Calcite crystals contain scattering planes separated by \(0.30 \mathrm{nm} .\) What is the angular separation between first and second-order diffraction maxima when X-rays of \(0.130 \mathrm{nm}\) wavelength are used?

X-ray diffraction (XRD) is a powerful technique used to study the structure of crystalline materials. When X-ray beams interact with the atoms within a crystal, they are scattered in various directions. Due to the uniform arrangement of atoms in a crystal lattice, these X-rays can constructively or destructively interfere with each other.

The conditions necessary for constructive interference are described by Bragg's Law, which connects the angle of these scattered rays, known as the diffraction angle, with the distance between the crystal planes and the wavelength of the X-rays. By measuring the angles and intensities of these diffracted beams, scientists can generate three-dimensional images of the electron densities within the crystal, allowing the position of the atoms to be established. This data is crucial for understanding the material's properties and functions.

The crystal lattice is a three-dimensional structure consisting of atoms or molecules arranged in a repeating pattern. The lattice is characterized by its geometry, including the length and angles between the lattice points. These points are mathematically precise locations where an atom, ion, or molecule can be found.

The spacing between parallel planes of atoms, known as the interplanar distance, directly affects how X-rays are diffracted as they travel through the crystal. When X-rays interact with these planes, they will either reinforce each other, creating a diffraction peak that can be detected, or cancel each other out if conditions for constructive interference are not met. By carefully analyzing the diffraction pattern, the details of the crystal lattice, including atomic spacing and lattice imperfections, can be understood.

Constructive interference is a phenomenon that occurs when two or more waves overlap and combine to produce a wave of greater amplitude. For this to happen in the context of X-ray diffraction, the path length difference between the waves reflected off different layers of atoms must be an integer multiple of the wavelength.

Bragg's Law provides a condition under which constructive interference occurs for x-rays reflecting off the crystal planes at specific angles. Each order of diffraction, represented by an integer 'n' in Bragg's formula, corresponds to a constructive interference scenario. Higher order maxima are a result of multiple wavelengths being in phase after scattering, and this principle is what makes it possible to calculate the angular positions of the diffracted rays and understand the crystal structure in depth.