A compound is made up of three elements X, Y and Z. X atoms are present at each corner of the unit cell, Y atoms are present on each edge and Z atoms are present in the centre of each face. The formula of the compound is:

Concept:

• Atoms at the edge of a unit cell:
  • ​In a crystal lattice, an edge of a unit cell is shared by four other unit cells.
  • One is the adjacent unit cell and the other two are on the layer just above the unit cell.
  • So each edge contributes to one-fourth of an atom in a unit cell.
  • There are 12 edges per unit cell, thus a total number of atoms becomes 12 × 1/4 = 3.

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• Atoms at the corners of a unit cell:
  • The corner atoms are shared via 8 cubes, so the contribution from each corner is 1/8.
  • Total number of atoms on the corner of a cube = 8
  • The total number of atoms from corners is 8 × 1/8 = 1.
• Atoms at the face of a unit cell:
  • ​Each face is shared by two unit cells.
  • the contribution from each face is thus 1/2.
  • There are six faces in a cube, so the total contribution from the faces of the cube = 3​

Calculation:

• The atoms present at the corners are X.
• Contribution from all the corners = 1.

So, the total number of X atoms = 1.

• Y atoms are present at the edges.
• Total number of atoms at edges = 3,

So the number of Y atoms = 3.

• The total contribution from all 6 faces is 3.
• Z atoms are present at the faces.

So total number of Z atoms = 3.

Hence, the formula of the compound is XY₃Z₃.