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Chapter 32 – STRUCTURE FACTORS FOR PARTICULATE SYSTEMS - PowerPoint PPT Presentation

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. h(r). c(r). Hard Sphere. Screened Coulomb. U(r). Square Well. r. 32:1. THE ORNSTEIN-ZERNIKE EQUATION.

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PARTICULATE SYSTEMS

32:1. THE ORNSTEIN-ZERNIKE EQUATION

32:2. THE PERCUS-YEVICK APPROXIMATION

32:3. THE MEAN SPHERICAL APPROXIMATION

c(r)

Hard Sphere

Screened Coulomb

U(r)

Square Well

r

32:1. THE ORNSTEIN-ZERNIKE EQUATION

The OZ equation:

Take Fourier transform:

Inter-particle structure factor:

Interaction potentials:

U(r)

r

32:2. THE PERCUS-YEVICK APPROXIMATION

PY closure relation:

Hard-sphere interaction potential:

PY solution of the OZ equation:

.

Parameters:

Result:

Inter-particle structure factor:

Scattering cross section:

U(r)

r

32:3. THE MEAN SPHERICAL APPROXIMATION

Screened Coulomb interaction potential:

MSA closure relation:

MSA solution of the OZ equation:

Take Fourier transform… to obtain tedious result… not reproduced here

Inter-particle structure factor:

Scattering cross section:

-- 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.