CRYSTALLOGRAPHIC CRYSTALLOGRAPHIC POINTS, DIRECTIONS, POINTS, DIRECTIONS,

PLANESPLANES& &

THETHE MILLER SYSTEM OF MILLER SYSTEM OF INDICESINDICES

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

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

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

Such identification is carried out by means Such identification is carried out by means of of

Miller Indices.Miller Indices.

POINT COORDINATESPOINT COORDINATESThe position of any The position of any point located within a point located within a unit cell is specified in unit cell is specified in terms of its coordinates terms of its coordinates as fractional multiplies as fractional multiplies of the unit cell edge of the unit cell edge lengths.lengths.

To determine the To determine the point coordinates of point coordinates of point P, the manner point P, the manner in which the q, r, s in which the q, r, s coordinates of point coordinates of point P within the unit P within the unit cell are determined. cell are determined.

The “q” coordinate (which is a fraction) corresponds to the distance “qa” along the x-axis where “a” is the unit cell length along x-axis.

CRYSTALLOGRAPHIC CRYSTALLOGRAPHIC DIRECTIONSDIRECTIONS

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

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

2.2. The length of the vector projection on each of the The length of the vector projection on each of the three axes is determined in terms of the unit cell three axes is determined in terms of the unit cell dimensions dimensions aa, , bb, and , and cc..

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

4.4. The tree indices are enclosed in brackets as [ The tree indices are enclosed in brackets as [uvwuvw]. ]. The The uu, , vv, and , and ww integers correspond to the reduced integers correspond to the reduced projections along x, y, and z-axes respectively.projections along x, y, and z-axes respectively.

1.1. The vector as drawn The vector as drawn passes through the passes through the origin of the coordinate origin of the coordinate system, and therefore no system, and therefore no translation is necessary. translation is necessary.

2.2. Projections of this Projections of this vector along x, y, and z vector along x, y, and z axes are a/2, b, and 0c. axes are a/2, b, and 0c. In terms of unit cell In terms of unit cell dimensions ½, 1, 0. dimensions ½, 1, 0.

3.3. Reduction of these Reduction of these numbers to the lowest numbers to the lowest set of integers could be set of integers could be done through done through multipliying these multipliying these numbers by 2 to yield 1, numbers by 2 to yield 1, 2, and 02, and 0

4.4. The crystallographic The crystallographic direction is then [120]direction is then [120]

Vector A Vector A → a, a, a 1/a, 1/a, 1/a → a, a, a 1/a, 1/a, 1/a [[1 1 11 1 1]] Vector B Vector B →→ [[1 1 01 1 0]] Vector C Vector C →→ [[1 1 11 1 1]]

z

x

ya

a

a

B

C

A

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

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

]100[],001[],010[],010[],001[],100[ As a convenience, equivalent directions are As a convenience, equivalent directions are

grouped into a “family” which are grouped grouped into a “family” which are grouped in angle brackets.in angle brackets.

010

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

Sometimes the angle between two Sometimes the angle between two directions may be necessary.directions may be necessary.

A A [[hh11 k k11 l l11]] and B and B [[hh22 k k22 l l22]] →→ the angle the angle

between them is between them is ..

cos =(h1

2+k12+l12)

(h22+k2

2+l22)

h1h2 + k1k2 + l1l2

A . B=|A| |B| cos

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

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

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

The following procedure is employed in the The following procedure is employed in the determining the determining the hh, , kk, and , and ll idex numbers of idex numbers of a plane:a plane:

CRYSTALLOGRAPHIC PLANESCRYSTALLOGRAPHIC PLANES

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

2.2. At this point the crystallographic plane either At this point the crystallographic plane either intersects or parallels each of the three axes; intersects or parallels each of the three axes; the length of the planar intercept for each axis the length of the planar intercept for each axis is determined in terms of the lattice parameters is determined in terms of the lattice parameters aa, , bb, and , and cc..

3.3. The reciprocals of these numbers are taken. The reciprocals of these numbers are taken. 4.4. If necessary, these three numbers are changed If necessary, these three numbers are changed

to the set of smallest integers by multiplication to the set of smallest integers by multiplication or division by a common factor.or division by a common factor.

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

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

2.2. The plane is parallel The plane is parallel to the x’-axis and the to the x’-axis and the intercept can be taken intercept can be taken as ∞a. The y’ and z’ as ∞a. The y’ and z’ intersections are –b intersections are –b and c/2. Lattice and c/2. Lattice parameters are parameters are ∞∞, , -1-1, , and and 1/21/2. .

3.3. Reciprocals are 0, -1, Reciprocals are 0, -1, 2.2.

4.4. All are integers no reduction is necessary. All are integers no reduction is necessary.5.5. The crystallographic plane is The crystallographic plane is (012)(012)

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

Example: Example: Faces of a cubic unit cell.Faces of a cubic unit cell.

(100)

(010)

(001)

Ξ {100}

(100) (010) (001)

PLANAR DENSITYPLANAR DENSITY When slip occurs under stress, it takes When slip occurs under stress, it takes

place on the planes on which the atoms place on the planes on which the atoms are most densely packed.are most densely packed.

δ(hkl) = area# of atoms in a plane

Example: FCC unit cell

a2

(100)

z

y

x

δ(100)=4*1/4+1 a2

= a2

2

a=1

r24 δ(100)= 4r2

LINEAR DENSITYLINEAR DENSITY When planes slip over each other, slip When planes slip over each other, slip

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

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

δ[h k l] = length# of atoms

Example: Example: [[100100]] of cubic unit cell of cubic unit cell

δδ[[100100]] = 1/a = 1/a

Example: Example: Calculate planar density of the face plane (100) and linear Calculate planar density of the face plane (100) and linear density on the face diagonal [011] of an FCC structure.density on the face diagonal [011] of an FCC structure.

a

a0

a0

2

2

2

0

]011[

20

)100(

a

a

(100)

[011]