Changeset caddb14 in sasmodels


Ignore:
Timestamp:
Oct 1, 2016 9:48:49 AM (5 years ago)
Author:
ajj
Branches:
master, core_shell_microgels, costrafo411, magnetic_model, release_v0.94, release_v0.95, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
69ef533
Parents:
a807206
Message:

Updated and re-parameterised Teubner-Strey model.

Closes #472
Closes #530

File:
1 edited

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  • sasmodels/models/teubner_strey.py

    r40a87fa rcaddb14  
    55This model calculates the scattered intensity of a two-component system 
    66using the Teubner-Strey model. Unlike :ref:`dab` this function generates 
    7 a peak. 
     7a peak. A two-phase material can be characterised by two length scales - 
     8a correlation length and a domain size (periodicity). 
     9 
     10The original paper by Teubner and Strey defined the function as: 
    811 
    912.. math:: 
    1013 
    11     I(q) = \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background} 
     14    I(q) \propto \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background} 
    1215 
    13 The parameters $a_2$, $c_1$ and $c_2$ can be used to determine the 
    14 characteristic domain size $d$, 
     16where the parameters $a_2$, $c_1$ and $c_2$ are defined in terms of the 
     17periodicity, $d$, and correlation length $\xi$ as: 
     18 
     19.. math:: 
     20 
     21    a_2 &= \biggl[1+\bigl(\frac{2\pi\xi}{d}\bigr)^2\biggr]\\ 
     22    c_1 &= -2\xi^2\bigl(\frac{2\pi\xi}{d}\bigr)^2+2\xi^2\\ 
     23    c_2 &= \xi^4 
     24 
     25and thus, the periodicity, $d$ is given by 
    1526 
    1627.. math:: 
    1728 
    1829    d = 2\pi\left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} 
    19                   + \frac14\frac{c_1}{c_2}\right]^{-1/2} 
     30                  - \frac14\frac{c_1}{c_2}\right]^{-1/2} 
    2031 
    21  
    22 and the correlation length $\xi$, 
     32and the correlation length, $\xi$, is given by 
    2333 
    2434.. math:: 
    2535 
    2636    \xi = \left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} 
    27                   - \frac14\frac{c_1}{c_2}\right]^{-1/2} 
     37                  + \frac14\frac{c_1}{c_2}\right]^{-1/2} 
    2838 
     39Here the model is parameterised in terms of  $d$ and $\xi$ and with an explicit 
     40volume fraction for one phase, $\phi_a$, and contrast, 
     41$\delta\rho^2 = (\rho_a - \rho_b)^2$ : 
     42 
     43.. math:: 
     44 
     45    I(q) = \frac{8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi} 
     46        {a_2 + c_1q^2 + c_2q^4} 
     47 
     48where :math:`8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi` is the constant of 
     49proportionality from the first equation above. 
     50 
     51In the case of a microemulsion, $a_2 > 0$, $c_1 < 0$, and $c_2 >0$. 
    2952 
    3053For 2D data, scattering intensity is calculated in the same way as 1D, 
     
    3457 
    3558    q = \sqrt{q_x^2 + q_y^2} 
    36  
    3759 
    3860References 
     
    4466*J. Chem. Phys.*, 101 (1994) 5343 
    4567 
     68H Endo, M Mihailescu, M. Monkenbusch, J Allgaier, G Gompper, D Richter, 
     69B Jakobs, T Sottmann, R Strey, and I Grillo, *J. Chem. Phys.*, 115 (2001), 580 
    4670""" 
    4771 
    4872import numpy as np 
    49 from numpy import inf 
     73from numpy import inf,power,pi 
    5074 
    5175name = "teubner_strey" 
    5276title = "Teubner-Strey model of microemulsions" 
    5377description = """\ 
    54    Scattering model class for the Teubner-Strey model given by 
    55     Provide F(x) = 1/( a2 + c1 q^2+  c2 q^4 ) + background 
    56     a2>0, c1<0, c2>0, 4 a2 c2 - c1^2 > 0 
     78    Calculates scattering according to the Teubner-Strey model 
    5779""" 
    5880category = "shape-independent" 
     
    6082#   ["name", "units", default, [lower, upper], "type","description"], 
    6183parameters = [ 
    62     ["a2", "", 0.1, [0, inf], "", "a2"], 
    63     ["c1", "1e-6/Ang^2", -30., [-inf, 0], "", "c1"], 
    64     ["c2", "Ang", 5000., [0, inf], "volume", "c2"], 
     84    ["volfraction_a", "", 0.5, [0, 1.0], "", "Volume fraction of phase a"], 
     85    ["sld_a", "1e-6/Ang^2", 0.3, [-inf, inf], "", "SLD of phase a"], 
     86    ["sld_b", "1e-6/Ang^2", 6.3, [-inf, inf], "", "SLD of phase b"], 
     87    ["d", "Ang", 100.0, [0, inf], "", "Domain size (periodicity)"], 
     88    ["xi", "Ang", 30.0, [0, inf], "", "Correlation length"], 
    6589    ] 
    6690 
    67 def Iq(q, a2, c1, c2): 
     91def Iq(q, volfraction, sld, sld_solvent,d,xi): 
    6892    """SAS form""" 
    69     return 1. / np.polyval([c2, c1, a2], q**2) 
     93    drho2 = (sld-sld_solvent)*(sld-sld_solvent) 
     94    a2 = power(1.0+power(2.0*pi*xi/d,2.0),2.0) 
     95    c1 = -2.0*xi*xi*power(2.0*pi*xi/d,2.0)+2*xi*xi 
     96    c2 = power(xi,4.0) 
     97    prefactor = 8.0*pi*volfraction*(1.0-volfraction)*drho2*c2/xi 
     98    #k2 = (2.0*pi/d)*(2.0*pi/d) 
     99    #xi2 = 1/(xi*xi) 
     100    #q2 = q*q 
     101    #result = prefactor/((xi2+k2)*(xi2+k2)+2.0*(xi2-k2)*q2+q2*q2) 
     102    return 1.0e-4*prefactor / np.polyval([c2, c1, a2], q**2) 
     103 
    70104Iq.vectorized = True  # Iq accepts an array of q values 
    71105 
    72 demo = dict(scale=1, background=0, a2=0.1, c1=-30.0, c2=5000.0) 
    73 tests = [[{}, 0.2, 0.145927536232]] 
     106demo = dict(scale=1, background=0, volfraction_a=0.5, 
     107                     sld_a=0.3, sld_b=6.3, 
     108                     d=100.0, xi=30.0) 
     109tests = [[{}, 0.06, 41.5918888453]] 
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