If effective number of particles along any body diagonal of unit cell in some cubical arrangement is 0.25 and along any face diagonal is 0.75. Find the effective number of particles per unit cell?

clearly,the arrangement is fcc. thereofore nao of particles per unit cell is 4

In a cube, there are 4 body diagonals and 2x6=12 face diagonals.

Effective no. of particles per unit cell = No. of particles at face diagonals + No. of particles at body diagonals

No. of particles at body diagonals = 0.25 x 4 x 1 = 1

(where 0.25 is the effective number of particles per body diagonal and I have multiplied it by 1 because each particle in a body diagonal is shared by one and only one unit cell)

No. of particles at face diagonals = 0.75 x 12 x 1/2 = 4.5

(where 0.75 is the effective number of particles per face diagonal and I have multiplied it by 1/2 because each particle in a face diagonal is shared by two unit cells)

Effective number of particles = 4.5 + 1 = 5.5 per unit cell