[23d3b41] | 1 | r""" |
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[513efc5] | 2 | This model describes the scattering from polymer chains subject to excluded |
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| 3 | volume effects and has been used as a template for describing mass fractals. |
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[23d3b41] | 4 | |
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| 5 | Definition |
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| 6 | ---------- |
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| 7 | |
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[513efc5] | 8 | The form factor was originally presented in the following integral form |
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| 9 | (Benoit, 1957) |
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[23d3b41] | 10 | |
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| 11 | .. math:: |
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| 12 | |
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| 13 | P(Q)=2\int_0^{1}dx(1-x)exp\left[-\frac{Q^2a^2}{6}n^{2v}x^{2v}\right] |
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| 14 | |
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| 15 | where $\nu$ is the excluded volume parameter |
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| 16 | (which is related to the Porod exponent $m$ as $\nu=1/m$ ), |
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| 17 | $a$ is the statistical segment length of the polymer chain, |
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| 18 | and $n$ is the degree of polymerization. |
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[a2ca6e5] | 19 | |
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[547c6f0] | 20 | This integral was put into an almost analytical form as follows |
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[513efc5] | 21 | (Hammouda, 1993) |
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[23d3b41] | 22 | |
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| 23 | .. math:: |
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| 24 | |
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[547c6f0] | 25 | P(Q)=\frac{1}{\nu U^{1/2\nu}} |
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| 26 | \left\{ |
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| 27 | \gamma\left(\frac{1}{2\nu},U\right) - |
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| 28 | \frac{1}{U^{1/2\nu}}\gamma\left(\frac{1}{\nu},U\right) |
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| 29 | \right\} |
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[a2ca6e5] | 30 | |
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[547c6f0] | 31 | and later recast as (for example, Hore, 2013; Hammouda & Kim, 2017) |
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[a2ca6e5] | 32 | |
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| 33 | .. math:: |
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| 34 | |
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| 35 | P(Q)=\frac{1}{\nu U^{1/2\nu}}\gamma\left(\frac{1}{2\nu},U\right) - |
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[23d3b41] | 36 | \frac{1}{\nu U^{1/\nu}}\gamma\left(\frac{1}{\nu},U\right) |
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| 37 | |
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| 38 | where $\gamma(x,U)$ is the incomplete gamma function |
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| 39 | |
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| 40 | .. math:: |
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| 41 | |
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[547c6f0] | 42 | \gamma(x,U)=\int_0^{U}dt\ \exp(-t)t^{x-1} |
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[23d3b41] | 43 | |
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| 44 | and the variable $U$ is given in terms of the scattering vector $Q$ as |
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| 45 | |
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| 46 | .. math:: |
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| 47 | |
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| 48 | U=\frac{Q^2a^2n^{2\nu}}{6} = \frac{Q^2R_{g}^2(2\nu+1)(2\nu+2)}{6} |
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| 49 | |
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[a2ca6e5] | 50 | The two analytic forms are equivalent. In the 1993 paper |
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| 51 | |
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| 52 | .. math:: |
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| 53 | |
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| 54 | \frac{1}{\nu U^{1/2\nu}} |
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| 55 | |
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| 56 | has been factored out. |
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| 57 | |
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| 58 | **SasView implements the 1993 expression**. |
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| 59 | |
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[23d3b41] | 60 | The square of the radius-of-gyration is defined as |
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| 61 | |
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| 62 | .. math:: |
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| 63 | |
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| 64 | R_{g}^2 = \frac{a^2n^{2\nu}}{(2\nu+1)(2\nu+2)} |
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| 65 | |
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[a2ca6e5] | 66 | .. note:: |
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| 67 | This model applies only in the mass fractal range (ie, $5/3<=m<=3$ ) |
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| 68 | and **does not apply** to surface fractals ( $3<m<=4$ ). |
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| 69 | It also does not reproduce the rigid rod limit (m=1) because it assumes chain |
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| 70 | flexibility from the outset. It may cover a portion of the semi-flexible chain |
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| 71 | range ( $1<m<5/3$ ). |
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[23d3b41] | 72 | |
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[513efc5] | 73 | A low-Q expansion yields the Guinier form and a high-Q expansion yields the |
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| 74 | Porod form which is given by |
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[23d3b41] | 75 | |
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| 76 | .. math:: |
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| 77 | |
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[513efc5] | 78 | P(Q\rightarrow \infty) = \frac{1}{\nu U^{1/2\nu}}\Gamma\left( |
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| 79 | \frac{1}{2\nu}\right) - \frac{1}{\nu U^{1/\nu}}\Gamma\left( |
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| 80 | \frac{1}{\nu}\right) |
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[23d3b41] | 81 | |
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| 82 | Here $\Gamma(x) = \gamma(x,\infty)$ is the gamma function. |
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| 83 | |
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| 84 | The asymptotic limit is dominated by the first term |
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| 85 | |
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| 86 | .. math:: |
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| 87 | |
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| 88 | P(Q\rightarrow \infty) \sim \frac{1}{\nu U^{1/2\nu}}\Gamma\left(\frac{1}{2\nu}\right) = |
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| 89 | \frac{m}{\left(QR_{g}\right)^m}\left[\frac{6}{(2\nu +1)(2\nu +2)} \right]^{m/2} |
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| 90 | \Gamma (m/2) |
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| 91 | |
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[bf59527] | 92 | The special case when $\nu=0.5$ (or $m=1/\nu=2$ ) corresponds to Gaussian chains for |
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[23d3b41] | 93 | which the form factor is given by the familiar Debye function. |
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| 94 | |
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| 95 | .. math:: |
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| 96 | |
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[547c6f0] | 97 | P(Q) = \frac{2}{Q^4R_{g}^4} \left[\exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right] |
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[23d3b41] | 98 | |
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| 99 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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| 100 | where the $q$ vector is defined as |
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| 101 | |
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| 102 | .. math:: |
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| 103 | |
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| 104 | q = \sqrt{q_x^2 + q_y^2} |
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| 105 | |
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| 106 | |
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| 107 | References |
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| 108 | ---------- |
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| 109 | |
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| 110 | H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247 |
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| 111 | |
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| 112 | B Hammouda, *SANS from Homogeneous Polymer Mixtures - A Unified Overview, |
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[a2ca6e5] | 113 | Advances in Polym. Sci.* 106 (1993) 87-133 |
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| 114 | |
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[547c6f0] | 115 | M Hore et al, *Co-Nonsolvency of Poly(n-isopropylacrylamide) in Deuterated |
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[a2ca6e5] | 116 | Water/Ethanol Mixtures* 46 (2013) 7894-7901 |
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| 117 | |
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| 118 | B Hammouda & M-H Kim, *The empirical core-chain model* 247 (2017) 434-440 |
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[23d3b41] | 119 | """ |
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| 120 | |
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[2d81cfe] | 121 | import numpy as np |
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[2c74c11] | 122 | from numpy import inf, power, errstate |
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[23d3b41] | 123 | from scipy.special import gammainc, gamma |
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| 124 | |
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| 125 | name = "polymer_excl_volume" |
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| 126 | title = "Polymer Excluded Volume model" |
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[513efc5] | 127 | description = """Compute the scattering intensity from polymers with excluded |
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| 128 | volume effects. |
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[23d3b41] | 129 | rg: radius of gyration |
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| 130 | porod_exp: Porod exponent |
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| 131 | """ |
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| 132 | category = "shape-independent" |
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| 133 | |
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[168052c] | 134 | # pylint: disable=bad-whitespace, line-too-long |
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[40a87fa] | 135 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 136 | parameters = [ |
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[404ebbd] | 137 | ["rg", "Ang", 60.0, [0, inf], "", "Radius of Gyration"], |
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| 138 | ["porod_exp", "", 3.0, [0, inf], "", "Porod exponent"], |
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[40a87fa] | 139 | ] |
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[168052c] | 140 | # pylint: enable=bad-whitespace, line-too-long |
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[23d3b41] | 141 | |
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| 142 | |
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[416609b] | 143 | def Iq(q, rg=60.0, porod_exp=3.0): |
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[23d3b41] | 144 | """ |
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| 145 | :param q: Input q-value (float or [float, float]) |
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| 146 | :param rg: Radius of gyration |
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| 147 | :param porod_exp: Porod exponent |
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| 148 | :return: Calculated intensity |
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| 149 | """ |
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[40a87fa] | 150 | usub = (q*rg)**2 * (2.0/porod_exp + 1.0) * (2.0/porod_exp + 2.0)/6.0 |
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[2c74c11] | 151 | with errstate(divide='ignore', invalid='ignore'): |
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[40a87fa] | 152 | upow = power(usub, -0.5*porod_exp) |
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[aa90015] | 153 | # Note: scipy gammainc is "regularized", being gamma(s,x)/Gamma(s), |
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| 154 | # so need to scale by Gamma(s) to recover gamma(s, x). |
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[2d81cfe] | 155 | result = (porod_exp*upow * |
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| 156 | (gamma(0.5*porod_exp)*gammainc(0.5*porod_exp, usub) - |
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| 157 | upow*gamma(porod_exp)*gammainc(porod_exp, usub))) |
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[40a87fa] | 158 | result[q <= 0] = 1.0 |
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[23d3b41] | 159 | |
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[40a87fa] | 160 | return result |
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[23d3b41] | 161 | |
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| 162 | Iq.vectorized = True # Iq accepts an array of q values |
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| 163 | |
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[404ebbd] | 164 | def random(): |
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[b297ba9] | 165 | """Return a random parameter set for the model.""" |
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[404ebbd] | 166 | rg = 10**np.random.uniform(0, 4) |
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| 167 | porod_exp = np.random.uniform(1e-3, 6) |
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| 168 | scale = 10**np.random.uniform(1, 5) |
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| 169 | pars = dict( |
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| 170 | #background=0, |
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| 171 | scale=scale, |
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| 172 | rg=rg, |
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| 173 | porod_exp=porod_exp, |
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| 174 | ) |
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| 175 | return pars |
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| 176 | |
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[07a6700] | 177 | tests = [ |
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[168052c] | 178 | # Accuracy tests based on content in test/polyexclvol_default_igor.txt |
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[6dd90c1] | 179 | [{'rg': 60, 'porod_exp': 3.0}, 0.001, 0.999801], |
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| 180 | [{'rg': 60, 'porod_exp': 3.0}, 0.105363, 0.0172751], |
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| 181 | [{'rg': 60, 'porod_exp': 3.0, 'background': 0.0}, 0.665075, 6.56261e-05], |
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[168052c] | 182 | |
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| 183 | # Additional tests with larger range of parameters |
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[6dd90c1] | 184 | [{'rg': 10, 'porod_exp': 4.0}, 0.1, 0.724436675809], |
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[168052c] | 185 | [{'rg': 2.2, 'porod_exp': 22.0, 'background': 100.0}, 5.0, 100.0], |
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| 186 | [{'rg': 1.1, 'porod_exp': 1, 'background': 10.0, 'scale': 1.25}, |
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| 187 | 20000., 10.0000712097] |
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| 188 | ] |
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