Guinier law (interpretation w/o a model)

1
Guinier law (interpretation w/o a model)
Regardless of particle shape, at small q
I(q) (?v)2 exp (-qRg)2/3
2
Guinier law (interpretation w/o a model)
Regardless of particle shape, at small q
I(q) (?v)2 exp (-qRg)2/3 ln (q) ln
(?v)2 - (Rg2/3) q2
3
Guinier law (interpretation w/o a model)
Regardless of particle shape, at small q
I(q) (?v)2 exp (-qRg)2/3 ln (q) ln
(?v)2 - (Rg2/3) q2
4
Guinier law (interpretation w/o a model)
Regardless of particle shape, at small q
I(q) (?v)2 exp (-qRg)2/3 ln (q) ln
(?v)2 - (Rg2/3) q2 Only holds if a. q lt
1/Rg b. dilute c. Isotropic d. matrix
or solvent scattering is removed
5
Guinier law (outline of derivation)
?(r) is scattering length distribution A(q)
??(r) exp (-iqr) dr Expand as a power
series A(q) ??(r) dr - i?qr ?(r) dr -
(1/2!)?(qr)2 ?(r) dr Origin at center of
mass
6
Guinier law (outline of derivation)
?(r) is scattering length distribution A(q)
??(r) exp (-iqr) dr Expand as a power
series A(q) ??(r) dr - i?qr ?(r) dr -
(1/2!)?(qr)2 ?(r) dr Origin at center of
mass (qr)2 (qxx qyy qzz)2 xy (1/
?v)?xy ?(r) dr , etc
?v
0
7
Guinier law (outline of derivation)
q ltlt ?v average intensity/particle, for large
randomly oriented particles I(q) (?v)2(1-
((qxx)2 (qyy)2 (qzz)2 2qxqyxy) Isotropic
average x2 average y2 average z2
Rg2/3 average xy average yz average zx
0 I(q) (?v)2(1- ((q Rg)2 /3 ) I(q)
(?v)2 exp (-qRg)2/3
8
Guinier law (outline of derivation)
For non-identical particles, Guinier law gives an
average R average v
9
Model structures - effect of dense packing
For N spherical particles, radius R, scattering
length density ? A(q) ??A1(q) ?????-iqrj) Rj
location of center of jth sphere, A1(q) form
factor for single sphere
N
j1
10
Model structures - effect of dense packing
For N spherical particles, radius R, scattering
length density ? A(q) ??A1(q) ?????-iqrj) Rj
location of center of jth sphere, A1(q) form
factor for single sphere I(q) I1(q)???
?????-iqrjk) I(q) I1(q)(N ??? ?????-iqrjk))
N
j1
N
N
j1
k1
N
N
j1
k?1
11
Model structures - effect of dense packing
For N spherical particles, radius R, scattering
length density ? A(q) ??A1(q) ?????-iqrj) Rj
location of center of jth sphere, A1(q) form
factor for single sphere I(q) I1(q)???
?????-iqrjk) I(q) I1(q)(N ??? ?????-iqrjk))
N
j1
N
N
j1
k1
N
N
j1
k?1
independent scattering from particles
correlated scattering betwn particles
12
Model structures - effect of dense packing
For N spherical particles, radius R, scattering
length density ? A(q) ??A1(q) ?????-iqrj) Rj
location of center of jth sphere, A1(q) form
factor for single sphere I(q) I1(q)???
?????-iqrjk) I(q) I1(q)(N ???
?????-iqrjk)) ltngt g(r) dr probability of
finding another particle in dr at distance r
from a given particle (ltngt average
density of particles) I(q) N I1(q)(1 ltngt
?g(r) ?????-iqr)dr) Or I(q) N I1(q)(1
ltngt?(g(r) - 1) ?????-iqr)dr)
N
j1
N
N
j1
k1
N
N
j1
k?1
13
Model structures - effect of dense packing
ltngt g(r) dr probability of finding another
particle in dr at distance r from a given
particle (ltngt average density of
particles) I(q) N I1(q)(1 ltngt ?g(r)
?????-iqr)dr) Or I(q) N I1(q)(1 ltngt?(g(r)
- 1) ?????-iqr)dr) Isotropic I(q) N
I1(q)(1 ltngt?4?r2(g(r) - 1) ?sin (qr))/(qr)
dr) One result as fraction of volume occupied
by spheres ltngtv ?????????
????8
0
14
Model structures - effect of dense packing
One result as fraction of volume occupied by
spheres ltngtv ?????????, low q intensity is
suppressed.
lo ltngtv
I(q)/N (?v)2
hi ltngtv
q
15
Model structures - effect of dense packing
Anisotropic particles give similar result, altho
more complicated.