Question

Can the periodicity in k-space of the electronic band structure be understood as a result of aliasing?

Answer 1

In the case of phonons in periodic solids, the picture is quite intuitive. The motion of the atoms in the lattice is described as a continuous wave (amplitude as a function of position and time). However, since it only makes sense to define an amplitude at the position of the atoms, a discrete sampling of the continuous wave is made, leading to ambiguity in the wavelength of the oscillation. This is because waves of different wavelengths can be used to describe the same set of samples (see figure). This aliasing effect thus naturally leads to the concept of Brillouin zones for phonons. To what extent can the periodicity of the electronic band structure be understood using the same picture? In the case of electrons, the wave under consideration is their quantum wavefunction. The difference with the phonon wave is that it is not only the value of the wavefunction at the atomic positions that is relevant: the wavefunction is equally real in between the atoms. Thus, it is not clear to me whether the same argument of discrete sampling can be applied.  

Answer 2

$<marquee><b><u>While the atoms are discrete, the forces between them are continuous in position. A phonon is clearly a wave disturbance propagating through the lattice, and must be explained using the same Bloch restrictions as the electronic wave functions.$