令： ⊙e=hv/k ⊙：爱因斯坦特征温度 ve:Einstein特征频率 U=3NKT2[olnq /OTlyN =3NKT2{(d/OT)In[e-hv:/2KT/(1-e-hve/KT)]}N.v 3NKT2 [0/OT(-hvE/2KT)-0/oTIn(1-e-hv/kT)] =3NkT2{(-1/T2)hve/2k)- [-e-hv:/kT/(1-e-hve/kT)](-hvE/k)(-1/T2)} =3NKT2(hve/2KT2+hv/kT2.[e-hve/KT/(1-e-hv:/kT)]) =3/2NhvE 3NhvE/(ehv/kT-1) U=3NhvE/(ehvr/KT-1)+Eo E。=3/2NhvE:振动的零点能。令: E=h/k E: 爱因斯坦特征温度 E: Einstein特征频率 U = 3NkT2 [∂lnq /∂T]V.N = 3NkT2 {(∂/∂T)ln[e–hE/2kT/(1－e –hE/kT)]}N,V = 3NkT2 [∂/∂T(–hE/2kT) – ∂/∂Tln(1－e –hE/kT)] =3NkT2 {(-1/T2 )·(-hE/2k)－ [-e –hE/kT/(1－e –hE/kT)](-hE/k)(-1/T2 )} = 3NkT2 {hE/2kT2 + h/kT2·[e–hE/kT/(1－e –hE/kT)]} = 3/2NhE + 3NhE/(ehE/kT－1) U = 3NhE/(ehE/kT－1)+Eo Eo= 3/2NhE ：振动的零点能