source: sasmodels/_sources/model/lamellarCailleHG.txt @ 68532f3

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