[dc02af0] | 1 | # Note: model title and parameter table are inserted automatically |
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| 2 | r""" |
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[12c810f] | 3 | This model provides the scattering intensity, $I(q) = P(q)S(q)$, for a lamellar |
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| 4 | phase where a random distribution in solution are assumed. Here a Caille $S(Q)$ |
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| 5 | is used for the lamellar stacks. |
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[dc02af0] | 6 | |
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[12c810f] | 7 | The scattering intensity $I(q)$ is |
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[dc02af0] | 8 | |
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[12c810f] | 9 | .. math:: |
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| 10 | |
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| 11 | I(q) = 2 \pi \frac{P(q)S(q)}{\delta q^2} |
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| 12 | |
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| 13 | |
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| 14 | The form factor $P(q)$ is |
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| 15 | |
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| 16 | .. math:: |
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[dc02af0] | 17 | |
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[12c810f] | 18 | P(q) = \frac{4}{q^2}\big\{ |
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| 19 | \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right] |
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| 20 | + \Delta\rho_T\sin(q\delta_T)\big\}^2 |
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[dc02af0] | 21 | |
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[12c810f] | 22 | and the structure factor $S(q)$ is |
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[dc02af0] | 23 | |
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[12c810f] | 24 | .. math:: |
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[dc02af0] | 25 | |
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[12c810f] | 26 | S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right) |
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| 27 | \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right) |
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[dc02af0] | 28 | |
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| 29 | where |
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| 30 | |
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[12c810f] | 31 | .. math:: |
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| 32 | |
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| 33 | \begin{eqnarray} |
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| 34 | \alpha(n) &=& \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right) \\ |
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| 35 | \gamma_E &=& 0.5772156649&&\text{Euler's constant} \\ |
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| 36 | \eta_{cp} &=& \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} && \text{Caille constant} |
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| 37 | \end{eqnarray} |
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| 38 | |
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[dc02af0] | 39 | |
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[12c810f] | 40 | $\delta_T$ is the tail length (or *tail_length*), $\delta_H$ is the head |
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| 41 | thickness (or *head_length*), $\Delta\rho_H$ is SLD(headgroup) - SLD(solvent), |
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| 42 | and $\Delta\rho_T$ is SLD(tail) - SLD(headgroup). Here $d$ is (repeat) spacing, |
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| 43 | $K$ is smectic bending elasticity, $B$ is compression modulus, and $N$ is the |
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| 44 | number of lamellar plates (*Nlayers*). |
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[dc02af0] | 45 | |
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[12c810f] | 46 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the |
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| 47 | assumptions of the model are incorrect.** And due to a complication of the |
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| 48 | model function, users are responsible for making sure that all the assumptions |
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| 49 | are handled accurately (see the original reference below for more details). |
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[dc02af0] | 50 | |
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[12c810f] | 51 | Non-integer numbers of stacks are calculated as a linear combination of |
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| 52 | results for the next lower and higher values. |
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[dc02af0] | 53 | |
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[12c810f] | 54 | The 2D scattering intensity is calculated in the same way as 1D, where |
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| 55 | the $q$ vector is defined as |
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[dc02af0] | 56 | |
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| 57 | .. math:: |
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| 58 | |
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[12c810f] | 59 | q = \sqrt{q_x^2 + q_y^2} |
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[dc02af0] | 60 | |
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| 61 | The returned value is in units of |cm^-1|, on absolute scale. |
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| 62 | |
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[12c810f] | 63 | .. image:: img/lamellarCailleHG_1d.jpg |
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[dc02af0] | 64 | |
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| 65 | *Figure. 1D plot using the default values (w/6000 data point).* |
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| 66 | |
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[12c810f] | 67 | Our model uses the form factor calculations implemented in a C library provided |
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| 68 | by the NIST Center for Neutron Research (Kline, 2006). |
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[dc02af0] | 69 | |
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| 70 | REFERENCE |
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| 71 | |
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| 72 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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| 73 | |
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| 74 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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| 75 | """ |
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| 76 | from numpy import pi, inf |
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| 77 | |
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| 78 | name = "lamellarCailleHG" |
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| 79 | title = "Random lamellar sheet with Caille structure factor" |
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| 80 | description = """\ |
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| 81 | [Random lamellar phase with Caille structure factor] |
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| 82 | randomly oriented stacks of infinite sheets |
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| 83 | with Caille S(Q), having polydisperse spacing. |
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| 84 | layer thickness =(H+T+T+H) = 2(Head+Tail) |
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| 85 | sld = Tail scattering length density |
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| 86 | sld_head = Head scattering length density |
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| 87 | sld_solvent = solvent scattering length density |
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| 88 | background = incoherent background |
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| 89 | scale = scale factor |
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| 90 | """ |
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[a5d0d00] | 91 | category = "shape:lamellae" |
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[dc02af0] | 92 | |
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| 93 | parameters = [ |
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| 94 | # [ "name", "units", default, [lower, upper], "type", |
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| 95 | # "description" ], |
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| 96 | [ "tail_length", "Ang", 10, [0, inf], "volume", |
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| 97 | "Tail thickness" ], |
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| 98 | [ "head_length", "Ang", 2, [0, inf], "volume", |
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| 99 | "head thickness" ], |
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| 100 | [ "Nlayers", "", 30, [0, inf], "", |
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| 101 | "Number of layers" ], |
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| 102 | [ "spacing", "Ang", 40., [0.0,inf], "volume", |
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| 103 | "d-spacing of Caille S(Q)" ], |
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| 104 | [ "Caille_parameter", "", 0.001, [0.0,0.8], "", |
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| 105 | "Caille parameter" ], |
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| 106 | [ "sld", "1e-6/Ang^2", 0.4, [-inf,inf], "", |
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| 107 | "Tail scattering length density" ], |
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| 108 | [ "head_sld", "1e-6/Ang^2", 2.0, [-inf,inf], "", |
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| 109 | "Head scattering length density" ], |
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| 110 | [ "solvent_sld", "1e-6/Ang^2", 6, [-inf,inf], "", |
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| 111 | "Solvent scattering length density" ], |
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| 112 | ] |
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| 113 | |
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| 114 | source = [ "lamellarCailleHG_kernel.c"] |
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| 115 | |
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| 116 | # No volume normalization despite having a volume parameter |
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| 117 | # This should perhaps be volume normalized? |
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| 118 | form_volume = """ |
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| 119 | return 1.0; |
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| 120 | """ |
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| 121 | |
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| 122 | Iqxy = """ |
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[bfb195e] | 123 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[dc02af0] | 124 | """ |
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| 125 | |
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| 126 | # ER defaults to 0.0 |
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| 127 | # VR defaults to 1.0 |
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| 128 | |
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| 129 | demo = dict( |
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| 130 | scale=1, background=0, |
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[12c810f] | 131 | Nlayers=20, |
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| 132 | spacing=200., Caille_parameter=0.05, |
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| 133 | tail_length=15,head_length=10, |
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| 134 | #sld=-1, head_sld=4.0, solvent_sld=6.0, |
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| 135 | sld=-1, head_sld=4.1, solvent_sld=6.0, |
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| 136 | tail_length_pd= 0.1, tail_length_pd_n=20, |
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| 137 | head_length_pd= 0.05, head_length_pd_n=30, |
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| 138 | spacing_pd= 0.2, spacing_pd_n=40 |
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| 139 | ) |
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[dc02af0] | 140 | |
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| 141 | oldname = 'LamellarPSHGModel' |
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| 142 | 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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