A compound formed by elements `X` and `Y` has a cubic structure in which `X` atoms are at the corner of the cube and also at the face centers. `Y` atoms are present at the body centre and at the edge centre of the cube.
a. Celculate (i) `Z_(eff')` (ii) total number of atoms in the cube, and (iii) formula of the compound.
b. If all the atoms are removed from one of the body diagonals of the cube, calculate (i) `Z_(eff')` (ii) total number of atoms in the cube, and (iii) formula of the compound.
c. If all the atoms from the diagonals of the one of the face of the cube are removed, calculate (i) `Z_(eff')` (ii) total number of atoms in the cube, and (iii) formula of the compound.
d. If all the atoms are removed from one of the plane passing through the middle of t he cube, calculate (i) `Z_(eff')` (ii) total number of atoms in the cube, and (iii) formula of the compound.
e. If all the atoms are removed from one of the axes passing through one of the face centres of the cube, calculate (i) `Z_(eff')` (ii) total number of atoms in the cube, and (iii) formula of the compound.

Explanation

a. `Z_(eff(X)) = (n_(c))/(8) + (n_(f))/(2) = (8)/(8) + (6)/(2) = 1 + 3 = 4`
`Z_(eff(Y)) = (n_(b))/(8) + (n_(e))/(4) = (1)/(1) + (12)/(4) = 1 + 3 = 4`.
`Z_(eff(X + Y)) = 4 + 4 = 8`
ii. Total number of atoms in the cube
ii. `= 8` (corners) `+ 6 ("face centre") + 1` (body centre)
`+ 12` (edge centre)
`= 27` atoms/units cube
(iii). Formula: `Z_(eff(X)) = 4, Z_(eff(Y)) = 4`
implies `X_(4) Y_(4) = 4XY`
b. i.

Atoms removed from one body diagonal
`= 2` atom form corner `+ 1` atom form body centre
`= 3` atoms. `Z_(eff(X)) = (n_(c))/(8) + (n_(f))/(2) = (8 - 2)/(8) + (6)/(2) = (15)/(4)`
`Z_(eff(Y)) = (n_(b))/(8) +(n_(c))/(4) = (1 - 1)/(1) + (12)/(4) = 0 + 3 = 3`
`Z_(eff(X + Y)) = (15)/(4) + 3 = (27)/(4)`
ii. Total number of atoms in the cube
`= 6 ("corner") + 6 ("face centre") +` Zero (body centre) `+ 12` (edge centres)
`= 24` atoms/unit cube
iii. `Z_(eff(X)) = (15)/(4), Z_(eff(Y)) = 3`
`implies X_((15/(4)) Y_(3)`.
Simplifying `X_(15) Y_(12)`

`Z_(eff(X)) = (n_(c)) /(8) + (n_(f))/(2) = (8 - 4)/(8) + (6 - 1)/(2) = 1 + 2 = 3`.
`Z_(eff(Y)) = (n_(b))/(8) +(n_(c))/(4) = (1 - 1)/(1) + (12)/(4) = 1 + 3 = 4`
`Z_(eff(X)) = 3 + 4 = 7`
ii. Total number of atoms in a cube
`= 4 ("corner") + 5 ("face centre") + 1` (body centre)
`+ 12` (edge centre)
`= 22` atoms/unit cube
iii. Formula: `Z_(eff(X)) = 3, Z_(eff(Y)) = 4`
`implies X_(3) Y_(4)`.
d. Plane passing through the middle of cube (other atoms are not shown)

i. `Z_(eff(X)) = (n_(c)) /(8) + (n_(f))/(2) = (8)/(8) + (6 - 4)/(2) = 1 + 1 = 2`
`Z_(eff(Y)) = (n_(b))/(8) +(n_(c))/(4) = ((1 - 1))/(1) + ((12 - 4))/(4) = 2`.
`Z_(eff(X + Y)) = 2 + 2 = 4`
ii. Total number of the atoms in the cube
`= 8 ("corner") + 2 ("face centres") +` zero (body centre) `+ 8` (edge centres)
`= 18` atoms/unit cube
iii. Formula: `Z_(eff(X)) = 2, Z_(eff(Y)) = 2`
e.
`Z_(eff(X)) = (n_(c)) /(8) + (n_(f))/(2) = (8)/(8) + ((6 - 2))/(2) = 1 + 2 = 3`
`Z_(eff(Y)) = (n_(b))/(8) +(n_(c))/(4) = ((1 - 1))/(1) + (12)/(4) = 0 + 3 = 3`.
`Z_(eff(X + Y)) = 3 + 3 = 6`.
ii. Total number atom in a cube `= 8` (corner) `+ 6` (face centre)
zero (body centre) `+ 12` (edge centre)
`= 24` atoms/unit cube
iii. Formula: `Z_(eff(X)) = 3, Z_(eff(Y)) = 3`
Formula is `X_(3) Y_(3)` or `3XY`.