Convert the \((010)\) and (101) planes into the four-index Miller-Bravais scheme for hexagonal unit cells.

Hexagonal crystal structures are unique due to their six-sided symmetry, which is visually evident in hexagonal-shaped crystals. Unlike cubic structures that are often taught first due to their simpler symmetry, hexagonal crystal structures have a more intricate arrangement of atoms.

The hexagonal lattice system is comprised of three equal vectors lying in a plane, each at 120 degrees to one another, with a fourth vector at a right angle to this plane. These vectors define the unit cell, the basic repeating unit of the crystal lattice that, when repeated throughout space, makes up the entire crystal structure.

Understanding these unit cells is key for delving into the Miller-Bravais indexing scheme, as they provide the framework used to describe crystallographic planes and directions. A proper grasp of hexagonal structures simplifies the comprehension of complex indexing systems used for their analysis.

Miller indices, denoted as (hkl), are a notation system in crystallography that quantitatively describes the orientation of crystal planes. They are essential for identifying and categorizing crystal faces and lattice planes. In cubic systems, three indices suffice because of the three mutually perpendicular axes.

However, the hexagonal system's unique symmetry requires an additional index, leading us to the four-index Miller-Bravais system (hkil). Here, the 'i' index is linked to the other two horizontal indices ('h' and 'k') through a specific relationship. Notably, the sum of the first three indices in Miller-Bravais indexing, for a correct hexagonal description, should always be zero (h + k + i = 0). This ensures that the plane is properly oriented within the constraints of hexagonal symmetry.

These indices are employed to solve practical problems in various scientific and engineering fields, and they are vital for the analysis, design, and synthesis of materials with hexagonal crystal structures.

Conversion of indices between the traditional three-index Miller system and the four-index Miller-Bravais system is an integral part of understanding hexagonal crystal structures. The fourth index is added in the Miller-Bravais system to account for the crystal symmetry more accurately and is specifically adapted for the hexagonal lattice.

To convert from the three-index (hkl) to the four-index (hkil) system, one uses known relationships: 'h + k = -i' and 'l' remains the same. When enhancing the learning process for students, it is important to break down these conversions into manageable steps and explain the mathematical logic behind these relations.

Through exercises that require the conversion of indices, such as the provided example, students begin to familiarize themselves with these relationships and the significance of each index in the system. They learn to appreciate the intricacies of hexagonal crystal structures and how proper indexing aids in the visualization and understanding of the material's crystallographic properties.