Changeset fc7bcd5 in sasmodels
 Timestamp:
 May 26, 2018 5:29:07 PM (4 years ago)
 Branches:
 master, core_shell_microgels, magnetic_model, ticket1257vesicleproduct, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
 Children:
 f89ec96
 Parents:
 96153e4
 Location:
 sasmodels/models
 Files:

 2 edited
Legend:
 Unmodified
 Added
 Removed

sasmodels/models/core_shell_parallelepiped.py
r96153e4 rfc7bcd5 8 8 parallelepiped (strictly here a cuboid) may be given in *any* size order as 9 9 long as the particles are randomly oriented (i.e. take on all possible 10 orientations see notes on 2D below). To avoid multiple fit solutions, e11 specially with MonteCarlo fit methods, it may be advisable to restrict their10 orientations see notes on 2D below). To avoid multiple fit solutions, 11 especially with MonteCarlo fit methods, it may be advisable to restrict their 12 12 ranges. There may be a number of closely similar "best fits", so some trial and 13 13 error, or fixing of some dimensions at expected values, may help. … … 17 17 .. math:: 18 18 19 I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle 19 I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle 20 20 + \text{background} 21 21 … … 110 110 ~~~~~~~~~~~~~ 111 111 112 There are many parameters in this model. Hold as many fixed as possible with 113 known values, or you will certainly end up at a solution that is unphysical. 114 115 116 NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated 117 based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ 118 and length $(C+2t_C)$ values, after appropriately sorting the three dimensions 119 to give an oblate or prolate particle, to give an effective radius 120 for $S(q)$ when $P(q) * S(q)$ is applied. 121 122 For 2d data the orientation of the particle is required, described using 123 angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details 124 of the calculation and angular dispersions see :ref:`orientation` . 125 126 The angle $\Psi$ is the rotational angle around the $C$ axis. 127 For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis 128 oriented parallel to the yaxis of the detector with $A$ along the xaxis. 129 For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated 130 $\theta$ degrees in the $zx$ plane and then $\phi$ degrees around the $z$ axis, 131 before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis 132 of the particle to obtain the final orientation of the parallelepiped. 112 #. There are many parameters in this model. Hold as many fixed as possible with 113 known values, or you will certainly end up at a solution that is unphysical. 114 115 #. The 2nd virial coefficient of the core_shell_parallelepiped is calculated 116 based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ 117 and length $(C+2t_C)$ values, after appropriately sorting the three 118 dimensions to give an oblate or prolate particle, to give an effective radius 119 for $S(q)$ when $P(q) * S(q)$ is applied. 120 121 #. For 2d data the orientation of the particle is required, described using 122 angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ 123 and $\phi$ define the orientation of the director in the laboratry reference 124 frame of the beam direction ($z$) and detector plane ($xy$ plane), while 125 the angle $\Psi$ is effectively the rotational angle around the particle 126 $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the 127 $B$ axis oriented parallel to the yaxis of the detector with $A$ along 128 the xaxis. For other $\theta$, $\phi$ values, the order of rotations 129 matters. In particular, the parallelepiped must first be rotated $\theta$ 130 degrees in the $xz$ plane before rotating $\phi$ degrees around the $z$ 131 axis (in the $xy$ plane). Applying orientational distribution to the 132 particle orientation (i.e `jitter` to one or more of these angles) can get 133 more confusing as `jitter` is defined **NOT** with respect to the laboratory 134 frame but the particle reference frame. It is thus highly recmmended to 135 read :ref:`orientation` for further details of the calculation and angular 136 dispersions. 133 137 134 138 .. note:: For 2d, constraints must be applied during fitting to ensure that the 135 inequality $A < B < C$ is not violated, and hence the correct definition 136 of angles is preserved. The calculation will not report an error, 137 but the results may be not correct. 139 order of sides chosen is not altered, and hence that the correct definition 140 of angles is preserved. For the default choice shown here, that means 141 ensuring that the inequality $A < B < C$ is not violated, The calculation 142 will not report an error, but the results may be not correct. 138 143 139 144 .. figure:: img/parallelepiped_angle_definition.png 140 145 141 146 Definition of the angles for oriented coreshell parallelepipeds. 142 Note that rotation $\theta$, initially in the $x z$ plane, is carried147 Note that rotation $\theta$, initially in the $xz$ plane, is carried 143 148 out first, then rotation $\phi$ about the $z$ axis, finally rotation 144 $\Psi$ is now around the axis of the particle. The neutron or Xray145 beam is along the $z$ axis .149 $\Psi$ is now around the $C$ axis of the particle. The neutron or Xray 150 beam is along the $z$ axis and the detecotr defines the $xy$ plane. 146 151 147 152 .. figure:: img/parallelepiped_angle_projection.png 
sasmodels/models/parallelepiped.py
r96153e4 rfc7bcd5 5 5  6 6 7 This model calculates the scattering from a rectangular parallelepiped7 This model calculates the scattering from a rectangular solid 8 8 (:numref:`parallelepipedimage`). 9 9 If you need to apply polydispersity, see also :ref:`rectangularprism`. For … … 72 72 applied. 73 73 74 NB: The 2nd virial coefficient of the parallelepiped is calculated based on 75 the averaged effective radius, after appropriately sorting the three 76 dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and 77 length $(= C)$ values, and used as the effective radius for 78 $S(q)$ when $P(q) \cdot S(q)$ is applied. 79 80 For 2d data the orientation of the particle is required, described using 81 angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details 82 of the calculation and angular dispersions see :ref:`orientation` . 83 84 The angle $\Psi$ is the rotational angle around the $C$ axis. 85 For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis 86 oriented parallel to the yaxis of the detector with $A$ along the xaxis. 87 For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated 88 $\theta$ degrees in the $zx$ plane and then $\phi$ degrees around the $z$ axis, 89 before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis 90 of the particle to obtain the final orientation of the parallelepiped. 91 92 .. note:: For 2d, constraints must be applied during fitting to ensure that the 93 inequality $A < B < C$ is not violated, and hence the correct definition 94 of angles is preserved. The calculation will not report an error, 95 but the results may be not correct. 96 97 .. _parallelepipedorientation: 98 99 .. figure:: img/parallelepiped_angle_definition.png 100 101 Definition of the angles for oriented parallelepiped, shown with $A<B<C$. 102 103 .. figure:: img/parallelepiped_angle_projection.png 104 105 Examples of the angles for an oriented parallelepiped against the 106 detector plane. 107 108 On introducing "Orientational Distribution" in the angles, "distribution of 109 theta" and "distribution of phi" parameters will appear. These are actually 110 rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, 111 perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ 112 these are parallel to the $Y$ and $X$ axes of the instrument.) The third 113 orientation distribution, in $\psi$, is about the $c$ axis of the particle, 114 perpendicular to the $a$ x $b$ face. Some experimentation may be required to 115 understand the 2d patterns fully as discussed in :ref:`orientation` . 116 117 For a given orientation of the parallelepiped, the 2D form factor is 118 calculated as 119 120 .. math:: 121 122 P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} 123 {2}qA\cos\alpha)}\right]^2 124 \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} 125 {2}qB\cos\beta)}\right]^2 126 \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} 127 {2}qC\cos\gamma)}\right]^2 128 129 with 130 131 .. math:: 132 133 \cos\alpha &= \hat A \cdot \hat q, \\ 134 \cos\beta &= \hat B \cdot \hat q, \\ 135 \cos\gamma &= \hat C \cdot \hat q 136 137 and the scattering intensity as: 138 139 .. math:: 140 141 I(q_x, q_y) = \frac{\text{scale}}{V} V^2 \Delta\rho^2 P(q_x, q_y) 74 For **oriented** particles, the 2D scattering intensity, $I(q_x, q_y)$, is 75 given as: 76 77 .. math:: 78 79 I(q_x, q_y) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 P(q_x, q_y) 142 80 + \text{background} 143 81 … … 151 89 with scale being the volume fraction. 152 90 91 Where $P(q_x, q_y)$ for a given orientation of the form factor is calculated as 92 93 .. math:: 94 95 P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} 96 {2}qA\cos\alpha)}\right]^2 97 \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} 98 {2}qB\cos\beta)}\right]^2 99 \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} 100 {2}qC\cos\gamma)}\right]^2 101 102 with 103 104 .. math:: 105 106 \cos\alpha &= \hat A \cdot \hat q, \\ 107 \cos\beta &= \hat B \cdot \hat q, \\ 108 \cos\gamma &= \hat C \cdot \hat q 109 110 111 FITTING NOTES 112 ~~~~~~~~~~~~~ 113 114 #. The 2nd virial coefficient of the parallelepiped is calculated based on 115 the averaged effective radius, after appropriately sorting the three 116 dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and 117 length $(= C)$ values, and used as the effective radius for 118 $S(q)$ when $P(q) \cdot S(q)$ is applied. 119 120 #. For 2d data the orientation of the particle is required, described using 121 angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ 122 and $\phi$ define the orientation of the director in the laboratry reference 123 frame of the beam direction ($z$) and detector plane ($xy$ plane), while 124 the angle $\Psi$ is effectively the rotational angle around the particle 125 $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the 126 $B$ axis oriented parallel to the yaxis of the detector with $A$ along 127 the xaxis. For other $\theta$, $\phi$ values, the order of rotations 128 matters. In particular, the parallelepiped must first be rotated $\theta$ 129 degrees in the $xz$ plane before rotating $\phi$ degrees around the $z$ 130 axis (in the $xy$ plane). Applying orientational distribution to the 131 particle orientation (i.e `jitter` to one or more of these angles) can get 132 more confusing as `jitter` is defined **NOT** with respect to the laboratory 133 frame but the particle reference frame. It is thus highly recmmended to 134 read :ref:`orientation` for further details of the calculation and angular 135 dispersions. 136 137 .. note:: For 2d, constraints must be applied during fitting to ensure that the 138 order of sides chosen is not altered, and hence that the correct definition 139 of angles is preserved. For the default choice shown here, that means 140 ensuring that the inequality $A < B < C$ is not violated, The calculation 141 will not report an error, but the results may be not correct. 142 143 .. _parallelepipedorientation: 144 145 .. figure:: img/parallelepiped_angle_definition.png 146 147 Definition of the angles for oriented parallelepiped, shown with $A<B<C$. 148 149 .. figure:: img/parallelepiped_angle_projection.png 150 151 Examples of the angles for an oriented parallelepiped against the 152 detector plane. 153 154 .. Comment by Paul Butler 155 I am commenting this section out as we are trying to minimize the amount of 156 oritentational detail here and encourage the user to go to the full 157 orientation documentation so that changes can be made in just one place. 158 below is the commented paragrah: 159 On introducing "Orientational Distribution" in the angles, "distribution of 160 theta" and "distribution of phi" parameters will appear. These are actually 161 rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, 162 perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ 163 these are parallel to the $Y$ and $X$ axes of the instrument.) The third 164 orientation distribution, in $\psi$, is about the $c$ axis of the particle, 165 perpendicular to the $a$ x $b$ face. Some experimentation may be required to 166 understand the 2d patterns fully as discussed in :ref:`orientation` . 167 153 168 154 169 Validation … … 158 173 to the angular average of the output of a 2D calculation over all possible 159 174 angles. 160 161 175 162 176 References
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