FCC Lattice Packing Fraction Calculation
The packing fraction (or packing efficiency) represents the fraction of the volume within a crystal structure unit cell that is occupied by atoms.
Determining Atoms in FCC Unit Cell
- A face-centered cubic (fcc) unit cell contains atoms at the 8 corners and at the center of each of the 6 faces.
- Total atoms per unit cell = (8 corners $\times$ 1/8 atom/corner) + (6 faces $\times$ 1/2 atom/face) = 1 + 3 = 4 atoms.
Relating Atomic Radius to Lattice Constant
- In an fcc structure, atoms touch along the face diagonal.
- Let 'a' be the lattice constant and 'r' be the atomic radius. The length of the face diagonal is $\sqrt{2}a$.
- This length equals four atomic radii ($4r$). Therefore, $\sqrt{2}a = 4r$.
- Solving for 'r', we get $r = \frac{\sqrt{2}a}{4} = \frac{a}{2\sqrt{2}}$.
Calculating Volume Occupied by Atoms
- The volume of a single atom is $\frac{4}{3}\pi r^3$.
- The total volume occupied by atoms in the fcc unit cell is $4 \times (\frac{4}{3}\pi r^3)$.
- Substituting $r = \frac{a}{2\sqrt{2}}$: Total Volume = $4 \times \frac{4}{3}\pi \left(\frac{a}{2\sqrt{2}}\right)^3$ Total Volume = $\frac{16}{3}\pi \left(\frac{a^3}{8 \times 2\sqrt{2}}\right)$ Total Volume = $\frac{16}{3}\pi \frac{a^3}{16\sqrt{2}}$ Total Volume = $\frac{\pi a^3}{3\sqrt{2}}$
Calculating Packing Fraction
- The volume of the unit cell is $a^3$.
- Packing Fraction (PF) = (Total Volume of Atoms) / (Volume of Unit Cell)
- PF = $\frac{\frac{\pi a^3}{3\sqrt{2}}}{a^3}$
- PF = $\frac{\pi}{3\sqrt{2}}$
- Evaluating numerically: PF $\approx \frac{3.14159}{3 \times 1.41421} \approx 0.74048$
The packing fraction for an fcc lattice is approximately 0.74.
Similar Questions
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- Which of the following relation gives the potential energy of a diatomic molecule ?
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- The shortest wavelength present in the X-rays at an accelerating potential of 50 KV is :
- Among the following which one term does not represent Gibbs free energy ?
- For an ideal hcp structure, where the atomic spheres touch each other, the ration of $\frac{c}{a}$ is :
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Important Questions from Mixed Topic (CUET PG)
- Who was the founder of Bolshevik Communist party?
- What is the key guide to statecraft in the realist tradition?
- Chronologically arrange the events in the Cold War period.
A. Berlin Wall is constructed
B. Communist China joins the UN
C. Soviet invasion of Czechoslovakia
D. Berlin Blockade
Choose the correct answer from the options given below: Morgenthau's principles of political realism are:
A. Politics is rooted in permanent and unchanging human nature which is basically self centred, self-regarding and self-interested
B. Politics is an autonomous sphere of action and cannot therefore be reduced to morals
C. International Politics is an arena of conflicting self-interests
D. The ethics of international relations is situational ethics which is very different from private morality
Choose the correct answer from the options given below:Who among the following political thinkers consider the anarchical self help system to be a compelling factor for States to maximise their relative power positions?
