Statistical Physics, Second Revised and Enlarged Edition
28 Two examples T 2 3 C 0.44 NkkkB 0 Fig. 3. 3 The variation ofheat capacitywithtemperatureforaspin-^12 solid,showingthe p ...
Aspin-^12 solid 29 T 2 3 S NkkkBln 2 0 Fig.3. 4 The variation of entropywith temperature for a spin-^12 solid. Note the tr ...
30 Two examples thesubstanceis paramagnetic(alignment ofpermanent momentsinanappliedfield) rather than ferromagnetic or antiferr ...
Aspin-^12 solid 31 T TTT 2 TTT 1 (Final temperature) (Precooling temperature) S C NkkkB ln 2 Low field,B 2 High field, B 1 B A F ...
32 Two examples simplelaw TTT 2 =T 1 (B 2 /B 1 ) (3.7) Thisisanice exampleoftheimportance ofentropy.Inthefirstlegwe reduce the e ...
Aspin-^12 solid 33 For nuclear spinsitwill bea‘dilution refrigerator’, adevice capableofreaching 3 –10 mK whose working substanc ...
34 Two examples interactionsleft, evenifthese are onlymagneticdipole–dipoleforcesbetween the Nspins. As remarked earlier, althou ...
Aspin-^12 solid 35 Withoutlabouringthepoint,itisnoaccident that the workingsubstance ofarefrigera- tor is also a possible candid ...
36 Two examples This resultmayreadily be verifiedbyexpandingthe exponentialsin (3.9),bearing in mind that the exponents are smal ...
Localized harmonic oscillators 37 enablesZtobe summedexplicitlyas ageometric progression. Wehave Z= ∑ j exp [ − ( j (( + 1 2 ) h ...
38 Two examples T 2 1 2 U 2 NkkkB NkkkB UUU(0) = Nk^12 kkB UU=NkkkBT 0 Fig. 3.9The variation ofinternalenergy withtempera ...
Localized harmonic oscillators 39 3 .2.2 The extreme quantum and the classical limits The assembly of localized oscillators form ...
40 Two examples 3 .2.3 Applications 1.Vibrations ofsolidsThe assembly of identical harmonic oscillators was used by Einsteininth ...
Summary 41 Forbothsubstances, the one-particle energystates are well describedbyquantum mechanics. A spin-^12 particlehasjus ...
...
4 Gases: the density of states In the last two chapters we have applied the statistical method as outlined in section 1. 5 to an ...
44 Gases: the density ofstates the particleis confinedwithinanotherwise emptybox ofvolumeV?For conve- nience,let us assume that ...
Fittingwaves into boxes 45 kkkyandkkky+δkkkyandwithz-componentbetweenkkzandkkz+δkkz.Callthis number g(kkx,kkky,kkz)δkkxδkkkyδkkz ...
46 Gases: the density ofstates Inthis method,theperiodic conditionisthatbothψanditsgradient should match up on opposite faces of ...
Other informationfor statistical physics 47 The correspondingone-dimensionalresultfor aline (a ‘quantum wire’) oflength Lis rath ...
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