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SAXS
4 X
• latter & Kratky (1982) Small Angle X-ray Scattering,
Academic Press • KEK 2010
, 2.3 X
X X SAXS
•
•
•
• •
SAXS
2θ
X-ray camera length
Sample in cuvette
q =4πsinθ/λ q h 2θ λ X
I(q) q �
(1 − 2 m)
0
0.5
1
0 0.1 0.2 0.3h ( A-1)
I (h)
=
2πλ
s!⋅r!− s0"!⋅r!( )
=2π s!− s0"!( )
λ⋅r!
= q!⋅r!
q!≡2π s!− s0"!( )
λ, q ≡ q
!= 4π sinθ
λ
1
F(q!) = ρ(r
!)
V∫ eiδ dV = ρ(r!)
V∫ eiq!⋅r!
dV
s!
s0!"!
r!
s!⋅r!
s0!"!⋅r"
o�
p�
q!
θθ
XV �
s0!"!, s"
ρ(r!)
i(q!) = F(q
!)F*(q!)
N
I(q!) ≈ N ⋅ i(q
!)
F(q!) = ρ(r
!)
V∫ eiq!⋅r!
dV1
*
F(q!) = ρ(r
!)
V∫ eiq!⋅r!
dV
I(q!) = F(q
!)F*(q!) = ρ(r1
"!)
V1∫ eiq!⋅r1"!
dV1 ρ(r2"!)
V2∫ e− iq!⋅r2"!"
dV2
= ρ(r1"!)
V2∫ ρ(r2"!)eiq!⋅(r1"!−r2"!") dV1 dV2V1∫ r
!≡ r1"!− r2"!
I(q!) = ρ(r2
"!+ r!)
V2∫ ρ(r2"!)eiq!⋅r!
dV dV2V∫= ρ(r2
"!)ρ(r2"!+ r!)dV2V2∫( )eiq!⋅r! dVV∫
γ (r!) ≡ ρ(r2
"!)ρ(r2"!+ r!)dV2V2∫
I(q!)= γ (r
!)eiq!⋅r!
dVV∫ ⇔ γ (r
!) = 1(2π )3
I(q!)e− iq
!⋅r!
dq!
V*∫Fourier
I(q!)= γ (r
!)eiq!⋅r!
dVV∫
I(q) ≡ I(q
!) Ω=
14π
I(q!)dΩ∫
dV = r2 sinθdθdϕdr = r2dωdr r!= (r,θ ,ϕ )
ω
x�
y�
z�
r!
ϕ
θ
= 14π
dr0
∞
∫ dω0
4π
∫ γ (r!)eiq⋅"!r!
r2∫ dΩ
= r2 dr0
∞
∫ γ (r!)dω
0
4π
∫14π
eiq⋅"!r!
dΩ∫= r2 dr
0
∞
∫ 4πγ (r) sin(qr)qr
= 4π r2γ (r) sin(qr)qr
dr0
∞
∫↑
2
I(q) = 4π r2γ (r) sin(qr)qr
dr0
∞
∫p(r) ≡ r2γ (r)
I(q)= 4π p(r) sin(qr)qr
dr0
∞
∫ ⇔ p(r) = 12π 2 I(q)qr sin(qr)dq
0
∞
∫Fourier
Debye
I(q) = I(q
!)
Ω= ρ(r1
"!)
V2∫ ρ(r2"!)eiq!⋅r!
dV1 dV2V1∫ Ω
r ≡ r!= r1"!− r2"!
= ρ(r1
!")
V2∫ ρ(r2!") sin(qr)
qrdV1 dV2V1∫
uinier
Debye
I(q) = ρ(r1
!")
V2∫ ρ(r2!") sin(qr)
qrdV1 dV2V1∫
sin(qr)qr
= 1− (qr)2
3!+ (qr)
4
5!−! Taylor
I(q) ≈ ρ(r1
!")
V2∫ ρ(r2!")dV1 dV2V1∫ − q
2
6ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫
= ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫ 1−
q2 ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫
6 ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= I(0) 1− q2
3⋅
ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫
2 ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫
⎛
⎝⎜⎜
⎞
⎠⎟⎟
= I(0) 1− q2
3⋅
ρ(r!1)
V1∫ r12dV1
ρ(r!1)
V1∫ dV1
⎛
⎝⎜⎜
⎞
⎠⎟⎟= I(0) 1− q
2
3Rg2⎛
⎝⎜⎞⎠⎟
← 1
I(q) ≈ I(0)e−Rg
2q2
3
e−Rg
2q2
3 = 1− Rg2q2
3+ 12!
Rg2q2
3⎛⎝⎜
⎞⎠⎟
2
−! Taylor
ln I(q) ≈ ln I(0)− Rg2
3q2
!Guinier
I(0) = ρ(r1
!")
V2∫ ρ(r2!")dV1 dV2V1∫ = ρ(r
"1)
V1∫ dV1( )2 = (ρV )2
Guinier lnI(q) vs. q2
0
2
4
6
8
10
0 0.0005 0.001 0.0015 0.002
1.4 mg/ml ln
I(q)
q 2 (A -2 )
→Rg I(0) Guinier ��Rg q < 1.3 Guinier Rg: I(0)
r
Rg2 =ρ(r!)
V∫ r2dV
ρ(r!)
V∫ dV=4πρ r4 dr
0
∞
∫ρ 43πr3
= 35r2 Rg = 3
5r ≈ 0.775r
I(0) = (ρV )2 ∝ cM c mg/mL M
Zimm
cI(0,c)
= 1K
1M
(1+ 2A2 Mc)
I(0)cM
= Ist (0)cstMst
K A2
I(0)c
Rg = p(r) ⋅r 2 dr
0
Dmax
∫ 2 p(r)dr0
Dmax
∫
1. 2
→ Dmax
p(r)
2. Guinier
p(r) = 0r > Dmax
3. p(r)p(r) = 1
2π 2 I(q)qr sin(qr)dq0
∞
∫i) termination effectp(r) ripple
ii) interparticle interference
iii) Fourier Δq ≤ π Dmax
1. imm
Guinier Rg I(0) Rg I(0)
Rg I(0)
2. SAXS
backgroud scattering Ibg
→ Porod Luzzati
limq→∞
I(q) = 1π
SV
Qq4 + Ibg
SV specific surface Q invariant
1. SAXS
2
SVD 2.
F(q!) =
ρ(r!)
V∫ eiq!⋅r!
dV
fi eiq!⋅r!
i∑
⎧⎨⎪
⎩⎪ I(q!) = F(q
!)F*(q!)
SAXS1. SAXS
− =
ρ Δρ = ρ − ρ01.0g / cm3 = 0.334e / A3
ρρ0
Δρ
2. SAXS
3 1.1
SAXS
→
c M(1) mol/L(2) 2
I(0)∝ cMcM I(0)∝ cMM
2
I(0)
(1) c(mg /mL) = c(g / L) = cM (mol / L) ⋅M (g /mol)I(0)∝ cMM( )M = cMM
2
( ) A2 → 2AI(0)
A2 cM� 2M cM(2M)2 = 4cMM2
A 2cM M (2cM)M2 = 2cMM2
I (0)
I(0) = (1−α )Idimer (0)+α Imonomer (0)∝ (1−α )4cMM
2 +α (2cMM2 ) = 2(2 −α )cMM
2
A2! 2A 2(1−α ) :α Keq =
α1−α
⎛⎝⎜
⎞⎠⎟
Guinier
1.
�
I(q) = ρ(r1!")
V2∫ ρ(r2!") sin(qr)
qrdV1 dV2V1∫ = ρ(r1
!")
V2∫ ρ(r2!") 1− (qr)
2
3!+ (qr)
4
5!−#
⎛⎝⎜
⎞⎠⎟dV1 dV2V1∫
= ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫ − q
2
6ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫ +#
≈ ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫ 1−
q2 ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫
6 ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫
⎛
⎝⎜⎜
⎞
⎠⎟⎟= I(0) 1− q
2
3⋅
ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫
2 ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫
⎛
⎝⎜⎜
⎞
⎠⎟⎟#�
r = r!= r1"!− r2"!
rG!"!
ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫
2 ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫
=ρ(r1!")
V2∫ ρ(r2!") r1!"− r2!" 2
dV1 dV2V1∫2 ρ(r1
!")
V2∫ ρ(r2!")dV1 dV2V1∫
=ρ(r1!")
V2∫ ρ(r2!") (r1!"− rG!"!)− (r2!"− rG!"!)2dV1 dV2V1∫
2 ρ(r1!")dV1 V2∫ ρ(r2
!")dV2V1∫
=ρ(r1!")
V2∫ ρ(r2!") r1!"− rG!"! 2
+ r2!"− rG!"! 2
− 2(r1!"− rG!"!) ⋅(r2!"− rG!"!)( )dV1 dV2V1∫
2V1∫ ρ(r1!")dV1( )2
∵V1∫ ρ(r1!")dV1 = V2∫ ρ(r2
!")dV2( )
=ρ(r1!")
V2∫ ρ(r2!") r1!"− rG!"! 2
dV1V1∫ dV2 + ρ(r1!")
V2∫ ρ(r2!") r2!"− rG!"! 2
dV1V1∫ dV2 − 2 ρ(r1!")
V2∫ ρ(r2!")(r1!"− rG!"!) ⋅(r2!"− rG!"!)dV1V1∫ dV2
2V1∫ ρ(r1!")dV1( )2
=ρ(r1!") r1!"− rG!"! 2
dV1V1∫ V2∫ ρ(r2!")dV2 + V1∫ ρ(r1
!")dV1 ρ(r2
!") r2!"− rG!"! 2
dV2V2∫ − 2 ρ(r1!")(r1!"− rG!"!)dV1V1∫( ) ⋅ V2∫ ρ(r2
!") ⋅(r2!"− rG!"!)dV2( )
2V1∫ ρ(r1!")dV1( )2
rG!"!
≡ρ(r"1)r1!"dV1V1∫
ρ(r"1)dV1V1∫
ρ(r1!")(r1!"− rG!"!)dV1V1∫ = 0
" r2!"
=2 ρ(r1
!") r1!"− rG!"! 2
dV1V1∫ V1∫ ρ(r1!")dV1 − 2 0
"⋅0"( )
2V1∫ ρ(r1!")dV1( )2
=ρ(r1!") r1!"− rG!"! 2
dV1V1∫V1∫ ρ(r1!")dV1
= Rg2 r1!"− rG!"!
I(q) ≈ I(0) 1− q2
3Rg2⎛
⎝⎜⎞⎠⎟
2.
1
r = r12 + r2
2 − 2r1r2 cosθ12
�
ρ(r1!")
V2∫ ρ(r2!")r2 dV1 dV2V1∫
2 ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫
=ρ(r1!")
V2∫ ρ(r2!")(r1
2 + r22 − 2r1r2 cosθ12 )dV1 dV2V1∫
2 ρ(r1!")
V2∫ ρ(r2!")dV1 dV2V1∫
=ρ(r1!")
V2∫ ρ(r2!")r1
2 dV1 dV2V1∫ + ρ(r1!")
V2∫ ρ(r2!")r2
2 dV1 dV2V1∫ − 2 ρ(r1!")
V2∫ ρ(r2!")r1r2 cosθ12 dV1 dV2V1∫
2 ρ(r1!")dV1 V2∫ ρ(r2
!")dV2V1∫
=ρ(r1!")r1
2 dV1 ρ(r2!")dV2V2∫V1∫ + ρ(r1
!")dV1 V2∫ ρ(r2
!")r2
2 dV2V1∫ − 2 ρ(r1!")
V2∫ ρ(r2!")r1r2 cosθ12 dV1 dV2V1∫
2V1∫ ρ(r1!")dV1( )2
=2 ρ(r1
!")r1
2 dV1 ρ(r1!")dV1V1∫V1∫ − 2 ρ(r1
!")
V2∫ ρ(r2!")r1r2 cosθ12 dV1 dV2V1∫
2V1∫ ρ(r1!")dV1( )2
=ρ(r1!")r1
2 dVV1∫V1∫ ρ(r1!")dV1
−ρ(r1!")
V2∫ ρ(r2!")r1r2 cosθ12 dV1 dV2V1∫
V1∫ ρ(r1!")dV1
#�
ρ(r1!")
V2∫ ρ(r2!")r1r2 cosθ12 dV1 dV2V1∫
1) r1r2 cosθ12 = r1!"⋅r2!"
r1, r2 r1!", r2!"
2) 1) r1!", r2!"
r1!", r2!"
θ12 r1!", r2!"
1
0
1 O
= Rg2 − 0 = Rg2
2
14π
eiq!⋅r!
dΩ∫ = sin(qr)qr
q!⋅r!= qr cosθqr
14π
eiq!⋅r!
dΩ∫ = 14π
eiqr cosθqr dΩ∫
eiqr cosθqr dΩ∫ r!
q = q!
eiq!⋅r!
1 r!
z q!
eiq!⋅r!
eiqr cosθqr dΩ∫ = dϕ0
2π
∫ eiqr cosθqr sinθqr dθqr0
π
∫ = 2π eiqr cosθ sinθ dθ0
π
∫
u = cosθ
2π eiqr cosθ sinθ dθ0
π
∫ = 2π eiqru du−1
1
∫ = 2π eiqru
iqr⎡
⎣⎢
⎤
⎦⎥−1
1
= 2πiqr
eiqr − e− iqr( )
eiqr = cosqr + isinqr Euler
2πiqr
eiqr − e− iqr( ) = 2πiqr ⋅2isinqr =4π sinqr
qr
14π
eiq!⋅r!
dΩ∫ = 14π
4π sinqrqr
= sinqrqr
14π
eiq!⋅r!
dΩ∫ =eiq!⋅r!
dΩ∫dΩ∫
= eiq!⋅r!
Ω
