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Title: Atomic bonding and crystal structure
Description: These notes contain whole chapter of Atomic bonding and crystal structure

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Atomic Bonding
and
Crystal Structure

Metallic Bond
F
F


F

Electron

F

F
Electron Charge Cloud

F F

F

Fก

F

F
F

กก Fก



F ก ก



F

F

F F
F
ก F Interatomic Distance

F

ก F

a0 = 2 R



Bonding Energy = E0 - Emin
F Bonding Energy

F

Crystal Structure

Principle Metallic Crystal Structures

Body Centered Cubic

Face Centered Cubic

Hexagonal Closed Pack

BCC

FCC

HCP

Body Centered Cubic

a 2 + a 2 = 2a 2
= 2a

a

a

( 2a ) + a
2

2

= 2a 2 + a 2
= 3a

a
2a

Atomic Packing Factor (APF)
APF = total sphere volume
total unit cell volume
=

Vatom

 4 3 1  4 3
= (1) πR  +  (8) πR 

 8  3
3
= 8
...
32 R 3
a=

3a = 4 R
3

Vatom 8
...
68
3
Vunit 12
...
35092 nm
...


Face Centered Cubic

Vatom

 1  4 3 
 1  4 3 
= (6 )  πR  + (8)  πR 

 8  3

 2  3
16 3
= πR
4
3
a=
R

Vunit = a

3

a3 =

2a = 4 R
Vatom
APF =
Vunit

3

R3

2
64 3
=
R
2 2
=

16 3
πR
= 3 3
= 0
...

2
...




F
F F

F

Fก 1

F F F กF 1
F

x, y
x,y,z
Cubic
F

z

Atomic Position in Cubic Unit Cell

Direction in Cubic Unit Cell
F

F

F

[ xyz ]
F F comma

ʽ F Square Bracket

Direction in Cubic Unit Cell

ก Destination
F

Origin
2
...
F F
Square Bracket
xyz
F comma
4
...


Direction in Cubic Unit Cell

Direction in Cubic Unit Cell
Example
( 1 ,1, 1 ) − (0,0,0) = [ 1 − 0,1 − 0, 1 − 0]
2
2
2
2
= [ 1 ,1, 1 ]
2
2
= 2 × [ 1 ,1, 1 ]
2
2
= [121]

Direction in Cubic Unit Cell
Example

(− 2 ,1, 1 ) − (0,0,0) = [− 2 ,1, 1 ]
3
3
3
3
= 3 × [− 2 ,1, 1 ]
3
3
= [ 2 31]

Direction in Cubic Unit Cell
Exercises
Draw the following direction vectors in cubic
unit cells

a)[100]and [110]
b)[112]
c)[ 1 10]
d )[ 3 2 1 ]

[???]

[???]

Crystallographic Plane in Cubic Unit Cells

Crystallographic Plane in Cubic Unit Cells

1
...

3
...


ก xyz
F
F

F
F
F

F

5
...
361 nm
...
361
2

2

2

(2) 2 + (2) 2 + (0) 2

= 0
...


Family of Direction and Plane
Some planes and directions are “Crystallographic equivalent”
Directions

[100], [010], [001], [100], [010], [001]

< 100 >

Planes

(100), (010), (001)

{100}

Comparison of FCC and HCP

Volume and Planar Density Calculations
Volume Density

mass / unit cell
ρv =
volume / unit cell
= g / cm3

Volume and Planar Density Calculations
Cu has FCC crystal structure and an atomic radius of 0
...
Calculate a
theoretical density of copper in g/cm3
...
54 g/mol

2a = 4 R
4 R (4)(0
...
361 nm
2
2
m
ρv =
v
(4 atom)(63
...
22 ×10 − 22 g
6
...
361×10 m) = 4
...
70 ×10

− 23

3

cm

− 29

m

3

3

− 22

m 4
...
98 3
− 23
3
v 4
...
of center of atom
ρp =
selected area
= atoms / mm 2

Volume and Planar Density Calculations
Calculate the planar atomic density on the (110) plane of α iron in
atoms/mm2
...
287 nm
...
2
2
nm 2
2 (0
...
2 × 10
mm 2

( )

Crystal Structure Analysis

Off-Phase

In-Phase

nλ = MP + PN
nλ = 2d hkl sin θ

λ = 2d hkl sin θ

X-Ray
Diffractometer

Example
A sample of BCC iron was placed in an x-ray diffractrometer using incoming
x-rays with a wavelength 0
...
Diffraction from the {110} planes was
obtained at 2θ = 44
...
Calculate a value for the lattice constant a of BCC
iron
...
704o

θ = 22
...
1541 nm
=
= 0
...
3803)
a = d hkl h 2 + k 2 + l 2
= 0
...
2026 nm)(1
...
287 nm


Title: Atomic bonding and crystal structure
Description: These notes contain whole chapter of Atomic bonding and crystal structure