The X-ray diffraction pattern of a cubic crystalline...

The X-ray diffraction pattern of a cubic crystalline substance shows data that correspond to reflections from the $110,200,220,310,222$, and 400 planes. What type of cubic unit cell does the substance have? (Hint: See the table in Problem 29-38.)

Key Concepts

X-ray Diffraction

X-ray diffraction is a powerful technique used to investigate the arrangement of atoms in a crystalline material. By measuring the angles and intensities of diffracted beams, we can determine the spatial distribution of atoms within the crystal. Bragg's law relates the diffraction angles to the distance between atomic layers, making it possible to deduce the geometry of the unit cell and, indirectly, identify the type of lattice present in the crystal.

Miller Indices

Miller indices are a notation system used in crystallography to denote the orientation of a crystal plane. They are determined by taking the reciprocals of the intercepts of the plane with the crystallographic axes and then clearing fractions. These indices are crucial in interpreting diffraction patterns because each set of indices corresponds to a particular family of planes within the crystal, and the conditions under which these planes diffract X-rays provide insight into the symmetry and centering of the unit cell.

Cubic Unit Cells – BCC and FCC

Cubic unit cells, particularly body-centered cubic (BCC) and face-centered cubic (FCC) structures, differ in the arrangement of atoms within the unit cell. In BCC structures, there is an additional atom at the center of the cube, whereas FCC structures have extra atoms at the centers of each face of the cube. These differences lead to distinct diffraction selection rules, which help in identifying the lattice type. For instance, in a BCC lattice, reflections are allowed only when the sum of the Miller indices is even—a rule that aids in differentiating it from the FCC lattice, where a different set of conditions applies.

Structure Factor and Reflection Conditions

The structure factor is a mathematical expression that accounts for the interference of X-rays scattered by the atoms in the unit cell. It determines the intensity of the diffracted beams observed in a diffraction experiment. Specific conditions based on the positions of atoms—such as centering in BCC or FCC lattices—result in systematic absences (or allowed reflections) for certain sets of Miller indices. By analyzing which reflections (families of lattice planes) are present or absent in the diffraction pattern, one can infer the type of cubic unit cell, such as when a pattern with reflections meeting the BCC condition is observed.

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