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CIVE 2700 – Civil Engineering Materials, Winter 2007
1
CRYSTALLOGRAPHIC
DIRECTIONS AND PLANES
Professor George Hadjisophocleous, Ph.D., P.Eng.
Carleton University
Department of Civil and Environmental Engineering
Ottawa, Ontario
CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.2
Outline
• Crystallographic directions
• Crystallographic planes

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CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.3
Objectives
• Learn to compute crystallographic
directions and planes
CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.4
Crystallographic Directions
• A crystallographic direction is defined as a line
between two points, or a vector.
• Three integers or indexes are used to
designate directions and planes
• Basis for determining index values is the unit
cell, with a coordinate system
(x,y,z)
axes
located at one corner of the cell and coinciding
with cell edges.

CIVE 2700 – Civil Engineering Materials, Winter 2007
3
CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.5
Unit Cell Coordinates
• The numbers refer to the distance from the origin
in terms of the numbers of lattice parameters.
CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.6
Crystallographic Directions
•
The following steps are used to determine
the three directional indices:
1.
A vector of convenient length is positioned such
that it passes through the origin of the coordinate
system. Any vector may be translated throughout
the crystal lattice without alteration, if parallelism
is maintained.
2.
The length of the vector projection on each of the
three axes is determined;
these are measured in
terms of the unit cell dimensions
a, b,
and
c.
3.
These three numbers are multiplied or divided by
a common factor to reduce them to the smallest
integer values.

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CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.7
Crystallographic Directions
4. The three indices, not separated by commas, are
enclosed in square brackets, thus:
[uv
w]. The u, v,
and
w
integers correspond to the reduced
projections along the x, y, and
z
axes,
respectively
.
•
For each of the three axes, there will exist
both positive and negative coordinates. Thus
negative indices are also possible, which are
represented by a bar over the appropriate
index.
CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.8
Crystallographic Directions
• For example, the [1
!
1] direction would have a
component in the
-y
direction.
• Changing the signs of all indices produces an
antiparallel direction; that is, [
!
1
!
] is directly
opposite to [1
!
1].
• If more than one direction or plane is to be

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