1 | r""" |
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2 | This model provides the form factor for a pearl necklace composed of two |
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3 | elements: *N* pearls (homogeneous spheres of radius *R*) freely jointed by *M* |
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4 | rods (like strings - with a total mass *Mw* = *M* \* *m*\ :sub:`r` + *N* \* *m*\ |
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5 | :sub:`s`, and the string segment length (or edge separation) *l* |
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6 | (= *A* - 2\ *R*)). *A* is the center-to-center pearl separation distance. |
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7 | |
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8 | .. figure:: img/pearl_necklace_geometry.jpg |
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9 | |
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10 | Pearl Necklace schematic |
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11 | |
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12 | Definition |
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13 | ---------- |
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14 | |
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15 | The output of the scattering intensity function for the pearl_necklace is |
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16 | given by (Schweins, 2004) |
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17 | |
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18 | .. math:: |
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19 | |
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20 | I(q)=\frac{ \text{scale} }{V} \cdot \frac{(S_{ss}(q)+S_{ff}(q)+S_{fs}(q))} |
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21 | {(M \cdot m_f + N \cdot m_s)^2} + \text{bkg} |
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22 | |
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23 | where |
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24 | |
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25 | .. math:: |
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26 | |
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27 | S_{ss}(q) &= sm_s^2\psi^2(q)[\frac{N}{1-sin(qA)/qA}-\frac{N}{2}- |
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28 | \frac{1-(sin(qA)/qA)^N}{(1-sin(qA)/qA)^2}\cdot\frac{sin(qA)}{qA}] \\ |
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29 | S_{ff}(q) &= sm_r^2[M\{2\Lambda(q)-(\frac{sin(ql/2)}{ql/2})\}+ |
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30 | \frac{2M\beta^2(q)}{1-sin(qA)/qA}-2\beta^2(q)\cdot |
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31 | \frac{1-(sin(qA)/qA)^M}{(1-sin(qA)/qA)^2}] \\ |
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32 | S_{fs}(q) &= m_r \beta (q) \cdot m_s \psi (q) \cdot 4[ |
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33 | \frac{N-1}{1-sin(qA)/qA}-\frac{1-(sin(qA)/qA)^{N-1}}{(1-sin(qA)/qA)^2} |
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34 | \cdot \frac{sin(qA)}{qA}] \\ |
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35 | \psi(q) &= 3 \cdot \frac{sin(qR)-(qR)\cdot cos(qR)}{(qR)^3} \\ |
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36 | \Lambda(q) &= \frac{\int_0^{ql}\frac{sin(t)}{t}dt}{ql} \\ |
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37 | \beta(q) &= \frac{\int_{qR}^{q(A-R)}\frac{sin(t)}{t}dt}{ql} |
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38 | |
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39 | where the mass *m*\ :sub:`i` is (SLD\ :sub:`i` - SLD\ :sub:`solvent`) \* |
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40 | (volume of the *N* pearls/rods). *V* is the total volume of the necklace. |
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41 | |
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42 | The 2D scattering intensity is the same as $P(q)$ above, regardless of the |
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43 | orientation of the *q* vector. |
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44 | |
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45 | The returned value is scaled to units of |cm^-1| and the parameters of the |
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46 | pearl_necklace model are the following |
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47 | |
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48 | NB: *num_pearls* must be an integer. |
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49 | |
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50 | References |
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51 | ---------- |
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52 | |
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53 | R Schweins and K Huber, *Particle Scattering Factor of Pearl Necklace Chains*, |
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54 | *Macromol. Symp.* 211 (2004) 25-42 2004 |
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55 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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56 | """ |
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57 | |
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58 | import numpy as np |
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59 | from numpy import inf, pi |
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60 | |
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61 | name = "pearl_necklace" |
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62 | title = "Colloidal spheres chained together with no preferential orientation" |
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63 | description = """ |
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64 | Calculate form factor for Pearl Necklace Model |
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65 | [Macromol. Symp. 2004, 211, 25-42] |
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66 | Parameters: |
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67 | background:background |
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68 | scale: scale factor |
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69 | sld: the SLD of the pearl spheres |
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70 | sld_string: the SLD of the strings |
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71 | sld_solvent: the SLD of the solvent |
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72 | num_pearls: number of the pearls |
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73 | radius: the radius of a pearl |
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74 | edge_sep: the length of string segment; surface to surface |
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75 | thick_string: thickness (ie, diameter) of the string |
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76 | """ |
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77 | category = "shape:cylinder" |
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78 | |
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79 | # ["name", "units", default, [lower, upper], "type","description"], |
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80 | parameters = [["radius", "Ang", 80.0, [0, inf], "volume", |
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81 | "Mean radius of the chained spheres"], |
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82 | ["edge_sep", "Ang", 350.0, [0, inf], "volume", |
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83 | "Mean separation of chained particles"], |
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84 | ["thick_string", "Ang", 2.5, [0, inf], "volume", |
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85 | "Thickness of the chain linkage"], |
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86 | ["num_pearls", "none", 3, [1, inf], "volume", |
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87 | "Number of pearls in the necklace (must be integer)"], |
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88 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", |
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89 | "Scattering length density of the chained spheres"], |
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90 | ["sld_string", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", |
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91 | "Scattering length density of the chain linkage"], |
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92 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", |
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93 | "Scattering length density of the solvent"], |
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94 | ] |
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95 | |
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96 | source = ["lib/sas_Si.c", "lib/sas_3j1x_x.c", "pearl_necklace.c"] |
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97 | single = False # use double precision unless told otherwise |
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98 | effective_radius_type = [ |
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99 | "equivalent volume sphere", |
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100 | ] |
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101 | |
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102 | def random(): |
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103 | radius = 10**np.random.uniform(1, 3) # 1 - 1000 |
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104 | thick_string = 10**np.random.uniform(0, np.log10(radius)-1) # 1 - radius/10 |
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105 | edge_sep = 10**np.random.uniform(0, 3) # 1 - 1000 |
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106 | num_pearls = np.round(10**np.random.uniform(0.3, 3)) # 2 - 1000 |
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107 | pars = dict( |
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108 | radius=radius, |
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109 | edge_sep=edge_sep, |
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110 | thick_string=thick_string, |
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111 | num_pearls=num_pearls, |
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112 | ) |
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113 | return pars |
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114 | |
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115 | # parameters for demo |
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116 | demo = dict(scale=1, background=0, radius=80.0, edge_sep=350.0, |
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117 | num_pearls=3, sld=1, sld_solvent=6.3, sld_string=1, |
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118 | thick_string=2.5, |
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119 | radius_pd=.2, radius_pd_n=5, |
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120 | edge_sep_pd=25.0, edge_sep_pd_n=5, |
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121 | num_pearls_pd=0, num_pearls_pd_n=0, |
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122 | thick_string_pd=0.2, thick_string_pd_n=5, |
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123 | ) |
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124 | # ER function is not being used here, not that it is likely very sensible to |
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125 | # include an S(Q) with this model, the default in sasview 5.0 would be to the |
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126 | # "unconstrained" radius_effective. |
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127 | #tests = [[{}, 0.001, 17380.245], [{}, 'ER', 115.39502]] |
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128 | tests = [[{}, 0.001, 17380.245]] |
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