fal test we also wat to cec f a Mawell eegy stbuto wll gve te coect esult. m p m p m p a eq. [..]: m m m m m P P Mawell s eegy stbuto! Hece te mea eegy s P

41 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: substtute: a e qe! e followg fgue sows Mawell s velocty stbuto vesus a mesoless abscssa omalse wt espect to v. v m.p. a v ms ae also cate..8.6 Mawell velocty stbuto v m.p. v v ms v P v [%].. v v e et gap sows Mawell s velocty stbuto fo gases vaous masses at K CO v..5 A Mawell stbuto of ffeet gases at K P v [%]..5 e He. v [m/s]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

42 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: e et fgue sows te tempeatue epeece of te Mawell stbuto. P v [%]. K K K Mawell stbuto fo A at ffeet tempeatues K 5 K. 5 v [m/s] 5.. otatoal Specfc Heat of atomc Molecules Oto/Paa-Hyoge I ts last secto of pat we wat to loe te fluece of quatum-mecacal effects o te specfc eat of omouclea atomc molecules. e most astc case s H. We tae ts as a pepaato fo te et pat wee we wat to stuy quatum effects moe etal. H I ucl ½ POO Femo combe wavefucto must be atsymmetc H I ucl UO oso combe wavefucto must be symmetc H I ucl ½ IIO Femo combe wavefucto must be atsymmetc We wll lea moe about Femos a osos te et pat!! fo H Hyoge-MOLCUL! e wavefuctos fo uclea sp σ a te combe otatoal wavefuctos fo te electo ψ must satsfy: σ ucl ot ucl ot ψ σ ψ Hece te molecula wavefucto s atsymmetcal Femo a obeys Paul s ecluso pcple. I ote wos: e total wavefucto cl. sps must be atsymmetcal wt espect to te tecage of posto. o two electos ca ave etcal set of quatum umbes. W. Paul. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

43 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: Summay of popetes fo H : ame uclea sp ucl σ ot ψ J * total wavefucto oto- H tplet symmetcal atsymm. o atsymm. paa- H sglet atsymm. symmetcal eve atsymm. * combe otatoal quatum umbe obtal sp fo electo XCUSUS: Lea combato a symmety of wavefuctos: e symmety of wavefuctos ca be efe a teste by toucg a ecage opeato Xˆ o swap-labelg opeato swaps te labels of a combe wavefucto ˆ Xˆ Xˆ Xˆ ψ a ψ ψ ψ X ψ wt Xˆ we ca ow test lea combatos of wavefuctos fo te symmety: ˆI symmetc: atsymmetc: ψ ψ ψ test : Xˆ ψ ψ ψ ψ ψ s a ψ ψ test : s Xˆ ψ s ψ ψ ψ s a e.g. sp I ½ symmetc: αα ββ αβ βα atsymmetc: αβ βα Geeally atomc molecules possess I of suc pouct fuctos. I of tese ae symmetc m I m I e.g. αα ββ. emag II of wc alf ae symmetc e.g. a te ote alf ae atsymmetc e.g. II atsymmetc II αβ βα αβ βα lea combatos. Hece symmetc I I I wt te ato: a s a s I fo yoge H wt I ½ I fo euteum H wt I lea combatos; s a Cocluso: mole of H at g tempeatues cossts of mole of oto tplet yoge o- H a mole of paa sglet yoge p- H. s a We te H -gas s coole ow fom oom tempeatue tese factos of o- H a p- H ae coseve because te ffeet types of yoge o ot easly covet to eac ote.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

44 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: Of couse at vey low tempeatues < K te molecules wat to ete te state of lowe eegy J o paa. Howeve ts meas tat te wavefuctos of oto someow ave to mutate to paa. s s ot ectly allowe o must lea ove a ssocate molecule a tus ts pocess of oto/paa coveso s vey slow ays. [e ate ca be cease by toucg catalysts o paamagetc e.g. O -taces mputes wc e.g. touce local magetc fels to alte te wavefucto va pola couplg]. s also las te etemely g caactestc otatoal tempeatue fo H of ot 88K see page 9. Heat of oto/paa coveso J J fo mole of H - J Heat fo lqufcato of mole of H 6 J e oto/paa coveso bols off easoable amouts of yoge!! Fo te calculato of temoyamc popetes of H we ave to eal wt two ffeet substaces o- H a p- H : t ot vb fo a eteouclea atomc molecule: fo a omouclea atomc molecule: Hee H H t ot vb oto oto oto!! t ot vb paa paa paa! aslatoal a vbatoal pats ae te same fo oto- a paa- H ece t oto t paa t vb oto vb paa a t vb ot H a! oto!! pefect gas ot paa! vb [..8] We ave to eteme see secto..7 a eq. [..6] ot ot ot J J J ot 6 e e paa 5 9 [..9] J eve... ot ot ot J J J ot e e oto 7 [..] J o Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

45 PH 65: emal & Statstcal Pyscs. Sem-Classcal Pyscs page: Ufotuately tese sums fte lmt caot be esse by a smple aalytcal esso at least I o t ow oe!. Hece we ave to compute tem see below. Fom eqs [..9] a [..] te eq. [..8] ca be compute a te l a fom tat C. s was pefome ee by a sot MAPL-pogam see below. e et fgue sows te esult. ~ C [ ] ot Fgue: Smulato Specfc eat of oto- blue a paa- e yoge. e oate s C uts of [ - ] te abscssa s mesoless uts of / ot. e : mtue of oto/paa s sow blac. ote tat all tee cuves covege te classcal lmt C see eq. [..7]. s appes at about / ot 5 o K. You ca owloa te MAPL-oute fom te couse-webpage: ttp://wwwm.uc.ac.u/m/staff/pb/teac/ph65/ph65.tml. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

46 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:5 Quatum Statstcs. Ieal Sols efoe we ae gog to eve te statstcal escpto of quatum systems we wat to see ow eal sols ca be escbe. s escpto s a goo compomse betwee sem-classcal teatmet a specal QM assumptos. e eal sols cystals ae ealse by assumg: avg o efects ot otato o taslato of te atoms ave oly vbatoal eegy Hece we ave to eve te vbatoal eegy of te cystal wc we wat to compae wt emetal esults... ste s eoy of a Ieal Cystal Assumptos: Cystal lattce of pefect symmety sotopc potetal fel Oly te moto vbato of a sgle patcle s cosee. It s egbous ae fe o vbatoal couplg e vbatoal splacemets ae vey small Hooe s law s val Hece eac atom te lattce as egees of vbatoal feeom. A cystal of atoms ca teefoe be escbe as amoc oscllatos wc soul ave a sgle caactestc ste- fequecy. Fom pat.. we ecogse te atoms te lattce as stgusable ece we ca ealy apply te solutos fom pat..8: Fom eq. [..]: vb [..] a fom eq. [..] vb [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

47 page:6. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy We we also efe a caactestc tempeatue te ste-tempeatue va cf. eq. [..6] [..] We get te followg essos fo te mola eegy of te cystal a t s specfc eat. ~ [..] a ~ ~ C [..5] see also eq. [..5] facto o test te esults of ste s teoy we wat to calculate te lmts at K a. We ect fo tat ~ C a fo tat C ~ ulog-pett s ule. Futemoe emetal ata see late ts secto eveal a epeece we appoacg K. a ste s teoy te low lmt we ca agumet tat te eomato appoaces fty faste ta te omato ece te lmt s as eque. Alteatvely we ca use L Hôsptal s ule tee tmes: oce: lm lm twce: lm lm tee tmes: lm lm lm ~ lm C

48 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:7 s tme we ae also teeste te fuctoalty wt wc ~ C appoaces K. eefoe we ca eglect te "" te eomato of eq. [..5] because te low lmt >> a we f: ~ C [..6] HIS OS O SML H XPIMAL AA epeece! b ste s teoy te g lmt s we ave aleay calculate fo te pefect gas te g lmt see page 8. e oly ffeece s te facto a we get te coect aswe ulog-pett s law. lm ~ C vb lm e L'Hosptal lm e gap o te gt sows pecte fom eq. [..5]. ~ C as e set s a magfcato fo low tempeatues. ~ C [ ] ste moel. Hece te poblem wt ste s teoy occus at low tempeatues. e followg table gves some values te ste-tempeatue fo some sols: Slve: Lea: amo: Ag K Pb 6 K C K / e ste tempeatue fo amo s etemely g a woul coespo to vbatos wt a fequecy of Hz wc s te fa-e! e assumpto o. tat all atoms vbate epeetly at must be wog! ecause ts eegy o fequecy o tempeatue coespos to elatvely g tempeatues see table tee must be a stbuto of eeges o fequeces.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

49 page:8. Quatum Statstcs PH 65: emal & Statstcal Pyscs.. ebye s eoy of a Ieal Cystal Impovemet of ste s teoy by toucto of a age of fequeces stea of a sgle wt a uppe lmt ebye -fequecy. Pyscal tepetato: At low tempeatues tee ae always some allowe fequeces so tat >. ese ca be ecte a tus C s lage ta ste s teoy. ee ae o loge epeet oscllatos but te fequecy stbuto escbes couple oscllatos. XCUSUS: Couple oscllatos see also couple peula eteso of te equato fo amoc moto by a couplg tem fo epee t oscllato s λ fo couple oscllato s ω λ a ω λ cotbuto of oscllato cotbuto of oscllato s ffeetal equato system ca be solve by toucg omal co-oates q a q q ω λ q a q ω λ q Fom te soluto we te get two epeet omal oscllato moes wt fequeces ω ± λ. Aalogous teatmet of couple oscllatos woul pove a geeal soluto but ca o loge be solve aalytcally Compute smulatos. Howeve Pete ebye fou a way to avo tese complcatos.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

50 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:9 eefoe e use te followg appoac: Stat wt a stbuto of oscllato fequeces stbuto of poo wavelegts of sou waves. e lattce s able to oscllate wt fequeces muc smalle ta. e pyscal ea be ts s tat te specfc eat of te cystal s stoe may stag poo waves omal moes. ae ste s appoac of quatzato of eeges a egees of feeom but ot all fequeces ae equally possble ete s t just oe stbuto of fequeces/wavelegts. e wavelegt λ s lage compae to te atomc staces λ >>. Hece te atomc stuctue ca be goe. e cystal beaves le a elastc cotuous boy. e followg gap compaes te stbuto of fequeces fo te ste a ebye moel. We o ot yet ow te sape of te ebye-stbuto but we ow tat ts aea must be omalse to te umbe of states a must ecay towas zeo a as a uppe lmt efg te ebye-fequecy. ste: ebye: esty of states aea? We ee to calculate te teal eegy of te cystal: vb vb [..7] wee s te stbuto o esty of fequeces.e. te umbe of allowe moes wt fequeces betwee a of wc we ow tat [..8]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

51 page:5. Quatum Statstcs PH 65: emal & Statstcal Pyscs e esty of fequeces ca be euce fom te esty of states cf. eq. [..7] wt a λ c λ Hece wt c a c [..9] c c c I tese equatos c s te velocty of te poos spee of sou. ut we emembe tat a sol te velocty of sou popagato epes o te ecto of popagato a tat we ave to stgus betwee: logtual waves tasvese waves compesso of lattce sea of lattce see fgue below s s compaable to ffeet polasato ectos of lgt waves see late blac-boy aato. Obvously tee ae ffeet foces actg o te lattce atoms fo tasvese a logtual movemets. Hece te ffeet eeges geeally also cause ffeet popagato spees.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

52 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:5 Oe ca efe a mea sou velocty c wegte by te egees of feeom of te logtual egee of feeom a tasvese egees of feeom: c [..] c c c logtual tasvese wc apples to all lattce vbatos of coseably low fequeces Hooe s law. ebye s smplfcato: o stcto betwee c logtual a c tasvese ece te spee of sou c S s ts apples to all fequeces somewat cotuous spectum c logtual ctasvese cs [..] quato [..7] esty of states was eve cf. secto.. sem-classcal agumet ue te assumpto tat te -values ae closely space. Small -values coespo to log λ loge ta te stace of te lattce pots. Fom a atomstc pot of vew tee must be a lowe lmt fo λ λ m o ma. s ca also be lae by te mmum wavelegt of te sou-waves te lattce ca popagate: A cystal wt a teatomc spacg of caot popagate waves wt a wavelegts less ta because te gest fequecy s eace we ectly egboug atoms move atpase see fgue.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

53 page:5. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy Alteatve laato va te samplg o yqust teoem see PH56*: e samplg teval of te atomc lattce s te atom spacg. Hece a wave wt λ < wll be sample as λ wee λ λ a see fgue. e samplg teoem also states λ m. ot agumets yel: c o ma m λ [..] ow we ave all te ecessay agumets to calculate : Fom eq. [..8]: c c c c c S S S S [.. ] [..9] Hece te aveage spee of sou a ebye sol s c S [..] ebye-fequecy c c S S [..] * ttp://wwwm.uc.ac.u/m/staff/pb/teac/ph56/ph56.tml

54 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:5 We ave to cec f fulfls te equemet of c ma : c S.6 S. S ma c We we substtute eq. [..] bac to eq. [..9] we eceve a esso fo te stbuto of fequeces te ebye-sol: c c [..] c S [..] 9 ebye s spectal stbuto fucto of a -sotopc eal cystal: 9 [..5] Smlaly we woul get fo a "two-mesoal lattce" cost. fo a "oe-mesoal lattce" e -spectal stbuto fucto s sow te et fgue: 8 [ / ] / Wat we ae afte s a esso fo te teal eegy cf. eq. [..7]. ut we ave to be caeful ow because aleay cotas te egees of feeom o te esty of states/egeeacy of te oscllatos. Hece we ee te vbatoal eegy of a -oscllato. vb. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

55 page:5. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy vb s gve by eq. [..] fo : vb Hece te mola eegy of te ebye cystal s esso smple o A A A A ~ A A ~ [..6] e tegal oes t ave a easy soluto. Howeve MAPL allows us to pobe te esso at seveal pot esultg te quas-cotuous fgue below. You ca owloa te MAPL-oute fom te couse-webpage: ttp://wwwm.uc.ac.u/m/staff/pb/teac/ph65/ph65.tml / ] [ ~ C

56 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:55. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy Le ste s moel we wat to vestgate te g a low lmt of ebye s teoy. eefoe te esso eq. [..6] s somewat smplfe by toucg a caactestc tempeatue te ebye tempeatue. [..7] ece: A ~ a ebye s teoy te g lmt fo t esults tat >> o << a <<. << A lm ~ lm e ctcal oet s muc smalle ta ts lmt ece we ca a t oly usg te fst two tems of te aso sees: aso te of tems ge 8 9 A A A ~ ± a ece tese but eact be to ~ ~ C We see tat ulog-pett s ule s fulflle by ebye s teoy te g lmt see also fgue o pevous page

57 page:56. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy b ebye s teoy te low lmt s s te moe teestg case because ste s moel t agee wt emetal values ee. A lm ~ lm o smplfy we substtute ece o a te tegato lmt ma >>. Hece te tegato lmt ma te low lmt. 5 A A ~ e e a ece 5 5 ~ ~ C So ebye s moel gves te ese epeece fou emetal ata. Of couse t also yels K ~ C. ebye s law fo te low tempeatue lmt.8 5 ~ C [..8] e followg gap sows a close-up of ~ C fo low tempeatues calculate wt te pevously metoe MAPL pogam. We compag to te set of te gap o page 7 fo te ste moel we ecogse tat ebye s moel pects a less steep appoac to K. ] [ ~ C /

58 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:57 e followg fgue ectly compaes bot teoes o te same emetal ata. We clealy ecogse tat bot moels smlaly well pect C at g tempeatue. ebye s teoy clealy s te bette matc we appoacg K. Mola specfc eat C [cal/k/mol]: pemetal values pots ae omalse wt espect to ste-tempeatue sow as a to ebye-tempeatue sow as. e teoetcal cuves ae also sow ste-fucto ebye-fucto. e set o te gt sows te supeoty of te ebye-teoy at low tempeatues. e followg table sows te ebye-tempeatues fo vaous substaces: substace [K] Ag 5 C amo 8 Fe 65 Pb 9.5 acl 8 CaF 7 FeS 65 e table sows vey ffeet values of fo ffeet elemets. If ebye s teoy s coect all te measue C values fo tese substaces must fall o te same pcpal cuve we s omalse by te substace specfc. Suc a cuve s calle "maste-cuve". s s sow te et fgue. We ecogse ecellet ageemet fo most substaces.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

59 page:58. Quatum Statstcs PH 65: emal & Statstcal Pyscs Fom te pevous table we ealse tat lgte elemets e.g. amo ave etemely g ebyetempeatues. e et table compaes ts wt te atomc mass a te mola specfc eat at C. ulog-pett s ule pect a C ~.9 Jmol - K - te classcal lmt g. Fom te table we see goo ageemet fo te eavy elemets. amo oweve oes ot. s s because te eegy of vbato amoc oscllato s: ħ ω s wt tus m Maste-cuve fo te ebye-teoy: empeatue epeece of C fo vaous sols vesus a abscssa omalse by te vual see table. substace [K] mass [u] m ω wee s te spg costat bo stegt eefoe te lgte a elemet te moe temal eegy s eee to ecte actvate oscllatos wt te same fequecy as fo te eave elemets at te same tempeatue. e oscllatos of te amo bos ae eve at oom tempeatue ot eally ecte o molte... ece tey ae g ece amo s ae ta lea at oom temp.* Howeve te fame of ebye s teoy ts s coecte va a substace-specfc caactestc tempeatue te all mateals sow te same tempeatue epeece of C maste-cuve. *to uesta as a ule of tumb. Of couse stuctual effects metal/o-metal play also a ole. ~9 K C [Jmol - K - ] C amo Ag Pb Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

60 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:59. Quatum Statstcs We oes quatum statstcs apply? O we o we leave te sem-classcal lmt? I secto.. we ave aleay eve a coto fo te valty of te sem- classcal appomato: eq. [..7] λ << o eq. [..9] e followg table summases elevat facts fo some teestg mateals: λ << / mateal mass [K] λ [m] [m] / λ A lqu He lqu e * * couctve electos Cu We clealy see tat fo lgt mateals o low tempeatues le electos o lqu elum te λ classcal coto >> gt colum table s o loge fulflle. I te tepetato of secto.. te e ogle wavelegt λ becomes about te sze of te te-atomc stace wavefuctos.. Hece we must clue tefeece effects of te patcle As a esult of ts te occupato umbes of ffeet eegy states ca o loge be cose abtaly. fo eample see secto.. Paul ecluso-pcple fo alf tege sps patcles e followg stctos ae esults of elatvstc quatum teoy wc oly ols we te symmety of te sp- wavefuctos obeys te followg estctos Wolfgag Paul ca Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

61 page:6. Quatum Statstcs PH 65: emal & Statstcal Pyscs Pcpally we ave to eal wt two vey ffeet classes of substace. Oe as to obey Paul s ecluso pcple a te ote as. s esults followg classfcato of matte a cosequeces ue to sp quatum umbe a te symmety of te total wavefucto. occupato umbe of a eegy state sp-quatum o. symmety of wavefucto classfcato statstcs o estctos... tege I... symmetc OSO ose- ste Paul-ecluso pcple: At least oe a patcula state alf-tege I at-symmetc FMIO Fem- ac amples: fo osos: potos I poos I -mesos I fo Femos: potos I ½ electos I ½ eutos I ½ Smla agumets ol fo teactg o combe patcles teactg atoms o combe atoms molecules. o see wc statstcs tey obey we ave to combe te agula mometum see secto.. et agula mometum of uclea a electo sp plus obtal mometum. Suc coseatos ca be summase te followg ules fo combe patcles: umbe of femos umbe of bosos composte sp o eve ay ay I alf tege Femo I tege oso Fo eample: He: p e eve 6 o. of femos oso He: p e o 5 o. of femos Femo A somewat qualtatve uestag about te symmety assocate wt bosos/femos ca be euce fom loog at lea combatos of wavefuctos cf. ecusus secto.. page.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

62 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:6 osos: suppose to ave symmetc wavefuctos ece symmetc combato: oso ψ φ a φ b φ a φ b sgle state wavefucto e.g. uclea sp at boso wt quatum umbe a. If tee s o estcto te occupato umbe assocate wt ts state we may just set a b volato of Paul s ecluso pcple wc oes t apply ee a oso see tat te total wavefucto s te: ψ φ a φ a te patcle ca ests ts state! Femos: suppose to ave at-symmetc wavefuctos ece at-symmetc combato: Femo ψ φ a φ b φ a φ b A volato of Paul s ecluso pcple two patcles te same state.e. same Femo quatum umbe e.g. by lettg a b causes ψ. e wavefucto collapses a te patcle caot est ts state! Plausble but o poof! As a cosequece te patcles quatum statstcs ave to be stgusable! s s because te eegy levels of a system of etcal patcles caot epe o wc patcle s wee wete o ot tey ae emetally stgusable!. Hece evey mco-state ca be tepete as a maco-state ts oes t mae sese te classcal teatmet! I ote wos: e Hamltoa escbg te ete system a eteme vew we ave total tefeece of all wavefuctos of all patcles must obey ts symmety ue ay pemutato swappg of patcle co-oates. s leas to te followg POSULA: e wavefucto fo a set of etcal patcles s ete totally symmetc o at-symmetc ue ay pemutato of patcle co-oates.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

63 page:6. Quatum Statstcs PH 65: emal & Statstcal Pyscs.. ose-ste Statstcs Satyea. ose a Albet ste 9 [ecap te evato of oltzma statstcs secto..] Fom te pevous toucto we ow tat we ave to eal wt stgusable patcles wt o lmtatos te occupato umbe. Fst we ave to eve a esso fo te statstcal wegt Ω. Fo ts pupose mage a le of patcles patcles a a set of A baes wc ve te A goups epesetg te eegy-states. As sow te followg setc. state... A - A patcle bae... A - A - So ow may ways est to aage tese A s gve by A!. objects? e total umbe of pemutatos Howeve ts also clues te pyscally meagless pemutatos of te etcal a stgusable patcles a baes. Hece we ave to coect te total umbe of pemutatos by tat of te pemutato of te A baes. patcles a Hece te total umbe of ffeet aagemets mco-states s: Ω A!! A!.. eample: A two baes a stgusable patcles ots! o. A I II III Ω!!! Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

64 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:6 ow we ae eay to eve te ose-ste abbevate by stbuto. e poceues we ae gog to apply ae completely aalogous to tose te evato of te oltzma statstc see secto.. Le tee we stat wt evaluatg te statstcal wegt of te mco-states. Ω Ω [..] A! A!! Howeve oe to smplfy te followg mats we ca assume tat tee ae eoug patcles a eegy levels to sp te - te esso. We ca of couse wtout lac geealsato see A ffeet levels. above smply equest Hece we get te followg cotos fo ou -system: a Ω A!! A! [..] b [..] c [..] et step s fg te mamal Ω o calculatg δlω. So we stat wt calculatg lω: l Ω l Ω [..] [..6] A! l! A! A l A A A l A A l A! l! l A! A l A l l A l A [..5] l ow we ave to f te most pobable statstcal wegt va Ω. Aga we calculate ts usg a small cage matematcal gou. δ l Ω A A δ δ l A δ l δ [ l A l ] δ δ l Ω δl Ω l A δ A l δ [..6]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

65 page:6. Quatum Statstcs PH 65: emal & Statstcal Pyscs Fom eqs. [..6] [..] a [..] we get te followg tee bouay cotos: l A δ Ω l δ [..7] δ [..8] δ [..9] We combe tem usg te meto of ueteme multples λ a β: δ A l λ β A l A λ β λ β β e e αe [..] Smla teatmet to tat o page 6 gves: β te mus sg occus because s te eomato of eq. [..] We wll vestgate te sgfcace of α late we just memose tat t s assocate wt bouay coto. eaagg eq. [..] te gves: ose-ste stbuto A α [..].. Fem-ac Statstcs co Fem a Paul. A. ac 96 Fom te toucto to ts secto we ow tat te patcles ae stgusable a ca oly ealse occupato umbes of ete o patcles pe avalable state. Hece tee smply must be moe o equal states ta patcles.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

66 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:65 o establs a esso fo te statstcal wegt of Fem-ac abbevate by F statstcs we woe ow may ways est to pc occupe by oe patcle cells fom A totally avalable cells wle A ema uoccupe? e total pemutato of all cells s A!. ut aga we must coect fo te pemutato of te occupe! a uoccupe A! cells te pemutato as o pyscal sgfcace. Hece te total umbe of ffeet aagemets mco-states s: A! Ω [..]! A! eample: A cells a stgusable patcles ots! o. A I II III Ω!!! 6 e teatmet s completely aalogous to te pevous secto ece a bt abbevate. We get te followg cotos fo ou F-system: a Ω F! A! A! [..] b [..] c [..5] F Fom eq. [..]: l Ω l! l! l A Usg Stlg s appomato eq. [..6]: l Ω F A l A A l A A l l A! A l A A l A A a mamsg Ω F va δl Ω F δ δ l δ l A δ A A. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

67 page:66. Quatum Statstcs PH 65: emal & Statstcal Pyscs Fom te pevous equato eqs. [..] a [..5] we get te followg tee bouay cotos: l F A δ Ω l δ [..6] δ [..7] δ [..8] wc we aga solve usg ueteme multples: δ A l λ β A l A λ β λ β β e e αe [..9] Smla teatmet to tat o page 6 gves ee also: β Hece we get Fem-ac stbuto A α [..].. Compaso of oltzma a F Statstcs I secto. we eve oltzma s stbuto wtout efeecg te A states wt eeges betwee a ove wc te patcles ca be stbute. We wat to matc up wt ts lac secto. oe to ectly compae te tee ffeet statstcs. I te classcal oltzma statstcs te stgusable! patcles ca be stbute ove te cells wtout ay estctos o lmtatos ecept tat swappg patcles occupyg te same cell oes t geeate a ew mco-state. Hece te fst patcle ca be stbute A ffeet ways ove te A cells a te same s tue fo te seco a all ote patcles.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy A

68 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:67 ecause eac assgmet of a patcle to a cell eegy state ca be combe wt eac ote to gve a ffeet mco-state we ave to cose A ffeet possbltes to stbute patcles A cells. et we ave to as ow may ways est to goup te patcles esembles of? e aswe s eq. [..]. Hece we get fo te total umbe of ealsatos Ω A! [..]! Applyg te same teatmet as secto. we get l Ω l l A δ l Ω A l l A δ δ l δ l a togete wt te usual bouay cotos δ a δ A l λ β δ A λ β e e β α e completely aalogous poceues to tose o page 5-6 gves: α A e β a β [..] a te usual fom fo oltzma's stbuto cf. eq. [..7] A A [..] s equato s etcal to eq. [..7] ecept fo te A. I te fst evato of oltzma's statstc secto. we a assume tat eac eegy level s efe by eactly oe quatum state. ow we ave etee te stbuto suc a way tat eac eegy s epesete by A egeeate states wt teatmet. A as te egee of egeeacy. s s of couse te moe geeal e aveage umbe of patcles te t state s te gve by A. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

69 page:68. Quatum Statstcs PH 65: emal & Statstcal Pyscs e stbuto fuctos: oltzma: A α [..] ose-ste: Fem-ac: [..5] A α F [..6] A α Caeful: e oet s ow postve o te eomato! Fo α >> te two quatum stbutos a F ae becomg etcal to te oltzma stbuto. s s te same as beg vey small wc s etcal to te scusso secto.. a..; o yet aote fomulato of te classcal lmt... etemato of a efoe te evato of te "ga patto fucto" te et secto we wat to ty to elate α to some macoscopc popetes. We emembe tat α was touce as a ueteme multple scalg te bouay coto wc state te costat umbe of patcles te system. Hece t must be elate to te umbe of patcles te system same as β elates to te eegy. We stat wt te fee Helmoltz eegy F S wee a S l Ω. Hece F F [ l Ω ] [..7] et step s to mmse F wt espect to j a. ut te ae ot epeet! We pc oe say [..8] j j ow we ca ewte eq. [..7]:. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

70 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:69 [ l Ω ] F j j l Ω j j [..9] j ow we mmse F wt espect to setg eq. [..9] gves j F j l Ω j but O fo j o j j ot j l Ω j j j j j Fom eq. [..8] we get a f we loo at eac vual state te equato above smplfes to Fo te momet we efe: l Ω l Ω j j j j ts esso as some fte valuefo all goupsbecause j was cose abtal y ζ l Ω [..] l Ω a ζ a wt β : β ζ l Ω ow we ca compae ts wt te stbuto fuctos eve eqs. [..]* o [..] [..] a [..9] a see cf. fo stace to te evato of eq. [..9] tat l Ω ** β ζ λ β ece λ βζ a α e.g.[..9] e λ e βζ [..] * but caeful wt te ffeet efto of β ts capte. Hee β - - ** fo all tee statstcs!. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

71 page:7. Quatum Statstcs PH 65: emal & Statstcal Pyscs We ow clam tat ζ s otg but te cemcal potetal µ see page. Mallett's scpt wc s a costat fo a system equlbum. F µ [..7] [..] [ ζ] ζ l Ω ζ ζ µ ece α e βµ µ [..]..5 Systems wt aable Patcle umbes ecap ga caocal esemble -. Mallett's lectue/scpt So fa we ave oly vestgate solate systems. Howeve some occasos patcles wll leave te system e.g. blac-boy aato see late. I suc stuatos we ca ete goe te paamete α te assocate stbuto fucto o we ave to eve a ffeet statstc wc allows patcles to leave te system. I te pevous secto we ave aleay fou a esso fo α cf. eq. [..] wt µ as te cemcal potetal: F S µ [..] We 't eve α va te stbutos cf. secto. fo - a F-statstcs ue to te fact tat ts volves etee mats a t s ot easly possble to l t ectly to macoscopc popetes. Howeve tee s a moe geeal appoac touce by Gbbs by attacg a patcle-esevo to te obseve system le a eat-bat wc geatly smplfes te mats a allows te te vaato of patcle umbes te system e.g. ue to eactos. e followg fgue sows a scematc epesetato of te ffeet systems. Ogally we ave stue solate systems Fgue a: wt a fe wc ca be ectly le to macoscopc popetes va te oltzma equato S l Ω see teatmet secto.. s coespos to a mco-caocal esemble. et we touce ecage of eegy wt a eat bat Fgue b wc ue sotemal cotos as a fe. Fo small eegy fluctuatos we ca evaluate te eegy va F S l. s coespos to a caocal esemble.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

72 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:7 ow we ae gog to cose a ego I ecage of eegy a patcles Fgue c wt a bat o esevo II wee fo te esevo ae fe. e ego wc efes te system I s abtaly cose wt a fe. s coespos to te ga caocal esemble. We also touce a paamete wc elates te cage of eegy a etopy ue to te ecage of patcles a ame t µ but we o ot assume to ow moe about t! e tc Gbbs use s to touce te vaato of patcle umbes of a system va beg a subsystem of a escbable lage a solate system wc we ca eteme...6 e Ga Patto Fucto Afte te toucto te pevous capte omeclatue follows pat c of te pevous fgue we ae ow eay to eteme a system wt vaable patcle umbes. e solate eat- a patcle-esevo II as: a e subsystem I of volume ca cota patcles Fo eac we efe: wee s te eegy of te t state of te patcles. We te system s a state assocate wt o we ave to "cut" te subsystem I away fom te esevo system II a aalyse system II: II II a II e maco-state of te esevo-system as te a statstcal wegt: ΩII e et questo s te to eteme te pobablty of fg te system te state.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

73 page:7. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy Fo ts pupose we use te Pcple of qual 'A-Po' Pobablty. Mallett's lectue wc s aote wo fo te tutve assumpto tat te pobablty of sometg occug s popotoal to te umbe of ways wc t ca occu. O P Ω. Hece we get te pobablty of fg te system te state va: II c p Ω wee c s a omalsato costat popotoalty facto. us II..] [ S c p [..] Howeve te subsystem s small compae to te esevo II a te ecage of a s te also small. Hece we ae allowe to a te etopy eq. [..] a aylo sees. µ... II eq.[..] / II / II II II II S S S S S S S S S µ costat II II II e fst two tems ae costat because te esevo s assume to be solate o ecage of etopy ca be touce c eq. [..] a te volume s also fe. Hece eq. [..] becomes: µ c p [..5] e omalsato costat s gve by p a so we get µ c wee s te ga caocal patto fucto.

74 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:7 wee Ga patto fucto: µ [..6] s a sgle patcle state of patcle eegy state. We wat to efomulate ts esso oe to get of te ouble sum. eefoe we efe bac to te occupato umbe of a eegy state. wt as te occupato umbe of te t eegy state. e te ouble sum eq. [..6] becomes a sum of poucts a we oly ave to sum ove all occupato umbes to cove all states a patcles: µ { [ β µ ] [ β µ ] [ β µ ]} wt β [..6a] wee all te factos ae epeet fom eac ote ece [ β µ ] [ β µ ] [ β µ ] [ β µ ] wee [ β µ ] [..7] ue to eq. [..5] te pobablty of fg te sgle state te sgle state etc. s te [ β µ ] [ β µ ] p. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

75 page:7. Quatum Statstcs PH 65: emal & Statstcal Pyscs p ow we ca calculate o fom [..7]: F [ β µ ] p [..8] fo te ffeet stbutos. Statg wt Fem-ac ece a fo ose-ste [ β µ ] [ ] [ β µ ] [ β µ ] [..9] F s s a geometc sees wc coveges oly fo [ β µ ] [ β µ ] [ β µ ] [ β µ ] < µ < s coto must ol fo all sgle patcle states. It oes so f t ols fo te gou-state because all ote > ece µ < [..] We ca speculate o elbeately set t to to efe a sutable eegy scale. us fo a covegg geometc sees we get [..] [ β µ ] Ga patto fuctos of te t state: F Fem-ac: [ β µ ] [..9] ose-ste: [..] [ β µ ] [ ] ± ± o { ± β µ } fo F a fo. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

76 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:75 We ow wat to cec ts esult by calculatg te mea occupato umbe as befoe sectos.. a.. of te t sgle patcle state: we use a lttle tc by calculatg µ [..8 ] p A [..7 ] µ [ β µ ] [ β µ ] β [ β µ ] µ β l β µ [..] Fem-ac: l µ F [..9 ] µ F { l [ β µ ]} [ β µ ] F β F F β µ a usg eq. [..]: e-substtutg eq. [..6a] ose-ste: l µ F [ β µ ] [ β µ ] β F F [ ] [ β µ ] β µ [ ] F eq. [..6] wt eq. [..] µ [ β µ ] [..] β { [ ] } [ β µ ] l β µ β[ β µ ] µ [ β µ ] a usg eq. [..]: e-substtutg eq. [..6a] [ β µ ] [ β µ ] β µ [ ] eq. [..5] wt eq. [..] µ. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

77 page:76. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Applcato to Femo/oso-Systems.. Fee electos metals We cose a metal as a egula tee-mesoal lattce cf. secto. of os a cotag a lage umbe of valece electos tat ae fee to move tougout te ete metal. Metals ae caactese by tese wealy bou electos wc ca float aou. pemetally oe fs about - of tese couctve electos pe metal-atom epeg o te mateal a ts stuctue. I te absece of a electc fel te electos move about te metal aomly vey muc te way le gas molecules a cotae. eefoe tey ae cosee as a "electo gas". I a classcal teatmet we woul ect ~ C. 5 ulogpett fo lattce vbatos taslato of te electos equpatto teoem Howeve te emets gve completely ffeet values wc ae muc lowe e.g. C ~. fo Au at oom tempeatue a te classcal lmt of.5 s ot eve eace at tempeatues wee metals evapoate at omal pessue -6K. ecause te electos ae Femos we ave to obey te Paul-ecluso pcple we escbg te electo gas. We wat to moel tem as a pefect gas obeyg Fem-ac statstcs cf. secto... Pefect gas meas toucg te followg smplfcatos: we eglect teactos wt te lattce fee popagatg electo-waves we assume tat te lattce sels te electos suffcetly fom eac ote ece collsos ca be eglecte. Fom eq. [..6] wt eq. [..] we obta te Fem-ac stbuto of te aveage occupato umbe pe avalable eegy level F wt β µ β[ µ ] e eegy of te electos te "electo gas" s cosee to be puely taslatoal. Hece t ca be esse by te esty of states cf. eq. [..7] a eq. [..] wt mometum a eegy as te vaables: a wt p pp p p. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

78 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:77 fo o-elatvstc veloctes te eegy s gve by p wee m e s te mass of te electo. m e Hece m e p me a p : m e me e m..a Howeve ts s ot coect because we ave eglecte te sp of te electos I ± ½ wc esults te same eegy fo ffeet quatum umbes. Hece two electos ca occupy te same eegy level wtout volatg Paul's ecluso pcple see gap o te gt. eefoe te coect esty of states tems of eeges s gve by : m e [..] We ca ow eve te facto of electos wt eeges betwee a m e β µ [ ] [..] F wee te mea occupato umbe s smply esse tems of te eegy. We fute ow tat omalsato of ts esso must gve te total umbe of electos te metal: me [..] β[ µ ] efoe we ca cotue we ave to que µ. Ote ta fo te evato of te ose-ste stbuto cf. page 7 eq. [..] we 't ave to mae ay assumptos about sg a value of te cemcal potetal to coclue to eq. [..9] fo Fem-ac systems. Geeally µ µ wt a fe t oly epes o te tempeatue. eefoe t s coveet to efe a lowe bouay o a eegy o caactestc tempeatue efe by te cemcal potetal of te substace at absolute zeo tempeatue. Fem-eegy: F µk [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

79 page:78. Quatum Statstcs PH 65: emal & Statstcal Pyscs We stll o't ow te sg of µ eq. [..]. eefoe we substtute β µ m e e µ [..5] ow we cec eq. [..5] te low lmt µ F fo ffeet sgs of F case F < : te F F lm a lm lm eq.[..5] ts s clealy osese! a ece { } case F > : te F F lm a lm o a vey bg umbe le ts maes sese! ow we ave a seco loo at eq. [..5] cecg te eomato te tegal: lm F fo > fo < F F a lm fo > F lm [..6] F fo < F quato [..8] s llustate pat a of te fgue above. s ca be lae because te system at K s seeg fo te lowest possble eegy state. Howeve ue to te Paul-ecluso pcple t caot coese te sgle lowest state cf. pat b of te fgue. eefoe te lowest / eegy states ae subsequetly occupe wt a cut-off at F above tey ae empty.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

80 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:79. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy e eegy stbuto of te Fem-"electo gas" at K s sow pat c of te fgue wc s a llustato of eq. [..] K multple wt te stbuto fucto fom fg. a. We ecogse tat a coseable amout of eegy s stoe te system eve at absolute zeo! Fom tese esults we ca ealy eve a esso fo F because ow we ow tat we oly ave to tegate up to F o electos ae above ts level at K a te oetal tem te eomato of eq. [..] s eglgble close to zeo cf. eq. [..6] te low lmt. F F e e m m [..6a] m e F 8 Fem-eegy a Fem-tempeatue F F e F a 8 m [..7] e total eegy of te "electo gas" at K aea ue fgue c s te gve by: F e [..7] F e e F F m m m Aveage eegy pe electo at K: F 5 [..8]

81 page:8. Quatum Statstcs PH 65: emal & Statstcal Pyscs We we plug some typcal values fo / of metals electos pe volume see weely questo to eqs. [..7] a [..8] we ca estmate typcal ages oes of magtue fo F a F F 8 9 J a F 7 7 K e followg table gves some specfc values fo ffeet metals: Metal / [cm ] F [e] F [K] L K...6 Cu Au We see tat F s muc ge ta te "omal" tempeatue of use actually at F te metals te table woul o loge be metals but plasma. We ecogse tat typcally "omal" tempeatues te electos metals ae a egeeate state wc s ot vey ffeet fom te completely egeeate state at absolute zeo tempeatue. o: K s vey col fo te electos almost as col as K! If we wat to calculate te eegy of te electo gas at ge tempeatues F we caot smply set ete to o as eq. [..6] but must solve: me [..9] µ s tegal s ot geeally aalytcal solvable. Howeve usg avace moels a estmate of µ s possble: F µ fo < << F F e umecal appomatos of eq. [..9] ca be geeate as splaye te et fgue fo / F. stll vey "col" fo te electos but "ot" fo us e.g. fo F K K. Hece te splaye stbuto wll coespo to ge tempeatues of use! We also see tat oly tose electos close to F ae ecte to ge states. Fom tat we ca calculate a oug estmate: y assumg tat cagg te tempeatue fom K to wll oly ecte tose electos a eegy age cete at F.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

82 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:8 Fgue: a e Fem-ac aveage occupato umbe at a tempeatue. µ/. b e Fem- ac eegy stbuto at a tempeatue. µ/. Gey sae aea s te evato to K cf. pevous gap I ts oug estmate te umbe of ecte electos ec ca te be estmate va ec F states esty at cete of F a teval aou F wt eq. [..6a]: [..] F me F me F F F F F a F [..7] ec [..] F Usually / F epesets oly a vey small facto e.g. fo K a F ec % wc valates ou pevous teatmet. I ts oug estmate te ectato eegy pe electo ca be estmate to be ec [..] F a C [..] F a avace teoy gves a bette estmate of C F.9 F. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

83 page:8. Quatum Statstcs PH 65: emal & Statstcal Pyscs e fgue below llustates te evato of te Fem-teoy fom te classcal teatmet at low tempeatues as appomate by te epeece eq. [..]. Hece fo a metal at Fgue: as appomate by eq. [..]. F Fem-ac moel of te electo gas F 5 g : lattce: C ulog-pett pe atom electos: C.9 usually eglgble low : lattce: C ebye moel see eq. [..8] electos: C wc omates at vey low see fgue F Fgue: Illustato of te omace of electo cotbutoc el to C at vey low tempeatues lattce C vb C C el C vb.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

84 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:8.. Paul-Paamagetsm ecap pat fom. Mallett page -5: Fo sgle spatally fe atoms sepaato s lage eoug to eglect sp-sp teactos wt sp I ½ te magetsato s gve by te Lagev fucto: µ M µ ta ow we ae gog to cose couctg electos metals applcato of Paul ecluso pcple a F statstcs a wat to cose all electos a te sps smultaeously. e magetc eegy pe electo s gve by eample: m - e << F 5e m µ [..] Hece te agumet about te electo sps beg a egeeate state smla to K at omal tempeatues s eve moe tue ta te taslatoal eegy of te electos because eve at vey g magetc fels te electo sps wll be close to te gou state. eatmet at K s justfe! We wat to stat wt pctug te stbuto of te sp-up / sp-ow states fo te electos: a a e sps ae fllg te lowest avalable states wt up a ow oete momets electo: I ±½. e cut-off s te same as te pevous secto Fem-eegy: F F b a We a eteal magetc fel s apple te eegy of eac sgle patcle state cages -µ fo a up movemet µ fo a ow movemet [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

85 page:8. Quatum Statstcs PH 65: emal & Statstcal Pyscs We ave to sft some electos fom ow to up ue to supple eegy by te magetc fel a eac of tese sfts cosumes - -µ. Howeve te Paul-ecluso pcple foces tese up-sfte states to ave eeges ge ta F because all lowe states ae aleay occupe as llustate te et fgue ese ecess sfts ae small m << F ece we ca stll assume te esty of te ow-states to F be about te same as te up-states ; a te umbe of sfte electos s to be appomately: s F F µ [..5] ut eac sfte electo ceases te magetc momet by µ. us te total magetc momet alog fo couctg electos s: M [..5] s F µ µ [..6] e total umbe of electos wt momets paallel o at-paallel - to s te: ± µ [..7] wee s te aveage umbe of electos tese ±states. Of couse ts aveage umbe s gve by a Fem-ac stbuto cf. eq. [..] a we ae allowe to eplace µ wt F because te system s smla to equlbum at K. µ [ ] β µ F fo fo µ µ < > ecause te states ae ete full o empty µ s ete o a we oly ave to tegate eq. [..7] fom to F ± µ te full states: F ±µ m ± e µ m ± F ± µ ± e F [..8] F F. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

86 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:85 ecause we ow tat µ << F we ca appomate ± eq. [..8] by aso: ± [..7] µ µ m ± ± e F F F e et magetc momet macoscopc magetsato of te sample s gve by: M µ µ µ µ F F Paul-magetsato µ M [..9] F We we compae ts wt te Lagev-fucto fo solate sps te low tempeatue lmt: M solate µ wc meas tat all te solate sps woul le up. µ ta a lm M solate µ.. e pefect poto gas - lac-oy aato We wat to loo at ts well ow poblem fom two ffeet pespectves. Pat a wll sow te stocal evato by M. Plac 9 a somewat classcal teatmet; wle pat b wll eal wt te bac-boy as a pefect oso gas poto sp obeyg ose-ste statstcs. a stocal appoac M. Plac We obseve potos temal equlbum emtte by oscllato of te atoms te cavty. We ow tat fom a quatum-mecacal teatmet a sgle amoc oscllato as cf. eq. [..8] s s wt s a zeo pot eegy : A ole a eate cavty appomates a eal blac-boy. teg aato wll be absobe emtte aato ece coespos to te tempeatue of te cavty. Plac just vetg Quatum-eoy a assume: cost. wee [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

87 page:86. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy fom eq. [..] we ow: l ece: l wc gves: C l wt C as te tegato costat s coespos to te patto fucto of a sgle moe of te oscllatg blac-boy cf. ebye teoy wc as te: C C [.. ] cost. Plac set all te costats to efeecg to zeo-pot eegy ca be calculate late va omalsato. Hece te patto fucto of all moes te blac-boy s gve by: all wee te last step was calculate usg a geometc sees <<. e eegy ca te be eve by e-substtutg to eq. [..] l l all moes all we toucg te esty of moes/states we get Plac's famous stbuto but ts s te same fo bot teatmets. b teatmet by oso-statstcs: Sce potos o ot possess a sp tey ae osos a escbe by eq. [..5]. Fo te etemato of te esty of states we ave to ealse tat lgt possesses two ectos of polasato wc - of couse- ave te same eegy ece tey ae egeeate a we ave to coect eq. [..7] by a facto of : 8 c c [..]

88 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:87. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy e aveage occupato umbe a ose-ste stbuto s gve by eq. [..5]: [.. ] µ α [..] ut we ae ealg wt a ope system! e umbe of potos s ot a costat tey ae emtte as a fucto of tempeatue te ge te moe potos ae emtte. Hece te cemcal potetal s "uefe" µ F o µ α. Hece te umbe of potos wt fequeces betwee a s cf. to pat a: 8 c [..] a te spectal eegy esty stbuto: 8 8 c c e spectal eegy esty stbuto pe volume s calle spectal aato esty of a blac-boy: ρ 8 c [..] s fucto s llustate te et fgue fo tee ffeet tempeatues. see et page e tempeatue epeece of te mamum of te stbuto ma ca be aalyse by ewtg eq. [..] accog to: ρ 8 c a substtutg ece e cost. ρ

89 page:88. Quatum Statstcs PH 65: emal & Statstcal Pyscs e mamum s fou fo ρ a esults te followg coto:. s s a tasceetal equato wt a soluto oot at ma ma ma ma.8 o ma HzK also ow as We's splacemet law! ty t youself! Itegato of eq. [..] gves Stefa-oltzma's law! see weely questo a b Plac's law: a aate powe vesus wavelegt! b egy esty ρ vesus fequecy eq. [..].. ose-ste coesato Sce tee s O Paul-ecluso pcple to obey fo osos tey wll all settle te gou state fo see gap. e aveage occupato umbe s gve by eq. [..5] combato wt eqs. [..] a [..]: µ wt eq.[..]: µ < [..5]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

90 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:89 a te esty of states s gve tems of eegy by eq. [..a] m e m m [..6] We ae ow gog to obseve wat wll appe to te system te lmts K a > K. a fo K: We ow tat all osos wll ty to mmse te eegy a settle te gou state Hece: K a lm o usg eq. [..5]: lm lm µ µ to ealse ts lmt µ lm o µ < as equeste by eq. [..] oe to gve a bg umbe le. Hece te oet must be small a we ca a t see secto..6 fo K: µ µ o lm µ vey small [..7] So µ as K a all wt > ae empte! we ave just pove ou tal assumptos but fou some ew essos fo tem! b fo > K but close: Fo ts pupose we combe eqs. [..5] a [..6]: µ m. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

91 page:9. Quatum Statstcs PH 65: emal & Statstcal Pyscs o C µ wt C m [..8] s equato elates te patcle esty to a tegal wc cotas te cemcal potetal a tempeatue as paametes. a ae suppose to be costat so te patcle esty / s a costat too tus fo vayg te tempeatue te tegal must also ema costat. s meas tat µ must cease as eceases because µ < eq. [..]. Howeve fo µ te tegato must gve a measuable esty ts mples tat µ s egatve; o µ must ecease fo eceasg. So we f a egatve µ at > K a fally eac µ at K. I oe to mae tgs moe coveet we ca tae te gou-state as te zeo of te eegy scale a µ. e tegal eq. [..8] te efes a mmum tempeatue suc tat fo c te cemcal potetal vases µ. ow we wat to calculate c fom eq. [..8] we substtute c C c m C c c e e.6... [..9a] ose-ste coesato tempeatue c c.57 m.67 m [..9] Howeve tee s sometg wog wt ts agumet. ee s o pot sayg "tee s a mmum tempeatue below wc te osos caot be coole"... we a calculate lm. c. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

92 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:9 So wat s wog? e tegal eq. [..8] scales wt wc gves te gou-state te wog wegt amely. We see tat ts causes te tegal to loo as f tee s a mmum tempeatue. We must coect fo ts wog wegt. Coecto: [..] patcles gou-state > ote states wt > C µ s s o poblem sce te gou-state gets te wog wegt. Hece we o ot cout te goustate twce > m [..] µ ow we ave to eteme µ aga. We ow tat fo < c a te lmt K µ Fom eq. [..7] we ow tat fo K: µ µ Hece fo aote eegy level : e aveage occupato umbe of suc a level s te wt te coto >> µ. s s te tue fo all states but te gou-state a we get a -stbuto wt µ. Let's loo at a eample: He a bo of volume L gou state: L λ p L p L. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

93 page:9. Quatum Statstcs PH 65: emal & Statstcal Pyscs fst ecte state λ L p p L L Hece p p m m L L fo L cm: -8 e - e s small! fo cm µ >> µ ece we ca teat µ fo < c Wt te assumpto µ fo < c we ca te solve eq. [..]: [..9a] m m.6... > [..9 ] > c tese ae te patcles wc ae ot te gou-state fo < < c. Wt eq. [..]: > c Facto of patcles te ose-ste coesate C: c [..] s equato meas tat below a ceta tempeatue c gve by eq. [..9] te osos stat to coese te gou-state wt amatcally ceasg umbe as K cf. followg fgue. We all o almost all patcles ae te same state wt almost o but zeo-pot eegy fo taslatoal eegy ts s zeo! tey must sae te same wavefucto. e patcles beave le a sgle patcle o te-patcle teactos!. If suc a system ca be ealse t wll ave completely ew popetes t s fact a ew mateal ose-ste coesate C.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

94 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:9 Fgue: Facto / of patcles te zeo-eegy gou-state as a fucto of tempeatue. c s te ose-ste coesato tempeatue. elow c : e -eal gas sepaates to a 'omal' gas o flu a a '-coese' gas o flu. e C as o cotbuto to eegy o pessue we ect 'o' beavou. e mea eegy te egme < < c s te gve by wt µ µ [.. ] c eglgble fo > c 5 [..] A appomate soluto s cf. page 8 eq. [..] ff.: > 5 fo c < < c a ~ 5 C fo < < c c e eact soluto see weely poblem s: ~.9... C fo < < c [..] c. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

95 page:9. Quatum Statstcs PH 65: emal & Statstcal Pyscs A umecal aalyss of fo < < c soluto of µ s sow te followg fgue. µ m fo > c c c µ Oce µ s eteme a C ~ ca be calculate va eq. [..] as sow te et fgue. C ~ c e pecewse beavou of µ esults a pase tasto of te mola specfc eat C ~ at ~ c. Moe etale aalyss eveals tat C c s cotuous but ot ffeetable. ~ Fo muc ge tempeatues C must appoac te classcal value of ase le fgue. e classcal lmt s typcally eace at about c. I te ego c < < classcal we must evaluate umecally.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

96 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:95..5 Supecouctvty a supefluty C I te pevous secto we ave eve te teoy fo ose-ste coesato. Ufotuately ts evato s oly val fo a eal gas. Ue omal ccumstaces o gas ca be coole to tempeatues ecessay to eac c. Most mateals wll be sol a ece ae ot subject to C because tey ae aleay coese a ave o taslatoal egees of feeom. A ecepto s He wc s a oso a -ue omal pessue- oes't solfy eve at vey low tempeatues. Howeve t s ot a gas but coeses to a lqu below.k esty.78 g/cm. ut te teactos a lqu ae stll wea a taslatoal moto s peset. We we teat lqu He as a eal gas we ca use eq. [..9] to calculate c. K. pemets ee sow a pase tasto at λ. K wee lqu He I cages to He II see fgue. C λ pemetal specfc eat of lqu He coestece wt ts vapou vesus [K] ecause of t's sape le te Gee lette λ te pot of pase tasto s calle "lamba-pot" λ. s fgue esembles te featues sow te teoetcal cuve te pevous capte. s s easo wy t s beleve tat te "lamba tasto" appes because of ose-ste coesato. e evato betwee c a λ ca be lae because He s a lqu a ot a gas. He II also sows te ecte "o" beavou. It as o eat capacty te ecease te above fgue s ue to te co-estece of He I a He II a o vscosty supeflu. He Femo oes't sow ts beavou utl < mk. see late. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

97 page:96. Quatum Statstcs PH 65: emal & Statstcal Pyscs Supefluty: HLIUM- He sows fascatg beavou at low tempeatues: Altoug t oes't solfy coseable pessue 5 ba as to be apple to get sol He temoyamc ata sow tat below λ t s a etemely oee state S fo He II lqu to He sol elow λ te eat couctvty of te lqu becomes so lage tat ot spots ecessay fo te fomato of bubbles of bolg caot occu. Ue pope cotos te temal couctvty of te supeflu ca be as lage as tmes tat of coppe at oom tempeatue. A op of tempeatue of oly K gog fom He I to He II causes a cease of te temal couctvty by a facto of seveal mllo. Heat ca flow supeflus fom of a wave ow as seco sou. Ue omal coto eat ffuses va aom molecule moto o s taspote va covecto. Howeve supeflu He a pulse eate causes a tempeatue pulse to tavel acoss te cotae. A eate tat s cycle susoally pouces a susoal tempeatue wave wc tavels toug te lqu wt a spee smla to tat of sou. Fouta of supeflu He potogape by Jac Alle te 97s at te Uvesty of St. Aews Scotla wo was te fst to obseve ts spectacula peomeo. Supeflu He sows eglgble vscosty. I a cotae te vapou above te lqu coats te walls wt a laye oly a few atomc layes tc ts s ot a specalty of supeflus. Howeve sce te vscosty s so small ts laye s able to flow upwas towas te m of te cotae a te ove te ege. s pocess wll empty a ope cotae. s etaoay flow popetes gve te substace te ame a also allows t to flow toug vey aow poes wc ae too aow fo omal flu He. Suc caels ae calle supeleas. e low vscosty also causes a supeflu flowg a ccle to cotue ts moto - pcple to etety as log as te velocty stays below a ctcal velocty. ubulece s easly aceve supeflus ue to low vscosty. e temomecacal o fouta effect supeflus s a specal aagemet of two cotaes of elum coecte by a supelea. If oe se s eate slgtly te lqu level of ts se ses at te ese of te lqu te ote cotae. It ca be aage suc a way tat te supeflu sputs out of cotae fom of a fouta see fgue o te left.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

98 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:97 HLIUM-: He Le He Helum- stays lqu ue omal pessues ow to absolute zeo. He sows eve moe bzae beavou ta He may of t s stll ot completely uestoo. e followg gap sows ts pase agam at low tempeatues. Pase agam fo He te absece of a eteal magetc fel. e pases A a ae supeflu ase aeas. Altoug a Femo F 6K He ca become supeflu at ca. mk... Oseoff. C. caso a. M. Lee scovee ts 97 obel pze 996 How ca tat be? We te couctg electos fom Coope-pas see below - oe to become a oso- te sps ae oete oppostely ece o et sp o magetc momet. He s ffeet. e atoms te supeflu pases A a see fgue also fom pas wc ae el togete by wea teatomc foces a te uclea sps ae alge paallel o at-paallel ece te oso-pa as a et sp of ħ a a et magetc momet o zeo pases A wt I a wt I fgue. ese supeflu A-pase espos supsgly stog to eteal magetc fels ceatg yet aote pase a te popetes of te supeflu ae te becomg asotopc elatve to te oetato of te magetc fel e.g. spee of sou s ffeet paallel o omal to te M-fel. He s use fo ceato of low tempeatues two ffeet types of efgeatos: a luto efgeato a b Pomeacu efgeato. a luto efgeato: elow.8 K te two elum sotopes He a He sepaate spotaeously fom a mtue. e sepaato s ot complete a te lowe pase eave He stll as ca. 6% of He t wle te He-pase floatg o top s pactcally pue see fgue. e tempeatue s too g fo He to be a supeflu ece avg a ozeo eat capacty but He s a ece as a eat capacty close to zeo.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

99 page:98. Quatum Statstcs PH 65: emal & Statstcal Pyscs We ca la te fucto of ts efgeato by loog at t as a bolg eal gas of He gt pat of fgue. e uppe pase s pue He wle te lowe pase as oly few He atoms a et atmospee of supeflu He wc ectly cools te gas ue to ts eomous eat coucto a almost o-estg eat capacty. So te uppe laye bols off He we te equlbum s stube by emovg stuff va a pump. e eat of evapoato s emove fom te uppe pase wc s coole by ts pocess. e ma avatage of a luto efgeato s tat t ca mata vey low tempeatues fo a log tme. He He He 6% He to pump to pump b Pomeacu efgeato ee s aote abomalty te beavou of He wc was pecte by I. Y. Pomeacu 95. It esults te fact tat sol He as a ge etopy ta te lqu. Hece te sol s moe soee ta te lqu at low tempeatues <. K. I te sol te atoms a te ucle ae locate at specfc lattce pots a ca be uestoo as actg appomately epeet fom eac ote. ey fom stgusable lattce pots a ae ot eally subject to te Paul ecluso pcple tey ae sa to be quas-epeet. Wle te lqu te stgusable atoms ae foce by te ecluso pcple to coelate te sps a te sp cotbuto to te etopy s ecease. A scematc etopy-tempeatue agam s sow te fgue o te gt. e lqu s equlbum wt ts vapou a te sol s ue pessue. If we tae lqu He pot a compess t aabatcally we ca eac pot wee t solfes wt te sow ecease tempeatue. O fom a ffeet pespectve we sol He melts te volume pe atom ceases. Hece we te lqu s compesse eceases a te sol foms but etopy ceases wc eques eat wc s emove fom te system wc s teefoe coole. s bzae beavou ca be summase te paao statemet: "o solfy He eat t!" S sol He lqu He. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

100 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:99 Supecouctvty: ecause of te lowe mass couctg electos sow smla beavou to C at muc ge tempeatues > 7K see table. elow suc ctcal tempeatues te electos become supecouctg zeo electcal esstace see gap wc we ca ow uesta as aote tem fo "supeflu" te esstace of a flu to a eteal foce potetal s vscosty wle tat of couctg electos s calle electcal esstace. lemet c [K] Ag - Cu - Au - Mo.9 Ga. Al. S.7 a. Pb 7. b 9. Po to te scovey of vey-g-tempeatue supecouctg oes g- c supecouctos te gest c obseve was tat of a b-alloy wt c K. e coveso of Femos to "oso-beavou" s lae by fomg pas wc ae ol togete by vey wea teactos poos. ue to ts pag Coope-pas see fgue te 'molecula' o pa sp becomes tege osos tus tey ca avo Paul's ecluso pcple a eac zeo-pot eegy. s so-calle CS-teoy was accomplse by J. aee L.. Coope a J.. Sceffe 956. s teoy s beyo te scope of ts lectue. Howeve te easo wy electos fom pas at low tempeatues ca be llustate by two mables a cotae wt a soft bottom e.g. a um. If te cotae Plot by K. Oes wo scovee supecouctvty of te esstace of mecuy vesus tempeatue sowg a sue ecease jump-tempeatue at.k. Smultaeously acque topogapc blue o top a spectoscopc gey o bottom mages of tee gaolum atoms o top of a supecouctg obum suface. I te ego ea te gaolum atoms te magetc popetes of tese vual efects bea up Coope electo pas a gs teefoe mofyg te supecouctvty of te obum s sae voletly g te mables move epeet of eac ote. If t s move getly low te mables move pas eac te small toug ceate te soft cove.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

101 page:. Quatum Statstcs PH 65: emal & Statstcal Pyscs e popetes of supecouctos ae well ow. A cuet oce state wll pesst eve afte te apple potetal ffeece as bee emove e.g. a sote col. s allows to "stoe" cuets ove a log tme wtout a sgfcat ecease a allows to bul supe-stable magets wc o ot ave to be coole because te esstace of te col geeates eat but to aceve a supecouctg state e.g. MI magets magetc popelle tas etc.. Aote fel of applcato uses te Josepso effect fo etectg vey small magetc fels wt SQUIs Supecouctg QUatum Itefeece evce wc measues te pase ffeece of te gly coeet electos te C tuelg toug a jucto. Fally supecouctos ae pefect amagets wc meas tat a eteal magetc flu s completely epelle fom tem wc maes tem to ove ove a maget Messe effect see fgues below. Messe-effect: Left: A supecouctg peulum s epelle fom a ose-soe maget. gt: A small pemaet maget levtates ove a s of a g- c supecoucto Yttum-aum Coppe Oe A eal ose-ste coesate! ose's a ste's pecto of te estece a popetes of a C ate bac to 9. Howeve utl 995 tee was o eal eample aaloges of supeflus a supecouctes est but te ete ot gas o eal osos. I 995. Coell a C. Wema wee te fst to emostate C a lute gas of 87 b atoms 87 b as eve umbe of eutos I ; ece s a oso. How tey aceve ts ew state of matte? a fom eq. [..9]: b fom eq. [..7]: < c.57 m. λ ecease cease esty /. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

102 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page: e successful appoac by Coell a Wema uses bot ways a a b: e b-atoms ae polase by cculaly polase lgt to a sp ħ. s tc pevets te vapou to eac temal equlbum at low tempeatues cf. aabatc emagetsato to fom a lqu o sol. I ts way tey ca coese a supe-satuate vapou. e vapou s coole by LASs fom oom tempeatue to ca. mk. s LAS-coolg s aceve by settg te LAS fequecy slgtly too low fo esoat absopto by te b-atoms at est. Hece oly te movg ot b atoms ae affecte a fom tem oly tose wo ae opple blue-sfte tose wo move towas te LAS see fgue. ey ae te oly oe wo ca absob a poto by a ea-o collso wc slows tem ow cools tem because tey te emt lgt wt o pefee ecto so o aveage tee s o ecol. o cofe te atoms magetc fels ae apple see fgue below. Afte te LAS-coolg of step te gaet of te magetc tap s cease a te vapou s compesse by a facto of ca.. e eat geeate by ts pocess s emove by fute LAS-coolg utl a tempeatue of ca. µk s eace. L AS I a covetoal tap te fel falls to zeo at te cete left allowg te atoms to lea out f tey ae cool eoug. e OP tap moulates te posto of te tap by atoal otato cete wc esults a aveage potetal wc s close gt. vetually te LAS-coolg s stoppe a te caactestcs of te magetc tap cage OP tap see fgue above. ow evapoatve coolg s use to aceve eve lowe tempeatues. e moe eegy te atoms possess te fute tey tavel fom te cete of te tap wt te lowest potetal. e mamum potetal s ow lowee to let te ot eegetc atoms escape wc cools te est. Sce te 87 b atoms wee sp polase step ao-fequecy M-emet ca be use to selectvely flp te sp of te tap ece ase te eegy ove a ctcal level a. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

103 page:. Quatum Statstcs PH 65: emal & Statstcal Pyscs emove tem fom te tap. e moe eegetc te atoms ae ge s te local fel tey eece te tap. ue to te Lamo coto ω f te moe eegetc atoms ca be selectvely emove wtout toucg te cool stuff. y ampg ow te f-fequecy ultacol tempeatues ca be aceve wle te mateal stay a gaseous state. e fst C of ca. atoms appeae at 7 K. Fute evapoatve coolg cause almost all atoms to settle te gou-state fomg a ew state of completely coeet matte ca. atoms a volume of ca. µm all te same quatum state. e C ca be obseve by aote tc: e LASs ae off ug evapoatve coolg step. Oce te state s eace wee te system soul be etecte te magetc taps ae emove. e te atoms ae gve some tme mll-secos to fly apat to allow te measuemet of te vual veloctes. s clou s te llumate by a LAS at esoace fequecy mamum scatteg fom te atoms a ts testy measue by a CC-cp. s poceue gave te followg mages. [M. H. Aeso J.. se M.. Mattews C.. Wema a. A. Coell Scece ] Obsevato of C: e mages epeset velocty stbutos of a supe-cool clou of 87 b atoms. a efoe te coesato te stbuto s sotopc as ecte fom a gas temal equlbum. b e coesate appeas te cete by atoms avg zeo velocty. c Fute coolg gves almost pue C. e stbuto s slgtly ellptcal because a ellptcal tap was use. ac mage s 5 µm a s eve fom LAS llumato afte a peo of 6 ms of fee aso.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

104 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page: Possble applcatos...wo ows?! Cs coul be use fo atomc lases wee te atoms ae pumpe to a sgle patcle state wt oe e ogle wavelegt see fgue below. Hece tey emt coeet matte ate ta just coeet lgt. s mgt ave applcatos fo g pecso ltogapy. ey coul also be use fo a ew type of tefeomete because te e ogle wavelegt ca be mae muc sote ta tat of lgt? Ay ote eas? Atomc LAS: esty stbuto of two supempose waves of coeet C-matte of ulta-col ubum gas. a o tefeece fo omal b-gas b tefeece appeas we some b-atoms ete C-state c fo almost all b-atoms fomg a C. e scale o te gt s stace mm! e stog cotast of te tefeece patte s aote poof tat te b-atoms fom a C because te atoms sae oe state wt oe e ogle wavelegt. [I. loc. W. Häsc a. sslge Spetum e Wssescaft 7.]..6 emoyamcs of Stas Some bascs fst! How oes a sta fom? Fom a gas clou fom clustes wc ae pulle togete by gavtato. Hece gavtato s te attactve foce. We wat to estmate te potetal eegy ue to gavtato. Acceleato of a mass elemet locate at a stace fom te cete: e matte o a specal sell of aus as a mass m see gap: m ρ [..5] wee ρ s te esty. e gavtatoal acceleato s te: Gm g [..6]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

105 page:. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy e patcles get close. ee as to be some couteactg pessue to evolve we o ot specfy at ts momet by wat meas to pevet te system fom collapse. o eac a stable state te foces ave to be equlbum: ρ ρ A P A g A P A P A g A P A P P m g A P [..6 ] G m P g ρ ρ [..7] e system s equlbum we eq. [..7] ols fo all ece we ave to tegate ove te volume to establs ts coto assumg a spee of aus a mass M: G m G m P G m P ρ ρ ρ substtute fom eq. [..5]: m ρ mass betwee a G m Gm G m P M m m ρ [..8] wee G eotes te gavtatoal potetal eegy.

106 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:5 e left se of eq. [..8] ca be obtae va tegato by pats: [ P ] P P P P wee [ ] P suface pessue combato wt eq. [..8] gves: P G wee P s te volume aveage pessue P volume aveage pessue: P G [..9] cf. al teoem. Howeve te patcles mgt become so eegetc a fast tat tey become elatvstc! e ma pyscal popetes of te su: Popety Mass aus Poto lumosty Suface tempeatue Cetal tempeatue Cetal pessue e questo wc teests us te cotet of ts lectue s: Wat ca pouce te ecessay pessue to avo gavtatoal collapse? Potos: lac-boy aato pessue: alue M.99 g m L.86 6 W S 578 K C.56 7 K P C.9 6 Pa Cetal esty ρ C.8 5 g m - Age t.55 9 yeas Fo te su see table above: Fst we ca calculate te aveage pessue ue to gavtato se te su: P G GM GM Pa [..]. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

107 page:6. Quatum Statstcs PH 65: emal & Statstcal Pyscs If we teat te su as a eal gas we ca estmate te aveage tempeatue o get t fom te table: wee P P ρ m M M ρ s te aveage esty a m te aveage mass pe su patcle e.g. ose H wc s te mass of a poto m p. [..] 6 p 7 m GM G M G M P m m ρ M m K I a weely set poblem we a calculate fom Plac's law see eq. [..] te aato pessue of a blac-boy to be: b P wt 5 8 b [..] 5c Fo te su: b Pa K - Suface: S 578 K P.8 Pa Ou estmate: 7 6 K P 6 Pa Cete: C.56 7 K P.5 Pa s s ot suffcet: Ou estmate fo te aveage pessue was Pa table gves 6 Pa te cete. Hece aato cotbutes oly ca.. % to te ecessay pessue tat stablses te su. b b G M M Howeve because of P m te aato pessue plays a sgfcat ole fo small a massve stas e.g. wte wafs! e gas pessue s M M M P g Hece: P M wc llustates tat aato pessue omates fo ese stas 5 P M g Pessue of te egeeate Fem-electo gas: ecause stas cosst of plasma te electos ca move aou fee. Altoug we mgt ave to alte te escpto fom secto.. ue to collsos. e teatmet of te electo gas close to absolute zeo tempeatue s stll val because of F ρ cf. eq. [..7] a stas ae muc ese see table et page ta metals wc aleay gave F of ca. K. /. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

108 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:7 So wat pessue s geeate by te electo gas? Fom etc gas teoy see fst wee poblem: P p v pv o P p m e [..8] 5 F [..7] 5m e 8 5 [..] s s te pessue of te egeeate electo-gas at K. We see tat o 5 P C ρ wt P 5 M C 8 8 5m e If we compae ts - fom of popotoaltes- wt te gavtatoal pessue fom eq. [..9] o eq. [..]: M C 5 5 M C 5 pessue of te electo gas GM gavtato pessue al We ealse tat te left se gows te fft powe as te aus eceases wle te left se oyl gows te fout powe. Hece te pessue of te electo gas s able to coute-act gavtatoal compesso. C We also f: M GM Hece te lage te mass te smalle te sta espte te electo pessue. A moe etale aalyss gves.57 M fo M M. e followg table compses some elevat o-elatvstc ata fo ts scusso fo wte wafs: ame M/ M / / [m] F [e] F [K] p F /m e c v F /c Sus e last colum sows te classcal assumptos about electo velocty a ece mass s ot coect because te electos stas eac spees of 5-8% of te spee of lgt. Hece we ave to teat tem elatvstcally.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy

109 page:8. Quatum Statstcs PH 65: emal & Statstcal Pyscs. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy e equato P pv s elatvstcally coect. Howeve we must avo toucg m e wc s te mass of te electo at est p m e v te elatvstc oma. I ts ew egme te electos ave a spee close to c ece to smplfy tgs -suffcet ts cotet- we say te spee s tat of lgt. If we assume te electos stll as completely egeeate we ca smply equate avog elatvstc coectos. pessue of elatvstc electos: c p P F [..] e mometum of a electo a Fem-gas at K s: F F m p wt F gve by eq. [..7]. 8 8 e e F m m p [..5] a eq. [..] gves: 8 c P Hece te eteme elatvstc egme electos ave spee of lgt te pessue of te egeeate Fem-gas gves a mass/aus ato of M M P ρ Compaso wt te gavtatoal pessue fom eq. [..9] o eq. [..] te gves: GM M C [..6] Oce te eteme elatvstc egme mamum spee of electos s eace fute cotacto oes't cease pessue faste ta gavtato. e balace oes't epe o aymoe! Howeve fute fomato ca be gatee fom eq. [..6]: G c G c M M M M m c GM e 9 8 C G c G c M [..7] If te mass of a sta oes't satsfy ts equato t caot ajust ts aus to pove a equalty betwee electo a gavtatoal pessue. e oute to stella equlbum s bloce.

110 PH 65: emal & Statstcal Pyscs. Quatum Statstcs page:9 I 9 S. Caasea oble pze 98 calculate eq. [..7]. e ctcal lmtg mass M C s also calle Caasea-lmt. Fute aalyss sows tat stas wt masses up to M M may evolve to wte wafs. Moe massve stas caot be suppote by te pessue of te egeeate electo gas a must seac fo ote outes to establs equlbum o collapse. Suc ote outes ca clue fo stace vete β-ecay.e. e - p e. s pocess emoves electos wc ops te pessue eve fute causg fute cotacto a moe vete β-ecay. s ca stops we te pessue of a egeeate euto gas ols up agast gavty euto stas. Scematc Hetzspug-ussell agam. Sapsot of te lumosty L a suface tempeatue of stas at ffeet stages te evoluto. Most of te obseve stas ae goupe alog te ma-sequece; tese ae yoge bug stas le te su yellow ccle. As stas evolve te cotacto of te cetal coe s accompae by a aso of te oute layes of te sta to fom lumous stas wt low suface tempeatue e.g. e gats. e e-pot of stella evoluto of a sta wt mass compaable to te su s a compact object suppote by egeeate electos a wte waf. e evoluto of moe massve stas M > M C ca lea to te fomato of euto stas o blc oles.. Pete lümle Scool of Pyscal Sceces Uvesty of Catebuy