Germanium has the same structure as silicon, but the unit cell size is different because Ge and Si atoms are not the same size. If you were to repeat the experiment described in the previous problem but replace the Si crystal with a Ge crystal, would you expect the X rays to be diffracted at a larger or smaller angle u?

Key Concepts

Bragg's Law

Bragg's Law relates the angle of diffraction of X-rays to the spacing between atomic planes in a crystal. It is expressed as nλ = 2d sin(θ), where n is an integer, λ is the wavelength of the X-rays, d is the distance between crystal planes, and θ is the angle of diffraction. Understanding this law is crucial for predicting how changes in atomic spacing affect diffraction angles.

Crystal Structure and Unit Cell

The crystal structure refers to the orderly arrangement of atoms in a crystalline solid, characterized by its unit cell, the smallest repeating unit that defines the entire structure. In this case, germanium (Ge) and silicon (Si) have similar structures but different unit cell sizes due to the differing atomic radii. This difference in size affects the spacing of atomic planes, which in turn influences the diffraction pattern.

Atomic Radius and Diffraction Angles

The atomic radius is the distance from the nucleus of an atom to the outermost electron shell. In the context of X-ray diffraction, a larger atomic radius in Ge compared to Si results in a larger unit cell and increased interplanar spacing. Consequently, this larger spacing leads to a change in the diffraction angle, typically resulting in smaller angles for larger unit cells, as per Bragg's Law.