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

## Density of states

`$$ D(W) $$`

Density of states of an electron gas - with free electrons that do not interact with each other.

The density of states represents the states per energy interval, in this case for both spin directions of the electron. To get the density of states for only one spin direction, multiply the density of states by \(\frac{1}{2}\). And, to get the density of states \(g(W)\) per energy interval AND per volume, multiply by \(\frac{1}{V}\):`\[ g(W) ~=~ \frac{1}{2\pi^2} \, \left(\frac{2m}{\hbar^2}\right)^{3/2} \, \sqrt{W} \]`

## Volume

`$$ V $$`Unit

`$$ \mathrm{m}^3 $$`

Box-shaped volume of a solid (for example, a metal) in which the electron gas is located:

`$$ V ~=~ L_{\text x} \, L_{\text y} \, L_{\text z} $$`## Mass

`$$ \class{brown}{m} $$`Unit

`$$ \mathrm{kg} $$`

Mass of the Fermi particle. In the case of an electron gas, it is the (effective) mass of the electron.

## Energy

`$$ W $$`Unit

`$$ \mathrm{J} $$`

Energy of a state which the electron can occupy. In the three-dimensional electron gas there are more states at higher energies because of \( D \sim \sqrt{W}\).

## Reduced Planck's constant

`$$ \hbar $$`Unit

`$$ \mathrm{Js} $$`

Reduced Planck's constant is a physical constant and has the value:

`$$ \hbar ~=~ \frac{h}{2\pi} ~=~ 1.054 \, 571 \, 817 \cdot 10^{-34} \, \text{Js} $$`