Sketch the (1\overline{1101) and (1120) planes in a hexag- } onal unit cell.

The Miller-Bravais indices provide a system for notating the orientation of planes and directions within hexagonal crystals. Unique from the Miller indices used in cubic systems, these indices employ a four-digit system, represented as \( hkil \), to account for the hexagonal symmetry. The first three indices \( h, k, i \) correspond to the three basal vectors while the last index \( l \) denotes the position relative to the vertical c-axis.

One unique feature of Miller-Bravais indices is the constraint \( h + k + i = 0 \), indicative of the hexagonal geometry. This redundancy ensures that the plane's orientation can be fully described despite the crystal's complexity. When students attempt to draw the planes as in the exercise, understanding these indices is crucial. For instance, the plane \(1\bar{1}01\) signifies a plane that intersects the a1 axis at 1, is parallel to the a2 axis, intersects the a3 axis at its negative reciprocal, and is one unit up the c-axis.

Lattice vectors are fundamental to understanding crystal structures as they describe the repeating pattern in a lattice. In a hexagonal unit cell, where our original problem is based, there are four lattice vectors: \( a_1, a_2, a_3, \) and \( c \). Vectors \( a_1, a_2, \) and \( a_3 \) lie in the horizontal plane of the hexagon, with \( a_3 \) being 120 degrees with respect to \( a_1 \) and \( a_2 \), and \( c \) pointing vertically, perpendicular to the plane formed by the other three.

The lengths of these vectors are tied to the physical dimensions of the crystal unit cell. For the hexagonal system, \( a_1 = a_2 = a_3 \), and their magnitudes are typically equal, while \( c \) can vary, giving the hexagonal prism its distinct shape. The correct use of lattice vectors allows one to accurately position planes within the cell, as showcased in the exercise solution, and they play an essential role in the calculation and visualization of the crystal structures.

Crystallographic planes are essential in characterizing the arrangement of atoms in a crystal. These are imaginary planes that slice through a crystal pattern and can be represented by Miller or Miller-Bravais indices in hexagonal systems.

In the context of our exercise, sketching the crystallographic planes \(1\bar{1}01\) and \(1120\) involves visualizing these cuts through the hexagonal lattice. Each index in the planes' notation corresponds to intersection points or parallelism to the crystallographic axes. The plane \(1\bar{1}01\) indicates intersections and non-intersections with the axes a1, a2, and c, respectively, as per the indices. Meanwhile, the plane \(1120\) slices through the crystal differently, according to its unique set of indices.

To aid with clarity and understanding for students, illustrating the location and orientation of these planes within the unit cell can be immensely beneficial. By doing so, we create a visual representation that can cement the theoretical aspects of crystallography into the minds of learners, allowing for a more intuitive grasp of such concepts.