Prove why body centered trigonal, and face centered trigonal are not part of the 14 bravais lattice. Which one of the bravais lattice could replace these two lattices?
Answers
A tetragonal crystal system has a defining symmetry of a single four-fold rotation axis.
A face-centered tetragonal (FCT) lattice does have this symmetry. So it must belong to the tetragonal crystal system. But there is no FCT Bravais lattice in the tetragonal system. We then explore to find a smaller body-centered tetragonal (BCT) unit cell as shown in the figure of the answer given by Jeanne Paquette.
Focussing only on the size of a possible smaller unit cell can lead to wrong conclusions. For example, a smaller BCT unit cell is also possible in a face-centered cubic (FCC) lattice. This can be seen using a figure similar to that shown by Paquette. But we cannot replace FCC by BCT because FCC has a higher symmetry (four three-fold axes along the body diagonals) which the BCT does not have. So although a smaller unit cell is possible in this case also we do not go for it. We retain the larger FCC unit cell in the interest of symmetry.
In fact why BCT unit cell, although smaller, cannot be used as a description of FCC lattice is a classic example of the priority of symmetry over unit cell size and shape.
A tetragonal crystal system has a defining symmetry of a single four-fold rotation axis.
A face-centered tetragonal (FCT) lattice does have this symmetry. So it must belong to the tetragonal crystal system. But there is no FCT Bravais lattice in the tetragonal system. We then explore to find a smaller body-centered tetragonal (BCT) unit cell as shown in the figure of the answer given by Jeanne Paquette.
Focussing only on the size of a possible smaller unit cell can lead to wrong conclusions. For example, a smaller BCT unit cell is also possible in a face-centered cubic (FCC) lattice. This can be seen using a figure similar to that shown by Paquette. But we cannot replace FCC by BCT because FCC has a higher symmetry (four three-fold axes along the body diagonals) which the BCT does not have. So although a smaller unit cell is possible in this case also we do not go for it. We retain the larger FCC unit cell in the interest of symmetry.
In fact why BCT unit cell, although smaller, cannot be used as a description of FCC lattice is a classic example of the priority of symmetry over unit cell size and shape.