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 路 a in the x-direction, r 路 v in the y-direction, and s 路 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 路 a in the x-direction, r 路 v in the y-direction, and s 路 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 路 a in the x-direction, 1 路 a

in the y-direction, and 1 路 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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