## Density of states (1d, 2d, 3d) of a free electron gas

Here you will learn how to derive the 1d, 2d, and 3d density of states of a free electron gas from the k-volume and the dispersion relation.

Structure, transport and interaction in solid matter

Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

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Lesson ## Density of states (1d, 2d, 3d) of a free electron gas

Here you will learn how to derive the 1d, 2d, and 3d density of states of a free electron gas from the k-volume and the dispersion relation.

## Related formulas

Formula ## Free electron gas in 3d (density of states)

`$$ D(W) ~=~ \frac{V}{2\pi^2} \, \left(\frac{2m}{\hbar^2}\right)^{3/2} \, \sqrt{W} $$`Formula ## Free electron gas in 2d (density of states)

`$$ D ~=~ \frac{A}{\pi} \, \frac{2m}{\hbar^2} $$`Formula ## Free electron gas in 1d (density of states)

`$$ D(W) ~=~ \frac{L}{\pi} \, \left(\frac{2m}{\hbar^2}\right)^{1/2} \, \frac{1}{\sqrt{W}} $$`

` $$ P(W) ~=~ \mathrm{e}^{ -\frac{ W - \mu }{ k_{\text B} \, T}} $$ `