# Density Of states derivation

In the aspect of density of state derivation or simply assuming the frequency of a solid as a continuous distribution we have to come up with an equation expressing the density of states. Its derived by the concept of wave vector $$k$$. It has introduced a 3D visualization of $$k$$ . The $$k$$ space volume taken up by each allowed state is π^(3 )/(L_x L_y L_z ). I really don't understand the previous line. How it came? I do understand the $$k$$ space but fail to realize what does it mean by the allowed states are separated by π/(L_x L_y L_z ). Im including the pdf file for better understanding. http://web.eecs.umich.edu/~fredty/public_html/EECS320_SP12/DOS_Derivation.pdf

From the PDF, $$k_xL_x = \pi n_x,\ k_yL_y = \pi n_y,\ k_zL_z = \pi n_z,\text{ for } n_x,n_y,n_z \text{ integers}$$
Then, $$k_x = \frac{\pi}{L_x}n_x,\ k_y = \frac{\pi}{L_y}n_y,\ k_z = \frac{\pi}{L_z}n_z$$
So the allowed values are separated by $$\frac{\pi}{L_x}$$ for $$k_x$$, $$\frac{\pi}{L_y}$$ for $$k_y$$, and $$\frac{\pi}{L_z}$$ for $$k_z$$. Now if you think this in a 3-D $$\mathbf{k}$$-space, a state, which is a cube-ish portion of this space, occupies the product of these separations:
$$\frac{\pi}{L_x}\cdot\frac{\pi}{L_y}\cdot\frac{\pi}{L_z}=\frac{\pi^3}{L_xL_yL_z}$$
The state is uniquely set by ($$k_x$$, $$k_y$$, $$k_z$$). But because of standing wave condition not every one ($$k_x$$, $$k_y$$, $$k_z$$) is available. And available combinations of ($$k_x$$, $$k_y$$, $$k_z$$) are the separated points in K-space and the nearby points separated by distance of $$\frac{π}{L_x}$$ on X-axis, $$\frac{π}{L_y}$$ on Y-axis etc. If you will draw it(in 2 dimensional case, for example), you will see that every single point "occupies" a square with area of $$(\frac{π}{L_x})(\frac{π}{L_x})$$, which means, that in area of $$\frac{π^2}{L_xL_y}$$ around of this one point there is no other points. And in 3 dimensional case your square with area of $$\frac{π^2}{L_xL_y}$$ will be multiplied by $$\frac{π}{L_z}$$ to become a cube with volume $$\frac{π^3}{L_xL_yL_z}$$.