Lattice Conditions for general absences
A k + l . 2n + 1 (i.e., the sum of k and l is odd)
B h + l . 2n + 1 (i.e., the sum of h and l is odd)
C h + k . 2n + 1 (i.e., the sum of h and k is odd)
F Reﬂections must have either all even or all odd indices to be observed
Mixed odd and even indices are not allowed
I h + k + l . 2n + 1 (i.e., the sum of the indices is odd)
Certain symmetry elements in the unit cell announce themselves in the diffraction pattern by causing specific reflections to be missing (intensity = 0). In particular, cell centering, screw axes, and glide plane operations can be identified by the fact that they cause certain groups of diffraction points to be systematically absent.
A twofold screw axis 21 along c causes all 00l reflections having odd values of l to be missing. In the International Tables, this is shown as "Conditions limiting possible reflections", and in the case of P21 the only condition is shown as l = 2n; this means that only even-numbered reflections are present along the l-axis (reflections 001, 003, 005, along with all other 00l reflections in which l is an odd number, are missing).
In space group P212121, the unit cell possesses twofold screw axes on all three edges, so odd-numbered reflections on all three principle axes of the reciprocal lattice (h00, 0k0, 00l) are missing. Therefore the presence of only even-numbered reflections on the reciprocal-lattice axes will announce a unit cell with P 212121 symmetry.
The equation for the structure factor for reflection Fhkl is:
If we consider that the c-axis of the unit cell is a twofold screw, the structure factor for reflections F00l is:
Consider that for every atom j with coordinates (x,y,z) in the unit cell there will be an identical atom j' at (-x,-y,z+1/2), where j and j' are called symmetry-related atoms. If we separate the contributions of atom j from that of their symmetry-related atoms j':
Because j and j' are identical, they have same scattering factor f, so we can write:
Now we substitute the values of j (that is z) and j' (z+1/2) in the above equation:
the fj terms are nonzero, so F00l is zero, and the corresponding 00l reflection is missing only if all the summed terms in square brackets equal zero.
Principles of X-ray Crystallography By Li-ling Ooi