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: http://www1.gantep.edu.tr/~bgonul/dersnotlari/.

Her homepage is Here: http://www1.gantep.edu.tr/~bgonul/

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: http://www1.gantep.edu.tr/~bgonul/dersnotlari/.

Her homepage is Here: http://www1.gantep.edu.tr/~bgonul/

• 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.

Illustration for the 2D parallelogram lattice.

Illustration for the 2D parallelogram lattice.

4. The volume enclosed is calleda Wigner-Seitz cell.

3

3 Dimensional Wigner-Seitz Cells

Face Centered Cubic (FCC)

Wigner-Seitz Cell

Body Centered Cubic (BCC)

Wigner-Seitz Cell

Body Centered Cubic (BCC)

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 = n1a1 + n2a2 + n3a3[111] direction

• In order to distinguish a Lattice Direction from a Lattice Point, (n1n2n3), the 3 integers are enclosed in square brackets [ ...] instead of parentheses (...), which are reserved to indicate a Lattice Point. In direction [n1n2n3], n1n2n3 are the smallest integers possible for the relative ratios.

Examples

X = ½ , Y = ½ , Z = 1[½ ½ 1] [1 1 2]

X = 1, Y = ½, Z = 0[1 ½ 0] [2 1 0]

210

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•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

X = -1, Y = -1, Z = 0[110]

X = 1, Y = 0, Z = 0[100]

Examples of Crystal Directions

X = -1, Y = 1, Z = -1/6[-1 1 -1/6] [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

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• 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

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Axis X Y Z

Intercept points 1 ∞ ∞

Reciprocals 1/1 1/ ∞ 1/ ∞Smallest

Ratio 1 0 0

Miller İndices (100)(1,0,0)

Example 1: (100) Plane

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Axis X Y Z

Intercept points 1 1 ∞

Reciprocals 1/1 1/ 1 1/ ∞Smallest

Ratio 1 1 0

Miller İndices (110)(1,0,0)

(0,1,0)

Example 2: (110) Plane

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Axis X Y Z

Intercept points 1 1 1

Reciprocals 1/1 1/ 1 1/ 1Smallest

Ratio 1 1 1

Miller İndices (111)(1,0,0)

(0,1,0)

(0,0,1)

Example 3: (111) Plane

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Axis X Y Z

Intercept points 1/2 1 ∞

Reciprocals 1/(½) 1/ 1 1/ ∞Smallest

Ratio 2 1 0

Miller İndices (210)

(1/2, 0, 0)

(0,1,0)

Example 4: (210) Plane

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Axis a b cIntercept

points 1 ∞ ½

Reciprocals 1/1 1/ ∞ 1/(½)

Smallest Ratio 1 0 2

Miller İndices (102)

Example 5: (102) Plane

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Example 6: (102) Plane

Reciprocal numbers are: 2

1 ,

2

1 ,

3

1Plane intercepts axes at cba 2 ,2 ,3

Miller Indices of the plane: (2,3,3)Indices of the direction: [2,3,3]

a3

2

2

bc

[2,3,3]

Examples of Miller Indices

• See the figure. • Consider the plane

shaded in yellow:

(100)

(200)

(110) (111)

(100)

Examples of Miller Indices

Examples of Miller Indices

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• 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 planesequivalent to the plane (hkl) through rotationalsymmetry.

)111(),111(),111(),111(),111(),111(),111(),111(}111{

)001(),100(),010(),001(),010(),100(}100{

Indices of a Family of Planes