[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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| 18 | - *sld* $-$ *solvent_sld* is the contrast $\Delta \rho$, |
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| 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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| 25 | $\langle D \rangle$, and |
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| 26 | |
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| 27 | - *spacing_polydisp* is the relative standard deviation of the Gaussian |
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| 28 | layer distance distribution $\sigma_D / \langle D \rangle$. |
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| 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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[d138d43] | 38 | infinite, planar bilayer of thickness $t$ |
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| 39 | |
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| 40 | .. math:: |
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[dc02af0] | 41 | |
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[d138d43] | 42 | P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2 |
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[dc02af0] | 43 | |
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[d138d43] | 44 | $Z_N(q)$ describes the interference effects for aggregates |
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[3c56da87] | 45 | consisting of more than one bilayer. The equations used are (3-5) |
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[d138d43] | 46 | from the Bergstrom reference: |
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| 47 | |
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| 48 | .. math:: |
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| 49 | |
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| 50 | |
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| 51 | Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)} |
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| 52 | + x_N S_N + (1 - x_N) S_{N+1} |
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| 53 | |
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| 54 | where |
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| 55 | |
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| 56 | .. math:: |
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| 57 | |
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| 58 | S_N(q) = \frac{a_N}{N}[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2 |
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| 59 | |
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| 60 | and |
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| 61 | |
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| 62 | .. math:: |
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| 63 | |
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| 64 | a_N = 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) |
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| 65 | - 4w^{N+2}\cos(Nq \langle D \rangle) |
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| 66 | + 2 w^{N+3}\cos[(N-1)q \langle D \rangle] |
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| 67 | + 2w^{N+1}\cos[(N+1)q \langle D \rangle] |
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| 68 | |
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| 69 | for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$. |
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[dc02af0] | 70 | |
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[3c56da87] | 71 | Non-integer numbers of stacks are calculated as a linear combination of |
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| 72 | the lower and higher values |
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[dc02af0] | 73 | |
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[d138d43] | 74 | .. math:: |
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| 75 | |
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| 76 | N_L = x_N N + (1 - x_N)(N+1) |
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[dc02af0] | 77 | |
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[3c56da87] | 78 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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[d138d43] | 79 | of the $q$ vector which is defined as |
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[dc02af0] | 80 | |
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| 81 | .. math:: |
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| 82 | |
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[d138d43] | 83 | q = \sqrt{q_x^2 + q_y^2} |
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[dc02af0] | 84 | |
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| 85 | |
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[d138d43] | 86 | .. figure:: img/lamellarPC_1d.jpg |
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[dc02af0] | 87 | |
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[d138d43] | 88 | 1D plot using the default values above (w/20000 data point). |
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[dc02af0] | 89 | |
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[d138d43] | 90 | Reference |
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| 91 | --------- |
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[dc02af0] | 92 | |
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[3c56da87] | 93 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, |
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| 94 | *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
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[dc02af0] | 95 | |
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| 96 | """ |
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| 97 | |
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[3c56da87] | 98 | from numpy import inf |
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[dc02af0] | 99 | |
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| 100 | name = "lamellarPC" |
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| 101 | title = "Random lamellar sheet with paracrystal structure factor" |
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| 102 | description = """\ |
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[3e428ec] | 103 | [Random lamellar phase with paracrystal structure factor] |
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[dc02af0] | 104 | randomly oriented stacks of infinite sheets |
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[3e428ec] | 105 | with paracrytal S(Q), having polydisperse spacing. |
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| 106 | sld = sheet scattering length density |
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| 107 | sld_solvent = solvent scattering length density |
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| 108 | background = incoherent background |
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| 109 | scale = scale factor |
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[dc02af0] | 110 | """ |
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[a5d0d00] | 111 | category = "shape:lamellae" |
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[dc02af0] | 112 | |
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[3e428ec] | 113 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 114 | parameters = [["thickness", "Ang", 33.0, [0, inf], "volume", |
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| 115 | "sheet thickness"], |
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| 116 | ["Nlayers", "", 20, [0, inf], "", |
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| 117 | "Number of layers"], |
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| 118 | ["spacing", "Ang", 250., [0.0, inf], "", |
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| 119 | "d-spacing of paracrystal stack"], |
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| 120 | ["spacing_polydisp", "Ang", 0.0, [0.0, inf], "", |
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[d138d43] | 121 | "d-spacing polydispersity"], |
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[3e428ec] | 122 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "", |
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| 123 | "layer scattering length density"], |
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| 124 | ["solvent_sld", "1e-6/Ang^2", 6.34, [-inf, inf], "", |
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| 125 | "Solvent scattering length density"], |
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| 126 | ] |
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| 127 | |
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| 128 | |
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| 129 | source = ["lamellarPC_kernel.c"] |
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[dc02af0] | 130 | |
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| 131 | form_volume = """ |
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| 132 | return 1.0; |
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| 133 | """ |
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| 134 | |
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| 135 | Iqxy = """ |
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[f734e7d] | 136 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[dc02af0] | 137 | """ |
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| 138 | |
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| 139 | # ER defaults to 0.0 |
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| 140 | # VR defaults to 1.0 |
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| 141 | |
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[3e428ec] | 142 | demo = dict(scale=1, background=0, |
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| 143 | thickness=33, Nlayers=20, spacing=250, spacing_polydisp=0.2, |
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| 144 | sld=1.0, solvent_sld=6.34, |
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| 145 | thickness_pd=0.2, thickness_pd_n=40) |
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[3c56da87] | 146 | |
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[dc02af0] | 147 | oldname = 'LamellarPCrystalModel' |
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[3e428ec] | 148 | oldpars = dict(spacing_polydisp='pd_spacing', sld='sld_layer', |
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| 149 | solvent_sld='sld_solvent') |
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