Wigner-Seitz Cell

The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to any of the other lattice points.

Crystal Structure 2

Wigner-Seitz Method

A simply way to find the primitivecell which is called Wigner-Seitzcell can be done as follows;

1. Choose a lattice point.2. Draw lines to connect these

lattice point to its neighbours.3. At the mid-point and normal

to these lines draw new lines.

The volume enclosed is called as a Wigner-Seitz cell.

Square lattice

Centred Rectangular lattice

Wigner-Seitz cells

Different kinds of CELLS

Unit cell

A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal.

Primitive unit cell

For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space.

Wigner-Seitz cell

A Wigner-Seitz cell is a particular kind of primitive cell

which has the same symmetry as the lattice.

Crystal Structure 5

Wigner-Seitz Cell - 3D

Crystal Structure 6

Lattice Sites in Cubic Unit Cell

CRYSTALLOGRAPHIC POINTS, DIRECTIONS, PLANES

& THE MILLER SYSTEM OF INDICES

Lecture 03

Fundamental properties of Solids

Crystal Structure 8

Crystal Directions

Fig. Shows [111] direction

• We choose one lattice point on the line as an origin, say the point O. Choice of origin is completely arbitrary, since every lattice point is identical.

• Then we choose the lattice vector joining O to any point on the line, say point T. This vector can be written as;

R = n1 a + n2 b + n3c

• To distinguish a lattice direction from a lattice point, the triple is enclosed in square brackets [ ...] is used.[n1n2n3]

• [n1n2n3] is the smallest integer of the same relative ratios.

Crystal Structure 9

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]

Examples

Crystal Structure 10

Negative directions• When we write the

direction [n1n2n3] depend on the origin, negative directions can be written as

• R = n1 a + n2 b + n3c

Direction must be

smallest integers.

Y direction

(origin) O

- Y direction

X direction

- X direction

Z direction

- Z direction

][ 321 nnn

][ 321 nnn

Crystal Structure 11

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

Examples of crystal directions

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

Crystal Structure 12

Examples

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

We can move vector to the origin.

When dealing with crsytalline materials, it is often necessary to specify a particular point within a unit cell, a particular direction or a particular plane of atoms.

Planes are important in crystals because if bonding is weak between a set of parallel planes, then brittle shear fracture may occur along these planes.

Therefore, it is necessary to be able to specify individual crystal planes and in the case of shear to specify directions within these planes.

Such identification is carried out by means of Miller Indices.

CRYSTALLOGRAPHIC DIRECTIONS

A crystallographic direction is defined as a line between two points (a vector).

1. A vector of convenient length is positioned such that it passes through the origin of the coordinate system. (Any vector can be translated throughout the crystal lattice, if parallelism is maintained).

2. The length of the vector projection on each of the three axes is determined in terms of the unit cell dimensions a, b, and c.

3. These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values.

4. The tree indices are enclosed in brackets as [uvw]. The u, v, and w integers correspond to the reduced projections along x, y, and z-axes respectively.

• Vector A → a, a, a 1/a, 1/a, 1/a [1 1 1]

• Vector B → [1 1 0]

• Vector C → [1 1 1]

z

x

ya

a

a

B

C

A

• For some crystal structures, several nonparallel directions with different indices are actually equivalent. (The spacing of atoms along each direction is the same)

• For example in cubic crystals, all the directions represented by the following indices are equivalent.

]100[],001[],010[],010[],001[],100[

• As a convenience, equivalent directions are grouped into a “family” which are grouped in angle brackets.

010

]100[],001[],010[],010[],001[],100[

Sometimes the angle between two directions may be necessary.A [h1 k1 l1] and B [h2 k2 l2] → the angle

between them is .

cos =(h1

2+k12+l12)

(h22+k2

2+l22)

h1h2 + k1k2 + l1l2

A . B=|A| |B| cos

Crystal Structure 18

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 density of lattice points on each plane of a set is the same and all lattice points are contained on each set of planes.

b

a

b

a

The set of planes in 2D lattice.

• The orientations of planes for a crystal structure are represented in a similar manner.

• In all but the hexagonal crystal system, crystallographic planes are specified by three Miller Indices as (hkl).

• Any two parallel planes are equivalent and have identical indices.

• The following procedure is employed in the determining the h, k, and l idex numbers of a plane:

CRYSTALLOGRAPHIC PLANES

1. If the plane passes through the selected origin, either another parallel plane must be constructed within the unit cell by an appropriate translation, or a new origin must be established at the corner of another unit cell.

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.

3. The reciprocals of these numbers are taken.

4. If necessary, these three numbers are changed to the set of smallest integers by multiplication or division by a common factor.

5. The integer indices are enclosed within parantheses as (hkl).

1. The plane passes through the selected origin O. Therefore, a new origin must be selected at the corner of an adjacent unit cell.

Various non-parallel planes may have similarities (crystallographically equivalent ). Such planes are referred to as “family of planes” and are designated as {h k l}

Example: Faces of a cubic unit cell.

(100)

(010)

(001)

Ξ {100}

(100) (010) (001)

Indices of Planes: Cubic Crystal

LINEAR DENSITYWhen planes slip over each other, slip

takes place in the direction of closest packing of atoms on the planes.

The linear density of a crystal direction [h k l] is determined as:

δ[h k l] = length# of atoms

26

ex: linear density of Al in [110] direction 

a = 0.405 nm

FCC: Linear Density• Linear Density of Atoms LD = 

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD

Adapted fromFig. 3.1(a),Callister & Rethwisch 8e.

BCC: Linear Density

Calculate the linear density for the following directions in terms of R:

a. [100]

b. [110]

c. [111]

28

Planar Density of (100) IronSolution:  At T < 912ºC iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a

Adapted from Fig. 3.2(c), Callister & Rethwisch 8e.

2D repeat unit

= Planar Density = a2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

12

R3

34area

2D repeat unit

P 3.55 (a): Planar Density for BCC

Derive the planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R.

30

Planar Density of BCC (111) IronSolution (cont):  (111) plane 1 atom in plane/ unit surface cell

333 2

2

R3

16R

34

2a3ah2area

atoms in plane

atoms above plane

atoms below plane

ah2

3

a 2

2D re

peat

uni

t

1

= = nm2

atoms7.0m2

atoms0.70 x 1019

3 2R3

16Planar Density =

atoms

2D repeat unit

area

2D repeat unit

P 3.54 (a): FCC

Derive planar density expressions for FCC (100), (110), and (111) planes.

P 3.56

32

3.56 (a) Derive the planar density expression for the HCP (0001) plane in terms of the atomic radius R.

(b) Compute the planar density value for this same plane for magnesium. (atomic radius for magnesium is

0.160 nm)