The interplanar spacing \(d_{h k l}\) for planes in a unit cell having orthorhombic geometry is given by $$ \frac{1}{d_{h k l}^{2}}=\frac{h^{2}}{a^{2}}+\frac{k^{2}}{b^{2}}+\frac{l^{2}}{c^{2}} $$ where \(a, b\), and \(c\) are the lattice parameters. (a) To what equation does this expression reduce for crystals having cubic symmetry? (b) For crystals having tetragonal symmetry?

Understanding the orthorhombic crystal structure is essential as it is one of the seven crystal systems in material science. It's characterized by its three mutually perpendicular axes which are of unequal length, labeled as 'a', 'b', and 'c'. These axes serve as references to describe the geometry of the crystal lattice and play a vital role in determining properties like interplanar spacing.

In the orthorhombic system, interplanar spacing, denoted as \(d_{hkl}\), is important for identifying how planes of atoms are spaced apart from one another. Understanding this spacing can help in determining various structural properties of materials, such as diffraction angles for X-ray crystallography. According to the formula provided, \(d_{hkl}\) is calculated based on the indices 'h', 'k', 'l' which represent the plane's orientation and the lattice parameters 'a', 'b', 'c'.

The interplanar spacing is inversely proportional to the square root of the sum of the squares of these indices divided by the corresponding lattice parameter squares. In concise terms, it allows us to predict how planes within the crystal are arranged and how they interact with external forces or radiation, a critical part of material analysis and engineering.

Cubic crystal symmetry is renowned for its simplicity and high degree of symmetry, making it one of the most frequently encountered crystal structures in both natural and synthetic materials. In a cubic structure, all three axes 'a', 'b', and 'c' are equal in length and are mutually perpendicular, similar to the dimensions of a cube.

When solving problems related to cubic crystals, the interplanar spacing formula simplifies considerably. Because the lattice parameters are identical (\(a_{cubic} = b_{cubic} = c_{cubic}\)), the equation provided in the exercise reduces to a single variable. This simplification underscores one key feature of cubic symmetry: its isotropy, meaning physical properties are identical in all directions.

The simplified equation allows for quicker and easier calculations of interplanar spacings and is fundamental in applications like determining the crystal's density or analyzing its electronic or optical properties. The equation also plays a crucial role in interpretive procedures like the calculation of X-ray diffraction patterns, which are pivotal for material identification and characterization.

Tetragonal crystal symmetry is another fundamental crystal structure, occupying a middle ground between the high symmetry of cubic structures and the lower symmetry of orthorhombic structures. In a tetragonal crystal, the lengths of the 'a' and 'b' axes are equal, but the 'c' axis is of a different length, and all are perpendicular to each other.

The formula adaptation for the tetragonal system reflects this arrangement. As 'a' and 'b' are equal, they can be combined into a single term in the equation (\(a_{tetragonal} = b_{tetragonal}\)), while 'c' remains distinct. This altered formula, as shown in the provided solution, is crucial for material scientists as it affects how interplanar spacings are calculated and, subsequently, how atoms are spatially distributed in the crystal.

The accurate determination of interplanar spacings in tetragonal crystals is essential for tailoring materials with specific directional properties, such as anisotropic thermal expansion or directional conductivity, and plays a role in the design of various electronic components, ceramics, and metals.