Crystal Structure

Structure of Crystalline Solids • surface of gold - (111) plane

Phenomenon • electron diffraction pattern of a single crystal GaAs (gallium arsenide)

• Atoms are periodically arranged in a crystalline solid 436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (2) Structure of a solid

Structure of crystals

• Structure of a building – building blocks: e. g. bricks – how bricks are laid and bound • Building blocks for a solid: atoms • How atoms are laid and bound • Types of possible structures – amorphous: random packing of atoms • e.g. glasses – crystalline: periodic packing of atoms (repeating patterns) • e.g. most engineering metals and ceramics • very important

• Approximation – atoms ~ hard spheres of a certain radius • Unit cell: the repeating unit

Reading: 3.1-3.6 (6th ed) 436-101 Unit 2: Engineering Materials

Dr. K. Xia

1

Structure of Crystalline Solids (3) • 4 (equivalent) atoms are completely contained in a FCC unit cell – each corner atom = 1/8 • 8 x 1/8 = 1 – each face center atom = 1/2 • 6 x 1/2 = 3

Basic types of crystal structures Face-Centered Cubic (FCC) • one atom at each corner of the cubic unit cell • one atom at each face certer • lattice: 3D array of center positions of atoms

IMSE: FCC Structure

436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (4) Face-Centered Cubic (FCC) • geometric relationships a = 2 2R

• total volume of atoms in a unit cell 4 16 FCC = 4 × πR 3 = πR 3 V atoms 3 3

• volume of the unit cell

• atomic packing factor (APF) V FCC APF FCC = atoms FCC = 0.74 VUC

FCC UC

V

= a = 16 2R 3

3

FCC is one of the closest packed structures. It is still only 3 quarters filled by atoms.

a

Materials with FCC: Al, Cu, Ni

Self Study: Density calculationExample 3.3

436-101 Unit 2: Engineering Materials

Dr. K. Xia

2

Structure of Crystalline Solids (5) Body-Centered Cubic (BCC) • one atom at each corner of the cubic unit cell • one atom at the body center

• 2 (equivalent) atoms are completely contained in a BCC unit cell – each corner atom = 1/8 • 8 x 1/8 = 1 – the body center atom = 1

IMSE: BCC Structure

436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (6) Body-Centered Cubic (BCC) • geometric relationships 4R a= 3 • volume of the unit cell 64 R 3 BCC VUC = a3 = 3 3

• total volume of atoms in a unit cell 4 8 BCC = 2 × πR 3 = πR 3 V atoms 3 3 • atomic packing factor (APF) V BCC APF BCC = atoms BCC = 0.68 VUC BCC is not as densely packed as FCC Materials with BCC: α–Fe, Cr, W

a 4R 4R

? 436-101 Unit 2: Engineering Materials

Self Study: Density calculationtutorial question

Dr. K. Xia

3

Structure of Crystalline Solids (7) Hexagonal Close-Packed (HCP) • unit cell is a hexagon • c/a = 1.633 (ideal value)

• 6 (equivalent) atoms are contained in a unit cell • as close packed as FCC (APF = 0.74)

IMSE: HCP Structure

Materials with HCP: Mg, α-Ti, Zn

436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (8) • coordinates Cubic crystal systems • way of describing unit cells – general α " β " γ; a " b " c – Cubic α = β = γ = 90° a=b=c

– right hand system: x, y, z – use a as one unit

011 001 111 101

111 222

000 100

010 110

Reading: 3.7-3.10 (excluding hexagonal)

436-101 Unit 2: Engineering Materials

Dr. K. Xia

4

Structure of Crystalline Solids (9) Indexing crystallographic directions • a system of describing directions in crystals - Miller indices

[111]

B

A

1/2

[021] [100]

[110]

• convention and procedure – the indices, uvw, of a direction are simply the components of the equivalent vector (projected lengths of the direction on x, y and z) – make uvw smallest integers – put any minus sign on top – use [uvw] to indicate a particular direction direction A • components of vector: 0, 1, 1/2 • smallest integers: 0, 2, 1 (x 2, i.e. double the length) • direction A = [021] direction B = [00 1 ]

436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (10) • family of directions – <100> includes [100], [010] and [001] as well as [1 00 ] [0 1 0 ] [00 1 ] – <110> includes [110], [101], [011], [1 1 0 ] [1 10 ] ……

– <111> includes …… homework: fill in the rest of the <110> and <111> family members

– members of a family have the same arrangement of atoms (for a cubic system) although they are different directions

[1 10 ] [110]

[111]

[1 1 1 ]

436-101 Unit 2: Engineering Materials

Dr. K. Xia

5

Structure of Crystalline Solids (11) Indexing crystallographic planes • a system of describing planes in crystals - Miller indices the intercept on z is "

(111)

(110)

2/3

1/2

A

(010)

• convention and procedure – the indices, hkl, of a plane are the reciprocals of the intercepts of the plane with x, y and z – make hklsmallest integers – put any minus sign on top – use (hkl) to indicate a particular plane plane A • intercepts: x = 1, y = 1/2 and z = 2/3 • reciprocals of intercepts: 1, 2, 3/2 • smallest integers (reciprocals x 2): 2, 4, 3 • plane A = (243)

the intercepts on x & z are "

436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (12) • family of planes – {100} includes (100), (010) and (001) – {110} includes (110), (101), (011), (1 1 0 ) (10 1 ) …… (001)

(010)

– {111} includes …… homework: fill in the rest of the {110} and {111} family members

– members of a family have the same arrangement of atoms (for a cubic system) although they are different planes • equivalent planes (they are exactly the same), e.g. (001) and (00 1 ) (001)

(100)

(00 1 ) 436-101 Unit 2: Engineering Materials

Dr. K. Xia

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Structure of Crystalline Solids (13) Packing of atoms in FCC and HCP • close-packed plane (e.g. layer A) • interstitial sites – B: e.g. the up right triangles – C: e.g. the upside down triangles • stacking of close-packed planes – first layer = A – second layer = B – third layer • = C: ABCABC ... packing • = A: ABAB … packing

C A

The third layer can be either C or A, but not both on the same layer!

Reading: 3.12 (6th ed)

436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (14) • ABC packing = FCC

Viewing direction

6 1

1

1 5

2

3

4

3

6

2

1 1

5 1 3

2 2

3

4

Projection along the viewing direction Video - Open Uni. T201/VC1:1

436-101 Unit 2: Engineering Materials

Dr. K. Xia

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Structure of Crystalline Solids (15) • AB packing = HCP

436-101 Unit 2: Engineering Materials

Dr. K. Xia

Structure of Crystalline Solids (16) Noncrystalline solids

Some questions

• amorphous packing of atoms is random

Why a material takes a certain structure, e.g. Al is FCC? So that the lowest potential energy is achieved Can a material takes more than one structure? – For some materials, yes e.g. Fe is BCC (ferrite) at room temperature, but transforms to FCC (austenite) at 912°C – Why? Again, to lower the potential energy corresponding to conditions (e.g. temperature)

Si

O

Crystalline (2D)

SiO2 Glass (2D) Cristobalite

Reading: 3.17 (6th ed)

436-101 Unit 2: Engineering Materials

Dr. K. Xia

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Structure of Crystalline Solids - Summary Phenomenon

Quantitative description

• Periodic diffraction patterns by Xray or electrons • Atomic images showing periodic patterns

• Miller indices for directions and planes – indexing directions or planes – plotting directions or planes from indices

Basic structures • FCC & BCC • Unit cells – geometric relationships – density – packing factor

436-101 Unit 2: Engineering Materials

Physical understanding • Structures are determined by atomic structure and bonding – minimum potential energy for the solid – structural changes

Dr. K. Xia

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