Crystallographic planes and directions

Crystallographic planes and directions

1. CRYSTALLOGRAPHIC PLANES AND DIRECTIONS 1 Author : Nicola Ergo

2. 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 www.agh.edu.pl 2

3. 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

4. 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.

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 . 5

6. 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

7. 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 a 1 , a 2 , and a 3 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 . 7 Coordinate axis system for a hexagonal unit cell (Miller– Bravais scheme ). Some examples of directions and planes within a hexagonal unit cell .

8. 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 ). 8 We specify the position of P in terms of the generalized coordinates q , r, and s , where q is some fractional length ( q a ) of a along the x axis, r is some fractional length ( r b ) 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.

9. 1.2 Example point coordinates 9 Locate the point ¼ 1 ½.

10. 1.2 Example point coordinates 10 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

11. 2. Crystallographic directions 2.1 Definition 11 Some 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

12. 2. Crystallographic directions 2.1 Definition 12 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 .

13. 13 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 .

14. 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. 14 2. Crystallographic directions 2.2 Examples O’

15. 15 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> .

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: 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)

17. 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

18. 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 3.1 Definition

19.   19 3. Crystallographic planes 3.2 Examples

20. Example of O translation. Determine the Miller indices for this plane: 20 3. Crystallographic planes 3.2 Examples

21. 21 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:

22. 22 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.

23. 23 3. Crystallographic planes 3.2 Examples

24. 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) 24 q r s

25. 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. C rystallographic direction 1. 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 planes 1. 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). 25

26. Summary 26

27. References 27 Materials Science and Engineering, An I ntroduction – William D. Callister, Jr.

28. THANK YOU DZIĘKUJĘ GRAZIE ΕΥΧΑΡΙΣΤΙΕΣ TEŞEKKÜRLER GRACIAS 28