.. _rpa:

rpa
=======================================================

Random Phase Approximation

========== ========================= ======= =============
Parameter  Description               Units   Default value
========== ========================= ======= =============
scale      Source intensity          None                1
background Source background         |cm^-1|         0.001
case_num   Component organization    None                1
N[4]       Degree of polymerization  None             1000
Phi[4]     volume fraction           None             0.25
v[4]       molar volume              mL/mol            100
L[4]       scattering length         fm                 10
b[4]       segment length            |Ang|               5
K12        A:B interaction parameter None          -0.0004
K13        A:C interaction parameter None          -0.0004
K14        A:D interaction parameter None          -0.0004
K23        B:C interaction parameter None          -0.0004
K24        B:D interaction parameter None          -0.0004
K34        C:D interaction parameter None          -0.0004
========== ========================= ======= =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


**Definition**

Calculates the macroscopic scattering intensity for a multi-component
homogeneous mixture of polymers using the Random Phase Approximation.
This general formalism contains 10 specific cases

Case 0: C/D binary mixture of homopolymers

Case 1: C-D diblock copolymer

Case 2: B/C/D ternary mixture of homopolymers

Case 3: C/C-D mixture of a homopolymer B and a diblock copolymer C-D

Case 4: B-C-D triblock copolymer

Case 5: A/B/C/D quaternary mixture of homopolymers

Case 6: A/B/C-D mixture of two homopolymers A/B and a diblock C-D

Case 7: A/B-C-D mixture of a homopolymer A and a triblock B-C-D

Case 8: A-B/C-D mixture of two diblock copolymers A-B and C-D

Case 9: A-B-C-D tetra-block copolymer

.. note::
    These case numbers are different from those in the NIST SANS package!

The models are based on the papers by Akcasu *et al.* and by
Hammouda assuming the polymer follows Gaussian statistics such
that $R_g^2 = n b^2/6$ where $b$ is the statistical segment length and $n$ is
the number of statistical segment lengths. A nice tutorial on how these are
constructed and implemented can be found in chapters 28 and 39 of Boualem
Hammouda's 'SANS Toolbox'.

In brief the macroscopic cross sections are derived from the general forms
for homopolymer scattering and the multiblock cross-terms while the inter
polymer cross terms are described in the usual way by the $\chi$ parameter.

USAGE NOTES:

* Only one case can be used at any one time.
* The RPA (mean field) formalism only applies only when the multicomponent
  polymer mixture is in the homogeneous mixed-phase region.
* **Component D is assumed to be the "background" component (ie, all contrasts
  are calculated with respect to component D).** So the scattering contrast
  for a C/D blend = [SLD(component C) - SLD(component D)]\ :sup:`2`.
* Depending on which case is being used, the number of fitting parameters can
  vary.

  .. Note::
    * In general the degrees of polymerization, the volume
      fractions, the molar volumes, and the neutron scattering lengths for each
      component are obtained from other methods and held fixed while The *scale*
      parameter should be held equal to unity.
    * The variables are normally the segment lengths ($b_a$, $b_b$,
      etc.) and $\chi$ parameters ($K_{ab}$, $K_{ac}$, etc).


.. figure:: img/rpa_autogenfig.png

    1D plot corresponding to the default parameters of the model.

**References**

A Z Akcasu, R Klein and B Hammouda, *Macromolecules*, 26 (1993) 4136.

B. Hammouda, *Advances in Polymer Science* 106 (1993) 87.

B. Hammouda, *SANS Toolbox*
https://www.ncnr.nist.gov/staff/hammouda/the_sans_toolbox.pdf.

**Authorship and Verification**

* **Author:** Boualem Hammouda - NIST IGOR/DANSE **Date:** pre 2010
* **Converted to sasmodels by:** Paul Kienzle **Date:** July 18, 2016
* **Last Modified by:** Paul Butler **Date:** March 12, 2017
* **Last Reviewed by:** Paul Butler **Date:** March 12, 2017

