[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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[fc7bcd5] | 7 | This model calculates the scattering from a rectangular solid |
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[b343226] | 8 | (:numref:`parallelepiped-image`). |
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| 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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[5bc6d21] | 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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[96153e4] | 21 | given in *any* size order as long as the particles are randomly oriented (i.e. |
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| 22 | take on all possible orientations see notes on 2D below). To avoid multiple fit |
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| 23 | solutions, especially with Monte-Carlo fit methods, it may be advisable to |
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| 24 | restrict their ranges. There may be a number of closely similar "best fits", so |
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| 25 | some trial and error, or fixing of some dimensions at expected values, may |
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| 26 | help. |
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[3330bb4] | 27 | |
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[5bc6d21] | 28 | The form factor is normalized by the particle volume and the 1D scattering |
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| 29 | intensity $I(q)$ is then calculated as: |
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[3330bb4] | 30 | |
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| 31 | .. Comment by Miguel Gonzalez: |
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| 32 | I am modifying the original text because I find the notation a little bit |
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| 33 | confusing. I think that in most textbooks/papers, the notation P(Q) is |
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| 34 | used for the form factor (adim, P(Q=0)=1), although F(q) seems also to |
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| 35 | be used. But here (as for many other models), P(q) is used to represent |
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| 36 | the scattering intensity (in cm-1 normally). It would be good to agree on |
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| 37 | a common notation. |
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| 38 | |
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| 39 | .. math:: |
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| 40 | |
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| 41 | I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 |
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[dbf1a60] | 42 | \left< P(q, \alpha, \beta) \right> + \text{background} |
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[3330bb4] | 43 | |
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| 44 | where the volume $V = A B C$, the contrast is defined as |
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[dbf1a60] | 45 | $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ |
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| 46 | is the form factor corresponding to a parallelepiped oriented |
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| 47 | at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ |
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[b343226] | 48 | (the angle between the projection of the particle in the $xy$ detector plane |
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[dbf1a60] | 49 | and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all |
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| 50 | orientations. |
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[3330bb4] | 51 | |
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| 52 | Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the |
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[dbf1a60] | 53 | form factor is given by (Mittelbach and Porod, 1961 [#Mittelbach]_) |
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[3330bb4] | 54 | |
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| 55 | .. math:: |
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| 56 | |
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| 57 | P(q, \alpha) = \int_0^1 \phi_Q\left(\mu \sqrt{1-\sigma^2},a\right) |
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| 58 | \left[S(\mu c \sigma/2)\right]^2 d\sigma |
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| 59 | |
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| 60 | with |
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| 61 | |
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| 62 | .. math:: |
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| 63 | |
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| 64 | \phi_Q(\mu,a) &= \int_0^1 |
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| 65 | \left\{S\left[\frac{\mu}{2}\cos\left(\frac{\pi}{2}u\right)\right] |
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| 66 | S\left[\frac{\mu a}{2}\sin\left(\frac{\pi}{2}u\right)\right] |
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[ca04add] | 67 | \right\}^2 du \\ |
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| 68 | S(x) &= \frac{\sin x}{x} \\ |
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[3330bb4] | 69 | \mu &= qB |
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| 70 | |
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[dbf1a60] | 71 | where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 \ u$ have been |
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| 72 | applied. |
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[3330bb4] | 73 | |
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[fc7bcd5] | 74 | For **oriented** particles, the 2D scattering intensity, $I(q_x, q_y)$, is |
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| 75 | given as: |
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[3330bb4] | 76 | |
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[fc7bcd5] | 77 | .. math:: |
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[3330bb4] | 78 | |
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[fc7bcd5] | 79 | I(q_x, q_y) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 P(q_x, q_y) |
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| 80 | + \text{background} |
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[3330bb4] | 81 | |
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[fc7bcd5] | 82 | .. Comment by Miguel Gonzalez: |
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| 83 | This reflects the logic of the code, as in parallelepiped.c the call |
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| 84 | to _pkernel returns $P(q_x, q_y)$ and then this is multiplied by |
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| 85 | $V^2 * (\Delta \rho)^2$. And finally outside parallelepiped.c it will be |
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| 86 | multiplied by scale, normalized by $V$ and the background added. But |
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| 87 | mathematically it makes more sense to write |
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| 88 | $I(q_x, q_y) = \text{scale} V \Delta\rho^2 P(q_x, q_y) + \text{background}$, |
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| 89 | with scale being the volume fraction. |
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[9802ab3] | 90 | |
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[fc7bcd5] | 91 | Where $P(q_x, q_y)$ for a given orientation of the form factor is calculated as |
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[3330bb4] | 92 | |
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| 93 | .. math:: |
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| 94 | |
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[dbf1a60] | 95 | P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} |
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| 96 | {2}qA\cos\alpha)}\right]^2 |
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| 97 | \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} |
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| 98 | {2}qB\cos\beta)}\right]^2 |
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| 99 | \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} |
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| 100 | {2}qC\cos\gamma)}\right]^2 |
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[3330bb4] | 101 | |
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| 102 | with |
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| 103 | |
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| 104 | .. math:: |
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| 105 | |
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[ca04add] | 106 | \cos\alpha &= \hat A \cdot \hat q, \\ |
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| 107 | \cos\beta &= \hat B \cdot \hat q, \\ |
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[3330bb4] | 108 | \cos\gamma &= \hat C \cdot \hat q |
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| 109 | |
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| 110 | |
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[fc7bcd5] | 111 | FITTING NOTES |
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| 112 | ~~~~~~~~~~~~~ |
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| 113 | |
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| 114 | #. The 2nd virial coefficient of the parallelepiped is calculated based on |
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| 115 | the averaged effective radius, after appropriately sorting the three |
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| 116 | dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and |
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| 117 | length $(= C)$ values, and used as the effective radius for |
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| 118 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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| 119 | |
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| 120 | #. For 2d data the orientation of the particle is required, described using |
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| 121 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ |
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| 122 | and $\phi$ define the orientation of the director in the laboratry reference |
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| 123 | frame of the beam direction ($z$) and detector plane ($x-y$ plane), while |
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| 124 | the angle $\Psi$ is effectively the rotational angle around the particle |
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| 125 | $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the |
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| 126 | $B$ axis oriented parallel to the y-axis of the detector with $A$ along |
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| 127 | the x-axis. For other $\theta$, $\phi$ values, the order of rotations |
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| 128 | matters. In particular, the parallelepiped must first be rotated $\theta$ |
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| 129 | degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ |
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| 130 | axis (in the $x-y$ plane). Applying orientational distribution to the |
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| 131 | particle orientation (i.e `jitter` to one or more of these angles) can get |
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| 132 | more confusing as `jitter` is defined **NOT** with respect to the laboratory |
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| 133 | frame but the particle reference frame. It is thus highly recmmended to |
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| 134 | read :ref:`orientation` for further details of the calculation and angular |
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| 135 | dispersions. |
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[3330bb4] | 136 | |
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[fc7bcd5] | 137 | .. note:: For 2d, constraints must be applied during fitting to ensure that the |
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| 138 | order of sides chosen is not altered, and hence that the correct definition |
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| 139 | of angles is preserved. For the default choice shown here, that means |
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| 140 | ensuring that the inequality $A < B < C$ is not violated, The calculation |
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| 141 | will not report an error, but the results may be not correct. |
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[ee60aa7] | 142 | |
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[fc7bcd5] | 143 | .. _parallelepiped-orientation: |
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[3330bb4] | 144 | |
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[fc7bcd5] | 145 | .. figure:: img/parallelepiped_angle_definition.png |
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| 146 | |
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| 147 | Definition of the angles for oriented parallelepiped, shown with $A<B<C$. |
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| 148 | |
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| 149 | .. figure:: img/parallelepiped_angle_projection.png |
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| 150 | |
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| 151 | Examples of the angles for an oriented parallelepiped against the |
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| 152 | detector plane. |
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| 153 | |
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| 154 | .. Comment by Paul Butler |
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| 155 | I am commenting this section out as we are trying to minimize the amount of |
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| 156 | oritentational detail here and encourage the user to go to the full |
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| 157 | orientation documentation so that changes can be made in just one place. |
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| 158 | below is the commented paragrah: |
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| 159 | On introducing "Orientational Distribution" in the angles, "distribution of |
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| 160 | theta" and "distribution of phi" parameters will appear. These are actually |
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| 161 | rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, |
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| 162 | perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ |
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| 163 | these are parallel to the $Y$ and $X$ axes of the instrument.) The third |
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| 164 | orientation distribution, in $\psi$, is about the $c$ axis of the particle, |
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| 165 | perpendicular to the $a$ x $b$ face. Some experimentation may be required to |
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| 166 | understand the 2d patterns fully as discussed in :ref:`orientation` . |
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[3330bb4] | 167 | |
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| 168 | |
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| 169 | Validation |
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| 170 | ---------- |
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| 171 | |
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| 172 | Validation of the code was done by comparing the output of the 1D calculation |
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| 173 | to the angular average of the output of a 2D calculation over all possible |
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| 174 | angles. |
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| 175 | |
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| 176 | References |
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| 177 | ---------- |
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| 178 | |
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[dbf1a60] | 179 | .. [#Mittelbach] P Mittelbach and G Porod, *Acta Physica Austriaca*, |
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| 180 | 14 (1961) 185-211 |
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| 181 | .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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[0507e09] | 182 | .. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659 |
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| 183 | |
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| 184 | Source |
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| 185 | ------ |
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| 186 | |
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| 187 | `parallelepiped.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/parallelepiped.py>`_ |
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| 188 | |
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| 189 | `parallelepiped.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/parallelepiped.c>`_ |
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[3fd0499] | 190 | |
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| 191 | Authorship and Verification |
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| 192 | ---------------------------- |
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| 193 | |
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[ef07e95] | 194 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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[3fd0499] | 195 | * **Last Modified by:** Paul Kienzle **Date:** April 05, 2017 |
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[f89ec96] | 196 | * **Last Reviewed by:** Miguel Gonzales and Paul Butler **Date:** May 24, |
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| 197 | 2018 - documentation updated |
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[0507e09] | 198 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[3330bb4] | 199 | """ |
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| 200 | |
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| 201 | import numpy as np |
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[b297ba9] | 202 | from numpy import inf |
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[3330bb4] | 203 | |
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| 204 | name = "parallelepiped" |
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| 205 | title = "Rectangular parallelepiped with uniform scattering length density." |
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| 206 | description = """ |
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| 207 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,alpha)+background |
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| 208 | P(q,alpha) = integral from 0 to 1 of ... |
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| 209 | phi(mu*sqrt(1-sigma^2),a) * S(mu*c*sigma/2)^2 * dsigma |
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| 210 | with |
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| 211 | phi(mu,a) = integral from 0 to 1 of .. |
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| 212 | (S((mu/2)*cos(pi*u/2))*S((mu*a/2)*sin(pi*u/2)))^2 * du |
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| 213 | S(x) = sin(x)/x |
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| 214 | mu = q*B |
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| 215 | V: Volume of the rectangular parallelepiped |
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[3fd0499] | 216 | alpha: angle between the long axis of the |
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[3330bb4] | 217 | parallelepiped and the q-vector for 1D |
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| 218 | """ |
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| 219 | category = "shape:parallelepiped" |
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| 220 | |
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| 221 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 222 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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| 223 | "Parallelepiped scattering length density"], |
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| 224 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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| 225 | "Solvent scattering length density"], |
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| 226 | ["length_a", "Ang", 35, [0, inf], "volume", |
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| 227 | "Shorter side of the parallelepiped"], |
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| 228 | ["length_b", "Ang", 75, [0, inf], "volume", |
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| 229 | "Second side of the parallelepiped"], |
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| 230 | ["length_c", "Ang", 400, [0, inf], "volume", |
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| 231 | "Larger side of the parallelepiped"], |
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[9b79f29] | 232 | ["theta", "degrees", 60, [-360, 360], "orientation", |
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| 233 | "c axis to beam angle"], |
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| 234 | ["phi", "degrees", 60, [-360, 360], "orientation", |
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| 235 | "rotation about beam"], |
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| 236 | ["psi", "degrees", 60, [-360, 360], "orientation", |
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| 237 | "rotation about c axis"], |
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[3330bb4] | 238 | ] |
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| 239 | |
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| 240 | source = ["lib/gauss76.c", "parallelepiped.c"] |
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[71b751d] | 241 | have_Fq = True |
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[a34b811] | 242 | radius_effective_modes = [ |
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[b297ba9] | 243 | "equivalent cylinder excluded volume", "equivalent volume sphere", |
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[99658f6] | 244 | "half length_a", "half length_b", "half length_c", |
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[ee60aa7] | 245 | "equivalent circular cross-section", "half ab diagonal", "half diagonal", |
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| 246 | ] |
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[31df0c9] | 247 | |
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| 248 | def random(): |
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[b297ba9] | 249 | """Return a random parameter set for the model.""" |
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[8f04da4] | 250 | length = 10**np.random.uniform(1, 4.7, size=3) |
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[31df0c9] | 251 | pars = dict( |
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[8f04da4] | 252 | length_a=length[0], |
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| 253 | length_b=length[1], |
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| 254 | length_c=length[2], |
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[31df0c9] | 255 | ) |
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| 256 | return pars |
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| 257 | |
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| 258 | |
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[3330bb4] | 259 | # parameters for demo |
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| 260 | demo = dict(scale=1, background=0, |
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| 261 | sld=6.3, sld_solvent=1.0, |
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| 262 | length_a=35, length_b=75, length_c=400, |
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| 263 | theta=45, phi=30, psi=15, |
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| 264 | length_a_pd=0.1, length_a_pd_n=10, |
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| 265 | length_b_pd=0.1, length_b_pd_n=1, |
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| 266 | length_c_pd=0.1, length_c_pd_n=1, |
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| 267 | theta_pd=10, theta_pd_n=1, |
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| 268 | phi_pd=10, phi_pd_n=1, |
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| 269 | psi_pd=10, psi_pd_n=10) |
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[2d81cfe] | 270 | # rkh 7/4/17 add random unit test for 2d, note make all params different, |
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| 271 | # 2d values not tested against other codes or models |
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[b297ba9] | 272 | qx, qy = 0.2 * np.cos(np.pi/6.), 0.2 * np.sin(np.pi/6.) |
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[3330bb4] | 273 | tests = [[{}, 0.2, 0.17758004974], |
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| 274 | [{}, [0.2], [0.17758004974]], |
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[3fd0499] | 275 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0089517140475], |
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| 276 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0089517140475]], |
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[3330bb4] | 277 | ] |
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| 278 | del qx, qy # not necessary to delete, but cleaner |
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