How are the indices for a crystallographic direction in a cubic unit cell determined?
The correct answer and explanation is:
Answer:
The indices for a crystallographic direction in a cubic unit cell, called direction indices or Miller indices for directions, are determined by the vector components along the unit cell edges expressed as the smallest set of integers with the same ratio.
Explanation (300 words):
In crystallography, directions within a crystal lattice are described using indices that give a standardized way to represent vectors between lattice points. For cubic systems, this is straightforward due to the orthogonal axes and equal edge lengths.
Steps to determine the crystallographic direction indices [uvw]:
- Identify the vector components along the unit cell edges:
The vector representing the direction is drawn from one lattice point (often the origin) to another point in the lattice. The vector components along the x, y, and z axes of the cubic unit cell are noted as fractional or decimal values relative to the unit cell edges a,b,ca, b, c. - Express in terms of the unit cell edges:
These components correspond to how far along each axis the vector extends. For example, if the vector extends 1 edge length in x, 2 in y, and 3 in z, the components are (1, 2, 3). - Clear fractions and reduce to smallest integers:
If the components are fractional, multiply by the least common denominator to convert them to whole numbers. For example, a vector with components (0.5, 1, 0.25) would be multiplied by 4 to get (2, 4, 1). Then simplify if possible. - Write the direction indices in square brackets [uvw]:
The resulting integers (u, v, w) are the direction indices. Negative values are denoted by a bar above the number, e.g., 1‾\overline{1}.
Example:
If a vector runs from (0,0,0) to (1/2, 1, 0), the components are (0.5, 1, 0). Multiply by 2 to get (1, 2, 0), so the direction indices are [120].
Why use indices?
These indices provide a universal language for describing directions in any crystal structure. For cubic cells, because axes are perpendicular and equal in length, the indices directly correspond to Cartesian components of the direction vector.
Summary:
- Find vector components along axes
- Convert components to smallest integer ratio
- Write as [uvw] with bars for negatives
This method ensures consistency and clarity when communicating directions in crystal lattices.