Today’s Outline - March 03, 2015segre/phys570/15S/lecture_13.pdf · Today’s Outline - March 03,...
Transcript of Today’s Outline - March 03, 2015segre/phys570/15S/lecture_13.pdf · Today’s Outline - March 03,...
Today’s Outline - March 03, 2015
• SAXS review
• Calculating Rg
• Porod regime
• Nucleation mechanism by SAXS
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Monday, March 10, 2015
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 1 / 15
Today’s Outline - March 03, 2015
• SAXS review
• Calculating Rg
• Porod regime
• Nucleation mechanism by SAXS
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Monday, March 10, 2015
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 1 / 15
Today’s Outline - March 03, 2015
• SAXS review
• Calculating Rg
• Porod regime
• Nucleation mechanism by SAXS
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Monday, March 10, 2015
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 1 / 15
Today’s Outline - March 03, 2015
• SAXS review
• Calculating Rg
• Porod regime
• Nucleation mechanism by SAXS
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Monday, March 10, 2015
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 1 / 15
Today’s Outline - March 03, 2015
• SAXS review
• Calculating Rg
• Porod regime
• Nucleation mechanism by SAXS
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Monday, March 10, 2015
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 1 / 15
Today’s Outline - March 03, 2015
• SAXS review
• Calculating Rg
• Porod regime
• Nucleation mechanism by SAXS
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Monday, March 10, 2015
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 1 / 15
Small Angle Scattering
Recall that there was an additional term in the scattering intensity whichbecomes important at small Q.
I SAXS( ~Q) = f 2∑n
∫Vρate
i ~Q·(~rn−~rm)dVm
= f 2∑n
e i~Q·~rn∫Vρate
−i ~Q·~rmdVm
= f 2∫Vρate
i ~Q·~rndVn
∫Vρate
−i ~Q·~rmdVm
=
∣∣∣∣∫Vρsle
i ~Q·~rdV
∣∣∣∣2Where we have assumed sufficient averaging and introduced ρsl = f ρat .This final expression looks just like an atomic form factor but the chargedensity that we consider here is on a much longer length scale than anatom.
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 2 / 15
Small Angle Scattering
Recall that there was an additional term in the scattering intensity whichbecomes important at small Q.
I SAXS( ~Q) = f 2∑n
∫Vρate
i ~Q·(~rn−~rm)dVm
= f 2∑n
e i~Q·~rn∫Vρate
−i ~Q·~rmdVm
= f 2∫Vρate
i ~Q·~rndVn
∫Vρate
−i ~Q·~rmdVm
=
∣∣∣∣∫Vρsle
i ~Q·~rdV
∣∣∣∣2Where we have assumed sufficient averaging and introduced ρsl = f ρat .This final expression looks just like an atomic form factor but the chargedensity that we consider here is on a much longer length scale than anatom.
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 2 / 15
Small Angle Scattering
Recall that there was an additional term in the scattering intensity whichbecomes important at small Q.
I SAXS( ~Q) = f 2∑n
∫Vρate
i ~Q·(~rn−~rm)dVm
= f 2∑n
e i~Q·~rn∫Vρate
−i ~Q·~rmdVm
= f 2∫Vρate
i ~Q·~rndVn
∫Vρate
−i ~Q·~rmdVm
=
∣∣∣∣∫Vρsle
i ~Q·~rdV
∣∣∣∣2Where we have assumed sufficient averaging and introduced ρsl = f ρat .This final expression looks just like an atomic form factor but the chargedensity that we consider here is on a much longer length scale than anatom.
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 2 / 15
Small Angle Scattering
Recall that there was an additional term in the scattering intensity whichbecomes important at small Q.
I SAXS( ~Q) = f 2∑n
∫Vρate
i ~Q·(~rn−~rm)dVm
= f 2∑n
e i~Q·~rn∫Vρate
−i ~Q·~rmdVm
= f 2∫Vρate
i ~Q·~rndVn
∫Vρate
−i ~Q·~rmdVm
=
∣∣∣∣∫Vρsle
i ~Q·~rdV
∣∣∣∣2Where we have assumed sufficient averaging and introduced ρsl = f ρat .This final expression looks just like an atomic form factor but the chargedensity that we consider here is on a much longer length scale than anatom.
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 2 / 15
Small Angle Scattering
Recall that there was an additional term in the scattering intensity whichbecomes important at small Q.
I SAXS( ~Q) = f 2∑n
∫Vρate
i ~Q·(~rn−~rm)dVm
= f 2∑n
e i~Q·~rn∫Vρate
−i ~Q·~rmdVm
= f 2∫Vρate
i ~Q·~rndVn
∫Vρate
−i ~Q·~rmdVm
=
∣∣∣∣∫Vρsle
i ~Q·~rdV
∣∣∣∣2
Where we have assumed sufficient averaging and introduced ρsl = f ρat .This final expression looks just like an atomic form factor but the chargedensity that we consider here is on a much longer length scale than anatom.
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 2 / 15
Small Angle Scattering
Recall that there was an additional term in the scattering intensity whichbecomes important at small Q.
I SAXS( ~Q) = f 2∑n
∫Vρate
i ~Q·(~rn−~rm)dVm
= f 2∑n
e i~Q·~rn∫Vρate
−i ~Q·~rmdVm
= f 2∫Vρate
i ~Q·~rndVn
∫Vρate
−i ~Q·~rmdVm
=
∣∣∣∣∫Vρsle
i ~Q·~rdV
∣∣∣∣2Where we have assumed sufficient averaging and introduced ρsl = f ρat .
This final expression looks just like an atomic form factor but the chargedensity that we consider here is on a much longer length scale than anatom.
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 2 / 15
Small Angle Scattering
Recall that there was an additional term in the scattering intensity whichbecomes important at small Q.
I SAXS( ~Q) = f 2∑n
∫Vρate
i ~Q·(~rn−~rm)dVm
= f 2∑n
e i~Q·~rn∫Vρate
−i ~Q·~rmdVm
= f 2∫Vρate
i ~Q·~rndVn
∫Vρate
−i ~Q·~rmdVm
=
∣∣∣∣∫Vρsle
i ~Q·~rdV
∣∣∣∣2Where we have assumed sufficient averaging and introduced ρsl = f ρat .This final expression looks just like an atomic form factor but the chargedensity that we consider here is on a much longer length scale than anatom.
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 2 / 15
The SAXS experiment
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 3 / 15
Scattering from a dilute solution
The simplest case is for a dilute solution of non-interacting molecules.
Assume that the scattering length density of each identical particle(molecule) is given by ρsl ,p and the scattering length density of the solventis ρsl ,0.
I SAXS( ~Q) =
∣∣∣∣∣∫Vp
ρslei ~Q·~rdVp
∣∣∣∣∣2
= (ρsl ,p − ρsl ,0)2
∣∣∣∣∣∫Vp
e i~Q·~rdVp
∣∣∣∣∣2
If we introduce the single-particleform factor F( ~Q):
F( ~Q) =1
Vp
∫Vp
e i~Q·~rdVp
I SAXS( ~Q) = ∆ρ2V 2p |F( ~Q)|2
Where ∆ρ = (ρsl ,p − ρsl ,0), and the form factor depends on themorphology of the particle (size and shape).
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 4 / 15
Scattering from a dilute solution
The simplest case is for a dilute solution of non-interacting molecules.
Assume that the scattering length density of each identical particle(molecule) is given by ρsl ,p and the scattering length density of the solventis ρsl ,0.
I SAXS( ~Q) =
∣∣∣∣∣∫Vp
ρslei ~Q·~rdVp
∣∣∣∣∣2
= (ρsl ,p − ρsl ,0)2
∣∣∣∣∣∫Vp
e i~Q·~rdVp
∣∣∣∣∣2
If we introduce the single-particleform factor F( ~Q):
F( ~Q) =1
Vp
∫Vp
e i~Q·~rdVp
I SAXS( ~Q) = ∆ρ2V 2p |F( ~Q)|2
Where ∆ρ = (ρsl ,p − ρsl ,0), and the form factor depends on themorphology of the particle (size and shape).
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 4 / 15
Scattering from a dilute solution
The simplest case is for a dilute solution of non-interacting molecules.
Assume that the scattering length density of each identical particle(molecule) is given by ρsl ,p and the scattering length density of the solventis ρsl ,0.
I SAXS( ~Q) =
∣∣∣∣∣∫Vp
ρslei ~Q·~rdVp
∣∣∣∣∣2
= (ρsl ,p − ρsl ,0)2
∣∣∣∣∣∫Vp
e i~Q·~rdVp
∣∣∣∣∣2
If we introduce the single-particleform factor F( ~Q):
F( ~Q) =1
Vp
∫Vp
e i~Q·~rdVp
I SAXS( ~Q) = ∆ρ2V 2p |F( ~Q)|2
Where ∆ρ = (ρsl ,p − ρsl ,0), and the form factor depends on themorphology of the particle (size and shape).
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 4 / 15
Scattering from a dilute solution
The simplest case is for a dilute solution of non-interacting molecules.
Assume that the scattering length density of each identical particle(molecule) is given by ρsl ,p and the scattering length density of the solventis ρsl ,0.
I SAXS( ~Q) =
∣∣∣∣∣∫Vp
ρslei ~Q·~rdVp
∣∣∣∣∣2
= (ρsl ,p − ρsl ,0)2
∣∣∣∣∣∫Vp
e i~Q·~rdVp
∣∣∣∣∣2
If we introduce the single-particleform factor F( ~Q):
F( ~Q) =1
Vp
∫Vp
e i~Q·~rdVp
I SAXS( ~Q) = ∆ρ2V 2p |F( ~Q)|2
Where ∆ρ = (ρsl ,p − ρsl ,0), and the form factor depends on themorphology of the particle (size and shape).
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 4 / 15
Scattering from a dilute solution
The simplest case is for a dilute solution of non-interacting molecules.
Assume that the scattering length density of each identical particle(molecule) is given by ρsl ,p and the scattering length density of the solventis ρsl ,0.
I SAXS( ~Q) =
∣∣∣∣∣∫Vp
ρslei ~Q·~rdVp
∣∣∣∣∣2
= (ρsl ,p − ρsl ,0)2
∣∣∣∣∣∫Vp
e i~Q·~rdVp
∣∣∣∣∣2
If we introduce the single-particleform factor F( ~Q):
F( ~Q) =1
Vp
∫Vp
e i~Q·~rdVp
I SAXS( ~Q) = ∆ρ2V 2p |F( ~Q)|2
Where ∆ρ = (ρsl ,p − ρsl ,0), and the form factor depends on themorphology of the particle (size and shape).
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 4 / 15
Scattering from a dilute solution
The simplest case is for a dilute solution of non-interacting molecules.
Assume that the scattering length density of each identical particle(molecule) is given by ρsl ,p and the scattering length density of the solventis ρsl ,0.
I SAXS( ~Q) =
∣∣∣∣∣∫Vp
ρslei ~Q·~rdVp
∣∣∣∣∣2
= (ρsl ,p − ρsl ,0)2
∣∣∣∣∣∫Vp
e i~Q·~rdVp
∣∣∣∣∣2
If we introduce the single-particleform factor F( ~Q):
F( ~Q) =1
Vp
∫Vp
e i~Q·~rdVp
I SAXS( ~Q) = ∆ρ2V 2p |F( ~Q)|2
Where ∆ρ = (ρsl ,p − ρsl ,0), and the form factor depends on themorphology of the particle (size and shape).
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 4 / 15
Scattering from a sphere
There are only a few morphologies which can be computed exactly and thesimplest is a constant density sphere of radius R.
F( ~Q) =1
Vp
∫ R
0
∫ 2π
0
∫ π
0e iQr cos θr2 sin θ dθ dφ dr
F( ~Q) =1
Vp
∫ R
04π
sin(Qr)
Qrr2 dr
= 3
[sin(QR)− QR cos(QR)
Q3R3
]≡ 3J1(QR)
QR
Where J1(x) is the Bessel functionof the first kind
0
0 5 10
j 1(x
)
x
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 5 / 15
Scattering from a sphere
There are only a few morphologies which can be computed exactly and thesimplest is a constant density sphere of radius R.
F( ~Q) =1
Vp
∫ R
0
∫ 2π
0
∫ π
0e iQr cos θr2 sin θ dθ dφ dr
F( ~Q) =1
Vp
∫ R
04π
sin(Qr)
Qrr2 dr
= 3
[sin(QR)− QR cos(QR)
Q3R3
]≡ 3J1(QR)
QR
Where J1(x) is the Bessel functionof the first kind
0
0 5 10
j 1(x
)
x
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 5 / 15
Scattering from a sphere
There are only a few morphologies which can be computed exactly and thesimplest is a constant density sphere of radius R.
F( ~Q) =1
Vp
∫ R
0
∫ 2π
0
∫ π
0e iQr cos θr2 sin θ dθ dφ dr
F( ~Q) =1
Vp
∫ R
04π
sin(Qr)
Qrr2 dr
= 3
[sin(QR)− QR cos(QR)
Q3R3
]≡ 3J1(QR)
QR
Where J1(x) is the Bessel functionof the first kind
0
0 5 10
j 1(x
)
x
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 5 / 15
Scattering from a sphere
There are only a few morphologies which can be computed exactly and thesimplest is a constant density sphere of radius R.
F( ~Q) =1
Vp
∫ R
0
∫ 2π
0
∫ π
0e iQr cos θr2 sin θ dθ dφ dr
F( ~Q) =1
Vp
∫ R
04π
sin(Qr)
Qrr2 dr
= 3
[sin(QR)− QR cos(QR)
Q3R3
]
≡ 3J1(QR)
QR
Where J1(x) is the Bessel functionof the first kind
0
0 5 10
j 1(x
)
x
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 5 / 15
Scattering from a sphere
There are only a few morphologies which can be computed exactly and thesimplest is a constant density sphere of radius R.
F( ~Q) =1
Vp
∫ R
0
∫ 2π
0
∫ π
0e iQr cos θr2 sin θ dθ dφ dr
F( ~Q) =1
Vp
∫ R
04π
sin(Qr)
Qrr2 dr
= 3
[sin(QR)− QR cos(QR)
Q3R3
]≡ 3J1(QR)
QR
Where J1(x) is the Bessel functionof the first kind
0
0 5 10
j 1(x
)
x
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 5 / 15
Scattering from a sphere
There are only a few morphologies which can be computed exactly and thesimplest is a constant density sphere of radius R.
F( ~Q) =1
Vp
∫ R
0
∫ 2π
0
∫ π
0e iQr cos θr2 sin θ dθ dφ dr
F( ~Q) =1
Vp
∫ R
04π
sin(Qr)
Qrr2 dr
= 3
[sin(QR)− QR cos(QR)
Q3R3
]≡ 3J1(QR)
QR
Where J1(x) is the Bessel functionof the first kind
0
0 5 10
j 1(x
)
x
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 5 / 15
Scattering from a sphere
There are only a few morphologies which can be computed exactly and thesimplest is a constant density sphere of radius R.
F( ~Q) =1
Vp
∫ R
0
∫ 2π
0
∫ π
0e iQr cos θr2 sin θ dθ dφ dr
F( ~Q) =1
Vp
∫ R
04π
sin(Qr)
Qrr2 dr
= 3
[sin(QR)− QR cos(QR)
Q3R3
]≡ 3J1(QR)
QR
Where J1(x) is the Bessel functionof the first kind
0
0 5 10
j 1(x
)
x
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 5 / 15
Scattering from a sphere
I (vectQ) = ∆ρ2V 2p
∣∣∣∣3J1(QR)
QR
∣∣∣∣2
= ∆ρ2V 2p
∣∣∣∣3sin(QR)− QR cos(QR)
Q3R3
∣∣∣∣2
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 6 / 15
Scattering from a sphere
I (vectQ) = ∆ρ2V 2p
∣∣∣∣3J1(QR)
QR
∣∣∣∣2 = ∆ρ2V 2p
∣∣∣∣3sin(QR)− QR cos(QR)
Q3R3
∣∣∣∣2
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 6 / 15
Scattering from a sphere
I (vectQ) = ∆ρ2V 2p
∣∣∣∣3J1(QR)
QR
∣∣∣∣2 = ∆ρ2V 2p
∣∣∣∣3sin(QR)− QR cos(QR)
Q3R3
∣∣∣∣2
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 6 / 15
Scattering from a sphere
I (vectQ) = ∆ρ2V 2p
∣∣∣∣3J1(QR)
QR
∣∣∣∣2 = ∆ρ2V 2p
∣∣∣∣3sin(QR)− QR cos(QR)
Q3R3
∣∣∣∣2
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
0 0.05 0.1 0.15 0.2
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 6 / 15
Guinier analysis
In the long wavelength limit QR → 0 we can approximate the scatteringfactor with the first terms of the sum
F(Q) ≈ 3
Q3R3
[QR − Q3R3
6+
Q5R5
120− · · ·
− QR
(1− Q2R2
2+
Q4R4
24− · · ·
)]this simplifies to F(Q) ≈ 1− Q2R2
10and
I SAXS(Q) ≈ ∆ρ2V 2p
[1− Q2R2
10
]2≈ ∆ρ2V 2
p
[1− Q2R2
5
]≈ ∆ρ2V 2
p e−Q2R2/5, QR � 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 7 / 15
Guinier analysis
In the long wavelength limit QR → 0 we can approximate the scatteringfactor with the first terms of the sum
F(Q) ≈ 3
Q3R3
[QR − Q3R3
6+
Q5R5
120− · · ·
− QR
(1− Q2R2
2+
Q4R4
24− · · ·
)]
this simplifies to F(Q) ≈ 1− Q2R2
10and
I SAXS(Q) ≈ ∆ρ2V 2p
[1− Q2R2
10
]2≈ ∆ρ2V 2
p
[1− Q2R2
5
]≈ ∆ρ2V 2
p e−Q2R2/5, QR � 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 7 / 15
Guinier analysis
In the long wavelength limit QR → 0 we can approximate the scatteringfactor with the first terms of the sum
F(Q) ≈ 3
Q3R3
[QR − Q3R3
6+
Q5R5
120− · · ·
− QR
(1− Q2R2
2+
Q4R4
24− · · ·
)]this simplifies to F(Q) ≈ 1− Q2R2
10and
I SAXS(Q) ≈ ∆ρ2V 2p
[1− Q2R2
10
]2≈ ∆ρ2V 2
p
[1− Q2R2
5
]≈ ∆ρ2V 2
p e−Q2R2/5, QR � 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 7 / 15
Guinier analysis
In the long wavelength limit QR → 0 we can approximate the scatteringfactor with the first terms of the sum
F(Q) ≈ 3
Q3R3
[QR − Q3R3
6+
Q5R5
120− · · ·
− QR
(1− Q2R2
2+
Q4R4
24− · · ·
)]this simplifies to F(Q) ≈ 1− Q2R2
10and
I SAXS(Q) ≈ ∆ρ2V 2p
[1− Q2R2
10
]2
≈ ∆ρ2V 2p
[1− Q2R2
5
]≈ ∆ρ2V 2
p e−Q2R2/5, QR � 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 7 / 15
Guinier analysis
In the long wavelength limit QR → 0 we can approximate the scatteringfactor with the first terms of the sum
F(Q) ≈ 3
Q3R3
[QR − Q3R3
6+
Q5R5
120− · · ·
− QR
(1− Q2R2
2+
Q4R4
24− · · ·
)]this simplifies to F(Q) ≈ 1− Q2R2
10and
I SAXS(Q) ≈ ∆ρ2V 2p
[1− Q2R2
10
]2≈ ∆ρ2V 2
p
[1− Q2R2
5
]
≈ ∆ρ2V 2p e
−Q2R2/5, QR � 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 7 / 15
Guinier analysis
In the long wavelength limit QR → 0 we can approximate the scatteringfactor with the first terms of the sum
F(Q) ≈ 3
Q3R3
[QR − Q3R3
6+
Q5R5
120− · · ·
− QR
(1− Q2R2
2+
Q4R4
24− · · ·
)]this simplifies to F(Q) ≈ 1− Q2R2
10and
I SAXS(Q) ≈ ∆ρ2V 2p
[1− Q2R2
10
]2≈ ∆ρ2V 2
p
[1− Q2R2
5
]≈ ∆ρ2V 2
p e−Q2R2/5, QR � 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 7 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Guinier analysis
In the long wavelength limit(QR → 0), the form factor be-comes
F(Q) ≈ 1− Q2R2
10
I (Q) ≈ ∆ρ2V 2p e
−Q2R2/5
and the initial slope of the log(I ) vsQ2 plot is −R2/5
In terms of the radius of gyration,Rg , which for a sphere is given by√
35R
I (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
0 0.001 0.002
Q2 (Å
-2)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
R=200Å
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 8 / 15
Calculation of Rg
R2g =
1
Vp
∫Vp
r2dVp
=
∫Vpρsl ,p(~r)r2dVp∫
Vpρsl ,p(~r)dVp
In terms of the scattering length den-sity, we have
after orientational averaging this ex-pression can be used to extract Rg fromexperimental data using
I SAXS1 (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
this expression holds for uniform and non-uniform densities
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 9 / 15
Calculation of Rg
R2g =
1
Vp
∫Vp
r2dVp
=
∫Vpρsl ,p(~r)r2dVp∫
Vpρsl ,p(~r)dVp
In terms of the scattering length den-sity, we have
after orientational averaging this ex-pression can be used to extract Rg fromexperimental data using
I SAXS1 (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
this expression holds for uniform and non-uniform densities
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 9 / 15
Calculation of Rg
R2g =
1
Vp
∫Vp
r2dVp
=
∫Vpρsl ,p(~r)r2dVp∫
Vpρsl ,p(~r)dVp
In terms of the scattering length den-sity, we have
after orientational averaging this ex-pression can be used to extract Rg fromexperimental data using
I SAXS1 (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
this expression holds for uniform and non-uniform densities
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 9 / 15
Calculation of Rg
R2g =
1
Vp
∫Vp
r2dVp
=
∫Vpρsl ,p(~r)r2dVp∫
Vpρsl ,p(~r)dVp
In terms of the scattering length den-sity, we have
after orientational averaging this ex-pression can be used to extract Rg fromexperimental data using
I SAXS1 (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
this expression holds for uniform and non-uniform densities
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 9 / 15
Calculation of Rg
R2g =
1
Vp
∫Vp
r2dVp
=
∫Vpρsl ,p(~r)r2dVp∫
Vpρsl ,p(~r)dVp
In terms of the scattering length den-sity, we have
after orientational averaging this ex-pression can be used to extract Rg fromexperimental data using
I SAXS1 (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
this expression holds for uniform and non-uniform densities
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 9 / 15
Calculation of Rg
R2g =
1
Vp
∫Vp
r2dVp
=
∫Vpρsl ,p(~r)r2dVp∫
Vpρsl ,p(~r)dVp
In terms of the scattering length den-sity, we have
after orientational averaging this ex-pression can be used to extract Rg fromexperimental data using
I SAXS1 (Q) ≈ ∆ρ2V 2p e
−Q2R2g/3
this expression holds for uniform and non-uniform densities
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 9 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]
≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]
I (Q) = 9∆ρ2V 2p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2
= 9∆ρ2V 2p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)
I (Q) =2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Porod analysis
In the short wavelength limit(QR � 1), the form factor for asphere can be approximated
F(Q) = 3
[sin(QR)
Q3R3− cos(QR)
Q2R2
]≈ 3
[−cos(QR)
Q2R2
]I (Q) = 9∆ρ2V 2
p
[−cos(QR)
Q2R2
]2= 9∆ρ2V 2
p
⟨cos2(QR)
⟩Q4R4
=9∆ρ2V 2
p
Q4R4
(1
2
)I (Q) =
2π∆ρ2
Q4Sp
10-1.4
10-1.2
10-1.0
10-0.8
Q (Å-1
)
10-6
10-5
10-4
10-3
10-2
10-1
100
I(Q
)
R=100ÅR=200Å
power law drop as Q−4 for spheres
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 10 / 15
Shape effect on scattering
The shape of the particle will have asignificant effect on the SAXS sincethe form factor is derived from anintegral over the particle volume,Vp.
If the particle is not spherical, thenits “dimensionality” is not 3 andthis will affect the form factor andintroduce a different power law inthe Porod regime.
dVp = 4πr2dr sphere
α = 4
dAp = 2πrdr disk
α = 2
dLp = dr rod
α = 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 11 / 15
Shape effect on scattering
The shape of the particle will have asignificant effect on the SAXS sincethe form factor is derived from anintegral over the particle volume,Vp.
If the particle is not spherical, thenits “dimensionality” is not 3 andthis will affect the form factor andintroduce a different power law inthe Porod regime.
dVp = 4πr2dr sphere
α = 4
dAp = 2πrdr disk
α = 2
dLp = dr rod
α = 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 11 / 15
Shape effect on scattering
The shape of the particle will have asignificant effect on the SAXS sincethe form factor is derived from anintegral over the particle volume,Vp.
If the particle is not spherical, thenits “dimensionality” is not 3 andthis will affect the form factor andintroduce a different power law inthe Porod regime.
dVp = 4πr2dr sphere
α = 4
dAp = 2πrdr disk
α = 2
dLp = dr rod
α = 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 11 / 15
Shape effect on scattering
The shape of the particle will have asignificant effect on the SAXS sincethe form factor is derived from anintegral over the particle volume,Vp.
If the particle is not spherical, thenits “dimensionality” is not 3 andthis will affect the form factor andintroduce a different power law inthe Porod regime.
dVp = 4πr2dr sphere
α = 4
dAp = 2πrdr disk
α = 2
dLp = dr rod
α = 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 11 / 15
Shape effect on scattering
The shape of the particle will have asignificant effect on the SAXS sincethe form factor is derived from anintegral over the particle volume,Vp.
If the particle is not spherical, thenits “dimensionality” is not 3 andthis will affect the form factor andintroduce a different power law inthe Porod regime.
dVp = 4πr2dr sphere
α = 4
dAp = 2πrdr disk
α = 2
dLp = dr rod
α = 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 11 / 15
Shape effect on scattering
The shape of the particle will have asignificant effect on the SAXS sincethe form factor is derived from anintegral over the particle volume,Vp.
If the particle is not spherical, thenits “dimensionality” is not 3 andthis will affect the form factor andintroduce a different power law inthe Porod regime.
dVp = 4πr2dr sphere α = 4
dAp = 2πrdr disk
α = 2
dLp = dr rod
α = 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 11 / 15
Shape effect on scattering
The shape of the particle will have asignificant effect on the SAXS sincethe form factor is derived from anintegral over the particle volume,Vp.
If the particle is not spherical, thenits “dimensionality” is not 3 andthis will affect the form factor andintroduce a different power law inthe Porod regime.
dVp = 4πr2dr sphere α = 4
dAp = 2πrdr disk α = 2
dLp = dr rod α = 1
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 11 / 15
Polydispersivity
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 12 / 15
Nucleation & growth of glycine
Can SAXS help us understand the nucleation and growth of a simplemolecule which is the prototype for pharmaceutical compounds?
initial studies at 12 keV show change but no crystallization
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 13 / 15
Glycine nucleation
0.04 0.06 0.08 0.1 0.2
q (Å-1
)
1.4×10-6
1.6×10-6
1.8×10-6
2.0×10-6
Inte
nsity
(ar
b. u
nits
)
0.04 0.06 0.08 0.1 0.2
q (Å-1
)
2.2×10-6
2.4×10-6
2.6×10-6
Inte
nsity
(ar
b. u
nits
)
(a)
(b)
change to 25 keV x-rays
study neutral (top) and acidic (bot-tom) solutions
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 14 / 15
Glycine nucleation
0.04 0.06 0.08 0.1 0.2
q (Å-1
)
1.4×10-6
1.6×10-6
1.8×10-6
2.0×10-6
Inte
nsity
(ar
b. u
nits
)
0.04 0.06 0.08 0.1 0.2
q (Å-1
)
2.2×10-6
2.4×10-6
2.6×10-6
Inte
nsity
(ar
b. u
nits
)
(a)
(b)
change to 25 keV x-rays
study neutral (top) and acidic (bot-tom) solutions
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 14 / 15
Glycine Rg
0 20 40 60 80Time (min)
2.0
3.0
4.0
Rg
(Å)
aqueous solution
acidic solution
in aqueous solution, Rg impliesdimerization and increases due toaggregation until crystallization
in acidic solution, Rg remains smalland implies that no dimerizationor aggregation occurs before nucle-ation
“Relationship between Self-Association of Glycine Molecules in Supersaturated Solution and Solid State Outcome”,D. Erdemir et al. Phys. Rev. Lett. 99, 115702 (2007)
C. Segre (IIT) PHYS 570 - Spring 2015 March 03, 2015 15 / 15