The metal rubidium has a BCC crystal structure. If the angle of diffraction for the (321) set of planes occurs at \(27.00^{\circ}\) (first-order reflection) when monochromatic \(\mathrm{x}\)-radiation having a wavelength of \(0.0711 \mathrm{~nm}\) is used, compute (a) the interplanar spacing for this set of planes and (b) the atomic radius for the rubidium atom.

Interplanar spacing, often symbolized as 'd', is a fundamental concept in crystallography representing the distance between two adjacent planes of atoms in a crystal lattice. This measurement is crucial as it influences the diffraction patterns observed when x-rays interact with the crystal lattice.

To calculate the interplanar spacing, we utilize Bragg's Law, which is an equation that correlates the incident wavelength of x-rays, the angle of diffraction, and the interplanar spacing. Bragg's Law is given by the formula \(n\lambda = 2d\sin{\theta}\), where:

• '\(n\)' represents the order of the reflected wave, which is usually the first order, hence '1'.
• '\(\lambda\)' is the wavelength of the incident x-ray.
• '\(\theta\)' is the angle of diffraction at which the peak reflection occurs.

For students struggling with this concept, stress the importance of correctly plugging in the given values and solving for 'd'. Understanding Bragg's Law not only helps with homework but also with visualizing the crystalline structure at the atomic level.

Miller indices are a shorthand notation to represent crystallographic planes and directions in a crystalline material. These indices, represented by the letters 'h', 'k', and 'l', reveal the orientation of the atomic planes within the crystal lattice.

Each index denotes the intercepts of a plane with the crystal's axes normalized to the smallest integer values. For example, a plane that cuts the crystal axes at a third, a half, and a fourth would be represented by Miller indices of (321). It's a way of expressing which facets of the crystal are being examined or are responsible for reflection or diffraction.

The indices are not just numerical labels; they play a crucial role in determining interplanar spacing through the formula \(d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}\), where 'a' is the unit cell length. To really grasp this concept, it's beneficial for students to practice with different Miller indices and calculate the corresponding 'd' values to see how crystal structure affects the properties and behavior of materials.

Atomic radius determination in crystallography is essential to understanding the physical properties of materials. The atomic radius, 'r', can be deduced from the geometry of the crystal lattice and is particularly straightforward for structures like Body-Centered Cubic (BCC) or Face-Centered Cubic (FCC).

In BCC crystals, where each corner of a cube is touched by an atom and another atom sits at the very center, a simple geometric relationship exists linking the atomic radius with the edge length of the unit cell. This relationship is given by the formula \(a = \frac{4r}{\sqrt{3}}\). By rearranging and solving for 'r', one can calculate the atomic radius once the unit cell dimension 'a' is known.

Students might find these calculations challenging because they involve abstract spatial reasoning. Encouraging the use of visual aids and 3D models can help in understanding how the atomic radii relate to the unit cell dimensions. Moreover, recognizing the difference in atomic radii in diverse crystal structures allows the prediction of material strength, ductility, and other mechanical properties.