Fundamentals of Neutron Diffraction

NCNR Summer School June 26, 2006

Norm Berk

@ .nberk nist gov

Basic Scattering TheorySANS

2θ

incident beam transmitted beam:unscattered beam

forward-scattered beam+

k

skdetector

neutronsource

scattered beam

element

θk

sk

neutronsource θ

θ2θ

detectorelement

2θ

incident beam transmitted beam:unscattered beam

forward-scattered beam+

k

sk

neutronsource

scattered beam

2sk k π

λ= =

Elastic scattering

sQ = k - k k

k 42 sinQ k πθ θλ

=

detectorelement

2 2 22 ( , ) ( , )) ( ,

2 2( )V k

m mk r krr k rψ ψ ψ− ∇ + =

( )22( ) ( )i i

i

V b gmπ δ= − +∑r r R µ B ri

Fermi scattering lengthsum over all nucleii at positions i

i

b =R

positive (repulsive interaction) =

negative (attractive interaction)ib⎧⎪⎪⎨⎪⎪⎩

neutron

nucleus

Shrodinger Equation

nuclear interaction: magnetic interaction:nucleus spinelectron spins

Fermi "contact"

2 24 ( )( , ) ( , ) ( , )kk r k r k rrψ π ψ ψρ−∇ + =

scattering length density (S( ) ( ) LD)i ii

br r Rρ δ= − =∑[ ]L 3L−⎡ ⎤⎢ ⎥⎣ ⎦

2L−⎡ ⎤⎢ ⎥⎣ ⎦

22

2( , ) 4 ( ) ( (2

, ) , )km

ψ πρ ψ ψ⎡ ⎤−∇ + =⎢ ⎥⎣ ⎦k r r k r k r

incident beam

scattered beam

2 2( , ) 4 ( ) ( , ) ( , )kψ πρ ψ ψ−∇ + =k r r k r k r

( , ) ieψ ⋅= k rk r

i

( ) e( , )sk r

s sfr

kk rψ =

spherical wave

sk

i 3( , )( ) ( )es

sV

f d rk rk rk rψ ρ − ⋅= ∫

transmitted beam

plane wavefront

Exact in scattererψ

sV

i 3( , )( ) ( )es

sV

f d rk rk rk rψ ρ − ⋅= ∫

k sk

ie inside( , scatt) erers rkk rψ ⋅≈

Born approximation ("Kinematic theory")

3ie( ) ( )s

sV

f d rQ rk rρ⋅∴ ≈ ∫

ˆ( ) FT ( )QQ rρ ρ= =

When is the BA valid?

k sk

k sk

distortion

phase shift ν

BA

exact (dynamical)

[ ] [ ]

[ ] [ ]

max

4 24 2 6 2

4 2max

For a uniform sphere of radius :

10 ( 10 10 )

5. ., for representative 10 5 1,

BA exact as 0; good for 1 (particles n

2

R

R m nm nmnm

e g nm nm R m

Rµ ρ λ

ρ ρ

ρ λ ν µ

ν

ν ν

ρλ− −

− − − −

− −

⎡ ⎤⎢ ⎥⎣ ⎦ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤ = × =⎢ ⎥⎣

→

∴

⎦<

= Å

ot too large)

What's measured?

scattered current d ensity sj

incident current densityij

2dS r d= Ωr

cross-sectional area

current: dNJ jAdt

= =

current density: / = "flux"dNj Adt

=

Differential Cross Section : dσ

A

s idj dS j σ=

What's measured?

scattered current d ensity sj

incident current densityij

2dS r d= Ωr

cross-sectional area A

/ ss i

i

dNj dS j AN

dd

σ∴ = =

d dσσ ∂= Ω∂Ω

d ddσ= ΩΩ

" "

What's calculated?

* *

2j

miψ ψ ψ ψ⎡ ⎤= ∇ − ∇⎢ ⎥⎣ ⎦

iincident: e , /kzi ij k mψ = =

2

2

( )scattered : ( ) ,

sik rss

s s s

fkef jr m r

kkψ = =

22 ( ) ss

i

k rj dS r ddj

σ = Ω∴ = =2

2

( )sf

kr

k

2( )sf kσ∂ =∂Ω

What's measured/calulated?

2 2( ) ( )s sf fk kσ∂ = →∂Ω thermal motions

random configurationsrandom orientationsspin degrees of freedom...other uncontrolled DOF

1

2

2

ˆ: we want to know ( ), since ( ) ( ),ˆand then ( ) FT ( );

ˆ but we directly measure ( ) .

ˆ ˆThe "phase problem": can we get ( ) from ( ) ?

s sf fNOTE k k Q

r Q

Q

Q Q

ρρ ρ

ρ

ρ ρ

−

≈≈

…

Coherent & Incoherent Scattering

nuccoh,spin

n nb b=

nucleusat nR

nuclearflucutating

spin

neutron

neutronspin

coherent

nucnucspinspin

22inc,n n nb b b= − incoherent

2inc, inc,4 n nbπ σ=

Coh/Inc Rule:2inc,n , coh, coh,m n m n m nb b b b bδ= +

(fluctuationvariance)

22 ˆ( ) ( )sf k Qσ ρ∂ = →∂Ω

2 2

iFT ( ) e nn n n

n n

b b Q RQ r Rδ= − =∑ ∑ i

( )i ( ) i ( )2inc,n ,nuc

scoh, coh,

, ,pine em n m n

m n m n m nm n m n

b b b b bQ R R Q R Rδ− −= = +∑ ∑i i

i ( )2inc, coh, coh,

,

e m nn m n

n m n

b b b Q R R−= +∑ ∑ i

(Rayleigh-Gansconstruction)

inc, icoh, coh

),

,

(e4

m nnm n

n m n

b b Q R Rσσπ

−∂ = +∂Ω ∑ ∑ i

inc

σ∂∂Ω coh

σ∂∂Ω

structure dependentstructure independent(a major contribution

to background)

coh / , "macroscopic" cross-sectionVσ = Σ1L−⎡ ⎤⎢ ⎥⎣ ⎦

3L⎡ ⎤⎢ ⎥⎣ ⎦2 /L⎡ ⎤⎢ ⎥⎣ ⎦

coh

, etc.dd

σ∂ Σ→∂Ω Ω

:Notation alert

icoh, coh

( ),

,coh

e m nm n

m n

b b Q R Rσ −∂ =∂Ω ∑ i

i ( )coh, coh,

,

e m nm n

m n

b b Q R R−= ∑ i

direction of Q

coh, coh,,

sinc m nm n

m n

Qb b

R Rπ

⎡ ⎤−⎢ ⎥= ⎢ ⎥⎣ ⎦∑

( ) ( ) ( )0

sinsinc x

xj x

xπ

ππ

= =

Isotropic scattering

... average over random orientations

random orientation

"Fundamental Theorem of Diffraction"

( ) ( ) ( ) ( )23 i 3 3

2. Convolution-product theorem:

ˆ 2 eV

d Q d rπ ρ ρ ρ− ′ ′ ′= +∫ ∫Q rQ r r ri

1. Born Approximation

( ) ( )2

i i 3

coh

ˆ3. ( ) e eV

d rσ ρ∂ ′= = Γ Γ∂Ω ∫Q r Q rQ r ri i

( ) ( ) ( )( )

( ) ( ) ( )

3

2

23

4.

0

V

V

d r

V

d r V

ρ ρ

ρ

ρ ρ ρ

′ ′ ′Γ = +

Γ =

′ ′ ′Γ ∞ = ∞+ =

∫

∫

r r r r

r rasymptotic independence

( ) ( ) ( )( )

( )

3

2

2

4.

0

d r

V

V

ρ ρ

ρ

ρ

′ ′ ′Γ = +

Γ =

Γ ∞ =

∫r r r r

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )

( )

22

2 22

5.0

0 1

0

,

contrast

V V ρ

γ

γ

ρ

γ

ρ

ρ ρ

γ

Γ − Γ ∞=

Γ − Γ ∞

=

∞ =

Γ ∆= +

∆ == −

rr

r r

Debye-Porod correlation function

( ) ( ) ( ) 3

fluctuation correlation function

d rγ ρ ρ′ ′ ′∴ = ∆ + ∆∫r r r r

( )i 3

coh

3. e d rσ∂ ′= Γ∂Ω ∫ Q r ri

( ) ( )

( ) ( )

32 2inc coh

2 i 3

6.

2

e

N b V

V d r

σ π ρ δ

ρ γ

∂ = +∂Ω

′+ ∆ ∫ Q r

Q

ri

incoherentbackground

forwardscattering

coherent scatteringfrom microstructure

( )needs contrast

( ) ( )22 , 1A B A B A Bρ ϕ ϕ ρ ρ ϕ ϕ∆ = − + =In 2-phase (A-B) system:

( ) ( )

( ) ( )

32 2inc coh

2 i 3

2

e

N b V

V d r

σ π ρ δ

ρ γ

∂ = +∂Ω

′+ ∆ ∫ Q r

Q

ri

( ) ( )2 i 3: eV d rσ ρ γ∂ ′∆∂Ω ∫ Q rNotation alert ri

( ) ( )

( ) ( )

2

i 3

where

e "scattering function"

V S

S d r

σ ρ

γ

∂∴ = ∆∂Ω

′= ∫ Q r

Q

Q ri

( ) ( ): more generally:

,S S dω ω= ∫Aside

Q Q

( ) ( )i 3eS d rγ= ′ Q rQ ri

( )( ) ( )

( )( ) ( )

33

233

coh

1 0 12

1 Porod invariant2

S d Q

d Q V

γπ

σρ

π

= =

∂∴ = ∆

∂Ω

∫

∫

Q

Q

The scattering function

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

i 3

0

2

0 0

Isotropic scattering:

e , sin

4 sin 4 sinc /

S r d r S Q r r Qr dr

r r Qr dr r r Qr drQ

γ γ

π γ π γ π

∞

∞ ∞

= =

= =

∫ ∫

∫ ∫

Q rQ i

( ) ( ) ( ) ( )2 2

0/ sinc / , Distance distribution functionp r r r r r Qr drγ γ π

∞= ∫

( )

( ) ( ) ( )

( ) ( )

3

0

2 2

0

for small , expand the Sin :

4 13!

10 1 ,6

n n

Q Qr

S Q r r Qr Qr drQ

S Q r r r p r dr

π γ∞

∞

= − + = − + =

The Laws

( ) ( ) 2 2G

10 13

S Q S Q R ∴ = − +

( )2 2 3G G0

, Guinier radius (radius of gyration)R r p r d r R∞

= =∫

( ) ( )( ) ( )

2 2G /3

G

0 e

: this implies only for .

Q RS Q SS Q S Q QR π

−≈

≈ ≤

Guinier's "Law"

Note

( ) [ ]

( ) 4

for large , first expand ( ) for small :

for a scatterer of volume and surface area ,1 / , 4 / Porod correlation length

Then from a basic theorem about Fourier transforms,/ 4

Q r r

V Sr r V S

S VS QQ

γ

γ ξ ξ= − + =

=

The Laws

∼

( ) ( )

4

2

4coh

/

, for 1.

Q

SQ

Q

ξ

ρσξ

∆∂= >>

∂Ω

Porod's Law

Q

Basic Scattering TheoryNSR

detectorelement

neutronsource

θ rθtθ

axisz −

k kzk zk−

Specular Reflection: rθ θ=

2 sin2z zQ k

k θ==−

incident beam reflected beam

transmitted beam

2θ

Shrodinger Equation

2 24 ( )( , ) ( , ) ( , )

1 wave equationz zz z zz kk z k z k z

D

πρψ ψ ψ−∂ + =

−

( ) ( , , ) ( )xy

x y z zrρ ρ ρ=Specular paradigm

ii( , ) e e ( , )yx k yxz

k k zk r ψψ =

2 24 ( )( , ) ( , ) ( , )kk r k r k rrψ π ψ ψρ−∇ + =

What's measured?

A dS A=

current: dNrecall Jdt

j A= = i

Reflectivity /rR j A j Ai i

rjj

What's calculated?

* *

2j

miψ ψ ψ ψ⎡ ⎤= ∇ − ∇⎢ ⎥⎣ ⎦

iincident: e , /i ij mψ = =k r ki

ireflected : e , r rr j rm

ψ −= =k r ki

2 2/ ( ) ( )r z zR r Q r Qj A j A∴ = = →i i

: 1RN O T E ≤

[ ]z Å

transmitting medium

→6[10 ]ρ -2Å

Lipid HBL

DMPC SAM

Lipid HBL

AuCr

L0

frontingbacking

incident medium

free film (air fronting & backing)

for the free film

( ) ( ) ( ) ( )i

0,4 e , 2

iz

L k zz z z

zzr Q z d

Qz z Q kkπ ρψ= ∫

i 3( ) ( )e for point scattere( , r) ss

sV

recall f d rρψ − ⋅= ∫ k rrk rk

( ) ( ) ( ) i

0

4 ei

, zL k z

zz

zk zr Q z dzQπ ρψ= ∫

( ) ( ) ( )i2BA

0

4 4 ˆei i

zL

k zz z

z z

r Q z dz QQ Qπ πρ ρ= =∫

( ) iBorn Approximati ,on: e zkz

zk zψ →

: BA breaks down as 0(unlike SANS case)zQ →WARNING

zQ

[ ]z Å

→

→L0

←

Free film

ie zk z

ie zk zr − ie zk zt

Transfer MatrixA B

MC D⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

( )i1 1

ei 1 i

zk Lr

tr

M⎛ ⎞ ⎛ ⎞+ ⎟ ⎟⎜ ⎜⎟= ⎟⎜ ⎜⎟ ⎟⎜ ⎜ ⎟⎜⎟⎟⎜ − ⎝ ⎠⎝ ⎠

Real numbers( )Det 1M AD BC= − =

giving two equations for and r t

( is unimodular)M

( ) ( )"transfers" and from 0 to z zM z z Lψ ψ′ = =

( )zψ

( )1 1

i 1 it

rM

r⎛ ⎞ ⎛ ⎞+ ⎟ ⎟⎜ ⎜⎟= ⎟⎜ ⎜⎟ ⎟⎜ ⎜ ⎟⎜⎟⎟⎜ − ⎝ ⎠⎝ ⎠

( )( )

ii

2i

B C D Ar

B C D Aα β γ

α β

+ + −=

− + +− +=−+

2 2

2 2

2 1

A CB DAB CD

αβγγ αβ

= += += −= −

2 2 2 2 22 ,2

R r A B C D α βΣ−= = Σ= + + + = +Σ+

[ ]The phase problem: doesn't determine , and .R α β α β γ+

1M 2M

Transfer Matrix

1 2M M M=

1M 2M 3M

1 2 3M M M M=

:historically, also is calledthe "optical" matrix.

MAside

jM 1 2 NM M M M=

NExact in lim

→∞

maxGood for N Q L≥

nd

nρ

( ) ( ) ( )( ) ( ) ( )cos sin /

sin cos

j z j z j z

j z j z j zn

k k n k

n k k k

ξ ξ

ξ ξ

⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎟⎜ ⎣ ⎦ ⎣ ⎦ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤ ⎡ ⎤⎜− ⎟⎟⎜⎝ ⎠⎢ ⎥ ⎣ ⎦⎣ ⎦jM =

2

4( ) 1 jj z

z

n kkπρ

= −

Transfer Matrix

↔

( ) ( ) ,j z j z z nk n k k dξ =

[ ]z Å

→

→

L0

←

Surrounded film

( )ie zf zn k k z

( )ie f zz zn k kr − ( )ie z zbn k k zt

A BM

C D⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

fronting: fρ ρ=backing: bρ ρ=

2ifb fb fb

fb fbr α β γα β− +=−+

, ,fb fb fbα α β β γ γ→ → →

Two prototypesvacuum

backing: bρ→)1 Fresnel Problem: backing only

( ) ( )( )

( ) ( )

b

b

b b b

1 / 21 / 2

/ 2 / 2 for 16i

zz

z

z z z c

n Qr Q

n Q

n Q n Q Q Qπρ

−=+

= ≤ =

cQ zQ

R1

41/ ( Porod's Law)zQ recall∼

0 z

( )cartoon

bb 2

16( ) 1zz

n kQπρ= −

Two prototypes

vacuum→)2 Free barrier

( ) ( )( )

( ) 2

ii

/ 2 1 16 /

B B B Bz

B B B B

B z B z

B C D Ar Q

B C D A

n Q Qπρ

+ + −=

− + +

= −

R1

41/ zQ∼

L0 z

Bρ vacuumBM M=

↔/ Lπ zQ

Kiessig fringes( )cartoon

Enjoy the rest of the course!