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  Å  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 multicomponent 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: CD diblock copolymer
Case 2: B/C/D ternary mixture of homopolymers
Case 3: C/CD mixture of a homopolymer B and a diblock copolymer CD
Case 4: BCD triblock copolymer
Case 5: A/B/C/D quaternary mixture of homopolymers
Case 6: A/B/CD mixture of two homopolymers A/B and a diblock CD
Case 7: A/BCD mixture of a homopolymer A and a triblock BCD
Case 8: AB/CD mixture of two diblock copolymers AB and CD
Case 9: ABCD tetrablock 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 and 39 of Boualem Hammouda’s ‘SANS Toolbox’[3].
In brief the macroscopic cross sections are derived from the general forms for homopolymer scattering and the multiblock crossterms 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 mixedphase 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)]^{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).
References
[1]  A Z Akcasu, R Klein and B Hammouda, Macromolecules, 26 (1993) 4136. 
[2] 

[3]  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