[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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[513efc5] | 19 | This integral was later put into an almost analytical form as follows |
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| 20 | (Hammouda, 1993) |
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[23d3b41] | 21 | |
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| 22 | .. math:: |
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| 23 | |
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| 24 | P(Q)=\frac{1}{\nu U^{1/2\nu}}\gamma\left(\frac{1}{2\nu},U\right) - |
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| 25 | \frac{1}{\nu U^{1/\nu}}\gamma\left(\frac{1}{\nu},U\right) |
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| 26 | |
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| 27 | where $\gamma(x,U)$ is the incomplete gamma function |
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| 28 | |
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| 29 | .. math:: |
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| 30 | |
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| 31 | \gamma(x,U)=\int_0^{U}dt\ exp(-t)t^{x-1} |
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| 32 | |
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| 33 | and the variable $U$ is given in terms of the scattering vector $Q$ as |
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| 34 | |
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| 35 | .. math:: |
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| 36 | |
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| 37 | U=\frac{Q^2a^2n^{2\nu}}{6} = \frac{Q^2R_{g}^2(2\nu+1)(2\nu+2)}{6} |
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| 38 | |
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| 39 | The square of the radius-of-gyration is defined as |
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| 40 | |
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| 41 | .. math:: |
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| 42 | |
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| 43 | R_{g}^2 = \frac{a^2n^{2\nu}}{(2\nu+1)(2\nu+2)} |
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| 44 | |
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| 45 | Note that this model applies only in the mass fractal range (ie, $5/3<=m<=3$ ) |
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| 46 | and **does not apply** to surface fractals ( $3<m<=4$ ). |
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[513efc5] | 47 | It also does not reproduce the rigid rod limit (m=1) because it assumes chain |
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| 48 | flexibility from the outset. It may cover a portion of the semi-flexible chain |
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| 49 | range ( $1<m<5/3$ ). |
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[23d3b41] | 50 | |
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[513efc5] | 51 | A low-Q expansion yields the Guinier form and a high-Q expansion yields the |
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| 52 | Porod form which is given by |
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[23d3b41] | 53 | |
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| 54 | .. math:: |
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| 55 | |
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[513efc5] | 56 | P(Q\rightarrow \infty) = \frac{1}{\nu U^{1/2\nu}}\Gamma\left( |
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| 57 | \frac{1}{2\nu}\right) - \frac{1}{\nu U^{1/\nu}}\Gamma\left( |
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| 58 | \frac{1}{\nu}\right) |
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[23d3b41] | 59 | |
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| 60 | Here $\Gamma(x) = \gamma(x,\infty)$ is the gamma function. |
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| 61 | |
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| 62 | The asymptotic limit is dominated by the first term |
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| 63 | |
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| 64 | .. math:: |
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| 65 | |
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| 66 | P(Q\rightarrow \infty) \sim \frac{1}{\nu U^{1/2\nu}}\Gamma\left(\frac{1}{2\nu}\right) = |
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| 67 | \frac{m}{\left(QR_{g}\right)^m}\left[\frac{6}{(2\nu +1)(2\nu +2)} \right]^{m/2} |
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| 68 | \Gamma (m/2) |
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| 69 | |
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| 70 | The special case when $\nu=0.5$ (or $m=2/\nu=2$ ) corresponds to Gaussian chains for |
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| 71 | which the form factor is given by the familiar Debye function. |
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| 72 | |
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| 73 | .. math:: |
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| 74 | |
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| 75 | P(Q) = \frac{2}{Q^4R_{g}^4} \left[exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right] |
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| 76 | |
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| 77 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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| 78 | where the $q$ vector is defined as |
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| 79 | |
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| 80 | .. math:: |
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| 81 | |
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| 82 | q = \sqrt{q_x^2 + q_y^2} |
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| 83 | |
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| 84 | This example dataset is produced using 200 data points, $qmin=0.001Ang^{-1}$, |
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| 85 | $qmax=0.2Ang^{-1}$ and the default values |
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| 86 | |
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| 87 | .. figure:: img/polymer_excl_volume_1d.jpg |
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| 88 | |
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| 89 | 1D plot using the default values (w/500 data point). |
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| 90 | |
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| 91 | |
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| 92 | References |
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| 93 | ---------- |
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| 94 | |
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| 95 | H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247 |
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| 96 | |
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| 97 | B Hammouda, *SANS from Homogeneous Polymer Mixtures - A Unified Overview, |
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| 98 | Advances in Polym. Sci.* 106(1993) 87-133 |
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| 99 | |
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| 100 | """ |
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| 101 | |
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| 102 | from math import sqrt |
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| 103 | from numpy import inf, power |
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| 104 | from scipy.special import gammainc, gamma |
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| 105 | |
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| 106 | name = "polymer_excl_volume" |
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| 107 | title = "Polymer Excluded Volume model" |
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[513efc5] | 108 | description = """Compute the scattering intensity from polymers with excluded |
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| 109 | volume effects. |
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[23d3b41] | 110 | rg: radius of gyration |
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| 111 | porod_exp: Porod exponent |
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| 112 | """ |
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| 113 | category = "shape-independent" |
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| 114 | |
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| 115 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 116 | parameters = [["rg", "Ang", 60.0, [0, inf], "", "Radius of Gyration"], |
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| 117 | ["porod_exp", "", 3.0, [-inf, inf], "", "Porod exponent"], |
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| 118 | ] |
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| 119 | |
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| 120 | |
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| 121 | def Iq(q, rg, porod_exp): |
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| 122 | |
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| 123 | """ |
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| 124 | :param q: Input q-value (float or [float, float]) |
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| 125 | :param rg: Radius of gyration |
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| 126 | :param porod_exp: Porod exponent |
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| 127 | :return: Calculated intensity |
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| 128 | """ |
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| 129 | nu = 1.0/porod_exp |
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| 130 | u = q*q*rg*rg*(2.0*nu+1.0) * (2.0*nu+2.0)/6.0 |
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| 131 | o2nu = 1.0/(2.0*nu) |
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| 132 | |
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| 133 | intensity = ((1.0/(nu*power(u, o2nu))) * (gamma(o2nu)*gammainc(o2nu, u) - |
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[513efc5] | 134 | 1.0/power(u, o2nu) * gamma(porod_exp) * |
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| 135 | gammainc(porod_exp, u))) * (q > 0) + 1.0*(q <= 0) |
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[23d3b41] | 136 | |
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| 137 | return intensity |
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| 138 | |
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| 139 | Iq.vectorized = True # Iq accepts an array of q values |
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| 140 | |
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| 141 | |
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| 142 | def Iqxy(qx, qy, *args): |
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| 143 | iq = Iq(sqrt(qx**2 + qy**2), *args) |
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| 144 | |
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| 145 | return iq |
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| 146 | |
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| 147 | Iqxy.vectorized = True # Iqxy accepts an array of qx, qy values |
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| 148 | |
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| 149 | |
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| 150 | demo = dict(scale=1, background=0.0, |
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| 151 | rg=60.0, |
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| 152 | porod_exp=3.0) |
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| 153 | |
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| 154 | oldname = "PolymerExclVolume" |
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| 155 | oldpars = dict(background='background', scale='scale', |
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| 156 | rg='rg', |
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| 157 | porod_exp='m') |
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| 158 | |
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| 159 | tests = [[{'rg': 10, 'porod_exp': 4.0}, 0.1, 0.723436675809], |
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| 160 | |
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| 161 | [{'rg': 2.2, 'porod_exp': 22.0, 'background': 100.0}, 5.0, 100.0], |
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| 162 | |
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[513efc5] | 163 | [{'rg': 1.1, 'porod_exp': 1, 'background': 10.0, 'scale': 1.25}, |
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| 164 | 20000., 10.0000712097] |
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[23d3b41] | 165 | ] |
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