Question

(i) Potassium chloride crystallizes with a face-centred cubic lattice similar to sodium chloride. If the X-ray powder pattern was recorded, what restrictions, if any, would be expected on \(h, k, l\) ? (ii) If the powder neutron diffraction pattern was recorded instead, would the restrictions change?

Short answer

For potassium chloride (KCl) crystallizing in a face-centered cubic (FCC) lattice, the restrictions on the Miller indices \((h, k, l)\) in both X-ray and neutron diffraction patterns are the same: all integers must be either all even or all odd. This is because both techniques rely on the diffraction of radiation by the periodic arrangement of atoms in the crystal lattice, which is related to the lattice's symmetry properties.

Step-by-step solution

Understanding the KCl crystal structure

Potassium chloride crystallizes in a face-centered cubic (FCC) lattice, which is similar to sodium chloride (NaCl). Each potassium ion (K+) is surrounded by six chloride ions (Cl-) in an octahedral arrangement. The Miller indices \((h, k, l)\) are integers used to define the crystal planes and are therefore important for understanding the diffraction patterns of various lattice types, especially those using X-ray and neutron diffraction.

Calculate the crystal planes in FCC lattice

For an FCC lattice, the characteristic restriction on the Miller indices for diffraction is that \(h, k, l\) must be either all even or all odd. In other words, \[h + k + l = 2n\] where \(n\) is an integer. This restriction comes from the systematic absence of certain diffraction peaks in mind due to the symmetry of the FCC lattice. Hence, if the X-ray powder pattern of KCl is recorded, the restrictions on the Miller indices \((h, k, l)\) would be: all integers must be either all even or all odd.

Determine restrictions in neutron diffraction pattern

The neutron diffraction pattern, unlike the X-ray diffraction pattern, is more sensitive to the atomic nucleus and less sensitive to the atomic electron cloud. Therefore, the restrictions on the Miller indices for a neutron diffraction pattern may differ from those for X-ray diffraction. However, for an FCC lattice like KCl, the neutron diffraction pattern will have similar restrictions on the Miller indices as X-ray diffraction. This is because both techniques are based on the diffraction of radiation (X-rays or neutrons) by the periodic arrangement of atoms in the crystal lattice, which is directly related to the symmetry properties of the lattice. So, if the powder neutron diffraction pattern of KCl is recorded, the restrictions on the Miller indices \((h, k, l)\) would be the same as for X-ray diffraction: all integers must be either all even or all odd. In conclusion, for both X-ray and neutron diffraction of potassium chloride, the restrictions on the Miller indices \((h, k, l)\) are that they must be either all even or all odd.