The Wigner-Seitz Method to Construct a Primitive Cell

1. A simple, geometric method to construct a Primitive Cell is called the Wigner-Seitz Method. The procedure is: The Wigner-Seitz Method to Construct a Primitive Cell 1. Choose a starting lattice point. 2. Draw lines to connect that point to its nearest neighbors. 3. At the mid-point & normal to these lines, draw new lines. 4. The volume enclosed is called a Wigner-Seitz cell. Illustration for the 2D parallelogram lattice.

2. 3 Dimensional Wigner-Seitz Cells Body Centered Cubic (BCC) Wigner-Seitz Cell Face Centered Cubic (FCC) Wigner-Seitz Cell 2

3. Lattice Sites in a Cubic Unit Cell • The standard notation is shown in the figure. It is understood that all distances are in units of the cubic lattice constant a, which is the length of a cube edge for the material of interest.

4. See Figure.Choose an origin, O. This choice is arbitrary, because every lattice point has identical symmetry. Then, consider the lattice vector joining O to any point in space, say point T in the figure. As we’ve seen, this vector can be written T = n1a1 + n2a2 + n3a3 Directions in a Crystal: Standard Notation [111] direction

5. A lattice Vector T = n1a1 + n2a2 + n3a3 Directions in a Crystal: Standard Notation • In order to distinguish a Lattice Direction from a Lattice Point,(n1n2n3), the 3 integers are enclosed in square brackets [ ...] [111] direction instead of parentheses (...), which are reserved to indicate a Lattice Point. In direction [n1n2n3], n1n2n3 are the smallest integerspossible for therelative ratios.

6. Examples 210 X = 1, Y = ½, Z = 0 [1 ½ 0] [2 1 0] X = ½ , Y = ½ , Z = 1 [½ ½ 1] [1 1 2]

7. When we write the direction [n1n2n3] depending on the origin, negative directions are written as R = n1a1 + n2a2 + n3a3 With a bar above the negative integers. To specify the direction, the smallest possible integers must be used. Negative Directions 7

8. Examples of Crystal Directions X = 1, Y = 0, Z = 0 [100] X = -1, Y = -1, Z = 0 [110]

9. Examples A vector can be moved to the origin. X =-1, Y = 1, Z = -1/6 [-1 1 -1/6] [6 6 1]

10. Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are called lattice planes. In the figure, the density of lattice points on each plane of a set is the same & all lattice points are contained on each set of planes. b b a a Crystal Planes The set of planes for a 2D lattice.

11. Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice & are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. To find the Miller indices of a plane, take the following steps: Determine the intercepts of the plane along each of the three crystallographic directions. Take the reciprocals of the intercepts. If fractions result, multiply each by the denominator of the smallest fraction. Miller Indices 11

12. (1,0,0) Example 1: (100) Plane 12

13. (0,1,0) (1,0,0) Example 2: (110) Plane 13

14. (0,0,1) (0,1,0) (1,0,0) Example 3: (111) Plane 14

15. (0,1,0) (1/2, 0, 0) Example 4: (210) Plane 15

16. Example 5: (102) Plane 16

17. Example 6: (102) Plane 17

18. [2,3,3] 2 Plane intercepts axes at 2 Reciprocal numbers are: 3 Examples of Miller Indices • See the figure. • Consider the plane shaded in yellow: Miller Indices of the plane: (2,3,3) Indices of the direction: [2,3,3]

19. (200) (111) (100) (100) (110) Examples of Miller Indices

20. Examples of Miller Indices

21. Sometimes. when the unit cell has rotational symmetry, several nonparallel planes may be equivalent by virtue of this symmetry, in which case it is convenient to lump all these planes in the same Miller Indices, but with curly brackets. Indices of a Family of Planes • So, indices {h,k,l}represent all of the planes equivalent to theplane (hkl) through rotational symmetry.