When X rays of wavelength \(0.090 \mathrm{nm}\) are diffracted by a metallic crystal, the angle of first-order diffraction \((n=1)\) is measured to be \(15.2^{\circ} .\) What is the distance (in pm) between the layers of atoms responsible for the diffraction?

X-ray diffraction (XRD) is a technique used to study the structure of crystalline materials. It relies on the principle that when X-rays hit a crystal, they are scattered in many different directions. If these scattered waves are in phase, they will constructively interfere and produce a diffracted beam of X-rays.

When an X-ray beam with a known wavelength, such as the one mentioned in the exercise with a wavelength of 90 pm, interacts with the crystal layers, a diffraction pattern is produced that can be used to determine various properties of the crystal, including the distances between the layers of atoms.

Crystallography is the science of determining the arrangement of atoms within a crystalline solid. One of the most important tools in crystallography is X-ray diffraction, which allows scientists to probe the internal structure of crystals. Through analyzing diffraction patterns, the position of atoms and the distances between layers of atoms in the crystal lattice can be measured. This process is pivotal for material science, chemistry, physics, and biology, as it aids in understanding material properties on an atomic level.

The exercise at hand involves using X-ray diffraction to determine the interlayer spacing of a metallic crystal, a common application of crystallography.

The term 'interlayer spacing' refers to the distance between layers of atoms within a crystal. This parameter is crucial because it is directly related to the material's properties, including strength, flexibility, and how it interacts with electromagnetic waves. In the given problem, the interlayer spacing is what we are trying to find using Bragg's Law.

During the solution process, we use the known wavelength and diffraction angle to calculate the unknown interlayer spacing. As shown, after applying the relevant equations, the result is that the spacing is 170 picometers.

The diffraction angle, commonly represented by the Greek letter theta (\( \theta \)), is the angle at which waves are scattered from the crystal lattice. It is a critical variable in Bragg's Law, which relates the angle to the wavelength of the incident waves and the interlayer spacing in the crystal.

In the problem statement, the angle of first-order diffraction is given as 15.2 degrees, which is then transformed into radians for the calculation, as most mathematical operations in trigonometry require the angle to be in radians. After determining the sine of the angle, it's used alongside the wavelength to calculate the interlayer spacing, showcasing the practical use of the diffraction angle in solving crystallographic problems.