Calculate the coefficient of thermal expansion of diamond given that the (111) reflection shifts from \(22^{\circ} 2\) ' 25" to 2157 ' 59 " on heating a crystal from \(100 \mathrm{~K}\) to \(300 \mathrm{~K}\) and \(154.0562 \mathrm{pm}\) X-rays are used.

Understanding Bragg's law is essential when exploring the properties of crystalline materials. Bragg's law reveals the conditions under which X-rays are diffracted by a crystal. It relates the wavelength of electromagnetic radiation to the crystal lattice spacing and the angle of incidence at which the wavefronts constructively interfere. The formula is represented as:
\[ n\lambda = 2d\sin(\theta) \]
In this equation, \( n \) is the order of the reflection, which is an integer, \( \lambda \) is the wavelength of the incident X-rays, \( d \) is the lattice spacing, and \( \theta \) is the angle of incidence, also known as the Bragg angle. When a crystal lattice undergoes thermal expansion, the lattice constant \( d \) changes, which in turn affects the angle \( \theta \) for a given wavelength of X-ray. By measuring the shift in the Bragg angle, one can determine the change in the lattice spacing due to thermal expansion.

The lattice constant of a material, often denoted by \( d \), is a physical property that signifies the distance between adjacent layers of atoms or ions in a crystal lattice. This constant is a critical parameter in characterizing the microscopic structure of a material and can be determined by X-ray diffraction techniques. Changes in the lattice constant are indicative of various physical processes, such as thermal expansion. As temperature increases, the atoms in a material move apart, leading to an increase in the lattice constant. Calculating the change in lattice constant is the first step towards computing the thermal expansion coefficient, as it directly influences the lattice's overall dimensional changes.

Thermal expansion refers to the tendency of matter to change its shape, area, volume, and density in response to a change in temperature. The amount by which a material expands upon heating is characterized by its thermal expansion coefficient, often denoted by \( \alpha \). This coefficient is a measure of how much a material's dimensions will change per unit temperature change. Mathematically, it's expressed as:
\[ \alpha = \frac{\Delta d}{d \times \Delta T} \]
where \( \Delta d \) is the change in the dimension (e.g., lattice constant), \( d \) is the original dimension, and \( \Delta T \) is the temperature change. Understanding thermal expansion is crucial for applications involving temperature changes, as it can affect the structural integrity and performance of materials.

X-ray diffraction (XRD) is a technique used to study crystal structures, providing information on the arrangement of atoms within a material. By directing X-rays of a known wavelength at a crystal and then detecting the angles at which these rays are scattered, scientists can reconstruct the positions of the atoms in the crystal lattice. XRD analysis utilizes Bragg's law to link these diffraction patterns to the lattice constants. Through careful measurement of the angles of diffracted beams, changes in the crystal structure due to thermal expansion can be deduced. In the given exercise, XRD is used to detect the minute shifting of diffraction patterns as the lattice expands, allowing for the calculation of the thermal expansion coefficient.