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Chapter 9

Q. 9.7

Calculate the Fermi energy and Fermi temperature for copper.

Strategy Equation (9.42) can be used to compute the Fermi energy, provided the number density of conduction electrons is known. That number for copper is given in Table 9.3. (See also Problem 27.)

E_{ F }=E_{1}\left(\frac{3 N}{\pi}\right)^{2 / 3}=\frac{h^{2}}{8 m}\left(\frac{3 N}{\pi L^{3}}\right)^{2 / 3} (9.42)

Table 9.3 Free-Electron Number Densities for Selected Elements at T = 300 K
Element N/V \left(\times 10^{28} m ^{-3}\right) lement N/V \left(\times 10^{28} m ^{-3}\right)
Cu 8.47 Mn (\alpha) 16.5
Ag 5.86 Zn 13.2
Au 5.9 Cd 9.27
Be 24.7 Hg (78 K) 8.65
Mg 8.61 Al 18.1
Ca 4.61 Ga 15.4
Sr 3.55 In 11.5
Ba 3.15 Sn 14.8
Nb 5.56 Pb 13.2
Fe 17
From N. W. Ashcroft and N. D. Mermin, Solid State Physics, Philadelphia: Saunders College (1976).

Step-by-Step

Verified Solution

The number density of conduction electrons in copper is given by Table 9.3 as 8.47 \times 10^{28} m ^{-3}. We use this value of N / L^{3} in Equation (9.42) to fi nd

 

\begin{aligned}E_{ F } &=\frac{\left(6.626 \times 10^{-34} J \cdot s \right)^{2}}{8\left(9.11 \times 10^{-31} kg \right)}\left[\frac{3\left(8.47 \times 10^{28} m ^{-3}\right)}{\pi}\right]^{2 / 3} \\&=1.13 \times 10^{-18} J =7.03 eV\end{aligned}

 

Within rounding errors, this result is equivalent to that given in Table 9.4.

 

T_{ F }=\frac{E_{ F }}{k}=\frac{7.03 eV }{8.62 \times 10^{-5} eV / K }=8.16 \times 10^{4} K

 

Fermi energies and Fermi temperatures for other common conductors are listed in Table 9.4. Note that E_{ F } changes little between T = 0 and room temperature.

 

Table 9.4 Fermi Energies (T = 300 K), Fermi Temperatures, and Fermi Velocities for Selected Metals
Element E_{ F }( eV ) T_{ F }\left(\times 10^{4} K \right) u_{ F }\left(\times 10^{6} m / s \right) Element E_{ F }( eV ) T_{ F }\left(\times 10^{4} K \right) u_{ F }\left(\times 10^{6} m / s \right)
Li 4.74 5.51 1.29 Fe 11.1 13 1.98
Na 3.24 3.77 1.07 Mn 10.9 12.7 1.96
K 2.12 2.46 0.86 Zn 9.47 11 1.83
Rb 1.85 2.15 0.81 Cd 7.17 8.68 1.62
Cs 1.59 1.84 0.75 Hg 7.13 8.29 1.58
Cu 7 8.16 1.57 Al 11.7 13.6 2.03
Ag 5.49 6.38 1.39 Ga 10.4 12.1 1.92
Au 5.53 6.42 1.4 In 8.63 10 1.74
Be 14.3 16.6 2.25 Tl 8.15 9.46 1.69
Mg 7.08 8.23 1.58 Sn 10.2 11.8 1.9
Ca 4.69 5.44 1.28 Pb 9.47 11 1.83
Sr 3.93 4.57 1.18 Bi 9.9 11.5 1.87
Ba 3.64 4.23 1.13 Sb 10.9 12.7 1.96
Nb 5.32 6.18 1.37
From N. W. Ashcroft and N. D. Mermin, Solid State Physics, Philadelphia: Saunders College (1976).