4. Crystallographic Points, Directions, and Planes

Definition

Fractional multiples of the unit cell edge lengths.

Crystallographic Direction: A line between two points, or a vector.
Any directions parallel to each other are equivalent and have identical indices.
Specific directions are indicated by square brackets, $[u, v, w]$ , and the negative direction is indicated by a bar on top of the corresponding index.

Any two planes parallel to each other are equivalent and have identical Miller indices.

If the plane passes the origin, then construct a parallel plane that doesn’t pass the origin, or select a new origin.
Find the intercepts of the plane along the axes, $x$ , $y$ , $z$ .
Take reciprocals of the intercepts.
Convert into smallest integers in the same exact ratio.
Enclose in parenthesis $(u, v, w)$ .

For cubic crystals (only), planes and directions having the same indices are perpendicular to one another.

Several nonparallel planes with different indices are crystallographic equivalent, i.e., a family of crystallographic planes.
Parenthesis ⇒ Braces
Caution has to be executed when the lattice is not cubic.

$L D = \frac{N u mb er o f a t o m s ce n t ere d o n d i rec t i o n v ec t or}{L e n g t h o f d i rec t i o n v ec t or}$

$P D = \frac{N u mb er o f a t o m s ce n t ere d o n a pl an e}{A re a o f pl an e}$

center

$(a)$ A portion of a close-packed plane of atoms; $A$ , $B$ , and $C$ position are indicated.

$(b)$ The $A B$ stacking for close-packed atomic planes.

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