Understanding Miller Indices
Miller indices are a notation system in crystallography for planes in crystal lattices. They are represented as (hkl), where h, k, and l are integers that reflect the orientation of the plane relative to the crystal axes.
Finding the Plane Equation
To find the Miller indices of the plane passing through the points (0,0,1), (1,0,0), and (1/2,1/2,1/4), follow these steps:
1. Determine Vectors
- Compute two vectors from the given points:
- Vector A = (1,0,0) - (0,0,1) = (1,0,-1)
- Vector B = (1/2,1/2,1/4) - (0,0,1) = (1/2,1/2,-3/4)
2. Calculate the Normal Vector
- The normal vector \( \mathbf{N} \) to the plane can be found using the cross product:
\( \mathbf{N} = \mathbf{A} \times \mathbf{B} \)
- After calculation:
\( \mathbf{N} = \left| \begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 0 & -1 \\
1/2 & 1/2 & -3/4 \end{array} \right| = (-1/4, 1/2, 1/2) \)
3. Convert to Miller Indices
- The Miller indices are derived from the normal vector components. The indices are normalized to the smallest integers:
- From (-1/4, 1/2, 1/2), we can multiply by -4 to obtain (1, -2, -2).
Final Miller Indices
- Thus, the Miller indices for the plane are (1, -2, -2).
These indices describe the orientation of the plane within the crystal lattice, providing essential information for understanding crystal structures.