[dc02af0] | 1 | # Note: model title and parameter table are inserted automatically |
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| 2 | r""" |
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[3c56da87] | 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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[dc02af0] | 8 | |
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[d138d43] | 9 | Definition |
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| 10 | ---------- |
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[dc02af0] | 11 | |
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[d138d43] | 12 | In the equations below, |
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[dc02af0] | 13 | |
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[d138d43] | 14 | - *scale* is used instead of the mass per area of the bilayer $\Gamma_m$ |
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| 15 | (this corresponds to the volume fraction of the material in the bilayer, |
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| 16 | *not* the total excluded volume of the paracrystal), |
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| 17 | |
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[6ab4ed8] | 18 | - *sld* $-$ *sld_solvent* is the contrast $\Delta \rho$, |
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[d138d43] | 19 | |
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| 20 | - *thickness* is the layer thickness $t$, |
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| 21 | |
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| 22 | - *Nlayers* is the number of layers $N$, |
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| 23 | |
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| 24 | - *spacing* is the average distance between adjacent layers |
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[d18f8a8] | 25 | $\langle D \rangle$, and |
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[d138d43] | 26 | |
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| 27 | - *spacing_polydisp* is the relative standard deviation of the Gaussian |
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[d18f8a8] | 28 | layer distance distribution $\sigma_D / \langle D \rangle$. |
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[d138d43] | 29 | |
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| 30 | |
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| 31 | The scattering intensity $I(q)$ is calculated as |
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| 32 | |
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| 33 | .. math:: |
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| 34 | |
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| 35 | I(q) = 2\pi\Delta\rho^2\Gamma_m\frac{P_\text{bil}(q)}{q^2} Z_N(q) |
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[dc02af0] | 36 | |
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[3c56da87] | 37 | The form factor of the bilayer is approximated as the cross section of an |
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[6ab4ed8] | 38 | infinite, planar bilayer of thickness $t$ (compare the equations for the |
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| 39 | lamellar model). |
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[d138d43] | 40 | |
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| 41 | .. math:: |
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[dc02af0] | 42 | |
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[d138d43] | 43 | P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2 |
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[dc02af0] | 44 | |
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[d138d43] | 45 | $Z_N(q)$ describes the interference effects for aggregates |
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[3c56da87] | 46 | consisting of more than one bilayer. The equations used are (3-5) |
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[d138d43] | 47 | from the Bergstrom reference: |
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| 48 | |
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| 49 | .. math:: |
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| 50 | |
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| 51 | |
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[d18f8a8] | 52 | Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)} |
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[d138d43] | 53 | + x_N S_N + (1 - x_N) S_{N+1} |
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| 54 | |
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| 55 | where |
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| 56 | |
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| 57 | .. math:: |
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| 58 | |
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[d18f8a8] | 59 | S_N(q) = \frac{a_N}{N}[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2 |
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[d138d43] | 60 | |
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| 61 | and |
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| 62 | |
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| 63 | .. math:: |
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| 64 | |
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[d18f8a8] | 65 | a_N &= 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) \\ |
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| 66 | &\quad - 4w^{N+2}\cos(Nq \langle D \rangle) |
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| 67 | + 2 w^{N+3}\cos[(N-1)q \langle D \rangle] |
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| 68 | + 2w^{N+1}\cos[(N+1)q \langle D \rangle] |
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[d138d43] | 69 | |
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| 70 | for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$. |
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[dc02af0] | 71 | |
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[3c56da87] | 72 | Non-integer numbers of stacks are calculated as a linear combination of |
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| 73 | the lower and higher values |
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[dc02af0] | 74 | |
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[d138d43] | 75 | .. math:: |
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| 76 | |
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| 77 | N_L = x_N N + (1 - x_N)(N+1) |
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[dc02af0] | 78 | |
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[3c56da87] | 79 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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[d138d43] | 80 | of the $q$ vector which is defined as |
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[dc02af0] | 81 | |
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| 82 | .. math:: |
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| 83 | |
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[d138d43] | 84 | q = \sqrt{q_x^2 + q_y^2} |
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[dc02af0] | 85 | |
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| 86 | |
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[d138d43] | 87 | Reference |
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| 88 | --------- |
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[dc02af0] | 89 | |
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[3c56da87] | 90 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, |
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| 91 | *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
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[dc02af0] | 92 | |
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| 93 | """ |
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| 94 | |
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[3c56da87] | 95 | from numpy import inf |
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[dc02af0] | 96 | |
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[6ab4ed8] | 97 | name = "lamellar_stack_paracrystal" |
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[dc02af0] | 98 | title = "Random lamellar sheet with paracrystal structure factor" |
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| 99 | description = """\ |
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[3e428ec] | 100 | [Random lamellar phase with paracrystal structure factor] |
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[dc02af0] | 101 | randomly oriented stacks of infinite sheets |
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[3e428ec] | 102 | with paracrytal S(Q), having polydisperse spacing. |
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| 103 | sld = sheet scattering length density |
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| 104 | sld_solvent = solvent scattering length density |
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| 105 | background = incoherent background |
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| 106 | scale = scale factor |
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[dc02af0] | 107 | """ |
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[a5d0d00] | 108 | category = "shape:lamellae" |
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[dc02af0] | 109 | |
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[13ed84c] | 110 | single = False |
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| 111 | |
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[3e428ec] | 112 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 113 | parameters = [["thickness", "Ang", 33.0, [0, inf], "volume", |
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| 114 | "sheet thickness"], |
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| 115 | ["Nlayers", "", 20, [0, inf], "", |
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| 116 | "Number of layers"], |
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| 117 | ["spacing", "Ang", 250., [0.0, inf], "", |
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| 118 | "d-spacing of paracrystal stack"], |
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| 119 | ["spacing_polydisp", "Ang", 0.0, [0.0, inf], "", |
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[d138d43] | 120 | "d-spacing polydispersity"], |
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[42356c8] | 121 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", |
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[3e428ec] | 122 | "layer scattering length density"], |
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[42356c8] | 123 | ["sld_solvent", "1e-6/Ang^2", 6.34, [-inf, inf], "sld", |
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[3e428ec] | 124 | "Solvent scattering length density"], |
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| 125 | ] |
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| 126 | |
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| 127 | |
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[6ab4ed8] | 128 | source = ["lamellar_stack_paracrystal_kernel.c"] |
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[dc02af0] | 129 | |
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| 130 | form_volume = """ |
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| 131 | return 1.0; |
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| 132 | """ |
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| 133 | |
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| 134 | # ER defaults to 0.0 |
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| 135 | # VR defaults to 1.0 |
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| 136 | |
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[3e428ec] | 137 | demo = dict(scale=1, background=0, |
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| 138 | thickness=33, Nlayers=20, spacing=250, spacing_polydisp=0.2, |
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[6ab4ed8] | 139 | sld=1.0, sld_solvent=6.34, |
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[3e428ec] | 140 | thickness_pd=0.2, thickness_pd_n=40) |
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[3c56da87] | 141 | |
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[7f47777] | 142 | # |
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| 143 | tests = [ |
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| 144 | [ {'scale': 1.0, 'background' : 0.0, 'thickness' : 33.,'Nlayers' : 20.0, 'spacing' : 250., 'spacing_polydisp' : 0.2, |
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[6ab4ed8] | 145 | 'sld' : 1.0, 'sld_solvent' : 6.34, |
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[7f47777] | 146 | 'thickness_pd' : 0.0, 'thickness_pd_n' : 40 }, [0.001, 0.215268], [21829.3, 0.00487686]] |
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| 147 | ] |
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[6ab4ed8] | 148 | # ADDED by: RKH ON: 18Mar2016 converted from sasview previously, now renaming everything & sorting the docs |
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