2.13: Periodic boundary conditions in 3-D

In three dimensions, again only discrete values of k are allowed. This time the volume of k-space per allowed state is

\[ \Delta k^{3}=\Delta k_{x}\Delta k_{y}\Delta k_{z}=\frac{2\pi}{L_{x}}\frac{2\pi}{L_{y}}\frac{2\pi}{L_{z}}=\frac{8\pi^{3}}{V} \nonumber \]

where V is the volume of the material.

In summary, the k-space occupied per state is

Table \(\PageIndex{1}\): The k-space occupied per state in 1, 2 and 3 dimensions.

1-d	\(\Delta k=\frac{2\pi}{L}\)
2-d	\(\Delta k^{2}=\frac{4\pi^{2}}{A}\)
3-d	\(\Delta k^{3}=\frac{8\pi^{3}}{V}\)

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