A. La Rosa Lecture Notes Portland State University APPLIED OPTICS. Semiconductor optoelectronic devices

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1 A. La Rsa Lctu ts Ptlad Stat Uvsty APPLID OPICS Smcduct ptlctc dvcs Quatum Statstcs pht ad ms patcls. Quatum Statstcs phts ad ms patcls. umb stats cmpatbl wth a gv gy Cas dstgushabl patcls.. Cas: Systm wth a hug umb patcls =. A hug systm thmal ctact wth a small systm Dt th abslut tmpatu h ltzma act.4 Rsv tactg wth a systm cmpsd multpl patcls.4.a Avag umb patcls th patcula -th stat.4. xampl: Statstcs llwd by pht patcls lctmagtc adat qulbum sd a cavty at tmpatu.4.c xampl: Statstcs llwd by m patcls s s Avag umb phts gy s s sd th cavty. Ma umb lcts th -lct lvl, gy, wh th -lct systm s thmal qulbum at tmpatu. Rcs R. Ashct ad Mm, Sld Stat Physcs, Sauds Cllg Publshg. dd las statstcal cdt: G. H.. hmps, Physcs smcduct Las Dvcs, Jh Wly & Ss A 7 45

2 a. umb stats cmpatbl wth a gv gy Cas dstgushabl patcls Csd 4 tactg -dtcal patcls u spgs, xampl. h gy that ach spg ca ta a,,,, Lt s assum that th ttal gy th u-patcl systm s ttal =5. A C D g. u dstgushabl tactg patcls whs ttal gy ttal s cstat. h tact allws ach mmb t dvdually adpt dt gy-lvl stats, whch d dt stats th ttal systm that a cmpatbl wth th xd ttal gy. Lt s call, umb dt stats avalabl th ttal systm t adpt whl stll bg cmpatbl wth th xd ttal gy. I u patcula cas, w wat t d ut hw may dstbuts th systm u patcls ca adpt whl stll bg cmpatbl wth th ttal gy 5. 5 =? Dstbut : O patcl th gy lvl-5 ad th patcls th gy lvl- g. Patcula ppulat gy-lvl dstbut cmpatbl wth th cdt ttal =5. A CD h a u pssbl dt stats cmpatbl wth th patcula ppulat dstbut-; thy a xplctly shw th gaph blw.

3 A C D A CD ACD AD AC CD g. u pssbl stats th systm ca adpt ud th ppulat dstbut-. hus, th multplcty th dstbut- s u ths cas. tc th gu abv that: A pmutat patcls lcatd at dt gy lvls lads t a w stat. I th thd stat, xampl, pmutg A ad C ds lad t a w stat; th latt s th st stat th gu. A pmutat patcls that a at th sam gy lvl, ds t lad t a w stat. I th thd stat, xampl, pmutg A ad D ds t lad t a w stat. I gal th a dt typs dstbuts cmpatbl wth th cdt that th ttal gy th systm s ttal =5s g. 4. g. 4 Dt dstbut cmpatbl wth th cdt ttal =5 Dstbut # # # #4 #5 #6 {,,,,, } {,,,,, } {,,,,, } {,,,,, } {,,,,, } {,,,,, } tc, ach dstbut ca b spcd by gvg th ppulat ach stat th mat {,,,, 4, 5 } Multplcty umb dt stats asscatd t ach dstbut: 4 4 t: h multplcty ass bcaus th atcls a dstgushabl thws th multplcty wuld b ach ths cass. h a ttal = 56 dt stats th systm ca adpt, ach cmpatbl wth th stct that th ttal gy s ttal =5.

4 hat s, w hav ud, 5 = 56 Pbablty ach dstbut t ccu W pstulat that ach ths 56 stats hav qually pbablty t ccu. Hc, w mtd th systm a suct lg tm, w ca assg th llwg pbablts t ach dstbut, Dstbut # # # #4 #5 # Cas: Systm wth a hug umb patcls = wh ~. A hug systm thmal ctact wth a small systm-a Lt s assum th gy th ttal systm s W a tstd calculatg th pbablty that th small systm s ud th stat gy. gy xchag Hug systm A Small systm A g. 5 Systm-A thmal ctact wth a hug systm calld h sv A. tc th P wll b pptal t th umb stats avalabl t th sv, Dt pbablty: I ut a ttal umb ttal vts.. thwg a dc, th umb tms w bta s, th th pbablty P s dd as th at P ttal. ut th ccpt pbablty t acqu a usul mag th ttal tal umb has t b vy lag. Symblcally, P x. x lm ttal ttal ttal

5 P ~ 4 Sc << w wll xpad l aud th valu. l l l... 5 Dt th abslut tmpatu l s a cstat dpdt. l chaactzs th sv; t s actually vlvd th dt th abslut tmpatu th sv. l Dt th abslut tmpatu A 6 h ltzma act Rplacg 6 5, 7 4 gvs l l 7 P P ~, whch ca b put as, C 8 wh C s a cstat pptalty dpdt. h cstat C ca b dtmd m th malzat cdt, P 9 P C xpss 8 bcms, C

6 P ' '.4 Rsv tactg wth a systm cmpsd multpl patcls Systm A s cmpsd patcls. A patcula stat- th systm A s spcd by, {,,, 4, 5, } wh s th umb patcls th stat Hug systm A Rsv gy xchag Small systm A g. 5 Systm-A cmpsd 4 patcls thmal tact wth a sv. h gu shws systm A a abtay patcula stat gy..4.a Avag umb patcls th patcula -th stat P 4 wh s th s th ccupat umb th -th stat wh th ull systm s th -th cguat: {,,, 4, 5, }. Usg btas,

7 Usg ' ' ' ' ' '... '... Captalzg th act that a stat gy s spcd by th ppulat umb ach stat {,,, 4, 5, } {,,...} { ', ',...}... ' ' xampl: Statstcs llwd by pht patcls lctmagtc adat qulbum sd a cavty at tmpatu All matt at tmpatu gat tha abslut z mts thmal adat. hmal adat s lctmagtc adat gatd by th thmal mt chagd patcls. A wd spctum adat s bsvd v at a sgl tmpatu. Radat gy K 5 K K 4 qucy 4 Hz g 6 At a gv tmpatu, a bdy mts lctmagtc adat. h spctal thmal adac shws stg dpdc th bdy s tmpatu.

8 Rsv Assmbly phts g. 7 Assmbly phts thmal qulbum sd a cavty at tmpatu. Lt: Phts dt qucs a ctuusly absbd a -mttd by th walls. Rght: A small aptu allws a small act phts t cm ut ad aalyz ts qucy ctt. Lt s csd lctmagtc adat assmbly phts that xsts thmal qulbum sd a clsu at tmpatu. I ths stuat phts a ctuusly absbd a mttd by th walls A patcula stat- th pht gas s spcd by {,,, 4, 5, }, phts qucy, phts qucy Cas: Ovtm th pht gas systm adpts may stats. W wat t calculat th avag ppulat lvl-. {,,...} { ', ', '...}... ' '... {,...} { ', '...}... ' '... 6 us ut, ths s a patcula cas pht statstcs th s stct t th valus ay ppulat phts

9 ay gy sc th wall th cavty adly absb ad mt gy 7 phts. h cavty acts as a lmtlss sv phts. xchag Hug systm A Rsv At tmpatu gy xchag Small systm A g. 8 Altatv pstat a gas pht thmal ctact wth a sv s als g.7. Phts a pstd pctally by balls. h umb pht ach lvl gy s s ulmtd. hst s, th s stct t th umb phts a gv gy lvl. Hc all th summats 6 u wthut stct up t ty. ' ' {,...} { ', '...} ' ' O th tms th umat cacls wth a cspdg th dmat, ' ' Avag umb pht gy 8 h pcdu culd hav b cad ut phts ay patcula qucy; thus

10 s s Avag umb phts gy s s sd th cavty xampl: Statstcs llwd by m patcls h m-dac Dstbut ad th Chmcal Pttal gu 9 shws a systm-a tactg wth a hug thmal sv. h ccpt thmal sv s a dalzat cvyg that matt what amut gy s tasd t ta m th sv ts tmpatu mas cstat. Paul s xclus pcpl I a systm cmpsd by m patcls lcts a a xampl gy xchag ly patcl s allwd t ppulat a patcula stat. Hug systm A Rsv gy xchag Small systm A g. 9 Systm-A cmpsd =4 lcts thmal tact wth a sv. h gu shws systm A a patcula stat gy h spcs th patcula stat. W spcalz t th cas whch w hav a systm cmpsd -tactg lcts; thy a dstbutd amg th dt pssbl -lct stats. gu shws tw pssbl patcula cguats cspdg t th cas =. Sc may cguats a pssbl, lt s call th gy asscatd t a gv -cguat stat.

11 gy xchag gy g. w pssbl statay stats th -lct systm. smplcty w hav chs =. h pbablty dg a -lct systm a patcula stat gy s gv by, P I th dmat th sum s all v th pssbl stats. h dmat s w as th patt uct ad t s latd t th Hlmhltz gy = U-S wh U s th tal gy ad S th tpy by, Dt hus, P - W ca mt th sub dx th. P W hav t cpat th Paul s xclus pcpl, whch allws at mst lct t ccupy ay sgl lct lvl. Accdgly, t cstuct a -lct stat must ll dt -lct lvls. ach -lct statay stat ca b spcd by lstg whch th -lct lvl a lld that stat. xampl [,5,7] ad [,,6] spctvly th xampls gv g. abv, =.

12 A vy usul quatty t calculat s, th pbablty t d a lct th patcula -lct lvl, wh th -lct systm s thmal qulbum. P 4 Summat xtds all v th -lct stats whch th s a lct th -lct lvl. tc, P 5 Summat xtds all v th -lct stats whch th s lct th -lct lvl. tc, y tag ay +-lct stat whch th s a lct th -lct lvl, w ca cstuct a -lct stat whch th s lct th lvl-, by smply mvg th lct th -th lvl, lavg th ccupat all th th lvls ualtd. uthm, Ay -lct stat wth lct th -lct lvl ca b cstuctd m ust +-lct stat wth lct th lvl. vdtly, th gs ay -lct stat ad th cspdg +-lct stat d by ust. hus, th st gs all -lct stats wth th lvl uccupd th stats dd 5 abv, s th sam as th st gs all +-lct stats wth th lvl ccupd lt s call thm th stats pvdd that ach gy th latt st s ducd by. R. Ashct ad Mm, Sld Stat Physcs hat s t say, P = P γ Summat xtds all v Summat xtds all v th -lct stats th +-lct stats whch th s lct whch th s a lct th -lct lvl th -lct lvl. Hc, 5,

13 χ χ ε P 6 Summat xtds all v th +-lct stats whch th s a lct th -lct lvl. Usg, Sc th summat xtds all v th +-lct stats whch th s a lct th -lct lvl, btas, - 7 h chmcal pttal at tmpatu s dd by = + 8 xpss 7 tas th m, 9 Sc s th d, addg lct wll hadly alt th pbablty. hus,

14 Hc, Pbablty t d a lct th patcula -lct lvl, gy, wh th -lct systm s thmal qulbum at tmpatu. tc s als th ma umb lcts th -lct lvl, = - Ma umb lcts th -lct lvl, gy, wh th -lct systm s thmal qulbum at tmpatu. Implctly, th chmcal pttal cas mat. I act, I subsqut xpsss, th xplct c t s dppd t wll b mplctly assumd. m-dac dstbut = > > g. h m-dac dstbut.