Changes in sasmodels/models/parallelepiped.py [ef07e95:b343226] in sasmodels
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sasmodels/models/parallelepiped.py
ref07e95 rb343226 2 2 # Note: model title and parameter table are inserted automatically 3 3 r""" 4 The form factor is normalized by the particle volume.5 For information about polarised and magnetic scattering, see6 the :ref:`magnetism` documentation.7 8 4 Definition 9 5 ---------- 10 6 11 This model calculates the scattering from a rectangular parallelepiped 12 (\:numref:`parallelepiped-image`\). 13 If you need to apply polydispersity, see also :ref:`rectangular-prism`. 7 This model calculates the scattering from a rectangular parallelepiped 8 (:numref:`parallelepiped-image`). 9 If you need to apply polydispersity, see also :ref:`rectangular-prism`. For 10 information about polarised and magnetic scattering, see 11 the :ref:`magnetism` documentation. 14 12 15 13 .. _parallelepiped-image: … … 26 24 error, or fixing of some dimensions at expected values, may help. 27 25 28 The 1D scattering intensity $I(q)$ is calculated as: 26 The form factor is normalized by the particle volume and the 1D scattering 27 intensity $I(q)$ is then calculated as: 29 28 30 29 .. Comment by Miguel Gonzalez: … … 39 38 40 39 I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 41 \left< P(q, \alpha ) \right> + \text{background}40 \left< P(q, \alpha, \beta) \right> + \text{background} 42 41 43 42 where the volume $V = A B C$, the contrast is defined as 44 $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, 45 $P(q, \alpha)$ is the form factor corresponding to a parallelepiped oriented 46 at an angle $\alpha$ (angle between the long axis C and $\vec q$), 47 and the averaging $\left<\ldots\right>$ is applied over all orientations. 43 $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ 44 is the form factor corresponding to a parallelepiped oriented 45 at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ 46 (the angle between the projection of the particle in the $xy$ detector plane 47 and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all 48 orientations. 48 49 49 50 Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the 50 form factor is given by (Mittelbach and Porod, 1961 )51 form factor is given by (Mittelbach and Porod, 1961 [#Mittelbach]_) 51 52 52 53 .. math:: … … 66 67 \mu &= qB 67 68 68 The scattering intensity per unit volume is returned in units of |cm^-1|. 69 where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 \ u$ have been 70 applied. 69 71 70 72 NB: The 2nd virial coefficient of the parallelepiped is calculated based on … … 120 122 .. math:: 121 123 122 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]^2 124 P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} 125 {2}qA\cos\alpha)}\right]^2 126 \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} 127 {2}qB\cos\beta)}\right]^2 128 \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} 129 {2}qC\cos\gamma)}\right]^2 125 130 126 131 with … … 160 165 ---------- 161 166 162 P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 163 164 R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854167 .. [#Mittelbach] P Mittelbach and G Porod, *Acta Physica Austriaca*, 168 14 (1961) 185-211 169 .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 165 170 166 171 Authorship and Verification
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