[cc3fac6] | 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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[48462b0] | 6 | and rods or between rods and platelets, and overcomes some of the |
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[6431056] | 7 | deficiencies of the (Beaucage) Unified_Power_Rg model (see Hammouda, 2010). |
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[cc3fac6] | 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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[40a87fa] | 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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[cc3fac6] | 20 | |
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| 21 | This is based on the generalized Guinier law for such elongated objects |
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[40a87fa] | 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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[cc3fac6] | 27 | |
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[40a87fa] | 28 | Enforcing the continuity of the Guinier and Porod functions and their |
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| 29 | derivatives yields |
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[cc3fac6] | 30 | |
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| 31 | .. math:: |
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[40a87fa] | 32 | |
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[cc3fac6] | 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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[40a87fa] | 39 | D &= G \ \exp{ \left[ \frac{-Q_1^2 R_g^2}{3-s} \right]} \ Q_1^{m-s} |
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[cc3fac6] | 40 | |
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[40a87fa] | 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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[cc3fac6] | 43 | |
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[45330ed] | 44 | |
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[40a87fa] | 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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[cc3fac6] | 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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[cdcebf1] | 62 | B Hammouda, *A new Guinier-Porod model, J. Appl. Cryst.*, (2010), 43, 716-719 |
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[6431056] | 63 | |
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[cdcebf1] | 64 | B Hammouda, *Analysis of the Beaucage model, J. Appl. Cryst.*, (2010), 43, 1474-1478 |
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[cc3fac6] | 65 | """ |
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| 66 | |
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[2d81cfe] | 67 | import numpy as np |
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[2c74c11] | 68 | from numpy import inf, sqrt, exp, errstate |
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[cc3fac6] | 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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[a807206] | 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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[cc3fac6] | 76 | List of parameters: |
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| 77 | scale = Guinier Scale |
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| 78 | s = Dimension Variable |
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[48462b0] | 79 | Rg = Radius of Gyration [A] |
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[a807206] | 80 | porod_exp = Porod Exponent |
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[cc3fac6] | 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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[a807206] | 89 | ["porod_exp", "", 3.0, [0, inf], "", "Porod exponent"]] |
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[cc3fac6] | 90 | # pylint: enable=bad-whitespace, line-too-long |
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| 91 | |
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[1ce6e82] | 92 | # pylint: disable=C0103 |
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[a807206] | 93 | def Iq(q, rg, s, porod_exp): |
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[cc3fac6] | 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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[a807206] | 98 | ms = 0.5*(porod_exp-s) # =(n-3+porod_exp)/2 |
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[2c74c11] | 99 | |
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| 100 | # preallocate return value |
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| 101 | iq = 0.0*q |
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[cc3fac6] | 102 | |
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| 103 | # Take care of the singular points |
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[2c74c11] | 104 | if rg <= 0.0 or ms <= 0.0: |
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| 105 | return iq |
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[cc3fac6] | 106 | |
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| 107 | # Do the calculation and return the function value |
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[2c74c11] | 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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[a807206] | 111 | iq[~idx] = q[~idx]**-porod_exp * (exp(-ms) * (n*ms/rg**2)**ms) |
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[1ce6e82] | 112 | return iq |
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[cc3fac6] | 113 | |
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[2c74c11] | 114 | Iq.vectorized = True # Iq accepts an array of q values |
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[cc3fac6] | 115 | |
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[48462b0] | 116 | def random(): |
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| 117 | rg = 10**np.random.uniform(1, 5) |
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| 118 | s = np.random.uniform(0, 3) |
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| 119 | porod_exp = s + np.random.uniform(0, 3) |
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| 120 | pars = dict( |
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| 121 | #scale=1, background=0, |
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| 122 | rg=rg, |
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| 123 | s=s, |
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| 124 | porod_exp=porod_exp, |
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| 125 | ) |
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| 126 | return pars |
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| 127 | |
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[a807206] | 128 | demo = dict(scale=1.5, background=0.5, rg=60, s=1.0, porod_exp=3.0) |
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[cc3fac6] | 129 | |
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| 130 | tests = [[{'scale': 1.5, 'background':0.5}, 0.04, 5.290096890253155]] |
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