[5933c7f] | 1 | # parallelepiped model |
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| 2 | # Note: model title and parameter table are inserted automatically |
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| 3 | r""" |
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| 4 | The form factor is normalized by the particle volume. |
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[9f60c06] | 5 | For information about polarised and magnetic scattering, see |
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[9a4811a] | 6 | the :ref:`magnetism` documentation. |
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[5933c7f] | 7 | |
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| 8 | Definition |
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| 9 | ---------- |
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| 10 | |
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[40a87fa] | 11 | | This model calculates the scattering from a rectangular parallelepiped |
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| 12 | | (:numref:`parallelepiped-image`). |
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[5933c7f] | 13 | | If you need to apply polydispersity, see also :ref:`rectangular-prism`. |
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| 14 | |
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| 15 | .. _parallelepiped-image: |
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| 16 | |
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| 17 | .. figure:: img/parallelepiped_geometry.jpg |
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| 18 | |
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| 19 | Parallelepiped with the corresponding definition of sides. |
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| 20 | |
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| 21 | .. note:: |
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| 22 | |
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| 23 | The edge of the solid must satisfy the condition that $A < B < C$. |
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| 24 | This requirement is not enforced in the model, so it is up to the |
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| 25 | user to check this during the analysis. |
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| 26 | |
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| 27 | The 1D scattering intensity $I(q)$ is calculated as: |
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| 28 | |
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| 29 | .. Comment by Miguel Gonzalez: |
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| 30 | I am modifying the original text because I find the notation a little bit |
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| 31 | confusing. I think that in most textbooks/papers, the notation P(Q) is |
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| 32 | used for the form factor (adim, P(Q=0)=1), although F(q) seems also to |
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| 33 | be used. But here (as for many other models), P(q) is used to represent |
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| 34 | the scattering intensity (in cm-1 normally). It would be good to agree on |
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| 35 | a common notation. |
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| 36 | |
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| 37 | .. math:: |
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| 38 | |
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[40a87fa] | 39 | I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 |
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| 40 | \left< P(q, \alpha) \right> + \text{background} |
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[5933c7f] | 41 | |
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| 42 | where the volume $V = A B C$, the contrast is defined as |
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[40a87fa] | 43 | $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, |
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[5933c7f] | 44 | $P(q, \alpha)$ is the form factor corresponding to a parallelepiped oriented |
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[40a87fa] | 45 | at an angle $\alpha$ (angle between the long axis C and $\vec q$), |
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[5933c7f] | 46 | and the averaging $\left<\ldots\right>$ is applied over all orientations. |
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| 47 | |
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| 48 | Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the |
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| 49 | form factor is given by (Mittelbach and Porod, 1961) |
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| 50 | |
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| 51 | .. math:: |
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| 52 | |
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| 53 | P(q, \alpha) = \int_0^1 \phi_Q\left(\mu \sqrt{1-\sigma^2},a\right) |
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| 54 | \left[S(\mu c \sigma/2)\right]^2 d\sigma |
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| 55 | |
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| 56 | with |
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| 57 | |
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| 58 | .. math:: |
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| 59 | |
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[40a87fa] | 60 | \phi_Q(\mu,a) &= \int_0^1 |
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[5933c7f] | 61 | \left\{S\left[\frac{\mu}{2}\cos\left(\frac{\pi}{2}u\right)\right] |
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| 62 | S\left[\frac{\mu a}{2}\sin\left(\frac{\pi}{2}u\right)\right] |
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| 63 | \right\}^2 du |
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| 64 | |
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[40a87fa] | 65 | S(x) &= \frac{\sin x}{x} |
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[5933c7f] | 66 | |
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[40a87fa] | 67 | \mu &= qB |
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[5933c7f] | 68 | |
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| 69 | |
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| 70 | The scattering intensity per unit volume is returned in units of |cm^-1|. |
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| 71 | |
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| 72 | NB: The 2nd virial coefficient of the parallelepiped is calculated based on |
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| 73 | the averaged effective radius $(=\sqrt{A B / \pi})$ and |
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| 74 | length $(= C)$ values, and used as the effective radius for |
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| 75 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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| 76 | |
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| 77 | To provide easy access to the orientation of the parallelepiped, we define |
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| 78 | three angles $\theta$, $\phi$ and $\Psi$. The definition of $\theta$ and |
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| 79 | $\phi$ is the same as for the cylinder model (see also figures below). |
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| 80 | |
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| 81 | .. Comment by Miguel Gonzalez: |
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| 82 | The following text has been commented because I think there are two |
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| 83 | mistakes. Psi is the rotational angle around C (but I cannot understand |
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| 84 | what it means against the q plane) and psi=0 corresponds to a||x and b||y. |
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| 85 | |
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| 86 | The angle $\Psi$ is the rotational angle around the $C$ axis against |
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| 87 | the $q$ plane. For example, $\Psi = 0$ when the $B$ axis is parallel |
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| 88 | to the $x$-axis of the detector. |
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| 89 | |
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| 90 | The angle $\Psi$ is the rotational angle around the $C$ axis. |
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| 91 | For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis |
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| 92 | oriented parallel to the y-axis of the detector with $A$ along the z-axis. |
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| 93 | For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated |
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| 94 | $\theta$ degrees around $z$ and $\phi$ degrees around $y$, |
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| 95 | before doing a final rotation of $\Psi$ degrees around the resulting $C$ to |
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| 96 | obtain the final orientation of the parallelepiped. |
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| 97 | For example, for $\theta = 0$ and $\phi = 90$, we have that $\Psi = 0$ |
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| 98 | corresponds to $A$ along $x$ and $B$ along $y$, |
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| 99 | while for $\theta = 90$ and $\phi = 0$, $\Psi = 0$ corresponds to |
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| 100 | $A$ along $z$ and $B$ along $x$. |
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| 101 | |
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| 102 | .. _parallelepiped-orientation: |
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| 103 | |
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| 104 | .. figure:: img/parallelepiped_angle_definition.jpg |
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| 105 | |
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| 106 | Definition of the angles for oriented parallelepipeds. |
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| 107 | |
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| 108 | .. figure:: img/parallelepiped_angle_projection.jpg |
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| 109 | |
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[40a87fa] | 110 | Examples of the angles for oriented parallelepipeds against the |
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| 111 | detector plane. |
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[5933c7f] | 112 | |
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[40a87fa] | 113 | For a given orientation of the parallelepiped, the 2D form factor is |
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| 114 | calculated as |
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[5933c7f] | 115 | |
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| 116 | .. math:: |
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| 117 | |
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[40a87fa] | 118 | P(q_x, q_y) = \left[\frac{\sin(qA\cos\alpha/2)}{(qA\cos\alpha/2)}\right]^2 |
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| 119 | \left[\frac{\sin(qB\cos\beta/2)}{(qB\cos\beta/2)}\right]^2 |
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| 120 | \left[\frac{\sin(qC\cos\gamma/2)}{(qC\cos\gamma/2)}\right]^2 |
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[5933c7f] | 121 | |
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| 122 | with |
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| 123 | |
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| 124 | .. math:: |
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| 125 | |
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[40a87fa] | 126 | \cos\alpha &= \hat A \cdot \hat q, |
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| 127 | |
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| 128 | \cos\beta &= \hat B \cdot \hat q, |
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| 129 | |
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| 130 | \cos\gamma &= \hat C \cdot \hat q |
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[5933c7f] | 131 | |
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| 132 | and the scattering intensity as: |
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| 133 | |
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| 134 | .. math:: |
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| 135 | |
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[40a87fa] | 136 | I(q_x, q_y) = \frac{\text{scale}}{V} V^2 \Delta\rho^2 P(q_x, q_y) |
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| 137 | + \text{background} |
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[5933c7f] | 138 | |
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| 139 | .. Comment by Miguel Gonzalez: |
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| 140 | This reflects the logic of the code, as in parallelepiped.c the call |
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[40a87fa] | 141 | to _pkernel returns $P(q_x, q_y)$ and then this is multiplied by |
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| 142 | $V^2 * (\Delta \rho)^2$. And finally outside parallelepiped.c it will be |
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| 143 | multiplied by scale, normalized by $V$ and the background added. But |
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| 144 | mathematically it makes more sense to write |
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| 145 | $I(q_x, q_y) = \text{scale} V \Delta\rho^2 P(q_x, q_y) + \text{background}$, |
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[5933c7f] | 146 | with scale being the volume fraction. |
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| 147 | |
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| 148 | |
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| 149 | Validation |
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| 150 | ---------- |
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| 151 | |
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| 152 | Validation of the code was done by comparing the output of the 1D calculation |
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| 153 | to the angular average of the output of a 2D calculation over all possible |
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[40a87fa] | 154 | angles. |
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[5933c7f] | 155 | |
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| 156 | This model is based on form factor calculations implemented in a c-library |
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| 157 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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| 158 | |
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| 159 | References |
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| 160 | ---------- |
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| 161 | |
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[76e5041] | 162 | P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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| 163 | |
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| 164 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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[5933c7f] | 165 | """ |
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| 166 | |
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| 167 | import numpy as np |
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| 168 | from numpy import pi, inf, sqrt |
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| 169 | |
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| 170 | name = "parallelepiped" |
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| 171 | title = "Rectangular parallelepiped with uniform scattering length density." |
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| 172 | description = """ |
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[76e5041] | 173 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,alpha)+background |
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[5933c7f] | 174 | P(q,alpha) = integral from 0 to 1 of ... |
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| 175 | phi(mu*sqrt(1-sigma^2),a) * S(mu*c*sigma/2)^2 * dsigma |
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| 176 | with |
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| 177 | phi(mu,a) = integral from 0 to 1 of .. |
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| 178 | (S((mu/2)*cos(pi*u/2))*S((mu*a/2)*sin(pi*u/2)))^2 * du |
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| 179 | S(x) = sin(x)/x |
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| 180 | mu = q*B |
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| 181 | V: Volume of the rectangular parallelepiped |
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| 182 | alpha: angle between the long axis of the |
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| 183 | parallelepiped and the q-vector for 1D |
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| 184 | """ |
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| 185 | category = "shape:parallelepiped" |
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| 186 | |
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| 187 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 188 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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[5933c7f] | 189 | "Parallelepiped scattering length density"], |
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[42356c8] | 190 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[5933c7f] | 191 | "Solvent scattering length density"], |
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[a807206] | 192 | ["length_a", "Ang", 35, [0, inf], "volume", |
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[5933c7f] | 193 | "Shorter side of the parallelepiped"], |
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[a807206] | 194 | ["length_b", "Ang", 75, [0, inf], "volume", |
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[5933c7f] | 195 | "Second side of the parallelepiped"], |
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[a807206] | 196 | ["length_c", "Ang", 400, [0, inf], "volume", |
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[5933c7f] | 197 | "Larger side of the parallelepiped"], |
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| 198 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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| 199 | "In plane angle"], |
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| 200 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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| 201 | "Out of plane angle"], |
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| 202 | ["psi", "degrees", 60, [-inf, inf], "orientation", |
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| 203 | "Rotation angle around its own c axis against q plane"], |
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| 204 | ] |
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| 205 | |
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| 206 | source = ["lib/gauss76.c", "parallelepiped.c"] |
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| 207 | |
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[a807206] | 208 | def ER(length_a, length_b, length_c): |
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[5933c7f] | 209 | """ |
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| 210 | Return effective radius (ER) for P(q)*S(q) |
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| 211 | """ |
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| 212 | |
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| 213 | # surface average radius (rough approximation) |
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[a807206] | 214 | surf_rad = sqrt(length_a * length_b / pi) |
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[5933c7f] | 215 | |
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[a807206] | 216 | ddd = 0.75 * surf_rad * (2 * surf_rad * length_c + (length_c + surf_rad) * (length_c + pi * surf_rad)) |
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[5933c7f] | 217 | return 0.5 * (ddd) ** (1. / 3.) |
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| 218 | |
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| 219 | # VR defaults to 1.0 |
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| 220 | |
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| 221 | # parameters for demo |
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| 222 | demo = dict(scale=1, background=0, |
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[76e5041] | 223 | sld=6.3e-6, sld_solvent=1.0e-6, |
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[a807206] | 224 | length_a=35, length_b=75, length_c=400, |
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[5933c7f] | 225 | theta=45, phi=30, psi=15, |
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[a807206] | 226 | length_a_pd=0.1, length_a_pd_n=10, |
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| 227 | length_b_pd=0.1, length_b_pd_n=1, |
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| 228 | length_c_pd=0.1, length_c_pd_n=1, |
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[5933c7f] | 229 | theta_pd=10, theta_pd_n=1, |
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| 230 | phi_pd=10, phi_pd_n=1, |
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| 231 | psi_pd=10, psi_pd_n=10) |
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| 232 | |
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| 233 | qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5) |
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| 234 | tests = [[{}, 0.2, 0.17758004974], |
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| 235 | [{}, [0.2], [0.17758004974]], |
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| 236 | [{'theta':10.0, 'phi':10.0}, (qx, qy), 0.00560296014], |
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| 237 | [{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.00560296014]], |
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| 238 | ] |
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| 239 | del qx, qy # not necessary to delete, but cleaner |
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