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
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