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Correlation Functions in Scattering Theory

Exploring the normalized amplitude and correlation functions in scattering theory as per Roe's section 1.5, including auto-correlation function and pair distribution function. Learn to relate amplitude and correlation using Fourier transforms.

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Correlation Functions in Scattering Theory

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

  2. Scattering Theory - Correlation (Read Roe, section 1.5) Normalized amplitude for multiple atom types: A(q) = bj exp(-iqr) Normalized amplitude for an electron density distribution A(q) = ∫ exp(-iqr) dr  = be n(r) (no polarization factor) j V

  3. Scattering Theory - Correlation I(q) = | A(q) | 2 = |∫ exp(-iqr) dr|2 = A*(q) A(q) V

  4. Scattering Theory - Correlation I(q) = | A(q) | 2 = |∫ exp(-iqr) dr|2 = A*(q) A(q) = [∫u' exp(iqu') du' ][∫u exp(-iqu) du] V

  5. Scattering Theory - Correlation I(q) = | A(q) | 2 = |∫ exp(-iqr) dr|2 = A*(q) A(q) = [∫u' exp(iqu') du' ][∫u exp(-iqu) du] Let u' = u + rAfter some algebra: I(q) = ∫[∫u+ u) du] exp(-iqr) dr = ∫p exp(-iqr) 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 2 1 0 u––> p (r) -3 -4 -2 -1 2 3 4 1 0 r––>

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

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

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

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