Scattering Theory - Correlation (Read Roe, section 1.5) Normalized amplitude for multiple atom types: A(q) = bj exp(-iq r) j

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

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

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

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

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

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

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 ––>

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

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

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

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)