% $ ( 3 2)R T >> T Fermi

Transcription

1 6 he gad caoical eemble theoy fo a ytem i equilibium with a heat/paticle eevoi Hiohi Matuoka I thi chapte we will dicu the thid appoach to calculate themal popetie of a micocopic model the caoical eemble appoach ha bee dicued i Ch2 ad 3 while the micocaoical eemble appoach ha bee biefly dicued i Sec5 aditioally thi appoach i called the gad caoical eemble theoy wo mai motivatio fo havig yet aothe appoach ae: to be able to deal with idetical o iditiguihable paticle without eotig to the N! tick ued fo the moatomic ad diatomic ideal ga model a thi tick doe ot wok at vey low tempeatue o at vey high deitie; 2 to be able to teat two type of idetical paticle boo ad femio withi a igle theoetical famewok with a miimum amout of wok he fit ad the ecod motivatio ae actually cloely coected becaue the diffeece betwee boo ad femio become mot otable at vey low tempeatue o at vey high deitie he gad caoical eemble appoach i thu tailoed to meet thee demad he micocopic model we will dicu i the ext two chapte he ideal phoo boo ga model fo lattice wave i a olid Ou mai goal i tudyig thi model i to calculate it mola heat capacity at cotat volume: c v = D3 <<! 3"R >>! whee! i the chaacteitic tempeatue fo the phoo o the Debye tempeatue that epaate the lowtempeatue egime fom the hightempeatue egime fo the phoo 2 he fee electo femio ga model fo coductio electo i a imple metal Ou mai goal i tudyig thi model i to calculate it mola heat capacity at cotat volume: " c v =! << Femi 3 2R >> Femi

2 2 whee F i the chaacteitic tempeatue fo the coductio electo o the Femi tempeatue that epaate the lowtempeatue egime fom the hightempeatue egime fo the electo 6 A ytem i equilibium with a heat/paticle eevoi I the gad caoical eemble appoach we coide a macocopic ytem of volume V that i attached to a heat/paticle eevoi at tempeatue ad with chemical potetial µ Chemical potetial: a iteive tate vaiable that cotol paticle umbe deity Jut a heat flow fom a hightempeatue to a lowtempeatue ytem paticle flow fom a high µ to a low µ ytem Whe two ytem ae i themal ad diffuive cotact with each othe o that they ca exchage eegy though heat ad paticle though diffuio they will evetually each themal ad diffuive equilibium which i chaacteized by the equal tempeatue ad 2 the equal chemical potetial i the ytem Jut a tempeatue i a iteive quatity chemical potetial i alo a iteive quatity Fo moe detail o chemical potetial ee Appedix 0 A we will ee below the mai ole of the chemical potetial i the gad caoical eemble appoach i to cotol the umbe deity o the mola volume of a ytem I othe wod the mola volume ca be egaded a a fuctio of ad µ : v = v µ O the macocopic level We meaue U ad N o a fuctio of V ad µ

3 3 O the micocopic level Eegy ad paticle ae beig exchaged betwee the ytem ad the eevoi o that the total eegy E ad the umbe ˆ N of paticle i the ytem fluctuate aoud thei aveage value U ad N which we obeve o the macocopic level 62 he gad caoical eemble theoy baed o a exteive tate vaiable called themodyamic potetial I the gad caoical eemble appoach we calculate a macocopic tate vaiable called the themodyamic potetial! fom a micocopic quatity called gad patitio fuctio! though the followig elatio: = "k B l V! V µ µ / N Avogado which povide a citical lik betwee the macocopic level ad the micocopic level jut a the followig elatio doe i the caoical eemble appoach F V =!k B l Z VN Avogado I macocopic themodyamic the themodyamic potetial i defied by! " U S µ A U S ad ae all exteive while ad µ ae iteive the themodyamic potetial i a exteive tate vaiable 63 hemal popetie ca be calculated diectly fom the themodyamic potetial Uig the exteded fom of the fudametal equatio of themodyamic we fid du = ds! PdV + µ d d! = "Sd " PdV " d µ Deivatio

4 4 d! = du " d S " d µ = ds " PdV + µ d + ds + Sd = "Sd " PdV " d µ + µ d + d µ Compaig thi equatio d! = "Sd " PdV " d µ with the followig calculu equatio we obtai ad d! = "! " V µ S V µ d + "! "V " =! " =! " P V µ V µ µ "V " µ =! " dv + "! " µ V µ µ V V d µ which allow u to calculate S P ad a fuctio of V ad µ We ca alo calculate the iteal eegy U a a fuctio of V ad µ by uig U V µ =! + S + µ It would be moe coveiet to expe all the themal popetie a fuctio of v ad I ode to achieve that goal we eed to eexpe the themodyamic potetial a PV he themodyamic potetial i othig but PV A the themodyamic potetial i exteive it mut cale with the ytem ize o that it ca be expeed i the followig fom: fom which we obtai = V"! V µ µ =! " P V µ "V µ =! V µ =! µ V

5 5 o that = "P V! V µ µ V he above equatio fo P i tem of! µ alo implie that we ca fid the peue a a fuctio of V ad µ diectly fom! o! Deivatio of! V µ = V" µ Beig exteive! mut atify! "V µ! = V we get = V! V µ! µ = "! V µ By chooig! to be o that = V! µ " V! V µ µ A! = U " S " µ the above eult! = "PV implie the followig Eule equatio U! S + PV! µ = 0 which imply mea that U S P V ad µ deped o each othe Chemical potetial a a fuctio of ad v: µ = µ v Uig the followig equatio =! " V µ " µ V we fid the mole umbe a a fuctio of V ad µ : = V µ Solvig thi equatio fo V we obtai V = v µ whee v i the mola volume a a fuctio of ad µ :

6 6 v = v µ Solvig thi equatio fo µ we the fid µ a a fuctio of ad v: µ = µ v State vaiable a fuctio of v ad Uig µ = µ v we ca eplace the idepedet vaiable µ fo P S ad U by ad v o that we ca expe thee tate vaiable a fuctio of V ad =! " P V µ 2 S =! " " V µ "V µ =!V " " o P =! " V µ V µ 3 U =! + S + µ = v" µ =!v " " µ + { v } + =! µ ad µ = µ v lead to P = P v ad µ = µ v lead to S = v µ v lead to U = u v O U =! + S + µ =! "! alo lead to U = u v V µ " µ! µ V = " + µ µ! ad µ = µ v Example: he moatomic ideal ga model a a miimal model fo the uiveality cla of lowdeity moatomic gae I the moatomic ideal ga model we model all the atom i the ga to be poit paticle that do ot exet foce o each othe We will late calculate the themodyamic potetial of thi model to be whee = " V! V µ Re " 2!h 2 = N Avogado mk B µ / R 3/ 2 0 V µ " µ =! " V = V e µ / R

7 7 o that v = e! µ / R ad µ v =!R l " v Alo! v = "R i P =! " V = R V = R v o P =! " "V = µ Re µ / R = R V = R v ii =! " S V µ " V µ = R V e µ / R! µ R + 5 2! v o that S = R l" HW63: how thi iii U =! + S + µ = 3 R HW632: how thi 2 o o that U =! + S + µ =! "! = 3 2 R V e µ / R U = 3 2 R = 3 2 PV V µ " µ! µ V HW633: how thi Awe fo the homewok quetio i Sec63 HW63 A! "3 2 dv Q =! 3 d 2! d Ax whee dx = A!x! " =! Ax! x i ued

8 8 =! " S V µ " V µ = R V e µ / R! R V e 2 = R V e µ / R! µ R d d + R V µ / R e µ / R! µ R 2 HW632 U =! + S + µ v = "R + R l + 5 / + 2 / " R l v = 3 2 R HW633 U =! + S + µ =! "! = " + µ µ! V µ " µ! µ = " + µ µ "R V e µ / R = " R V e µ / R d + 2 d + µ "R V e = 3 2 R V e µ / R µ / R / R / V " R V e µ / R " µ R 2 64 Maco fom mico: the themodyamic potetial of a micocopic model fom it Gibb um o the gad patitio fuctio he lik betwee a micocopic model ad it themodyamic potetial! i povided by what we call the Gibb um o the gad patitio fuctio! which i a fuctio of the

9 9 tempeatue the volume V ad the chemical potetial pe paticle µ of the model ad i diectly elated to the themodyamic potetial by = "k B l V! V µ µ / N Avogado A with F =!k B l Z we ca deive thi elatio baed o the micocopic defiitio of etopy fo moe detail ee Appedix 0 hi elatio o the bidge betwee the macocopic quatity the themodyamic potetial ad the micocopic quatity the Gibb um o the gad patitio fuctio a well a F =!k B l Z ad S = k B lw ae the mot impotat equatio i tatitical mechaic Example: he moatomic ideal ga model a the miimal model fo the uiveality cla of lowdeity moatomic gae calculate it Gibb um o gad patitio fuctio to be l! Vµ " Fo the moatomic ideal ga model we will late V Q V eµ / k B whee the quatum volume i give by " V Q = 2!h 2 mk B 3/ 2 A we will ee late the above Gibb um i valid oly whe the ditictio betwee boo ad femio i ielevat he themodyamic potetial of thi model i the give by = " V! V µ Re µ / R HW64: how thi whee we have ued N Avogado k B = R ad the quatum mola volume i elated to the quatum volume V Q by " 2!h = N Avogado V Q 2 = N Avogado mk B 3/ 2

10 0 Awe fo the homewok quetio i Sec64 HW64 = "k B l V! V µ = "N Avogado k B = " V Re µ / R V µ / N Avogado = "k B V Q e µ / N Avogado k B V N Avogado V Q e µ / N Avogado k B 65 Eegy eigevalue of a micocopic model Befoe we calculate the Gibb um o the gad patitio fuctio fo a micocopic model we eed to obtai the eegy eigevalue fo the model by olvig the followig Schödige equatio fo the model: = E V N H! 2 N! 2 N whee the idex i the label o the quatum umbe fo the model eegy eigetate each of which i epeeted by the quatum wavefuctio! 2 N which deped o the poitio of the N micocopic paticle i the model ad i accompaied by a coepodig eegy eigevalue E V N which deped o the volume V of the model ytem ad the umbe N of the micocopic paticle while the Hamiltoia opeato H i a opeato that opeate o the wavefuctio ad coepod to the total eegy of the model: N H =! i + " 2 N i= whee i i the kietic eegy opeato fo the ith paticle ad i give by while! 2 H i =! h2 2m " 2 "x i 2 + " 2 "y i 2 + " 2 i the potetial eegy fuctio fo the paticle he potetial eegy N coit of two pat oe of which i due to the itemolecula foce amog the cotituet molecule while the othe pat i ued to keep the molecule iide the box of volume V "z i 2

11 66 he Gibb um o the gad patitio fuctio! the Gibb facto he Gibb um o the gad patitio fuctio i baed o a quatity called Gibb facto ad i defied a + = exp " E V ˆ N! Vµ " µ N ˆ N ˆ = 0 k B whee µ i the chemical potetial pe paticle elated to the chemical potetial µ by µ = µ N Avogado he idex i the label o the quatum umbe fo the eegy eigetate of a micocopic model ad u ove all the eegy eigetate each with it coepodig eegy eigevalue E which omally deped o the volume V ad the umbe N of micocopic paticle eg atom makig up the ytem he expoetial facto exp {! E! µ N ˆ / k B } i called the Gibb facto Note that the Gibb um o the gad patitio fuctio i dimeiole becaue the Gibb facto i dimeiole 67 Macocopic quatitie a tatitical aveage of micocopic quatitie A i the caoical eemble appoach the macocopic themal quatitie uch a the iteal eegy the peue the umbe of paticle i a macocopic ytem ae tatitical aveage of thei coepodig micocopic quatitie Iteal eegy a a tatitical aveage of eegy eigevalue We ca how thi coectio betwee the iteal eegy ad the micocopic eegy eigevalue of a micocopic model by ubtitutig the elatio betwee it themodyamic potetial ad gad patitio fuctio = "k B l V! V µ µ / N Avogado ito the equatio fo the iteal eegy i tem of the Helmholtz fee eegy

12 2 We the obtai U =! " " + µ " " µ =! " " + µ " "µ U =! + + N ˆ =0 E = E V N ˆ N ˆ =0 " µ N ˆ V N ˆ exp " E V ˆ N k B! exp " E V ˆ N " µ N ˆ k B / 0 which ca be expeed a HW67: how thi whee p i defied by U = E =!! p E " N ˆ = 0 p! " exp E V ˆ N µ N ˆ k B which implie that the iteal eegy i a tatitical aveage E of the eegy eigevalue E ove all the eegy eigetate each of which cotibute to the aveage accodig to tatitical weight p which i popotioal to the Gibb facto Macocopic peue a a tatitical aveage of micocopic peue We ca alo how a imila coectio betwee the macocopic peue ad it micocopic coutepat: " P = P =!! p P HW672: how thi N ˆ = 0 whee the micocopic peue P of the model ytem that i i eegy eigetate i defied by P! " E V N ˆ

13 3 Macocopic umbe of paticle a a tatitical aveage of micocopic umbe of paticle Sice the chemical potetial µ ad the chemical potetial pe paticle µ ae elated by µ = N Avogado µ = µ N Avogado = µn the themodyamic potetial i alo elated to the chemical potetial pe paticle by! = U " S " µ = U " S " µn fom which we obtai o that d! = "Sd " PdV " Ndµ =! " N Vµ "µ V Uig thi equatio wee ca the how a imila coectio betwee the macocopic umbe of paticle ad it micocopic coutepat: N = ˆ N = "!! p N ˆ HW673: how thi N ˆ =0 Awe fo the homewok quetio i Sec67 HW67 U =! " " + µ " " µ =! " " + µ " "µ = k B " " + µ " "µ l = k B " " + µ " "µ = k B = 3! µ ˆ! µ N ˆ + exp! E V N ˆ N / E V N ˆ N ˆ = 0 k B 6 k B N ˆ =0 E! µ N ˆ + V N ˆ exp! E V ˆ N k B / 0 / 0 + µ N ˆ 7 9 k B 9 89

14 4 HW672 HW673 P =! " "V = k B = 3 3 µ 22 N ˆ = 0 22 N ˆ =0 N =! " "µ = k B = N ˆ = 0 22 N ˆ =0 = k B " "V l + exp! E +! "E "V N ˆ = k B µ V N ˆ! µ N ˆ k B / + 0 exp! E V ˆ N k B " "V µ / k B! "E 6 56 "V! µ N ˆ / 0 = k B " "µ l = k B " V "µ V + exp! E V ˆ N! µ N ˆ / N ˆ 0 k B k B + N ˆ exp! E V ˆ N! µ N ˆ / 0 k B V N ˆ / Oveall cheme fo the gad caoical eemble appoach he oveall cheme fo the gad caoical eemble appoach i vey imila to that fo the caoical eemble appoach N Cotuct a micocopic model with a Hamiltoia: H =! i + " 2 N i= 2 By olvig the Schödige equatio fo the model = E V N H! 2 calculate it eegy eigevalue: E V N N! 2 { } N 3 Calculate the Gibb um o gad patitio fuctio! Vµ = exp " E V ˆ N " µ N ˆ ˆ k B N = 0

15 5 4 Calculate the themodyamic potetial:! V µ = "k B l V µ / N Avogado 5 Calculate the tate vaiable fom the themodyamic potetial: =! " " µ V P =! " "V µ S =! " " ad U =! + S + µ V µ 69 State vaiable of a quatum ideal ga model Oepaticle tate Coide a igle paticle eg a atom a electo a photo etc i a box of volume V Fid the eegy eigetate ad eigevalue fo the paticle hee eigetate ae called oepaticle tate Each oepaticle tate i uually labeled by a itege vecto = x y z ad the coepodig eegy eigevalue i! Example: the igleatom tate! = h 2 " 2 2mV 2/ 3 2 x y + z! = 2 3 "! = x y z Eegy eigetate fo a quatum ideal ga of ˆ N paticle Each eegy eigetate fo a quatum ideal ga of ˆ N paticle i epeeted by the umbe of paticle i all the oepaticle tate whee! {! } :all the oepaticle tate i the umbe of paticle i oepaticle tate ad i called the occupatio umbe hi way we ae ot ditiguihig o labelig the paticle i the ga All the paticle i the ga ae idetical o that we ca oly pecify how may of them ae i each oepaticle tate Boo ad femio Quatum paticle ae claified ito two goup: boo ad femio Boo: paticle with itegal pi o! = itege h

16 6 Example: photo W ± Z gluo pio phoo 4 He 2e! 2p + 2 Ay umbe of boo ca occupy each oepaticle tate o that! = " Femio: paticle with halfitegal pi o! = odd itege h 2 Example: electo poto euto eutio quak 3 He 2e! 2p + Accodig to the Pauli excluio piciple a femio caot hae a oepaticle tate with aothe o that whee! = 0 o 0 mea that oepaticle tate i empty; mea that oepaticle tate i occupied he total eegy eigevalue {! } = : all the oe paticle tate E "! he total umbe of paticle { } = N ˆ! "! : all the oepaticle tate he Gibb um o the gad patitio fuctio

17 7! Vµ = : all the oepaticle tate " exp + E " = " exp + + µ " k B = exp + + µ " 0 / 2 k B / 2 " = / e + / " + µ / k B { } " { } + µ N ˆ {" } k B Fo oiteactig paticle the oepaticle tate ae idepedet of each othe o that the Gibb um become whee! =! =!! 2! """ = e" " µ / k B i the Gibb um fo oepaticle tate! hi eult i aalogou to the decompoitio fomula fo the patitio fuctio fo N oiteactig ditiguihable paticle Z = Z Z 2 Z 3!!! Z N = Z N whee Z i the patitio fuctio fo a igle paticle Note that fo N oiteactig iditiguihable paticle we caot apply the decompoitio fomula to thei patitio fuctio becaue we mut impoe a cotait that the total umbe of the paticle i N while we um ove all poible value fo the occupatio umbe! of each oepaticle tate i the patitio fuctio: Z = whee becaue of the cotait " exp E! { } k B! "! = N! { } " = N the um ove the occupatio umbe ae ot idepedet of each othe I cotat the Gibb um atifie the decompoitio fomula! = "! ad thi i exactly why the gad caoical eemble appoach i moe uitable fo the quatum ideal ga model tha the caoical eemble appoach

18 8 = Boo:! " Vµ e " =0 { "µ / k B } + = " e " "µ / k B whee we have aumed o o that we ca ue e!"!µ / k B <! > µ fo all " x!!= 0 = x x < Sice uually! > 0 ad the lowet oepaticle eegy i cloe to zeo o! 00 ~ 0 the! > µ implie µ! 0 fo boo Femio:! + Vµ = e " { " µ / k B } + = + e " + [ "µ / k B ] =0 he themodyamic potetial Boo:! " Vµ Femio:! + Vµ [ "µ / k B ] = "k B l " = k B l " e " = "k B l + = "k B l + e " [ " µ / k B ] he claical egime he ditictio betwee boo ad femio become ielevat if o that e!"!µ / k B <<! ± " k B e µ / k B = k B e µ / k B e / k B whee we have ued l + x! x x <<

19 9 Example: he themodyamic potetial fo the moatomic ideal ga model i the claical egime i the give by! Vµ " k B e µ / k B V Z = V Q k e µ / k B B whee Z i the patitio fuctio fo a igle atom calculated i Sec32: ad Z = e!" / kb = " V Q = 2!h 2 mk B 3/ 2 V Q V hi themodyamic potetial i exactly the oe we have ued i Sec63 to deive the tate vaiable fo the moatomic ideal ga model icludig the chemical potetial: o that µ = " v µ =!R l µ =! N Avogado R " v " l N Avogado =!k l v B whee we have ued R = N Avogado k B he coditio fo the claical egime e!"!µ / k B << implie! " µ >> k A B the lowet eegy eigevalue i vey cloe to zeo! 00 ~ 0 thi i tu implie that µ <<!k B o that o which i equivalet to " v!k B l <<!k B! v l " >>

20 20 v! = " Q v 3 2 >> which i atified fo a High tempeatue: >> Q v! 2"h2 mk B b Low deitie: v >> N Avogado v 2 /3 he BoeEitei ditibutio fuctio a the aveage occupatio umbe fo boo he aveage umbe fo a quatum ideal ga of boo i give by =! "! N Vµ "µ V = + e! µ / k B! If we defie the BoeEitei ditibutio fuctio by f! " µ e "!µ / k B! the we fid o that N =! ˆ N = "! = "! = " f µ = = f " µ e "µ / k B " he BoeEitei ditibutio fuctio i theefoe the aveage occupatio umbe! of oepaticle tate ad it tell u how may boo o the aveage ae foud i oepaticle tate f! deceae a! i iceaed o that oepaticle tate with lowe eegy ae occupied by moe boo A the tempeatue appoache = 0 moe ad moe boo ae foud i the loweteegy oepaticle tate util all the boo ae accommodated i the loweteegy oe

21 2 paticle tate at = 0 hi accumulatio of boo i the loweteegy oepaticle tate i called the Boe Eitei codeatio he figue below how f! a a fuctio of! µ ad k B µ the top cuve i fo k B µ = 0 the middle oe i fo k B µ = 005 ad the bottom oe i fo k B µ = 00 f! " µ k B µ =! e " µ! / k B µ he FemiDiac ditibutio fuctio a the aveage occupatio umbe of femio he aveage umbe fo a quatum ideal ga of boo i give by =! " + N Vµ "µ V = + e! µ / k B + If we defie the FemiDiac ditibutio fuctio by f +! µ " e! µ / k B + the we fid o that N =! ˆ N = "! = "! = " f + µ = = f + " µ e " µ / k B +

22 22 he FemiDiac ditibutio fuctio i theefoe the aveage occupatio umbe! of oepaticle tate ad it tell u how may femio o the aveage ae foud i oepaticle tate Baic popetie of the FemiDiac ditibutio fuctio! Becaue of the Pauli excluio piciple each oepaticle tate ca be i at mot oe a 0! f + " µ occupied by oe paticle at mot hat i why f +!µ b f +! µ deceae a! i iceaed o that oepaticle tate with lowe eegy ae occupied by moe femio tha thoe with highe eegy c f + µµ = / 2 d At = 0 : f +! µ i a tep fuctio: "! < µ 0! > µ f +!0 µ Phyically thi mea that at = 0 all the oepaticle tate with eegy le tha µ ae all occupied while thoe with eegy above µ ae all empty e Fo! << µ " k B : f +! µ " f Fo! >> µ " k B : f +! µ " 0 he figue below how f + a a fuctio of! µ ad k B µ fo! µ jut below the top cuve i fo k B µ = 0 the middle oe i fo k B µ = 00 ad the bottom oe i fo k B µ = 0 f +! µ k B µ = e! µ "/ k B µ +

23 23 I the claical egime: e!"!µ / k B << o e! " µ / k B >> o that f ± = e! "µ / k B e "! ± "µ / k B f claical << heefoe both the BoeEitei ditibutio fuctio ad the FemiDiac ditibutio fuctio pactically coicide with exp! "! µ / k B " µ >> k B { } whe! { / k B } v "! µ / k B f ± exp! "! µ he top cuve i the BoeEitei ditibutio fuctio he middle cuve i the claical ditibutio fuctio he bottom cuve i FemiDiac ditibutio fuctio

24 24 he iteal eegy U = E {! } = "! = "! = " f ± " µ! Boo: U = e! "µ / k B "! e! "µ / k B + Femio: U = We will ue thee equatio to calculate the iteal eegy fo photo phoo ad electo i a imple metal he peue i cotolled by the iteal eegy Boo ad Femio +: =! " ± P Vµ "V µ =! + " "V e! µ / k B " =! ± + "V Fo a mateial paticle eg a atom o a electo i a box of volume V we fid o that We the fid! " L 2 = V 2/3!" = 2 "!V µ 3 V P = 2 3V "! = 2 3V U which implie that the equatio of tate follow fom the iteal eegy U = U V We alo fid U = 3 2 PV

25 25 Fo phoo ad photo we will fid late o that We the fid! " V / 3!" = "!V µ 3 V P = 3V "! = 3V U which implie that the equatio of tate follow fom the iteal eegy U = U V We alo fid U = 3PV