Packing Efficiency of Metal Crystal in Face-centered Cubic Lattice
The steps below are used to achieve Face-centered Cubic Lattice’s Packing Efficiency of Metal Crystal:
Step 1: Radius of sphere
The corner particles are expected to touch the face ABCD’s central particle, as indicated in the figure below.
According to Pythagoras Theorem, the triangle ABC has a right angle.
AC2 = AB2 + BC2
∴ AC2 = a2 + a2
∴ AC2 = 2a2
∴ AC = √2 a …(Equation 1)
We can rewrite the equation as since the radius of each sphere equals r.
AC = 4r …(Equation 2)
From equation 1 and 2, we get
√2 a = 4r
∴ r = a / 2√2 …(Equation 3)
Step 2: Volume of sphere:
Volume of sphere particle = 4/3 πr3. Substitution for r from equation 3, we get
∴ Volume of one particle = 4/3 π(a / 2√2)3
∴ Volume of one particle = 4/3 πa3 × (1/2√2)3
∴ Volume of one particle = πa3 / 12√2
Step 3: Total volume of particles:
Unit cell bcc contains 4 particles. Hence,
volume occupied by particles in FCC unit cell = 4 × πa3 / 12√2
∴ volume occupied by particles in FCC unit cell = πa3 / 3√2
Step 4: Packing Efficiency:
We have,
Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100
∴ Packing efficiency = πa3 / 3√2 a3 × 100
∴ Packing efficiency = 74 %
As a result, particles occupy 74% of the entire volume in the FCC, CCP, and HCP crystal lattice, whereas void volume, or empty space, makes up 26% of the total volume.
Importance of Efficient Packing
Packing Efficiency is important as:
- The object’s sturdy construction is shown through packing efficiency.
- It shows various solid qualities, including isotropy, consistency, and density.
- Numerous characteristics of solid structures can be obtained with the aid of packing efficiency.
Following table lists the packing efficiency of several solid architectures
Unit Cell | Relation between a and r | Number of Atoms in Unit Cell | Coordination Number of atoms | Packing efficiency | Free space |
---|---|---|---|---|---|
SCC | 0.5000a | 1 | 6 | 52.4% | 47.6% |
BCC | 0.4330a | 2 | 8 | 68% | 32% |
FCC | 0.3535a | 4 | 12 | 74% | 26% |
Packing Efficiency of Unit Cell
A crystal lattice is made up of a relatively large number of unit cells, each of which contains one constituent particle at each lattice point. A three-dimensional structure with one or more atoms can be thought of as the unit cell. Regardless of the packing method, there are always some empty spaces in the unit cell. so the question is, What Is Unit Cell Packing Efficiency? The packing fraction of the unit cell is the percentage of empty spaces in the unit cell that is filled with particles. In this article, we shall learn about packing efficiency.
Table of Content
- Packing Efficiency
- Packing Efficiency Formula
- Packing Fraction Formula
- Packing Efficiency of Metal Crystal in Simple Cubic Lattice
- Packing Efficiency of Metal Crystal in Body-centered Cubic Lattice
- Packing Efficiency of Metal Crystal in Face-centered Cubic Lattice
- Unit Cell Packing Efficiency
- Solved Examples of Packing Efficiency