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CRYSTALLOGRAPHICPLANES AND DIRECTIONS
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Author: Nicola Ergo
Plan1. 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
www.agh.edu.pl
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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 right-handed 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). 4
Lattice parameters of crystal structure.
1. IntroductionOn 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. IntroductionOn 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. IntroductionA 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.
7Coordinate axis system for a hexagonal unit cell (Miller–Bravais scheme).
Some examples of directions and planes within a hexagonal unit cell.
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).
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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.
1.2 Example point coordinates
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• Locate the point ¼ 1 ½.
1.2 Example point coordinates
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• 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
2. Crystallographic directions 2.1 Definition
11Some crystallographic directions. Example of vector translation.
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
2. Crystallographic directions 2.1 Definition
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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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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].
2. Crystallographic directions 2.1 Definition
Example of a crystallographic direction.
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.
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
O’
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Family of directions <100> in a cubic crystal structure.
z
y
x a
a
a
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>.
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).
16(a) (b)
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 1z-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 3.1 Definition
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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
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3. Crystallographic planes 3.2 Examples
Solution:Since the plane passes through the selected origin O, a new origin must bechosen 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 crystallographicallyequivalent—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.
( 1 0 0 ) (1 1 1 )(1 1 0 )
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3. Crystallographic planes 3.2 Examples
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).
[u v w]
(h k l)
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q r s
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 direction1. The vector must pass through the origin.2. Projections.3. Projections in term of a, b, and c.4. Reductions to the smaller integer value.5. Enclosure [u v w].
• Crystallographic planes1. The plane must not pass through the origin.2. Intersections.3. Intersections in term of a, b, and c.4. Reciprocals.5. Enclosure (h k l).
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Summary
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References
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Materials Science and Engineering, An Introduction – William D. Callister, Jr.
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