1 | import numpy as np |
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2 | from numpy import exp, sin, cos, pi, radians, degrees |
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3 | |
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4 | from sasmodels.weights import Dispersion as BaseDispersion |
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5 | |
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6 | class Dispersion(BaseDispersion): |
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7 | r""" |
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8 | Cyclic gaussian dispersion on orientation. |
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9 | |
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10 | .. math: |
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11 | |
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12 | w(\theta) = e^{-\frac{\sin^2 \theta}{2 \sigma^2}} |
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13 | |
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14 | This provides a close match to the gaussian distribution for |
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15 | low angles (with $\sin \theta \approx \theta$), but the tails |
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16 | are limited to $\pm 90^\circ$. For $\sigma$ large the |
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17 | distribution is approximately uniform. The usual polar coordinate |
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18 | projection applies, with $\theta$ weights scaled by $\cos \theta$ |
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19 | and $\phi$ weights unscaled. |
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20 | |
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21 | This is closely related to a Maier-Saupe distribution with order |
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22 | parameter $P_2$ and appropriate scaling constants, and changes |
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23 | between $\sin$ and $\cos$ as appropriate for the coordinate system |
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24 | representation. |
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25 | """ |
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26 | type = "cyclic_gaussian" |
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27 | default = dict(npts=35, width=1, nsigmas=3) |
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28 | |
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29 | # Note: center is always zero for orientation distributions |
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30 | def _weights(self, center, sigma, lb, ub): |
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31 | # Convert sigma in degrees to the approximately equivalent Maier-Saupe "a" |
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32 | sigma = radians(sigma) |
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33 | a = -0.5/sigma**2 |
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34 | |
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35 | # Limit width to +/-90 degrees; use an open interval since the |
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36 | # pattern at +90 is the same as that at -90. |
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37 | width = min(self.nsigmas*sigma, pi/2) |
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38 | x = np.linspace(-width, width, self.npts+2)[1:-1] |
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39 | wx = P(x, a) |
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40 | |
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41 | # Return orientation in degrees with Maier-Saupe weights |
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42 | return degrees(x), exp(a*sin(x)**2) |
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43 | |
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