II Crystal Structure
2-1 Basic concept
 Crystal structure = lattice structure + basis
 Lattice points: positions (points) in the
structure which are identical.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
lattice point
Not a lattice point
 Lattice translation vector
 Lattice plane
 Unit cell
 Primitive unit cell【1 lattice point/unit
cell】
Examples : CsCl
 S.C
Fe (ferrite) b.c.c (not primitive)
Al f.c.c (not primitive)
Mg h.c.p simple hexagonal lattice
Si diamond f.c.c (not primitive)
'
T2

T2

T

T1



 T  2T1  1T2 Rational direction
integer

'


'
T1  T2
T  2.43T1  1.03T2
Cartesian coordinate
Use lattice net to describe is much easier!
Rational direction & Rational plane are chosen to describe
the crystal structure!
2-2 Miller Indices in a crystal (direction, plane)
2-2-1 direction
The direction [u v w] is expressed as a

vector
r  uaˆ  vbˆ  wcˆ
The direction <u v w>are all the [u v w]
types of direction, which are
crystallographic equivalent.
2-2-2 plane
The plane (h k l) is the Miller index of the
plane in the figure below.
z
c/l
(hkl)
c
a/ h
y
a
x
a/h
b/k
x
y
z


1
a/h b/k
c/l
{h k l} are the (h k l) types of planes which
are crystllographic equivalent.
You can practice indexing directions and planes
in the following website
http://www.materials.ac.uk/elearning/matter/Crystall
ography/IndexingDirectionsAndPlanes/index.html
Crystallographic equivalent?
(hkl) Individual plane
{hkl} Symmetry related set
Example:
{100}
z
(100) (1 00)
(010) (0 1 0)
y (001) (001)
x
z
{100}
(100) (1 00)
(001) (001)
x
y
Different
Symmetry
related set
2-2-3 meaning of miller indices
(200)
d 200 
1
d100
2
(100)
>Low index planes are widely spaced.
x
(110)
[120]
>Low index directions correspond to short
lattice translation vectors.
y
x
[120]
[110]
>Low index directions and planes are
important for slip, cross slip, and electron
mobility.
2-3 Miller Indices and Miller - Bravais Indices
(h k l)
(h k i l)
2-3-1 in cubic system
(1)Direction [h k l] is perpendicular to (h k l)
plane in the cubic system, but not true for
other crystal systems.
x
y[110]
(110)
|x| = |y|
x
y
[110]
(110)
|x|  |y|
2-3-2 In hexagonal system
using Miller - Bravais indexing system:
(hkil) and [hkil]
y
x
[100]
[010]
Miller indces
[110]
Reason (i): Type [110] does not equal to
[010] , but these directions are
crystallographic equivalent.
Reason (ii): z axis is [001],
crystallographically distinct from [100] and
[010]. (This is not a reason!)
If crystal planes in hexagonal systems are
indexed using Miller indices, then
crystallographically equivalent planes have
indices which appear dissimilar.
 the Miller-Bravais indexing system
(specific for hexagonal system)
http://www.materials.ac.uk/elearning/matter/
Crystallography/IndexingDirectionsAndPlan
es/indexing-of-hexagonal-systems.html
2-3-3 Miller-Bravais indices
(a) direction
The direction [h k i l] is expressed as a

r  hxˆ  kyˆ  iuˆ  lzˆ
vector
[1120] uˆ
yˆ
[2110]
[1210]
xˆ
𝑎
Note : [2110] is the shortest translation
3
vector on the basal plane.
uˆ
yˆ
𝑎
[2110]
3
xˆ
[100]
𝑎
[1210]
3
[010]
You can check 𝑢
(b) planes (h k i l) ; h + k + i = 0
Plane (h k l)  (h k i l)
uˆ
uˆ
(010)  (0110)
(100)
yˆ


plane
xˆ
(100)  (1010)
xˆ
plane
yˆ
uˆ
? plane
xˆ

yˆ
(h k l)
(h k i l)
(210)
(2110)
Proof for general case:
For plane (h k l), the
Check back to page 5
intersection with the basal
uˆ
plane (001) is a line that is
expressed as
𝑥 𝑦
line equation : + = 1
1 1
xˆ
ℎ 𝑘
Where we set the lattice constant a = b = 1
in the hexagonal lattice for simplicity.
line equation :
yˆ
The line along the 𝑢 axis can be expressed as
line equation : 𝑥 = 𝑦 or 𝑥 − 𝑦 = 0
Intersection points of these two lines
ℎ𝑥 + 𝑘𝑦 = 1 and 𝑥 − 𝑦 = 0
is at
1
1
(
,
)
ℎ+𝑘 ℎ+𝑘
uˆ
The vector from origin to the point can be
expressed along the 𝑢 axis as
1
1
1
𝑥 𝑥 + 𝑦𝑦 =
𝑥+
𝑦=
ℎ+𝑘
ℎ+𝑘
ℎ+𝑘
1
1
𝑢 = 𝑢  i = - (h + k) xˆ
− ℎ+𝑘
𝑖
𝑥+𝑦 =
yˆ
uˆ
1
1
(
,
)
ℎ+𝑘 ℎ+𝑘
1
ℎ+𝑘
xˆ
?
1
ℎ+𝑘
yˆ
1
ℎ
1
𝑘
1
ℎ+𝑘
xˆ
1
1
−1
:
:
= ℎ: 𝑘: −(ℎ + 𝑘)
1/ℎ 1/𝑘 1/(ℎ + 𝑘)
yˆ
(c) Transformation from Miller [x y z] to
Miller-Bravais index [h k i l]
uˆ
rule
2𝑥 − 𝑦
ℎ=
3
2𝑦 − 𝑥
𝑘=
3
−(𝑥 + 𝑦)
𝑖=
3
𝑧=𝑙
yˆ
xˆ
Proof:
The same vector is expressed as [x y z] in
miller indices and as [h k I l] in
Miller-Bravais indices!
𝑥 𝑦 𝑧 = 𝑥 𝑥 + 𝑦𝑦 + 𝑧𝑧
ℎ 𝑘 𝑖 𝑙 = ℎ𝑥 + 𝑘 𝑦 + 𝑖 𝑢 + 𝑙 𝑧
= ℎ𝑥 + 𝑘𝑦 + 𝑖 −𝑥 − 𝑦 + 𝑙 𝑧
= ℎ − 𝑖 𝑥 + 𝑘 − 𝑖 𝑦 + 𝑙𝑧

𝑥=ℎ−𝑖
𝑦=𝑘−𝑖
𝑧=𝑙
Moreover, ℎ + 𝑘 = −𝑖
𝑥 = ℎ − 𝑖 = ℎ + ℎ + 𝑘 = 2ℎ + 𝑘
 𝑦 = 𝑘 − 𝑖 = 𝑘 + ℎ + 𝑘 = ℎ + 2𝑘
𝑧=𝑙
2𝑥 − 𝑦
ℎ=
3
2𝑦 − 𝑥
 𝑘=
3
𝑧=𝑙
−(𝑥 + 𝑦)
 𝑖=
3
2-4 Stereographic projections
N
2-4-1 direction
[ℎ 𝑘 𝑙]
[𝑢 𝑣 𝑤]
Horizontal
plane
[𝑢 𝑣 𝑤]
[ℎ 𝑘 𝑙]
[𝑢 𝑣 𝑤]
S
[𝑢 𝑣 𝑤]
Representation of relationship of planes and
directions in 3D on a 2D plane. Useful for the
orientation problems.
A line (direction)  a point.
(100)
2-4-2 plane
Great circle: the plane passing through the
center of the sphere.
N
S
http://courses.eas.
ualberta.ca/eas23
3/0809winter/EA
S233Lab03notes.
pdf
A plane (Great Circle)  trace
Small circle: the plane not passing through
the center of the sphere.
N
S
B.D. Cullity
Example: [001] stereographic projection; cubic
Zone
axis
B.D. Cullity
2-4-2 Stereographic projection of different
Bravais systems
Cubic
How about a standard (011) stereographic
projection of a cubic crystal?
𝑥
𝑦
𝑧
Start with what you know!
What does (011) look like?
[011]
[100]
𝑥
(011)
[100]
𝑧
𝑦
[011]
[111]
[111]
109.47o
70.53o
(011)
[011] [001] [011]
[011]
[100]
𝑥
(011)
[100]
𝑧
𝑦
45o
[011]
[001]
[011]
[011] [111] [100]
[111]
[011]
[100]
35.26o
[100]
𝑥
(011)
𝑧
𝑦
[011]
[011]
110
111
Trigonal
Hexagonal
3a
[2111]
[0001]
tan
−1
3𝑎
𝑐
c
3a
[2110]
Orthorhombic
Monoclinic
2-5 Two convections used in stereographic
projection
(1) plot directions as poles and planes as
great circles
(2) plot planes as poles and directions as
great circles
2-5-1 find angle between two directions
(a) find a great circle going through them
(b) measure angle by Wulff net
Meridians: great circle
Parallels except the
equator are small
circles
Equal angle
with respect
to N or S pole
Measure the angle
between two points:
Bring these two points on
the same great circle;
counting the latitude angle.
(i) If two poles up
angle
(ii) If one pole up, one pole down
angle
2-5-2 measuring the angle between planes
This is equivalent to measuring angle between
poles
Pole and trace
http://en.wikipedia.org/wiki/Pole_figure
Angle between the planes of two zone circles is the angle
between the poles of the corresponding
use of stereographic projections
(i) plot directions as poles
---- used to measure angle between
directions
---- use to establish if direction lie in a
particular plane
(ii) plot planes as poles
---- used to measure angles between
planes
---- used to find if planes lies in the
same zone