[263daec] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[48462b0] | 5 | This model employs the empirical multiple level unified Exponential/Power-law |
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| 6 | fit method developed by Beaucage. Four functions are included so that 1, 2, 3, |
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| 7 | or 4 levels can be used. In addition a 0 level has been added which simply |
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[6431056] | 8 | calculates |
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[263daec] | 9 | |
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[948db69] | 10 | .. math:: |
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[263daec] | 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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[48462b0] | 18 | The model works best for mass fractal systems characterized by Porod exponents |
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| 19 | between 5/3 and 3. It should not be used for surface fractal systems. Hammouda |
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| 20 | (2010) has pointed out a deficiency in the way this model handles the |
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| 21 | transitioning between the Guinier and Porod regimes and which can create |
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[6431056] | 22 | artefacts that appear as kinks in the fitted model function. |
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| 23 | |
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| 24 | Also see the Guinier_Porod model. |
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| 25 | |
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[948db69] | 26 | The empirical fit function is: |
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[263daec] | 27 | |
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[948db69] | 28 | .. math:: |
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[263daec] | 29 | |
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| 30 | I(q) = \text{background} |
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[eb574d7] | 31 | + \sum_{i=1}^N \Bigl[ |
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| 32 | G_i \exp\Bigl(-\frac{q^2R_{gi}^2}{3}\Bigr) |
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| 33 | + B_i \exp\Bigl(-\frac{q^2R_{g(i+1)}^2}{3}\Bigr) |
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| 34 | \Bigl(\frac{1}{q_i^*}\Bigr)^{P_i} \Bigr] |
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[263daec] | 35 | |
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| 36 | where |
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| 37 | |
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[948db69] | 38 | .. math:: |
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[263daec] | 39 | |
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[66ca2a6] | 40 | q_i^* = q \left[\operatorname{erf} |
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| 41 | \left(\frac{q R_{gi}}{\sqrt{6}}\right) |
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| 42 | \right]^{-3} |
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[263daec] | 43 | |
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| 44 | |
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| 45 | For each level, the four parameters $G_i$, $R_{gi}$, $B_i$ and $P_i$ must |
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| 46 | be chosen. Beaucage has an additional factor $k$ in the definition of |
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| 47 | $q_i^*$ which is ignored here. |
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| 48 | |
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[40a87fa] | 49 | For example, to approximate the scattering from random coils (Debye equation), |
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[263daec] | 50 | set $R_{gi}$ as the Guinier radius, $P_i = 2$, and $B_i = 2 G_i / R_{gi}$ |
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| 51 | |
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| 52 | See the references for further information on choosing the parameters. |
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| 53 | |
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| 54 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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| 55 | where the $q$ vector is defined as |
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| 56 | |
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[948db69] | 57 | .. math:: |
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[263daec] | 58 | |
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| 59 | q = \sqrt{q_x^2 + q_y^2} |
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| 60 | |
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| 61 | |
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| 62 | References |
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| 63 | ---------- |
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| 64 | |
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| 65 | G Beaucage, *J. Appl. Cryst.*, 28 (1995) 717-728 |
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| 66 | |
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| 67 | G Beaucage, *J. Appl. Cryst.*, 29 (1996) 134-146 |
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| 68 | |
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[cdcebf1] | 69 | B Hammouda, *Analysis of the Beaucage model, J. Appl. Cryst.*, (2010), 43, 1474-1478 |
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[263daec] | 70 | """ |
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| 71 | |
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| 72 | from __future__ import division |
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| 73 | |
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| 74 | import numpy as np |
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[2c74c11] | 75 | from numpy import inf, exp, sqrt, errstate |
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[2d81cfe] | 76 | from scipy.special import erf, gamma |
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[263daec] | 77 | |
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[40a87fa] | 78 | category = "shape-independent" |
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[948db69] | 79 | name = "unified_power_Rg" |
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| 80 | title = "Unified Power Rg" |
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| 81 | description = """ |
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| 82 | The Beaucage model employs the empirical multiple level unified |
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| 83 | Exponential/Power-law fit method developed by G. Beaucage. Four functions |
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| 84 | are included so that 1, 2, 3, or 4 levels can be used. |
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| 85 | """ |
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[40a87fa] | 86 | |
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| 87 | # pylint: disable=bad-whitespace, line-too-long |
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[263daec] | 88 | parameters = [ |
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[48462b0] | 89 | ["level", "", 1, [1, 6], "", "Level number"], |
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[42356c8] | 90 | ["rg[level]", "Ang", 15.8, [0, inf], "", "Radius of gyration"], |
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[263daec] | 91 | ["power[level]", "", 4, [-inf, inf], "", "Power"], |
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| 92 | ["B[level]", "1/cm", 4.5e-6, [-inf, inf], "", ""], |
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| 93 | ["G[level]", "1/cm", 400, [0, inf], "", ""], |
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| 94 | ] |
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[40a87fa] | 95 | # pylint: enable=bad-whitespace, line-too-long |
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[263daec] | 96 | |
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[42356c8] | 97 | def Iq(q, level, rg, power, B, G): |
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[0542fe1] | 98 | level = int(level + 0.5) |
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| 99 | if level == 0: |
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[2c74c11] | 100 | with errstate(divide='ignore'): |
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| 101 | return 1./q |
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| 102 | |
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| 103 | with errstate(divide='ignore', invalid='ignore'): |
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[ec77322] | 104 | result = np.zeros(q.shape, 'd') |
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[0542fe1] | 105 | for i in range(level): |
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[2c74c11] | 106 | exp_now = exp(-(q*rg[i])**2/3.) |
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| 107 | pow_now = (erf(q*rg[i]/sqrt(6.))**3/q)**power[i] |
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[0542fe1] | 108 | if i < level-1: |
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[b3f2a24] | 109 | exp_next = exp(-(q*rg[i+1])**2/3.) |
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| 110 | else: |
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| 111 | exp_next = 1 |
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[2c74c11] | 112 | result += G[i]*exp_now + B[i]*exp_next*pow_now |
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[b3f2a24] | 113 | |
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[0542fe1] | 114 | result[q == 0] = np.sum(G[:level]) |
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[263daec] | 115 | return result |
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[2c74c11] | 116 | |
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[263daec] | 117 | Iq.vectorized = True |
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| 118 | |
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[48462b0] | 119 | def random(): |
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| 120 | level = np.minimum(np.random.poisson(0.5) + 1, 6) |
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| 121 | n = level |
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| 122 | power = np.random.uniform(1.6, 3, n) |
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| 123 | rg = 10**np.random.uniform(1, 5, n) |
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| 124 | G = np.random.uniform(0.1, 10, n)**2 * 10**np.random.uniform(0.3, 3, n) |
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| 125 | B = G * power / rg**power * gamma(power/2) |
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| 126 | scale = 10**np.random.uniform(1, 4) |
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| 127 | pars = dict( |
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| 128 | #background=0, |
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| 129 | scale=scale, |
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| 130 | level=level, |
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| 131 | ) |
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| 132 | pars.update(("power%d"%(k+1), v) for k, v in enumerate(power)) |
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| 133 | pars.update(("rg%d"%(k+1), v) for k, v in enumerate(rg)) |
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| 134 | pars.update(("B%d"%(k+1), v) for k, v in enumerate(B)) |
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| 135 | pars.update(("G%d"%(k+1), v) for k, v in enumerate(G)) |
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| 136 | return pars |
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| 137 | |
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[263daec] | 138 | demo = dict( |
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| 139 | level=2, |
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[42356c8] | 140 | rg=[15.8, 21], |
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[263daec] | 141 | power=[4, 2], |
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| 142 | B=[4.5e-6, 0.0006], |
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| 143 | G=[400, 3], |
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| 144 | scale=1., |
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| 145 | background=0., |
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[40a87fa] | 146 | ) |
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