The interference cube

The atomic arrangement within the mean elastic free path is resolved by the electrons at the Fermi surface employing the concept of planes and cubes. Hence, the volume per atom is a cube of the side length $a_0=N_0^{-1/3}$ and the average interplanar distance is $a_0$. Take such a stack of atomic planes which are perpendicular to a direction say $\bf e$. Then, the condition for constructive interference of the scattered waves $\bf k_s$ emerging from the incident wave $\bf k_i$ reads as $\bf q e$ = $2\pi/a_0$ where $\bf q$=$\bf k_s - k_i$ is the momentum transfer.

We are interested in transitions with $\vert\bf k_i\vert$=$\vert\bf k_s\vert$= $k_F$ and normal vectors $\bf e$ which correspond to the supposed macroscopic isotropy of the system. This means that any three directions which are mutually perpendicular to one another have to be fully equivalent. Hence, an "interference cube" (fig.1a) which is concentric with the Fermi sphere helps visualizing the interference conditions for scattering between states at the Fermi surface. According to the interference condition mentioned above, transitions from $\bf k_i$ to $\bf k_s$ are accompanied by constructive interference the more the closer components of $\bf q = k_s - k_i$ are approaching the side length of the interference cube, $2\pi/a_0$. This makes the difference to the Brillouin zone of a cubical crystal where extinction results from any misfit to the exact Bragg conditions.