1 | r""" |
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2 | Calculates the scattering for a generalized Guinier/power law object. |
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3 | This is an empirical model that can be used to determine the size |
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4 | and dimensionality of scattering objects, including asymmetric objects |
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5 | such as rods or platelets, and shapes intermediate between spheres |
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6 | and rods or between rods and platelets, and overcomes some of the |
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7 | deficiencies of the (Beaucage) Unified_Power_Rg model (see Hammouda, 2010). |
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8 | |
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9 | Definition |
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10 | ---------- |
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11 | |
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12 | The following functional form is used |
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13 | |
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14 | .. math:: |
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15 | |
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16 | I(q) = \begin{cases} |
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17 | \frac{G}{Q^s}\ \exp{\left[\frac{-Q^2R_g^2}{3-s} \right]} & Q \leq Q_1 \\ |
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18 | D / Q^m & Q \geq Q_1 |
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19 | \end{cases} |
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20 | |
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21 | This is based on the generalized Guinier law for such elongated objects |
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22 | (see the Glatter reference below). For 3D globular objects (such as spheres), |
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23 | $s = 0$ and one recovers the standard Guinier formula. For 2D symmetry |
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24 | (such as for rods) $s = 1$, and for 1D symmetry (such as for lamellae or |
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25 | platelets) $s = 2$. A dimensionality parameter ($3-s$) is thus defined, |
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26 | and is 3 for spherical objects, 2 for rods, and 1 for plates. |
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27 | |
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28 | Enforcing the continuity of the Guinier and Porod functions and their |
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29 | derivatives yields |
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30 | |
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31 | .. math:: |
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32 | |
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33 | Q_1 = \frac{1}{R_g} \sqrt{(m-s)(3-s)/2} |
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34 | |
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35 | and |
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36 | |
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37 | .. math:: |
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38 | |
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39 | D &= G \ \exp{ \left[ \frac{-Q_1^2 R_g^2}{3-s} \right]} \ Q_1^{m-s} |
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40 | |
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41 | &= \frac{G}{R_g^{m-s}} \ \exp \left[ -\frac{m-s}{2} \right] |
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42 | \left( \frac{(m-s)(3-s)}{2} \right)^{\frac{m-s}{2}} |
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43 | |
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44 | |
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45 | Note that the radius of gyration for a sphere of radius $R$ is given |
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46 | by $R_g = R \sqrt{3/5}$. For a cylinder of radius $R$ and length $L$, |
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47 | $R_g^2 = \frac{L^2}{12} + \frac{R^2}{2}$ from which the cross-sectional |
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48 | radius of gyration for a randomly oriented thin cylinder is $R_g = R/\sqrt{2}$ |
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49 | and the cross-sectional radius of gyration of a randomly oriented lamella |
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50 | of thickness $T$ is given by $R_g = T / \sqrt{12}$. |
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51 | |
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52 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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53 | where the q vector is defined as |
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54 | |
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55 | .. math:: |
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56 | q = \sqrt{q_x^2+q_y^2} |
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57 | |
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58 | |
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59 | Reference |
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60 | --------- |
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61 | |
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62 | B Hammouda, *A new Guinier-Porod model, J. Appl. Cryst.*, (2010), 43, 716-719 |
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63 | |
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64 | B Hammouda, *Analysis of the Beaucage model, J. Appl. Cryst.*, (2010), 43, 1474-1478 |
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65 | |
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66 | """ |
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67 | |
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68 | from numpy import inf, sqrt, exp, errstate |
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69 | |
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70 | name = "guinier_porod" |
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71 | title = "Guinier-Porod function" |
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72 | description = """\ |
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73 | I(q) = scale/q^s* exp ( - R_g^2 q^2 / (3-s) ) for q<= ql |
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74 | = scale/q^porod_exp*exp((-ql^2*Rg^2)/(3-s))*ql^(porod_exp-s) for q>=ql |
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75 | where ql = sqrt((porod_exp-s)(3-s)/2)/Rg. |
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76 | List of parameters: |
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77 | scale = Guinier Scale |
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78 | s = Dimension Variable |
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79 | Rg = Radius of Gyration [A] |
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80 | porod_exp = Porod Exponent |
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81 | background = Background [1/cm]""" |
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82 | |
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83 | category = "shape-independent" |
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84 | |
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85 | # pylint: disable=bad-whitespace, line-too-long |
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86 | # ["name", "units", default, [lower, upper], "type","description"], |
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87 | parameters = [["rg", "Ang", 60.0, [0, inf], "", "Radius of gyration"], |
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88 | ["s", "", 1.0, [0, inf], "", "Dimension variable"], |
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89 | ["porod_exp", "", 3.0, [0, inf], "", "Porod exponent"]] |
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90 | # pylint: enable=bad-whitespace, line-too-long |
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91 | |
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92 | # pylint: disable=C0103 |
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93 | def Iq(q, rg, s, porod_exp): |
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94 | """ |
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95 | @param q: Input q-value |
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96 | """ |
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97 | n = 3.0 - s |
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98 | ms = 0.5*(porod_exp-s) # =(n-3+porod_exp)/2 |
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99 | |
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100 | # preallocate return value |
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101 | iq = 0.0*q |
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102 | |
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103 | # Take care of the singular points |
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104 | if rg <= 0.0 or ms <= 0.0: |
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105 | return iq |
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106 | |
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107 | # Do the calculation and return the function value |
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108 | idx = q < sqrt(n*ms)/rg |
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109 | with errstate(divide='ignore'): |
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110 | iq[idx] = q[idx]**-s * exp(-(q[idx]*rg)**2/n) |
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111 | iq[~idx] = q[~idx]**-porod_exp * (exp(-ms) * (n*ms/rg**2)**ms) |
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112 | return iq |
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113 | |
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114 | Iq.vectorized = True # Iq accepts an array of q values |
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115 | |
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116 | def random(): |
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117 | import numpy as np |
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118 | rg = 10**np.random.uniform(1, 5) |
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119 | s = np.random.uniform(0, 3) |
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120 | porod_exp = s + np.random.uniform(0, 3) |
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121 | pars = dict( |
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122 | #scale=1, background=0, |
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123 | rg=rg, |
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124 | s=s, |
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125 | porod_exp=porod_exp, |
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126 | ) |
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127 | return pars |
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128 | |
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129 | demo = dict(scale=1.5, background=0.5, rg=60, s=1.0, porod_exp=3.0) |
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130 | |
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131 | tests = [[{'scale': 1.5, 'background':0.5}, 0.04, 5.290096890253155]] |
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