Determine the Miller indices of the plane that intersects the crystal axes at (a) (a, 2 b, 3 c),

step 1 Looking at the first part of this question, the intersecting points of the planes to the axis, w

Determine the Miller indices of the plane that intersects...

Key Concepts

Negative Indices

When a plane intercepts an axis at a negative coordinate, the corresponding reciprocal will also be negative. In crystallography, negative Miller indices are typically represented with a bar over the number to clearly indicate the negative orientation of the intercept relative to the crystal axis.

Miller Indices

Miller indices are a notation system used to denote the orientation of crystallographic planes in a lattice by a set of three integers. They are derived by taking the reciprocals of the intercepts a given plane makes with the crystal axes, followed by clearing any fractions to express the indices as the smallest set of whole numbers.

Intercepts

The intercepts of a plane with the crystal axes are the specific points where the plane crosses each axis. These intercepts, typically given in units relative to the lattice parameters, are the starting point for calculating the Miller indices by first converting them into fractional coordinates.

Reciprocal Transformation and Normalization

After determining the intercepts, the next step is to take the reciprocal of each intercept, which transforms the spatial positions into a form that can be directly related to the plane’s orientation. The resulting values are then multiplied by a common factor to eliminate any fractions, yielding the smallest possible integer values for the Miller indices. A plane parallel to an axis is assigned an intercept of infinity, leading to a Miller index of zero for that axis.

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