---
title: "Fermi energy of copper at 0 K"
subject: "Electrical Engineering Material"
chapter: 2
key: "semester_3/electrical_engineering_material:fermi-energy-copper-0k"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/fermi-energy-copper-0k
site: BCT Engineering
---

# Fermi energy of copper at 0 K

Electrical Engineering Material · Chapter 2: Free Electron Theory of Conduction in Metal

- Asked in: 2080 Baisakh, 4 marks; 2068 Shrawan, 8 marks (asked 2 times)
- Video: [Density of states and Derivation of Fermi Energy at 0 K](https://www.youtube.com/watch?v=M7Z6B9MJOb4) — Ravi C.B
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/fermi-energy-copper-0k
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-2

## Question

Calculate the Fermi energy at 0 K for copper. Given its density as 8.96 g/cc and atomic mass as 63.5 g/mol.

## Short answer

Electron concentration n = ρ N_A / M = 8960 kg/m³ × 6.022×10²³ mol⁻¹ / 0.0635 kg/mol = 8.50×10²⁸ m⁻³. Fermi energy at 0 K: E_F0 = (h²/8m_e)(3n/π)^(2/3) = 1.13×10⁻¹⁸ J = 7.05 eV.

## Answer

The Fermi energy at 0 K for a free-electron metal is

$$
E_{F0} = \frac{h^2}{8 m_e}\left(\frac{3n}{\pi}\right)^{2/3}
$$

where $n$ is the free electron concentration.

**Origin of the formula:** at 0 K all states up to $E_{F0}$ are filled. Integrating the density of states $g(E) = \frac{8\pi\sqrt{2}m_e^{3/2}}{h^3}E^{1/2}$ from 0 to $E_{F0}$ gives $n = \frac{8\pi}{3}\left(\frac{2m_e E_{F0}}{h^2}\right)^{3/2}$; solving for $E_{F0}$ gives the expression above.

**Given:** $\rho = 8.96\ \text{g/cm}^3 = 8960\ \text{kg/m}^3$, $M = 63.5\ \text{g/mol} = 0.0635\ \text{kg/mol}$, $N_A = 6.022 \times 10^{23}\ \text{mol}^{-1}$. Copper is monovalent, so each atom gives **one** free electron.

#### Electron concentration

$$
n = \frac{\rho N_A}{M} = \frac{8960 \times 6.022 \times 10^{23}}{0.0635} = 8.50 \times 10^{28}\ \text{m}^{-3}
$$

#### Fermi energy

$$
\begin{aligned}
\left(\frac{3n}{\pi}\right)^{2/3} &= \left(\frac{3 \times 8.50 \times 10^{28}}{\pi}\right)^{2/3} = 1.874 \times 10^{19}\ \text{m}^{-2} \\
\frac{h^2}{8m_e} &= \frac{(6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31}} = 6.025 \times 10^{-38}\ \text{J m}^2 \\
E_{F0} &= 6.025 \times 10^{-38} \times 1.874 \times 10^{19} = 1.13 \times 10^{-18}\ \text{J} \\
&= \frac{1.13 \times 10^{-18}}{1.602 \times 10^{-19}} = 7.05\ \text{eV}
\end{aligned}
$$

The corresponding Fermi speed is $v_F = \sqrt{2E_{F0}/m_e} \approx 1.57 \times 10^{6}\ \text{m/s}$, showing that even at 0 K the top electrons move very fast.

**Answer:** $n = 8.50 \times 10^{28}\ \text{m}^{-3}$, $E_{F0} \approx 1.13 \times 10^{-18}\ \text{J} = 7.05\ \text{eV}$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which formula gives E_F at 0 K?
   A. h^2/8m (3n/pi)^(2/3)
   B. h^2/8m (3n/pi)^(3/2)
   C. h^2/2m (n)^(2/3)
   D. (3n/pi)^(2/3) m/h
   Answer: A. Integrating g(E) up to E_F0 and solving for E_F0 gives this. (Key point: Fermi energy formula)

2. How many free electrons does each copper atom give?
   A. 1
   B. 2
   C. 3
   D. 0.5
   Answer: A. Copper is monovalent. (Key point: Free electrons per atom)

3. How is n computed?
   A. n = rho N_A / M
   B. n = M/(rho N_A)
   C. n = rho M / N_A
   D. n = N_A/(rho M)
   Answer: A. Atoms per volume = density x Avogadro / molar mass = 8.50e28 per m^3. (Key point: Concentration)

4. What is E_F0 for copper?
   A. 7.05 eV
   B. 0.705 eV
   C. 70.5 eV
   D. 1.13 eV
   Answer: A. E_F0 = 1.13e-18 J = 7.05 eV. (Key point: Result)

## Related in Electrical Engineering Material

- [Copper FCC: packing, density and Fermi energy](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-fermi-energy) — Electrical Engineering Material, chapter 1
- [Fermi energy of copper at 0 K and 15°C](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afermi-energy-copper-15c) — Electrical Engineering Material, chapter 2
- [Copper FCC: packing density and atomic concentration](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-packing-concentration) — Electrical Engineering Material, chapter 1
- [Copper FCC: atoms, packing, concentration and density](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-atoms-density) — Electrical Engineering Material, chapter 1
- [Degenerate state, Fermi energy and density of states](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Adegenerate-state-density-of-states) — Electrical Engineering Material, chapter 1
- [Population density and temperature-independent Fermi energy](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afermi-energy-temperature-independent) — Electrical Engineering Material, chapter 2

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