2-1 Basic concept Crystal structure = lattice structure + basis Lattice points: positions (points) in the structure which are identical.

II Crystal Structure

X X X X

X X X X

X X X X

X X X X

Not a lattice point?lattice point

Lattice translation vector Lattice plane Unit cell Primitive unit cell【 1 lattice point/unit cell】 Examples : CsCl

CS.

Fe (ferrite) b.c.c (not primitive)Al f.c.c (not primitive)Mg h.c.p simple hexagonal latticeSi diamond f.c.c (not primitive)

T

1T

2T

21 12 TTT

'21 03.143.2 TTT

'

2T

'21 TT

Cartesian coordinate

Rational directioninteger

Use lattice net to describe is much easier!

Rational direction & Rational plane are chosen to describethe crystal structure!

2-2 Miller Indices in a crystal (direction, plane) 2-2-1 direction

The direction <u v w>are all the [u v w] types of direction, which are crystallographic equivalent.

The direction [u v w] is expressed as a vector cwbvaur ˆˆˆ

2-2-2 plane The plane (h k l) is the Miller index of the plane in the figure below.

x

ha /

c

y

z)(hkl

a

ha /

lc /

kb /

1///

lc

zkb

yha

x

{h k l} are the (h k l) types of planes which are crystllographic equivalent.

http://www.materials.ac.uk/elearning/matter/Crystallography/IndexingDirectionsAndPlanes/index.html

You can practice indexing directions and planesin the following website

(hkl) Individual planeSymmetry related set{hkl}

DifferentSymmetry related set

)100()010()001(

)001()010()100(

)100( )001()001( )100(

xy

z

x y

z

{100}

{100}

Crystallographic equivalent?

Example:

2-2-3 meaning of miller indices

)200(

)100(

100200 21 dd

>Low index planes are widely spaced.

x

(110)

[120]

>Low index directions and planes areimportant for slip, cross slip, and electronmobility.

>Low index directions correspond to short lattice translation vectors.

xy

[110]

[120]

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

|x| = |y|

[110] (110) [110]

|x| |y|

(110)yx

2-3-2 In hexagonal system using Miller - Bravais indexing system: (hkil) and [hkil]

[100]

[010]x

y

[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!)

Miller indces

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/IndexingDirectionsAndPlanes/indexing-of-hexagonal-systems.html

2-3-3 Miller-Bravais indices

The direction [h k i l] is expressed as a vector zluiykxhr ˆˆˆˆ

x

y

u

[21 1 0]

[11 20 ]

[121 0]

𝑎3 [2 110 ]Note : is the shortest translation

vector on the basal plane.

(a) direction

u

x

y𝑎3 [2 110 ] 𝑎

3 [1 210 ]

You can check

[100] [010]

(b) planes (h k i l) ; h + k + i = 0

Plane (h k l) (h k i l)

(100)

(100) x

y

u

x

y

u

(010)

planeplane

x

y

u

? plane(2 110)(10)

(h k l) (h k i l)

Proof for general case:

x

y

uFor plane (h k l), the intersection with the basal plane (001) is a line that is expressed as

𝑥1h

+𝑦1𝑘

=1line equation :

Where we set the lattice constant a = b = 1 in the hexagonal lattice for simplicity.

line equation :

Check back to page 5

The line along the axis can be expressed as𝑥=𝑦line equation : or 𝑥− 𝑦=0

h𝑥+𝑘𝑦=1 𝑥− 𝑦=0Intersection points of these two lines

and

()is at

The vector from origin to the point can be expressed along the axis as

i = - (h + k) x

y

u

x

y

u()

1h

1𝑘

1h+𝑘

11/h : 1

1/𝑘 : − 11/(h+𝑘)

=h :𝑘 :−(h+𝑘)

xy

1h+𝑘

1h+𝑘

?

(c) Transformation from Miller [x y z] to Miller-Bravais index [h k i l]

rule

h=2𝑥− 𝑦

3

𝑘=2 𝑦−𝑥

3

𝑖=−(𝑥+𝑦 )3

𝑧=𝑙

x

y

u

The same vector is expressed as [x y z] inmiller indices and as [h k I l] inMiller-Bravais indices!

Proof:

[ 𝑥 𝑦 𝑧 ]=𝑥 ��+𝑦 ��+ 𝑧 ��[ h𝑘 𝑖𝑙 ]=h ��+𝑘 ��+𝑖 ��+𝑙 ��

¿h ��+𝑘 ��+𝑖 (− ��− �� )+𝑙 ��¿ (h − 𝑖 ) ��+(𝑘− 𝑖 ) ��+𝑙 ��

𝑥=h −𝑖=h+h+𝑘=2h+𝑘𝑦=𝑘−𝑖=𝑘+h+𝑘=h+2𝑘𝑧=𝑙

Moreover,

h=2𝑥− 𝑦

3𝑘=

2 𝑦−𝑥3 𝑖=−(𝑥+𝑦 )

3

𝑧=𝑙

2-4 Stereographic projectionsN

S

Horizontalplane

[h𝑘𝑙 ]

[𝑢𝑣𝑤 ]

[𝑢𝑣𝑤 ]

[h𝑘𝑙 ]

[𝑢𝑣𝑤 ]

[𝑢𝑣𝑤 ]

2-4-1 direction

Representation of relationship of planes and directions in 3D on a 2D plane. Useful for theorientation problems.

A line (direction) a point.

(100)

2-4-2 planeGreat circle: the plane passing through the center of the sphere.

S

N

A plane (Great Circle) trace

http://courses.eas.ualberta.ca/eas233/0809winter/EAS233Lab03notes.pdf

Small circle: the plane not passing through the center of the sphere.

S

N

B.D. Cullity

Example: [001] stereographic projection; cubic

B.D. Cullity

Zoneaxis

2-4-2 Stereographic projection of different Bravais systemsCubic

How about a standard (011) stereographicprojection of a cubic crystal?

��

�� ��

Start with what you know!

What does (011) look like?

��

�� ��

(011)

[100]

[01]

[]

[]

(011)

[][11]109.47o

70.53o

��

�� ��[100]

[01]

[] (011)

[011][][001]45o

[011] [001] [01]

��

�� ��

[01]

[] (011)

[011] []

35.26o

[011] [111] [100]

[100][111]

1 10

111

HexagonalTrigonal 3a

3ac

[0001]

[110]

[111]

tan−1( 3𝑎𝑐 )

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 theequator are small circles

Equal anglewith respectto N or S pole

Measure the anglebetween two points:

Bring these two points onthe same great circle; counting the latitude angle.

(i) If two poles up

(ii) If one pole up, one pole down

angle

angle

2-5-2 measuring the angle between planes This is equivalent to measuring angle between poles

http://en.wikipedia.org/wiki/Pole_figure

Pole and trace

Angle between the planes of two zone circles is the anglebetween 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