1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | The Beaucage model employs the empirical multiple level unified |
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6 | Exponential/Power-law fit method developed by G. Beaucage. Four functions |
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7 | are included so that 1, 2, 3, or 4 levels can be used. In addition a 0 level |
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8 | has been added which simply calculates |
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9 | |
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10 | .. math:: |
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11 | |
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12 | I(q) = \text{scale} / q + \text{background} |
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13 | |
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14 | The Beaucage method is able to reasonably approximate the scattering from |
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15 | many different types of particles, including fractal clusters, random coils |
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16 | (Debye equation), ellipsoidal particles, etc. |
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17 | |
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18 | The empirical fit function is: |
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19 | |
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20 | .. math:: |
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21 | |
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22 | I(q) = \text{background} |
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23 | + \sum_{i=1}^N \Bigl[ |
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24 | G_i \exp\Bigl(-\frac{q^2R_{gi}^2}{3}\Bigr) |
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25 | + B_i \exp\Bigl(-\frac{q^2R_{g(i+1)}^2}{3}\Bigr) |
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26 | \Bigl(\frac{1}{q_i^*}\Bigr)^{P_i} \Bigr] |
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27 | |
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28 | where |
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29 | |
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30 | .. math:: |
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31 | |
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32 | q_i^* = \frac{q}{\operatorname{erf}^3(q R_{gi}/\sqrt{6}} |
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33 | |
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34 | |
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35 | For each level, the four parameters $G_i$, $R_{gi}$, $B_i$ and $P_i$ must |
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36 | be chosen. Beaucage has an additional factor $k$ in the definition of |
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37 | $q_i^*$ which is ignored here. |
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38 | |
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39 | For example, to approximate the scattering from random coils (Debye equation), |
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40 | set $R_{gi}$ as the Guinier radius, $P_i = 2$, and $B_i = 2 G_i / R_{gi}$ |
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41 | |
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42 | See the references for further information on choosing the parameters. |
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43 | |
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44 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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45 | where the $q$ vector is defined as |
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46 | |
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47 | .. math:: |
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48 | |
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49 | q = \sqrt{q_x^2 + q_y^2} |
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50 | |
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51 | |
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52 | References |
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53 | ---------- |
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54 | |
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55 | G Beaucage, *J. Appl. Cryst.*, 28 (1995) 717-728 |
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56 | |
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57 | G Beaucage, *J. Appl. Cryst.*, 29 (1996) 134-146 |
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58 | |
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59 | """ |
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60 | |
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61 | from __future__ import division |
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62 | |
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63 | import numpy as np |
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64 | from numpy import inf, exp, sqrt, errstate |
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65 | from scipy.special import erf |
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66 | |
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67 | category = "shape-independent" |
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68 | name = "unified_power_Rg" |
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69 | title = "Unified Power Rg" |
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70 | description = """ |
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71 | The Beaucage model employs the empirical multiple level unified |
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72 | Exponential/Power-law fit method developed by G. Beaucage. Four functions |
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73 | are included so that 1, 2, 3, or 4 levels can be used. |
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74 | """ |
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75 | |
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76 | # pylint: disable=bad-whitespace, line-too-long |
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77 | parameters = [ |
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78 | ["level", "", 1, [0, 6], "", "Level number"], |
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79 | ["rg[level]", "Ang", 15.8, [0, inf], "", "Radius of gyration"], |
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80 | ["power[level]", "", 4, [-inf, inf], "", "Power"], |
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81 | ["B[level]", "1/cm", 4.5e-6, [-inf, inf], "", ""], |
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82 | ["G[level]", "1/cm", 400, [0, inf], "", ""], |
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83 | ] |
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84 | # pylint: enable=bad-whitespace, line-too-long |
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85 | |
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86 | def Iq(q, level, rg, power, B, G): |
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87 | ilevel = int(level) |
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88 | if ilevel == 0: |
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89 | with errstate(divide='ignore'): |
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90 | return 1./q |
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91 | |
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92 | with errstate(divide='ignore', invalid='ignore'): |
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93 | result = np.zeros(q.shape, 'd') |
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94 | for i in range(ilevel): |
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95 | exp_now = exp(-(q*rg[i])**2/3.) |
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96 | pow_now = (erf(q*rg[i]/sqrt(6.))**3/q)**power[i] |
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97 | exp_next = exp(-(q*rg[i+1])**2/3.) if i < ilevel-1 else 1. |
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98 | result += G[i]*exp_now + B[i]*exp_next*pow_now |
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99 | result[q == 0] = np.sum(G[:ilevel]) |
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100 | return result |
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101 | |
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102 | Iq.vectorized = True |
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103 | |
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104 | demo = dict( |
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105 | level=2, |
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106 | rg=[15.8, 21], |
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107 | power=[4, 2], |
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108 | B=[4.5e-6, 0.0006], |
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109 | G=[400, 3], |
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110 | scale=1., |
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111 | background=0., |
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112 | ) |
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