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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

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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:

$$\frac{\pi}{L_x}\cdot\frac{\pi}{L_y}\cdot\frac{\pi}{L_z}=\frac{\pi^3}{L_xL_yL_z}$$

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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 $\frac{π^3}{L_xL_yL_z}$.

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