Systematic absences

Lattice   Conditions for general absences

P           None
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 centeringscrew 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 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 21212symmetry.

The equation for the structure factor for reflection 
Fhkl is:

F_{hkl}=\sum_{j=1}^{n} f_{j} e^{2\pi i(hx_{j}+ky_{j}+lz_{j})

If we consider that the 
c-axis of the unit cell is a twofold screw, the structure factor for reflections F00is:

F_{00l}=\sum_{j} f_{j} e^{2\pi i(lz_{j})

Consider that for every atom 
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 from that of their symmetry-related atoms j':

F_{00l}=\sum_{j} f_{j} e^{2\pi i(lz_{j})}+\sum_{j'} f_{j'} e^{2\pi i(lz_{j'})

j and j' are identical, they have same scattering factor f, so we can write:

F_{00l}=\sum_{j} f_{j} (\sum_{j}e^{2\pi i(lz_{j})}+\sum_{j'}e^{2\pi i(lz_{j'})})

Now we substitute the values of j (that is z) and j' (z+1/2) in the above equation:

F_{00l}=\sum_{j} f_{j} (\sum_{j}e^{2\pi ilz}+\sum_{j'}e^{2\pi i(lz+1/2)})

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.

F_{00l}=\sum_{j} f_{j} (\sum_{j}[e^{2\pi ilz}+e^{2\pi i(lz+1/2)}])

Principles of X-ray Crystallography By Li-ling Ooi