Diffraction from CrystalsDiffraction from Crystals““Structure FactorStructure Factor””

Lecture 5Lecture 5

Coherent elastic scatteringCoherent elastic scattering

kkdd

DetectorDetector

Ψincident= exp 2πi(

rk

i⋅ ′rr − ωt)⎡⎣ ⎤⎦ = exp 2πi(

rk

i⋅ ′rr )⎡⎣ ⎤⎦Incident plane wave:Incident plane wave:

Scattered wave:Scattered wave:

Ψ

scattered= f k

i,k

d( )exp 2πi

rk

d⋅

rr − ′

rr( )⎡

⎣⎤⎦r

r − ′rr

= f θ( )exp 2πi

rk

d⋅

rr − ′

rr( )⎡

⎣⎤⎦r

r − ′rr

rr’’ss -- position vectorsposition vectors

kk’’ss -- wave vectorswave vectors

kkii

rr’’

r r -- rr’’rr

--kkii

kkddKK

θθ

Scattering from a latticeScattering from a lattice

r R j[ ]- atom locations

r R n

′ r r

′ r r −

r R n

Vatom

r ′ r ( )= Vat

r ′ r −

r R j( )∑

Ψscattered

r K ,

r r ( )=

−m

2πh2

exp 2πir k d ⋅

r r ( )[ ]

r r

Vat ′ r r −

r R j( )∑ exp −2πi

v K ⋅ ′

r r ( )[ ]∫ d3 ′

r r

fel

r R j,

r K ( )≈ fel

r R j( )

Scattering from a latticeScattering from a lattice

r r ≡

r ′ r −

v R j (i.e.

r ′ r = r + ′ R j)

Ψscatt

r K ( )= −m

2πh2Vat,

r R j

r( )∑ exp −2πiv K ⋅

r r +

r R j( )( )[ ]∫ d3 ′

r r

= −m

2πh2Vat,

r R j

r( )∫ exp −2πiv K ⋅

r r ( )[ ]d3 ′

r r

⎛

⎝ ⎜

⎞

⎠ ⎟ exp −2πi

v K ⋅

r R j( )[ ]

v R j

∑

Ignore the 1/rIgnore the 1/r22 term to give:term to give:

r R j[ ]- atom locations

r R n

′ r r

′ r r −

r R n

Scattering from a latticeScattering from a lattice

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

Notes:Notes:–– Now have Now have ΨΨ(K) only, (K) only, RRjj are fixed. Means scattered wave are fixed. Means scattered wave

depends on incident wave vector.depends on incident wave vector.–– This is a Fourier Transform (FT) This is a Fourier Transform (FT) Diffracted intensity is proportional to the Fourier Transform ofDiffracted intensity is proportional to the Fourier Transform of the the

scattering factor distribution in the materialscattering factor distribution in the material

Ψscatt

r K ( )= fel

r R j( )exp −2πi

v K ⋅

r R j( )[ ]

j

N

∑

r R j[ ]- atom locations

r R n

′ r r

′ r r −

r R n

Scattering from a latticeScattering from a lattice

In simplest form, a crystal is:In simplest form, a crystal is:

For now, consider a defect free crystal:For now, consider a defect free crystal:

Scattered wave:Scattered wave:

Basis must be same for all unit cells:Basis must be same for all unit cells:

Crystal = lattice + basis + defect displacementsr R j =

r r lattice +

r r basis +

r r defects =

r r l +

r r b +

r r d

Crystal = lattice + basis r R j =

r r lattice +

r r basis =

r r l +

r r b

Ψscatt

r K ( )= fel

r r l +

r r b( )exp −2πi

v K ⋅

r r l +

r r b( )( )[ ]

rb

∑rl

∑

Ψscatt

r K ( )= exp −2πi

v K ⋅

r r l( )[ ]

rl

∑ fel

r r b( )exp −2πi

v K ⋅

r r b( )[ ]

rb

∑ fr r l +

r r b( )= f

r r b( )

Scattering from a latticeScattering from a lattice

Repeating:Repeating:

where:where:

FF(K) same for all lattice points:(K) same for all lattice points:

Ψscatt

r K ( )= exp −2πi

v K ⋅

r r l( )[ ]

r r l

∑ fel rb( )exp −2πiv K ⋅

r r b( )[ ]

r r b

∑

Ψscatt

r K ( )= S

r K ( )F

r K ( )

Sv K ( )= exp −2πi

v K ⋅

r r l( )[ ]

r r l

lattice

∑

Fv K ( )= fel rb( )exp −2πi

v K ⋅

r r b( )[ ]

r r b

basis

∑

““Shape FactorShape Factor””

““Structure FactorStructure Factor””

Ψ

v K ( )= F

v K ( )exp −2πi

v K ⋅

r r l( )[ ]

r r l

lattice

∑

Structure FactorStructure Factor

Structure Factor:Structure Factor:–– Scattering from all atoms in the unit cellScattering from all atoms in the unit cell

F

v K ( )= fel

r r b( )exp −2πi

v K ⋅

r r b( )[ ]

r r b

basis

∑

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

This means our prior condition This means our prior condition -- K = g K = g -- is a is a necessary condition, but is not sufficientnecessary condition, but is not sufficient

–– There may not be intensity at a given g!There may not be intensity at a given g!

Structure factorStructure factorcalculationcalculation

F

v K ( )= fi exp −2πi

v K ⋅

r r i( )[ ]

i

∑

Consider i atoms in the unit cellConsider i atoms in the unit cell

r r i = xi

r a + yi

r b + zi

r c

Location of Location of iithth atom (with respect to real lattice):atom (with respect to real lattice):

Diffraction condition (K = g):Diffraction condition (K = g):

r K = h

r a * +k

r b * +l

r c *

General from:General from:

F

v K ( )= fi exp −2πi hxi +kyi + lzi( )[ ]

i

∑

Using Using defdefnn of a*, of a*, b* & c*b* & c*

Structure FactorStructure Factorsimple cubicsimple cubic

F = fi exp −2πi hxi +kyi + lzi( )[ ]i

∑

One atom basis : r = 0,0,0( )F = f exp −2πi 0 ⋅h+ 0 ⋅k + 0 ⋅ l( )[ ]{ } = f exp −2πi 0( )[ ]{ } = f exp 0[ ]{ } = f

Simple result: intensity at every reciprocal lattice pointSimple result: intensity at every reciprocal lattice point

Structure FactorStructure Factorbodybody--centered cubiccentered cubic

Two atom basis : r = 0,0,0( ) & r =12

,12

,12

⎛

⎝ ⎜

⎞

⎠ ⎟

F = f exp −2πi 0 ⋅h+ 0 ⋅k + 0 ⋅ l( )[ ]+ exp −2πi12

⋅h+12

⋅k +12

⋅ l⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

= f 1+ exp −πi h+k + l( )[ ]{ }h, k & l are integers, so h +k + l = N (where N is an integer)

The exponential can then take one of two values:

exp −πi h+k + l( )[ ]= +1 if N = even

exp −πi h+k + l( )[ ]= −1 if N = odd

So :

F = 2f if N = even

F = 0 if N = odd

Structure FactorStructure Factorbodybody--centered cubiccentered cubic

Allowed low order Allowed low order reflections are:reflections are:

–– 110, 200, 220, 310, 222, 321, 110, 200, 220, 310, 222, 321, 400, 330, 411, 420 400, 330, 411, 420 ……

Forbidden reflections are:Forbidden reflections are:–– 100, 111, 210100, 111, 210

Origin of forbidden Origin of forbidden reflections?reflections?

–– Identical plane of atoms Identical plane of atoms halfway between causes halfway between causes destructive interferencedestructive interference

Real bcc lattice has an Real bcc lattice has an fccfccreciprocal latticereciprocal lattice

002002 022022

220220

020020

200200

202202

112112

101101000000

011011

110110

211211

121121

Structure FactorStructure Factorface centered cubicface centered cubic

Four atom basis: r = 0,0,0( ), r =12

,12

,0⎛

⎝⎜⎞

⎠⎟, r =

12

,0,12

⎛

⎝⎜⎞

⎠⎟ & r = 0,

12

,12

⎛

⎝⎜⎞

⎠⎟

F = f 1+ exp −πi h + k( )⎡⎣

⎤⎦ + exp −πi k + l( )⎡

⎣⎤⎦ + exp −πi h + l( )⎡

⎣⎤⎦{ }

So:

F=4f if h,k,l all even or odd

F=0 if h,k,l are mixed even or odd

Structure FactorStructure Factorface centered cubicface centered cubic

Allowed low order Allowed low order reflections are:reflections are:

–– 111, 200, 220, 311, 222, 400, 111, 200, 220, 311, 222, 400, 331, 310331, 310

Forbidden reflections:Forbidden reflections:–– 100, 110, 210, 211100, 110, 210, 211

Real fcc lattice has a bcc Real fcc lattice has a bcc reciprocal latticereciprocal lattice

002002 022022

220220

020020

200200

202202

000000 111111

Structure Factor Structure Factor NaClNaCl (rock salt) structure(rock salt) structure

If h,k,l all odd have If h,k,l all odd have ‘‘chemically chemically sensitivesensitive’’ reflectionsreflections

Na on each fcc site, but with a two atom basis :

rNa = 0,0,0( ) & rCl =12

,12

,12

⎛

⎝ ⎜

⎞

⎠ ⎟

F = fNa + fCl exp −πi h+k + l( )[ ]{ }×

1+ exp −πi h+k( )[ ]+ exp −πi k + l( )[ ]+ exp −πi h+ l( )[ ]{ }F = 4 fNa + fCl( ) if h,k,l all even

F = 4 fNa − fCl( ) if h,k,l all odd

F = 0 if h,k,l mixed

Structure FactorStructure FactorNiNi33Al (L1Al (L122) structure) structure

rAl

= 0,0,0( ), rNi=

1

2,1

2,0

⎛

⎝⎜⎞

⎠⎟ , r

Ni=

1

2,0,

1

2

⎛

⎝⎜⎞

⎠⎟ & r

Ni= 0,

1

2,1

2

⎛

⎝⎜⎞

⎠⎟

F = fal

+ fNi

exp −πi h + k( )⎡⎣

⎤⎦ + exp −πi k + l( )⎡

⎣⎤⎦ + exp −πi h + l( )⎡

⎣⎤⎦

So :

F=fAl

+3fNi

if h,k,l all even or odd

F=fAl

-fNi

if h,k,l are mixed even or odd

Simple cubic lattice, with at four atom basisSimple cubic lattice, with at four atom basis

Again, since simple cubic, intensity at all points. Again, since simple cubic, intensity at all points.

But each point is But each point is ‘‘chemically sensitivechemically sensitive’’..

Chemically sensitive reflectionsChemically sensitive reflections

Dark field images formed from chemically sensitive Dark field images formed from chemically sensitive reflections produce a real space map of (highly) local reflections produce a real space map of (highly) local chemistrychemistry

Long period Long period superlatticessuperlattices