r""" Definition ---------- This model calculates an empirical functional form for SAS data characterized by a broad scattering peak. Many SAS spectra are characterized by a broad peak even though they are from amorphous soft materials. For example, soft systems that show a SAS peak include copolymers, polyelectrolytes, multiphase systems, layered structures, etc. The d-spacing corresponding to the broad peak is a characteristic distance between the scattering inhomogeneities (such as in lamellar, cylindrical, or spherical morphologies, or for bicontinuous structures). The scattering intensity $I(q)$ is calculated as .. math:: I(q) = \frac{A}{q^n} + \frac{C}{1 + (|q - q_0|\xi)^m} + B Here the peak position is related to the d-spacing as $q_0 = 2\pi / d_0$. $A$ is the Porod law scale factor, $n$ the Porod exponent, $C$ is the Lorentzian scale factor, $m$ the exponent of $q$, $\xi$ the screening length, and $B$ the flat background. For 2D data the scattering intensity is calculated in the same way as 1D, where the $q$ vector is defined as .. math:: q = \sqrt{q_x^2 + q_y^2} References ---------- None. Authorship and Verification ---------------------------- * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Last Modified by:** Paul kienle **Date:** July 24, 2016 * **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016 """ import numpy as np from numpy import inf, errstate name = "broad_peak" title = "Broad Lorentzian type peak on top of a power law decay" description = """\ I(q) = scale_p/pow(q,exponent)+scale_l/ (1.0 + pow((fabs(q-q_peak)*length_l),exponent_l) )+ background List of default parameters: porod_scale = Porod term scaling porod_exp = Porod exponent lorentz_scale = Lorentzian term scaling lorentz_length = Lorentzian screening length [A] peak_pos = peak location [1/A] lorentz_exp = Lorentzian exponent background = Incoherent background""" category = "shape-independent" # pylint: disable=bad-whitespace, line-too-long # ["name", "units", default, [lower, upper], "type", "description"], parameters = [["porod_scale", "", 1.0e-05, [-inf, inf], "", "Power law scale factor"], ["porod_exp", "", 3.0, [-inf, inf], "", "Exponent of power law"], ["lorentz_scale", "", 10.0, [-inf, inf], "", "Scale factor for broad Lorentzian peak"], ["lorentz_length", "Ang", 50.0, [-inf, inf], "", "Lorentzian screening length"], ["peak_pos", "1/Ang", 0.1, [-inf, inf], "", "Peak position in q"], ["lorentz_exp", "", 2.0, [-inf, inf], "", "Exponent of Lorentz function"], ] # pylint: enable=bad-whitespace, line-too-long def Iq(q, porod_scale=1.0e-5, porod_exp=3.0, lorentz_scale=10.0, lorentz_length=50.0, peak_pos=0.1, lorentz_exp=2.0): """ :param q: Input q-value :param porod_scale: Power law scale factor :param porod_exp: Exponent of power law :param lorentz_scale: Scale factor for broad Lorentzian peak :param lorentz_length: Lorentzian screening length :param peak_pos: Peak position in q :param lorentz_exp: Exponent of Lorentz function :return: Calculated intensity """ z = abs(q - peak_pos) * lorentz_length with errstate(divide='ignore'): inten = (porod_scale / q ** porod_exp + lorentz_scale / (1 + z ** lorentz_exp)) return inten Iq.vectorized = True # Iq accepts an array of q values def random(): """Return a random parameter set for the model.""" pars = dict( scale=1, porod_scale=10**np.random.uniform(-8, -5), porod_exp=np.random.uniform(1, 6), lorentz_scale=10**np.random.uniform(0.3, 6), lorentz_length=10**np.random.uniform(0, 2), peak_pos=10**np.random.uniform(-3, -1), lorentz_exp=np.random.uniform(1, 4), ) pars['lorentz_length'] /= pars['peak_pos'] pars['lorentz_scale'] *= pars['porod_scale'] / pars['peak_pos']**pars['porod_exp'] #pars['porod_scale'] = 0. return pars