Statistical Physics, Second Revised and Enlarged Edition
108 Bose–Einsteingases back to (4.5) and usinga computer to fold in the complicated dispersion relation. An approximate fudge (t ...
Summary 109 9 .4 A note about coldatoms We have seen that quantum statistics are needed at high densities and low temper- atures ...
110 Bose–Einsteingases ModifiedBE statisticsdescribes theproperties ofassemblies ofmasslessbosons, for which there is no conser ...
1 0 Entropyin other situations Asthechartinthefront ofthebookshows, wehave now completedour elementary studyofthethermalequilibr ...
112 Entropyin other situations The calculationofSfor thissituationgoes asfollows: S=kkkBln definition(1.5) =kkkB ( lnN!− ∑ L ln ...
Entropyand disorder 113 exactlyparallelsthederivation of(10.2), the resultbeing S=−NkkkB ∑ j PPjlnPPj (10.3) wherePPj =nnj/N(str ...
114 Entropyin other situations The secondtermsintheFDandBE expressions areintriguing. Equation (10.4) displays that in an FD gas ...
An assembly atfixed temperature 115 temperature. This canbe readilyshown toimplythat thefree energyofthesystem must not increase ...
116 Entropyin other situations Asweshouldexpect, thisisagaintheBoltzmanndistribution. However, thereis animportant difference fr ...
Vacancies in solids 117 theformation energyofa vacancyis,andthat the vacancies are sufficiently dilute that they are weakly int ...
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1 1Phasetransitions Changes ofphase are ofgreatinterest, notleastbecause oftheir surprise value. In this chapter we examinehow s ...
120 Phase transitions T (and P) G A B 2 ' 1 ' 1 2 Fig. 11. 1 The Gibbs free energyGaround a first-order phase transition. In reg ...
Ferromagnetism ofa spin-^12 solid 121 fullyalignedvalueM=Nμ,the saturation magnetization. The spontaneous mag- netizationM( 0 )o ...
122 Phase transitions answeris M=Nμtanh(μB/kkkBT) =Nμtanh[μ(B 0 +λM)/kkkBT] (11.3) The spontaneous magnetizationisobtainedbysett ...
Ferromagnetism ofa spin-^12 solid 123 1 T m TTTc Fig. 11. 3 Variation ofthe magnitudeoftheorder parametermwithtemperatureT.Note ...
124 Phase transitions T < TTTc T = 0 1 F^ + 1 m T = TTTc T > TTTTc Fig. 11.4Variation ofFwithmat various temperatures. T ...
Ferromagnetism ofa spin-^12 solid 125 T S,C S =NkkkBln 2 C S TTTc Fig.11. 5 Entropyand heat capacityin the mean field approximat ...
126 Phase transitions line upallthespins. Thisrelates to theflatF(m)curves,since theaddition ofan applied fieldB 0 to Fig. 11.4 ...
Order–disorder transformations in alloys 127 In practice spin wavesdoexist, andtheyare seeninthelow-temperature regionboth inman ...
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