德拜模型

推导

$\lambda_n = {2L\over n}\,,$

$E_n\ =h\nu_n\,,$

$E_n=h\nu_n={hc_s\over\lambda_n}={hc_sn\over 2L}\,,$

$E_n^2=E_{nx}^2+E_{ny}^2+E_{nz}^2=\left({hc_s\over2L}\right)^2\left(n_x^2+n_y^2+n_z^2\right)\,.$

$U = \sum_n E_n\,\bar{N}(E_n)\,,$

$U = \sum_{n_x}\sum_{n_y}\sum_{n_z}E_n\,\bar{N}(E_n)\,.$

$\lambda_{\rm min} = {2L \over \sqrt[3]{N}}\,,$

$n_{\rm max} = \sqrt[3]{N}\,.$

$U = \sum_{n_x}^{\sqrt[3]{N}}\sum_{n_y}^{\sqrt[3]{N}}\sum_{n_z}^{\sqrt[3]{N}}E_n\,\bar{N}(E_n)\,.$

$U \approx\int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}} E(n)\,\bar{N}\left(E(n)\right)\,dn_x\, dn_y\, dn_z\,.$

$\langle N\rangle_{BE} = {1\over e^{E/kT}-1}\,.$

$\bar{N}(E) = {3\over e^{E/kT}-1}\,.$

$U = \int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}}\int_0^{\sqrt[3]{N}} E(n)\,{3\over e^{E(n)/kT}-1}\,dn_x\, dn_y\, dn_z\,.$

$\ (n_x,n_y,n_z)=(n\cos \theta \cos \phi,n\cos \theta \sin \phi,n\sin \theta )$

$U \approx\int_0^{\pi/2}\int_0^{\pi/2}\int_0^R E(n)\,{3\over e^{E(n)/kT}-1}n^2 \sin\theta\, dn\, d\theta\, d\phi\,,$

$N = {1\over8}{4\over3}\pi R^3\,,$

$R = \sqrt[3]{6N\over\pi}\,.$

$U = {3\pi\over2}\int_0^R \,{hc_sn\over 2L}{n^2\over e^{hc_sn/2LkT}-1} \,dn$

$U = {3\pi\over2} kT \left({2LkT\over hc_s}\right)^3\int_0^{hc_sR/2LkT} {x^3\over e^x-1}\, dx$

$T_D\ \stackrel{\mathrm{def}}{=}\ {hc_sR\over2Lk} = {hc_s\over2Lk}\sqrt[3]{6N\over\pi} = {hc_s\over2k}\sqrt[3]{{6\over\pi}{N\over V}}$

$\frac{U}{Nk} = 9T \left({T\over T_D}\right)^3\int_0^{T_D/T} {x^3\over e^x-1}\, dx = 3T D_3 \left({T_D\over T}\right)\,,$

$T$微分，我们便得到无量纲热容：

$\frac{C_V}{Nk} = 9 \left({T\over T_D}\right)^3\int_0^{T_D/T} {x^4 e^x\over\left(e^x-1\right)^2}\, dx\,.$

德拜的推导

$n \sim {1 \over 3} \nu^3 V F\,,$

$U = \int_0^\infty \,{h\nu^3 V F\over e^{h\nu/kT}-1}\, d\nu\,,$

$3N = {1 \over 3} \nu_m^3 V F \,.$

$U = \int_0^{\nu_m} \,{h\nu^3 V F\over e^{h\nu/kT}-1}\, d\nu\,,$
$= V F kT (kT/h)^3 \int_0^{T_D/T} \,{x^3 \over e^x-1}\, dx\,,$

$= 9 N k T (T/T_D)^3 \int_0^{T_D/T} \,{x^3 \over e^x-1}\, dx\,,$
$= 3 N k T D_3(T_D/T)\,,$

低温极限

$\frac{C_V}{Nk} \sim 9 \left({T\over T_D}\right)^3\int_0^{\infty} {x^4 e^x\over \left(e^x-1\right)^2}\, dx$

$\frac{C_V}{Nk} \sim {12\pi^4\over5} \left({T\over T_D}\right)^3$

高温极限

$\frac{C_V}{Nk} \sim 9 \left({T\over T_D}\right)^3\int_0^{T_D/T} {x^4 \over x^2}\, dx$
$\frac{C_V}{Nk} \sim 3\,.$

参考文献

1. ^ 'Zur Theorie der spezifischen Waerme', Annalen der Physik (Leipzig) 39(4), p. 789 (1912)
• CRC Handbook of Chemistry and Physics, 56th Edition (1975-1976)
• Schroeder, Daniel V. An Introduction to Thermal Physics. Addison-Wesley, San Francisco, Calif. (2000). Section 7.5.
• Kittel, Charles, Introduction to Solid State Physics, 7th Ed., Wiley, (1996)