Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan
Ovrviw
Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl
h Hat Capacity
Hat capacity h hat capacity of a systm hld at a constant volum V is dfind as C V U whr U is th total intrnal nrgy of th systm and is th absolut tmpratur. Not that this is an xtrinsic quantity, dpnding on th physical dimnsions of th sampl. Of mor fundamntal significanc ar spcific hats, which corrspond to th ta up of hat pr unit mass or pr particl.,
h Dulong-Ptit Law
Vibrational nrgy h total nrgy of a vibrating atom E tot is givn by 1 2 m v 2 1 2 u 2 E tot, whr is th spring constant and u is th displacmnt from quilibrium. Not that th two componnts of nrgy constitut two dgrs of frdom along ach spatial axis.
Vibrational nrgy z x y h vibration of th atom has thr dgrs of frdom (along ach spatial axis).
Vibrational nrgy hus w hav a total of six dgrs of frdom (two componnts of nrgy pr spatial dirction). Now th avrag nrgy of a particl pr dgr of frdom is givn by th thrmal nrgy E thrm 1 2. Hnc, th avrag vibrational nrgy is E vib 3.
Vibrational nrgy h total nrgy U of N vibrating atoms is thn U 3N. (compar this to th total nrgy of an idal gas). h similarity btwn this and th idal gas quation allows us to put U 3nR, whr n is th numbr of mols in th sampl and R is th univrsal gas constant.
pcific hat pr mol W wish to obtain an xprssion for th chang in th total nrgy of a solid as th tmpratur changs. o this nd, w may dfin th spcific hat pr mol (for a constant volum) c V Q n, whr Q is th chang in nrgy in th form of hat transfrrd. his has I units of JK -1 mol -1.
h Dulong-Ptit Law From U 3nR, w hav Q U 3nR, so Q n c V 3R. his is nown as th Dulong-Ptit Law. his has a constant valu of c V 24.94JK mol 1 1.
Limits of th Dulong-Ptit Law h Dulong-Ptit Law (I) wors wll at high tmpraturs but fails of low tmpraturs.
om masurd spcific hats ubstanc c (J/gm K) Molar C (J/mol K) Aluminum 0.900 24.3 ismuth 0.123 25.7 Coppr 0.386 24.5 Gold 0.126 25.6 Lad 0.128 26.4 ilvr 0.233 24.9 ungstn 0.134 24.8 Zinc 0.387 25.2 Mrcury 0.140 28.3 Alcohol(thyl) 2.400 111 Watr 4.186 75.2 Ic (-10 C) 2.050 36.9
h Einstin modl
h quantum harmonic oscillator In th Einstin modl w ta a stp closr to th phonon pictur of solids. A solid is nvisiond as bing composd of N indpndnt simpl harmonic oscillators, ach with 3 dgrs of frdom and th sam angular frquncy. Hnc, th numbr of stats will b N = 3N. h quantisd nrgy ignvalus of th oscillators ar thn givn by lmntary quantum mchanics as E n n 1. whr n is an intgral quantum numbr. 2
h hat capacity Earlir, w dfind th hat capacity of a systm hld at a constant volum V as C V U If no wor is don on or by th systm, th chang in intrnal nrgy is qual to th hat Q. his is rlatd to th ntropy of th systm by Q..
oltzmann s Entropy Equation Howvr, ntropy may altrnativly b dfind via oltzmann s Entropy Equation lnw, whr W is th numbr of microstats of th systm. It will turn out that w can us this to find th hat capacity of th systm by finding how W dpnds on th numbr of mods r.
h distribution of quanta W may thrfor considr ach atom of th systm to hav som numbr of nrgy quanta givn by E. W lt th numbr of nrgy quanta of th systm b r. r quanta...... N stats
h distribution of quanta h numbr of distributions of ths r quanta ovr th N stats is givn by th numbr of arrangmnts of r quanta and N - 1 stats (for just on stat, all th quanta must fall in it). For instanc q q q q q q q q q q q q q q q q q
Calculation of th ntropy h numbr of ways of choosing r quanta from N +r-1 arrangmnts is givn by th binomial cofficint N r1 C r N N r 1!. 1! r! his is thrfor th numbr of microstats of th systm W N N r 1!. 1! r!
h quantum harmonic oscillator h ntropy of th systm is thrfor ln N N r 1! 1! r! ln N r!, N! r! sinc N 1. Hnc ln N r! ln N! ln r!,
Calculation of th ntropy Using trling s approximation ln n! nln n n, this bcoms N r ln N r N r N ln ln N N r r r, N rlnn r N ln N r ln r.
h total nrgy h total nrgy may b writtn in trms of th numbr of quanta as U r, whr th ground stat nrgy has bn omittd as it disappars on diffrntiation. W can combin this with th xprssion for th ntropy via U 1.
h total nrgy Now Diffrntiating th ntropy. 1 r r U r U, 1 ln r N r so. 1 1 ln 1 r N r U
h total nrgy Rarranging this r N /. 1 ut from th xprssion for th total nrgy U r, N. 1 /
h hat capacity Diffrntiating this with rspct to,. 1 2 / / 2 V N C U In trms of th numbr of particls N this is. 1 3 2 / / 2 V N C
h hat capacity Maing th substitution nr = N as bfor. 1 3 2 / / 2 V nr C Hnc, th molar spcific hat is. 1 3 2 / / 2 V R c
Limit as As, c V 2 1 / 3R 1 / 12 3R. Hnc, th Einstin modl rproducs th Dulong-Ptit modl in th high tmpratur limit. hr ar still unralistic aspcts of th Einstin modl. Not last that thr is only on frquncy. his is rmdid in th Dby modl.
h Dby modl
h os-einstin factor In th cours of th drivation of th Einstin modl, w passd through th rsult which w could r-writ as f r r N N /, 1 1. / 1
h os-einstin factor his is nown as th os-einstin factor, which in th contxt w mt it, may b intrprtd as th numbr of quanta pr stat (with frquncy ). Mor commonly, in is intrprtd as th mod occupation numbr. W can intrprt it gnrally as th numbr of quanta in a stat q n q f 1. q r N / q 1
Linar disprsion h total intrnal nrgy du to ths quanta may thn b writtn as U q n q qq q / q. 1 W add a stp of gnralisation to this modl by taing th mods to hav a linar disprsion rlation givn by q vq. Not that this approximats th low nrgy acoustic branchs of th phonons of a solid.
Rciprocal volum of a q-stat h priodic boundary conditions on th solid imply that th rciprocal volum of a stat in q-spac is (2p) 3 /V. Forcing th stats of th systm to occupy a sphrical volum, w thrfor impos N 2p 3 D V 4p q 3, whr N is th numbr of stats in th systm and q D is th radius of th sphr in q-spac.
Rciprocal volum of a q-stat Dfining th numbr of stats pr unit volum n = N/V, w hav n 3 q D 6p 2 W can convrt th summation ovr q in to an intgral via. q V 2p 3 d 3 q.
otal nrgy h xprssion for th total nrgy thn bcoms U q q q / 3 1 V 2p / q q d 1 3 q. Convrting to sphrical polar coordinats, w hav d 3 q 4pq 2 dq. inc w also hav q vq,
otal nrgy th total nrgy bcoms U 3 V q D vq dq. 2 vq p 1 2 0 / Using our prvious rsults, this bcoms U 3 3N q D vq dq. 3 0 vq / q 1 D
Hat capacity Diffrntiating this with rspct to, w hav U C V 3N q 3 D 3N q Maing th substitutions 3 D v 2 q q 0 vq / D D vq vq / 3 dq, 1 4 1. vq / 0 2 q dq D D vqd.
Hat capacity. 1 3, 1 3 / 0 2 4 3 / 0 2 4 3 3 x x D x x D D D dx x N dx x q q N U, vq x and w gt
Hat capacity In trms of spcific hat capacity, w thrfor hav c V 3R D 3 4 / x x dx 0 2 1. x his is th Dby modl of hat capacity. Hr, Q is nown as th Dby tmpratur and is a masur of th tmpratur at which th mods bgin to b xcitd. Not that this rproducs th Dulong-Ptit Law at high tmpratur.
Hat capacity a comparison