1. Non-particulate 2-phase systems.
(See Roe Sects 5.3, 1.6)
Remember:
I(q) = ∫(r) exp(-iqr) dr
In this form, integral does not converge. So
deviation from the mean <n> = (r) = (r) - <n>

2. Non-particulate 2-phase systems.
(See Roe Sects 5.3, 1.6)
Remember:
I(q) = (r) exp(-iqr) dr
In this form, integral does not converge. So
deviation from the mean <> = (r) = (r) - <>
Then
(r) =∫(u) (u + r) du
(r) =∫((u) + <>) ((u + r) + <>) du

3. Non-particulate 2-phase systems.
deviation from the mean <> = (r) = (r) - <>
Then
(r) =∫(u) (u + r) du
(r) =∫((u) + <>) ((u + r) + <>) du
(r) =∫(u) (u + r) du + <>2∫du + <>∫(u) du
+ <>∫(u + r) du

4. Non-particulate 2-phase systems.
deviation from the mean <> = (r) = (r) - <>
Then
(r) =∫(u) (u + r) du
(r) =∫((u) + <>) ((u + r) + <>) du
(r) =∫(u) (u + r) du + <>2∫du + <>∫(u) du
+ <>∫(u + r) du

5. Non-particulate 2-phase systems.
deviation from the mean <> = (r) = (r) - <>
Then
(r) =∫(u) (u + r) du
(r) =∫((u) + <>) ((u + r) + <>) du
(r) =∫(u) (u + r) du + <>2∫du + <>∫(u) du
+ <>∫(u + r) du
(r) = (r) + <>2V
volume of specimen

6. Non-particulate 2-phase systems.
deviation from the mean <> = (r) = (r) - <>
Then
(r) =∫(u) (u + r) du
(r) =∫((u) + <>) ((u + r) + <>) du
(r) =∫(u) (u + r) du + <>2∫du + <>∫(u) du
+ <>∫(u + r) du
(r) = (r) + <>2V
leads to null scattering

7. Non-particulate 2-phase systems.
(r) = (r)
Finally
I(q) = ∫(r) exp(-iqr) dr
Normalization of (r):
(r) = (r)/ (0)
where
(0) =∫(u) (u + 0) du = <2>V
Then:
I(q) = <2>V∫(r) exp(-iqr) dr

8. Non-particulate 2-phase systems.
I(q) = <2>V∫(r) exp(-iqr) dr
Remember the invariant (total scattering power of specimen):
Q =∫I(S) dS = (1/(2π)3)∫I(q) dq
= total scattering over the whole of
reciprocal space
For isotropic material:
Q = 4∫S2 I(S) dS = (1/(2π2))∫q2 I(q) dq
(S = (2 sin )/ = q/2) ∞ ∞ o o

9. Non-particulate 2-phase systems.
For isotropic material:
Q = 4∫S2 I(S) dS = (1/(2π2))∫q2 I(q) dq
Q = ∫(r) [(1/(2π2))∫exp(-iqr) dq ] dr = (0) = <2>V ∞ ∞ o o

10. Non-particulate 2-phase systems.
Now for the model -
Ideal 2-phase system:
only 2 regions or phases
each phase has constant scattering length density
sharp phase boundary
randomly mixed
isotropic

11. Non-particulate 2-phase systems.
Now for the model -
Ideal 2-phase system:
only 2 regions or phases
each phase has constant scattering length density
sharp phase boundary
randomly mixed
isotropic
Can get:
volume fractionsi of each phase & interface area

12. Non-particulate 2-phase systems.
Ideal 2-phase system:
<> = 1 12 2
1 = 1 - <> = 2
2 = 2 - <> = 1
 = 1 - 2 = 1 + 2 1 2
<> 1 2

13. Non-particulate 2-phase systems.
Ideal 2-phase system:
<> = 1 12 2
1 = 1 - <> = 2
2 = 2 - <> = 1
 = 1 - 2 = 1 + 2
Q = <2>V = V (12
if 1, 2 known, Q --> 1, 2
if 1, 2 known, Q --> 1 - 2 1 2
<> 1 2

14. Non-particulate 2-phase systems.
Ideal 2-phase system:
Most important result - Porod law:
I(q) --> (2()2 S)/q4 at large q
S = total boundary area betwn 2 phases
log I(q) = const – 4 x log q

15. Non-particulate 2-phase systems.
Ideal 2-phase system:
Most important result - Porod law:
I(q) --> (2()2 S)/q4 at large q
S = total boundary area betwn 2 phases
log I(q) = const – 4 x log q

16. Non-particulate 2-phase systems.
Ideal 2-phase system:
Most important result - Porod law:
I(q) --> (2()2 S)/q4 at large q
S = total boundary area betwn 2 phases
log I(q) = const – 4 x log q
Need absolute Is
If relative Is, can get
invariant Q --> S/V
I(q)/Q = (2S)/(1 2Vq4)

17. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Ideal 2-phase system:
only 2 regions or phases
each phase has constant scattering length density
sharp phase boundary
randomly mixed
isotropic

18. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Ideal 2-phase system:
only 2 regions or phases
each phase has constant scattering length density
sharp phase boundary
randomly mixed
isotropic
Density fluctuations (from thermal motion):
I(q) = Io(q) + I1(q) + I2(q) + I12(q)
uniform density density fluctuations interaction betwn phases

19. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
I(q) = Io(q) + I1(q) + I2(q) + I12(q)
uniform density density fluctuations interaction betwn phases
= intensity from pure phases short range correlations -
weighted by vol %s small, neglected

20. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
I(q) = Io(q) + I1(q) + I2(q) + I12(q)
uniform density density fluctuations interaction betwn phases
= intensity from pure phases short range correlations -
weighted by vol %s small, neglected
To correct, then, measure scattering from the 2 pure phases

21. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Al otro lado:
can make empirical correction for I1(q) + I2(q)
intensity
high angle region q

22. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Diffuse boundary betwn phase regions:
let (r) = scattering length density in 2-phase matl w/
diffuse boundaries
id(r) = scattering length density in the same 2-phase matl w/ sharp boundaries (hypothetical)

23. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Diffuse boundary betwn phase regions:
let (r) = scattering length density in 2-phase matl w/
diffuse boundaries
id(r) = scattering length density in the same 2-phase matl w/ sharp boundaries (hypothetical)
Then
(r) = id(r) * g(r)
g(r) = fcn which characterizes diffuse boundaries

24. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Diffuse boundary betwn phase regions:
let (r) = scattering length density in 2-phase matl w/
diffuse boundaries
id(r) = scattering length density in the same 2-phase matl w/ sharp boundaries (hypothetical)
Then
(r) = id(r) * g(r)
g(r) = fcn which characterizes diffuse boundaries
I(q) = Iid(q) G2(q) G(q) = Fourier transform of g(r)
(Fourier transform of convolution of 2 fcns = product of their transforms)

25. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Diffuse boundary betwn phase regions:
(r) = id(r) * g(r)
g(r) = fcn which characterizes diffuse boundaries
I(q) = Iid(q) G2(q) G(q) = Fourier transform of g(r)
(Fourier transform of convolution of 2 fcns = product of their transforms)
Common:
G(q) = exp(–2/2)q & for small bdy width ():
I(q) = Iid(q) (1 - 2 q2)

26. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Diffuse boundary betwn phase regions:
I(q) = Iid(q) (1 - 2 q2)
q4 I(q) = (2()2 S) - (2()2 S) (2 q2)
q4 I(q) q2

27. Non-particulate 2-phase systems.
Deviations from ideal 2-phase system
Diffuse boundary betwn phase regions: