Question

Why do hexagonal and trigonal lattices not have bcc and fcc variations?

Answer 1

Hint: Crystal structure is a description of the orderly organisation of atoms, ions, or molecules in a crystalline substance used in crystallography. The inherent nature of the component particles results in symmetric patterns that recur along the major directions of three-dimensional space in matter, forming ordered structures. The unit cell of the structure is the smallest group of particles in the substance that makes up this repeating pattern.

Complete answer:
The symmetry and structure of the whole crystal are perfectly reflected in the unit cell, which is built up by recurrent translation of the unit cell along its major axes. The nodes of the Bravais lattice are defined by the translation vectors. The lattice constants, also known as lattice parameters or cell parameters, are the lengths of the unit cell's main axes, or edges, and the angles between them. The idea of space groups is used to characterise the crystal's symmetry characteristics. The 230 space groups may describe all conceivable symmetric groupings of particles in three-dimensional space.
For a trigonal system crystal parameters are said to be as $ \mathrm{a}=\mathrm{b}=\mathrm{c} $ and $ \alpha=\beta= $ $ \gamma \neq 90^{\circ} $ . So, the length of sides and angles of the unit cell are equal but the shape is even harder to visualize because $ \alpha, \beta, \gamma $ can be anything. For a hexagonal system, the crystal parameters are said as $ \mathrm{a}=\mathrm{b} \neq \mathrm{c} ; \alpha=\beta= $ $ 90^{\circ} $ and $ \gamma=120^{\circ} . $ So, we got one restriction to have $ \gamma=120^{\circ} $
Because the cubic unit cell has more symmetry, we can have bcc and fcc variations, but we can't have such variations in trigonal and hexagonal systems because of the absence of symmetry.

Note:
The geometry of particle arrangement in the unit cell is used to characterise crystal structure. The unit cell is the smallest repeating unit in the crystal structure that has complete symmetry. The unit cell's shape is parallelepiped, with six lattice parameters representing the lengths of the cell edges (a, b, c) and the angles between them ( $ \alpha, \beta, \gamma $ ).