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Understanding Scattering Theory: Correlation Functions and Amplitude Normalization

This document examines scattering theory with a focus on correlation functions and normalized amplitudes for various atom types. It dives into the mathematical representation of scattering, including the normalized amplitude, A(q), and the correlation function, Γp(r). The interrelations between these functions are explored, revealing insights into electron density distributions and the complexities of the scattering process. Key examples illustrate autocorrelation functions and introduce pair distribution and normalization concepts, enhancing the understanding of scattering phenomena in various materials.

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Understanding Scattering Theory: Correlation Functions and Amplitude Normalization

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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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