Calculating Packing Efficiency in BCC Lattice

Packing efficiency is a crucial concept in understanding the structure of crystalline solids. It represents the percentage of the total volume of a crystal lattice unit cell that is occupied by the constituent particles, which are often approximated as spheres. A higher packing efficiency indicates a more tightly packed structure.

Understanding the Body-Centered Cubic (BCC) Lattice Structure

In a body-centered cubic (bcc) lattice, atoms are located at each corner of the cube and one atom is present at the center of the body of the cube. Atoms touch along the body diagonal of the cube.

Let 'a' be the edge length of the cubic unit cell and 'r' be the radius of the sphere (atom).

In a bcc structure, the atoms touch along the body diagonal. The length of the body diagonal is related to the edge length 'a' by the Pythagorean theorem in three dimensions. Consider a face diagonal first, which has length \(\sqrt{a^2 + a^2} = a\sqrt{2}\). The body diagonal connects opposite corners and passes through the center atom. This forms a right-angled triangle with one side being the face diagonal (\(a\sqrt{2}\)) and the other side being an edge (a). The body diagonal is the hypotenuse.

Length of body diagonal = \(\sqrt{(a\sqrt{2})^2 + a^2} = \sqrt{2a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3}\).

Along the body diagonal, the corner atom (radius r), the center atom (radius r), and the opposite corner atom (radius r) are in contact. Thus, the length of the body diagonal is equal to \(r + 2r + r = 4r\).

Therefore, the relationship between the edge length 'a' and the atomic radius 'r' in a bcc lattice is:

\( a\sqrt{3} = 4r \)

From this, we can express the edge length 'a' in terms of 'r':

\( a = \frac{4r}{\sqrt{3}} \)

Calculating the Volume of the BCC Unit Cell

The volume of a cubic unit cell with edge length 'a' is \(V_{cell} = a^3\). Substituting the expression for 'a' in terms of 'r':

\( V_{cell} = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{(4r)^3}{(\sqrt{3})^3} = \frac{64r^3}{3\sqrt{3}} \)

Calculating the Number of Spheres (Atoms) per BCC Unit Cell

A bcc unit cell contains:

• 8 atoms at the corners, each contributing \(\frac{1}{8}\) of an atom to the unit cell. Total contribution from corners = \(8 \times \frac{1}{8} = 1\) atom.
• 1 atom at the body center, which is entirely within the unit cell. Total contribution from the center = \(1 \times 1 = 1\) atom.

Total number of atoms per bcc unit cell = \(1 + 1 = 2\) atoms.

Calculating the Volume Occupied by Spheres

The volume of one sphere with radius 'r' is \(\frac{4}{3}\pi r^3\). Since there are 2 atoms (spheres) in a bcc unit cell, the total volume occupied by spheres is:

\( V_{spheres} = 2 \times \left(\frac{4}{3}\pi r^3\right) = \frac{8}{3}\pi r^3 \)

Calculating the Packing Efficiency for BCC Lattice

Packing efficiency is calculated using the formula:

\( \text{Packing Efficiency} = \left(\frac{\text{Volume occupied by spheres in the unit cell}}{\text{Total volume of the unit cell}}\right) \times 100\% \)

Substitute the values we calculated:

\( \text{Packing Efficiency} = \left(\frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}}\right) \times 100\% \)

Simplify the expression:

\( \text{Packing Efficiency} = \left(\frac{8\pi r^3}{3} \times \frac{3\sqrt{3}}{64r^3}\right) \times 100\% \)

The terms \(r^3\) and 3 cancel out:

\( \text{Packing Efficiency} = \left(\frac{8\pi \sqrt{3}}{64}\right) \times 100\% \)

Simplify further:

\( \text{Packing Efficiency} = \left(\frac{\pi \sqrt{3}}{8}\right) \times 100\% \)

Now, substitute the approximate values for \(\pi \approx 3.14159\) and \(\sqrt{3} \approx 1.73205\):

\( \text{Packing Efficiency} \approx \left(\frac{3.14159 \times 1.73205}{8}\right) \times 100\% \)

\( \text{Packing Efficiency} \approx \left(\frac{5.4396}{8}\right) \times 100\% \)

\( \text{Packing Efficiency} \approx 0.67995 \times 100\% \)

\( \text{Packing Efficiency} \approx 67.995\% \)

Rounding to the nearest whole number, the packing efficiency is approximately 68%.

Comparing with Options

Let's look at the given options:

Our calculated value of approximately 68% matches option 2.

Revision Table: Crystal Lattice Packing Efficiencies

Additional Information on Crystal Packing

Crystal packing efficiency is a measure of how closely spheres are packed together in a crystal lattice. Different types of unit cells result in different packing efficiencies. Understanding these values helps predict properties like density and stability of solid materials.

• The simple cubic (SC) lattice has the lowest packing efficiency (52.4%) among the common cubic structures, leaving the most empty space.
• The body-centered cubic (BCC) lattice is more efficiently packed than simple cubic but less efficient than face-centered cubic or hexagonal close-packed structures.
• Face-centered cubic (FCC) and hexagonal close-packed (HCP) structures represent the most efficient ways to pack spheres of equal size, both having a packing efficiency of 74%. These are often referred to as close-packed structures.
• The empty space in a crystal lattice is called voids or interstitial sites. The size and shape of these voids are important for understanding the structures of ionic compounds or interstitial alloys where smaller atoms occupy these spaces.