A compound formed by elements `X` and `Y` has a cubic structure in which `X` atoms are at the face centres. `Y` atoms are present at the body center and also at the alternate edge center of the cube.
a. Calculate: (i) `Z_(eff')` (ii) total number of atoms in a cube, and (iii) formula of the compound.
b. If all the atoms are removed from a single axis passing through the centre of the cube, calculate
(i) `Z_(eff')` (ii) total number of atoms in a cube, and (iii) formula of the compound.

Explanation

a. i. Number of effective X atoms
`= 8` corners `xx (1)/(8)` per corner atom share
`= 1` atom/unit cell
Number of effective Y atoms
`= 8` corners `xx (1)/(2)` per face atom share
`= 3` atom/unit cell
Number of effective `Y` atoms
`=8` corners `xx (1)/(2)` per face atom share
`= 3` atom/unit cell
`Z_(eff (8X + Y)) = 1 + 3 = 4`
ii. Total atoms in a cube
`= 8X` atoms at corner `+ 6Y` atoms at faces
`= 8 + 6 = 14` atoms/units cube
iii. Formula of the compound:
`Z_(eff (x)) = 1, Z_(eff(Y)) = 3`
Thus, formula of the compound `= XY_(3)`
b. i. Two atoms from two face centres are removed by passing an axis as shown in the figure below.
`:.` Face centre atom left `= 6 - 2 = 4`
`Z_(eff(Y)) = 4` Face atoms `xx (1)/(2)` Per face atom share
`= 2` atoms/units cell
`Z_(eff(X)) = 8` corners `xx (1)/(8)` Per corner atom share
`= 1` atom/units cell
`:. Z_(eff (X + Y)) = 2 + 1 = 3` atoms/units cube.
ii. Total atoms in a cube `= 8 ("corners") + 4 ("Face centre")`
`= 12` atoms/units cube
iii. Formula of the compound:
`Z_(eff(X)) = 1, Z_(eff(Y)) = 2`
`:.` Formula `= XY_(2)`