Rationalizing Systematic Absences in Diffraction patterns.
Let us see how Eq (19.1) works in detail. The short table below (Table 19.1) gives the values of h2 + k2 + l 2 for different Miller planes.
h k l
100
110
111
200
210
211
220
300
221
310
h2 + k2 + l 2
1
2
3
4
5
6
8
9
9
10
Table (19.1) h2 + k2 + l 2 for Miller planes of a simple cubic lattice
Continuing the table for higher values of h k l we will see that the integers 7 and 15 are absent for h2 + k2 + l 2 because the sums of squares of three integers can not be 7 or 15. Of course, there are other higher missing integers for the sums too!
We conclude from the above discussion that if values corresponding to integer square sums of 7 and 15 are absent, but all other integers are present between 1 to 15, then the unit cell must correspond to a simple cubic (sc) lattice.
In Fig 19.1 the Schematic diffraction patterns for the three cubic lattices (SC, BCC and FCC) are shown. It is seen that in the case of BCC, all reflections with h + k + l = odd are absent and that in the case of FCC, only the reflections with all h, k, l even or all h k l odd are present.
Figure 19.1 Diffraction patterns for SC, BCC and FCC lattices.
values corresponding to integer square sums of 7 and 15 are absent, but all other integers are present between 1 to 15, then the unit cell must correspond to a simple cubic (sc) lattice. 
