r""" For information about polarised and magnetic scattering, see the :ref:magnetism documentation. Definition ---------- The 1D scattering intensity is calculated in the following way (Guinier, 1955) .. math:: I(q) = \frac{\text{scale}}{V} \cdot \left[ 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} \right]^2 + \text{background} where *scale* is a volume fraction, $V$ is the volume of the scatterer, $r$ is the radius of the sphere and *background* is the background level. *sld* and *sld_solvent* are the scattering length densities (SLDs) of the scatterer and the solvent respectively, whose difference is $\Delta\rho$. Note that if your data is in absolute scale, the *scale* should represent the volume fraction (which is unitless) if you have a good fit. If not, it should represent the volume fraction times a factor (by which your data might need to be rescaled). The 2D scattering intensity is the same as above, regardless of the orientation of $\vec q$. Validation ---------- Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). References ---------- #. A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) Authorship and Verification ---------------------------- * **Author:** * **Last Modified by:** * **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06 """ import numpy as np from numpy import inf name = "sphere" title = "Spheres with uniform scattering length density" description = """\ P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) /(qr)^3]^2 + background r: radius of sphere V: The volume of the scatter sld: the SLD of the sphere sld_solvent: the SLD of the solvent """ category = "shape:sphere" # ["name", "units", default, [lower, upper], "type","description"], parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Layer scattering length density"], ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", "Solvent scattering length density"], ["radius", "Ang", 50, [0, inf], "volume", "Sphere radius"], ] source = ["lib/sas_3j1x_x.c", "sphere.c"] have_Fq = True radius_effective_modes = ["radius"] #single = False def random(): """Return a random parameter set for the model.""" radius = 10**np.random.uniform(1.3, 4) pars = dict( radius=radius, ) return pars #2345678901234567890123456789012345678901234567890123456789012345678901234567890 tests = [ [{}, 0.2, 0.726362], # each test starts with default parameter values # inside { }, unless modified. Then Q and expected value of I(Q) # putting None for an expected result will pass the test if there are no # errors from the routine, but without any check on the value of the result [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., "radius": 120.}, [0.01, 0.1, 0.2], [1.34836265e+04, 6.20114062e+00, 1.04733914e-01]], [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., # careful tests here R=120 Pd=.2, then with S(Q) at default Reff=50 # (but this gets changed to 120) phi=0,2 "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45}, [0.01, 0.1, 0.2], [1.74395295e+04, 3.68016987e+00, 2.28843099e-01]], # a list of Q values and list of expected results is also possible [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45}, 0.01, 335839.88055473, 1.41045057e+11, 120.0, 8087664.122641933, 1.0], # the longer list here checks F1, F2, R_eff, volume, volume_ratio [{"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45}, 0.1, 482.93824329, 29763977.79867414, 120.0, 8087664.122641933, 1.0], [{"radius": 120., "radius_pd": 0.2, "radius_pd_n": 45}, 0.2, 1.23330406, 1850806.1197361, 120.0, 8087664.122641933, 1.0], # But note P(Q) = F2/volume # F and F^2 are "unscaled", with for n S(q) or for beta approx # I(q) = n [ + (S(q) - 1)] # for n the number density and <.> the orientation average, and # F = integral rho(r) exp(i q . r) dr. # The number density is volume fraction divided by particle volume. # Effectively, this leaves F = V drho form, where form is the usual # 3 j1(qr)/(qr) or whatever depending on the shape. # @S RESULTS using F1 and F2 from the longer test string above: # # I(Q) = (F2 + F1^2*(S(Q) -1))*volfraction*scale/Volume + background # # with by default scale=1.0, background=0.001 # NOTE currently S(Q) volfraction is also included in scaling # structure_factor_mode 0 = normal decoupling approx, # 1 = beta(Q) approx # radius_effective_mode 0 is for free choice, # 1 is use radius from F2(Q) # (sphere only has two choices, other models may have more) [{"@S": "hardsphere", "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45, "volfraction": 0.2, #"radius_effective":50.0, # hard sphere structure factor "structure_factor_mode": 1, # mode 0 = normal decoupling approx, # 1 = beta(Q) approx "radius_effective_mode": 0 # this used default hardsphere Reff=50 }, [0.01, 0.1, 0.2], [1.32473756e+03, 7.36633631e-01, 4.67686201e-02]], [{"@S": "hardsphere", "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45, "volfraction": 0.2, "radius_effective": 45.0, # explicit Reff over rides either 50 or 120 "structure_factor_mode": 1, # beta approx "radius_effective_mode": 0 # }, 0.01, 1316.2990966463444], [{"@S": "hardsphere", "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45, "volfraction": 0.2, "radius_effective": 120.0, # over ride Reff "structure_factor_mode": 1, # beta approx "radius_effective_mode": 0 # (mode=1 here also uses 120) }, [0.01, 0.1, 0.2], [1.57928589e+03, 7.37067923e-01, 4.67686197e-02]], [{"@S": "hardsphere", "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45, "volfraction": 0.2, #"radius_effective": 120.0, # hard sphere structure factor "structure_factor_mode": 0, # normal decoupling approximation "radius_effective_mode": 1 # this uses 120 from the form factor }, [0.01, 0.1, 0.2], [1.10112335e+03, 7.41366536e-01, 4.66630207e-02]], [{"@S": "hardsphere", "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45, "volfraction": 0.2, #"radius_effective": 50.0, # hard sphere structure factor "structure_factor_mode": 0, # normal decoupling approximation "radius_effective_mode": 0 # this used 50 the default for hardsphere }, [0.01, 0.1, 0.2], [7.82803598e+02, 6.85943611e-01, 4.71586457e-02]], # Check returned intermediate results. # Note: Target values come from double precision dll calculation. # TODO: Cross check results against other software. [{"@S": "hardsphere", "radius": 120., "radius_pd": 0.2, "radius_pd_n": 45, "volfraction": 0.2, "radius_effective": 120.0, # hard sphere structure factor "structure_factor_mode": 1, # normal decoupling approximation "radius_effective_mode": 0, # mode 0 says ignore Reff from P }, [0.01, 0.1, 0.2], [1.57928589e+03, 7.37067923e-01, 4.67686197e-02], {"P(Q)": [3487.905895219423, 0.7360339734027279, 0.04576861975646704], "S(Q)": [0.31569726516764035, 1.005886362143737, 0.9976927625183415], #"beta(Q)": [0.7996623765645325, 0.007835960247334845, 8.218250904154009e-07], "beta(Q)": None, # Single precision not good enough for 5 digits of beta "S_eff(Q)": [0.4527888487743462, 1.0000461252997597, 0.9999999981038543], "volume": 8087664.122641933, "volume_ratio": 1.0, "radius_effective": 0.0, # zero since mode is 0, and Reff isn't computed }], ]