# Free Electron Gas in Crystals with Unequal Dimensions

| If we consider a crystal with dimensions Lx, Ly, Lz, it has the volume V = Lx· Ly· Lz.
|  | All we have to do is to replace the periodic boundary conditions y(x + L) = y(x) by:
y(x + Lx, y, z) | = | y(x , y + Ly, z) | = | y (x, y, z + Lz) | = | y(x, y, z)
--- | --- | --- | --- | --- | --- | ---
| This leads to simple expressions for the allowed wave vectors k:
kx | = | 0, ± | 2p Lx | , ± | 4p Lx | , ...
--- | --- | --- | --- | --- | --- | ---
ky |  | 0, ± | 2p Ly | , ± | 4p Ly | , ...
kz |  | 0, ± | 2p Lz | , ± | 4p Lz | , ..
|  | The pre-exponential factor, which was (1/L)3/2, now changes to (1/V) 1/2.
| Since all relevant quantities are usually expressed as densities, i.e. divided by V, and the quantization of k is usually given up in favor of a continuous range of k's, we may just as well stick to the more simple description of a crystal with equal sides - the results are the same.

With frame

2.1.1 Essentials of the Free Electron Gas

© H. Föll (Semiconductors - Script)