r""" This model provides the form factor, $P(q)$, for a flexible cylinder where the form factor is normalized by the volume of the cylinder. **Inter-cylinder interactions are NOT provided for.** .. math:: P(q) = \text{scale} \left/V + \text{background} where the averaging $\left<\ldots\right>$ is applied only for the 1D calculation The 2D scattering intensity is the same as 1D, regardless of the orientation of the q vector which is defined as .. math:: q = \sqrt{q_x^2 + q_y^2} Definitions ----------- .. figure:: img/flexible_cylinder_geometry.jpg The chain of contour length, $L$, (the total length) can be described as a chain of some number of locally stiff segments of length $l_p$, the persistence length (the length along the cylinder over which the flexible cylinder can be considered a rigid rod). The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain. In the parameters, the sld and sld\_solvent represent the SLD of the cylinder and solvent respectively. Our model uses the form factor calculations in reference [1] as implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). This states: 'Method 3 With Excluded Volume' is used. The model is a parametrization of simulations of a discrete representation of the worm-like chain model of Kratky and Porod applied in the pseudocontinuous limit. See equations (13,26-27) in the original reference for the details. .. note:: There are several typos in the original reference that have been corrected by WRC [2]. Details of the corrections are in the reference below. Most notably - Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$ - Equations (23) and (24) are incorrect; WRC has entered these into Mathematica and solved analytically. The results were then converted to code. - Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of $max(a3*b(Rg^2)^{1/2},3)$ - The scattering function is negative for a range of parameter values and q-values that are experimentally accessible. A correction function has been added to give the proper behavior. **This is a model with complex behaviour depending on the ratio of** $L/b$ **and the reader is strongly encouraged to read reference [1] before use.** References ---------- .. [#] J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612 Correction of the formula can be found in .. [#] W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles.* Langmuir, 22(15) 2006 6539-6548 Source ------ `flexible_cylinder.py `_ `flexible_cylinder.c `_ `wrc_cyl.c `_ Authorship and Verification ---------------------------- * **Author:** * **Last Modified by:** * **Last Reviewed by:** Steve King **Date:** March 26, 2019 * **Source added by :** Steve King **Date:** March 25, 2019 """ import numpy as np from numpy import inf name = "flexible_cylinder" title = "Flexible cylinder where the form factor is normalized by the volume " \ "of the cylinder." description = """Note : scale and contrast = (sld - sld_solvent) are both multiplicative factors in the model and are perfectly correlated. One or both of these parameters must be held fixed during model fitting. """ category = "shape:cylinder" single = False # double precision only! # pylint: disable=bad-whitespace, line-too-long # ["name", "units", default, [lower, upper], "type", "description"], parameters = [ ["length", "Ang", 1000.0, [0, inf], "volume", "Length of the flexible cylinder"], ["kuhn_length", "Ang", 100.0, [0, inf], "volume", "Kuhn length of the flexible cylinder"], ["radius", "Ang", 20.0, [0, inf], "volume", "Radius of the flexible cylinder"], ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Cylinder scattering length density"], ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], ] # pylint: enable=bad-whitespace, line-too-long source = ["lib/polevl.c", "lib/sas_J1.c", "lib/wrc_cyl.c", "flexible_cylinder.c"] def random(): """Return a random parameter set for the model.""" length = 10**np.random.uniform(2, 6) radius = 10**np.random.uniform(1, 3) kuhn_length = 10**np.random.uniform(-2, 0)*length pars = dict( length=length, radius=radius, kuhn_length=kuhn_length, ) return pars tests = [ # Accuracy tests based on content in test/utest_other_models.py [{'length': 1000.0, # test T1 'kuhn_length': 100.0, 'radius': 20.0, 'sld': 1.0, 'sld_solvent': 6.3, 'background': 0.0001, }, 0.001, 3509.2187], # Additional tests with larger range of parameters [{'length': 1000.0, # test T2 'kuhn_length': 100.0, 'radius': 20.0, 'sld': 1.0, 'sld_solvent': 6.3, 'background': 0.0001, }, 1.0, 0.000595345], [{'length': 10.0, # test T3 'kuhn_length': 800.0, 'radius': 2.0, 'sld': 6.0, 'sld_solvent': 12.3, 'background': 0.001, }, 0.1, 1.55228], [{'length': 100.0, # test T4 'kuhn_length': 800.0, 'radius': 50.0, 'sld': 0.1, 'sld_solvent': 5.1, 'background': 0.0, }, 1.0, 0.000938456] ] # There are a few branches in the code that ought to have test values: # # For length > 4 * kuhn_length # if length > 10 * kuhn_length then C is scaled by 3.06 (L/b)^(-0.44) # q*kuhn_length <= 3.1 => Sexv_new # dS/dQ < 0 has different behaviour from dS/dQ >= 0 # T2 q*kuhn_length > 3.1 => a_long # # For length <= 4 * kuhn_length # q*kuhn_length <= max(1.9/Rg_short, 3.0) => Sdebye((q*Rg)^2) # q*Rg < 0.5 uses Pade approx, q*Rg > 1.0 uses math lib # T3,T4 q*kuhn_length > max(1.9/Rg_short, 3.0) => a_short # # Note that the transitions between branches may be abrupt. You can see a # several percent change around length=10*kuhn_length and length=4*kuhn_length # using the following: # # sascomp flexible_cylinder -calc=double -sets=10 length=10*kuhn_length,10.000001*kuhn_length # sascomp flexible_cylinder -calc=double -sets=10 length=4*kuhn_length,4.000001*kuhn_length # # The transition between low q and high q around q*kuhn_length = 3 seems # to be good to 4 digits or better. This was tested by computing the value # on each branches near the transition point and reporting the relative error # for kuhn lengths of 10, 100 and 1000 and a variety of length:kuhn_length # ratios.