Miller Indices

Report
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 passing from the origin to a lattice point:
r = r1 a + r2 b + r3 c
a, b, c → fundamental translation vectors
Miller Indices for directions
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(4,3)
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5a + 3b
(0,0)
b
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Miller indices → [53]
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a
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[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
[]
[uvw]
<>
<uvw>
()
(hkl)
{}
{hkl}
..
.xyz.
::
:xyz:
→
Particular direction
→
Family of directions
→
Particular plane
(( ))
→
Family of planes
[[ ]]
→
Particular point
→
Family of point
Direction
[[ ]]
Plane
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 pass through
h even
midpoint of a
(k + l) even
(h + k + l) even
face centre (001)
midpoint of face diagonal (001)
body centre
midpoint of body diagonal
Index
(100)
(110)
(111)
(210)
(211)
(221)
(310)
(311)
(320)
(321)
Number of
members in a
cubic lattice
6
12
8
24
24
24
24
24
24
48
dhkl
d100  a
d110  a / 2  a 2 / 2
The (110) plane bisects the
face diagonal
d111  a / 3  a 3 / 3
The (111) plane trisects the
body diagonal
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)

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