[dc02af0] | 1 | # Note: model title and parameter table are inserted automatically |
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| 2 | r""" |
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| 3 | This model provides the scattering intensity, *I(q)* = *P(q)* \* *S(q)*, for a lamellar phase where a random |
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| 4 | distribution in solution are assumed. Here a Caille S(Q) is used for the lamellar stacks. |
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| 5 | |
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| 6 | The scattering intensity *I(q)* is |
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| 7 | |
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| 8 | .. image:: img/lamellarCailleHG_139.PNG |
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| 9 | |
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| 10 | The form factor is |
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| 11 | |
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| 12 | .. image:: img/lamellarCailleHG_143.PNG |
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| 13 | |
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| 14 | and the structure factor is |
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| 15 | |
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| 16 | .. image:: img/lamellarCailleHG_140.PNG |
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| 17 | |
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| 18 | where |
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| 19 | |
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| 20 | .. image:: img/lamellarCailleHG_141.PNG |
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| 21 | |
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| 22 | where |delta|\ T = tail length (or *tail_length*), |delta|\ H = head thickness (or *h_thickness*), |
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| 23 | |drho|\ H = SLD(headgroup) - SLD(solvent), and |drho|\ T = SLD(tail) - SLD(headgroup). |
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| 24 | Here *d* = (repeat) spacing, *K* = smectic bending elasticity, *B* = compression modulus, and N = number of lamellar |
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| 25 | plates (*n_plates*). |
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| 26 | |
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| 27 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect.** |
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| 28 | And due to a complication of the model function, users are responsible for making sure that all the assumptions are |
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| 29 | handled accurately (see the original reference below for more details). |
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| 30 | |
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| 31 | Non-integer numbers of stacks are calculated as a linear combination of results for the next lower and higher values. |
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| 32 | |
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| 33 | The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector 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 returned value is in units of |cm^-1|, on absolute scale. |
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| 40 | |
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| 41 | ============== ======== ============= |
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| 42 | Parameter name Units Default value |
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| 43 | ============== ======== ============= |
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| 44 | background |cm^-1| 0.001 |
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| 45 | sld_head |Ang^-2| 2e-06 |
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| 46 | scale None 1 |
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| 47 | sld_solvent |Ang^-2| 6e-06 |
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| 48 | deltaH |Ang| 2 |
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| 49 | deltaT |Ang| 10 |
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| 50 | sld_tail |Ang^-2| 0 |
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| 51 | n_plates None 30 |
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| 52 | spacing |Ang| 40 |
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| 53 | caille |Ang^-2| 0.001 |
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| 54 | ============== ======== ============= |
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| 55 | |
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| 56 | .. image:: img/lamellarCailleHG_142.jpg |
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| 57 | |
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| 58 | *Figure. 1D plot using the default values (w/6000 data point).* |
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| 59 | |
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| 60 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
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| 61 | (Kline, 2006). |
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| 62 | |
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| 63 | REFERENCE |
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| 64 | |
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| 65 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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| 66 | |
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| 67 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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| 68 | """ |
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| 69 | from numpy import pi, inf |
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| 70 | |
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| 71 | name = "lamellarCailleHG" |
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| 72 | title = "Random lamellar sheet with Caille structure factor" |
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| 73 | description = """\ |
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| 74 | [Random lamellar phase with Caille structure factor] |
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| 75 | randomly oriented stacks of infinite sheets |
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| 76 | with Caille S(Q), having polydisperse spacing. |
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| 77 | layer thickness =(H+T+T+H) = 2(Head+Tail) |
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| 78 | sld = Tail scattering length density |
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| 79 | sld_head = Head scattering length density |
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| 80 | sld_solvent = solvent scattering length density |
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| 81 | background = incoherent background |
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| 82 | scale = scale factor |
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| 83 | """ |
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[a5d0d00] | 84 | category = "shape:lamellae" |
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[dc02af0] | 85 | |
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| 86 | parameters = [ |
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| 87 | # [ "name", "units", default, [lower, upper], "type", |
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| 88 | # "description" ], |
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| 89 | [ "tail_length", "Ang", 10, [0, inf], "volume", |
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| 90 | "Tail thickness" ], |
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| 91 | [ "head_length", "Ang", 2, [0, inf], "volume", |
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| 92 | "head thickness" ], |
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| 93 | [ "Nlayers", "", 30, [0, inf], "", |
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| 94 | "Number of layers" ], |
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| 95 | [ "spacing", "Ang", 40., [0.0,inf], "volume", |
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| 96 | "d-spacing of Caille S(Q)" ], |
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| 97 | [ "Caille_parameter", "", 0.001, [0.0,0.8], "", |
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| 98 | "Caille parameter" ], |
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| 99 | [ "sld", "1e-6/Ang^2", 0.4, [-inf,inf], "", |
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| 100 | "Tail scattering length density" ], |
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| 101 | [ "head_sld", "1e-6/Ang^2", 2.0, [-inf,inf], "", |
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| 102 | "Head scattering length density" ], |
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| 103 | [ "solvent_sld", "1e-6/Ang^2", 6, [-inf,inf], "", |
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| 104 | "Solvent scattering length density" ], |
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| 105 | ] |
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| 106 | |
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| 107 | source = [ "lamellarCailleHG_kernel.c"] |
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| 108 | |
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| 109 | # No volume normalization despite having a volume parameter |
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| 110 | # This should perhaps be volume normalized? |
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| 111 | form_volume = """ |
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| 112 | return 1.0; |
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| 113 | """ |
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| 114 | |
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| 115 | Iqxy = """ |
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[bfb195e] | 116 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[dc02af0] | 117 | """ |
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| 118 | |
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| 119 | # ER defaults to 0.0 |
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| 120 | # VR defaults to 1.0 |
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| 121 | |
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| 122 | demo = dict( |
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| 123 | scale=1, background=0, |
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| 124 | Nlayers=20,spacing=200., |
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| 125 | Caille_parameter=0.05, |
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| 126 | tail_length=15,head_length=10, |
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| 127 | sld=-1, head_sld=4.0, solvent_sld=6.0, |
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| 128 | tail_length_pd= 0.1, tail_length_pd_n=20, |
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| 129 | head_length_pd= 0.05, head_length_pd_n=30, |
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[cd55ac3] | 130 | spacing_pd= 0.2, spacing_pd_n=40 |
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[dc02af0] | 131 | ) |
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| 132 | |
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| 133 | oldname = 'LamellarPSHGModel' |
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| 134 | oldpars = dict(tail_length='deltaT',head_length='deltaH',Nlayers='n_plates',Caille_parameter='caille', sld='sld_tail', head_sld='sld_head',solvent_sld='sld_solvent') |
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