.. _rpa:

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

Random Phase Approximation

========== =============================== ======= =============
Parameter  Description                     Units   Default value
========== =============================== ======= =============
scale      Scale factor or Volume fraction 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.


.. warning:: This model is not functioning correctly in SasView and it
             appears it has not done so for some time. Whilst the
             problem is investigated, a workaround for Case 0 below
             (the most common use case) is to use the binary_blend
             model available on the `Model Maketplace
             <https://marketplace.sasview.org/models/124/>`_ . For further
             information, please email help@sasview.org . *The
             SasView Developers. February 2022.*

**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.* [1] and by
Hammouda [2] 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, 31 and 34, and Part H,
of Hammouda's 'SANS Toolbox' [3].

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 $\rho_{C/D} = [\rho_C - \rho_D]$\ :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.


**Source**

:download:`rpa.py <src/rpa.py>`
$\ \star\ $ :download:`rpa.c <src/rpa.c>`

**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:** Steve King **Date:** March 27, 2019

