Dstrbuto Boltzm he gves d dstrbuto Boltzm s hs bth het wth cotct system tht probblty stte prtculr be should temperture t. low At system. sttes ll lbel

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1 Dr Roger Beett Rm. 3 Xt Lecture 19

2 Dstrbuto Boltzm he gves d dstrbuto Boltzm s hs bth het wth cotct system tht probblty stte prtculr be should temperture t. low At system. sttes ll lbels r chce y hve sttes lowest oly temperture rsed s temperture As occuped. beg to lkely more d more become sttes lyg hgher occuped. be ll bth, het wth cotct cse, ths I be to lkely eqully ot refore re mcrosttes populted. r r e p 1 r r e

3 he Boltzm p Dstrbuto r sully re re huge umbers mcrosttes tht c ll hve sme eergy. hs s clled d egeercy. I ths cse we c do dvdul e ergy level dvdul m crostte. r 1 e r e r our summtos bove over rr th sum over ech r 1 p( r ) g( r ) e g( r ) e r ech he summto s ow over ll dfferet eerges r d g(r ) s umber sttes possessg eergy r. he probblty s tht fdg system wth eergy r. r

4 esembles Etropy clled s bth het system embedded Our ow ts system o solted (our esemble cocl termed s 16 rom Lecture f l mcrococ esemble). solted he W l crococ m hs esemble probblty tht so eergy terl defed s mcrostte prtculr system fdg mcrostte. or y s sme system fluctutes eergy bth het I prtculr y fdg probblty d equl. ot s crostte m ow clculte we C derve hece system d such for etropy propertes? from sttstcl vrbles rmodymc

5 Etropy cocl esemble Embed our system het bth mde up (M- 1) replc subsystems to oe were terested. Ech subsystem my be oe my mcrosttes. he umber subsystems t h mcrostte s. he umber wys rrgg 1 systems µ stte 1, s ystems µ stte, 3. M! W!

6 Etropy cocl esemble S k p l p hs s geerl defto etropy d holds probbltes ech dvdul mcrostte re dfferet. If ll mcrosttes re eqully probble p = 1/W ( m crococl esemble) eve f S k p l p k W 1 1 W 1 l W k l W Whch brgs us cely bck to Boltzm relto

7 Etropy S k p l p cocl p 1 e esemble e he geerl defto etropy, combto wth Boltzm dstrbuto llows us to clculte rel propertes system. l p l S k p l p k p l 1 S p k l p k l

8 Eergy Free Helmholtz Ū eergy terl vlue verge s system. (Ū vlue verge s S) free Helmholtz ergy, e F brefly we tht stte fucto s hs. sttstcl to cetrl s It lectures. erler metoed mechcs. fucto Prtto he result our ppered s h t fctor. ormlsg mere th more much be to seems world mcroscopc lkg brdge s cts d eergy free to (qutum sttes) mcrosttes system. propertes scle lrge ll to hece k S l S F l

9 Helmholtz Free Eergy F S l F s stte fucto. C we ow clculte rml propertes our system? Lets gore Ū s verge d ssume = Ū. We c ow use our rmodymcs kowledge to defe our rmodymc vrbles. For smll r eversble chge: F S df d ds Sd df dq PdV ds R df ds PdV ds Sd Sd PdV Sd

10 Helmholtz Free Eergy df PdV Sd F S l Employg prtl dfferetls we fd: - F V P P F V l k V F V S S F We hve tmtely relted eerges mcrosttes system to pressure d etropy. V k l V

11 R espose free eergy fuctos d dervtves M odul elstcty re stress/str or force/ut re dvded by frctol deformto (Lecture 8): - K V P V V F V Most usefully volume: - C V het cpcty t costt Q V S V F V

12 Eergy Me Ū eergy terl vlue verge s fluctutes eergy terl ctul he system. temperture defed hve we becuse bth. het fluctutos? re ow bg H y re mportt? V S F ) l ( l V l l l l

13 Eergy Iterl Fluctutos by vlue me from deprtures mesure We devtos stdrd se u - wth would we s dstrbuto. y ) ( V r V V r e e C

14 Eergy Iterl Fluctutos ) ( 1 e e C V r V r C V ) (

15 Fluctutos Iterl Eergy W e hve clculted vrce reltve / Ū s most use: - fluctuto C V But Ū d CV re extesve propertes proportol to sze system ~ N, umber prtcles system. tesve d s sze depedet. 1 N

16 Fluctutos Iterl Eergy or typcl mcroscopc systems wth ~10 fluctutos / Ū ~ F 3 1 N prtcles Fluctutos re ty d hece d Ū c be cosdered detcl for ll prctcl purposes. Mcroscopc systems het bth effectvely hve eergy determed. Smlr reltoshps c be foud for or reltve fluctutos propertes mcroscopc systems. r

17 Dr Roger Beett Rm. 3 Xt Lecture 0

18 S ummry Sttstcl Mechcs M crostte he stte system defed mcroscopclly complete descrpto o tomc scle. M crostte he stte s tystem mcroscopc sze specfed by few mcroscopclly observble quttes oly. Sttstcl Weght (W or ) o f mcrostte s umber mcrosttes c ompsg mcrostte. Postulte equl pror p robbltes for solted system defte mcrostte, W mcrosttes comprsg ths mcrostte occur wth equl probblty. E qulbrum Postulte for solted mcroscopc system, defed by,v,n (whch re fxed) d vrble prmeters, equlbrum correspods to those vlues for whch sttstcl weght W(,V,N, ) tts ts m xmum.

19 S ummry Sttstcl Mechcs Boltzm defto Etropy S(, V, N, ) k lw (, V, N, ) emperture defto Pressure defto Geerl defto Etropy S k V, N P 1 S(, V, N ) p l p S(, V, N ) V, N

20 S ummry Sttstcl Mechcs B oltzm Dstrbuto p s probblty tht t temperture s stte wth eergy. P rtto Fucto p summed 1 e e over ll mcrosttes system Me Eergy l H elmholtz free eergy F = - S

21 mterls Prmgetc our develop to model smple s hs proves t but Mechcs stt. uderstdg sgfct. very be to hve whch toms cot Prmgets ( momets dpole getc m ot do hese ). to respod c but or ech wth terct ( mgetc exterl ppled B. feld ) depedet s thought be c dpoles he crystl rrged mgets br (tomc) lttce.

22 mterls Prmgetc (tomc) depedet s thought be c dpoles he lttce. crystl rrged mgets br hk feld B I e dpol ch e sttes two oe exst c ( feld wth lged p u p s ( lged t or ) sp ow d. ) eergy hve dpoles up p S - + dow sp, B. B mgetsto how out fd to wt We feld. ppled d temperture o depeds mterl

23 mterls Prmgetc or ech depedet re dpoles ll As verge t look to eed oly relly we ll use c We dpole. oe propertes bth het s dpoles r o Cocl esemble. d mcrosttes possble two hve We lredy erges e Fucto Prtto et g 1 for dpole. sgle our x B e e e k B B h cos cosh B x

24 Prmgetc mterls We c ow clculte versus sp dow. p 1 e p cosh probblty sp up 1 x x e p e x 1 cosh x Probblty Sp lged ( = - B) Sp tlged ( = + B) X= B/

25 Prmgetc mterls We c ow clculte me mgetc momet our dvdul dpole. x 1 p e p e cosh x cosh x p p ( e cosh x 1 x Ad hece me eergy dvdul x th x B th e x x ) cosh x sh x

26 mterls Prmgetc dpole dvdul for ecessry vlues hve We or ll terct ot do dpoles our becuse d smlrly. behve must dpoles : eergy d hs refore N dpoles sold A drecto ( momet mgetc me Ad : feld) ppled tht ote N = - B M x N N M h t x NB N h t

27 mterls Prmgetc mgetsto he L momet mgetc s volume ut per temperture hgh or feld wek lmt I d <<1 x h t x : so x V x N V M L th B V N V Nx L

28 Prmgetc mterls Probblty Sp p Sp Dow Mgetsto M getsto / N /V X= b/

29 Ref, Fgure Sturto M getzto C ure s Lw P rmgetsm: M 0 χh χ N ( 0 μ )/() ure Susceptblty C

30 P rmgetc mterls C ure s Lw he susceptblty s mgetsto per ppled feld testy whch for smll mgetstos s gve by H = B/ 0. 1 L 0 H N V I s Cure s lw. It holds very well for prmgetc mterls wth wekly terctg dpoles. So well tht c be used for temperture clbrto for exmple Cerum Mgesum trte obeys cures lw to 0.01K! t

31 Cure Lw M 1 NgBgμ BkB Ng μ B 4k B CH where C = Ng B /(4kB) s Cure costt Sce mgetc susceptblty s defed s = M/ H Cure Lw results: C

32 v s. plot 1/ = /C gves strght le grdet C-1 d tercept zero = C gves strght le prllel to X- xs t costt vlue showg temperture depedece mgetc momet.

33 C ure L w 1. As sgle electros re mgets, f you plce m mgetc f eld, y ll lg wth feld. However eergy dfferece b etwee lged wth feld d gst feld s << rml eergy t room temperture. Get r dom oretto equl popultos lgmet wth/gst feld.. As you lower, eergy dfferece becomes more mportt d populto chges more lg t- prllel to feld. 3. o expl ths behvor, Cure veted prmeter clled M getc Susceptblty, χ, whch s mesure how ttrcted smple s to mgetc feld. Normlly mesured s ppret mss crese. As more electros lg t- prllel to feld t low temperture, χ creses. I fct χ s versely proportol to t he feld: ths s C ure L w ( 1/ χ ) = C C = he Cure Costt

34 Dr Roger Beett Rm. 3 Xt Lecture 1

35 Cpcty Het upo depeds tht eergy hs sold prmgetc Our emperture t het mgetc hve must refore t cpcty. t cpcty het mesures oe Expermetlly H. testy feld mgetc costt mgetc wekly s compoud prmgetc our s H = B/ 0. costt lso s B so B NB x NB N h t h t B NB dq C H H H h t

36 Het Cpcty C H NB th B Nk B sech B 0.4 Mgetc Het Cpcty C H / N k /X = / B

37 Schottky Het C H Cpcty B Nk sech B hs s fct geerl result for het cpcty y two level system. Oe exmple we hve lredy ecoutered s S chottky defect. 0.4 Mgetc Het Cpcty C H / N k /X = / B

38 Sold Prmgetc Isolted treted prevously oe problem to smlr Very eergy totl (fx) costr we Here bth. het system. solted sp dpoles, totl N - d fel B ppled wth lged up. fucto clerly s umber gve to correspods () eergy gve A weght: sttstcl wth toms up p s - ) ( ) ( ) ( N B B B N! )! (! ) ( N N W

39 Isolted Prmgetc Hece Etropy s gve by: - Sold S( ) k lw ( ) k l ( N N! )!! or lrge N (~10 F 3 c use Strl g S hould look fmlr t s S chottky Vccy formto. ) s Approxmto S( ) k N l N l ( N )l( N ) 1 S S( ) ( ) sme S( ) problem s d d 1 S( )

40 Sold Prmgetc Isolted : toms up sp desty fd to, for olve S - up sp fdg probblty to detcl s hs lecture! lst wth strted we bth het tom N B k S B l ) ( 1 1 x e N 1 1 d d S S S ) ( ) ( ) ( 1 ) ( ) ( N B

41 Negtve emperture From our prevous dervto we hd 1 k B l N k B l N If < N/ more th hlf dpoles re t- prllel d becomes egtve! Wht s egtve temperture? We kow tht s temperture popultos sp-up d sp- dow oly become equl! A egtve temperture stte must refore be hotter th = s ts s more eergetc stte system.

42 emperture Negtve sttstcl d etropy temperture egtve For E. fuctos decresg be must weght fte stte system possess f hppe c hs xmum eergy m wth prmget our s such =N. B prtculr ll for hppes ths where exst systems No eerges electroc eerges, vbrtol (I.e. spects or spect such oe f However, eerges). mgetc d so ors, from decoupled effectvely subsystem s cosdered be subsystem my tht terct, ot do y equlbrum beg equlbrum wthout terl rech to ors. wth where systems mgetc for cse s hs qucker much s sps tomc betwee tmes relxto vbrtol d sps betwee relxto th lttce. modes

43 Negtve emperture I prmget lowest possble eergy = -N B d hghest =+ N B. hese re b oth uque mcrosttes so S=0. I betwee we postve eergy c oly rech sttes wth wth egtve temperture. s System Eergy 0.5 System Eergy E ergy / N B E ergy / N B emperture / emperture