The first order diffraction of \(X\) -rays from a certain set of crystal planes oceurs at an angle of \(11.8^{\circ}\) from the planes. If the planes are \(0.281 \mathrm{~nm}\) apart, what is the wavelength of \(X\) -rays?

X-ray diffraction (XRD) is a powerful tool widely used to analyze the structure of crystalline materials. At its core, XRD is based on the phenomenon that occurs when X-rays are shined on a crystal, causing them to scatter in many directions. However, constructive interference (when the waves amplify each other) only happens at specific angles, determined by the spacing between the crystal planes and the X-ray's wavelength. This is known as Bragg's Law. When X-rays hit the planes within the crystal at the correct angle, they are diffracted, and this diffraction pattern can be measured.

XRD is crucial for determining various properties of the material, such as lattice parameters, crystal structure, and even the presence of defects or impurities within the crystal. As students delve into the world of crystallography and materials science, understanding the principles of X-ray diffraction becomes a foundational aspect of their studies.

Crystal planes are, in effect, imaginary planes that pass through crystal lattice points. Crystals have a periodic structure, and the orientation of these planes is critical for understanding X-ray diffraction. Imagine crystal planes as the slices in a loaf of bread, with each slice having a specific orientation and spacing.

These planes are defined by three indices, known as Miller indices, which denote the orientation of the planes in the crystal structure. The distance between these planes, denoted as 'd' in Bragg's Law, affects the path and hence the diffraction pattern of the X-rays passing through them. The closer the planes are, the higher the angle required for X-rays to meet the conditions of Bragg's Law. Scientists and engineers utilize the concept of crystal planes to interpret the XRD data and gain insights into the material's structure.

Wavelength calculation is an integral part of XRD analysis and is systematically executed using Bragg's Law. Let's break down the steps to derive the wavelength from the given data. When the first-order diffraction pattern is observed at an angle, you have the variables needed to solve for the X-ray's wavelength. The law's formula,
\[\begin{equation} n\textbackslashlambda = 2d\textbackslashsin(\textbackslashtheta) \end{equation}\]

provides a clear equation to follow, where the known diffraction order (n), crystal plane spacing (d), and diffraction angle (θ) can be plugged in to solve for the unknown wavelength (λ). The exercise involves converting units and angles into compatible formats for calculation. Angles are converted to radians for this purpose, as it's a requirement of the sine function when using the formula. After the variables are substituted, the rearranged Bragg's Law equation allows for the computation of the wavelength. This calculated wavelength is significant because it reveals details about the X-ray source and can also be used to further interpret the crystal structure.