A cubic lattice has A atoms at the body center, B atoms at the corners and C atoms at half of the face centers. The formula of the lattice is:
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For unit-cell calculations:
\[
\text{Corner atom contribution}=\frac{1}{8}
\]
\[
\text{Face-centered atom contribution}=\frac{1}{2}
\]
\[
\text{Body-centered atom contribution}=1
\]
Always calculate the effective number of atoms first and then convert them into the simplest whole-number ratio.
Step 1: Count contributions of each atom type. A at body center: 1 per unit cell (body center belongs fully). B at corners: 8 corners \( imes rac{1}{8}\) = 1. C at half of face centers: 6 faces total, half = 3 face centers \( imes rac{1}{2}\) = 1.5.
Step 2: Find the simplest whole number ratio. A:B:C = 1:1:1.5. Multiply by 2: A:B:C = 2:2:3. Formula = A\(_2\)B\(_2\)C\(_3\). \[ oxed{A_2B_2C_3} \]