University of Kentucy Deprtment of Pysics nd stronomy PHY : Solid Stte Pysics II Fll Finl Emintion Solutions Dte: December, (Mondy ime llowed: minutes. nswer ll questions.. Het cpcity of ferromgnets. ( Use te pproimtion mgnon dispersion reltion to find te leding term in te et cpcity of treedimensionl ferromgnet t low tempertures <<J. e result is. ( / /, per unit d [ ( ] volume. e definite intergrl ep is given s ζ wit ζ. pproimtely. (b How does te et cpcity of metl depend on temperture t low (just write down te result, you do not need to do ny derivtion? Free electron s similr dispersion reltion s E ;, wy te et cpcities of metl nd ferromgnet depend differently on temperture t low? Solution. D( d U ep o clculte D( : d D( d V d For, d D( d D( d 8 V d D( 8 d d D( d ( (
Given:: int (^(//(ep(,..infinity; Zet(.;.877 ( [ ] ep d U, Let ep d U : tis into ( Substitute ( (.. C Numericlly, 6 U C U ζ ζ ζ ζ ζ
. Mgnetic field penetrtion in plte. e penetrtion eqution my be written s, were is te penetrtion dept. ( Sow tt ( inside superconding plte perpendiculr to te is nd of ticness is given by cos( / (, cos( / were is te field outside te plte nd prllel to it; ere is t te center of te plte. (b e effective mgnetiztion M( in te plne is defined by ( M(. Sow tt, in CGS, M( (/ 8 (, for <<. In SI, we replce te by µ. Solution. ( / e Ce t / nd /, e e / / Ce Ce / / / ( (
( e ( e C(e / / C( e C e e / / / Substitute tis into (, e ( e / Ce / (e e / / / / Ce Ce Ce e / / ( e / Ce / / e cos / e / cos / ( e e cos cos cos / / e / / e e / e e / e e / / e e cos / cos (e / / / e e / / (b M( ( cos cos cos cos ( ( ( << [ ] ( ( ( ( ( [ ] 8 (
. Susceptibility of d nd f electrons. ( pply Hund rule to determine te ground stte of (i Dy in te configurtion of f 9 s p 6, nd (ii Cu in te configurtion of d 9. Epress your nswer in stndrd tomic nottion. (b e Curie constnt C is given s Ng [J{J ] µ C for felectrons. Estimte te Curie constnts for Dy nd Cu. vogdro s number is given s 6. mol. Note tt te gfctor is given by te Lndé eqution: J(J S(S L(L g J(J (c Wt is te rtio χ(k/χ(.k of te sme prmgnetic mteril, for smll field? Solution. ( Dy : L ΣL (i.e. H ΣS /, S 6 ΣJ LS / e ground stte is 6 H /. Cu : L ΣL (i.e. D ΣS /, S ΣJ LS / e ground stte is D /.
(b Np µ C, p g j(j (i For Dy, J /, S /, L (ii J(J S(S g J(J p. Np µ 6..6 (9.7 C 6.8.9 cm Kmol For Cu, J /, 7. 8..6 J(J S(S g J(J L(L S /, L L(L 7. 8... 6. 7. 8........ Since tere is orbitl ngulr momentum quencing in d electrons, it is replce te J by S nd te g fctor by in clculting p. p...7 Np µ 6..7 (9.7 C 6.8.7 cm Kmol better to (c C For prmgnetism, χ χ(k For te sme type of ion, χ(.k..
. Moleculr rottion in solid. We cn model te polriztion vector P lwys reles towrds n equilibrium vlue P E t rte clled reltion time τ. P E depends on te electric field E s. e reltion eqution cn ten be written s dp PE P i t. ( y considering E E, sow tt dt τ e P E χ(e χ( iχ( ε'( nd ε"( τ τ (b t wt is te dielectric loss mimum? Wt re te reltion times t frequencies Hz, Hz, nd Hz? (c Rel nd imginry prts of te dielectric constnt of ice s function of temperture t different frequencies re given in te figure. Sow tt te temperture dependence is consistent wit τ τ ep( / nd find vlues of τ nd. (d Wt is te pysicl bsis for temperture dependence of te ind given in prt (c?
Solution. ( ii Wit E Ee, ii P χ( E P χ( E e ii PE χ(e P χ(ee Substitute tese into te reltion eqution, i( χe E D P Substitute tis into (, ε( [ χ( χ( ] E εe χe ε χ χ( ( iτ χ( [ iτ] τ E χ( ( iτ χ( ( iτ χ( iχ( τ τ τ χ( ε'( nd ε"( τ τ χ( χ( ( ( χ( τ τ (b Dielectric loss is mimum if ε"is mimum, dε" d χ( τ τ d d ( τ ( τ τ τ, or τ f For f Hz, τ.µ s For f Hz, τ 9. µ s For f Hz, τ.8 µ s
(c Since tere re only tree curves sowing pe in te given dt of ε", nd tere re two undetermined prmeters nd τ. o demonstrte tt te dt is consistent wit te given eqution, we cn use te Hz nd Hz curves to determine te unnow prmeters first, nd ten cec weter tese prmeters ve resonble greement wit te Hz curve. o For te Hz curve, te pe occurs t C or K : (. τ ep 6.7 n l τ 6.7 ln( τ.67 o For te Hz curve, te pe occurs t C or K : ( ( 9. τ ep..8 679K bove prmeters require τ.µ s,.7 ln ( τ 6 679 τ 6.8 ep 67 6 τ 6.8.6 ( τ ( τ.98 Substitute tis pc into (, 6.7 ln.67 679 ln( τ. 6 τ 6.8 µ s 6.8 s o For te Hz curve, te pe occurs t 6 C or 67K :.7 ln tis is close to te result in prt (b. ( ( (d Reling time increses s temperture is lowered. is mens longer time for te rottion to rec equilibrium t low temperture. ogeter wit te eponentil reltion, we now tt tere is brrier defined by for te rottionl motion. Only wen te terml energy is iger tn te brrier, te molecules cn rotte fst enoug to reduce te reltion time eponentilly.