Ting-Yun Huang, 2012/09/18Page 1

3.14/3.40J/22.71J Recap

Basic Crystallography and Miller Indices

Ting-Yun Huang

2012 Sep. 18th

Outline• Basic Structures

– Cubic: FCC/BCC

– HCP

• Miller Indices

– From 3D to 4D: Miller indices for hexagonal crystals

– From 3D to 2D: Stereographic Projection

– Zone axis

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BASIC STRUCTURESBCC/FCC /HCP

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Basic Structures: BCC

• Coordination number = 8

• Number of atoms in unit cell = 2

• α-Fe(room temperature) , δ-Fe(high T), β-Ti(high T), Zr (high T)

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Basic Structures: FCC• Coordination number =

12

• Number of atoms in unit cell = 4

• Closest packing

– ABCABC packing

• γ-Fe, Cu, Ag, Au, Al, Ni, Pb, Pt

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Basic Structures: HCP

•Coordination number = 12

•Number of atoms in unit cell = 2(*)

•Closest packing

– ABAB packing

•Mg, Zn, α-Ti, Be, Cd, Zr (low T)

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MILLER INDICESNotations/Hexagonal Systems/Stereographic Projection

Presentation Author, 2003Page 7

Miller Indices-Notations• Directions

– Specific Direction: [u1 u2 u3]

– Family of Directions: <u1 u2 u3>

• Planes

– Reciprocal of the intercepts

– Specific Plane: (u1 u2 u3)

– Family of Planes: {u1 u2 u3}

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From 3D to 4D-

Miller indices for HCP structures• Constraints:

– [u1u2u3v]→u1a1+u2a2+u3a3+vc

– u1+u2+u3 = 0

• Redundant coordinate system which is not orthogonal

• But we can still perform inner product.

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From 3D to 2D-Stereographic Projection

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Zone axis

In this TEM diffraction, the structure

is FCC and the zone axis is [112].

The (11-1) and (2-20) plane have the

same zone axis [112].

Presentation Author, 2003Page 11

_(111) _

(220)

From 3D to 2D-Stereographic Projection

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Thank you!

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