1 | # Note: model title and parameter table are inserted automatically |
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2 | r""" |
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3 | This model provides the scattering intensity, *I(q)* = *P(q)* \* *S(q)*, for a lamellar phase where a random |
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4 | distribution in solution are assumed. Here a Caille S(Q) is used for the lamellar stacks. |
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5 | |
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6 | The scattering intensity *I(q)* is |
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7 | |
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8 | .. image:: img/lamellarCailleHG_139.PNG |
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9 | |
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10 | The form factor is |
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11 | |
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12 | .. image:: img/lamellarCailleHG_143.PNG |
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13 | |
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14 | and the structure factor is |
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15 | |
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16 | .. image:: img/lamellarCailleHG_140.PNG |
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17 | |
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18 | where |
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19 | |
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20 | .. image:: img/lamellarCailleHG_141.PNG |
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21 | |
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22 | where |delta|\ T = tail length (or *tail_length*), |delta|\ H = head thickness (or *h_thickness*), |
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23 | |drho|\ H = SLD(headgroup) - SLD(solvent), and |drho|\ T = SLD(tail) - SLD(headgroup). |
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24 | Here *d* = (repeat) spacing, *K* = smectic bending elasticity, *B* = compression modulus, and N = number of lamellar |
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25 | plates (*n_plates*). |
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26 | |
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27 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect.** |
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28 | And due to a complication of the model function, users are responsible for making sure that all the assumptions are |
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29 | handled accurately (see the original reference below for more details). |
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30 | |
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31 | Non-integer numbers of stacks are calculated as a linear combination of results for the next lower and higher values. |
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32 | |
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33 | The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector 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 returned value is in units of |cm^-1|, on absolute scale. |
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40 | |
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41 | ============== ======== ============= |
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42 | Parameter name Units Default value |
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43 | ============== ======== ============= |
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44 | background |cm^-1| 0.001 |
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45 | sld_head |Ang^-2| 2e-06 |
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46 | scale None 1 |
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47 | sld_solvent |Ang^-2| 6e-06 |
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48 | deltaH |Ang| 2 |
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49 | deltaT |Ang| 10 |
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50 | sld_tail |Ang^-2| 0 |
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51 | n_plates None 30 |
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52 | spacing |Ang| 40 |
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53 | caille |Ang^-2| 0.001 |
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54 | ============== ======== ============= |
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55 | |
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56 | .. image:: img/lamellarCailleHG_142.jpg |
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57 | |
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58 | *Figure. 1D plot using the default values (w/6000 data point).* |
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59 | |
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60 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
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61 | (Kline, 2006). |
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62 | |
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63 | REFERENCE |
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64 | |
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65 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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66 | |
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67 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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68 | """ |
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69 | from numpy import pi, inf |
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70 | |
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71 | name = "lamellarCailleHG" |
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72 | title = "Random lamellar sheet with Caille structure factor" |
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73 | description = """\ |
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74 | [Random lamellar phase with Caille structure factor] |
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75 | randomly oriented stacks of infinite sheets |
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76 | with Caille S(Q), having polydisperse spacing. |
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77 | layer thickness =(H+T+T+H) = 2(Head+Tail) |
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78 | sld = Tail scattering length density |
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79 | sld_head = Head scattering length density |
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80 | sld_solvent = solvent scattering length density |
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81 | background = incoherent background |
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82 | scale = scale factor |
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83 | """ |
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84 | |
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85 | parameters = [ |
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86 | # [ "name", "units", default, [lower, upper], "type", |
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87 | # "description" ], |
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88 | [ "tail_length", "Ang", 10, [0, inf], "volume", |
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89 | "Tail thickness" ], |
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90 | [ "head_length", "Ang", 2, [0, inf], "volume", |
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91 | "head thickness" ], |
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92 | [ "Nlayers", "", 30, [0, inf], "", |
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93 | "Number of layers" ], |
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94 | [ "spacing", "Ang", 40., [0.0,inf], "volume", |
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95 | "d-spacing of Caille S(Q)" ], |
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96 | [ "Caille_parameter", "", 0.001, [0.0,0.8], "", |
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97 | "Caille parameter" ], |
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98 | [ "sld", "1e-6/Ang^2", 0.4, [-inf,inf], "", |
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99 | "Tail scattering length density" ], |
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100 | [ "head_sld", "1e-6/Ang^2", 2.0, [-inf,inf], "", |
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101 | "Head scattering length density" ], |
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102 | [ "solvent_sld", "1e-6/Ang^2", 6, [-inf,inf], "", |
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103 | "Solvent scattering length density" ], |
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104 | ] |
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105 | |
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106 | source = [ "lamellarCailleHG_kernel.c"] |
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107 | |
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108 | # No volume normalization despite having a volume parameter |
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109 | # This should perhaps be volume normalized? |
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110 | form_volume = """ |
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111 | return 1.0; |
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112 | """ |
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113 | |
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114 | Iqxy = """ |
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115 | // never called since no orientation or magnetic parameters. |
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116 | return -1.0; |
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117 | """ |
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118 | |
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119 | # ER defaults to 0.0 |
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120 | # VR defaults to 1.0 |
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121 | |
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122 | demo = dict( |
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123 | scale=1, background=0, |
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124 | Nlayers=20,spacing=200., |
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125 | Caille_parameter=0.05, |
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126 | tail_length=15,head_length=10, |
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127 | sld=-1, head_sld=4.0, solvent_sld=6.0, |
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128 | tail_length_pd= 0.1, tail_length_pd_n=20, |
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129 | head_length_pd= 0.05, head_length_pd_n=30, |
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130 | spacing_pd= 0, spacing_pd_n=40 |
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131 | ) |
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132 | |
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133 | oldname = 'LamellarPSHGModel' |
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134 | oldpars = dict(tail_length='deltaT',head_length='deltaH',Nlayers='n_plates',Caille_parameter='caille', sld='sld_tail', head_sld='sld_head',solvent_sld='sld_solvent') |
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