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For a free electron gas the procedure for determining the density of states is as follows.
Apply periodic boundary conditions to the free electron waves over a cube of side L. This gives us that there is one state per volume 2[itex]\pi[/itex]/L3=2[itex]\pi[/itex]/V
And from there we can find the number of states at a given energy E by multiplying by the volume of a sphere at E in k space.
One big problem with this is however: Why do we assume that material is necessarily a cube? What if we worked with a ball of metal?
Apply periodic boundary conditions to the free electron waves over a cube of side L. This gives us that there is one state per volume 2[itex]\pi[/itex]/L3=2[itex]\pi[/itex]/V
And from there we can find the number of states at a given energy E by multiplying by the volume of a sphere at E in k space.
One big problem with this is however: Why do we assume that material is necessarily a cube? What if we worked with a ball of metal?