Using the data for molybdenum in Table \(3.1\), compute the interplanar spacing for the (111) set of planes.

Miller indices are a notation system in crystallography for planes in crystal lattices. The indices are presented as a set of three numbers, \(hkl\), which are derived from the reciprocals of the fractional intercepts the plane makes with the crystallographic axes. To find these indices for a plane:

• Identify the points where the plane intersects the axes.
• Take the reciprocal of these intersect points.
• Clear the fractions to get the smallest set of whole numbers.

In the case of the (111) plane, the intercepts are one unit cell length away from the origin on each of the three axes, making the Miller indices \(h=1, k=1, l=1\). The simplicity of this notation allows for quick identification of plane orientations and plays a crucial role in understanding and calculating properties such as interplanar spacing.

The lattice constant, often symbolized as \(a\), is a physical dimension that represents the unit cell edge length within a crystal structure. For cubic structures, it's the length of one side of the cube. This measurement is fundamental as it relates to the overall size of a crystal lattice and is a key factor in determining various properties of the material, such as the density, interplanar spacing, and thermal expansion.
When performing calculations involving the crystal lattice, such as computing interplanar spacing, the lattice constant serves as a critical input value. In materials science and engineering, it is vital for validating theoretical models with experimental data and for designing new materials with desired properties.

In materials science, the cubic lattice structure is one of the simplest and most symmetric crystal arrangements encountered. It is characterized by atoms located at each corner of a cube. There are three types of cubic lattices: simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC).

• Simple cubic has one atom per unit cell;
• BCC has one additional atom at the center of the cube;
• FCC has atoms in the centers of each face in addition to the corners.

The arrangement of atoms within the unit cell can significantly influence the material's physical properties, including strength, ductility, and electrical conductivity. For example, molybdenum, as mentioned in our exercise, crystallizes in a BCC cubic lattice, which dictates its characteristic behavior in various applications.

Materials Science and Engineering is an interdisciplinary field focused on designing and discovering new materials, as well as understanding the processing-structure-properties relationships in materials. It combines principles from chemistry, physics, and engineering to understand the fundamental building blocks of materials and how they can be manipulated to enhance performance for specific applications.
Key tasks in this field include the analysis and development of metals, ceramics, polymers, and composites, aiming to create more durable, sustainable, and functional materials. This scientific domain empowers advancements in technology and industry by pushing the boundaries of what materials can do, as illustrated by the precision required when calculating interplanar spacings and understanding crystal structures, as exemplified in the exercise with molybdenum.