# Points of symmetry in $k$-space

Can you relate a point in the reciprocal space with a vector in real space?

How do I find the family of planes that represent a point of symmetry in the Brillouin zone?

For example, germanium has its characteristic gap at the center of the first Brillouin zone, that is, at the gamma point. Which will be the family of planes that will present this characteristic gap?

• Germanium is an indirect gap material - while the top of the valence band is at the $\Gamma$ point, the bottom of the conduction band is towards the L point (but not at it). As for real space vs reciprocal space, no you can't relate them one-to-one in the same way you can't connect a point in one waveform to a point in its Fourier transform. – Jon Custer Oct 24 '18 at 17:03
• k-space and real space are two different spaces. It's like the position and the frequency domains during a Fourier transform. So technically, you relate each point in one of the spaces to the whole reciprocal space. – fgoudra Oct 25 '18 at 18:31
• Thanks to you. Yes, I understand. A point in k-space represent a whole set of periodic planes. But, going back to the example of Ge. If you see this indirect gap in an energy band plot, how i can decide the cutting directions of wafers in a monocrystalline ingot? – Diego Cartaya Espina Oct 25 '18 at 22:45

The gamma point represents waves with $$k=0$$, infinite wavelength. In the tight-binding approximation, this means a constant value of the phase factor for the atomic orbitals.
Germanium is not an easy example. It has an indirect band gap. But many salts have direct band gaps at the gamma point, for example MgO. In a tight-binding picture, at $$\Gamma$$ the phase factors are the same throughout the crystal. The lower band corresponds to electron wave functions with mostly oxygen $$2p$$ character, the upper band to wave functions with mainly magnesium $$3sp$$ character.
When waves have the same periodicity as the lattice in some direction, this will give rise to standing waves. There will be a gap at the zone boundary between waves with the same $$k$$ but locations of nodes and antinodes interchanged. So then there is a direction involved: along cubic directions for gaps at the X point, along the body diagonals for gaps at the L point. This may be relevant for surface properties at different cuts (image states on silver single crystals come to mind), but it does not affect bulk phenomena. And there may be complications due to surface reconstruction. Is this about a photoemission experiment?