[3330bb4] | 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 | Definition |
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| 5 | ---------- |
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| 6 | |
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[3401a7a] | 7 | This model calculates the scattering from a rectangular parallelepiped |
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[5bc373b] | 8 | (:numref:`parallelepiped-image`). |
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[5bc6d21] | 9 | If you need to apply polydispersity, see also :ref:`rectangular-prism`. For |
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| 10 | information about polarised and magnetic scattering, see |
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| 11 | the :ref:`magnetism` documentation. |
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[3330bb4] | 12 | |
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| 13 | .. _parallelepiped-image: |
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| 14 | |
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[3fd0499] | 15 | |
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[3330bb4] | 16 | .. figure:: img/parallelepiped_geometry.jpg |
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| 17 | |
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| 18 | Parallelepiped with the corresponding definition of sides. |
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| 19 | |
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[30b60d2] | 20 | The three dimensions of the parallelepiped (strictly here a cuboid) may be |
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| 21 | given in *any* size order. To avoid multiple fit solutions, especially |
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| 22 | with Monte-Carlo fit methods, it may be advisable to restrict their ranges. |
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| 23 | There may be a number of closely similar "best fits", so some trial and |
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| 24 | error, or fixing of some dimensions at expected values, may help. |
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[3330bb4] | 25 | |
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[5bc6d21] | 26 | The form factor is normalized by the particle volume and the 1D scattering |
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| 27 | intensity $I(q)$ is then calculated as: |
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[3330bb4] | 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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| 39 | I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 |
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[dbf1a60] | 40 | \left< P(q, \alpha, \beta) \right> + \text{background} |
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[3330bb4] | 41 | |
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| 42 | where the volume $V = A B C$, the contrast is defined as |
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[dbf1a60] | 43 | $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ |
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| 44 | is the form factor corresponding to a parallelepiped oriented |
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| 45 | at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ |
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| 46 | ( the angle between the projection of the particle in the $xy$ detector plane |
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| 47 | and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all |
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| 48 | orientations. |
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[3330bb4] | 49 | |
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| 50 | Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the |
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[dbf1a60] | 51 | form factor is given by (Mittelbach and Porod, 1961 [#Mittelbach]_) |
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[3330bb4] | 52 | |
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| 53 | .. math:: |
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| 54 | |
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| 55 | P(q, \alpha) = \int_0^1 \phi_Q\left(\mu \sqrt{1-\sigma^2},a\right) |
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| 56 | \left[S(\mu c \sigma/2)\right]^2 d\sigma |
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| 57 | |
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| 58 | with |
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| 59 | |
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| 60 | .. math:: |
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| 61 | |
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| 62 | \phi_Q(\mu,a) &= \int_0^1 |
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| 63 | \left\{S\left[\frac{\mu}{2}\cos\left(\frac{\pi}{2}u\right)\right] |
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| 64 | S\left[\frac{\mu a}{2}\sin\left(\frac{\pi}{2}u\right)\right] |
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[ca04add] | 65 | \right\}^2 du \\ |
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| 66 | S(x) &= \frac{\sin x}{x} \\ |
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[3330bb4] | 67 | \mu &= qB |
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| 68 | |
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[dbf1a60] | 69 | where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 \ u$ have been |
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| 70 | applied. |
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[3330bb4] | 71 | |
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| 72 | NB: The 2nd virial coefficient of the parallelepiped is calculated based on |
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[afd4692] | 73 | the averaged effective radius, after appropriately sorting the three |
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| 74 | dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and |
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[3330bb4] | 75 | length $(= C)$ values, and used as the effective radius for |
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| 76 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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| 77 | |
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[2d81cfe] | 78 | For 2d data the orientation of the particle is required, described using |
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| 79 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details |
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[eda8b30] | 80 | of the calculation and angular dispersions see :ref:`orientation` . |
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[3330bb4] | 81 | |
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| 82 | .. Comment by Miguel Gonzalez: |
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| 83 | The following text has been commented because I think there are two |
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| 84 | mistakes. Psi is the rotational angle around C (but I cannot understand |
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| 85 | what it means against the q plane) and psi=0 corresponds to a||x and b||y. |
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| 86 | |
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| 87 | The angle $\Psi$ is the rotational angle around the $C$ axis against |
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| 88 | the $q$ plane. For example, $\Psi = 0$ when the $B$ axis is parallel |
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| 89 | to the $x$-axis of the detector. |
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| 90 | |
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| 91 | The angle $\Psi$ is the rotational angle around the $C$ axis. |
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| 92 | For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis |
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[eda8b30] | 93 | oriented parallel to the y-axis of the detector with $A$ along the x-axis. |
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[3330bb4] | 94 | For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated |
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[eda8b30] | 95 | $\theta$ degrees in the $z-x$ plane and then $\phi$ degrees around the $z$ axis, |
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| 96 | before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis |
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| 97 | of the particle to obtain the final orientation of the parallelepiped. |
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[3330bb4] | 98 | |
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| 99 | .. _parallelepiped-orientation: |
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| 100 | |
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[3fd0499] | 101 | .. figure:: img/parallelepiped_angle_definition.png |
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[3330bb4] | 102 | |
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[afd4692] | 103 | Definition of the angles for oriented parallelepiped, shown with $A<B<C$. |
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[3330bb4] | 104 | |
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[3fd0499] | 105 | .. figure:: img/parallelepiped_angle_projection.png |
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[3330bb4] | 106 | |
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[3fd0499] | 107 | Examples of the angles for an oriented parallelepiped against the |
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[3330bb4] | 108 | detector plane. |
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| 109 | |
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[2d81cfe] | 110 | On introducing "Orientational Distribution" in the angles, "distribution of |
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| 111 | theta" and "distribution of phi" parameters will appear. These are actually |
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| 112 | rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, |
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| 113 | perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ |
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| 114 | these are parallel to the $Y$ and $X$ axes of the instrument.) The third |
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| 115 | orientation distribution, in $\psi$, is about the $c$ axis of the particle, |
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| 116 | perpendicular to the $a$ x $b$ face. Some experimentation may be required to |
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| 117 | understand the 2d patterns fully as discussed in :ref:`orientation` . |
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[9802ab3] | 118 | |
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[3330bb4] | 119 | For a given orientation of the parallelepiped, the 2D form factor is |
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| 120 | calculated as |
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| 121 | |
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| 122 | .. math:: |
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| 123 | |
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[dbf1a60] | 124 | P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} |
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| 125 | {2}qA\cos\alpha)}\right]^2 |
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| 126 | \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} |
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| 127 | {2}qB\cos\beta)}\right]^2 |
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| 128 | \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} |
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| 129 | {2}qC\cos\gamma)}\right]^2 |
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[3330bb4] | 130 | |
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| 131 | with |
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| 132 | |
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| 133 | .. math:: |
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| 134 | |
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[ca04add] | 135 | \cos\alpha &= \hat A \cdot \hat q, \\ |
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| 136 | \cos\beta &= \hat B \cdot \hat q, \\ |
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[3330bb4] | 137 | \cos\gamma &= \hat C \cdot \hat q |
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| 138 | |
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| 139 | and the scattering intensity as: |
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| 140 | |
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| 141 | .. math:: |
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| 142 | |
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| 143 | I(q_x, q_y) = \frac{\text{scale}}{V} V^2 \Delta\rho^2 P(q_x, q_y) |
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| 144 | + \text{background} |
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| 145 | |
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| 146 | .. Comment by Miguel Gonzalez: |
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| 147 | This reflects the logic of the code, as in parallelepiped.c the call |
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| 148 | to _pkernel returns $P(q_x, q_y)$ and then this is multiplied by |
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| 149 | $V^2 * (\Delta \rho)^2$. And finally outside parallelepiped.c it will be |
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| 150 | multiplied by scale, normalized by $V$ and the background added. But |
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| 151 | mathematically it makes more sense to write |
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| 152 | $I(q_x, q_y) = \text{scale} V \Delta\rho^2 P(q_x, q_y) + \text{background}$, |
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| 153 | with scale being the volume fraction. |
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| 154 | |
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| 155 | |
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| 156 | Validation |
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| 157 | ---------- |
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| 158 | |
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| 159 | Validation of the code was done by comparing the output of the 1D calculation |
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| 160 | to the angular average of the output of a 2D calculation over all possible |
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| 161 | angles. |
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| 162 | |
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| 163 | |
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| 164 | References |
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| 165 | ---------- |
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| 166 | |
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[dbf1a60] | 167 | .. [#Mittelbach] P Mittelbach and G Porod, *Acta Physica Austriaca*, |
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| 168 | 14 (1961) 185-211 |
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| 169 | .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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[3fd0499] | 170 | |
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| 171 | Authorship and Verification |
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| 172 | ---------------------------- |
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| 173 | |
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[ef07e95] | 174 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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[3fd0499] | 175 | * **Last Modified by:** Paul Kienzle **Date:** April 05, 2017 |
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| 176 | * **Last Reviewed by:** Richard Heenan **Date:** April 06, 2017 |
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[3330bb4] | 177 | """ |
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| 178 | |
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| 179 | import numpy as np |
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[3fd0499] | 180 | from numpy import pi, inf, sqrt, sin, cos |
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[3330bb4] | 181 | |
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| 182 | name = "parallelepiped" |
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| 183 | title = "Rectangular parallelepiped with uniform scattering length density." |
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| 184 | description = """ |
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| 185 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,alpha)+background |
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| 186 | P(q,alpha) = integral from 0 to 1 of ... |
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| 187 | phi(mu*sqrt(1-sigma^2),a) * S(mu*c*sigma/2)^2 * dsigma |
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| 188 | with |
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| 189 | phi(mu,a) = integral from 0 to 1 of .. |
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| 190 | (S((mu/2)*cos(pi*u/2))*S((mu*a/2)*sin(pi*u/2)))^2 * du |
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| 191 | S(x) = sin(x)/x |
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| 192 | mu = q*B |
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| 193 | V: Volume of the rectangular parallelepiped |
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[3fd0499] | 194 | alpha: angle between the long axis of the |
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[3330bb4] | 195 | parallelepiped and the q-vector for 1D |
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| 196 | """ |
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| 197 | category = "shape:parallelepiped" |
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| 198 | |
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| 199 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 200 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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| 201 | "Parallelepiped scattering length density"], |
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| 202 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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| 203 | "Solvent scattering length density"], |
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| 204 | ["length_a", "Ang", 35, [0, inf], "volume", |
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| 205 | "Shorter side of the parallelepiped"], |
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| 206 | ["length_b", "Ang", 75, [0, inf], "volume", |
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| 207 | "Second side of the parallelepiped"], |
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| 208 | ["length_c", "Ang", 400, [0, inf], "volume", |
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| 209 | "Larger side of the parallelepiped"], |
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[9b79f29] | 210 | ["theta", "degrees", 60, [-360, 360], "orientation", |
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| 211 | "c axis to beam angle"], |
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| 212 | ["phi", "degrees", 60, [-360, 360], "orientation", |
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| 213 | "rotation about beam"], |
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| 214 | ["psi", "degrees", 60, [-360, 360], "orientation", |
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| 215 | "rotation about c axis"], |
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[3330bb4] | 216 | ] |
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| 217 | |
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| 218 | source = ["lib/gauss76.c", "parallelepiped.c"] |
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| 219 | |
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| 220 | def ER(length_a, length_b, length_c): |
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| 221 | """ |
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[3fd0499] | 222 | Return effective radius (ER) for P(q)*S(q) |
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[3330bb4] | 223 | """ |
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[2d81cfe] | 224 | # now that axes can be in any size order, need to sort a,b,c |
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| 225 | # where a~b and c is either much smaller or much larger |
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[3fd0499] | 226 | abc = np.vstack((length_a, length_b, length_c)) |
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| 227 | abc = np.sort(abc, axis=0) |
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| 228 | selector = (abc[1] - abc[0]) > (abc[2] - abc[1]) |
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| 229 | length = np.where(selector, abc[0], abc[2]) |
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[3330bb4] | 230 | # surface average radius (rough approximation) |
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[2d81cfe] | 231 | radius = sqrt(np.where(~selector, abc[0]*abc[1], abc[1]*abc[2]) / pi) |
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[3330bb4] | 232 | |
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[3fd0499] | 233 | ddd = 0.75 * radius * (2*radius*length + (length + radius)*(length + pi*radius)) |
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[3330bb4] | 234 | return 0.5 * (ddd) ** (1. / 3.) |
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| 235 | |
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| 236 | # VR defaults to 1.0 |
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| 237 | |
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[31df0c9] | 238 | |
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| 239 | def random(): |
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[8f04da4] | 240 | length = 10**np.random.uniform(1, 4.7, size=3) |
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[31df0c9] | 241 | pars = dict( |
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[8f04da4] | 242 | length_a=length[0], |
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| 243 | length_b=length[1], |
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| 244 | length_c=length[2], |
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[31df0c9] | 245 | ) |
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| 246 | return pars |
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| 247 | |
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| 248 | |
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[3330bb4] | 249 | # parameters for demo |
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| 250 | demo = dict(scale=1, background=0, |
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| 251 | sld=6.3, sld_solvent=1.0, |
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| 252 | length_a=35, length_b=75, length_c=400, |
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| 253 | theta=45, phi=30, psi=15, |
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| 254 | length_a_pd=0.1, length_a_pd_n=10, |
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| 255 | length_b_pd=0.1, length_b_pd_n=1, |
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| 256 | length_c_pd=0.1, length_c_pd_n=1, |
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| 257 | theta_pd=10, theta_pd_n=1, |
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| 258 | phi_pd=10, phi_pd_n=1, |
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| 259 | psi_pd=10, psi_pd_n=10) |
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[2d81cfe] | 260 | # rkh 7/4/17 add random unit test for 2d, note make all params different, |
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| 261 | # 2d values not tested against other codes or models |
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[3fd0499] | 262 | qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) |
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[3330bb4] | 263 | tests = [[{}, 0.2, 0.17758004974], |
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| 264 | [{}, [0.2], [0.17758004974]], |
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[3fd0499] | 265 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0089517140475], |
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| 266 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0089517140475]], |
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[3330bb4] | 267 | ] |
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| 268 | del qx, qy # not necessary to delete, but cleaner |
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