Chem 253, UC, Berkeley What we will see in XRD of simple ... · PDF file1 Chem 253, UC, Berkeley What we will see in XRD of simple cubic, BCC, FCC? 0.5 ( ); 0.75 sin sin 2 2 BCC FCC

1

Chem 253, UC, Berkeley

What we will see in XRD of simple cubic, BCC, FCC?

)(75.0);(5.0sin

sin2

2

FCCBCCB

A

222 lkh

adhkl


Reciprocal Lattice

n

'nd

R

cnbnanR 321Path Difference:

mnnRnRnR

)( ''

mnnR

2)(2 '

2


Reciprocal Lattice

d

R

cnbnanR 321

mnnR

2)(2 '

mkkR 2)( '

Correspond to plane wave:12)( '

mikkRi ee

nk2

'k


Reciprocal Lattice

n

'nd

R

cnbnanR 321

1)( '

kkRie Laue Condition

Reciprocal lattice vector

For all R in the Bravais Lattice

'k

k

kkK'

1 RiKe

3


Reciprocal Lattice

d

R

1)( '

kkRie Laue Condition

Reciprocal lattice vector


'k

k

kkK'

1 RiKe

K


Reciprocal Lattice


A reciprocal lattice is defined with reference to a particular Bravias Lattice.

a

b

c Primitive vectors

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

1 RiKe

4


Reciprocal Lattice


)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

Verify:

0

0

2

ca

ba

aa

For any

cnbnanR

ckbkakK

321

321

)(2 332211 nknknkRK

1 RiKe


Reciprocal Lattice

Simple cubic

Reciprocal lattice is always one of 14 Bravais Lattice.

xaa

yab

zac

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

ya

2

za

2

xa

2

Simple cubic

5


Primitive Cell of FCC

•Angle between a1, a2, a3: 60o

)(2

11

yxaa

)(2

12

yzaa

)(2

13

xzaa


Primitive Cell of BCC

)(2

12

zyxaa

•Primitive Translation Vectors:

•Rhombohedron primitive cell

0.53a

109o28’)(2

11

zyxaa

)(2

13

zyxaa

6


Reciprocal Lattice

FCC

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac

Reciprocal lattice is always one of 14 Bravais Lattice.

BCC

)(2

zya

a

)(2

zxa

b

)(2

yxa

c

)(2

14

xzya

)(2

14

yzxa

)(2

14

zyxa


Reciprocal Lattice

BCC FCC

Simple hexagonal Simple hexagonal

7



8



9


Wigner-Seitz cells of reciprocal lattice


10


X (a 0 0)

L


11


Theorem:

For any family of lattice planes separated by distance d, there are

reciprocal lattice vectors perpendicular to the planes, the shortest

being 2/d.


Orientation of plane is determined by a normal vector

The miller indices of a lattice plane are the coordination at the reciprocal lattice vector normal to the plane.

12


Plane (hkl)

lckbhaK

A

B

C ax1

bx2

cx3

)1

,1

,1

()(321 xxx

hkl

cx

bx

ax 321

222

)()( 3121 cxaxbxaxACABK

cx

bx

ax 321

111

lckbhaK

)(2

cba

cba

)(2

cba

acb

)(2

cba

bac


13



PositionIntensity


Structure Factor:adds up all scattered X-ray from each lattice points in crystal

czbyaxd

lckbhaK

j

2

)( khkl SI

n

j

diKk

jeS1

14


X-ray scattered from each primitivecell interfere constructively when:

nd sin2

For n-atom basis:

sum up the X-ray scattered from the whole basis

1 RiKe


d

Rid jd

k

'k

)( ji ddK Phase difference:

The amplitude of the two rays differ: )( ji ddiKe

kkK'

15


The amplitude of the rays scattered at d1, d2, d3…. are in

the ratios :

The net ray scattered by the entire cell:

2

)( khkl SI

jdiKe

n

j

diKk

jeS1


For simple cubic: (0,0,0)

10 iKk eS

16


For BCC: (0,0,0), (1/2, ½, ½)…. Two point basis

lkh )1(1

S=2, when h+k+l evenS=0, when h+k+l odd, systematical absence

)(

)(2

10

2

1

1 lkhi

zyxiKiK

j

diKk

e

eeeS j


For BCC: (0,0,0), (1/2, ½, ½)…. Two point basis

S=2, when h+k+l evenS=0, when h+k+l odd, systematical absence

(100): destructive

(200): constructive

17


Observable diffraction peaks

222 lkh Ratio

Simple cubic

SC: 1,2,3,4,5,6,8,9,10,11,12..

BCC: 2,4,6,8,10, 12….

FCC: 3,4,8,11,12,16,24….

222 lkh

adhkl

nd sin2


S=4 when h+k, k+l, h+lall even (h,k, l all even/odd)

S=0, otherwise

)()()(1 klilhikhik eeeS

FCC

18



222 lkh Ratio

Simple cubic

SC: 1,2,3,4,5,6,8,9,10,11,12..

BCC: 2,4,6,8,10, 12, 14….(1,2,3,4,5,6,7…)

FCC: 3,4,8,11,12,16,24….

222 lkh

adhkl

nd sin2



222 lkh Ratio

SC: 1,2,3,4,5,6,8,9,10,11,12..

BCC: 2,4,6,8,10, 12, 14….(1,2,3,4,5,6,7…)

FCC: 3,4,8,11,12,16,24….

222 lkh

adhkl

nd sin2

)(4

sin 2222

22 lkh

a

19



)(75.0);(5.0sin

sin2

2

FCCBCCB

A

222 lkh

adhkl


Ex: An element, BCC or FCC, shows diffraction peaks at 2: 40, 58, 73, 86.8,100.4 and 114.7. Determine:(a) Crystal structure?(b) Lattice constant?(c) What is the element?

2theta theta (hkl)

40 20 0.117 1 (110)

58 29 0.235 2 (200)

73 36.5 0.3538 3 (211)

86.8 43.4 0.4721 4 (220)

100.4 50.2 0.5903 5 (310)

114.7 57.35 0.7090 6 (222)

2sin 222 lkh

a =3.18 Å, BCC, W

20


n

j

dKijk

jekfS1

)(

Polyatomic Structures

fj: atomic scattering factor No. of electrons

czbyaxd

lckbhaK

j

Reading : West Chapter 5

n

j

lzkyhxij

n

j

diKjk efefS j

1

)(2

1


Simple Cubic Lattice

Caesium Chloride (CsCl) is primitive cubic

Cs (0,0,0)

Cl (1/2,1/2,1/2)

What about CsI?

)( lkhiClCsk effS

21


(hkl) CsCl CsI

(100) (110) (111) (200) (210) (211) (220) (221) (300) (310) (311)


FCC Lattices

Sodium Chloride (NaCl)

Na: (0,0,0)(0,1/2,1/2)(1/2,0,1/2)(1/2,1/2,0)

Cl: (1/2,1/2,1/2) (1/2,0,0)(0,1/2,0)(0,0,1/2)

Add (1/2,1/2,1/2)

22


S=4 (fNa +fCl) when h, k, l, all even;

S=4(fNa-fCl) when h, k, l all odd

S=0, otherwise

]1][[ )()()()( klilhikhilkhiClNak eeeeffS


(hkl) NaCl KCl

(100)

(110)

(111) (200) (210)

(211)

(220) (221)

(300)

(310)

(311)

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Chem 253, UC, Berkeley What we will see in XRD of simple ... · PDF file1 Chem 253, UC, Berkeley What we will see in XRD of simple cubic, BCC, FCC? 0.5 ( ); 0.75 sin sin 2 2 BCC FCC