[44bd2be] | 1 | r""" |
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[5810f00] | 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[44bd2be] | 5 | Calculates the form factor for a rectangular solid with a core-shell structure. |
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[96153e4] | 6 | The thickness and the scattering length density of the shell or "rim" can be |
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| 7 | different on each (pair) of faces. The three dimensions of the core of the |
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| 8 | parallelepiped (strictly here a cuboid) may be given in *any* size order as |
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| 9 | long as the particles are randomly oriented (i.e. take on all possible |
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[fc7bcd5] | 10 | orientations see notes on 2D below). To avoid multiple fit solutions, |
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| 11 | especially with Monte-Carlo fit methods, it may be advisable to restrict their |
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[96153e4] | 12 | ranges. There may be a number of closely similar "best fits", so some trial and |
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| 13 | error, or fixing of some dimensions at expected values, may help. |
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[cb0dc22] | 14 | |
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[500128b] | 15 | The form factor is normalized by the particle volume $V$ such that |
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[44bd2be] | 16 | |
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[500128b] | 17 | .. math:: |
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| 18 | |
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[fc7bcd5] | 19 | I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle |
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[367886f] | 20 | + \text{background} |
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[44bd2be] | 21 | |
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[500128b] | 22 | where $\langle \ldots \rangle$ is an average over all possible orientations |
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[dbf1a60] | 23 | of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be |
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| 24 | pulled out of the form factor term due to the multiple slds in the model. |
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[44bd2be] | 25 | |
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[96153e4] | 26 | The core of the solid is defined by the dimensions $A$, $B$, $C$ here shown |
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| 27 | such that $A < B < C$. |
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[44bd2be] | 28 | |
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[5bc373b] | 29 | .. figure:: img/parallelepiped_geometry.jpg |
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| 30 | |
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[331870d] | 31 | Core of the core shell parallelepiped with the corresponding definition |
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[5bc373b] | 32 | of sides. |
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| 33 | |
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[44bd2be] | 34 | |
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[500128b] | 35 | There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension |
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| 36 | (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ |
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[dbf1a60] | 37 | $(=t_C)$ faces. The projection in the $AB$ plane is |
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[44bd2be] | 38 | |
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[5bc373b] | 39 | .. figure:: img/core_shell_parallelepiped_projection.jpg |
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| 40 | |
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[331870d] | 41 | AB cut through the core-shell parallelipiped showing the cross secion of |
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| 42 | four of the six shell slabs. As can be seen, this model leaves **"gaps"** |
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[dbf1a60] | 43 | at the corners of the solid. |
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[44bd2be] | 44 | |
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[dbf1a60] | 45 | |
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| 46 | The total volume of the solid is thus given as |
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[44bd2be] | 47 | |
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| 48 | .. math:: |
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| 49 | |
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| 50 | V = ABC + 2t_ABC + 2t_BAC + 2t_CAB |
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| 51 | |
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[5810f00] | 52 | The intensity calculated follows the :ref:`parallelepiped` model, with the |
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| 53 | core-shell intensity being calculated as the square of the sum of the |
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[dbf1a60] | 54 | amplitudes of the core and the slabs on the edges. The scattering amplitude is |
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| 55 | computed for a particular orientation of the core-shell parallelepiped with |
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| 56 | respect to the scattering vector and then averaged over all possible |
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| 57 | orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis |
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| 58 | of the parallelepiped, and $\beta$ is the angle between the projection of the |
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| 59 | particle in the $xy$ detector plane and the $y$ axis. |
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[44bd2be] | 60 | |
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[5810f00] | 61 | .. math:: |
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[4493288] | 62 | |
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[dbf1a60] | 63 | P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha |
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| 64 | \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta} |
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[367886f] | 65 | |
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| 66 | and |
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| 67 | |
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| 68 | .. math:: |
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| 69 | |
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[dbf1a60] | 70 | F(q,\alpha,\beta) |
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[4493288] | 71 | &= (\rho_\text{core}-\rho_\text{solvent}) |
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| 72 | S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ |
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| 73 | &+ (\rho_\text{A}-\rho_\text{solvent}) |
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[5bc373b] | 74 | \left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\ |
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[4493288] | 75 | &+ (\rho_\text{B}-\rho_\text{solvent}) |
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| 76 | S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ |
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| 77 | &+ (\rho_\text{C}-\rho_\text{solvent}) |
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| 78 | S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right] |
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[393facf] | 79 | |
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| 80 | with |
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[5810f00] | 81 | |
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[393facf] | 82 | .. math:: |
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[5810f00] | 83 | |
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[331870d] | 84 | S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L} |
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[4493288] | 85 | |
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| 86 | and |
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| 87 | |
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| 88 | .. math:: |
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[5810f00] | 89 | |
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[5bc373b] | 90 | Q_A &= q \sin\alpha \sin\beta \\ |
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| 91 | Q_B &= q \sin\alpha \cos\beta \\ |
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| 92 | Q_C &= q \cos\alpha |
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[4493288] | 93 | |
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| 94 | |
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| 95 | where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$ |
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[331870d] | 96 | are the scattering lengths of the parallelepiped core, and the rectangular |
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[4493288] | 97 | slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$ |
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| 98 | is the scattering length of the solvent. |
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[44bd2be] | 99 | |
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[dbf1a60] | 100 | .. note:: |
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| 101 | |
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| 102 | the code actually implements two substitutions: $d(cos\alpha)$ is |
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| 103 | substituted for -$sin\alpha \ d\alpha$ (note that in the |
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| 104 | :ref:`parallelepiped` code this is explicitly implemented with |
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| 105 | $\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that |
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| 106 | $du = \pi/2 \ d\beta$. Thus both integrals go from 0 to 1 rather than 0 |
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| 107 | to $\pi/2$. |
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| 108 | |
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[44bd2be] | 109 | FITTING NOTES |
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[4493288] | 110 | ~~~~~~~~~~~~~ |
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| 111 | |
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[fc7bcd5] | 112 | #. There are many parameters in this model. Hold as many fixed as possible with |
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| 113 | known values, or you will certainly end up at a solution that is unphysical. |
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| 114 | |
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| 115 | #. The 2nd virial coefficient of the core_shell_parallelepiped is calculated |
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| 116 | based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ |
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| 117 | and length $(C+2t_C)$ values, after appropriately sorting the three |
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| 118 | dimensions to give an oblate or prolate particle, to give an effective radius |
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| 119 | for $S(q)$ when $P(q) * S(q)$ is applied. |
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| 120 | |
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| 121 | #. For 2d data the orientation of the particle is required, described using |
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| 122 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ |
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| 123 | and $\phi$ define the orientation of the director in the laboratry reference |
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| 124 | frame of the beam direction ($z$) and detector plane ($x-y$ plane), while |
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| 125 | the angle $\Psi$ is effectively the rotational angle around the particle |
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| 126 | $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the |
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| 127 | $B$ axis oriented parallel to the y-axis of the detector with $A$ along |
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| 128 | the x-axis. For other $\theta$, $\phi$ values, the order of rotations |
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| 129 | matters. In particular, the parallelepiped must first be rotated $\theta$ |
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| 130 | degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ |
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| 131 | axis (in the $x-y$ plane). Applying orientational distribution to the |
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| 132 | particle orientation (i.e `jitter` to one or more of these angles) can get |
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| 133 | more confusing as `jitter` is defined **NOT** with respect to the laboratory |
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| 134 | frame but the particle reference frame. It is thus highly recmmended to |
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| 135 | read :ref:`orientation` for further details of the calculation and angular |
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| 136 | dispersions. |
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[44bd2be] | 137 | |
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[dbf1a60] | 138 | .. note:: For 2d, constraints must be applied during fitting to ensure that the |
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[fc7bcd5] | 139 | order of sides chosen is not altered, and hence that the correct definition |
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| 140 | of angles is preserved. For the default choice shown here, that means |
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| 141 | ensuring that the inequality $A < B < C$ is not violated, The calculation |
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| 142 | will not report an error, but the results may be not correct. |
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[393facf] | 143 | |
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[15a90c1] | 144 | .. figure:: img/parallelepiped_angle_definition.png |
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[44bd2be] | 145 | |
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| 146 | Definition of the angles for oriented core-shell parallelepipeds. |
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[fc7bcd5] | 147 | Note that rotation $\theta$, initially in the $x-z$ plane, is carried |
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[2d81cfe] | 148 | out first, then rotation $\phi$ about the $z$ axis, finally rotation |
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[fc7bcd5] | 149 | $\Psi$ is now around the $C$ axis of the particle. The neutron or X-ray |
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| 150 | beam is along the $z$ axis and the detecotr defines the $x-y$ plane. |
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[44bd2be] | 151 | |
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[1916c52] | 152 | .. figure:: img/parallelepiped_angle_projection.png |
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[44bd2be] | 153 | |
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| 154 | Examples of the angles for oriented core-shell parallelepipeds against the |
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| 155 | detector plane. |
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| 156 | |
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[5bc6d21] | 157 | |
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| 158 | Validation |
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| 159 | ---------- |
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| 160 | |
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| 161 | Cross-checked against hollow rectangular prism and rectangular prism for equal |
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| 162 | thickness overlapping sides, and by Monte Carlo sampling of points within the |
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| 163 | shape for non-uniform, non-overlapping sides. |
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| 164 | |
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| 165 | |
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[aa2edb2] | 166 | References |
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| 167 | ---------- |
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[44bd2be] | 168 | |
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[5810f00] | 169 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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| 170 | Equations (1), (13-14). (in German) |
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| 171 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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[fc0b7aa] | 172 | lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available |
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[5810f00] | 173 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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| 174 | =26379>`_ |
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| 175 | |
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| 176 | Authorship and Verification |
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| 177 | ---------------------------- |
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[44bd2be] | 178 | |
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[5810f00] | 179 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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[331870d] | 180 | * **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016 |
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[97be877] | 181 | * **Last Modified by:** Paul Kienzle **Date:** October 17, 2017 |
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[f89ec96] | 182 | * **Last Reviewed by:** Paul Butler **Date:** May 24, 2018 - documentation |
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| 183 | updated |
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[44bd2be] | 184 | """ |
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| 185 | |
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| 186 | import numpy as np |
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[14207bb] | 187 | from numpy import pi, inf, sqrt, cos, sin |
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[44bd2be] | 188 | |
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| 189 | name = "core_shell_parallelepiped" |
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| 190 | title = "Rectangular solid with a core-shell structure." |
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| 191 | description = """ |
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[8f04da4] | 192 | P(q)= |
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[44bd2be] | 193 | """ |
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| 194 | category = "shape:parallelepiped" |
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| 195 | |
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| 196 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 197 | parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[44bd2be] | 198 | "Parallelepiped core scattering length density"], |
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[42356c8] | 199 | ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 200 | "Parallelepiped A rim scattering length density"], |
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[42356c8] | 201 | ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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[44bd2be] | 202 | "Parallelepiped B rim scattering length density"], |
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[42356c8] | 203 | ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 204 | "Parallelepiped C rim scattering length density"], |
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[42356c8] | 205 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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[44bd2be] | 206 | "Solvent scattering length density"], |
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[2222134] | 207 | ["length_a", "Ang", 35, [0, inf], "volume", |
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[44bd2be] | 208 | "Shorter side of the parallelepiped"], |
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[2222134] | 209 | ["length_b", "Ang", 75, [0, inf], "volume", |
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[44bd2be] | 210 | "Second side of the parallelepiped"], |
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[2222134] | 211 | ["length_c", "Ang", 400, [0, inf], "volume", |
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[44bd2be] | 212 | "Larger side of the parallelepiped"], |
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[2222134] | 213 | ["thick_rim_a", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 214 | "Thickness of A rim"], |
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[2222134] | 215 | ["thick_rim_b", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 216 | "Thickness of B rim"], |
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[2222134] | 217 | ["thick_rim_c", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 218 | "Thickness of C rim"], |
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[9b79f29] | 219 | ["theta", "degrees", 0, [-360, 360], "orientation", |
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| 220 | "c axis to beam angle"], |
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| 221 | ["phi", "degrees", 0, [-360, 360], "orientation", |
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| 222 | "rotation about beam"], |
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| 223 | ["psi", "degrees", 0, [-360, 360], "orientation", |
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| 224 | "rotation about c axis"], |
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[44bd2be] | 225 | ] |
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| 226 | |
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[43b7eea] | 227 | source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] |
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[44bd2be] | 228 | |
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| 229 | |
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[2222134] | 230 | def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c): |
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[44bd2be] | 231 | """ |
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| 232 | Return equivalent radius (ER) |
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| 233 | """ |
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[10ee838] | 234 | from .parallelepiped import ER as ER_p |
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[44bd2be] | 235 | |
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[10ee838] | 236 | a = length_a + 2*thick_rim_a |
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| 237 | b = length_b + 2*thick_rim_b |
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| 238 | c = length_c + 2*thick_rim_c |
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| 239 | return ER_p(a, b, c) |
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[44bd2be] | 240 | |
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| 241 | # VR defaults to 1.0 |
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| 242 | |
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[8f04da4] | 243 | def random(): |
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| 244 | outer = 10**np.random.uniform(1, 4.7, size=3) |
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| 245 | thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1 |
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| 246 | length = outer - thick |
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| 247 | pars = dict( |
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| 248 | length_a=length[0], |
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| 249 | length_b=length[1], |
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| 250 | length_c=length[2], |
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| 251 | thick_rim_a=thick[0], |
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| 252 | thick_rim_b=thick[1], |
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| 253 | thick_rim_c=thick[2], |
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| 254 | ) |
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| 255 | return pars |
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| 256 | |
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[44bd2be] | 257 | # parameters for demo |
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| 258 | demo = dict(scale=1, background=0.0, |
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[14838a3] | 259 | sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, |
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[2222134] | 260 | length_a=35, length_b=75, length_c=400, |
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| 261 | thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, |
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[44bd2be] | 262 | theta=0, phi=0, psi=0, |
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[2222134] | 263 | length_a_pd=0.1, length_a_pd_n=1, |
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| 264 | length_b_pd=0.1, length_b_pd_n=1, |
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| 265 | length_c_pd=0.1, length_c_pd_n=1, |
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| 266 | thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, |
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| 267 | thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, |
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| 268 | thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, |
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[44bd2be] | 269 | theta_pd=10, theta_pd_n=1, |
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| 270 | phi_pd=10, phi_pd_n=1, |
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[14838a3] | 271 | psi_pd=10, psi_pd_n=1) |
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[44bd2be] | 272 | |
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[4493288] | 273 | # rkh 7/4/17 add random unit test for 2d, note make all params different, |
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| 274 | # 2d values not tested against other codes or models |
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[fa70e04] | 275 | if 0: # pak: model rewrite; need to update tests |
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| 276 | qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) |
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| 277 | tests = [[{}, 0.2, 0.533149288477], |
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[2d81cfe] | 278 | [{}, [0.2], [0.533149288477]], |
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| 279 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222], |
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| 280 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]], |
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[fa70e04] | 281 | ] |
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| 282 | del qx, qy # not necessary to delete, but cleaner |
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