r""" Definition ---------- Calculates the form factor for a rectangular solid with a core-shell structure. The thickness and the scattering length density of the shell or "rim" can be different on each (pair) of faces. The form factor is normalized by the particle volume $V$ such that .. math:: I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle + \text{background} where $\langle \ldots \rangle$ is an average over all possible orientations of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be pulled out of the form factor term due to the multiple slds in the model. The core of the solid is defined by the dimensions $A$, $B$, $C$ such that $A < B < C$. .. figure:: img/parallelepiped_geometry.jpg Core of the core shell parallelepiped with the corresponding definition of sides. There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ $(=t_C)$ faces. The projection in the $AB$ plane is .. figure:: img/core_shell_parallelepiped_projection.jpg AB cut through the core-shell parallelipiped showing the cross secion of four of the six shell slabs. As can be seen, this model leaves **"gaps"** at the corners of the solid. The total volume of the solid is thus given as .. math:: V = ABC + 2t_ABC + 2t_BAC + 2t_CAB The intensity calculated follows the :ref:`parallelepiped` model, with the core-shell intensity being calculated as the square of the sum of the amplitudes of the core and the slabs on the edges. The scattering amplitude is computed for a particular orientation of the core-shell parallelepiped with respect to the scattering vector and then averaged over all possible orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis of the parallelepiped, and $\beta$ is the angle between the projection of the particle in the $xy$ detector plane and the $y$ axis. .. math:: P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta} and .. math:: F(q,\alpha,\beta) &= (\rho_\text{core}-\rho_\text{solvent}) S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ &+ (\rho_\text{A}-\rho_\text{solvent}) \left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\ &+ (\rho_\text{B}-\rho_\text{solvent}) S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ &+ (\rho_\text{C}-\rho_\text{solvent}) S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right] with .. math:: S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L} and .. math:: Q_A &= q \sin\alpha \sin\beta \\ Q_B &= q \sin\alpha \cos\beta \\ Q_C &= q \cos\alpha where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$ are the scattering lengths of the parallelepiped core, and the rectangular slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$ is the scattering length of the solvent. .. note:: the code actually implements two substitutions: $d(cos\alpha)$ is substituted for -$sin\alpha \ d\alpha$ (note that in the :ref:`parallelepiped` code this is explicitly implemented with $\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that $du = \pi/2 \ d\beta$. Thus both integrals go from 0 to 1 rather than 0 to $\pi/2$. FITTING NOTES ~~~~~~~~~~~~~ If the scale is set equal to the particle volume fraction, $\phi$, the returned value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. However, **no interparticle interference effects are included in this calculation.** There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end up at a solution that is unphysical. The returned value is in units of |cm^-1|, on absolute scale. NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ and length $(C+2t_C)$ values, after appropriately sorting the three dimensions to give an oblate or prolate particle, to give an effective radius for $S(q)$ when $P(q) * S(q)$ is applied. For 2d data the orientation of the particle is required, described using angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below. For further details of the calculation and angular dispersions see :ref:`orientation`. The angle $\Psi$ is the rotational angle around the *long_c* axis. For example, $\Psi = 0$ when the *short_b* axis is parallel to the *x*-axis of the detector. .. note:: For 2d, constraints must be applied during fitting to ensure that the inequality $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, but the results may be not correct. .. figure:: img/parallelepiped_angle_definition.png Definition of the angles for oriented core-shell parallelepipeds. Note that rotation $\theta$, initially in the $xz$ plane, is carried out first, then rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the particle. The neutron or X-ray beam is along the $z$ axis. .. figure:: img/parallelepiped_angle_projection.png Examples of the angles for oriented core-shell parallelepipeds against the detector plane. Validation ---------- Cross-checked against hollow rectangular prism and rectangular prism for equal thickness overlapping sides, and by Monte Carlo sampling of points within the shape for non-uniform, non-overlapping sides. References ---------- .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 Equations (1), (13-14). (in German) .. [#] D Singh (2009). *Small angle scattering studies of self assembly in lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available from Proquest `_ Authorship and Verification ---------------------------- * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016 * **Last Modified by:** Paul Kienzle **Date:** October 17, 2017 """ import numpy as np from numpy import pi, inf, sqrt, cos, sin name = "core_shell_parallelepiped" title = "Rectangular solid with a core-shell structure." description = """ P(q)= """ category = "shape:parallelepiped" # ["name", "units", default, [lower, upper], "type","description"], parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Parallelepiped core scattering length density"], ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", "Parallelepiped A rim scattering length density"], ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Parallelepiped B rim scattering length density"], ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", "Parallelepiped C rim scattering length density"], ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", "Solvent scattering length density"], ["length_a", "Ang", 35, [0, inf], "volume", "Shorter side of the parallelepiped"], ["length_b", "Ang", 75, [0, inf], "volume", "Second side of the parallelepiped"], ["length_c", "Ang", 400, [0, inf], "volume", "Larger side of the parallelepiped"], ["thick_rim_a", "Ang", 10, [0, inf], "volume", "Thickness of A rim"], ["thick_rim_b", "Ang", 10, [0, inf], "volume", "Thickness of B rim"], ["thick_rim_c", "Ang", 10, [0, inf], "volume", "Thickness of C rim"], ["theta", "degrees", 0, [-360, 360], "orientation", "c axis to beam angle"], ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], ["psi", "degrees", 0, [-360, 360], "orientation", "rotation about c axis"], ] source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c): """ Return equivalent radius (ER) """ from .parallelepiped import ER as ER_p a = length_a + 2*thick_rim_a b = length_b + 2*thick_rim_b c = length_c + 2*thick_rim_c return ER_p(a, b, c) # VR defaults to 1.0 def random(): outer = 10**np.random.uniform(1, 4.7, size=3) thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1 length = outer - thick pars = dict( length_a=length[0], length_b=length[1], length_c=length[2], thick_rim_a=thick[0], thick_rim_b=thick[1], thick_rim_c=thick[2], ) return pars # parameters for demo demo = dict(scale=1, background=0.0, sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, length_a=35, length_b=75, length_c=400, thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, theta=0, phi=0, psi=0, length_a_pd=0.1, length_a_pd_n=1, length_b_pd=0.1, length_b_pd_n=1, length_c_pd=0.1, length_c_pd_n=1, thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, theta_pd=10, theta_pd_n=1, phi_pd=10, phi_pd_n=1, psi_pd=10, psi_pd_n=1) # rkh 7/4/17 add random unit test for 2d, note make all params different, # 2d values not tested against other codes or models if 0: # pak: model rewrite; need to update tests qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) tests = [[{}, 0.2, 0.533149288477], [{}, [0.2], [0.533149288477]], [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222], [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]], ] del qx, qy # not necessary to delete, but cleaner