A simple model for metals is the free electron model where the potential energy of the electrons is zero and the electron-electron interactions are ignored. This is equivalent to any system of noninteracting fermions with zero potential energy. In this model the thermodynamic properties only depend on one parameter, the particle density n. In the table below, n denotes the number of particles per meter in one-dimension, the number of particle per square meter in two-dimensions, and the number of particles per cubic meter in three dimensions. In the table many quantities are given twice: once in terms of the density of states at the Fermi energy D(EF) and once in terms of the electron density n. The expressions with D(EF) were derived using the Sommerfeld expansion and should be more accurate than the simple free electron model which only depends on the electron density n. Derivations (in German) of these results can be downloaded for 1-D, 2-D, and 3-D.
1-D Schrödinger equation for a free particle | 2-D Schrödinger equation for a free particle | 3-D Schrödinger equation for a free particle | |
| Eigenfunction solutions |
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Eigenvalues of the translation operator |
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| Dispersion relation |
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| Density of states |
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Density of states |
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Fermi energy EF |
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Chemical potential μ |
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Internal energy distribution |
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Internal energy |
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Specific heat |
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Entropy |
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Helmholtz free energy |
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Pressure |
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Bulk modulus |
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| Enthalpy \begin{align} h &= u + P \end{align} |
\begin{align} h &\approx \left(nE_F+\frac{\pi^2D(E_F)}{2}(k_BT)^2\right) \quad \mathrm{J\,m^{-1}}\\ &\approx \left(\frac{\pi^2 \hbar^2 n^3}{8m}+\frac{2m}{\hbar^2 n}(k_BT)^2\right)\quad\mathrm{J\,m^{-1}} \end{align} | \begin{align} h &\approx \left(nE_F+\frac{\pi^2D(E_F)}{3}(k_BT)^2\right) \quad \mathrm{J\,m^{-2}}\\ &\approx \left(\frac{\pi \hbar^2 n^2}{m}+\frac{\pi m}{3\hbar^2}(k_BT)^2\right) \quad\mathrm{J\,m^{-2}} \end{align} | \begin{align} h &\approx \left(nE_F+\frac{5\pi^2D(E_F)}{18}(k_BT)^2\right) \quad \mathrm{J\,m^{-3}}\\ &\approx \left(\frac{\hbar^2}{2m}(3\pi^2)^{\frac{2}{3}}n^{\frac{5}{3}}+\frac{5 m (3\pi^2n)^{\frac{1}{3}}}{18\hbar^2}(k_BT)^2\right) \quad\mathrm{J\,m^{-3}} \end{align} |
| Specific heat \begin{align} c_p &= \left(\frac{\partial h}{\partial T}\right)_p \end{align} |
\begin{align} c_p &\approx \pi^2 D(E_F)k_B^2 T \qquad\mathrm{J\,K^{-1}\,m^{-1}} \\ &\approx \frac{4m}{\hbar^2 n}k_B^2 T \qquad\mathrm{J\,K^{-1}m^{-1}} \end{align} | \begin{align} c_p &\approx \frac{2\pi^2 D(E_F)}{3}k_B^2 T \qquad\mathrm{J\,K^{-1}m^{-2}} \\ &\approx \frac{2\pi m}{3\hbar^2}k_B^2 T \qquad\mathrm{J\,K^{-1}m^{-2}} \end{align} | \begin{align} c_p &\approx \left(\frac{5\pi^2D(E_F)}{9}k_B^2 T\right) \qquad \mathrm{J\,K^{-1}m^{-3}}\\ &\approx \left(\frac{5 m (3\pi^2n)^{\frac{1}{3}}}{9\hbar^2}k_B^2 T\right)\qquad\mathrm{J\,K^{-1}m^{-3}} \end{align} |
| Gibbs energy \begin{align} g &= u + P - Ts \\ &= h - Ts \notag \end{align} |
\begin{align} g &\approx \left(nE_F+\frac{\pi^2D(E_F)}{6}(k_BT)^2\right) \quad \mathrm{J\,m^{-1}}\\ &\approx \left(\frac{\pi^2 \hbar^2 n^3}{8m}+\frac{2m}{3\hbar^2 n}(k_BT)^2\right)\quad\mathrm{J\,m^{-1}} \end{align} | \begin{align} g &\approx nE_F \qquad \mathrm{J\,m^{-2}}\\ &\approx \frac{\pi \hbar^2 n^2}{m}\qquad\mathrm{J\,m^{-2}} \end{align} | \begin{align} g &\approx \left(nE_F-\frac{\pi^2 D(E_F)}{18}(k_BT)^2\right) \quad \mathrm{J\,m^{-3}}\\ &\approx \left(\frac{\hbar^2}{2m}(3\pi^2)^{\frac{2}{3}}n^{\frac{5}{3}}-\frac{m (3\pi^2n)^{\frac{1}{3}}}{18\hbar^2}(k_BT)^2\right)\quad\mathrm{J\,m^{-3}} \end{align} |
