Changes in / [9616dfe:bf94e6e] in sasmodels


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  • sasmodels/models/core_shell_parallelepiped.c

    rdbf1a60 re077231  
    5959 
    6060    // outer integral (with gauss points), integration limits = 0, 1 
    61     // substitute d_cos_alpha for sin_alpha d_alpha 
    6261    double outer_sum = 0; //initialize integral 
    6362    for( int i=0; i<GAUSS_N; i++) { 
    6463        const double cos_alpha = 0.5 * ( GAUSS_Z[i] + 1.0 ); 
    6564        const double mu = half_q * sqrt(1.0-cos_alpha*cos_alpha); 
     65 
     66        // inner integral (with gauss points), integration limits = 0, pi/2 
    6667        const double siC = length_c * sas_sinx_x(length_c * cos_alpha * half_q); 
    6768        const double siCt = tC * sas_sinx_x(tC * cos_alpha * half_q); 
    68  
    69         // inner integral (with gauss points), integration limits = 0, 1 
    70         // substitute beta = PI/2 u (so 2/PI * d_(PI/2 * beta) = d_beta) 
    7169        double inner_sum = 0.0; 
    7270        for(int j=0; j<GAUSS_N; j++) { 
    73             const double u = 0.5 * ( GAUSS_Z[j] + 1.0 ); 
     71            const double beta = 0.5 * ( GAUSS_Z[j] + 1.0 ); 
    7472            double sin_beta, cos_beta; 
    75             SINCOS(M_PI_2*u, sin_beta, cos_beta); 
     73            SINCOS(M_PI_2*beta, sin_beta, cos_beta); 
    7674            const double siA = length_a * sas_sinx_x(length_a * mu * sin_beta); 
    7775            const double siB = length_b * sas_sinx_x(length_b * mu * cos_beta); 
    … …  
    9391            inner_sum += GAUSS_W[j] * f * f; 
    9492        } 
    95         // now complete change of inner integration variable (1-0)/(1-(-1))= 0.5 
    9693        inner_sum *= 0.5; 
    9794        // now sum up the outer integral 
    9895        outer_sum += GAUSS_W[i] * inner_sum; 
    9996    } 
    100     // now complete change of outer integration variable (1-0)/(1-(-1))= 0.5 
    10197    outer_sum *= 0.5; 
    10298 

  • sasmodels/models/core_shell_parallelepiped.py

    rdbf1a60 r97be877  
    1111.. math:: 
    1212 
    13     I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle  
    14     + \text{background} 
     13    I(q) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background} 
    1514 
    1615where $\langle \ldots \rangle$ is an average over all possible orientations 
    17 of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be 
    18 pulled out of the form factor term due to the multiple slds in the model. 
    19  
     16of the rectangular solid. 
     17 
     18The function calculated is the form factor of the rectangular solid below. 
    2019The core of the solid is defined by the dimensions $A$, $B$, $C$ such that 
    2120$A < B < C$. 
    2221 
    23 .. figure:: img/parallelepiped_geometry.jpg 
    24  
    25    Core of the core shell Parallelepiped with the corresponding definition 
    26    of sides. 
    27  
     22.. image:: img/core_shell_parallelepiped_geometry.jpg 
    2823 
    2924There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension 
    3025(on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ 
    31 $(=t_C)$ faces. The projection in the $AB$ plane is 
    32  
    33 .. figure:: img/core_shell_parallelepiped_projection.jpg 
    34  
    35    AB cut through the core-shell parllelipiped showing the cross secion of 
    36    four of the six shell slabs. As can be seen This model leaves **"gaps"** 
    37    at the corners of the solid. 
    38  
    39  
    40 The total volume of the solid is thus given as 
     26$(=t_C)$ faces. The projection in the $AB$ plane is then 
     27 
     28.. image:: img/core_shell_parallelepiped_projection.jpg 
     29 
     30The volume of the solid is 
    4131 
    4232.. math:: 
    4333 
    4434    V = ABC + 2t_ABC + 2t_BAC + 2t_CAB 
     35 
     36**meaning that there are "gaps" at the corners of the solid.** 
    4537 
    4638The intensity calculated follows the :ref:`parallelepiped` model, with the 
    4739core-shell intensity being calculated as the square of the sum of the 
    48 amplitudes of the core and the slabs on the edges. The scattering amplitude is 
    49 computed for a particular orientation of the core-shell parallelepiped with 
    50 respect to the scattering vector and then averaged over all possible 
    51 orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis 
    52 of the parallelepiped, and $\beta$ is the angle between the projection of the 
    53 particle in the $xy$ detector plane and the $y$ axis. 
    54  
    55 .. math:: 
    56  
    57     P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha 
    58     \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta} 
    59  
    60 and 
    61  
    62 .. math:: 
    63  
    64     F(q,\alpha,\beta) 
     40amplitudes of the core and the slabs on the edges. 
     41 
     42the scattering amplitude is computed for a particular orientation of the 
     43core-shell parallelepiped with respect to the scattering vector and then 
     44averaged over all possible orientations, where $\alpha$ is the angle between 
     45the $z$ axis and the $C$ axis of the parallelepiped, $\beta$ is 
     46the angle between projection of the particle in the $xy$ detector plane 
     47and the $y$ axis. 
     48 
     49.. math:: 
     50 
     51    F(Q) 
    6552    &= (\rho_\text{core}-\rho_\text{solvent}) 
    6653       S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ 
    6754    &+ (\rho_\text{A}-\rho_\text{solvent}) 
    68         \left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\ 
     55        \left[S(Q_A, A+2t_A) - S(Q_A, Q)\right] S(Q_B, B) S(Q_C, C) \\ 
    6956    &+ (\rho_\text{B}-\rho_\text{solvent}) 
    7057        S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ 
    … …  
    7663.. math:: 
    7764 
    78     S(Q_X, L) = L \frac{\sin \tfrac{1}{2} Q_X L}{\tfrac{1}{2} Q_X L} 
     65    S(Q, L) = L \frac{\sin \tfrac{1}{2} Q L}{\tfrac{1}{2} Q L} 
    7966 
    8067and 
    … …  
    8269.. math:: 
    8370 
    84     Q_A &= q \sin\alpha \sin\beta \\ 
    85     Q_B &= q \sin\alpha \cos\beta \\ 
    86     Q_C &= q \cos\alpha 
     71    Q_A &= \sin\alpha \sin\beta \\ 
     72    Q_B &= \sin\alpha \cos\beta \\ 
     73    Q_C &= \cos\alpha 
    8774 
    8875 
    … …  
    9279is the scattering length of the solvent. 
    9380 
    94 .. note::  
    95  
    96    the code actually implements two substitutions: $d(cos\alpha)$ is 
    97    substituted for -$sin\alpha \ d\alpha$ (note that in the 
    98    :ref:`parallelepiped` code this is explicitly implemented with 
    99    $\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that 
    100    $du = \pi/2 \ d\beta$.  Thus both integrals go from 0 to 1 rather than 0 
    101    to $\pi/2$. 
    102  
    10381FITTING NOTES 
    10482~~~~~~~~~~~~~ 
    … …  
    11795and length $(C+2t_C)$ values, after appropriately sorting the three dimensions 
    11896to give an oblate or prolate particle, to give an effective radius, 
    119 for $S(q)$ when $P(q) * S(q)$ is applied. 
     97for $S(Q)$ when $P(Q) * S(Q)$ is applied. 
    12098 
    12199For 2d data the orientation of the particle is required, described using 
    … …  
    125103$\Psi = 0$ when the *short_b* axis is parallel to the *x*-axis of the detector. 
    126104 
    127 .. note:: For 2d, constraints must be applied during fitting to ensure that the 
    128    inequality $A < B < C$ is not violated, and hence the correct definition 
    129    of angles is preserved. The calculation will not report an error, 
    130    but the results may be not correct. 
     105For 2d, constraints must be applied during fitting to ensure that the 
     106inequality $A < B < C$ is not violated, and hence the correct definition 
     107of angles is preserved. The calculation will not report an error, 
     108but the results may be not correct. 
    131109 
    132110.. figure:: img/parallelepiped_angle_definition.png 

  • sasmodels/models/parallelepiped.c

    rdbf1a60 r108e70e  
    3838            inner_total += GAUSS_W[j] * square(si1 * si2); 
    3939        } 
    40         // now complete change of inner integration variable (1-0)/(1-(-1))= 0.5 
    4140        inner_total *= 0.5; 
    4241 
    … …  
    4443        outer_total += GAUSS_W[i] * inner_total * si * si; 
    4544    } 
    46     // now complete change of outer integration variable (1-0)/(1-(-1))= 0.5 
    4745    outer_total *= 0.5; 
    4846 

  • sasmodels/models/parallelepiped.py

    rdbf1a60 ref07e95  
    1010 
    1111 This model calculates the scattering from a rectangular parallelepiped 
    12  (:numref:`parallelepiped-image`). 
     12 (\:numref:`parallelepiped-image`\). 
    1313 If you need to apply polydispersity, see also :ref:`rectangular-prism`. 
    1414 
    … …  
    3939 
    4040    I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 
    41            \left< P(q, \alpha, \beta) \right> + \text{background} 
     41           \left< P(q, \alpha) \right> + \text{background} 
    4242 
    4343where the volume $V = A B C$, the contrast is defined as 
    44 $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ 
    45 is the form factor corresponding to a parallelepiped oriented 
    46 at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ 
    47 ( the angle between the projection of the particle in the $xy$ detector plane 
    48 and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all 
    49 orientations. 
     44$\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, 
     45$P(q, \alpha)$ is the form factor corresponding to a parallelepiped oriented 
     46at an angle $\alpha$ (angle between the long axis C and $\vec q$), 
     47and the averaging $\left<\ldots\right>$ is applied over all orientations. 
    5048 
    5149Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the 
    52 form factor is given by (Mittelbach and Porod, 1961 [#Mittelbach]_) 
     50form factor is given by (Mittelbach and Porod, 1961) 
    5351 
    5452.. math:: 
    … …  
    6866    \mu &= qB 
    6967 
    70 where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 \ u$ have been 
    71 applied. 
     68The scattering intensity per unit volume is returned in units of |cm^-1|. 
    7269 
    7370NB: The 2nd virial coefficient of the parallelepiped is calculated based on 
    … …  
    123120.. math:: 
    124121 
    125     P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} 
    126                    {2}qA\cos\alpha)}\right]^2 
    127                   \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} 
    128                    {2}qB\cos\beta)}\right]^2 
    129                   \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} 
    130                    {2}qC\cos\gamma)}\right]^2 
     122    P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1}{2}qA\cos\alpha)}\right]^2 
     123                  \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1}{2}qB\cos\beta)}\right]^2 
     124                  \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1}{2}qC\cos\gamma)}\right]^2 
    131125 
    132126with 
    … …  
    166160---------- 
    167161 
    168 .. [#Mittelbach] P Mittelbach and G Porod, *Acta Physica Austriaca*, 
    169    14 (1961) 185-211 
    170 .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 
     162P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 
     163 
     164R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 
    171165 
    172166Authorship and Verification 
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