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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