CRYSTALLOGRAPHIC PLANES AND DIRECTIONS Author: Nicola Ergo 1 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 2 www.agh.edu.pl 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. 3 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. 4 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. 5 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. 6 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. 7 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. 8 1.2 Example point coordinates • Locate the point ¼ 1 ½. 9 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 10 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. 11 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. 12 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. 13 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. 14 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 15 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) 16 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 17 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) 18 3. Crystallographic planes 3.2 Examples 19 3. Crystallographic planes 3.2 Examples Example of O translation. Determine the Miller indices for this plane: 20 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: 21 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. 22 3. Crystallographic planes 3.2 Examples ( 1 0 0) (1 1 0) (1 1 1) 23 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) 24 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). 25 Summary 26 References Materials Science and Engineering, An Introduction – William D. Callister, Jr. 27 THANK YOU DZIĘKUJĘ GRAZIE ΕΥΧΑΡΙΣΤΙΕΣ TEŞEKKÜRLER GRACIAS 28