Changeset a5d0d00 in sasmodels for sasmodels/models/lamellarCaille.py


Ignore:
Timestamp:
Feb 27, 2015 1:16:23 PM (9 years ago)
Author:
Paul Kienzle <pkienzle@…>
Branches:
master, core_shell_microgels, costrafo411, magnetic_model, release_v0.94, release_v0.95, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
61ba623
Parents:
529b8b4
Message:

doc fixups: add doc category to model def, convert equations to latex for barbell and bcc

File:
1 edited

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  • sasmodels/models/lamellarCaille.py

    rdc02af0 ra5d0d00  
    11# Note: model title and parameter table are inserted automatically 
    22r""" 
    3 This model provides the scattering intensity, *I(q)* = *P(q)* \* *S(q)*, for a lamellar phase where a random 
    4 distribution in solution are assumed. Here a Caille S(Q) is used for the lamellar stacks. 
     3This model provides the scattering intensity, $I(q) = P(q) S(q)$, for a 
     4lamellar phase where a random distribution in solution are assumed. 
     5Here a Caille $S(Q)$ is used for the lamellar stacks. 
    56 
    6 The scattering intensity *I(q)* is 
     7The scattering intensity $I(q)$ is 
    78 
    8 .. image:: img/lamellarCaille_139.PNG 
     9.. math: 
     10 
     11    I(q) = 2\pi \frac{P(q)S(q)}{\delta q^2} 
    912 
    1013The form factor is 
    1114 
    12 .. image:: img/lamellarCaille_134.PNG 
     15.. math: 
     16 
     17    P(q) = \frac{2\Delta\rho^2}{q^2}\left(1-\cos q\delta \right) 
    1318 
    1419and the structure factor is 
    1520 
    16 .. image:: img/lamellarCaille_.PNG 
     21.. math: 
     22 
     23    S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right) 
     24           \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right) 
    1725 
    1826where 
    1927 
    20 .. image:: img/lamellarCaille_.PNG 
     28.. math: 
    2129 
    22 Here *d* = (repeat) spacing, |delta| = bilayer thickness, the contrast |drho| = SLD(headgroup) - SLD(solvent), 
    23 K = smectic bending elasticity, B = compression modulus, and N = number of lamellar plates (*n_plates*). 
     30    \begin{eqnarray} 
     31    \alpha(n) &=& \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right)  \\ 
     32    \gamma_E &=& 0.5772156649 && \text{Euler's constant} \\ 
     33    \eta_{cp} &=& \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} && \text{Caille constant} 
     34    \end{eqnarray} 
    2435 
    25 NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect.** 
    26 And due to a complication of the model function, users are responsible for making sure that all the assumptions are 
    27 handled accurately (see the original reference below for more details). 
     36Here $d$ = (repeat) spacing, $\delta$ = bilayer thickness, 
     37the contrast $\Delta\rho$ = SLD(headgroup) - SLD(solvent), 
     38$K$ = smectic bending elasticity, $B$ = compression modulus, and 
     39$N$ = number of lamellar plates (*n_plates*). 
    2840 
    29 Non-integer numbers of stacks are calculated as a linear combination of results for the next lower and higher values. 
     41NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the 
     42assumptions of the model are incorrect.** And due to a complication of the 
     43model function, users are responsible for making sure that all the assumptions 
     44are handled accurately (see the original reference below for more details). 
    3045 
    31 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
     46Non-integer numbers of stacks are calculated as a linear combination of 
     47results for the next lower and higher values. 
     48 
     49The 2D scattering intensity is calculated in the same way as 1D, where the 
     50$q$ vector is defined as 
    3251 
    3352.. math:: 
    3453 
    35     Q = \sqrt{Q_x^2 + Q_y^2} 
     54    q = \sqrt{q_x^2 + q_y^2} 
    3655 
    3756The returned value is in units of |cm^-1|, on absolute scale. 
    3857 
    39 ==============  ========  ============= 
    40 Parameter name  Units     Default value 
    41 ==============  ========  ============= 
    42 background      |cm^-1|   0.0 
    43 contrast        |Ang^-2|  5e-06 
    44 scale           None      1 
    45 delta           |Ang|     30 
    46 n_plates        None      20 
    47 spacing         |Ang|     400 
    48 caille          |Ang^-2|  0.1 
    49 ==============  ========  ============= 
    50  
    51 .. image:: img/lamellarPS_142.jpg 
     58.. image:: img/lamellarCaille_1d.jpg 
    5259 
    5360*Figure. 1D plot using the default values (w/6000 data point).* 
    5461 
    55 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 
    56 (Kline, 2006). 
     62Our model uses the form factor calculations as implemented in a c library 
     63provided by the NIST Center for Neutron Research (Kline, 2006). 
    5764 
    5865REFERENCE 
     66--------- 
    5967 
    6068F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 
    … …  
    7583                scale = scale factor 
    7684""" 
     85category = "shape:lamellae" 
    7786 
    7887parameters = [ 
    … …  
    8594    [ "spacing", "Ang", 400., [0.0,inf], "volume", 
    8695      "d-spacing of Caille S(Q)" ], 
    87     [ "Caille_parameter", "Ang^-2", 0.1, [0.0,0.8], "", 
     96    [ "Caille_parameter", "1/Ang^2", 0.1, [0.0,0.8], "", 
    8897      "Caille parameter" ], 
    8998    [ "sld", "1e-6/Ang^2", 6.3, [-inf,inf], "", 
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