Miller Indices and Interplanar Spacings

Oct 13, 2025 by windailys.com 35 views

Introduction

In the field of solid-state physics, the study of crystal lattices is crucial for understanding the properties of materials. One of the fundamental concepts in this field is the relationship between Miller indices and interplanar spacings. In this article, we will explore this relationship for a sample cubic lattice, specifically focusing on the ratios of d100, d110, and d111.

Miller Indices and Interplanar Spacings

Miller indices are a way to describe the orientation of a plane within a crystal lattice. They are represented by a set of three numbers (hkl) that correspond to the reciprocals of the fractional intercepts of the plane with the crystal axes. Interplanar spacings, on the other hand, are the distances between adjacent planes of atoms within the crystal lattice.

The Cubic Lattice

A cubic lattice is a type of crystal lattice that has a three-dimensional arrangement of atoms with a cubic symmetry. In a cubic lattice, the atoms are arranged in a way that the lattice parameters (a, b, and c) are equal, and the angles between the lattice axes are 90 degrees.

Calculating Interplanar Spacings

The interplanar spacings in a cubic lattice can be calculated using the following formula:

d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}

where d_{hkl} is the interplanar spacing, a is the lattice parameter, and h, k, and l are the Miller indices.

Calculating the Ratios of d100, d110, and d111

Using the formula above, we can calculate the interplanar spacings for the planes with Miller indices (100), (110), and (111).

For the (100) plane:

d_{100} = \frac{a}{\sqrt{1^2 + 0^2 + 0^2}} = \frac{a}{1} = a

For the (110) plane:

d_{110} = \frac{a}{\sqrt{1^2 + 1^2 + 0^2}} = \frac{a}{\sqrt{2}}

For the (111) plane:

d_{111} = \frac{a}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{a}{\sqrt{3}}

Calculating the Ratios

Now that we have calculated the interplanar spacings for the (100), (110), and (111) planes, we can calculate the ratios of d100, d110, and d111.

d_100} d_{110 : d_111} = a \frac{a\sqrt{2}} \frac{a{\sqrt{3}}

To simplify the ratios, we can multiply both sides by \sqrt{2} \sqrt{3} to get:

d_100} d_{110 : d_111} = a \sqrt{2} \sqrt{3} a \sqrt{3 : a \sqrt{2}

Simplifying further, we get:

d_100} d_{110 : d_111} = 6^{1/2} 3^{1/2 : 2^{1/2}

Conclusion

In this article, we have explored the relationship between Miller indices and interplanar spacings in a cubic lattice. We have calculated the interplanar spacings for the planes with Miller indices (100), (110), and (111) and have shown that the ratios of d100, d110, and d111 are indeed 6^1/2} 3^{1/2 : 2^{1/2}. This relationship is a fundamental concept in solid-state physics and is essential for understanding the properties of materials.

References

Kittel, C. (2005). Introduction to Solid State Physics. John Wiley & Sons.
Ashcroft, N. W., & Mermin, N. D. (1976). Solid State Physics. Holt, Rinehart and Winston.