Using schematics, explain clearly why body-centered tetragonal is a Bravais Lattice, but based-centered tetragonal is not?
To understand why the body-centered tetragonal lattice is a Bravais lattice, while the base-centered tetragonal lattice is not, let's start by examining the characteristics and requirements of a Bravais lattice.
A Bravais lattice is a regular, three-dimensional arrangement of points in space. It is defined by a set of translation vectors that generate the entire lattice structure when repetitively applied. Bravais lattices have a high degree of translational symmetry, meaning that the lattice is invariant under translations by any of the lattice vectors.
In a two-dimensional lattice, the Bravais lattice types are limited to square, rectangular, hexagonal, oblique (parallelogram), and centered rectangular. However, in three dimensions, there are 14 possible Bravais lattices, categorized into seven crystal systems.
Now, let's examine the body-centered and base-centered tetragonal lattices.
1. Body-centered tetragonal (BCT) lattice:
- In a BCT lattice, the unit cell contains lattice points at each of its corners (like a conventional tetragonal lattice) and an additional lattice point at the center of the cell.
- The translation vectors connecting the lattice points in the BCT lattice generate the entire lattice structure when repeated.
- This lattice has translational symmetry in all directions and satisfies the requirements of a Bravais lattice.
- Therefore, the BCT lattice is a Bravais lattice.
2. Base-centered tetragonal (BCTg) lattice:
- In a BCTg lattice, the unit cell contains lattice points at each of its corners (like a conventional tetragonal lattice) and two additional lattice points: one at the base center of the cell and another at the body center of the cell.
- The translation vectors connecting the lattice points in the BCTg lattice do not solely generate the entire lattice structure when repeated.
- While translations by the lattice vectors generate the tetragonal lattice, the presence of additional lattice points at the base and body centers disrupts the necessary translational symmetry in all directions.
- Therefore, the BCTg lattice does not satisfy the requirements of a Bravais lattice.
In summary, the body-centered tetragonal lattice is a Bravais lattice because it possesses translational symmetry in all directions. On the other hand, the base-centered tetragonal lattice is not a Bravais lattice because it lacks the necessary translational symmetry due to the presence of additional lattice points at the base and body centers.