Non-particulate 2-phase systems.

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 ltngt
?(r) ?(r) - ltngt
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 lt?gt
?(r) ?(r) - lt?gt Then ??(r) ??(u) ?(u r)
du ??(r) ?(?(u) lt?gt) (?(u r) lt?gt) du
3
Non-particulate 2-phase systems.
deviation from the mean lt?gt ?(r) ?(r) -
lt?gt Then ??(r) ??(u) ?(u r) du ??(r)
?(?(u) lt?gt) (?(u r) lt?gt) du ??(r)
??(u) ?(u r) du lt?gt2?du lt?gt??(u) du
lt?gt??(u r) du
4
Non-particulate 2-phase systems.
deviation from the mean lt?gt ?(r) ?(r) -
lt?gt Then ??(r) ??(u) ?(u r) du ??(r)
?(?(u) lt?gt) (?(u r) lt?gt) du ??(r)
??(u) ?(u r) du lt?gt2?du lt?gt??(u) du
lt?gt??(u r) du
0
0
5
Non-particulate 2-phase systems.
deviation from the mean lt?gt ?(r) ?(r) -
lt?gt Then ??(r) ??(u) ?(u r) du ??(r)
?(?(u) lt?gt) (?(u r) lt?gt) du ??(r)
??(u) ?(u r) du lt?gt2?du lt?gt??(u) du
lt?gt??(u r) du ??(r) ??(r) lt?gt2V
0
0
volume of specimen
6
Non-particulate 2-phase systems.
deviation from the mean lt?gt ?(r) ?(r) -
lt?gt Then ??(r) ??(u) ?(u r) du ??(r)
?(?(u) lt?gt) (?(u r) lt?gt) du ??(r)
??(u) ?(u r) du lt?gt2?du lt?gt??(u) du
lt?gt??(u r) du ??(r) ??(r) lt?gt2V
0
0
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
lt?2gtV Then I(q) lt?2gtV??(r) exp(-iqr) dr
8
Non-particulate 2-phase systems.
I(q) lt?2gtV??(r) exp(-iqr) dr Remember the
invariant (total scattering power of
specimen) Q ?I(S) dS (1/(2p)3)?I(q) dq
total scattering over the whole of
reciprocal space For isotropic material Q
4??S2 I(S) dS (1/(2p2))?q2 I(q) dq (S (2
sin ?)/? q/2?)
8
8
o
o
9
Non-particulate 2-phase systems.
For isotropic material Q 4??S2 I(S) dS
(1/(2p2))?q2 I(q) dq Q ???(r)
(1/(2p2))?exp(-iqr) dq dr ??(0) lt?2gtV
8
8
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 lt?gt ?1 ?1????2 ?2 ?1
?1 - lt?gt ????2 ?2 ?2 - lt?gt ????1 ??
?1 - ?2 ?1 ?2
13
Non-particulate 2-phase systems.
Ideal 2-phase system lt?gt ?1 ?1????2 ?2 ?1
?1 - lt?gt ????2 ?2 ?2 - lt?gt ????1 ??
?1 - ?2 ?1 ?2 Q lt?2gtV V
(??????1?2 if ?1, ?2 known, Q --gt ?1, ?2
if ?1, ?2 known, Q --gt ?1 - ?2
14
Non-particulate 2-phase systems.
Ideal 2-phase system Most important result -
Porod law I(q) --gt (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) --gt (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) --gt (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 --gt 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)q2
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