198

Analytical Methods for Materials

Lesson 8Lattice Planes and Directions

Suggested Reading• Chapters 2 and 6 in Waseda

199

Directions and Miller Indices • Draw vector and define the tail

as the origin.

• Determine the length of the vector projection in unit cell dimensions

– a, b, and c.

• Remove fractions by multiplying by the smallest possible factor.

• Enclose in square brackets

• Negative indices are written with a bar over the number..

[111]

[110]

x

y

z

[201]

1 12 3

13

12

10 0 0

1[ 26 3 6 ]

P

a b c

OriginPoint P

ab

c

O

200

Families of Directions(i.e., directions of a form)

• In cubic systems, directions that have the same indices areequivalent regardless of their order or sign.

[10The

0],famil

[100]y of <100> d

[010], [010]

ir

[0

ections

01], [

is:

001]

[001]

[010]

[100] [001]

[100]

x

z

y[010]

We enclose indices in carats rather than brackets to indicate

a family of directions

All of these vectors have the same

“size” and# lattice

points/length

201

[100]

[100] [010] [001]

<100>

CUBIC <aaa>

[100] [010] [001]

TETRAGONAL <aac>[100] [010][100] [010]

ORTHORHOMBIC <abc>[100]

In non-cubic systems,

directions with the same

indices may notbe equivalent.

202

Directions in Crystals

Directions and their multiples are identical

x

z y

[010]

[020]

[030]

[110]

[220]

Ex.:[220] 2 [110]

Vectors and multiples of vectors

have the same # lattice points/length

203

Miller Indices for Planes Specific crystallographic plane: (hkl)

Family of crystallographic planes: {hkl}

– Ex.: (hkl), (lkh), (hlk) … etc.

– In cubic systems, planes having the same indices are equivalent regardless of order or sign.

• In hexagonal crystals, we use a four index system (h k i l).– We can convert from three to four indices

• h+k = -i

204

ALL MEMBERS HAVE SAME ARRANGEMENT OF LATTICE POINTS

{h k l}

We use Miller indices to denote planes

FAMILY OF PLANES

205

1. Identify the coordinate intercepts of the plane (i.e., the coordinates at whichthe plane intersects the x, y, and z axes).

If plane is parallel to an axis, the intercept is taken as infinity ().

If the plane passes through the origin, consider an equivalent plane in an adjacentunit cell or select a different origin for the same plane.

2. Take reciprocals of the intercepts.

3. Clear fractions to the lowest integers.

4. Cite specific planes in parentheses, (h k l), placing bars over negative indices.

PROCEDURES FOR INDICES OF PLANES(Miller indices)

206Slide 206

MILLER INDICES FOR A SINGLE PLANE

x

y

z

x y zIntercept 1 1

Reciprocal 1/1 1/1 1/Clear 1 1 0

INDICES 1 1 0

The {110} family of planes(110), (011), (101), (110), (011), (101)(110), (110), (101), (101), (011), (011)

(110)

207Slide 207

MILLER INDICES FOR A SINGLE PLANE – cont’d

x

y

zx y z

Intercept 1 1 1Reciprocal 1/1 1/1 1/1

Clear 1 1 1INDICES 1 1 1

x y zIntercept -1 -1 -1

Reciprocal -1/1 -1/1 -1/1Clear -1 -1 -1

INDICES 1 1 11111 1 1 (( ) )

208

2 2 0

2 2 0

2/1 2/1 1/

1/2 1/2

MILLER INDICES FOR A SINGLE PLANE – cont’d

x y zIntercept

Reciprocal

Clear

INDICES

(220)

x

y

z

Planes and their multiples are not identical( (110)220)

209

Planes in Unit CellsSome important aspects of Miller indices for planes:

1. Planes and their negatives are identical.This was NOT the case for directions.

2. Planes and their multiples are NOT identical.This is opposite to the case for directions.

3. In cubic systems, a direction that has the same indices as a plane is to that plane. This is not always true for non-cubic systems.

210

211

Planes of a Zone

• A zone is a direction [uvw]

• Planes belonging to a particular zone share a direction.

This direction is known as a zoneaxis.

y

z

(220)

(010)

ZONE AXIS [uvw] = [001]

(110)

001 1 10

001 220

110 001 1 0 1 0 0 1 0

220 001 2 0 2 0 0 1 0

0

lies on

lies on

hkl uvw

Weiss Zone Law

• If a direction [uvw] lies in a plane {hkl}:

[uvw]·{hkl} = uh + vk + wl = 0

212

ZONE AXIS

(110)

(220)

(010)

[ ] [001]uvw

This rule holds for all

crystal systems

213

How to Determine the Zone Axis

• Take the cross product of the intersecting planes.

y

z ZONE AXIS [uvw] = [001]

(110)

(h1k1l1) × (h2k2l2) = [uvw]

1 1 0 1 12 2 0 2 2

1 0 0 2 1 2 1 0 0 2 1 2[0 ] [ 040 0 1]

u u

u u

v w

w w

v

v v

(220)

(010)

214

Indexing in Hexagonal Systems

• The regular 3 index system is not suitable.

• Planes with the same indices do not necessarily look like.

• 4 index system introduced.– Miller-Bravais indices

a1

a2

a3

c

120°

120° 120°

a1

a2

a3

c (0001)

(1210)(1100)

(10 11)[100]

[010]

[001][110]

215

Indexing in Hexagonal Systems

• Planes:– (hkl) becomes (hkil)– i = -(h+k)

• Directions:– [UVW] becomes [uvtw]– U = u-t ; u = (2U – V)/3– V = v-t ; v = (2V – U)/3– W=w ; t=-(U+V)

216

a1

a2

a3

a1

a2

a3

PLANES

(100)

(010) (110)

(010)(110)

(100)

(1010)

(1100)(0110)

(1100) (0110)

(1010)

DIRECTIONS

a1

a2

a3

a1

a2

a3

[120] [110]

[100]

[210]

[110]

[1010]

[2110][1120]

[1100][0110]

Miller Indices Miller-Bravais Indices(hkil)(hkl)

(uvtw)(UVW)

217

a1

c

a2

a3

010 12 10

100 21 10

110 1120

Some typical directions in an HCP unit cell using three- and four-axis systems.

218

Inter-planar Spacings

• The inter-planar spacing in a particular direction is the distance between equivalent planes of atoms.

Each material has a set of characteristic inter-planar spacings.They are directly related to crystal size (i.e. lattice parameters) and atom location.

a

x

y(100)

(110)

(210)

z

Assuming no intercept on z-axis

d210

219

Interplanar Spacing – cont’d

2 2 2

2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2 2

2 2 2 3

2 2 2

2 2 2 2

2

1

1 43

1

sin 2 cos cos11 3cos 2cos

1

1 1si

h k ld a

h hk k ld a c

h k ld a c

h hk k hk kl hld a

h k ld a b c

d

CUBIC:

HEXAGONAL:

TETRAGONAL:

RHOMBOHEDRAL:

ORTHORHOMBIC:

MONOCLINIC:

2 2 2 2

2 2 2 2

2 2 211 22 3 12 23 132 2

sin 2 cosn

1 1 2 2 2

h k l hla b c ac

S h S k S l S hk S kl S hld V

TRICLINIC*:

2 2 2 2 2 2 2 2 211 22 33

2 2 212 23 13

2 2 2

sin ; sin ; sin

cos cos cos ; cos cos cos ; cos cos cos

1 cos cos cos 2cos cos cos

S b c S a c S a b

S abc S a bc S ab c

V abc