why the end centered cubic lattice is basically primitive tertragonal lattice?

The end-centered cubic lattice is essentially a primitive tetragonal lattice due to the way the lattice points are arranged.

1. In an end-centered cubic lattice, there are lattice points at each corner of the cube and at the center of two opposite faces.

2. If you take this cube and stretch it along the diagonal running from one of the corners with a face-centered point to the opposite corner, you will transform the cube into a tetragonal prism.

3. In this tetragonal prism, there are still lattice points at each corner and at the center of two opposite faces. However, because the prism is no longer a cube, these face-centered points are now in the middle of the top and bottom faces of the prism.

4. This arrangement of lattice points is exactly the definition of a primitive tetragonal lattice.

5. Therefore, an end-centered cubic lattice is essentially a primitive tetragonal lattice.

This transformation is purely geometric and does not affect the symmetry or the periodicity of the lattice.