# Changeset d682f66 in sasmodels

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Timestamp:
Jun 19, 2018 8:57:56 AM (4 years ago)
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
5c36bf1
Parents:
ce156e3 (diff), 052d4c5 (diff)
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Merge branch 'master' of github.com:sasview/sasmodels

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1 deleted
4 edited

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Removed
• ## sasmodels/models/core_shell_parallelepiped.c

 re077231 // outer integral (with gauss points), integration limits = 0, 1 // substitute d_cos_alpha for sin_alpha d_alpha double outer_sum = 0; //initialize integral for( int i=0; i
• ## sasmodels/models/core_shell_parallelepiped.py

 r97be877 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 thickness and the scattering length density of the shell or "rim" can be different on each (pair) of faces. The three dimensions of the core of the parallelepiped (strictly here a cuboid) may be given in *any* size order as long as the particles are randomly oriented (i.e. take on all possible orientations see notes on 2D below). To avoid multiple fit solutions, especially with Monte-Carlo fit methods, it may be advisable to restrict their ranges. There may be a number of closely similar "best fits", so some trial and error, or fixing of some dimensions at expected values, may help. The form factor is normalized by the particle volume $V$ such that .. math:: I(q) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background} 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. The function calculated is the form factor of the rectangular solid below. The core of the solid is defined by the dimensions $A$, $B$, $C$ such that $A < B < C$. .. image:: img/core_shell_parallelepiped_geometry.jpg 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 $A$, $B$, $C$ here shown 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 then .. image:: img/core_shell_parallelepiped_projection.jpg The volume of the solid is $(=t_C)$ 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 **meaning that there are "gaps" at the corners of the solid.** 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, $\beta$ is the angle between projection of the particle in the $xy$ detector plane and the $y$ axis. .. math:: F(Q) 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, Q)\right] S(Q_B, B) S(Q_C, C) \\ \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) \\ .. math:: S(Q, L) = L \frac{\sin \tfrac{1}{2} Q L}{\tfrac{1}{2} Q L} S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L} and .. math:: Q_A &= \sin\alpha \sin\beta \\ Q_B &= \sin\alpha \cos\beta \\ Q_C &= \cos\alpha 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 length of the parallelepiped core, and the rectangular 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. 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. #. 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 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, where $\theta$ and $\phi$ define the orientation of the director in the laboratry reference frame of the beam direction ($z$) and detector plane ($x-y$ plane), while the angle $\Psi$ is effectively the rotational angle around the particle $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis oriented parallel to the y-axis of the detector with $A$ along the x-axis. For other $\theta$, $\phi$ values, the order of rotations matters. In particular, the parallelepiped must first be rotated $\theta$ degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ axis (in the $x-y$ plane). Applying orientational distribution to the particle orientation (i.e  jitter to one or more of these angles) can get more confusing as jitter is defined **NOT** with respect to the laboratory frame but the particle reference frame. It is thus highly recmmended to read :ref:orientation for further details of the calculation and angular dispersions. .. note:: For 2d, constraints must be applied during fitting to ensure that the order of sides chosen is not altered, and hence that the correct definition of angles is preserved. For the default choice shown here, that means ensuring that the inequality $A < B < C$ is not violated,  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 Note that rotation $\theta$, initially in the $x-z$ plane, is carried out first, then rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the cylinder. The neutron or X-ray beam is along the $z$ axis. $\Psi$ is now around the $C$ axis of the particle. The neutron or X-ray beam is along the $z$ axis and the detecotr defines the $x-y$ plane. .. 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 * **Author:** NIST IGOR/DANSE **Date:** pre 2010 * **Converted to sasmodels by:** Miguel Gonzales **Date:** February 26, 2016 * **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016 * **Last Modified by:** Paul Kienzle **Date:** October 17, 2017 * 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. * **Last Reviewed by:** Paul Butler **Date:** May 24, 2018 - documentation updated """
• ## sasmodels/models/parallelepiped.c

 r108e70e inner_total += GAUSS_W[j] * square(si1 * si2); } // now complete change of inner integration variable (1-0)/(1-(-1))= 0.5 inner_total *= 0.5; outer_total += GAUSS_W[i] * inner_total * si * si; } // now complete change of outer integration variable (1-0)/(1-(-1))= 0.5 outer_total *= 0.5;
• ## sasmodels/models/parallelepiped.py

 ref07e95 # Note: model title and parameter table are inserted automatically r""" The form factor is normalized by the particle volume. For information about polarised and magnetic scattering, see the :ref:magnetism documentation. Definition ---------- This model calculates the scattering from a rectangular parallelepiped (\:numref:parallelepiped-image\). If you need to apply polydispersity, see also :ref:rectangular-prism. This model calculates the scattering from a rectangular solid (:numref:parallelepiped-image). If you need to apply polydispersity, see also :ref:rectangular-prism. For information about polarised and magnetic scattering, see the :ref:magnetism documentation. .. _parallelepiped-image: The three dimensions of the parallelepiped (strictly here a cuboid) may be given in *any* size order. To avoid multiple fit solutions, especially with Monte-Carlo fit methods, it may be advisable to restrict their ranges. There may be a number of closely similar "best fits", so some trial and error, or fixing of some dimensions at expected values, may help. The 1D scattering intensity $I(q)$ is calculated as: given in *any* size order as long as the particles are randomly oriented (i.e. take on all possible orientations see notes on 2D below). To avoid multiple fit solutions, especially with Monte-Carlo fit methods, it may be advisable to restrict their ranges. There may be a number of closely similar "best fits", so some trial and error, or fixing of some dimensions at expected values, may help. The form factor is normalized by the particle volume and the 1D scattering intensity $I(q)$ is then calculated as: .. Comment by Miguel Gonzalez: I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 \left< P(q, \alpha) \right> + \text{background} \left< P(q, \alpha, \beta) \right> + \text{background} where the volume $V = A B C$, the contrast is defined as $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha)$ is the form factor corresponding to a parallelepiped oriented at an angle $\alpha$ (angle between the long axis C and $\vec q$), and the averaging $\left<\ldots\right>$ is applied over all orientations. $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ is the form factor corresponding to a parallelepiped oriented at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ (the angle between the projection of the particle in the $xy$ detector plane and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all orientations. Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the form factor is given by (Mittelbach and Porod, 1961) form factor is given by (Mittelbach and Porod, 1961 [#Mittelbach]_) .. math:: \mu &= qB The scattering intensity per unit volume is returned in units of |cm^-1|. NB: The 2nd virial coefficient of the parallelepiped is calculated based on the averaged effective radius, after appropriately sorting the three dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and length $(= C)$ values, and used as the effective radius for $S(q)$ when $P(q) \cdot S(q)$ 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 . .. Comment by Miguel Gonzalez: The following text has been commented because I think there are two mistakes. Psi is the rotational angle around C (but I cannot understand what it means against the q plane) and psi=0 corresponds to a||x and b||y. The angle $\Psi$ is the rotational angle around the $C$ axis against the $q$ plane. For example, $\Psi = 0$ when the $B$ axis is parallel to the $x$-axis of the detector. The angle $\Psi$ is the rotational angle around the $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis oriented parallel to the y-axis of the detector with $A$ along the x-axis. For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated $\theta$ degrees in the $z-x$ plane and then $\phi$ degrees around the $z$ axis, before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis of the particle to obtain the final orientation of the parallelepiped. .. _parallelepiped-orientation: .. figure:: img/parallelepiped_angle_definition.png Definition of the angles for oriented parallelepiped, shown with \$A
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