[d1fe925] | 1 | # Note: model title and parameter table are inserted automatically |
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| 2 | r""" |
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| 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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| 8 | |
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| 9 | *2.1.33.1. Definition* |
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| 10 | |
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| 11 | The scattering intensity *I(q)* is calculated as |
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| 12 | |
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| 13 | .. image:: img/image145.jpg |
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| 14 | |
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| 15 | The form factor of the bilayer is approximated as the cross section of an |
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| 16 | infinite, planar bilayer of thickness *t* |
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| 17 | |
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| 18 | .. image:: img/image146.jpg |
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| 19 | |
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| 20 | Here, the scale factor is used instead of the mass per area of the |
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| 21 | bilayer (*G*). The scale factor is the volume fraction of the material in |
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| 22 | the bilayer, *not* the total excluded volume of the paracrystal. |
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| 23 | *Z*\ :sub:`N`\ *(q)* describes the interference effects for aggregates |
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| 24 | consisting of more than one bilayer. The equations used are (3-5) |
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| 25 | from the Bergstrom reference below. |
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| 26 | |
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| 27 | Non-integer numbers of stacks are calculated as a linear combination of |
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| 28 | the lower and higher values |
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| 29 | |
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| 30 | .. image:: img/image147.jpg |
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| 31 | |
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| 32 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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| 33 | of the *q* vector which 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 parameters of the model are *Nlayers* = no. of layers, and |
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| 40 | *pd_spacing* = polydispersity of spacing. |
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| 41 | |
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| 42 | ============== ======== ============= |
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| 43 | Parameter name Units Default value |
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| 44 | ============== ======== ============= |
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| 45 | background |cm^-1| 0 |
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| 46 | scale None 1 |
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| 47 | Nlayers None 20 |
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| 48 | pd_spacing None 0.2 |
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| 49 | sld_layer |Ang^-2| 1e-6 |
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| 50 | sld_solvent |Ang^-2| 6.34e-6 |
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| 51 | spacing |Ang| 250 |
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| 52 | thickness |Ang| 33 |
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| 53 | ============== ======== ============= |
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| 54 | |
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| 55 | .. image:: img/image148.jpg |
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| 56 | |
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| 57 | *Figure. 1D plot using the default values above (w/20000 data point).* |
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| 58 | |
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| 59 | Our model uses the form factor calculations implemented in a C library |
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| 60 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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| 61 | |
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| 62 | REFERENCE |
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| 63 | |
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| 64 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, |
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| 65 | *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
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| 66 | |
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| 67 | """ |
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| 68 | |
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| 69 | from numpy import inf |
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| 70 | |
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| 71 | name = "lamellarPC" |
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| 72 | title = "Random lamellar sheet with paracrystal structure factor" |
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| 73 | description = """\ |
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| 74 | [Random lamellar phase with paracrystal structure factor] |
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| 75 | randomly oriented stacks of infinite sheets |
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| 76 | with paracrytal S(Q), having polydisperse spacing. |
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| 77 | sld = sheet scattering length density |
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| 78 | sld_solvent = solvent scattering length density |
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| 79 | background = incoherent background |
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| 80 | scale = scale factor |
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| 81 | """ |
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| 82 | category = "shape:lamellae" |
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| 83 | |
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| 84 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 85 | parameters = [["thickness", "Ang", 33.0, [0, inf], "volume", |
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| 86 | "sheet thickness"], |
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| 87 | ["Nlayers", "", 20, [0, inf], "", |
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| 88 | "Number of layers"], |
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| 89 | ["spacing", "Ang", 250., [0.0, inf], "", |
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| 90 | "d-spacing of paracrystal stack"], |
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| 91 | ["spacing_polydisp", "Ang", 0.0, [0.0, inf], "", |
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| 92 | "d-spacing of paracrystal stack"], |
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| 93 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "", |
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| 94 | "layer scattering length density"], |
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| 95 | ["solvent_sld", "1e-6/Ang^2", 6.34, [-inf, inf], "", |
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| 96 | "Solvent scattering length density"], |
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| 97 | ] |
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| 98 | |
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| 99 | |
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| 100 | source = ["lamellarPC_kernel.c"] |
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| 101 | |
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| 102 | form_volume = """ |
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| 103 | return 1.0; |
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| 104 | """ |
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| 105 | |
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| 106 | Iqxy = """ |
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| 107 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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| 108 | """ |
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| 109 | |
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| 110 | # ER defaults to 0.0 |
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| 111 | # VR defaults to 1.0 |
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| 112 | |
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| 113 | demo = dict(scale=1, background=0, |
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| 114 | thickness=33, Nlayers=20, spacing=250, spacing_polydisp=0.2, |
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| 115 | sld=1.0, solvent_sld=6.34, |
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| 116 | thickness_pd=0.2, thickness_pd_n=40) |
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| 117 | |
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| 118 | oldname = 'LamellarPCrystalModel' |
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| 119 | oldpars = dict(spacing_polydisp='pd_spacing', sld='sld_layer', |
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| 120 | solvent_sld='sld_solvent') |
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