Lecture 3

Diffraction from CrystalsDiffraction from Crystals

Lecture 3

Diffraction from Crystals“Structure Factor”

Diffraction from Crystals“Structure Factor”

Fultz & Howe, Chap. 5

AMSE 609 Spring 2010

Scattering from a Latticeg

the distance of r’ from the center of the nth atom

j

at jR

V r V r R

32

exp, exp

2scatt atomm

V r d rh r

ik rk r i k r

22 h r

32

exp, exp

2scatt at jm

V r R d rh r

ik rk r i k r

AMSE 609 Spring 2010

22

j

jRh r

Scattering from a Latticeg

,el j el jf R k f R

(i.e. )j jr r R r r R• Define the new coordinate:

• Ignore the 1/r2 term to give:Ignore the 1/r term to give:

3

, 2 exp2 j

j

scatt at RR

mV r d r

h jk i k r +R

3

, 2 exp exp2

j

j

R

at RR R

mV r d r

h ji k r i k R

AMSE 609 Spring 2010

2

j jR Rh

Scattering from a Latticeg

• So scattered wave from an array of N atoms:y

expN

scatt el jf R jk i k Rj

Amplitude of wavelets from atoms at Rj

Relative phase of the wavelet (Phase factor)

• Notes:

- Now have ψ(Δk) only, Rj are fixed. Mean scattered wave depends on incident wave vector.

“Diff t d i t it i ti l t th F i T f ti

AMSE 609 Spring 2010

“Diffracted intensity is proportional to the Fourier Transformation of the scattering factor distribution in the material”.

Scattering from a Latticeg• In simplest form, a crystal is:

C t l l tti b i d f t di l tCrystal = lattice + basis + defect displacements

, g k g k j lattice basis defectsR r r r r r r

• For now, consider a defect free crystal:

Crystal = lattice + basisCrystal lattice basis

g k j lattice basisR r r r r

S tt d • Scattered wave:

expscatt atf j jk R i k R

exp 2

j

scatt atR

atf

j j

g k g kr r i k r r

AMSE 609 Spring 2010

g kr r

Scattering from a Lattice• Basis is identical for all unit cells: g k kf f r r r

g

exp 2 exp 2g k

scatt atr r

f g k kk i k r r i k r

S k F k

lattice

• Where: exp 2g

lattice

r

gS k i k r “Shape Factor”

exp 2k

basis

atr

f k kF k r i k r “Structure Factor”

• F(Δk) same for all lattice points:

exp 2lattice

k F k i k r

AMSE 609 Spring 2010

exp 2gr

gk F k i k r

Structure Factor• Structure Factor:

- Scattering from all atoms in the unit cell

exp 2basis

f F k r i k r exp 2k

atr

f k kF k r i k r

• In essence Structure Factor determines whether or In essence, Structure Factor determines whether or not constructive interference occurs at a given reciprocal lattice point:

• This means our prior condition – Δk = g – is a necessary d b ffcondition, but is not sufficient:

- There may not be intensity at a given g.

AMSE 609 Spring 2010

Structure Factorcalculation

• Consider i atoms in the unit cell:• Consider i atoms in the unit cell:

exp 2ii

f iF k i k ri

• Location of ith atom (with respect to real lattice):

i i ix y z ir a b c

• Diffraction condition (Δk = g):Diffraction condition (Δk g):

* * *h k l K a b c

• General form:

exp 2i i i if hx ky lz F k i

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pi i i ii

y

Structure FactorSimple cubic

• Simple result: intensity at every reciprocal point

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p y y p p

Structure FactorBody-centered cubic

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Structure FactorBody-centered cubic

• Allowed low order reflections are:Allowed low order reflections are:- 110, 200, 220, 310, 222, 321,

400, 330, 411, 420, …

• Forbidden reflections are:- 100, 111, 210, …

• Origin of forbidden reflections:- Identical plane of atoms Identical plane of atoms

halfway between causes destructive interference

• Real bcc lattice has an fccreciprocal lattice

AMSE 609 Spring 2010

Structure FactorFace centered cubic

AMSE 609 Spring 2010

Structure FactorFace centered cubic

• Allowed low order reflections are:Allowed low order reflections are:- 111, 200, 220, 311, 222,

400, 331, 310, …

• Forbidden reflections are:- 100, 110, 210, 211…

• Real fcc lattice has a bcc reciprocal latticereciprocal lattice

AMSE 609 Spring 2010

Structure FactorNaCl (rock salt) structure

AMSE 609 Spring 2010

Structure FactorNi3Al (L12) structure

• Simple cubic lattice, with at four atom basis:Simple cubic lattice, with at four atom basis:

A i i i l bi i t it t ll i t• Again, since simple cubic, intensity at all points.

But each point is ‘chemically sensitive’.

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Chemically Sensitive Reflectionsy

• Dark field images formed from chemically sensitive reflections produces a real space map of (highly) local chemistry

AMSE 609 Spring 2010

chemistry.

Long Period Superlatticesg p

• Effect of “Shape factor”

AMSE 609 Spring 2010