Scattering Theory - Correlation

1
Scattering Theory - Correlation
(Read Roe, section 1.5) Normalized amplitude for
multiple atom types A(q) ??bj exp(-iq
r)
j
2
Scattering Theory - Correlation
(Read Roe, section 1.5) Normalized amplitude for
multiple atom types A(q) ??bj exp(-iq
r) Normalized amplitude for an electron density
distribution A(q) ????? exp(-iq r)
dr ???? be n(r) (no polarization
factor)
j
V
3
Scattering Theory - Correlation
I(q) A(q) 2 ????? exp(-iq r) dr
2 A(q) A(q)
V
4
Scattering Theory - Correlation
I(q) A(q) 2 ????? exp(-iq r) dr
2 A(q) A(q) ???u'? exp(iq u') du'
???u? exp(-iq u) du
V
5
Scattering Theory - Correlation
I(q) A(q) 2 ????? exp(-iq r) dr
2 A(q) A(q) ???u'? exp(iq u') du'
???u? exp(-iq u) du Let u' u r After
some algebra I(q) ????u?? ??u) du exp(-iq
r) dr ??p exp(-iq r) dr
V
6
Scattering Theory - Correlation
?p (r) ???u?? ??u) du
?p (r) is known as ????????????????????????
correlation function pair correlation
function fold of ? into itself self-convolutio
n function pair distribution function radial
distribution function Patterson function
7
Scattering Theory - Correlation
?p (r) ???u?? ??u) du
?p (r) is known as ????????????????????????
correlation function But, whatever it's called,
it represents correlation betwn pair
correlation function ??u') and ??u) on
avg. fold of ? into itself self-convolution
function pair distribution function radial
distribution function Patterson function
8
Scattering Theory - Correlation
?p (r) ???u?? ??u) du
Simple example
??u)
-2
1
2
0
u gt
?p (r)
-3
-4
-2
-1
1
2
3
4
0
r gt
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Scattering Theory - Correlation
?p (r) ???u?? ??u) du
More If we know A(S), then ??r) ?A(S)
exp (-2pi S rk) dS
10
Scattering Theory - Correlation
?p (r) ???u?? ??u) du
More If we know A(S), then ??r) ?A(S)
exp (-2pi S rk) dS If we know I(S), then
??r) ?I(S) exp (-2pi S rk) dS
11
Scattering Theory - Correlation
?p (r) ???u?? ??u) du
More If we know A(S), then ??r) ?A(S)
exp (-2pi S rk) dS If we know I(S),
then ??r) ?I(S) exp (-2pi S rk)
dS Note Since A(S) is complex, can only get
??r) from A(S) ??r) from I(S)
inverses..nothing else
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Scattering Theory - Correlation
Since A(S) is complex, can only get ??r) from
A(S) ??r) from I(S) inverses..nothing
else This means must model in some way to get
??r) from I(S) or ??r) Saxs uses ??r)