Overview on Scattering

Scattering pattern of nanocrystal

2for constructive interference: 2k s k⋅ =

r rr

ki kf

d

2 sinn dλ θ=

elastic scattering: ki=kf

-19

/1/ = =10 s

c vE hvp h v

λ

ν τ

===

Properties of x-rays and neutrons

x-rays:

electromagneticradiation

2

-13

v depends on v 2v

1/ = =10 s

E mp m

λ

ν τ

==

neutron:

an uncharged elementary particle

magnetic moment

v distribution from reactor:-dep. on moderatorMaxwell-Boltzmann distribution

-kT (at RT) ~ 20 meVinvestigation on dyanamics is also availabe.

v ( / )km s

cold neutron source(liq D2 25 K, 6 A, 2 meV)

thermal neutron source (D2O 330 K, 1.7 A, 28 meV)

hot neutron source(graphite 2000 K, 0.6 A, 172 meV)

num

ber

of p

artic

les

0

tot

number of particles scattered into a unit solid angle per secondflux of incident beam

total number of particles scattered in all direction per secondflux of incident beam

J dJ d

σ

σ

=Ω

=

=

• differential scattering cross-section:the probability that a photon or a neutron impinging on the sample into a unit solid angle in the given direction

2 *J A AA= =0

J dJ d

σ=

Ω

2θ

dΩ

2for constructive interference: 2k s k⋅ =

r rr

ki kf

d

2 sinn dλ θ=

elastic scattering: ki=kf

2for constructive interference: 2k s k⋅ =

r rr

ki kf

d

2 sinn dλ θ=

elastic scattering: ki=kf

02 2 ( )

2

S r S r

s r

πδ πφλ λ

π

Δ = = ⋅ − ⋅

= − ⋅

ur r ur r

rr

2 sinn dλ θ=

coherent scattering:

0S Ssλ−

=

ur urr

S0 S

d

2 ( )1 0

i vt xA A be π λ−=

2 1

2 ( ) 21

i

i vt x i s r

A Abe

Abe e

φ

π λ π

Δ

− − ⋅

=

=rr

S0 S

d

2 2 2 ( ) 21 2 0 (1 )i vt x i s rA A A A b e eπ λ π− − ⋅= + = +

rrat the detector:

2 2 2 ( ) 21 2 0 (1 )i vt x i s rA A A A b e eπ λ π− − ⋅= + = +

rr

at the detector:

2 2 2 20* (1 )(1 )i s r i s rJ AA A b e eπ π⋅ − ⋅= = + +

r rr r

20

20 0

1 1

0

(1 )

when there are N identical scatteres,

( )

j j

i s r

N Niq ri s r

j j

iq r

V

A A b e

A A b e A b e

A A b n r e dr

π

π

− ⋅

− ⋅− ⋅

= =

− ⋅

= +

= =

=

∑ ∑

∫

rr

r uurrr

r rr r

i j ij2

when , 0when , 2

b a

i ji j

πδ

π

⋅ =

≠=

2 31

2 3 1

2 a aba a a

π ×=

⋅ ×

3 12

3 1 2

2 a aba a a

π ×=

⋅ ×

1 23

1 2 3

2 a aba a a

π ×=

⋅ ×

1 1 2 2 3 3q v b v b v b= + +r r rr

1 1 2 2 3 3r u a u a u a= + +r r r r

Real and inverse lattice

2 s qπ =r r

1ar

2ar3ar

3br

beam center

diffracted spot

L2

1/L1

this axis is normal to L1, and the length is inverse of L1

1/L2

L1

real lattice

unit cell from a certain angle

j

j j j

j j

( ) ( ) exp( )

( ) exp( )

( ) exp[ ( )] exp( )

exp( )

j

Vj

j

F q r ir q dr

r r ir q dr

r r iq r r dr iq r

f iq r

ρ

ρ

ρ

= − ⋅

= − − ⋅

⎡ ⎤= − − ⋅ − − ⋅⎣ ⎦

= − ⋅

∫∑∫

∑ ∫

∑

r r r r

r r r r r

r r r r r r r r

r r

Atomic form factor:intensity determination

Position determination

( ) ~ ( )A q F q

Structure factor or form factor

j j( ) ( )j

r r rρ ρ= −∑r r r

jrrrr

2

( ) ( ) exp( )

assuming spherical symmetry,sin( ) 4 ( )

F q r ir q dr

r qr r drr q

ρ

π ρ

= − ⋅

⋅=

⋅

∫

∫

r r r r

r r

r r

( ) ~ ( )A q F q

Structure factor or form factor

j j( ) ( )j

r r rρ ρ= −∑r r r

see Fig 1.6

•Scattering length of a single nucleus

-interaction w/ nucleus: highly penetrating

-scattering occurs due to

1. structure2. randomness of spin state

and distribution of isotope

•Coherent and incoherent scattering length

2( 1/ 2) 1 2 2i i+ + = +

2( 1/ 2) 1 2i i− + = 24 2 2 1

i ifi i

− = =+ +

2 2 14 2 2 1i ifi i

+ + += =

+ +

1/ 2i −

1/ 2i +

Ewald sphere and reciprocal scattering

u( ) ( )* ( )r r z rρ ρ=r r r FT

IFT( ) ( ) ( )A q F q Z q=r r r

( )I qrFT

IFT( )rρΓr

SLDD Scattering amplitude

X squa

ring

Autocorrelation ftn

Form factor lattice factor

auto

corr

elat

ion

( ) ( ) ir qA q r e drρ − ⋅= ∫r rr r r

( ) ( ) ir qI q r e drρ− ⋅= Γ∫r rr r r

*

( )( ) ( )

I qA q A q= ⋅

r

r r

( )

( ) ( ')

( ) ( )

r

V u u

r u r du

ρ

ρ ρ

ρ ρ

Γ

=

= +∫

r

r r

r r r r

u( ) ( )* ( )r r z rρ ρ=r r r FT

IFT( ) ( ) ( )A q F q Z q=r r r

( )I qr

SLDD Scattering amplitude

X squa

ring

Form factor lattice factor

( ) ( ) ir qA q r e drρ − ⋅= ∫r rr r r

*

( )( ) ( )

I qA q A q= ⋅

r

r r

u( ) ( )* ( )r r z rρ ρ=r r r FT

IFT( ) ( ) ( )A q F q Z q=r r r

Scattering amplitude

Form factor lattice factor

SLDD

aa− 0

b

a 2a 3aa−2a−3a− 0

b

u ( ) ( )r z rρ ×r r

a 2aa−2a− 0

b

u( ) ( )* ( )r r z rρ ρ=r r r

0

0

( ) ( )

( ) ( )

( ) (( ) )

since ( ) (( ) ) ( ),

( )

nx na

n

x na

n

x z x

x x na

u x na u du

u x na u du f x na

x na

ρ

ρ δ

ρ δ

ρ δ

ρ

∞

=−∞

−∞

=−∞

−

∞

=−∞

∗

= ∗ −

= − −

⎡ ⎤− − = −⎢ ⎥

⎣ ⎦

= −

∑

∑ ∫

∫

∑

u( ) ( )* ( )r r z rρ ρ=r r r

( )n

x naρ∞

=−∞

= −∑

a 2aa−2a− 0

b

u( ) ( )* ( )r r z rρ ρ=r r r

( )I qrFT

IFT( )rρΓr

SLDDau

toco

rrel

atio

n

Autocorrelation ftn( ) ( ) ir qI q r e drρ

− ⋅= Γ∫r rr r r

( )

( ) ( ')

( ) ( )

r

V u u

r u r du

ρ

ρ ρ

ρ ρ

Γ

=

= +∫

r

r r

r r r r

''

u r udu dr= +=

r r r

r r

2 *

2*

'

( )

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ') '

( ) ( )

( ) ( )

( )

ir q

iu q iu q

iu q i r u q

ir q

ir q

I q

A q A q A q

r e dr A q A q

u e du u e du

u e du r u e dr

u r u du e dr

r eρ

ρ

ρ ρ

ρ ρ

ρ ρ

⋅

⋅ − ⋅

⋅ − + ⋅

− ⋅

− ⋅

= = ⋅

= = ⋅

⎡ ⎤ ⎡ ⎤= ⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦⎡ ⎤= +⎣ ⎦

= Γ

∫

∫ ∫∫ ∫∫ ∫

r r

r r r r

r r r r r

r r

r r

r

r r r

r r r r

r r r r

r r r r r

r r r r r

r dr∫r

( )

( ) ( ')

( ) ( )

r

V u u

r u r du

ρ

ρ ρ

ρ ρ

Γ

=

= +∫

r

r r

r r r r

rr u r+r r

ur

Physical meaning of auto correlation ftn

see also p.96

2

2

( )

( ) ( ')

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

~ ( ) ( )

~ ( )

r

V u u

r u r du

u u r du

u u r du du u du u r du

u u r du V

r

ρ

η

ρ ρ

ρ ρ

η ρ η ρ

η η ρ ρ η ρ η

η η ρ

Γ

=

= +

= ⎡ + ⎤ ⎡ + + ⎤⎣ ⎦ ⎣ ⎦

= + + + + +

+ +

Γ

∫∫∫ ∫ ∫ ∫∫

r

r r

r r r r

r r r r

r r r r r r r r r r

r r r r

rnull scattering or scattering at q=0Experimentally unobservable

( ) ( )r rη ρ ρ= −r r

=0

macroscopic dimension

u( ) ( )* ( )r r z rρ ρ=r r r FT

IFT( ) ( ) ( )A q F q Z q=r r r

( )I qrFT

IFT( )rρΓr

SLDD Scattering amplitude

X squa

ring

Autocorrelation ftn

Form factor lattice factor

auto

corr

elat

ion

( ) ( ) ir qA q r e drρ − ⋅= ∫r rr r r

( ) ( ) ir qI q r e drρ− ⋅= Γ∫r rr r r

*

( )( ) ( )

I qA q A q= ⋅

r

r r

( )

( ) ( ')

( ) ( )

r

V u u

r u r du

ρ

ρ ρ

ρ ρ

Γ

=

= +∫

r

r r

r r r r

Structure factor for a uniform sphere

[ ]2( ) ~ ( )P q F q ρ2R

~ 0ρ

( ) ( ) exp( )

= HOMEWORK!!! (by sep/22/2005)

F q r iq r drρ= − ⋅∫r r r r

2 s qπ =r r

Structure factors for several structures

4 cos( ) sin( )cos 2

qLF qqL

Θ=

Θ

Thin rod

0

2 sin 1 cos( )qL u qLP q du

qL u qL⎡ ⎤−

= −⎢ ⎥⎣ ⎦∫

At a certain orientations,

Random orientations,

Θ a

L

V=aL

Homework!!!

Structure factors for several structures

R

Circular disk

12 2

(2 )2( ) 1 J qRP qq R qR

⎡ ⎤= −⎢ ⎥

⎣ ⎦

Homework!!!

Size of chain moleculesSize of chain molecules--synthetic polymer, DNA, proteinsynthetic polymer, DNA, protein……

2220

22

6

6/

NbRR

nlCR

g

g

==

= ∞

Number of Kuhn segment

Kuhn segment length

Charateristic ratio

Number of repeat unit

How about this?

Random coilOr Gaussian coil

[ ]2( ) ~ ( )P q F q

constant form factor?

2 2 2 2

4 4

exp( ) 1( ) ~ 2 g g

g

q R q RP q

q R− − +

http://www.ncnr.nist.gov/programs/sans/pdf/polymer_tut.pdf

Not only lattice scatteringbut also shape of the single object is important

beam center

diffracted spot

L2

1/L1

this axis is normal to L1, and the length is inverse of L1

1/L2

L1

real lattice

unit cell from a certain angle