Why do end centred lattices exist only in orthorhombic and monoclinic bravais lattices

The end centred lattices exist only in orthorhombic and monoclinic bravais lattices because the end centred lattices are not having all the symmetrical elements like body centred, face centred and simple cubic unit cell.

It can be treated as a special type of unit cell with one or more symmetry elements missing. Hence they are forming only with orthorhombic and monoclinic bravais lattices.

BODY CENTRED

FACE CENTRED UNIT CELL

END CENTRED UNIT CELL

BCC:- a body centred cubic unit cell has an atom at each of its corner and also one atom at its body centre .total no. of atoms in bcc

1. 8 corners x 1/8 per corner atom= 8x 1/8 =1
2. 1 body center atom =1 x1 =1
3. Total no of atoms per unit cell = 2 atoms

FCC :- a face centred cubic unit cell contain atoms at all the corner and at the center of all the face of the cube . total no. of atoms in FCC

1. 8 corners x 1/8 per corner atom= 8x 1/8 =1
2. 6 face centred atom x ½ atom per unit cell = 3
3. Total no of atoms per unit cell = 4

End centred: -a end centered cubic unit cell contain atoms at all the corner and at the centers of the opposite faces .total no. of atoms inend centered cubic unit cell

1. 8 corners x 1/8 per corner atom= 8x 1/8 =1
2. 2 face centered atom x ½ atom per unit cell = 1
3. Total no of atoms per unit cell = 2