Miller Indices 3k626t

MILLER INDICES  PLANES  DIRECTIONS

From the law of rational indices developed by French Physicist and mineralogist Abbé René Just Haüy and popularized by William Hallowes Miller

Vector r ing from the origin to a lattice point: r = r 1 a + r2 b + r3 c a, b, c → fundamental translation vectors

Miller Indices for directions

(4,3)

(0,0) 5a + 3b

b a

Miller indices → [53]

[001]

[011]

[101]

[010] [111]

[1 10] [100]

[110]

• Coordinates of the final point  coordinates of the initial point • Reduce to smallest integer values

Family of directions Index

Number in the family for cubic lattice

<100>

→

3x2=6

<110>

→

6 x 2 = 12

<111>

→

4x2=8

Symbol

Alternate symbol

[] <>

[[ ]]

→

Particular direction

→

Family of directions

Miller Indices for planes (0,0,1)

(0,3,0)

(2,0,0)

 Find intercepts along axes → 2 3 1  Take reciprocal → 1/2 1/3 1  Convert to smallest integers in the same ratio → 3 2 6  Enclose in parenthesis → (326)

Intercepts → 1   Plane → (100) Family → {100} → 3

Intercepts → 1 1  Plane → (110) Family → {110} → 6

Intercepts → 1 1 1 Plane → (111) Family → {111} → 8 (Octahedral plane)

(111) Family of {111} planes within the cubic unit cell

d111  a / 3  a 3 / 3 The (111) plane trisects the body diagonal

(111) Plane cutting the cube into two polyhedra with equal volumes

Points about (hkl) planes For a set of translationally equivalent lattice planes will divide:

Entity being divided (Dimension containing the entity) Cell edge (1D)

Diagonal of cell face (2D)

Body diagonal (3D)

Direction

number of parts

a

[100]

h

b

[010]

k

c

[001]

l

(100)

[011]

(k + l)

(010)

[101]

(l + h)

(001)

[110]

(h + k)

[111]

(h + k + l)

The (111) planes:

The portion of the central (111) plane as intersected by the various unit cells

Tetrahedron inscribed inside a cube with bounding planes belonging to the {111} family

8 planes of {111} family forming a regular octahedron

Summary of notations

Alternate symbols

Symbol Direction

Plane

Point

[]

[uvw]

<>



()

(hkl)

{}

{hkl}

..

.xyz.

::

:xyz:

→

Particular direction

→

Family of directions

→

Particular plane

(( ))

→

Family of planes

[[ ]]

→

Particular point

→

Family of point

[[ ]]

A family is also referred to as a symmetrical set

 Unknown direction → [uvw]  Unknown plane → (hkl)  Double digit indices should be separated by commas → (12,22,3)  In cubic crystals [hkl]  (hkl)

d hkl 

a h k l 2

2

2

Condition

(hkl) will through

h even

midpoint of a

(k + l) even

face centre (001) midpoint of face diagonal (001)

(h + k + l) even

body centre midpoint of body diagonal

Index

Number of in a cubic lattice

(100)

6

d100  a

(110)

12

d110  a / 2  a 2 / 2

The (110) plane bisects the face diagonal

(111)

8

d111  a / 3  a 3 / 3

The (111) plane trisects the body diagonal

(210)

24

(211)

24

(221)

24

(310)

24

(311)

24

(320)

24

(321)

48

dhkl

Multiplicity factor

Cubic Hexagonal Tetragonal Orthorhombic Monoclinic Triclinic

hkl 48* hk.l 24* hkl 16* hkl 8 hkl 4 hkl 2

hhl 24 hh.l 12* hhl 8 hk0 4 h0l 2

hk0 24* h0.l 12* h0l 8 h0l 4 0k0 2

hh0 12 hk.0 12* hk0 8* 0kl 4

hhh 8 hh.0 6 hh0 4 h00 2

h00 6 h0.0 6 h00 4 0k0 2

* Altered in crystals with lower symmetry (of the same crystal class)

00.l 2 00l 2 00l 2

Hexagonal crystals → Miller-Bravais Indices a3

Intercepts → 1 1 - ½  Plane → (1 1 2 0)

(h k i l) i = (h + k)

a2

a1

The use of the 4 index notation is to bring out the equivalence between crystallographically equivalent planes and directions

Examples to show the utility of the 4 index notation

a3

a2

a1 Intercepts → 1 -1  

Intercepts →  1 -1 

Miller → (1 1 0 )

Miller → (0 1 0)

Miller-Bravais → (1 1 0 0 )

Miller-Bravais → (0 1 1 0)

Examples to show the utility of the 4 index notation

a3

a2

Intercepts → 1 -2 -2  Plane → (2 1 1 0 )

a1

Intercepts → 1 1 - ½  Plane → (1 1 2 0)

Intercepts → 1 1 - ½ 1 Plane → (1 1 2 1)

Intercepts → 1   1 1 Plane → (1 0 1 1)

Directions

Directions are projected onto the basis vectors to determine the components

[1120]

a1

a2

a3

Projections

a/2

a/2

−a

Normalized wrt LP

1/2

1/2

−1

Factorization

1

1

−2

Indices

[1 1  2 0]

Transformation between 3-index [UVW] and 4-index [uvtw] notations

U  u t

V  v t

1 u  (2U  V ) 3

W w

1 v  (2V  U ) t  (u  v) 3

w W

 Directions in the hexagonal system can be expressed in many ways  3-indices: By the three vector components along a1, a2 and c: rUVW = Ua1 + Va2 + Wc  In the three index notation equivalent directions may not seem equivalent  4-indices:

Directions  Planes  Cubic system: (hkl)  [hkl]  Tetragonal system: only special planes are  to the direction with same indices: [100]  (100), [010]  (010), [001]  (001), [110]  (110) ([101] not  (101))  Orthorhombic system: [100]  (100), [010]  (010), [001]  (001)  Hexagonal system: [0001]  (0001) (this is for a general c/a ratio; for a Hexagonal crystal with the special c/a ratio = (3/2) the cubic rule is followed)  Monoclinic system: [010]  (010)  Other than these a general [hkl] is NOT  (hkl)