Caluclation of number of atoms in different types of unit cells

Explanation

There are seven primitive crystal systems
Cubuic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic and rhombohedral. They differ in the arrangement of their crystallographic axes and angles.
(i) Cubic `:` The cubic unit cell is the smallest repeating unit when all angles are `90^(@)C` and all lengths are equal. As such , each axis is defined by a Cartesian coordinate. Each cubic cell has 8 atoms in each corner of the cube, and that atom is shared with 8 neighboring cells. This is a unit cell with parameters a=b=c, `alpha= beta = gamma = 90^(@)` .

(ii)Tetragonal `:` Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base. This is a unit cell with parameters `a =b cancel(=)c, alpha = beta = gamma = 90^(@)`.

(iii) Orthorhombic `:` Orthorhombic lattice result from stretching a cubic lattic along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base ( a by b ) and height ( c ) , such that a,b, and c are distinct . All three bases intersect at `90^(@)C` angles, so the three lattice vectors remain mutually orthogonal. This is a unit cell with parameters `a cancel(=) b cancel(=) c, alpha = beta = gamma = 90^(@)` .

(iv) Hexagonal`:` This is a unit cell with parameters `a = b cancel(=) c , alpha = beta = 90^(@), gamma = 120^(@)`. The hexagonal closest packed ( hcp ) has a coordination number of 12 and contains 6 atoms per unit cell.

(v) Monoclinic `:` In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base. This is a unit cell with parameters `a cancel(=) b cancel(=) c, alpha = gamma= 90^(@), beta cancel(=) 90^(@)`.

(vi) Triclinic `:` The triclinic is also called as anorthic crystal system. A crystal system is described by the basis vectors. In the triclinic system, the crystal is described by vectors of unequal length,as in the orthorhombic system. In addition, the angles between these vectors must all be different and may include `90^(@)`. This is a unit cell with parameters `a cancel(=) b cancel(=) c, alpha cancel(=) beta cancel(=) gamma = 90^(@)` .
The triclinic lattice is the least symmetric of the 14 three-dimentional Bravais lattices. It has ( itself) the minimum symmetry all lattices have `:` points of inversion at each lattice point and at 7 more points for each lattice point `:` at the midpoint of the edges and the faces, and at the center points. It is the only lattice type that itself has no mirror planes.

(vii) Rhombohedral `:` The unit cellis a rhombohedron ( which gives the name for the rhombohedral lattice system ) . This is a unit cell with parameters a=b=c, `alpha = beta = gamma cancel(=) 90^(@)`. In geometry, a rhombohedron is a three dimensional figure like a cube except that its faces are not square but rhombi. It is a special case of a parallelpiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells.