Chapter 32 – STRUCTURE FACTORS FOR PARTICULATE SYSTEMS 32:1. THE ORNSTEIN-ZERNIKE EQUATION 32:2. THE PERCUS-YEVICK APPROXIMATION 32:3. THE MEAN SPHERICAL.

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Presentation transcript:

Chapter 32 – STRUCTURE FACTORS FOR PARTICULATE SYSTEMS 32:1. THE ORNSTEIN-ZERNIKE EQUATION 32:2. THE PERCUS-YEVICK APPROXIMATION 32:3. THE MEAN SPHERICAL APPROXIMATION

The OZ equation: 32:1. THE ORNSTEIN-ZERNIKE EQUATION c(r) h(r) U(r) r Hard Sphere Screened Coulomb Square Well Take Fourier transform: Inter-particle structure factor: Interaction potentials:

32:2. THE PERCUS-YEVICK APPROXIMATION. PY closure relation: Hard-sphere interaction potential: PY solution of the OZ equation: Parameters: Result: U(r) r Hard Sphere

Inter-particle structure factor: Scattering cross section: RESULT

FORM FACTOR P(Q) AND STRUCTURE FACTOR S I (Q)

32:3. THE MEAN SPHERICAL APPROXIMATION MSA closure relation: MSA solution of the OZ equation: Screened Coulomb interaction potential: Take Fourier transform… to obtain tedious result… not reproduced here U(r) r Screened Coulomb

Inter-particle structure factor: Scattering cross section: RESULT

COMMENTS -- The OZ equation along with one of the closure relations (the PY or the MSA for charged systems) is used to calculate the structure factor for scattering particles. -- Structure factors are needed to mode the cross section for concentrated systems.