Systematic absences are ‘missing’ diffraction intensities on a powder Xray diffraction spectrum of intensity versus 2θ. Consider the possible values of h, k and l that satisfy Bragg’s equation of for a primitive cubic unit cell:
hkl  100  110  111  200  210  211  220  300  221 
h^{2 }+ k^{2 }+ l^{2}  1  2  3  4  5  6  8  9 
9 
7 is missing in the h^{2 }+ k^{2 }+ l^{2} sequence, as h, k and l are integers. An extension of the sequence reveals that 15, 23, etc. are missing as well. Hence, the spectrum for a primitive cubic unit cell has a specific set of missing intensities. Likewise, different missing intensity sets are expected for other unit cells. For example, a sample with grains that are composed of facecentred cubic unit cells (figure I below) has scattered rays from {200} planes that are out of phase with rays from {100} (figure II), resulting in destructive interference with no {100} intensity peak on the spectrum.
In fact, for all facecentred cubic unit cells (FCC), intensity peaks are observed when the integers h, k and l are either all odd or all even, e.g. {111}, {200}, {220}, …. A mathematical treatment of this requires the knowledge of the structure factor, which is described in a later section. The structure factor for a bodycentred cubic unit cell (BCC) also shows that systematic absences do not appear when h + k + l = 2n, i.e. when the sum of the Miller indices is even, e.g. {110}, {200}, {211}, ….
Question
Identify the cubic unit cells of polonium and aluminium and determine their dimensions using the following powder Xray diffraction spectral data:
2θ_{i}  2θ_{1}  2θ_{2}  2θ_{3}  2θ_{4}  2θ_{5}  2θ_{6}  2θ_{7}  2θ_{8} 
Po 
12.1^{o}  17.1^{o}  21.0^{o}  24.3^{o}  27.2^{o}  29.9^{o}  34.7^{o} 
36.9^{o} 
Al 
38.4^{ o}  44.7^{ o}  65.0^{ o}  78.1^{ o}  82.3^{ o}  98.9^{ o}  111.8^{ o}  116.4^{ o} 
λ_{Al }= 71.0 pm and λ_{Po }= 154.1 pm
Answer
Let’s tabulate the systematic absences of the three cubic unit cells for reference, knowing that i) 7 is missing for a primitive cubic cell, ii) the integers h, k and l for an FCC cell are either all odd or all even, and iii) that h + k + l = 2n for a BCC cell.
hkl  100  110  111  200  210  211  220  300 
221 

h^{2}+ k^{2} + l^{2} 
Primitive 
1  2  3  4  5  6  8  9 
9 
BCC 
–  2  –  4  –  6  8  – 
– 

FCC 
–  –  3  4  –  –  8  –  – 
Squaring both sides of eq13, we have
If the unit cell is primitive, the successive ratios of will produce the sequence 1,2,3,4,5,6,8,9… since from eq14, and . If the unit cell is BCC, we need to multiply the successive ratios of by the factor 2 to obtain the fingerprint sequence of 2,4,6,8… since . By the same logic, the multiplication factor is 3 if the unit cell is FCC. Tabulating the ratios,


Po 
1.00  1.99  2.99  3.99  4.98  5.99  8.01 
9.02 
Al 
1.00  1.34  2.67  3.67  4.00  5.34  6.34 
6.68 
it is clear that the unit cell for polonium is primitive cubic while that for aluminium is FCC. To verify the fingerprint sequence of an FCC unit cell, we multiply the aluminium ratios by the factor of 3:


Al x3 
3.00  4.01  8.01  11.01  12.01  16.01  19.02 
20.04 
Finally, the dimensions of the unit cell of each sample is obtained by substituting any corresponding set of θ and hkl values in eq14:

a 
Po 
337 pm 
Al 
406 pm 
As mentioned above, to mathematically show that the integers h, k and l for an FCC cell are either all odd or all even, and that h + k + l = 2n for a BCC cell, we have to explain what structure factor is. To do that, we need to first understand the Laue equations and the scattering factor.