1. 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:
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

2. The Wigner-Seitz Method to Construct a Primitive Cell
A simple, geometric method to construct a Primitive Cell is called the Wigner-Seitz Method. The procedure is:
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.

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

4. 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.
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

5. 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 = n1a1 + n2a2 + n3a3
[111] direction
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

6. Directions in a Crystal: Standard Notation
A lattice Vector
T = n1a1 + n2a2 + n3a3
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 integers possible for the relative ratios.
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

7. Examples X = 1, Y = ½, Z = 0 [1 ½ 0] [2 1 0] X = ½ , Y = ½ , Z = 1
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]
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

8. Negative Directions R = n1a1 + n2a2 + n3a3
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. 8 8

9. Examples of Crystal Directions
X = 1, Y = 0, Z = 0
[100]
X = -1, Y = -1, Z = 0
[110] 9

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

11. These are called lattice planes.
Crystal Planes
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.

12. Miller Indices
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. 12

13. Example 1: (100) Plane (1,0,0) Axis 1 ∞ 1/1 1/ ∞ Miller İndices (100)Y Z
Intercept points 1 ∞
Reciprocals
1/1
1/ ∞
Smallest Ratio
Miller İndices (100) 13

14. Example 2: (110) Plane Axis 1 ∞ 1/1 1/ 1 1/ ∞ Miller İndices (110)
(1,0,0)
(0,1,0)
Axis X Y Z
Intercept points 1 ∞
Reciprocals
1/1
1/ 1
1/ ∞
Smallest Ratio
Miller İndices (110) 14

15. Example 3: (111) Plane Axis 1 1/1 1/ 1 Miller İndices (111)
(1,0,0)
(0,1,0)
(0,0,1)
Axis X Y Z
Intercept points 1
Reciprocals
1/1
1/ 1
Smallest Ratio
Miller İndices (111) 15

16. Example 4: (210) Plane Axis 1/2 1 ∞ 1/(½) 1/ 1 1/ ∞ 2
(1/2, 0, 0)
(0,1,0)
Axis X Y Z
Intercept points
1/2 1 ∞
Reciprocals
1/(½)
1/ 1
1/ ∞
Smallest Ratio 2
Miller İndices (210) 16

17. Example 5: (102) Plane Axis a b c 1 ∞ ½ 1/1 2 Miller İndices (102)
Intercept points 1 ∞
Reciprocals
1/1
1/ ∞
1/(½)
Smallest Ratio 2
Miller İndices (102) 17

18. Example 6: (102) Plane Axis a b c -1 ∞ ½ 1/-1 2 Miller İndices (102)
Intercept points -1 ∞
Reciprocals
1/-1
1/ ∞
1/(½)
Smallest Ratio 2
Miller İndices (102) 18

19. Examples of Miller Indices3 2
[2,3,3]
See the figure.
Consider the plane shaded in yellow:
Plane intercepts axes at
Reciprocal numbers are:
Miller Indices of the plane: (2,3,3)
Indices of the direction: [2,3,3]

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

21. Examples of Miller Indices

22. Indices of a Family of Planes
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.
So, indices {h,k,l} represent all of the planes
equivalent to the plane (hkl) through rotational
symmetry. 22