Crystallographic Points, Directions and Planes
Step-by-Step Guide to Crystallographic Points,
Directions, and Planes
Kelsey Jorgensen, Materials 100A
December 13, 2015
Naming points, directions, and planes in a unit cell can seem overwhelming at 铿乺st, but will be-
come easy as you practice and follow the following procedures. Indices of crystallographic points,
directions, and planes are given in terms of the lattice constants of the unit cell. For points and
directions, you can consider the indices to be coef铿乧ients of the lattice constants. Remember that
you only need to invert the indices for planes. It is essential to label axes, lattice constants, and
identifying information for directions (vector arrow head) and planes (axes intercepts) in order to
receive full credit.
For extra practice with indexing directions, visit the University of Liverpool鈥檚 interactive site
for crystallographic directions. For extra practice with planes, visit the University of Liverpool鈥檚
interactive site for crystallographic planes. This website from Cambridge has an excellent website
about crystallographic planes along with plug-ins that will draw a plane given its indices and will
let you match indices to a range of planes.
Points
Labeling points in a unit cell follows the same procedure for listing points in any Cartesian coordi-
nate system. The indices used to refer to points are q, r, and s. They are listed without commas,
parentheses, or brackets. Consider point P in Figure 1a. If you were standing at the origin of the
unit cell, you could travel q a in the x-direction, r v in the y-direction, and s c in the z-direction
to get to point P. Thus we would say that point P corresponds to the qrs point coordinates.
To 铿乶d q, r, and s when you are shown a drawing with a point:
1. Start with your pencil at the origin.
2. Count the number lattice constants you must move in the x-, y-, and z-directions to reach the
point.
3. Write the point as qrs without commas, parentheses, or brackets. Do not convert the coor-
dinates
to
reduced
integers.
The
111
222
point
in
the
BCC
structure
is
not
the
same
as
the
111
point.
To draw a point given qrs:
1. Start with your pencil at the origin.
2. Count q a in the x-direction, r v in the y-direction, and s c in the z-direction.
3. Draw and label the point.
In the BCC system shown in Figure 1b, the path to point 9 would be 0 a in the x-direction, 1 a
in the y-direction, and 1 a in the z-direction. Thus the point coordinates of point 9 are 011. The
point
coordinates
for
point
5
are
1
2
1
2
1
2
.
1
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