CRYSTALLOGRAPHIC
PLANES AND DIRECTIONS
Author: Nicola Ergo
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Plan
1. Introduction
1.1 Point coordinates
1.2 Example point coordinates
2. Crystallographic directions
2.1 Definition
2.2 Examples
3. Crystallographic planes
3.1 Definition
3.2 Examples
4. Summary
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1. Introduction
When dealing with crystalline materials, it often becomes necessary
to specify a particular point within a unit cell, a crystallographic
direction, or some crystallographic plane of atoms.
Three numbers or indices are used to designate point locations,
directions, and planes.
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1. Introduction
The basis for determining index values is the unit cell, with a righthanded coordinate system consisting of three (x, y, and z) axes
situated at one of the corners and coinciding with the unit cell edges,
as shown in figure.
A unit cell with x, y, and z coordinate axes,
showing axial lengths (a, b, and c) and
interaxial angles (α, β, and γ).
Lattice parameters of crystal structure.
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1. Introduction
On this basis there are seven different possible combinations of a, b, and
c, and α, β, and γ, each of which represents a distinct crystal system.
These seven crystal systems are cubic, tetragonal, hexagonal,
orthorhombic, rhombohedral, monoclinic, and triclinic.
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1. Introduction
On this basis there are seven different possible combinations of a, b, and
c, and α, β, and γ, each of which represents a distinct crystal system.
These seven crystal systems are cubic, tetragonal, hexagonal,
orthorhombic, rhombohedral, monoclinic, and triclinic.
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1. Introduction
A problem arises for crystals having hexagonal symmetry in that some
crystallographic equivalent directions will not have the same set of indices.
This is circumvented by utilizing a four-axis, or Miller–Bravais, coordinate
system. The three a1, a2, and a3 axes are all contained within a single
plane (called the basal plane) and are at 120° angles to one another.
The z axis is perpendicular to this basal plane.
Coordinate axis system for a hexagonal
unit cell (Miller–Bravais scheme).
Some examples of directions and planes
within a hexagonal unit cell.
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1.1 Point coordinates
The position of any point located within a unit cell may be specified in
terms of its coordinates as fractional multiples of the unit cell edge
lengths (i.e., in terms of a, b, and c).
We specify the position of P in terms of
the generalized coordinates q, r, and s,
where q is some fractional length (qa)
of a along the x axis, r is some
fractional length (rb) of b along the y
axis, and similarly for s. Thus, the
position of P is designated using
coordinates q r s with values that are
less than or equal to unity.
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1.2 Example point coordinates
• Locate the point ¼ 1 ½.
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1.2 Example point coordinates
• The lengths of a, b, and c are 0.48nm, 0.46nm, and
0.40nm, respectively.
• The indices (1/4;1;1/2) should be multiplied to give the
coordinates within the unit cell:
• x coordinate: 1/4xa=1/4x0.48= 0,12nm
• y coordinate: 1xb=1x0.46=
0.46nm
• z coordinate: 1/2xc=1/2x0.40= 0.20nm
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2. Crystallographic directions
2.1 Definition
A crystallographic direction is defined as a line between two points or a
vector.
The following steps are utilized in the determination of the three
directional indices:
1. A vector of convenient length is positioned such that it passes through
the origin O of the coordinate system. Any vector may be translated
throughout the crystal lattice without alteration, if parallelism is
maintained.
O
Some crystallographic directions.
Example of vector translation.
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2. Crystallographic directions
2.1 Definition
A crystallographic direction is defined as a line between two points or a
vector.
The following steps are utilized in the determination of the three
directional indices:
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.
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2. Crystallographic directions
2.1 Definition
A crystallographic direction is defined as a line between two points or a
vector.
The following steps are utilized in the determination of the three
directional indices:
3. These three numbers are multiplied or divided by a common factor to
reduce them to the smallest integer values.
4. The three indices, not separated by commas, are enclosed in square
brackets, thus: [u v w].
Example of a crystallographic direction.
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2. Crystallographic directions
2.2 Examples
If any of the indices is negative, a bar is placed in top of that index.
Example: Draw a [110] direction within a cubic unit cell.
O’
When one index is negative, it’s also possible to translate the origin O to
the position O’, in order to have the direction within the unit cell.
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2. Crystallographic directions
2.2 Examples
For some crystal structures, several nonparallel directions with different
indices are actually equivalent; this means that the spacing of atoms
along each direction is the same. For example, in cubic crystal, all the
direction represented by the following indices are equivalent: [100],
[100], [010], [010], [001], and [001]. As a convenience, equivalent
directions are grouped together into a family of directions, which are
enclosed in angle brackets, thus: <100>.
z
a
a
y
x
Family of directions <100> in a
cubic crystal structure.
a
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3. Crystallographic planes
3.1 Definition
Crystallographic planes are specified by three Miller indices as (h k l).
The procedure employed in determination of the h, k, and l index numbers
is as follows:
1. If the plane passes through the selected origin O, either another
parallel plane must be constructed within the unit cell by an appropriate
translation (a), or a new origin O’ must be established at the corner of
another unit cell (b).
(a)
(b)
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3. Crystallographic planes
3.1 Definition
Crystallographic planes are specified by three Miller indices as (h k l).
The procedure employed in determination of the h, k, and l index numbers
is as follows:
2. At this point the crystallographic plane either intersects or parallels
each of the three axes; the length of the planar intercept for each axis is
determined in terms of the lattice parameters a, b, and c.
Intersections:
x-axis ∞
y-axis 1
z-axis 1/2
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3. Crystallographic planes
3.1 Definition
3. The reciprocals of these numbers are taken. A plane that parallels an
axis may be considered to have an infinite intercept, and, therefore, a zero
index.
4. If necessary, these three numbers are changed to the set of smallest
integers by multiplication or division by a common factor.
5. Finally, the integer indices, not separated by commas, are enclosed
within parentheses, thus: (h k l).
Intersections: (∞ 1 ½)
Reciprocals:
(0 1 2)
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3. Crystallographic planes
3.2 Examples
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3. Crystallographic planes
3.2 Examples
Example of O translation.
Determine the Miller indices for this plane:
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3. Crystallographic planes
3.2 Examples
Solution:
Since the plane passes through the selected origin O, a new origin must be
chosen at the corner of an adjacent unit cell, taken as O’ and shown in
sketch (b). This plane is parallel to the x axis, and the intercept may be
taken as ∞a. The y and z axes intersections, referenced to the new origin
O’, are –b and c/2, respectively. Thus, in terms of the lattice parameters a,
b, and c, these intersections are ∞, -1, and ½. The reciprocals of these
numbers are 0, -1, and 2; and since all are integers, no further
reduction is necessary. Finally, enclosure in parentheses yields (012).
These steps are briefly summarized below:
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3. Crystallographic planes
3.2 Examples
A family of planes contains all the planes that are crystallographically
equivalent—that is, having the same atomic packing; and a family is
designated by indices that are enclosed in braces—such as {100}. For
example, in cubic crystals the (111), (111), (111), (111), (111), (111),
(111), and (111) planes all belong to the {111} family.
(a) Reduced-sphere BCC unit cell with (110)
plane.
(b) Atomic packing of a BCC (110) plane.
Corresponding atom positions from (a) are
indicated.
(a) Reduced-sphere FCC unit cell with (110)
plane.
(b) Atomic packing of an FCC (110) plane.
Corresponding atom positions from (a) are
indicated.
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3. Crystallographic planes
3.2 Examples
( 1 0 0)
(1 1 0)
(1 1 1)
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Summary
• Coordinates of points
We can locate certain points, such as
atom positions, in the lattice or unit
cell by constructing the right-handed
coordinate system.
• A crystallographic direction
is defined as a line between two
points, or a vector.
• Crystallographic planes
are specified by three Miller indices
as (h k l).
qrs
[u v w]
(h k l)
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Summary
• Coordinates of points
The position of any point located within a unit
cell may be specified in terms of a, b, and c as
fractional multiples of the unit cell edge lengths.
• Crystallographic direction
1.
2.
3.
4.
5.
The vector must pass through the origin.
Projections.
Projections in term of a, b, and c.
Reductions to the smaller integer value.
Enclosure [u v w].
• Crystallographic planes
1.
2.
3.
4.
5.
The plane must not pass through the origin.
Intersections.
Intersections in term of a, b, and c.
Reciprocals.
Enclosure (h k l).
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Summary
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References
Materials Science and Engineering, An Introduction – William D. Callister, Jr.
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THANK YOU
DZIĘKUJĘ
GRAZIE
ΕΥΧΑΡΙΣΤΙΕΣ
TEŞEKKÜRLER
GRACIAS
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