The structure of the NaCl crystal forms reflecting planes 0.541 nm apart. What is the smallest angle, measured from these planes, at which X-ray diffraction can be observed, if X-rays of wavelength 0.085 nm are used?

Bragg's Law is a fundamental principle that is used to determine the positions of atoms within a crystal based on how it diffracts X-rays. When X-ray beams hit a crystalline material, they are scattered in various directions. For certain angles, known as Bragg angles, the scattered waves reinforce each other leading to a strong reflected beam. This phenomenon is known as constructive interference.

The law is mathematically represented by the equation \(n\lambda = 2d\sin{\theta}\), where \(n\) is the order of the diffraction pattern, \(\lambda\) is the wavelength of the X-ray, \(d\) is the distance between crystal planes, and \(\theta\) is the angle of incidence that results in a diffracted beam. To find the smallest angle \(\theta\) at which diffraction can be observed, the first-order diffraction (where \(n=1\)) is considered because it requires the least path difference and hence the smallest angle. By rearranging Bragg's Law and using the given values, one can calculate the required angle for X-ray diffraction to occur.

The crystal structure refers to the orderly, repeating arrangement of atoms, ions, or molecules in a solid. The geometrical pattern, which extends in all three spatial dimensions, is known as a crystal lattice. Different substances have unique crystal structures characterized by a specific lattice type. For example, common table salt (NaCl) forms a cubic lattice, where sodium and chlorine ions alternate, creating a regular and repeating structure.

The distance between planes of atoms in the lattice is a critical parameter for X-ray diffraction and Bragg's Law. In the given exercise, the reflecting planes in the NaCl crystal structure are separated by 0.541 nm. This spacing affects the angle at which X-rays will be diffracted and thus is crucial in determining the angles that satisfy Bragg's condition for visible diffraction patterns.

Constructive interference occurs when two or more waves meet and their displacements combine to produce a wave of greater amplitude. In the context of X-ray crystallography, this happens when waves reflected off different layers of atoms within a crystal coincide in phase. When the path difference between these waves is equal to an integer multiple of their wavelength, the result is an amplified signal, which is what Bragg’s Law describes.

The occurrence of constructive interference is critical for the detection of X-ray diffraction patterns. It is precisely these interference patterns that provide information about the crystal structure. The patterns created by constructive interference can be captured using detectors, allowing scientists to deduce the positions of atoms within the crystal based on the diffraction pattern. Understanding this concept is crucial when working with Bragg's Law and in the field of crystallography.