Crystal Structure Continued! NOTE!! Again, much discussion & many figures in what follows was constructed from lectures posted on the web by Prof. Beşire GÖNÜL (Turkey). She has done an excellent job covering many details of crystallography & she illustrates with many very nice pictures of lattice structures. Her lectures are posted Here: Her homepage is Here:

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 2 dimensional parallelogram lattice.

3 3 Dimensional Wigner-Seitz Cells Face Centered Cubic Wigner-Seitz Cell Body Centered Cubic Wigner-Seitz Cell

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.

Directions in a Crystal: Standard Notation 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 = n 1 a 1 + n 2 a 2 + n 3 a 3 [111] direction In order to distinguish a Lattice Direction from a Lattice Point, (n 1 n 2 n 3 ), the 3 integers are enclosed in square brackets [...] instead of parentheses (...), which are reserved to indicate a Lattice Point. In direction [n 1 n 2 n 3 ], n 1 n 2 n 3 are the smallest integers possible for the relative ratios.

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

7 When we write the direction [n 1 n 2 n 3 ] depending on the origin, negative directions are written as R = n 1 a 1 + n 2 a 2 + n 3 a 3 To specify the direction, the smallest possible integers must be used. Y direction (origin) O - Y direction X direction - X direction Z direction - Z direction Negative Directions

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

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

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 a b a The set of planes for a 2D lattice. Crystal Planes

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: 1.Determine the intercepts of the plane along each of the three crystallographic directions. 2.Take the reciprocals of the intercepts. 3.If fractions result, multiply each by the denominator of the smallest fraction. Miller Indices

12 Axis XYZ Intercept points 1∞∞ Reciprocals 1/11/ ∞ Smallest Ratio 100 Miller İndices (100) (1,0,0) Example 1

13 Axis XYZ Intercept points 11∞ Reciprocals 1/1 1/ ∞ Smallest Ratio 110 Miller İndices (110) (1,0,0) (0,1,0) Example 2

14 Axis XYZ Intercept points 111 Reciprocals 1/1 Smallest Ratio 111 Miller İndices (111) (1,0,0) (0,1,0) (0,0,1) Example 3

15 Axis XYZ Intercept points 1/21∞ Reciprocals 1/(½)1/ 11/ ∞ Smallest Ratio 210 Miller İndices (210) (1/2, 0, 0) (0,1,0) Example 4

16 Axisabc Intercept points 1∞½ Reciprocals 1/1 1/ ∞1/(½) Smallest Ratio 102 Miller İndices (102) Example 5

17 Axisabc Intercept points ∞½ Reciprocals 1/-1 1/ ∞1/(½) Smallest Ratio 02 Miller İndices (102) Example 6

Reciprocal numbers are: Plane intercepts axes at Miller Indices of the plane: (2,3,3) (100) (200) (110) (111) (100) Indices of the direction: [2,3,3] [2,3,3] Examples of Miller Indices

Crystal Structure19

20 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. Thus indices {h,k,l} represent all the planes equivalent to the plane (hkl) through rotational symmetry. Indices of a Family of Planes