[68532f3] | 1 | .. _lamellarCailleHG: |
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| 2 | |
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| 3 | Lamellarcaillehg |
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| 4 | ======================================================= |
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| 5 | |
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| 6 | Random lamellar sheet with Caille structure factor |
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| 7 | |
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| 8 | ================ ================================= ============ ============= |
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| 9 | Parameter Description Units Default value |
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| 10 | ================ ================================= ============ ============= |
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| 11 | scale Source intensity None 1 |
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| 12 | background Source background |cm^-1| 0 |
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| 13 | tail_length Tail thickness |Ang| 10 |
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| 14 | head_length head thickness |Ang| 2 |
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| 15 | Nlayers Number of layers None 30 |
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| 16 | spacing d-spacing of Caille S(Q) |Ang| 40 |
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| 17 | Caille_parameter Caille parameter None 0.001 |
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| 18 | sld Tail scattering length density |1e-6Ang^-2| 0.4 |
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| 19 | head_sld Head scattering length density |1e-6Ang^-2| 2 |
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| 20 | solvent_sld Solvent scattering length density |1e-6Ang^-2| 6 |
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| 21 | ================ ================================= ============ ============= |
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| 22 | |
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| 23 | The returned value is scaled to units of |cm^-1|. |
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| 24 | |
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| 25 | |
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| 26 | This model provides the scattering intensity, $I(q) = P(q)S(q)$, for a lamellar |
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| 27 | phase where a random distribution in solution are assumed. Here a Caille $S(Q)$ |
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| 28 | is used for the lamellar stacks. |
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| 29 | |
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| 30 | The scattering intensity $I(q)$ is |
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| 31 | |
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| 32 | .. math:: |
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| 33 | |
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| 34 | I(q) = 2 \pi \frac{P(q)S(q)}{\delta q^2} |
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| 35 | |
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| 36 | |
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| 37 | The form factor $P(q)$ is |
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| 38 | |
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| 39 | .. math:: |
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| 40 | |
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| 41 | P(q) = \frac{4}{q^2}\big\{ |
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| 42 | \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right] |
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| 43 | + \Delta\rho_T\sin(q\delta_T)\big\}^2 |
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| 44 | |
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| 45 | and the structure factor $S(q)$ is |
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| 46 | |
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| 47 | .. math:: |
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| 48 | |
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| 49 | S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right) |
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| 50 | \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right) |
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| 51 | |
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| 52 | where |
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| 53 | |
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| 54 | .. math:: |
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| 55 | |
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| 56 | \begin{eqnarray} |
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| 57 | \alpha(n) &=& \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right) \\ |
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| 58 | \gamma_E &=& 0.5772156649&&\text{Euler's constant} \\ |
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| 59 | \eta_{cp} &=& \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} && \text{Caille constant} |
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| 60 | \end{eqnarray} |
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| 61 | |
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| 62 | |
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| 63 | $\delta_T$ is the tail length (or *tail_length*), $\delta_H$ is the head |
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| 64 | thickness (or *head_length*), $\Delta\rho_H$ is SLD(headgroup) - SLD(solvent), |
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| 65 | and $\Delta\rho_T$ is SLD(tail) - SLD(headgroup). Here $d$ is (repeat) spacing, |
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| 66 | $K$ is smectic bending elasticity, $B$ is compression modulus, and $N$ is the |
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| 67 | number of lamellar plates (*Nlayers*). |
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| 68 | |
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| 69 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the |
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| 70 | assumptions of the model are incorrect.** And due to a complication of the |
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| 71 | model function, users are responsible for making sure that all the assumptions |
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| 72 | are handled accurately (see the original reference below for more details). |
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| 73 | |
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| 74 | Non-integer numbers of stacks are calculated as a linear combination of |
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| 75 | results for the next lower and higher values. |
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| 76 | |
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| 77 | The 2D scattering intensity is calculated in the same way as 1D, where |
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| 78 | the $q$ vector is defined as |
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| 79 | |
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| 80 | .. math:: |
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| 81 | |
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| 82 | q = \sqrt{q_x^2 + q_y^2} |
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| 83 | |
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| 84 | The returned value is in units of |cm^-1|, on absolute scale. |
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| 85 | |
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| 86 | .. image:: img/lamellarCailleHG_1d.jpg |
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| 87 | |
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| 88 | *Figure. 1D plot using the default values (w/6000 data point).* |
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| 89 | |
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| 90 | Our model uses the form factor calculations implemented in a C library provided |
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| 91 | by the NIST Center for Neutron Research (Kline, 2006). |
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| 92 | |
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| 93 | REFERENCE |
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| 94 | |
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| 95 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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| 96 | |
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| 97 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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| 98 | |
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