2.9.1 Heat capacity for solids: Einstein- and Debye-model

We will repeat the details of this discussion in a later chapter. Here we will just show some experimental findings which can be well understood within the Debye model and partly within the easier Einstein model.

• The Einstein model assumes \(3 N\) oscillators each with the frequency \(\nu\). The capacity related to the vibrational energy is found to be

  \begin{equation*} C_{vib}=3R \frac{\left(\frac{\Theta}{T}\right)^2 \exp(\Theta/T)}{\left(\exp(\Theta/T)-1 \right)^2} \quad \mbox{with}\quad \Theta = \frac{h\nu}{k} \label{eq:Cvib_Einstein} \end{equation*}

  (2.25)

• The Debye model assumes \(3 N\) oscillators each with the frequency \(\nu\). The capacity related to the vibrational energy is found to be

  \begin{equation*} \begin{split} C_{vib} & =9 R \frac{T^3}{\Theta_D^3}\int_0^{\Theta_D/T} \frac{x^4\,e^x}{\left(\exp(x)-1 \right)^2}\; dx\\ & \quad\;\mbox{with}\quad x = \frac{h \nu}{kT} \quad\mbox{and}\quad \Theta_D = \frac{h\nu_{max}}{k} \\ \mbox{For} &\quad T \ll \Theta_D: \qquad C_{vib} = \frac{12 \pi^4 R}{5} \frac{T^3}{\Theta_D^3}\\ \mbox{For} &\quad T \gg \Theta_D: \qquad C_{vib} = 3 R\\ \end{split} \label{eq:Cvib_Debye} \end{equation*}

  (2.26)

Fig. 2.6 shows a comparison between measured heat capacity data and corresponding points using the Debye and the Einstein model. Obviously the Debye model allows for a very good description of the measured data.

With frame

Experimental data on the heat capacity

© J. Carstensen (TD Kin I)