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_fig.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 PearlNecklaceModel 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: *number_of_pearls* must be an integer. |
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49 | |
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50 | .. figure:: img/pearl_plot.jpg |
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51 | |
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52 | 1D plot using the default values (w/1000 data point). |
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53 | |
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54 | REFERENCE |
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55 | |
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56 | R Schweins and K Huber, *Particle Scattering Factor of Pearl Necklace Chains*, |
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57 | *Macromol. Symp.* 211 (2004) 25-42 2004 |
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58 | """ |
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59 | |
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60 | from numpy import inf |
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61 | |
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62 | name = "pearl_necklace" |
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63 | title = "Colloidal spheres chained together with o preferential orientation" |
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64 | description = """ |
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65 | Calculate form factor for Pearl Necklace Model |
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66 | [Macromol. Symp. 2004, 211, 25-42] |
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67 | Parameters: |
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68 | background:background |
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69 | scale: scale factor |
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70 | sld: the SLD of the pearl spheres |
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71 | sld_string: the SLD of the strings |
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72 | sld_solvent: the SLD of the solvent |
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73 | number_of_pearls: number of the pearls |
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74 | radius: the radius of a pearl |
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75 | edge_separation: the length of string segment; surface to surface |
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76 | string_thickness: thickness (ie, diameter) of the string |
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77 | """ |
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78 | category = "shape:cylinder" |
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79 | |
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80 | # ["name", "units", default, [lower, upper], "type","description"], |
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81 | parameters = [["radius", "Angstrom", 80.0, [0, inf], "volume", |
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82 | "Mean radius of the chained spheres"], |
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83 | ["edge_separation", "Angstrom", 350.0, [0, inf], "volume", |
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84 | "Mean separation of chained particles"], |
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85 | ["string_thickness", "Angstrom", 2.5, [0, inf], "volume", |
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86 | "Thickness of the chain linkage"], |
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87 | ["number_of_pearls", "none", 3, [0, inf], "volume", |
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88 | "Mean number of pearls in each necklace"], |
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89 | ["sld", "Angstrom^2", 1.0, [-inf, inf], "", |
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90 | "Scattering length density of the chained spheres"], |
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91 | ["string_sld", "Angstrom^2", 1.0, [-inf, inf], "", |
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92 | "Scattering length density of the chain linkage"], |
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93 | ["solvent_sld", "Angstrom^2", 6.3, [-inf, inf], "", |
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94 | "Scattering length density of the solvent"], |
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95 | ] |
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96 | |
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97 | source = ["lib/Si.c", "pearl_necklace.c"] |
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98 | |
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99 | # parameters for demo |
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100 | demo = dict(scale=1, background=0, radius=80.0, edge_separation=350.0, |
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101 | number_of_pearls=3, sld=1, solvent_sld=6.3, string_sld=1, |
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102 | string_thickness=2.5, |
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103 | radius_pd=.2, radius_pd_n=5, |
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104 | edge_separation_pd=25.0, edge_separation_pd_n=5, |
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105 | number_of_pearls_pd=0, number_of_pearls_pd_n=0, |
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106 | string_thickness_pd=0.2, string_thickness_pd_n=5, |
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107 | ) |
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108 | |
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109 | # For testing against the old sasview models, include the converted parameter |
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110 | # names and the target sasview model name. |
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111 | oldname = 'PearlNecklaceModel' |
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112 | oldpars = dict(scale='scale',background='background',radius='radius', |
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113 | number_of_pearls='num_pearls', solvent_sld='sld_solv', |
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114 | string_thickness='thick_string', sld='sld_pearl', |
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115 | string_sld='sld_string', edge_separation='edge_separation') |
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