Products Rewards
from HOLOOLY

We are determined to provide the latest solutions related to all subjects FREE of charge!

Enjoy Limited offers, deals & Discounts by signing up to Holooly Rewards Program

HOLOOLY

HOLOOLY
TABLES

All the data tables that you may search for.

HOLOOLY
ARABIA

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

HOLOOLY
TEXTBOOKS

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

HOLOOLY
HELP DESK

Need Help? We got you covered.

## 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).

## 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).