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 |
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4 | phase where a random distribution in solution are assumed. Here a Caille $S(q)$ |
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5 | is used for the lamellar stacks. |
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6 | |
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7 | The scattering intensity $I(q)$ is |
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8 | |
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9 | .. math:: |
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10 | |
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11 | I(q) = 2 \pi \frac{P(q)S(q)}{\delta q^2} |
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12 | |
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13 | |
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14 | The form factor $P(q)$ is |
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15 | |
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16 | .. math:: |
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17 | |
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18 | P(q) = \frac{4}{q^2}\big\{ |
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19 | \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right] |
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20 | + \Delta\rho_T\sin(q\delta_T)\big\}^2 |
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21 | |
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22 | and the structure factor $S(q)$ is |
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23 | |
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24 | .. math:: |
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25 | |
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26 | S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right) |
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27 | \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right) |
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28 | |
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29 | where |
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30 | |
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31 | .. math:: |
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32 | :nowrap: |
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33 | |
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34 | \begin{align*} |
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35 | \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right) |
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36 | && \\ |
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37 | \gamma_E &= 0.5772156649 |
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38 | && \text{Euler's constant} \\ |
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39 | \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} |
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40 | && \text{Caille constant} |
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41 | \end{align*} |
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42 | |
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43 | |
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44 | $\delta_T$ is the tail length (or *tail_length*), $\delta_H$ is the head |
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45 | thickness (or *head_length*), $\Delta\rho_H$ is SLD(headgroup) - SLD(solvent), |
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46 | and $\Delta\rho_T$ is SLD(tail) - SLD(headgroup). Here $d$ is (repeat) spacing, |
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47 | $K$ is smectic bending elasticity, $B$ is compression modulus, and $N$ is the |
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48 | number of lamellar plates (*Nlayers*). |
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49 | |
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50 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the |
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51 | assumptions of the model are incorrect.** And due to a complication of the |
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52 | model function, users are responsible for making sure that all the assumptions |
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53 | are handled accurately (see the original reference below for more details). |
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54 | |
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55 | Non-integer numbers of stacks are calculated as a linear combination of |
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56 | results for the next lower and higher values. |
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57 | |
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58 | The 2D scattering intensity is calculated in the same way as 1D, where |
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59 | the $q$ vector is defined as |
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60 | |
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61 | .. math:: |
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62 | |
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63 | q = \sqrt{q_x^2 + q_y^2} |
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64 | |
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65 | .. figure:: img/lamellarCailleHG_1d.jpg |
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66 | |
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67 | 1D plot using the default values (w/6000 data point). |
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68 | |
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69 | References |
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70 | ---------- |
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71 | |
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72 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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73 | |
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74 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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75 | """ |
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76 | from numpy import inf |
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77 | |
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78 | name = "lamellarCailleHG" |
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79 | title = "Random lamellar sheet with Caille structure factor" |
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80 | description = """\ |
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81 | [Random lamellar phase with Caille structure factor] |
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82 | randomly oriented stacks of infinite sheets |
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83 | with Caille S(Q), having polydisperse spacing. |
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84 | layer thickness =(H+T+T+H) = 2(Head+Tail) |
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85 | sld = Tail scattering length density |
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86 | sld_head = Head scattering length density |
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87 | sld_solvent = solvent scattering length density |
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88 | background = incoherent background |
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89 | scale = scale factor |
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90 | """ |
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91 | category = "shape:lamellae" |
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92 | |
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93 | single = False |
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94 | parameters = [ |
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95 | # [ "name", "units", default, [lower, upper], "type", |
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96 | # "description" ], |
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97 | ["tail_length", "Ang", 10, [0, inf], "volume", |
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98 | "Tail thickness"], |
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99 | ["head_length", "Ang", 2, [0, inf], "volume", |
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100 | "head thickness"], |
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101 | ["Nlayers", "", 30, [0, inf], "", |
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102 | "Number of layers"], |
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103 | ["spacing", "Ang", 40., [0.0, inf], "volume", |
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104 | "d-spacing of Caille S(Q)"], |
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105 | ["Caille_parameter", "", 0.001, [0.0, 0.8], "", |
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106 | "Caille parameter"], |
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107 | ["sld", "1e-6/Ang^2", 0.4, [-inf, inf], "", |
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108 | "Tail scattering length density"], |
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109 | ["head_sld", "1e-6/Ang^2", 2.0, [-inf, inf], "", |
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110 | "Head scattering length density"], |
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111 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "", |
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112 | "Solvent scattering length density"], |
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113 | ] |
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114 | |
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115 | source = ["lamellarCailleHG_kernel.c"] |
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116 | |
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117 | # No volume normalization despite having a volume parameter |
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118 | # This should perhaps be volume normalized? |
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119 | form_volume = """ |
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120 | return 1.0; |
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121 | """ |
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122 | |
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123 | Iqxy = """ |
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124 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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125 | """ |
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126 | |
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127 | # ER defaults to 0.0 |
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128 | # VR defaults to 1.0 |
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129 | |
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130 | demo = dict( |
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131 | scale=1, background=0, |
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132 | Nlayers=20, spacing=200., Caille_parameter=0.05, |
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133 | tail_length=15, head_length=10, |
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134 | #sld=-1, head_sld=4.0, solvent_sld=6.0, |
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135 | sld=-1, head_sld=4.1, solvent_sld=6.0, |
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136 | tail_length_pd=0.1, tail_length_pd_n=20, |
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137 | head_length_pd=0.05, head_length_pd_n=30, |
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138 | spacing_pd=0.2, spacing_pd_n=40, |
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139 | ) |
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140 | |
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141 | oldname = 'LamellarPSHGModel' |
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142 | |
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143 | oldpars = dict( |
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144 | tail_length='deltaT', head_length='deltaH', Nlayers='n_plates', |
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145 | Caille_parameter='caille', sld='sld_tail', head_sld='sld_head', |
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146 | solvent_sld='sld_solvent') |
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147 | # |
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148 | tests = [[{'scale': 1.0, 'background': 0.0, 'tail_length': 10.0, 'head_length': 2.0, |
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149 | 'Nlayers': 30.0, 'spacing': 40., 'Caille_parameter': 0.001, 'sld': 0.4, |
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150 | 'head_sld': 2.0, 'solvent_sld': 6.0, 'tail_length_pd': 0.0, |
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151 | 'head_length_pd': 0.0, 'spacing_pd': 0.0}, [0.001], [6838238.571488]]] |
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