For a cubic crystal the diffraction line from the planes with (h^2+k^2+l^2)=8 is observed at the

step 1 Okay, nice and easy question. We need to calculate the distance between the plans of atom respon

For a cubic crystal the diffraction line from the planes...

For a cubic crystal the diffraction line from the planes with $\left(h^{2}+k^{2}+l^{2}\right)=8$ is observed at the angle of diffraction $10.23^{\circ}$. If only one line is observed at an angle lower than this, what is the crystal structure? Assuming the wavelength of $x$ -rays used as $0.71 \AA$, calculate the lattice parameters.

Key Concepts

Bragg’s Law

Bragg’s law (2d sin? = ?) is the fundamental relation in x?ray diffraction that connects the wavelength of the x-rays, the diffraction angle, and the spacing between crystallographic planes. In the context of cubic crystals, it is used along with the relation for d-spacing to determine lattice parameters by equating the calculated d with a measured diffraction angle.

Interplanar Spacing in Cubic Crystals

For cubic crystals the spacing d between crystal planes indexed by (h k l) is given by d = a/?(h²+k²+l²), where a is the lattice parameter. This relation allows one to directly relate the observed diffraction peaks (characterized by their Miller indices sum of squares) to the dimensions of the unit cell.

Structure Factor and Systematic Absences

The structure factor takes into account the arrangement of atoms within the unit cell (the basis) and determines which reflections are present or absent in the diffraction pattern. Due to the interference from atoms at different positions, certain reflections vanish (systematic absences) depending on the symmetry and basis of the crystal structure. This concept is crucial for distinguishing among different cubic structures based on which diffraction peaks appear.

NaCl (Rock Salt) Structure

In the NaCl structure, which is based on a face?centered cubic lattice with a two?atom basis, the structure factor results in the extinction of reflections where h, k, l are not all even. Consequently, the lowest observable reflection comes from the (200) planes, and the next allowed reflection (e.g. from the (220) set with h²+k²+l² = 8) appears at a slightly higher angle. The observation of just one diffraction line at an angle lower than that from the planes with h²+k²+l² = 8 is a signature of the NaCl structure, and the lattice parameter can be calculated using the corresponding d–spacing.

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