How do the spacings of the three planes (001) (011) and (111) of fcc lacttice vary?

To understand how the spacings of the three planes—(001), (011), and (111)—in a face-centered cubic (FCC) lattice vary, we need to delve into the concept of Miller indices and the geometry of the FCC structure. Each set of Miller indices corresponds to a specific crystallographic plane, and the spacing between these planes can be calculated using a formula that relates the indices to the lattice parameters.

Miller Indices and Plane Spacing

Miller indices are a notation system in crystallography for planes in crystal lattices. For an FCC lattice, the lattice parameter \( a \) is the length of the cube's edge. The spacing \( d \) between the planes can be determined using the formula:

d = a / √(h² + k² + l²)

Here, \( h \), \( k \), and \( l \) are the Miller indices of the plane. Let's break down the spacings for the three planes you mentioned.

Spacing for the (001) Plane

For the (001) plane, the Miller indices are \( h = 0 \), \( k = 0 \), and \( l = 1 \). Plugging these values into the formula gives:

d_{001} = a / √(0² + 0² + 1²) = a

This indicates that the spacing between the (001) planes is equal to the lattice parameter \( a \).

Spacing for the (011) Plane

Next, consider the (011) plane, where \( h = 0 \), \( k = 1 \), and \( l = 1 \). The calculation becomes:

d_{011} = a / √(0² + 1² + 1²) = a / √2

This shows that the spacing between the (011) planes is \( a / √2 \), which is shorter than the spacing for the (001) planes.

Spacing for the (111) Plane

Finally, for the (111) plane, we have \( h = 1 \), \( k = 1 \), and \( l = 1 \). The spacing is calculated as follows:

d_{111} = a / √(1² + 1² + 1²) = a / √3

This indicates that the spacing between the (111) planes is \( a / √3 \), which is even shorter than the spacings for both the (001) and (011) planes.

Summary of Plane Spacings

• (001) Plane: Spacing \( d_{001} = a \)
• (011) Plane: Spacing \( d_{011} = a / √2 \)
• (111) Plane: Spacing \( d_{111} = a / √3 \)

In summary, the spacings of the (001), (011), and (111) planes in an FCC lattice decrease as the Miller indices increase. This variation in spacing is crucial for understanding various properties of materials, such as their diffraction patterns and mechanical behavior. Each plane's unique spacing contributes to the overall characteristics of the crystal structure, influencing how it interacts with light, heat, and other materials.