systematic absences

Look at a real diffraction pattern and you notice something odd: whole rows of reflections that should be there — perfectly good (hkl) triples — are simply missing, and they go missing in a regular, patterned way. These are the systematic absences (also called extinctions). Far from a nuisance, they are a gift: the exact set of reflections that vanish is a fingerprint that reveals the crystal's centring and much of its space-group symmetry, without your ever needing to solve the full structure.

The mechanism is the structure factor going exactly to zero. Some symmetry element places a second atom in a position whose scattered wave is always precisely out of step with the first for a certain class of (hkl), so the two cancel completely. Lattice centring is the simplest case: a body-centred lattice (extra atom at 1/2,1/2,1/2) kills every reflection with h+k+l odd; a face-centred lattice kills all reflections where h, k, l are mixed even-and-odd, leaving only all-even or all-odd. Screw axes and glide planes cause absences too, but restricted to special rows or zones — a 2_1 screw along b removes 0k0 reflections with k odd; an a-glide removes h0l reflections with h odd.

Reading the absences is how crystallographers deduce the lattice type and narrow the space group before refining anything. The general absences (which affect all hkl) tell you the centring; the special ones (confined to axes or planes) betray screw axes and glide planes. It is genuine detective work with a caveat: some space groups leave identical absence patterns and cannot be told apart by absences alone (and a reflection can also be accidentally near-zero for reasons unrelated to symmetry), so absences narrow the possibilities rather than always pinning down a unique answer.