Øystein Prytz

Introduction to diffraction 2

Øystein Prytz

Øystein Prytz

Summary from last time

rkiAer 2)(k

'k

k

θ

ndhkl )sin(2

*** clbkahg

)(*,

)(*,

)(*

bac

bac

acb

acb

cba

cba

hkldg

gk

1

Vector normal to a plane

Øystein Prytz

The structure factor

B

A

Origin

k

k'k

'k

Ar

Br

rkier 2)(

Incoming wave:

rkirkiA

rkrkrkiA

rrkirkiAA

eef

efeefr

A

AAAA

'22

)''(2)('22)(

Wave scattered from atom A:

Equivalent expression for atom B

Total scattered wave detected far away:

rkirkiB

rkiA

rkirkiB

rkirkiABA

eefef

eefeefrrr

BA

BA

'222

'22'22

}{

)()()('

j

rkijk

jefF

2 Can be calculated for all scattering vectors (angles) Δk, but

usually we only concern ourselves with Δk=g (destructive interference in all other cases)

Øystein Prytz

The structure factor for copper

• fcc lattice with one atom in [000] relative to the lattice point.

• r1= [0,0,0], r2= [1/2,1/2,0], r3= [1/2,0,1/2], r4= [0,1/2,1/2]

j

rgijg

jefF

2

Øystein Prytz

The structure factor for copper

• Result:

• hkl even: F=fCu(1+1+1+1)=4fCu

• hkl odd: F=fCu(1+1+1+1)=4fCu

• h odd, k and l even: F=fCu(1-1+1-1)=0

• h and k odd, l even: F=fCu(1+1-1-1)=0

• Conclusion: hkl even or odd gives allowed reflections, hkl mixed gives forbidden reflections

)()()(1 lkilhikhiCuhkl eeefF

1

1)12(

2

in

in

e

e

Øystein Prytz

The reciprocal lattice for bcc iron?

• Body centered cubic lattice • One atom per lattice point, [000] relative to the lattice

point• What is the reciprocal lattice?