Which of the 14 Bravais lattices other than face-centered cubic and body-centered cubic do not have reciprocal lattices of the same kind?
The Correct Answer and Explanation is :
The 14 Bravais lattices represent all possible distinct three-dimensional lattice structures in crystallography, each of which can be classified by their symmetry and unit cell dimensions. These lattices are divided into several types based on the positions of atoms within the cell.
Now, the question specifically asks about lattices other than face-centered cubic (FCC) and body-centered cubic (BCC) that do not have reciprocal lattices of the same kind.
The reciprocal lattice of a Bravais lattice describes a lattice in the Fourier space, where the position vectors of the points are inversely related to the original lattice. For most Bravais lattices, the reciprocal lattice is of the same type as the original lattice (meaning that it retains its basic symmetry).
Answer:
Among the 14 Bravais lattices, the two that do not have reciprocal lattices of the same kind (other than FCC and BCC) are:
- Hexagonal
- Rhombohedral (Trigonal)
Explanation:
- Hexagonal lattice: The hexagonal Bravais lattice has a six-fold symmetry in the plane and is characterized by the
a = b ≠ cunit cell dimensions. Its reciprocal lattice is trigonal rather than hexagonal. The six-fold symmetry is lost when moving to the reciprocal space, and it takes a different form. - Rhombohedral lattice: The rhombohedral lattice also has three-fold symmetry along a single axis but is not cubic. In this case, the reciprocal lattice is not rhombohedral; instead, it forms a triclinic lattice, where all angles are non-right and the axes are of unequal length.
Both these lattices represent special cases where the symmetry in real space does not map directly to the reciprocal lattice space as would happen with FCC and BCC lattices.
Image:
Let me generate a diagram showing the relationship between the different lattices, including the reciprocal lattice for these special cases.
Here is an illustration of the 14 Bravais lattices, highlighting how the reciprocal lattices differ, especially for the hexagonal and rhombohedral types, in contrast to FCC and BCC. It shows the transformation of these lattices from real space to reciprocal space. I hope this helps clarify the relationship! Let me know if you’d like more details.
