[dc02af0] | 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 structures. The stacks of lamellae (infinite |
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| 4 | in lateral dimension) are treated as a paracrystal to account for the repeating spacing. The repeat distance is further |
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| 5 | characterized by a Gaussian polydispersity. **This model can be used for large multilamellar vesicles.** |
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| 6 | |
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| 7 | *2.1.33.1. Definition* |
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| 8 | |
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| 9 | The scattering intensity *I(q)* is calculated as |
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| 10 | |
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| 11 | .. image:: img/image145.jpg |
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| 12 | |
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| 13 | The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness *t* |
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| 14 | |
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| 15 | .. image:: img/image146.jpg |
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| 16 | |
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| 17 | Here, the scale factor is used instead of the mass per area of the bilayer (*G*). The scale factor is the volume |
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| 18 | fraction of the material in the bilayer, *not* the total excluded volume of the paracrystal. *Z*\ :sub:`N`\ *(q)* |
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| 19 | describes the interference effects for aggregates consisting of more than one bilayer. The equations used are (3-5) |
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| 20 | from the Bergstrom reference below. |
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| 21 | |
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| 22 | Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values |
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| 23 | |
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| 24 | .. image:: img/image147.jpg |
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| 25 | |
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| 26 | The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as |
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| 27 | |
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| 28 | .. math:: |
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| 29 | |
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| 30 | Q = \sqrt{Q_x^2 + Q_y^2} |
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| 31 | |
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| 32 | The parameters of the model are *Nlayers* = no. of layers, and *pd_spacing* = polydispersity of spacing. |
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| 33 | |
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| 34 | ============== ======== ============= |
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| 35 | Parameter name Units Default value |
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| 36 | ============== ======== ============= |
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| 37 | background |cm^-1| 0 |
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| 38 | scale None 1 |
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| 39 | Nlayers None 20 |
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| 40 | pd_spacing None 0.2 |
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| 41 | sld_layer |Ang^-2| 1e-6 |
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| 42 | sld_solvent |Ang^-2| 6.34e-6 |
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| 43 | spacing |Ang| 250 |
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| 44 | thickness |Ang| 33 |
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| 45 | ============== ======== ============= |
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| 46 | |
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| 47 | .. image:: img/image148.jpg |
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| 48 | |
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| 49 | *Figure. 1D plot using the default values above (w/20000 data point).* |
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| 50 | |
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| 51 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
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| 52 | (Kline, 2006). |
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| 53 | |
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| 54 | REFERENCE |
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| 55 | |
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| 56 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
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| 57 | |
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| 58 | """ |
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| 59 | |
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| 60 | from numpy import pi, inf |
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| 61 | |
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| 62 | name = "lamellarPC" |
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| 63 | title = "Random lamellar sheet with paracrystal structure factor" |
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| 64 | description = """\ |
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| 65 | [Random lamellar phase with paracrystal structure factor] |
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| 66 | randomly oriented stacks of infinite sheets |
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| 67 | with paracrytal S(Q), having polydisperse spacing. |
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| 68 | sld = sheet scattering length density |
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| 69 | sld_solvent = solvent scattering length density |
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| 70 | background = incoherent background |
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| 71 | scale = scale factor |
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| 72 | """ |
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[a5d0d00] | 73 | category = "shape:lamellae" |
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[dc02af0] | 74 | |
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| 75 | parameters = [ |
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| 76 | # [ "name", "units", default, [lower, upper], "type", |
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| 77 | # "description" ], |
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| 78 | [ "thickness", "Ang", 33.0, [0, inf], "volume", |
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| 79 | "sheet thickness" ], |
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| 80 | [ "Nlayers", "", 20, [0, inf], "", |
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| 81 | "Number of layers" ], |
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| 82 | [ "spacing", "Ang", 250., [0.0,inf], "", |
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| 83 | "d-spacing of paracrystal stack" ], |
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| 84 | [ "spacing_polydisp", "Ang", 0.0, [0.0,inf], "", |
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| 85 | "d-spacing of paracrystal stack" ], |
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| 86 | [ "sld", "1e-6/Ang^2", 1.0, [-inf,inf], "", |
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| 87 | "layer scattering length density" ], |
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| 88 | [ "solvent_sld", "1e-6/Ang^2", 6.34, [-inf,inf], "", |
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| 89 | "Solvent scattering length density" ], |
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| 90 | ] |
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| 91 | |
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| 92 | |
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| 93 | source = [ "lamellarPC_kernel.c"] |
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| 94 | |
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| 95 | form_volume = """ |
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| 96 | return 1.0; |
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| 97 | """ |
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| 98 | |
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| 99 | Iqxy = """ |
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[f734e7d] | 100 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[dc02af0] | 101 | """ |
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| 102 | |
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| 103 | # ER defaults to 0.0 |
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| 104 | # VR defaults to 1.0 |
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| 105 | |
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| 106 | demo = dict( |
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| 107 | scale=1, background=0, |
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| 108 | thickness=33,Nlayers=20,spacing=250,spacing_polydisp=0.2, |
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| 109 | sld=1.0, solvent_sld=6.34, |
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| 110 | thickness_pd= 0.2, thickness_pd_n=40 |
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| 111 | ) |
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| 112 | oldname = 'LamellarPCrystalModel' |
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| 113 | oldpars = dict(spacing_polydisp='pd_spacing', sld='sld_layer',solvent_sld='sld_solvent') |
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| 114 | |
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| 115 | |
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