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!
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