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. I(q) = | A(q) | 2 = |∫ exp(-iq r) dr | 2
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-convolution 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
u ––>
p (r) -4 -3 -2 -1 1 2 3 4
r ––>

9. Scattering Theory - Correlation
p (r) = ∫u+ u) du
More:
If we know A(S), then
r) = ∫A(S) exp (-2πi S rk) dS

10. Scattering Theory - Correlation
p (r) = ∫u+ u) du
More:
If we know A(S), then
r) = ∫A(S) exp (-2πi S rk) dS
If we know I(S), then
r) = ∫I(S) exp (-2πi S rk) dS

11. Scattering Theory - Correlation
p (r) = ∫u+ u) du
More:
If we know A(S), then
r) = ∫A(S) exp (-2πi S rk) dS
If we know I(S), then
r) = ∫I(S) exp (-2πi S rk) dS
Note:
Since A(S) is complex, can only get r) from A(S) &
r) from I(S) & inverses…..nothing else

12. 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)