Model Answer
0 min readIntroduction
Miller Indices are a system of notation used in crystallography to denote directions and planes in crystal lattices. They represent the reciprocals of the intercepts of a crystal plane with the crystallographic axes (a, b, and c). This notation is crucial for understanding and predicting the physical properties of crystals, such as cleavage, diffraction, and optical behavior. The concept was introduced by William Hallowes Miller in 1838, providing a standardized way to describe crystal faces. Understanding Miller Indices is fundamental to interpreting X-ray diffraction patterns and predicting anisotropic properties of materials.
Calculating Miller Indices: A Step-by-Step Approach
The calculation of Miller Indices involves the following steps:
- Determine the intercepts: Identify the points where the crystal plane intersects each of the crystallographic axes (a, b, and c).
- Take the reciprocals: Calculate the reciprocal of each intercept.
- Clear the fractions: Find the smallest integer values that, when multiplied by the reciprocals, result in whole numbers.
- Enclose in parentheses: The resulting whole numbers are the Miller Indices, enclosed in parentheses (hkl).
Calculating Miller Indices for Given Crystal Faces
(i) A face intersects all three crystallographic axes at 3-unit distance.
- Intercepts: a = 3, b = 3, c = 3
- Reciprocals: 1/3, 1/3, 1/3
- Clear fractions: Multiplying by 3 gives 1, 1, 1
- Miller Indices: (111)
(ii) A face intersects a-axis at 4-unit distance and is parallel to b and c axes.
A plane parallel to an axis means it intersects that axis at infinity. The reciprocal of infinity is zero.
- Intercepts: a = 4, b = ∞, c = ∞
- Reciprocals: 1/4, 0, 0
- Clear fractions: Multiplying by 4 gives 1, 0, 0
- Miller Indices: (100)
Handling Special Cases
When a plane is parallel to an axis, its intercept is considered to be infinity, and its reciprocal is zero. If a plane intersects an axis at the origin, the reciprocal is undefined. In such cases, the plane is assumed to intersect the axis at infinity, resulting in a zero reciprocal.
Importance of Miller Indices
Miller Indices are not just a mathematical notation; they are directly related to the atomic arrangement within the crystal. They are used to predict the density of atoms on a particular plane, which influences surface energy and reactivity. They are also essential for understanding X-ray diffraction patterns, as the angles of diffraction are determined by the spacing between crystal planes, which is described by the Miller Indices.
Conclusion
In conclusion, Miller Indices provide a systematic and unambiguous way to describe crystal planes. The calculation involves determining intercepts, taking reciprocals, and simplifying to the smallest integer values. The examples demonstrate the application of these steps to specific crystal faces, including those parallel to crystallographic axes. A thorough understanding of Miller Indices is crucial for anyone studying crystallography, materials science, or related fields, enabling the prediction and interpretation of crystal properties.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.