# Changeset dbf1a60 in sasmodels

Ignore:
Timestamp:
Mar 11, 2018 2:29:41 PM (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:
9616dfe
Parents:
367886f
Message:

parallelepiped noting the change of integration varialbes in the
computation. Cleaned up and final corrections to the core shell
documentation and did some cleaning of parallelipiped. In particular
tried to bring a bit more consistency between the docs.

Location:
sasmodels/models
Files:
4 edited

Unmodified
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

 r367886f .. math:: I(q) = \text{scale}\frac{\langle P(q,\alpha,\beta) \rangle}{V} 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. 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 $A < B < C$. 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 $(=t_C)$ faces. The projection in the $AB$ plane is .. figure:: img/core_shell_parallelepiped_projection.jpg AB cut through the core-shell parllelipiped showing the cross secion of four of the six shell slabs The volume of the solid is 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:: P(q)=\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ cos\alpha \ d\alpha \ d\beta 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) F(q,\alpha,\beta) &= (\rho_\text{core}-\rho_\text{solvent}) S(Q_A, A) S(Q_B, B) 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 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 $\Psi = 0$ 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. .. 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
• ## 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

 r5bc373b 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|. where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 \ u$ have been applied. NB: The 2nd virial coefficient of the parallelepiped is calculated based on .. math:: P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1}{2}qA\cos\alpha)}\right]^2 \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1}{2}qB\cos\beta)}\right]^2 \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1}{2}qC\cos\gamma)}\right]^2 P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} {2}qA\cos\alpha)}\right]^2 \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} {2}qB\cos\beta)}\right]^2 \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} {2}qC\cos\gamma)}\right]^2 with ---------- P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 .. [#Mittelbach] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 Authorship and Verification
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