9. Statstcal Mechancs of non-localzed systems Physcs 337 9 Statstcal Mechancs of non-localzed systems Next we turn to delocalzed systems. In ths course we ll only consder gases, but smlar deas hold for lquds; except that lquds are strongly nteractng and approxmatons nvolvng the sngle partcle partton functon aren t very useful. We ll frst consder the classcal case, whch s vald at hgh temperatures. The paradgm we ll take s a gas of dentcal partcles. The outstandng dfference between a gas of dentcal partcles and a sold of dentcal partcles s that, n the sold, each partcle s dstngushed by ts poston, and has ts own prvate energy levels, whle the energy levels n a gas are open, n prncple, to occupancy by any partcle. Hence the countng of dstngushable states s qute dfferent. 9. Classcal Maxwell-Boltzmann Statstcs Consder a sold and a gas, each wth 5 partcles, one of whch s n an excted state. In ths case, there are 5 dstngushable mcrostates possble n the sold, whle there s only dstngushable state n the gas!! Sold (5 dstngushable states) ε ε ε 2 ε 3 ε 4 Gas ( dstngushable state) Consder a gas of non-nteractng partcles wth zero charge. If we treat them as non-nteractng, then each of them separately satsfes the Schrödnger equaton n a box wth sze L. The energes are quantzed accordng to n ( E n p2 n 2m 2 π ) 2 n 2, (9.) 2m L (recall p k nπ/l). The separaton between adjacent energy levels s of order n2 δe n δ ( (p2 n) k B 2m 2 π ) 2 2E n n n 6 K, mk B L k B (9.2) for a mole of helum atoms at room temperature and pressure (volume (3 cm) 3 )). The levels of a partcle n a sold may depend on the energy levels of the neghbors, n a typcal case: we ve treated the non-nteractng case of ndependent energy levels. n3 6
9. Classcal Maxwell-Boltzmann Statstcs Physcs 337 At room temperature of order ( 9 ) 3 27 states are accessble: Ths s a lot of nformaton! To avod worryng about detal, we ll artfcally splt all the states up nto bns, wth δe k B T. E 3 Even f there are dvsons, there are stll of w 2 order states wthn each bn, so ths sn t too E 2 hard. If the temperature gets smaller we are free w to choose the energy wdth δe to be a bt smaller, E as desred. Let n # partcles n energy bn E w # states wth energes between E and E + E As wth the localzed case, our goal s to maxmze the statstcal weght ln Ω or, equvalently, maxmze the entropy S k B ln Ω. To do ths we maxmze S over the dstrbuton numbers {n } of partcles occupyng the energy bns E, subject as before to constrants on total partcle number N and total energy E: w N E n (9.3) n E. (9.4) 9.. The Maxwell-Boltzmann Dstrbuton Functon The frst task s to fnd the statstcal weght of each bn; that s, how many ways can we dstrbute n objects among w levels? Consder the th bn. Here are a few possbltes: (n 6, w ) possblty (n 6, w 2) 7 possbltes or dfferent parttons. (n 6, w 3) 8 7 42 dfferent parttons. 2 To see the countng, magne orderng the 2 parttons amongst 8 objects: there are 8 places to choose the frst, and 7 to choose the second, and then we dvde by 2 because the parttons are dentcal. Thus, Ω s the number of ways we can permute w boundares and n partcles, whch can be wrtten compactly as Ω (n + w )! n!(w )! total permutatons (overcountng n n )(overcountng n w ) (9.5) The total statstcal weght s thus the product of the weghts n all states: Ω Ω (n + w )! n!(w )! (9.6) Now we make some approxmatons: 62
9. Classcal Maxwell-Boltzmann Statstcs Physcs 337. Assume w. We have a huge number of states n each bn; as we estmated before, we can choose δe/k B T and stll have w of order. Hence, w w, and we have Ω (n + w )! n!w! (9.7) 2. We assume that the partcles are sparsely dstrbuted among the bns, w n. Ths requres a hgh temperature so that hgh enough energes can be explored. In ths lmt (w + n )! w! (w + n )(w + n )... (w + ) w n. (9.8) Ths leads to the statstcal weght for a hgh temperature (Maxwell-Boltzmann) gas, Ω MB w n n! (9.9) ln Ω [n ln w (n ln n n )] (Strlng s Approx) (9.) If we maxmze S k B ln Ω over the dstrbutons {n }, subject to constant N and E, we obtan (maxmzng ln Ω) [ ( ) ( )] ln Ω α n N β E n E ln w j ln n j α βe j, (9.) n j where α and β Lagrange multplers that have the same nterpretaton as before: α µ/k B T and β /k B T. respectvely enforce constant partcle number N n and total energy E n E. Ths can be solved to fnd the equlbrum dstrbuton, n w e α βe N w e βe w e, (9.2) βe where we have solved for α as before ( n N e α N/( w e βe )). Now, n s the equlbrum occupancy of a gven energy bn. If we dvde through by w then we have the equlbrum occupancy per quantum state, where f MB n w Ne βe Z e α+βe Z (9.3) w e βe (9.4) s, as before, the sngle partcle Partton Functon. It s essentally the same as we had before: n ths case a sum over all quantum states of Boltzmann factors, wth care to count for the degeneracy of each energy level E. Hence the probablty of a gven (quantum) mcrostate occurrng s proportonal to a Boltzmann factor, as wth localzed systems. Note that these 63
9. Classcal Maxwell-Boltzmann Statstcs Physcs 337 statstcs are usually called Maxwell-Boltzmann statstcs, because they have been derved wth a non-localzed gas n mnd, rather than a localzed sold. Gbbs Paradox Compare the statstcal weghts for the localzed and non-localzed systems: N! (localzed) n! Ω (9.5) w n (non-localzed) n! Apart from the degeneracy factor, the non-localzed system has a factor of roughly N! fewer dstngushable states. Recall the smple example at the begnnng of the dscusson. Gbbs noted that the entropy of a gas, f the N! was kept, s not extensve n the partcle number: hence the Gbbs Paradox. Boltzmann argued that the N! should be dropped for non-localzed systems to recover extensvty. Roughly ths s because all energy levels are avalable to all partcles n a gas; whle only prvate energy levels are avalable to each partcle n a sold. That s, the number of states avalable to a gven partcle n a gas s a factor of order N! more than the states avalable to a gven partcle n a sold. Ths extra number of states s accounted for by dvdng by N!, recognzng that n fact all partcles are ndstngushable, n contrast to the sold where the partcles are dstngushed by ther postons. Ultmately one must appeal to experment to valdate such an assumpton: and t turns out to be correct for every experment so far! 9..2 The Classcal Gas Let s say a few rather bref (unfortunately!) words about classcal gases and the use of the Maxwell-Boltzmann dstrbuton.. Free Energy and Partton Functon Frst, we ll calculate the entropy. Substtutng the MB dstrbuton, n Nw e βe nto the entropy S k B ln Ω, we have S k B ln Ω MB k B ln w n n! k B [ w e βe Nk B Z [n ln w n ln n + n ] (9.6a) ln w w e βe Z (ln N + ln w βe ln Z ) ] (9.6b) k B [βe + N ln Z N ln N + N] (9.6c) (usng w e βe /Z and E N E w e βe /Z ). As before, we can dentfy the Helmholtz Free Energy F E T S, ths tme as F k B T N ln Z + N ln N N k B T ln Z, (9.7) where we have denttfed the partton functon as (rememberng Strlng s Approxmaton ln N! N ln N N) Z MB gas ZN N!. (9.8) 64
9. Classcal Maxwell-Boltzmann Statstcs Physcs 337 We can compare ths wth the partton functon for a localzed sold. For the sold we had F sold Nk B T ln Z, or F sold k B T ln Z, wth Z classcal sold Z N. (9.9) Ths s the same N! that we saw before: t makes sure we don t overcount ndstngushable states. We wll fnd that ths factor s also necessary to make the entropy a proper extensve quantty (.e. proportonal to N). 2. Sngle Partcle Partton Functon for an Ideal Gas Let s now study Z for a classcal deal gas. We wrte the sum over sngle-partcle states as follows: Z momenta p postons r α:vbratonal states, rotatonal states, spn levels,... { } w(p, r, α) e p 2 k B T 2m +U(r)+E nt(α) (9.2a) w(p, r, α) degeneracy U(r) external potental (electrc feld, gravty,... ) E nt nternal energy. (9.2b) (9.2c) (9.2d) Ths s an dealzed form where all degrees of freedom are decoupled from one another. Ths s true n many, but not all, cases (for example, the spn degrees of freedom can couple to rotatonal states of datomc molecules). In ths case the partton functon factorzes over all degrees of freedom, and we can wrte: Z Z nt p { w(p) e k B T } p 2 2m r e U(r) 2k B T (9.2) where Z nt carres all the nformaton about nternal degrees of freedom. Now we can convert the sums ntegrals. The result s Z Z nt d 3 p e p 2 2mk B T d 3 r e U(r) 2k B T. (9.22a) h } 3 {{ } } {{ } Z p Z r Z nt Z p Z r, (9.22b) where we have ntroduced Planck s constant h, whch has unts of momentum poston and provdes the densty of states n phase space (poston and momentum) [We wll revst a smlar argument later n the context of Fermons]. Ths can be understood heurstcally from the uncertanty prncple: h δr δp. For postons between r and dr dp dr dp δrδp h r + dr and momenta between p and p + dp there are possble quantum states. Note that, when Boltzmann frst performed ths calculate, h was just a factor needed to get the dmensons correct! Most thermodynamc propertes don t depend on t, but the entropy does, for example. The valdty of the classcal approxmaton requres that the quantum states be sparsely occuped. One way of expressng ths s n terms of the de Brogle wavelength λ db. 65
9. Classcal Maxwell-Boltzmann Statstcs Physcs 337 From the uncertanty prncple, we have p λ db h λ db h 2mk B T, (9.23) where we have used p 2 /2m k B T for a classcal partcle. Ths wavelength must be nconsequental compared to the separaton between partcles for the classcal approxmaton to be obeyed. For ntrogen n ar at room temperature, λ db. nm, and the nterpartcle spacng s of order 3 nm, so that very few quantum states are occuped. 3. Use of the Partton Functon/Boltzmann Factor Exactly as n the non-nteractng case, we can use the Partton functon to calculate thermodynamc quanttes, and use the Boltzmann factor to calculate any average quanttes. The nce thng for the nonnteractng gas, where the energes are all ndependent, s that averages can be taken ndependently. To calculate any average quantty we calculate: { [ ]} d 3 pd 3 r α Q exp p 2 + U(r) + E nternal h Q 3 k B T 2m nt(α) (9.24a) Z nt Z p Z r In many cases ths smplfes. To calculate the mean square momentum, we need { [ ]} d 3 pd 3 r p 2 α p 2 exp p 2 + U(r) + E nternal h 3 k B T 2m nt(α) (9.25a) Z nt Z p Z r e E nt(α)/k B T d 3 re U(r)/k BT d 3 p p 2 e p 2 h 3 2mk B T α nternal } {{ } } {{ } Z Z r nt (9.25b) Z nt Z p Z r d 3 p h 3 p 2 e p Z p 2 2mk B T (9.25c) The nternal and spatal partton functons Z nt and Z r cancelled top and bottom! Ths happens qute frequently. If we need the averages of spns, for example, the only relevant part of the partton functon s the nternal partton functon Z nt, whle f we want the mean poston we only need that part of the partton functon nvolvng the potental U(r), whch appears n Z r. The dstrbuton of momenta that enters Z p s often called the Maxwell-Boltzmann speed dstrbuton. Snce p mv, where v s the velocty and v v s the speed, the 66
9. Classcal Maxwell-Boltzmann Statstcs Physcs 337 dstrbuton can be rewrtten n terms of speeds as P (v) dv 4π Z p h 3 p2 dp e p2 /(2mk B T ) ( ) 3/2 2πmkB T h } 2 {{ } /Z p ( m 2πk B T 4π h 3 m3 v 2 dv e mv2 /(2k B T ) (9.26a) (9.26b) ) 3/2 4πv 2 dv e mv2 /(2k B T ). (9.26c) P (v)dv s the probablty that a partcle has speed between v and v +dv. Z p s just the entre ntegral over p, whch can be done explctly. By constructon ths s a normalzed dstrbuton, P (v)dv. 4. Partton functon and Thermodynamcs n zero external potental The momentum ntegral s doable, and result for Z s Z V Z V N ( ) 3/2 2πmkB T Z nt (9.27a) N! h 2 ( 2πmkB T h 2 ) 3N/2 Z N nt. (9.27b) Now we can calculate the Helmholtz Free Energy, F k B T ln Z, or [ ( ) 3/2 ev 2πmkB T F Nk B T ln Z nt(t )], (9.28) N h 2 where we have used Strlng s Approxmaton to wrte N! (N/e) N. Notce that the free energy (and hence the entropy) s properly extensve and s proportonal to N. If we hadn t had the /N! to account for overcountng, we would have had a term N ln V, whch s not extensve. The entropy can be easly calculated from S ( ) F (we T V won t examne ths here), and the pressure calculated from p ( ) F. For the V T pressure we fnd: p Nk BT V, (9.29) whch s just the deal gas law! Hence we see that nternal degrees of freedom don t contrbute to the equaton of state, but they wll contrbute to thermodynamc [ dervatve ( ) of ] F whch depend on temperature, such as the heat capacty, C V T 2 F/T. T T V Alternatvely, we can wrte C V T ( ) S T 2 F V, usng S ( ) F (these are n T V T 2 T V fact the same expressons, as can be easly verfed). 67
9.2 Quantum Statstcs Physcs 337 9.2 Quantum Statstcs The next task s to tackle quantum statstcs of ndstngushable partcles. If we have a many-partcle quantum state of ndstngushable partcles, we wll generally fnd a set of quantum states, wth correspondng energy levels (whch may be degenerate). The presence of spn (Paul Excluson Prncple) tells us that there are n fact two knds of ndstngushable quantum partcles. The dfference between these s the occupancy of states n the manypartcle wavefuncton. Consder a two-partcle wavefuncton: ψ(r, r 2 ) ψ(2). If we permute two dentcal partcles, the two-partcle wavefuncton becomes ψ(2). Snce the partcles are ndstngushable and dentcal, the probablty densty at any pont, ψ 2 ψ ψ must reman unchanged, where the denotes complex conjugaton. Hence, ψ(2)ψ (2) ψ(2)ψ (2) (9.3) Ths can only be true f ψ(2) e δ ψ(2), where δ s an arbtrary phase factor. There are a few cases + (δ ) Bosons (nteger spn) e δ (δ π) Fermons (half-nteger spn) (9.3) ± (δ, π) Anyons (other fractonal spn: 2 dmensons only!) In three dmensons the phase s ± because f you swap the partcles agan, then you re back to the begnnng, and there can be no dfference. In 2 dmensons, the presence of a magnetc feld means that the drecton you transport the partcles when swappng them matters, and a dfferent phase factor can result, dependng on whether a flux s enclosed upon transportng the partcle n a crcle.. Bosons can have any number of partcles n a gven quantum state. Examples are photons, even-spn partcles, and phonons. 2. Conversely, Fermons can only have sngle (or zero) occupancy of a gven quantum state. Examples are electrons, postrons, quarks, or other partcles wth half-nteger spn (spn, 3, etc). Ths prohbton on multple occupancy follows drectly from the 2 2 mnus sgn pcked up under partcle exchange. 3. Anyons mght seem to be a strange mathematcal object; these objects have fractonal spns, 3,.... However, they have been found n nature, as exctatons above the 3 5 ground state n the Fractonal Quantum Hall Effect! We ll only study Bosons and Fermons here! Because of the dstnct dfference n the occupancy of quantum states, we ll study these n turn. 68
9.2 Quantum Statstcs Physcs 337 9.2. Bose-Ensten Dstrbuton As n the classcal system, we ll assume a set of energy levels {E }, whch we group such that there are, n prncple, w quantum states for each energy level. We wll assume a dstrbuton {n }, whch specfes the occupances of the states for each energy level. The statstcal weght s exactly as we derved for the classcal case, before we started makng approxmatons: Ω Ω (n + w )! n!(w )! We can stll make the approxmaton w w, yeldng (9.32) Ω BE (n + w )! n! w! (9.33) Now, however, we cannot necessarly make the dlute approxmaton w n ; for example, at low temperatures the lower energy levels are lkely to be densely populated. So we are stuck wth ths statstcal weght ( BE stands for Bose-Ensten ). As before, we fnd the dstrbuton whch maxmzes ln Ω, and assume we have large enough numbers that ths dstrbuton descrbes the entre statstcal weght. Usng Strlng s Approxmaton, ln Ω BE [(n + w ) ln(n + w ) (n + w ) (n ln n n + w ln w w )] (9.34) [(n + w ) ln(n + w ) n ln n w ln w ] (9.35) As before, we maxmze ln Ω BE for a gven fxed number N and total energy E, ntroducng Lagrange multplers as before: [ ( ) ( )] ln Ω BE α n N β E n E (9.36a) n j ln (n j + w j ) ln n j α βe j (9.36b) n BE w j j. e α+βe j (9.36c) Ths s the equlbrum dstrbuton functon for partcles whch obey Bose-Ensten statstcs. [Recall that, because de T ds p dv + µ dn, and S k B ln Ω, we have α µ/k B T and β /k B T ]. As we dd wth the Maxwell-Boltzmann dstrbuton, f dvde n j by the degeneracy of each level, w j, we recover the equlbrum occupancy per quantum state, n BE w f BE e α+βe e (E µ)/k BT (9.37) In the Maxwell-Boltzmann case we could solve exactly for α usng the condton n N. Here we can t, so we must leave α (.e. µ) n the dstrbuton. 69
9.2 Quantum Statstcs Physcs 337 9.2.2 Ferm-Drac Dstrbuton Our last case s the statstcs of odd-spn partcles, such as electrons. Paul s Excluson Prncple states that, n a gven system, no two partcles may occupy the same quantum state. For example, f we quantze non-nteractng partcles n a box wth non-penetratng sdes, we fnd waves wth nodes at the wall. The Paul prncple states that two electrons n the box may not occupy, for example, the state wth a sngle half-wave n all three dmensons x, y, and z. As before, we dvde our energy levels up nto artfcal dvsons (bns) accordng to energy E, wth degeneraces correspondng to w quantum states n each bn. Now, we can place ether or partcle n each quantum state. We agan parametrze the dstrbuton of partcles among energy levels by n, the number of partcles wth approxmate energy E occupyng w levels. Now, obvously, there can be no more than n w partcles n each bn. If we consder n 2 partcles n w 3 quantum states, there are three ways of arrangng ths: Hence the statstcal weght s Ω w! n! (w n )!. (9.38) There are w! permutatons of w boxes, and then we dvde by the approprate factors to denote that flled boxes all look the same and unflled boxes all look the same. Hence the total statstcal weght for a gven dstrbuton of dentcal fermons s Ω F D w! n! (w n )!. (9.39) We can maxmze Ω F D, subject to constant partcle number N and energy E (wth Lagrange multplers α and β), to fnd the equlbrum dstrbuton [For the reader: check ths!] n w f F D e α+βe + e (E µ)/k BT +. (9.4) 9.2.3 Summary of Dstrbutons for Gases In all cases, f s the probablty that a gven quantum state wth energy E s occuped, and w s the number of quantum states wth a gven energy E. Ferm-Drac Bose-Ensten Maxwell-Boltzmann (Hgh T, sparse occupancy) w! Statstcal Weght Ω n! (w n )! Occupancy f e (E µ)/k BT + (w + n )! n!w! e (E µ)/k BT e (E µ)/k B T w n n! e E/kBT Z 7
9.3 Ferm-Drac Statstcs: The Electron Gas Physcs 337 Chemcal Potental The chemcal potental must be determned n all cases by the condton N w f. (9.4) In the MB case each state s very sparsely occuped, so that f. To have ths for all energes requres µ to be a large negatve number, µ. In the BE case we must always have a postve f (there s no sense n a negatve number of partcles n a gven state). For ths to be compatble wth the n the denomnator of f BE requres E µ > for all energes E. Snce the lowest possble energy s very close to zero (or less), we requres µ for BE statstcs. In the FD case there are no such restrctons, and n fact µ can be ether postve or negatve. In fact, t s postve at low temperatures and becomes negatve at hgher temperatures, as t should to recover the classcal MB lmt. The postve µ at low temperatures s a reflecton of the Paul excluson prncple: t costs energy to put another partcle nto the system, because all the low energy states are already full. Hgh Temperature Lmt In the hgh temperature lmt both the FD and BE dstrbutons reduce to the MB lmt. Ths s not as straghtforward as t looks, for f we send T and gnore µ, there s obvously a dfference. The key s that µ µ(t ), and n the hgh temperature lmt µ s a large negatve number (equvalently, α s a large postve number). Consder ether a BE or FD gas at hgh temperatures. The dstrbuton s f e (E µ)/k BT ±. (9.42) To obtan the hgh temperature lmt we requre f, as more hgh energy states become occuped. In ths case, the ± wll become neglgble, for all energes, relatve to the exponental, and we thus recover the MB classcal dstrbuton. 9.3 Ferm-Drac Statstcs: The Electron Gas Dstrbuton Functon Let s examne the Ferm-Drac Dstrbuton functon, f F D (E) e (E µ)/k BT + e (E E F )/k BT + (9.43) (9.44) where we have wrtten µ E F n the second lne, as s conventonal for Ferm systems. E F s known as the Ferm Energy. f(e) s the probablty of occupancy of a quantum state wth gven energy E, and s shown here for hgh and low temperatures. f(e).8.6.4.2 T~ δe~k B T T> Energy EE F 7
9.3 Ferm-Drac Statstcs: The Electron Gas Physcs 337 As T, the dstrbuton approaches a step functon, wth all states below the Ferm Energy E F occuped wth equal probablty. Ths s a necessary consequence of the Paul prncple: as the temperature s lowered all partcles fll states begnnng wth the lowest energy states. At zero temperature the lowest energy states are occuped, whle at fnte temperature hgher energy states are occuped. The Ferm energy (or chemcal potental) s a fundamental quantty of a Ferm-Drac system, and underles all dscusson of the physcs of such systems. In the lmt where very few states above the Ferm level are excted, the gas s sad to be degenerate. Densty of States To calculate the Ferm energy we must add up the occupancy of all states to satsfy the gven total partcle number N: N w f w e (E E F )/k BT + (9.45) Now we need to fnd w. We ll label the states by E. The number of quantum states w(e) n the energy nterval between E and E + δe s defned by w(e) g(e) δe. (9.46) The functon g(e) s known as the densty of states; that s, the number of states per energy nterval (we multply by energy δe to obtan a [dmensonless] number of states). g(e) contans all the nformaton about the system: once we have the densty of states we can evaluate the dstrbuton of varous quanttes. For nteractng systems g(e) s dffcult to calculate, whle for non-nteractng systems, where the energy s just that of a free partcle n a box, g(e) s straghtforward. The energy of quantum level n (n x, n y, n z ) s E n h2 8mL 2 (n2 x + n 2 y + n 2 z) h2 8mL 2 n2 (9.47) where n n 2 x + n 2 y + n 2 z and n x, n y, n z, 2, 3,.... Hence δe s related to δn, by δe h2 nδn. (9.48) 4mL2 In three dmensons, the number of states between n and n + δn s the number of states w(n) wthn a sphercal shell n postve n -space (we have to thnk of n as havng three components, n three dmensons). That s, w(e) s, for large n, exactly the volume of ths sphercal shell: w(n) }{{} 2 4πn 8 spn }{{} postve n x, n y, n z 2 δn π 8mL2 E h 2 } {{ } n 4mL 2 δe } h 2 {{ } n δn (9.49a) ce /2 δe, (9.49b) (where the factor /8 restrcts the sum to /8 th of the sphere, or postve quantum numbers n x, n y, n z ). 72
9.3 Ferm-Drac Statstcs: The Electron Gas Physcs 337 n y E + δe E where ( ) 2mL 2 3/2 c 4π. (9.5) h 2 Hence the densty of states s n x g(e) ce /2 (9.5a) In two dmensons w(n) nδn δe, so g(e) constant!! Ths s one reason two dmensonal electron physcs can sometmes be qute pecular. Ferm Energy Now we are ready to calculate the Ferm energy (or, n our prevous notatons, ether α or µ). The Ferm energy E F (or chemcal potental µ) s determned by the constrant on the total number of partcles: N w f de g(e)f(e) w e (E E F )/k BT + (9.52) g(e) de e (E E F )/k BT +, (9.53) where we have converted the sum to an ntegral over energes, and ntroduced the densty of states. fll states to E F f(e), T E F E E F g(e) Ths ntegral s dffcult n general, but s easy n the zero temperature lmt. { (E < E F ()) f(e) (E > E F ()), (9.54) 73
9.3 Ferm-Drac Statstcs: The Electron Gas Physcs 337 where E F () denotes the Ferm energy at zero temperature. Hence we have N E F () EF () ( 3 2 g(e) de ) 2/3 ( N 3 c π N V EF () ) 2/3 h 2 c E /2 de 2 3 ce F () 3/2 (9.55) 8m, (9.56) where N/V s smply the partcle densty. Hence we can wrte the densty of states (n 3D) as g(e) 3N ( ) /2 E. (9.57) 2E F () E F () The Ferm Energy s enormous for typcal electron gases, as are found n metals. Some typcal values are below: Element E F ()/ev T F E F ()/k B v F /ms L 4.7 5.5 4 K.3 6 K 2. 2.4 4 K.85 6 Cu 7. 8.2 4 K.56 6 Au 5.5 6.4 4 K.39 6 E F () k B T at most normal temperatures; ths mples that the electrons occupy hgh energy states, and are movng qute fast (the Ferm velocty v F s the velocty assocated wth the momentum of the hghest energy occuped state). Ths s all due to the Paul prncple. Moreover, snce E F () s so hgh, at hgh temperatures only a few electrons are excted about the Ferm energy. Ths means that there wll be only a slght change to E F f we evaluate the complete ntegral to determne N (Eq. 9.53). To lowest order, the Ferm energy s gven by [ ( ) ] 2 E F (T ) E F () π2 T +.... (9.58) 2 For most Ferm systems at normal temperatures, we can gnore the addtonal temperature dependence. Note that E F decreases wth ncreasng temperature. In the classcal lmt the Ferm energy, or chemcal potental, becomes negatve and the partcles are well-descrbed by Maxwell-Boltzmann statstcs. Classcal Lmt We have seen before that, whle µ F D >, n the classcle MB lmt we must have µ MB <. Let s try and understand what s gong on when µ F D becomes negatve. As T ncreases µ ncreases, untl a temperature s reached at whch µ. The total partcle number s gven by T F N de g(e)f(e) de g(e) e E/k BT +, 3N 2 (E F ()) 3/2 E /2 de e E/k BT +. (9.59) 74
9.3 Ferm-Drac Statstcs: The Electron Gas Physcs 337 whch determnes the temperature at whch µ. Scalng the ntegral, we can wrte N 3N 2 ( ) 3/2 kb T x /2 dx E F () e x + 2 ( ) 3/2 3 kb T.678, (9.6) E F () dong the ntegral numercally. Hence, µ at T T F. Ths s the temperature at whch all states have a non-zero probablty of beng empty (recall that states of order k B T are excted above the zero temperature Ferm level). The chemcal potental changes from postve to negatve at ths pont, because there are no fully occuped states, on average, and hence the temperature s too hgh for the Paul prncple to have effect. Energy In the T lmt (typcal of most metals!) we can calculate the energy of a degenerate Ferm system: E EF 2 5 ce5/2 F ε g(ε)f(ε) dε (9.6a) ε c ε /2 dε (use T lmt of f(e)) (9.6b) 3NE 5 F 3 ( ) 2/3 3N 5 N h 2 4πV 8m. (9.6c) Heat Capacty We are now n a poston to estmate the heat capacty assocated wth a Ferm system. Strctly, ths should be calculated from C V ( ) E, wth E calculated T V accordng to E εf(ε, T )g(ε) dε, so that ( ) f(e, T ) C V E g(e) de, (9.62) T where the dependence of E F on T must be ncluded. We wll content ourselves wth estmatng C V. The ncrease n energy at fnte temperature s due to the exctaton of states above the zero-temperature Ferm level. The energy of these excted states s de k B T. Hence the number of excted electrons s gven by Hence, the energy s N exc g(e)de 3 N 2 E F () 3Nk BT 2E F () 3 2 N T. T F V ( ) /2 E k B T (9.63a) E F () (9.63b) E E(T ) + 3 2 N T T F k B T, (9.64) and the heat capacty for T T F s C V ( ) E 3Nk T V BT/T F. The more accurate answer s C V π 2 Nk B T/(2T F ), only off by a factor of.6 or so. 75
9.4 Bose-Ensten Condensaton Physcs 337 9.4 Bose-Ensten Condensaton 9.4. Ground State Occupancy Let s now turn to the low temperature lmt of an deal Bose gas, rather than an deal Ferm gas. In an deal Bose gas wth no nteractons n a box of sze L, the wavefuncton s and the energy levels are gven by ( π ) 3/2 lπx ψ(r) sn L L E n,l,m mπy sn L nπz sn L, (9.65) h2 8mL 2 (l2 + m 2 + n 2 ), (9.66) The chemcal potental must be less than the ground state energy ε 3h 2 /(2mL 2 ) so that f e (E µ)/k BT (9.67) s always postve. In the lmt T the number of partcles n the ground state becomes qute large. The exponental must be close to one, whch means µ s close to, but a lttle less than, ε, so we can expand the exponental: N e (ε µ)/k BT + ε µ k BT ε k B T µ. (9.68) When T we must have µ ε ; whle for very small T we must have µ just a lttle bt less than ε so that most partcles are n the ground state. At these low temperatures, let s examne the condton on the total number of partcles, that must be satsfed to determne µ: N w e (E µ)/k BT 2 ( ) 3/2 2πm V π h 2 εdε e (ε µ)/k BT, (9.69) convertng nto an ntegral wth the densty of states (the same as for electrons, except for the factor of two for the electron spn degeneracy). For all the hgh energy states µ s a good approxmaton, so we wll be sloppy and evaluate the ntegral at µ (and wth lmts from nstead of ε ). The ntegral can be done for µ, yeldng ( ) 3/2 N V 2.62 2πmkB T. (9.7) h 2 Ths can t be correct! At fxed volume, ths says that the number of partcles depends on temperature! What has happened s that the ntegral verson of the sum doesn t nclude the ground state: notce that the densty of states ε vanshes at ε. The pont s that, when many partcles have condensed nto the ground state the sum s not a smoothly varyng We use the notaton ε to denote a number slghtly smaller than ε 76
9.4 Bose-Ensten Condensaton Physcs 337 functon energy precsely at the ground state. So, we should really approxmate the sum by the ground state and the excted states separately: N e ε /k BT + 2 ( ) 3/2 2πm εdε V π h 2 e (ε µ)/k BT. (9.7) We take µ ε, snce we have already seen that ths must be the case to have a macroscopc number of partcles n the ground state. The ntegral should really be taken from ε, whch s close enough to zero for the densty of states to allow us to change the lmt to zero. Hence the number of partcles s gven by ( ) 3/2 2πmkB T N N + 2.62V. (9.72) At T all partcles are n the ground state. At fnte temperature the number of partcles n the ground state decreases, untl the crtcal temperature T c gven by ( ) 3/2 2πmkB T c N 2.62V. (9.73) Above ths temperature there are essentally no partcles n the ground state. Hence, the number of partcles n the ground state s gven by (T > T c ) N [ ( ) ] 3/2 (9.74) T N T c (T < T c ) h 2 h 2 9.4.2 The Transton The transton looks somethng lke ths, N Nexcted T c µ 2 3 T c 77
9.4 Bose-Ensten Condensaton Physcs 337 where µ ε s essentally zero below T c. For T > T c the chemcal potental can be determned as a functon of temperature by solvng N 2 ( ) 3/2 2πm V π h 2 εdε e (ε µ)/k BT (9.75) for µ. Ths s straghtforward to calculate numercally. We can begn to understand what s happenng by examnng the transton temperature. If recall the de Brogle wavelength λ db we see that the transton temperature s h 2mk B T that temperature for whch the nterpartcle spacng l (V/N) /3 s roughly equal to the de Brogle wavelength. So: as the temperature s reduced, the wavefuncton for each partcle spreads out, and condensaton occurs roughly when the wavefunctons overlap. At ths pont the wavefunctons jump to encompass the entre contaner. The partcles n the condensate occupy the same quantum state, and have the same wavefuncton! How does the transton temperature compare to the lowest energy ε? If there was a sngle partcle, then the ground state would occur at a temperature k B T ε. However, for the Bose gas, the transton temperature can be wrtten as k B T c ( ) 2/3 N h 2 2.62 2πmL N 2/3 ε 2. (9.76) The collectve behavor nduces partcles to drop nto the ground state at a much hgher temperature than one mght have thought! 9.4.3 Expermental Relevance Ths calculaton of BEC assumes an deal, weakly-nteractng gas. In practce most systems nteract strongly at low temperatures, so ths calculaton can be expected to be a cartoon. For example, Helum 4 s a strongly nteractng Bose lqud: t also undergoes Bose condensaton, as well as a transton to a superflud state (the superflud state s a result of nteractons). Another example s a superconductor: n ths case the boson s a composte object called a Cooper par, comprsng two electrons that dance together to form an effectve boson. Another composte boson system that Bose-condenses s a gas of exctons: an excton s an electron-hole par n a semconductor that crcle each other n an effectve hydrogen-atom-lke state. The composte object has even spn, and s a boson. By exctng a cloud of electrons and holes (by blastng a semconductor wth a laser), an effectve Bose Condensaton can be produced from the gas of exctons, before they decay back to the ground state. Exctons are weakly nteractng, and the calculaton above s closely relevant. More recently, of course, Bose Condensaton has been produced n dlute gases of frst rubdum-87 (n 995), and then sodum, lthum, and hydrogen (and probably more). These gases are contaned n an optcal trap and cooled to extremely low temperatures by removng momenta through collsons wth lasers tuned to absorbton frequences of the atoms. The fgure shows the emergence of the condensate upon coolng a gas of rubdum atoms: what s shown s the dstrbuton n velocty space, measured by turnng off the trap at gven temperatures and observng the absorbton and hence the velocty dstrbuton. Above T c 78
9.4 Bose-Ensten Condensaton Physcs 337 there s a Maxwell-Boltzmann dstrbuton of veloctes. Below T c the condensate occupes the lowest energy state, whch s spread out n momentum and poston space, accordng to the uncertantly prncple, and mrrors the shape and sze of the trap. We are actually 79
9.5 Black Body Radaton Physcs 337 observng the macroscopc wave functon! 9.5 Black Body Radaton The fnal topc we ll menton s black body radaton. Black-body radaton refers to the equlbrum dstrbuton of photons radatng wthn a box at a gven prescrbed temperature. Photons are spn-zero partcles, and are thus descrbed by Bose-Ensten statstcs. However, they have a pecular characterstc that dstngushes them from bosons such as Helum-4 or other even-spn partcles: the number of photons s not a conserved quantty. Helum-4 and other atoms are made of matter, whch IS conserved! Expermental evdence that the number of photons s not conserved are many, such as the smple emsson of a photon when an electron drops from an excted state to a lower energy level. Of course a photon carres momentum, whch must be conserved. Let s consder a gas of photons n thermal equlbrum. Bose statstcs apply, except n ths case there s no constrant on partcle number; hence we don t have to ntroduce the Lagrange multpler α, or equvalently the chemcal potental vanshes, µ. Hence the dstrbuton of photons at fnte temperature s gven by f e E /k BT. (9.77) We can calculate the energy n a box of photons much lke we calculate the energy n a box of electrons. In cavty of volume V L 3 the momenta of photons are quantzed. Ths follows from the classcal Maxwell wave equatons, and also from quantum electrodynamcs. The momenta and energes are p h n 2 x + n 2 y + n 2 z 2L (n x, n y, n z, 2, 3,... ) (9.78) ε pc hnc, (energes) (9.79) 2L where n n 2 x + n 2 y + n 2 z and c s the speed of lght. We can wrte the total energy of a box of photons n thermal equlbrum as E ε g(ε) dε ε w f e ε/k BT, (9.8) where we have ntroduced the densty of states g(ε). We can calculate g(ε)dε as before: w 2 }{{} π polarzatons ( 2Lε hc g(ε) dε. 8 4πn2 dn (sphercal shell n n-space) (9.8a) ) 2 2L dε hc π ( ) 3 2L ε 2 dε. (9.8b) hc (9.8c) 8
9.5 Black Body Radaton Physcs 337 Hence we can wrte the total energy as E 8πV (hc) 3 ε 3 dε e ε/k BT. (9.82) The energy densty per unt energy, u(ε), s defned by U/V u(ε) dε and s gven by u(ε) 8π (hc) 3 ε 3 e ε/k BT 8πc 3 f 3 e hf/k BT, (9.83a) (9.83b) where the second form s n terms of frequency f, usng ε hf. Ths s the spectrum of energes nsde a cavty n thermal equlbrum at temperature T. If you measure the spectrum of energes comng out of your oven by countng the photons as they emerge from a hole, ths s what you ll fnd. Ths s called the Planck Spectrum. Ths s also the spectrum of energes emtted by a black box,.e. a box that s perfectly absorbng, hence non-reflectng or black. Consder placng a black body n equlbrum wth a hole n a box n thermal equlbrum. If the two are to reman at the same temperature, the amount emtted by the hole must be balanced by the emsson from the black body. If the amounts dffer, then they can t reman n thermal equlbrum. Ths argument holds for any part of the spectra, snce one could magne restrctng the radaton transfer to certan frequences usng flters. If the blackbody s NOT perfectly absorbng, then some s reflected back n addton to that emtted, and the spectrum of the black body emsson s where e < s the emssvty. u B B(ε) eu P (ε), (9.84) Cosmc Background Radaton The unverse t a pretty good black box! The Nobel prze was awarded for measurments of the black body spectrum of the unverse n 965 (by accdent), whch turns out to be sotropc to a very good approxmaton, and characterstc of a temperature T 2.735 K. The characterstc wavelength s n the far nfrared (about a mllmeter). Temperature-Dependence We can ntegrate the Planck spectrum over all energes (frequences) to obtan the temperature dependence: U V 8π ε 3 (hc) 3 8π(k B T ) 4 hc 3 e ε/k BT x 3 e x. (9.85a) (9.85b) Hence the total energy radated scales T 4. Ths forms the bass for for the Stefan-Boltzmann Law (whch we won t derve), whch says that the power radated per unt surface area s P/A σt 4, wth σ the Stefan-Boltzmann constant. 8