Periodic systems.

1
Periodic systems.
(See Roe Sect 5.5) Repeat periods of 10-1000
Å Degree of order usually low gt smears
reciprocal lattice spots Ordered domain size
frequently small gt smears reciprocal lattice
spots
2
Periodic systems.
(See Roe Sect 5.5) Repeat periods of 10-1000
Å Degree of order usually low gt smears
reciprocal lattice spots Ordered domain size
frequently small gt smears reciprocal lattice
spots Extracting info from saxs pattern a.
periodic character - use std high angle
techniques b. disorder - analysis more complex,
similar to previous saxs discussions
3
Periodic systems.
Repeat periods of 10-1000 Å Degree of order
usually low gt smears reciprocal lattice
spots Ordered domain size frequently small gt
smears reciprocal lattice spots Extracting info
from saxs pattern a. periodic character - use
std high angle techniques b. disorder - analysis
more complex, similar to previous saxs
discussions
4
Periodic systems.
Repeat periods of 10-1000 Å Degree of order
usually low gt smears reciprocal lattice
spots Ordered domain size frequently small gt
smears reciprocal lattice spots Extracting info
from saxs pattern a. periodic character - use
std high angle techniques b. disorder - analysis
more complex, similar to previous saxs
discussions 1st ?(r) ?u(r)
z(r) ?u(r) scattering length density for
single repeated motif z(r) fcn which describes
ordering or periodicity
5
Periodic systems.
1st ?(r) ?u(r) z(r) ?u(r) scattering
length density for single repeated motif z(r)
fcn which describes ordering or
periodicity and I(q) F(q)2
Z(q)2 (Fourier transform of convolution of 2
fcns product of their transforms) Z(q)
describes the reciprocal lattice F(q) is the
"structure factor"
6
Periodic systems.
Consider lamellar morphology (membranes,
folded-chain crystallites, etc.) If one stack
of well-ordered lamellae saxs gt array of
points along line in reciprocal space If many
randomly oriented stacks saxs gt isotropic
pattern (rings)
7
Periodic systems.
Ideal 2-phase lamellar structure ?(x)
?u(x) z(x/d) z(x) ???(x - n)
da
db
?a
?b
d
n8
n-8
8
Periodic systems.
Ideal 2-phase lamellar structure ?(x)
?u(x) z(x/d) z(x) ???(x - n) I(q)
F(q)2 Z(q)2 gt I(q) ? F(q)2
z(dq/2?) F(q)2 4(??/q)2 sin2 (daq/2)
da
db
?a
?b
d
n8
n-8
9
Periodic systems.
Ideal 2-phase lamellar structure ?(x) ?u(x)
z(x/d) z(x) ???(x - n) I(q) F(q)2
Z(q)2 gt I(q) ? F(q)2 z(dq/2?) F(q)2
4(??/q)2 sin2 (daq/2) Result (Fourier
transform)2 of z(x/d) is reciprocal lattice w/
period 2?/d Sharp Bragg-type peaks occur at q
2?n/d w/ intensity (n)-2sin2 (?n?a) (can get
volume fractions of phases from intensities)
n8
n-8
10
Periodic systems.
Ideal 2-phase lamellar structure Result (Fourie
r transform)2 of z(x/d) is reciprocal lattice w/
period 2?/d Sharp Bragg-type peaks occur at q
2?n/d w/ intensity (n)-2sin2 (?n?a) (can get
volume fractions of phases from
intensities) Peaks are ?-fcn sharp if
structure is ideal Non-ideal structures gt peak
broadening
11
Periodic systems.
2-phase structure w/ variable thickness
lamellae lamellae parallel lamellae thickness
varies no correlation in thickness of
neighboring lamellae
12
Periodic systems.
2-phase structure w/ variable thickness
lamellae lamellae parallel lamellae thickness
varies no correlation in thickness of
neighboring lamellae probability of thickness of
A lamellae betwn a a da is pa(a)
da similar for B lamellae Then, as
before A(q) ? Aj(q)
N
j1
13
Periodic systems.
2-phase structure w/ variable thickness
lamellae Then, as before A(q) ?
Aj(q) Aj(q) ????exp (-iqx) dx (???iq) exp
(-iqxj)(1 exp (-iqaj))
N
j1
xjaj
xj
14
Periodic systems.
2-phase structure w/ variable thickness
lamellae Then, as before A(q) ?
Aj(q) Aj(q) ????exp (-iqx) dx (???iq) exp
(-iqxj)(1 exp (-iqaj)) I(q) ? Aj ?
Ak I(q) ? Aj Aj ? ? ?Aj Ajm Ajm Aj)
(N very large)
N
j1
xjaj
xj
N
N
j1
k1
N
N
N
j1
m1
j1
15
Periodic systems.
2-phase structure w/ variable thickness
lamellae Then, as before A(q) ?
Aj(q) Aj(q) ????exp (-iqx) dx (???iq) exp
(-iqxj)(1 exp (-iqaj)) I(q) ? Aj ?
Ak I(q) ? Aj Aj ? ? ?Aj Ajm Ajm Aj)
(N very large) ? Aj Aj (??/q)2 ??(2 exp
(-iqaj) exp (iqaj)) ? Aj Aj (??/q)2 N?(2
ltexp (-iqaj)gt ltexp (iqaj)gt)
N
j1
xjaj
xj
N
N
j1
k1
N
N
N
j1
m1
j1
16
Periodic systems.
2-phase structure w/ variable thickness
lamellae Now ltexp (-iqaj)gt ? exp (-iqaj)
pa(a) da Pa(q) Fourier transform
of pa(a) After further similar
manipulations I(q) 2N (??/q)2 Re (1 Pa) (1
Pb)/ (1 Pa Pb) If pa(a) known or assumed, can
calc I(q)
8
-8
17
Periodic systems.
2-phase structure w/ variable thickness
lamellae Example Suppose pa, pb Gaussian
I 1/(n4 (?a2 ?b2))
18
Periodic systems.
2-phase lamellar structures - correlation
functions Can get ?obs from Fourier transform of
I(q) compare with ?model(x) ??(u) ?(u x)
du ?a ?b - lt?gt ????b ?b ?b - lt?gt
????a
19
Periodic systems.
2-phase lamellar structures - correlation
functions Can get ?obs from Fourier transform of
I(q) compare with ?model(x) ??(u) ?(u x)
du If thicknesses of lamellae vary (Gaussian)
smeared
20
Periodic systems.
2-phase lamellar structures - correlation
functions Can get ?obs from Fourier transform of
I(q) compare with ?model(x) ??(u) ?(u x)
du If lamellae/lamellae transitions not sharp,
self correlation peak rounds
smeared