[dc02af0] | 1 | # Note: model title and parameter table are inserted automatically |
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| 2 | r""" |
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[12c810f] | 3 | This model provides the scattering intensity, $I(q) = P(q)S(q)$, for a lamellar |
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[eb69cce] | 4 | phase where a random distribution in solution are assumed. Here a Caille $S(q)$ |
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[12c810f] | 5 | is used for the lamellar stacks. |
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[dc02af0] | 6 | |
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[12c810f] | 7 | The scattering intensity $I(q)$ is |
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[dc02af0] | 8 | |
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[12c810f] | 9 | .. math:: |
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| 10 | |
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[6ab4ed8] | 11 | I(q) = 2 \pi \frac{P(q)S(q)}{q^2\delta } |
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[12c810f] | 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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[dc02af0] | 17 | |
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[12c810f] | 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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[dc02af0] | 21 | |
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[12c810f] | 22 | and the structure factor $S(q)$ is |
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[dc02af0] | 23 | |
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[12c810f] | 24 | .. math:: |
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[dc02af0] | 25 | |
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[12c810f] | 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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[dc02af0] | 28 | |
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| 29 | where |
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| 30 | |
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[12c810f] | 31 | .. math:: |
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[d18f8a8] | 32 | :nowrap: |
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[12c810f] | 33 | |
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[d18f8a8] | 34 | \begin{align*} |
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[eb69cce] | 35 | \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right) |
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[d18f8a8] | 36 | && \\ |
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[eb69cce] | 37 | \gamma_E &= 0.5772156649 |
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[d18f8a8] | 38 | && \text{Euler's constant} \\ |
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[eb69cce] | 39 | \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} |
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[d18f8a8] | 40 | && \text{Caille constant} |
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| 41 | \end{align*} |
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[12c810f] | 42 | |
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[dc02af0] | 43 | |
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[a807206] | 44 | $\delta_T$ is the tail length (or *length_tail*), $\delta_H$ is the head |
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| 45 | thickness (or *length_head*), $\Delta\rho_H$ is SLD(headgroup) - SLD(solvent), |
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[12c810f] | 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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[dc02af0] | 49 | |
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[12c810f] | 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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[dc02af0] | 54 | |
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[12c810f] | 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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[dc02af0] | 57 | |
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[6ab4ed8] | 58 | Be aware that the computations may be very slow. |
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| 59 | |
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[12c810f] | 60 | The 2D scattering intensity is calculated in the same way as 1D, where |
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| 61 | the $q$ vector is defined as |
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[dc02af0] | 62 | |
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| 63 | .. math:: |
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| 64 | |
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[12c810f] | 65 | q = \sqrt{q_x^2 + q_y^2} |
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[dc02af0] | 66 | |
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| 67 | |
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[eb69cce] | 68 | References |
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| 69 | ---------- |
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[dc02af0] | 70 | |
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| 71 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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| 72 | |
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| 73 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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| 74 | """ |
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[2d81cfe] | 75 | |
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| 76 | import numpy as np |
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[3c56da87] | 77 | from numpy import inf |
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[dc02af0] | 78 | |
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[6ab4ed8] | 79 | name = "lamellar_hg_stack_caille" |
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| 80 | title = "Random lamellar head/tail/tail/head sheet with Caille structure factor" |
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[dc02af0] | 81 | description = """\ |
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[3e428ec] | 82 | [Random lamellar phase with Caille structure factor] |
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[dc02af0] | 83 | randomly oriented stacks of infinite sheets |
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[3e428ec] | 84 | with Caille S(Q), having polydisperse spacing. |
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| 85 | layer thickness =(H+T+T+H) = 2(Head+Tail) |
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| 86 | sld = Tail scattering length density |
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| 87 | sld_head = Head scattering length density |
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| 88 | sld_solvent = solvent scattering length density |
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| 89 | background = incoherent background |
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| 90 | scale = scale factor |
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[dc02af0] | 91 | """ |
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[a5d0d00] | 92 | category = "shape:lamellae" |
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[dc02af0] | 93 | |
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[7f1ee79] | 94 | single = False # TODO: check |
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[dc02af0] | 95 | parameters = [ |
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[d4666ca] | 96 | # [ "name", "units", default, [lower, upper], "type", |
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| 97 | # "description" ], |
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[a807206] | 98 | ["length_tail", "Ang", 10, [0, inf], "volume", |
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[d4666ca] | 99 | "Tail thickness"], |
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[a807206] | 100 | ["length_head", "Ang", 2, [0, inf], "volume", |
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[d4666ca] | 101 | "head thickness"], |
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[a57b31d] | 102 | ["Nlayers", "", 30, [1, inf], "", |
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[d4666ca] | 103 | "Number of layers"], |
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[7c57861] | 104 | ["d_spacing", "Ang", 40., [0.0, inf], "volume", |
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| 105 | "lamellar d-spacing of Caille S(Q)"], |
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[d4666ca] | 106 | ["Caille_parameter", "", 0.001, [0.0, 0.8], "", |
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| 107 | "Caille parameter"], |
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[42356c8] | 108 | ["sld", "1e-6/Ang^2", 0.4, [-inf, inf], "sld", |
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[d4666ca] | 109 | "Tail scattering length density"], |
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[42356c8] | 110 | ["sld_head", "1e-6/Ang^2", 2.0, [-inf, inf], "sld", |
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[d4666ca] | 111 | "Head scattering length density"], |
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[42356c8] | 112 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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[d4666ca] | 113 | "Solvent scattering length density"], |
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[dc02af0] | 114 | ] |
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| 115 | |
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[0bef47b] | 116 | source = ["lamellar_hg_stack_caille.c"] |
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[dc02af0] | 117 | |
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| 118 | # No volume normalization despite having a volume parameter |
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| 119 | # This should perhaps be volume normalized? |
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| 120 | form_volume = """ |
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| 121 | return 1.0; |
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| 122 | """ |
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| 123 | |
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| 124 | # ER defaults to 0.0 |
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| 125 | # VR defaults to 1.0 |
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| 126 | |
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[404ebbd] | 127 | def random(): |
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| 128 | total_thickness = 10**np.random.uniform(2, 4.7) |
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| 129 | Nlayers = np.random.randint(2, 200) |
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| 130 | d_spacing = total_thickness / Nlayers |
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| 131 | thickness = d_spacing * np.random.uniform(0, 1) |
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| 132 | length_head = thickness * np.random.uniform(0, 1) |
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| 133 | length_tail = thickness - length_head |
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| 134 | Caille_parameter = np.random.uniform(0, 0.8) |
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| 135 | pars = dict( |
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| 136 | length_head=length_head, |
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| 137 | length_tail=length_tail, |
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| 138 | Nlayers=Nlayers, |
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| 139 | d_spacing=d_spacing, |
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| 140 | Caille_parameter=Caille_parameter, |
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| 141 | ) |
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| 142 | return pars |
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| 143 | |
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[dc02af0] | 144 | demo = dict( |
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[d4666ca] | 145 | scale=1, background=0, |
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[7c57861] | 146 | Nlayers=20, d_spacing=200., Caille_parameter=0.05, |
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[a807206] | 147 | length_tail=15, length_head=10, |
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[6ab4ed8] | 148 | #sld=-1, sld_head=4.0, sld_solvent=6.0, |
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| 149 | sld=-1, sld_head=4.1, sld_solvent=6.0, |
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[a807206] | 150 | length_tail_pd=0.1, length_tail_pd_n=20, |
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| 151 | length_head_pd=0.05, length_head_pd_n=30, |
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[7c57861] | 152 | d_spacing_pd=0.2, d_spacing_pd_n=40, |
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[d4666ca] | 153 | ) |
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[dc02af0] | 154 | |
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[7f47777] | 155 | # |
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[a807206] | 156 | tests = [[{'scale': 1.0, 'background': 0.0, 'length_tail': 10.0, 'length_head': 2.0, |
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[7c57861] | 157 | 'Nlayers': 30.0, 'd_spacing': 40., 'Caille_parameter': 0.001, 'sld': 0.4, |
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[a807206] | 158 | 'sld_head': 2.0, 'sld_solvent': 6.0, 'length_tail_pd': 0.0, |
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[7c57861] | 159 | 'length_head_pd': 0.0, 'd_spacing_pd': 0.0}, [0.001], [6838238.571488]]] |
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[40a87fa] | 160 | # ADDED by: RKH ON: 18Mar2016 converted from sasview previously, now renaming everything & sorting the docs |
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