[cc3fac6] | 1 | # pylint: disable=line-too-long |
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| 2 | r""" |
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| 3 | Calculates the scattering for a generalized Guinier/power law object. |
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| 4 | This is an empirical model that can be used to determine the size |
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| 5 | and dimensionality of scattering objects, including asymmetric objects |
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| 6 | such as rods or platelets, and shapes intermediate between spheres |
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| 7 | and rods or between rods and platelets. |
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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 | I(q) = \frac{G}{Q^s} \ \exp{\left[ \frac{-Q^2R_g^2}{3-s} \right]} \textrm{ for } Q \leq Q_1 |
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| 16 | \\ |
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| 17 | I(q) = D / Q^m \textrm{ for } Q \geq Q_1 |
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| 18 | |
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| 19 | This is based on the generalized Guinier law for such elongated objects |
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| 20 | (see the Glatter reference below). For 3D globular objects (such as spheres), $s = 0$ |
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| 21 | and one recovers the standard Guinier formula. |
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| 22 | For 2D symmetry (such as for rods) $s = 1$, |
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| 23 | and for 1D symmetry (such as for lamellae or platelets) $s = 2$. |
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| 24 | A dimensionality parameter ($3-s$) is thus defined, and is 3 for spherical objects, |
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| 25 | 2 for rods, and 1 for plates. |
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| 26 | |
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| 27 | Enforcing the continuity of the Guinier and Porod functions and their derivatives yields |
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| 28 | |
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| 29 | .. math:: |
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| 30 | Q_1 = \frac{1}{R_g} \sqrt{(m-s)(3-s)/2} |
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| 31 | |
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| 32 | and |
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| 33 | |
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| 34 | .. math:: |
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| 35 | D = G \ \exp{ \left[ \frac{-Q_1^2 R_g^2}{3-s} \right]} \ Q_1^{m-s} |
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| 36 | = \frac{G}{R_g^{m-s}} \ \exp{\left[ -\frac{m-s}{2} \right]} \left( \frac{(m-s)(3-s)}{2} \right)^{\frac{m-s}{2}} |
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| 37 | |
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| 38 | |
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| 39 | Note that the radius-of-gyration for a sphere of radius R is given by $R_g = R \sqrt(3/5)$. |
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| 40 | |
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| 41 | The cross-sectional radius-of-gyration for a randomly oriented cylinder |
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| 42 | of radius R is given by $R_g = R / \sqrt(2)$. |
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| 43 | |
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| 44 | The cross-sectional radius-of-gyration of a randomly oriented lamella |
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| 45 | of thickness $T$ is given by $R_g = T / \sqrt(12)$. |
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| 46 | |
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| 47 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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| 48 | where the q vector is defined as |
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| 49 | |
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| 50 | .. math:: |
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| 51 | q = \sqrt{q_x^2+q_y^2} |
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| 52 | |
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| 53 | .. image:: img/guinier_porod_model.jpg |
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| 54 | |
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| 55 | Figure 1: Guinier-Porod model for $R_g=100$ |Ang|, $s=1$, $m=3$, and $background=0.1$. |
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| 56 | |
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| 57 | |
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| 58 | Reference |
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| 59 | --------- |
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| 60 | |
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| 61 | A Guinier, G Fournet, Small-Angle Scattering of X-Rays, John Wiley and Sons, New York, (1955) |
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| 62 | |
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| 63 | O Glatter, O Kratky, Small-Angle X-Ray Scattering, Academic Press (1982) |
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| 64 | Check out Chapter 4 on Data Treatment, pages 155-156. |
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| 65 | """ |
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| 66 | |
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| 67 | from numpy import inf, sqrt, power, exp |
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| 68 | |
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| 69 | name = "guinier_porod" |
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| 70 | title = "Guinier-Porod function" |
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| 71 | description = """\ |
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| 72 | I(q) = scale/q^s* exp ( - R_g^2 q^2 / (3-s) ) for q<= ql |
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| 73 | = scale/q^m*exp((-ql^2*Rg^2)/(3-s))*ql^(m-s) for q>=ql |
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| 74 | where ql = sqrt((m-s)(3-s)/2)/Rg. |
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| 75 | List of parameters: |
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| 76 | scale = Guinier Scale |
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| 77 | s = Dimension Variable |
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| 78 | Rg = Radius of Gyration [A] |
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| 79 | m = Porod Exponent |
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| 80 | background = Background [1/cm]""" |
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| 81 | |
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| 82 | category = "shape-independent" |
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| 83 | |
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| 84 | # pylint: disable=bad-whitespace, line-too-long |
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| 85 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 86 | parameters = [["rg", "Ang", 60.0, [0, inf], "", "Radius of gyration"], |
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| 87 | ["s", "", 1.0, [0, inf], "", "Dimension variable"], |
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| 88 | ["m", "", 3.0, [0, inf], "", "Porod exponent"]] |
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| 89 | # pylint: enable=bad-whitespace, line-too-long |
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| 90 | |
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| 91 | |
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| 92 | def Iq(q, rg, s, m): |
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| 93 | """ |
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| 94 | @param q: Input q-value |
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| 95 | """ |
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| 96 | n = 3.0 - s |
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| 97 | |
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| 98 | # Take care of the singular points |
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| 99 | if rg <= 0.0: |
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| 100 | return 0.0 |
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| 101 | if (n-3.0+m) <= 0.0: |
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| 102 | return 0.0 |
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| 103 | |
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| 104 | # Do the calculation and return the function value |
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| 105 | q1 = sqrt((n-3.0+m)*n/2.0)/rg |
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| 106 | if q < q1: |
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| 107 | F = (1.0/power(q,(3.0-n)))*exp((-q*q*rg*rg)/n) |
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| 108 | else: |
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| 109 | F = (1.0/power(q, m))*exp(-(n-3.0+m)/2.0)*power(((n-3.0+m)*n/2.0), |
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| 110 | ((n-3.0+m)/2.0))/power(rg,(n-3.0+m)) |
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| 111 | return F |
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| 112 | |
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| 113 | Iq.vectorized = False # Iq accepts an array of q values |
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| 114 | |
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| 115 | def Iqxy(qx, qy, *args): |
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| 116 | """ |
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| 117 | @param qx: Input q_x-value |
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| 118 | @param qy: Input q_y-value |
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| 119 | @param args: Remaining arguments |
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| 120 | """ |
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| 121 | return Iq(sqrt(qx ** 2 + qy ** 2), *args) |
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| 122 | |
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| 123 | Iqxy.vectorized = False # Iqxy accepts an array of qx, qy values |
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| 124 | |
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| 125 | demo = dict(scale=1.5, background=0.5, rg=60, s=1.0, m=3.0) |
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| 126 | |
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| 127 | oldname = "GuinierPorodModel" |
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| 128 | oldpars = dict(scale='scale', background='background', s='dim', m='m', rg='rg') |
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| 129 | |
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| 130 | tests = [[{'scale': 1.5, 'background':0.5}, 0.04, 5.290096890253155]] |
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