[dc02af0] | 1 | r""" |
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[a5d0d00] | 2 | This model provides the scattering intensity, $I(q) = P(q) S(q)$, for a |
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| 3 | lamellar phase where a random distribution in solution are assumed. |
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[eb69cce] | 4 | Here a Caille $S(q)$ is used for the lamellar stacks. |
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| 5 | |
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| 6 | Definition |
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| 7 | ---------- |
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[dc02af0] | 8 | |
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[a5d0d00] | 9 | The scattering intensity $I(q)$ is |
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[dc02af0] | 10 | |
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[eb69cce] | 11 | .. math:: |
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[a5d0d00] | 12 | |
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| 13 | I(q) = 2\pi \frac{P(q)S(q)}{\delta q^2} |
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[dc02af0] | 14 | |
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| 15 | The form factor is |
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| 16 | |
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[eb69cce] | 17 | .. math:: |
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[a5d0d00] | 18 | |
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| 19 | P(q) = \frac{2\Delta\rho^2}{q^2}\left(1-\cos q\delta \right) |
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[dc02af0] | 20 | |
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| 21 | and the structure factor is |
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| 22 | |
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[eb69cce] | 23 | .. math:: |
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[a5d0d00] | 24 | |
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| 25 | S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right) |
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| 26 | \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right) |
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[dc02af0] | 27 | |
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| 28 | where |
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| 29 | |
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[eb69cce] | 30 | .. math:: |
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[d18f8a8] | 31 | :nowrap: |
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[dc02af0] | 32 | |
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[d18f8a8] | 33 | \begin{align*} |
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[eb69cce] | 34 | \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right) |
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[d18f8a8] | 35 | && \\ |
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[eb69cce] | 36 | \gamma_E &= 0.5772156649 |
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[d18f8a8] | 37 | && \text{Euler's constant} \\ |
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[eb69cce] | 38 | \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} |
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[d18f8a8] | 39 | && \text{Caille constant} |
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| 40 | \end{align*} |
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[dc02af0] | 41 | |
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[a5d0d00] | 42 | Here $d$ = (repeat) spacing, $\delta$ = bilayer thickness, |
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| 43 | the contrast $\Delta\rho$ = SLD(headgroup) - SLD(solvent), |
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| 44 | $K$ = smectic bending elasticity, $B$ = compression modulus, and |
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| 45 | $N$ = number of lamellar plates (*n_plates*). |
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[dc02af0] | 46 | |
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[a5d0d00] | 47 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the |
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| 48 | assumptions of the model are incorrect.** And due to a complication of the |
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| 49 | model function, users are responsible for making sure that all the assumptions |
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| 50 | are handled accurately (see the original reference below for more details). |
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[dc02af0] | 51 | |
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[a5d0d00] | 52 | Non-integer numbers of stacks are calculated as a linear combination of |
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| 53 | results for the next lower and higher values. |
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| 54 | |
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| 55 | The 2D scattering intensity is calculated in the same way as 1D, where the |
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| 56 | $q$ vector is defined as |
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[dc02af0] | 57 | |
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| 58 | .. math:: |
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| 59 | |
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[a5d0d00] | 60 | q = \sqrt{q_x^2 + q_y^2} |
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[dc02af0] | 61 | |
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[eb69cce] | 62 | .. figure:: img/lamellarCaille_1d.jpg |
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[dc02af0] | 63 | |
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[eb69cce] | 64 | 1D plot using the default values (w/6000 data point). |
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[dc02af0] | 65 | |
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[eb69cce] | 66 | References |
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| 67 | ---------- |
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[dc02af0] | 68 | |
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| 69 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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| 70 | |
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| 71 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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| 72 | """ |
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[3c56da87] | 73 | from numpy import inf |
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[dc02af0] | 74 | |
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| 75 | name = "lamellarPS" |
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| 76 | title = "Random lamellar sheet with Caille structure factor" |
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| 77 | description = """\ |
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[ec2ca99] | 78 | [Random lamellar phase with Caille structure factor] |
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| 79 | randomly oriented stacks of infinite sheets |
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| 80 | with Caille S(Q), having polydisperse spacing. |
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| 81 | sld = sheet scattering length density |
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| 82 | sld_solvent = solvent scattering length density |
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| 83 | background = incoherent background |
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| 84 | scale = scale factor |
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[dc02af0] | 85 | """ |
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[a5d0d00] | 86 | category = "shape:lamellae" |
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[dc02af0] | 87 | |
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[13ed84c] | 88 | single = False |
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[ec2ca99] | 89 | # pylint: disable=bad-whitespace, line-too-long |
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[3e428ec] | 90 | # ["name", "units", default, [lower, upper], "type","description"], |
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[ec2ca99] | 91 | parameters = [ |
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| 92 | ["thickness", "Ang", 30.0, [0, inf], "volume", "sheet thickness"], |
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| 93 | ["Nlayers", "", 20, [0, inf], "", "Number of layers"], |
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| 94 | ["spacing", "Ang", 400., [0.0,inf], "volume", "d-spacing of Caille S(Q)"], |
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| 95 | ["Caille_parameter", "1/Ang^2", 0.1, [0.0,0.8], "", "Caille parameter"], |
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| 96 | ["sld", "1e-6/Ang^2", 6.3, [-inf,inf], "", "layer scattering length density"], |
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| 97 | ["solvent_sld", "1e-6/Ang^2", 1.0, [-inf,inf], "", "Solvent scattering length density"], |
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| 98 | ] |
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| 99 | # pylint: enable=bad-whitespace, line-too-long |
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[3e428ec] | 100 | |
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| 101 | source = ["lamellarCaille_kernel.c"] |
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[dc02af0] | 102 | |
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| 103 | # No volume normalization despite having a volume parameter |
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| 104 | # This should perhaps be volume normalized? |
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| 105 | form_volume = """ |
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| 106 | return 1.0; |
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| 107 | """ |
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| 108 | |
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| 109 | Iqxy = """ |
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[bfb195e] | 110 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[dc02af0] | 111 | """ |
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| 112 | |
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| 113 | # ER defaults to 0.0 |
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| 114 | # VR defaults to 1.0 |
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| 115 | |
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[3e428ec] | 116 | demo = dict(scale=1, background=0, |
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[ec2ca99] | 117 | thickness=67., Nlayers=3.75, spacing=200., |
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| 118 | Caille_parameter=0.268, sld=1.0, solvent_sld=6.34, |
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| 119 | thickness_pd=0.1, thickness_pd_n=100, |
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| 120 | spacing_pd=0.05, spacing_pd_n=40) |
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[dc02af0] | 121 | |
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| 122 | oldname = 'LamellarPSModel' |
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[ec2ca99] | 123 | oldpars = dict(thickness='delta', Nlayers='N_plates', |
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| 124 | Caille_parameter='caille', |
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| 125 | sld='sld_bi', solvent_sld='sld_sol') |
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[7f47777] | 126 | # |
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| 127 | tests = [ |
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[ec2ca99] | 128 | [{'scale': 1.0, 'background': 0.0, 'thickness': 30., 'Nlayers': 20.0, |
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| 129 | 'spacing': 400., 'Caille_parameter': 0.1, 'sld': 6.3, |
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| 130 | 'solvent_sld': 1.0, 'thickness_pd': 0.0, 'spacing_pd': 0.0}, |
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| 131 | [0.001], [28895.13397]] |
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| 132 | ] |
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