Electron gas is a collection of non-interacting electrons. If these electrons are confined to certain volume (for example, cube of metal), their behavior can be described by the wavefunction which is a solution to the particle in a box problem in quantum mechanics. Allowed energy states for any electron is given by: $$ \epsilon = \frac {h^2} {8mL^2} ({n_x}^2 + {n_y}^2 + {n_z}^2) = \frac {h^2} {8mL^2} n^2 $$
where $n_x$, $n_y$ and $n_z$ are quantum numbers which determine allowed momentum components of the electron:
$$p_x = \frac {hn_x}{2L}, p_y = \frac {hn_y}{2L}, p_z = \frac {hn_z}{2L}$$
Allowed states of the electrons can be visualized in the n-space, where the states can be shown graphically in the $n_x, n_y, n_z$ coordinate system. Every coordinate in the $n$-space actually represents two possible states for certain energy level because there are two electrons with opposite spin states for certain energy level. If we know maximum occupied energy level (Fermi energy), we can calculate maximum quantum number $n_{max}$ from the first equation.
To determine the number of points in the $n$-space for known Fermi energy, we can first count the ''volume'' of the one electron state in the $n$-space (particular $n_x, n_y, n_z$ coordinate). To do so, we note that the difference between two ''neighboring'' quantum numbers is equal to one (quantum numbers are positive integers) which means that the ''volume'' of any cube composed of neighboring points in the n-space is equal to 1. Each cube is enclosed by eight points in the n-space, but since any point is shared by the 8 neighboring cubes, there is actually only one point per each cube since only 1/8 of each point belongs to the individual cube. This means that the ''volume'' of the one electron state in the n-space is equal to 1.
When the ''volume'' per electron state in the $n$-space is known, we can calculate the total number of points in the $n$-space (for certain Fermi energy, $E_f$) by taking the total volume in the $n$-space and dividing it by the ''volume'' per electron state (which is equal to 1, so total number of points equals the total volume).
What I don't understand is that the total volume taken is equal to the 1/8 volume of the sphere with the ''radius'' equal to $n_{max}$: $$V_f = \frac {1} {8}\frac {4 \pi n_{max}^3} {3}$$ According to my understanding, it doesn't really make sense to take the sphere as the total volume because we determined the volume per state as being equal to the volume of the cube with sides having a length 1. Because of this, we should know the total volume of all the cubes for certain $E_f$ and since the volume per state is equal to 1, total number of points is equal to the total volume as mentioned previously.
If this is the case, why is volume of the sphere taken as the paramount when total volume of the cubes should be taken?
