CLASSICAL STATISTICS OF PARAMAGNETISM

Transcription

1 Prof. Dr. I. assr Phys Dc_0 CLASSICAL STATISTICS OF PARAMAGETISM Th most famous typs of Magntc matrals ar: () Paramagntc: A proprty xhbt by substancs whch, whn placd n a magntc fld, ar magntd paralll to th fld to an xtnt proportonal to th fld (xcpt at vry low tmpraturs or n xtrmly larg magntc flds). () Frromagntc: A proprty, xhbtd by crtan matrals, alloys, and compound of th transton (ron group), rar-arth, and actnd lmnts, n whch th ntrnal magntc momnts ontanously organd n a common drcton; gvs rs to a prmablty consdrably gratr than that of vacuum, and to magntc hystrss. () Damagntc: Havng a magntc prmablty lss than on; matrals wth ths proprty ar rplld by a magnt and tnd to poston thmslvs at rght angls to magntc lns of forc. Typ Exampl Suscptblty ( ) Prmablty Commnts Paramagntc Alumnum postv >, (.00) Tmpratur dpndant Frromagntc Iron, nckl 0 4 Magntc doman, hystrss Damagntc bsmuth ngatv <, (0.999) 0 Magntc suscptblty rprsnts th rons of a systm to th xtrnal fld. Hystrss mans th dpndnc of th polaraton of frromagntc matrals not only on th appld (magntc) fld but also on thr prvous hstory. Magntc doman, a rgon of frromagntc matral wthn whch atomc or molcular magntc momnts ar algnd paralll. Prmablty, a factor, charactrstc of a matral, that s proportonal to th magntc nducton producd n a matral dvdd by th magntc fld strngth; t s a tnsor whn ths quantts ar not paralll. Consult Phys-0 book for mor dtals dscusson.

2 Prof. Dr. I. assr Phys Dc_0 Typ of Magntsm Suscptblty Atomc / Magntc havour Exampl / Suscptblty Damagntsm Small & ngatv. Atoms hav no magntc momnt Au Cu -.74x x0-6 Paramagntsm Small & postv. Atoms hav randomly orntd magntc momnts β-sn Pt Mn 0.9x0-6.04x x0-6 Frromagntsm Larg & postv, functon of appld fld, mcrostructur dpndnt. Antfrromagntsm Small & postv. Frrmagntsm Larg & postv, functon of appld fld, mcrostructur dpndnt Atoms hav paralll algnd magntc momnts Atoms hav mxd paralll and antparalll algnd magntc momnts Atoms hav antparalll algnd magntc momnts Tabl : Summary of dffrnt typs of magntc bhavour. F ~00,000 Cr 3.6x0-6 a frrt ~3

3 Prof. Dr. I. assr Phys Dc_0 Modl: Consdr dntcal, locald (.. dstngushabl), practcally statc, mutually nonntractng and frly orntabl dpols at absolut tmpratur T and placd n an xtrnal magntc fld H pontng along drcton. Thn th torqu actng on th dpol s gvn by: H H sn, and th (magntc) potntal nrgy can b wrttn as:. H H cos, whr θ s th angl btwn th magntc fld and th dpol and μ s th magntc dpol momnt. μ H d H snd H cos = H, / / - Qualtatv Dscrpton: A non-ntractng atom wth magntc dpol momnt ( s postv) could b pont thr paralll or ant-paralll to an xtrnal magntc fld H. At tmpratur T, w hav th quston: Q: What s th man magntc momnt H (n th drcton H) of such an atom? A: Thr ar two possbl stats, and thy ar: stat condton Magntc nrgy probablty (+) H (lowr H H ) P C (-) H (hghr H ) P C H H s lowr nrgy s hghr nrgy H atom s mor lkly to b found atom s lss lkly to b found Dfn H (magntc) potntal nrgy H kt (thrmal) translatonal nrgy s a dmnsonlss paramtr whch masur th rato of th magntc nrgy tnds to algn th magntc momnt, to th thrmal nrgy randomly orntd. Thn kt, whch, whch tnds to kp t Cas T P P 0 s randomly orntd (dsordr) Cas T P, P 0 H (ordrd) - Qualtatv Dscrpton: th man magntc momnt H s gvn by: P ( ) P H tanh P P Th magntaton M o, or man magntc momnt pr unt volum, s thn n th drcton of H and rads n H for << (hgh tmpratur) M o nh n for >> (low tmpratur) whr n s th total numbr of magntc atoms pr unt volum. Th abov rsults agr wth n th qualtatv dscrptons. Hr, s th ''magntc suscptblty''. Th rsult s k T T H H H cos H d 3

4 Prof. Dr. I. assr Phys Dc_0 known as Cur's law. At vry low tmpratur bcoms ndpndnt of H and qual to th maxmum (or ''saturaton magntaton ) magntaton whch th substanc can xhbt. Saturaton magntaton mans complt algnmnt of th magntc dpols n th fld drcton. M o n M o Magntc saturaton Fgur: Total magntc momnt of a n ½ paramagnt. Classcally, th numbr of dpols, dn, havng axs wthn th sold angl d lyng btwn two hollow cons on sm-angls and s gvn by: d H cos H cos dn C d C d C sn d Whr C s a constant. Each on of ths dpols contrbuts a componnt of magntc momnt cos to th magntaton, whr as th componnts prpndcular to th fld drcton cancl ach othr. Hnc th avrag componnt of th magntc momnt of ach atom along th fld drcton multpld by th numbr of atoms pr unt volum,, gvs th magntaton,.., M cos H cos dn d cos cos sn 0 0 H cos dn sn d 0 0 H (magntc) potntal nrgy Lt us dfn th rato H, and kt (thrmal) translatonal nrgy y cos sn dy, thrfor: y y dy M coth y dy L( ) T larg, hgh dsordr T small, low dsordr H largr mor ordrd M ( ) for << (hgh tmpratur) o L H 0 3 H H for >> (low tmpratur) H 0 Exampl: A -monatomc oltmann dal gas of n ½ atoms n a unform magntc fld, n addton to ts usual kntc nrgy, a magntc nrgy of pr atom, whr s th magntc 4

5 Prof. Dr. I. assr Phys Dc_0 momnt. (It s assumd that th gas s so dlut that th ntracton of magntc momnts may b nglctd.) a- What s th partton functon of th systm? cosh( ), E tanh( ) cosh( ) and th total nrgy U E tanh( ). In summary: Quantty Partton functon Formula cosh( ) cosh ( ) Hlmholt fr nrgy F k T ln( ) k T ln{cosh( )} Entropy Intrnal nrgy F S k ln cosh( ) tanh( ) T V, ln U H tanh( ) Hat capacty U F k CV T T H T cosh V, Total magntc momnt F M ( n n tanh( ) H V, otc: U M H V, Fgur: Hat capacty of n ½ paramagnt. (Schottky anomaly) C k Fgur: Total magntc momnt of a n ½ paramagnt. M n T 5

6 Prof. Dr. I. assr Phys Dc_0 Commnts: - For th ntrnal nrgy: At low T, all th ns ar algnd wth th fld and th nrgy pr n s clos to H. Howvr as ncrass, thrmal fluctuatons start to flp som of th ns; ths s notcabl whn s of th ordr of H. As T gts vry larg, th nrgy kt P H tnds to ro as th numbr of up and down ns bcom mor narly qual. C, P so t nvr xcds on. W can say that: at hgh tmpratur, th thrmal nrgy s suffcnt to dsordr th magntc dpol orntaton. - Th hat capacty tnds to ro both at hgh and low T. At low T th hat capacty s small bcaus kt s much smallr than th nrgy gap H, so thrmal fluctuatons whch flp ns ar rar and t s hard for th systm to absorb hat. Ths bhavor s unvrsal; quantaton mans that thr s always mnmum xctaton nrgy of a systm and f th tmpratur s low nough, th systm can no longr absorb hat. Th hgh-t bhavor arss bcaus th numbr of down-ns nvr xcds th numbr of up-ns, and th nrgy has a maxmum of ro. As th tmpratur gts vry hgh, that lmt s clos to bng rachd, and rasng th tmpratur stll furthr maks vry lttl dffrnc. Ths bhavor s not unvrsal, but only occurs whr thr s a fnt numbr of nrgy lvls (hr, thr ar only two). Most systms hav an nfnt towr of nrgy lvls, thr s no maxmum nrgy and th hat capacty dos not fall off. 6

7 Prof. Dr. I. assr Phys Dc_0 3- It s not asy to attan th maxmum of th paramagntc hat capacty curv as th followng calculaton shows. Th paramagntc hat capacty bcoms mportant only at μ H vry low tmpraturs. Th maxmum occurs at. ow th magntud of kt μ k.380 so to gt th maxmum w must hav H maxmum attanabl flds ar 4 3,.. so w nd a maxmum tmpratur of T K 4- At ro tmpratur, th magntaton gos to and all th ns ar up. Thr s an ordr, and so th ntropy s ro. Th strongr th fld, th hghr th tmpratur has to b bfor th ns start to b apprcably dsordrd. At hgh tmpraturs th ns ar narly as lkly to b up as down; th magntaton falls to ro and th ntropy rachs a maxmum. Th ntropy of ths stat s. Rmmbr that, s th total numbr of mcrostats. 5- If t s possbl to xct all th partcls to th uppr nrgy stat so n th systm T H Tsla k ln would agan b compltly ordrd and n stat of ro ntropy. Accordng th quaton n. Ths stuaton could only b achvd f th tmpratur T approachd a valu of ro from th ngatv tmpratur sd,.. at T 0. /( kt ) Whl a ngatv tmpratur of ths magntud s not obtanabl n practc, t s possbl to obtan fnt ngatv tmpraturs as dfnd by th abov quaton. 7

8 Prof. Dr. I. assr Phys Dc_0 Modl: Consdr dntcal, locald (.. dstngushabl), practcally statc, mutually nonntractng and frly orntabl dpols at absolut tmpratur T and placd n an xtrnal magntc fld H pontng along drcton. Thn th torqu actng on th dpol s gvn by: H H sn, and th (magntc) potntal nrgy can b wrttn as: E. H H cos, whr θ s th angl btwn th magntc fld and th dpol and μ s th magntc dpol momnt. Th partton functon of th systm,, s gvn by whr th frst summaton for on can hav: H cos { }, gos ovr all sts of orntatons of th systm. Classcally H cos H cos sn d snd H Whr H. Th man magntc momnt M of th systm wll ndd b n th kt drcton of th fld H; for ts magntud w hav: ln ln M cos cos H H. Hnc, w obtan th man magntc momnt pr dpol as: M coth / L( ), whr L(x) s th Langvn s functon, L ( x) coth x / x. Th dmnsonlss paramtr H dtrmns th strngth of th (magntc) potntal nrgy aganst th (thrmal) kntc nrgy kt. S th plottng of L(x) functon. H If w hav n dpols pr unt volum n th systm, thn th magntaton of th systm, v. th man magntc momnt pr unt volum, wll b gvn by: M n nl(). n For (, L ( ) ),.. th magntc flds so strong (or tmpratur so low), w hav th magntc saturaton:,. M n n For η <<,.. th magntc flds so wak (or tmpratur so hgh), w hav: 3 n M n n nl( ) nl( ) H, kT to th lowst ordr of approxmaton. So, th hgh tmpratur suscptblty of th systm s, thrfor, gvn by: 8

9 Prof. Dr. I. assr Phys Dc_0 M n n C lm. H H 0 3kT T Th last quaton s th Cur s law of paramagntsm and th paramtr C bng th Cur s constant Appndx () Gnral Calculaton of Magntaton Th problm of paramagntsm could b tratd classcally (Langvn's thory) or quantum-mchancally. Hr, w ar followng th quantum mchancal tratmnt. Consdr a systm consstng of non-ntractng dpols at absolut tmpratur T and placd n an xtrnal magntc fld H pontng along - drcton (ot that: H s th local magntc fld actng on th atom,.. t ncluds both xtrnal and fld producd by all othr atoms). Thn th (magntc) potntal nrgy of a dpol can b wrttn as: μ H H H Hr g g o, g g ( o s th ohr magnton = m, and th charg and th rst mass of th lctron, rctvly) and s th Land` g-factor,.. 3 S ( S ) L( L ) g ( ) S and L bng, rctvly, th n and orbtal quantum numbrs of th dpol and s th total angular momntum of th atom. In quantum mchancs, th valus of ar dscrt and ar gvn by: m, m,,,, Thus thr ar possbl valus of m corrondng to that many possbl projctons of th angular momntum vctor along th - axs. Th probablty that an atom s n a stat labld by m s gvn by P C H, m Th man m m g ar componnt of th magntc momnt of an atom s thrfor: whch could b smplfd as: To calculat Thn m H, lt us ntroduc th rato, ln H H P m m H H, thus kt H ( ) snh( ) m m snh( ) m H 9

10 Prof. Dr. I. assr Phys Dc_0 ln g ( ) H Whr ( ) s th rlloun functon and s gvn by ( ) =, ( ) ( )snh( ) coth( ), 3 If thr ar n atoms pr unt volum, th man magntc momnt pr unt volum (or magntaton) bcoms ng, M n n ng ( ) ng = H, 3 ( ) and ng s th ''magntc suscptblty''. Th rsult s known as Cur's 3kT law. In th cas ndpndnt of and g 0 but g and g and dntcal wth Langvn s functon stay constants, L( ). T ( ) tnds to bcom 0

11 Prof. Dr. I. assr Phys Dc_0 Appndx () Statstcs of varous nsmbls - Th mcrocanoncal nsmbl:- Systms wth fxd and V, and an nrgy lyng wthn th ntrval ( E, E ), whr E. Th total numbr of dstnct mcrostats accssbl to a systm s ( E, V, ; ). From th qual a pror probablts p k ( E, V, ; ) Thus all th stats n th mcrocanoncal nsmbl appar wth th sam wght whch mpls that ˆ p E wth th dscrt gnvalus ( ) lyng wthn th rang E E E. ˆ mn m p n m n pn mn ( E ) wth, for ach of th accssbl stats p ( ) n E 0 for all othr stats Th ntropy S k ln ( E ) whr ( E ) s now calculatd quantum mchancally, takng nto account th ndstngushablty of th partcls, whch mpls no paradox, such as Gbbs' paradox. Also, as T 0, systm gos to th ground stat whch gvs ( E ).. S 0 (thrd law of thrmodynamcs) pur cas, p p ( E ) > mxd cas (dgnrat), p p, S 0 - Th mcrocanoncal nsmbl:- Systms wth fxd, V and T and dffrnt nrgy E. Th probablty that a systm, chosn at random from th nsmbl, posssss an nrgy s n E dtrmnd by oltmaan factor, and th dnsty matrx n th nrgy rprsntaton s thrfor takn as p ˆ mn n mn wth E n pn, n 0,,, Thus dnsty oprator n th canoncal nsmbl could b wrttn as: E ˆ H ˆ p Tr Hˆ Hˆ Th xpctaton valu A ˆ of a physcal quantty A s now gvn by E n

12 Prof. Dr. I. assr Phys Dc_0 th suffx fxd. Aˆ ˆ H Tr A Tr A ˆ ˆ = ˆ Tr H hr mphass h fact that th avragng has bn don ovr an nsmbl wth 0 Exampl: If Hˆ ˆ fnd 0 Answr: Hˆ ˆ 3 3 ˆ ( ) ˆ ( ) ˆ ( ) ˆ! 3! ˆ 3 ( ) ˆ ( ) ( )! 3! ˆ cosh ˆ ˆ snh ( ) ( ) cosh( ) 0 snh( ) 0 0 cosh( ) 0 snh( ) 0 ot that: Wth th dfnton thn 0 ˆ 0, on fnds 0 ˆ ˆ ˆ ˆ, ˆ ˆ ˆ ˆ, Hˆ Tr( ) cosh Hˆ ( ) 0 ˆ Hˆ Tr( ) cosh( ) 0. ˆ ˆ 0 0 Tr ˆ ˆ Tr cosh( ) Tr cosh( ) 0 snh( ) tanh( ) cosh( ) cosh( ) a a E n a na (snh ) a a a n0 n0