1 | # Note: model title and parameter table are inserted automatically |
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2 | r""" |
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3 | This model calculates the scattering from a stack of repeating lamellar |
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4 | structures. The stacks of lamellae (infinite in lateral dimension) are |
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5 | treated as a paracrystal to account for the repeating spacing. The repeat |
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6 | distance is further characterized by a Gaussian polydispersity. **This model |
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7 | can be used for large multilamellar vesicles.** |
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8 | |
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9 | *2.1.33.1. Definition* |
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10 | |
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11 | The scattering intensity *I(q)* is calculated as |
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12 | |
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13 | .. image:: img/image145.jpg |
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14 | |
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15 | The form factor of the bilayer is approximated as the cross section of an |
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16 | infinite, planar bilayer of thickness *t* |
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17 | |
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18 | .. image:: img/image146.jpg |
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19 | |
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20 | Here, the scale factor is used instead of the mass per area of the |
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21 | bilayer (*G*). The scale factor is the volume fraction of the material in |
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22 | the bilayer, *not* the total excluded volume of the paracrystal. |
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23 | *Z*\ :sub:`N`\ *(q)* describes the interference effects for aggregates |
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24 | consisting of more than one bilayer. The equations used are (3-5) |
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25 | from the Bergstrom reference below. |
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26 | |
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27 | Non-integer numbers of stacks are calculated as a linear combination of |
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28 | the lower and higher values |
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29 | |
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30 | .. image:: img/image147.jpg |
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31 | |
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32 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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33 | of the *q* vector which is defined as |
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34 | |
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35 | .. math:: |
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36 | |
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37 | Q = \sqrt{Q_x^2 + Q_y^2} |
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38 | |
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39 | The parameters of the model are *Nlayers* = no. of layers, and |
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40 | *pd_spacing* = polydispersity of spacing. |
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41 | |
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42 | ============== ======== ============= |
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43 | Parameter name Units Default value |
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44 | ============== ======== ============= |
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45 | background |cm^-1| 0 |
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46 | scale None 1 |
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47 | Nlayers None 20 |
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48 | pd_spacing None 0.2 |
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49 | sld_layer |Ang^-2| 1e-6 |
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50 | sld_solvent |Ang^-2| 6.34e-6 |
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51 | spacing |Ang| 250 |
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52 | thickness |Ang| 33 |
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53 | ============== ======== ============= |
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54 | |
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55 | .. image:: img/image148.jpg |
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56 | |
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57 | *Figure. 1D plot using the default values above (w/20000 data point).* |
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58 | |
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59 | Our model uses the form factor calculations implemented in a C library |
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60 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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61 | |
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62 | REFERENCE |
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63 | |
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64 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, |
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65 | *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
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66 | |
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67 | """ |
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68 | |
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69 | from numpy import inf |
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70 | |
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71 | name = "lamellarPC" |
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72 | title = "Random lamellar sheet with paracrystal structure factor" |
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73 | description = """\ |
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74 | [Random lamellar phase with paracrystal structure factor] |
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75 | randomly oriented stacks of infinite sheets |
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76 | with paracrytal S(Q), having polydisperse spacing. |
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77 | sld = sheet scattering length density |
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78 | sld_solvent = solvent scattering length density |
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79 | background = incoherent background |
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80 | scale = scale factor |
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81 | """ |
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82 | category = "shape:lamellae" |
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83 | |
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84 | # ["name", "units", default, [lower, upper], "type","description"], |
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85 | parameters = [["thickness", "Ang", 33.0, [0, inf], "volume", |
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86 | "sheet thickness"], |
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87 | ["Nlayers", "", 20, [0, inf], "", |
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88 | "Number of layers"], |
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89 | ["spacing", "Ang", 250., [0.0, inf], "", |
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90 | "d-spacing of paracrystal stack"], |
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91 | ["spacing_polydisp", "Ang", 0.0, [0.0, inf], "", |
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92 | "d-spacing of paracrystal stack"], |
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93 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "", |
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94 | "layer scattering length density"], |
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95 | ["solvent_sld", "1e-6/Ang^2", 6.34, [-inf, inf], "", |
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96 | "Solvent scattering length density"], |
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97 | ] |
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98 | |
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99 | |
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100 | source = ["lamellarPC_kernel.c"] |
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101 | |
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102 | form_volume = """ |
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103 | return 1.0; |
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104 | """ |
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105 | |
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106 | Iqxy = """ |
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107 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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108 | """ |
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109 | |
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110 | # ER defaults to 0.0 |
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111 | # VR defaults to 1.0 |
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112 | |
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113 | demo = dict(scale=1, background=0, |
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114 | thickness=33, Nlayers=20, spacing=250, spacing_polydisp=0.2, |
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115 | sld=1.0, solvent_sld=6.34, |
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116 | thickness_pd=0.2, thickness_pd_n=40) |
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117 | |
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118 | oldname = 'LamellarPCrystalModel' |
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119 | oldpars = dict(spacing_polydisp='pd_spacing', sld='sld_layer', |
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120 | solvent_sld='sld_solvent') |
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