## 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)

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

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

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

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Derivation ## Dispersion Relation of the Monatomic Lattice Vibration

Here you will learn how to derive the dispersion relation for a linear monoatomic chain using Hooke's law.

## Related formulas

Formula ## Dispersion Relation (Crystal with Monatomic Basis)

`$$ \omega(k) ~=~ \sqrt{\frac{4 D}{\class{brown}{m}} \sin^2\left(\frac{ka}{2}\right)} $$`Formula ## Group Velocity (Oscillation of a Monatomic Chain)

`$$ v_{\text g} ~=~ \sqrt{\frac{D \, a^2}{\class{brown}{m}}} \, \cos\left(\frac{1}{2} \, k \, a\right) $$` - 3
Derivation ## Dispersion Relation of the Diatomic Lattice Vibration

Here you will find the derivation of the acoustic and optical dispersion relation for a crystal with a diatomic basis.

## Questions & Answers

## Related formulas

Formula ## Dispersion Relation (Crystal with Diatomic Basis)

`$$ \begin{align} \omega_{\pm}^2 ~&=~ D \, \left( \frac{1}{m_1} + \frac{1}{m_2} \right) \\\\ ~&\pm~ D \, \sqrt{\left(\frac{1}{m_1} + \frac{1}{m_2}\right)^2 ~-~ \frac{4}{m_1 \, m_2}\,\sin^2\left(\frac{k\,a}{2}\right) } \end{align} $$`## Related Illustrations

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Lesson ## Epitaxy and 3 Important Techniques for Fabrication of Heterostructures

Here you will learn the basics of epitaxy (crystal growth), as well as different growth models and epitaxial techniques.

## Questions & Answers

## Related Illustrations

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Lesson ## Photolithography: How to Make a Circuit in 7 Easy Steps

Here you will learn how integrated circuits in semiconductor technology are manufactured step by step using lithography technique.

## Related Illustrations