Ⅰ.Crystal structure is the real arrangement of atom in crystals Crystal structure ＝ Space lattice + Basis or structure unit

§2.4 Crystal Structure and §2.4 Crystal Structure and Complex LatticeComplex Lattice

++ ==

Fe : Al = 1 : 1FeFe

AlAl

The difference between space lattice and crystal structure

2×3 atoms / cell

1. BCC

Example: α-Fe , V, Nb, Ta, Cr, Mo, W, alkali metals

n = 2 atoms/cell

CN=8

The number of nearest neighbours around each atom is called — Coordination Number.

ⅡⅡ.Typical crystal structures of metals.Typical crystal structures of metals

Packing fraction

＝

To determineξ, The atom is looked as a hard sphere, and the nearest neighbours touch each other.

∴ For BCC, 68.0

)43(

342

3

3

a

a

Volume of atoms / cellVolume of atoms / cell

Volume of unit cellVolume of unit cell

2. FCC

Example: γ-Fe , Al , Ni , Pb , Cu ,

Ag , Au , stainless steal

n= 8×1/8+6×1/2=4 atoms/cell

CN=12

74.0)

42(

344

3

3

a

3. HCP

74.012CN

633.138

23

32)

2(

222

ac

aca•••• • ••

•••• • ••

• • •

Example: Be, Mg, Zn, Cd, Zr, Hf

Ti( low temperature)

n ＝ CN ＝ 12

ξ＝ 0.74

6322112

61

Structure a0 vs. r Atoms per cell

Coordination Number

Packing factor Examples

SC 1 6 0.52 Polonium (Po),α-Mn

BCC 2 8 0.68Fe,Ti,W,Mo,Nb,Ta,K,Na,

V,Zr,Cr

FCC 4 12 0.74 Fe,Cu,Au,Pt,Ag,Pb,Ni

HCP 2 12 0.74 Ti,Mg,Zn,Be,Co,Zr,Cd

ra 20

ra3

40

ra2

40

00

0

633.12

acra

4. Summary4. Summary

§2.5 Interstices in typical crystals §2.5 Interstices in typical crystals of metalsof metals

ⅠⅠ. Two types of Interstitials in . Two types of Interstitials in typical crystalstypical crystals Octahedral interstitial Tetrahedral interstitial

Definition:

In any of the crystal structures, there are small holes between the usual atoms into which smaller atoms may be placed. These locations are called interstitial sites.

1. Octahedral interstitial

BCC

a23

2a

2a

FCC

2a

2a

HCP

2a

2. Tetrahedral interstitial

BCC

a23

a45

a

FCC

2a

a43

HCP

Ⅱ.Determination of the sizes of interstitialsDefinition: By size of an interstitial we mean diameter of the maximum hard sphere which can be accommodated in the interstitial without distorting the lattice.

di

da

diameter of interstitial atomdiameter of atom in lattice point＝＝

1. Octahedral interstitial

condition for touching

add ai

1aa

i

da

dd

For BCC

For FCC

15.01

23

23

add

ad

a

i

a

41.01

22

22

add

ad

a

i

a

2. Tetrahedral interstitial

222 )2

()2

()2

( HLdd ia

22 HLdd ai 122

aa

i

dHL

dd

HH

LL

2ia dd

AA

DDCC

BBinterstitialinterstitial

host atomhost atom

For BCC

ada 23

aL 2aH

29.0a

i

dd

22.0a

i

dd

aL22

ada 22

2aH

For FCC

3. Summary

n CN ξ interstices di/da

oct. tete. oct. tete.

BCC 2 8 0.68 6 6/2=3 12 12/2=6 0.15 0.29

FCC 4 12 0.74 4 4/4=1 8 8/4=2 0.41 0.22

HCP 6 12 0.74 6 6/6=1 12 12/6=2 0.41 0.22

Examples and Discussions

1. Both FCC and BCC are close-packed structures while BCC is more open?

2. The interstitial atoms most likely occupy the oct. interstitial position in FCC and HCP, while in BCC two types of interstitial can be occupied equally.

3. The solid solubility in BCC is much lower than in FCC.

4. Diffusion of interstitial atoms in BCC diffusion is much faster than in FCC or HCP at same temperature.

5. Determine the relationship between the atomic radius and the lattice parameter in SC, BCC, and FCC structures when one atom is located at each lattice point.

6.6. Determine the density of BCC iron, which has Determine the density of BCC iron, which has a lattice parameter of 0.2866nm.a lattice parameter of 0.2866nm.

Solution:Solution: For a BCC cell, Atoms/cell = 2

a0 = 0.2866nm = 2.866×10 － 8cm

Atomic mass = 55.847g/mol

Volume of unit cell = a03 = 23.54×10 － 24cm3/cell

Density

32324- cm/g882.7

)10)(6.0210(23.54)847.55)(2(

)numbersAvogadro')(cellunitofvolume()ironofmassatomic)(cell/atomsofnumber(

Exercise1. Determine the coordinates of centers of both

the octahedral and the tetrahedral interstitials in HCP referred to a, b and c.

aabb

cc

120120oo

4. Prove that the A-face-centered hexagonal lattice is not a new type of lattice in addition to the 14 space lattices.

5. Draw a primitive cell for BCC lattice.

Thanks for your attention !