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  • sasmodels/models/multilayer_vesicle.py

    r925ad6e r5d23de2  
    33---------- 
    44 
    5 This model is a trivial extension of the core_shell_sphere function to include 
    6 *N* shells where the core is filled with solvent and the shells are interleaved 
    7 with layers of solvent. For *N = 1*, this returns the same as the vesicle model, 
    8 except for the normalisation, which here is to outermost volume. 
    9 The shell thicknessess and SLD are constant for all shells as expected for 
    10 a multilayer vesicle. 
     5This model is a trivial extension of the core_shell_sphere function where the 
     6core is filled with solvent and is surrounded by $N$ shells of material 
     7(such as lipids) interleaved with $N - 1$ layers of solvent. For $N = 1$, this 
     8returns the same as the vesicle model, except for the normalisation, which here 
     9is to outermost volume. The shell thicknesses and SLD are constant for all 
     10shells as expected for a multilayer vesicle. 
    1111 
    1212.. figure:: img/multi_shell_geometry.jpg 
    … …  
    1919 
    2020.. math:: 
    21  
    22     P(q) = \frac{\text{scale.volfraction}}{V_t} F^2(q) + \text{background} 
     21    P(q) = \text{scale} \cdot \frac{\phi}{V(R_N)} F^2(q) + \text{background} 
    2322 
    2423where 
    2524 
    2625.. math:: 
    27  
    28      F(q) = (\rho_{shell}-\rho_{solv}) \sum_{i=1}^{n\_pairs} \left[ 
    29      3V(R_i)\frac{\sin(qR_i)-qR_i\cos(qR_i)}{(qR_i)^3} \\ 
    30       - 3V(R_i+t_s)\frac{\sin(q(R_i+t_s))-q(R_i+t_s)\cos(q(R_i+t_s))}{(q(R_i+t_s))^3} 
     26     F(q) = (\rho_\text{shell}-\rho_\text{solv}) \sum_{i=1}^{N} \left[ 
     27     3V(r_i)\frac{\sin(qr_i) - qr_i\cos(qr_i)}{(qr_i)^3} 
     28     - 3V(R_i)\frac{\sin(qR_i) - qR_i\cos(qR_i)}{(qR_i)^3} 
    3129     \right] 
    3230 
     31for 
    3332 
    34 where $R_i = r_c + (i-1)(t_s + t_w)$ 
    35     
    36 where $V_t$ is the volume of the whole particle, $V(R)$ is the volume of a sphere 
    37 of radius $R$, $r_c$ is the radius of the core, $\rho_{shell}$ is the scattering length  
    38 density of a shell, $\rho_{solv}$ is the scattering length density of the solvent. 
     33.. math:: 
    3934 
     35     r_i &= r_c + (i-1)(t_s + t_w) && \text{ solvent radius before shell } i \\ 
     36     R_i &= r_i + t_s && \text{ shell radius for shell } i 
     37 
     38$\phi$ is the volume fraction of particles, $V(r)$ is the volume of a sphere 
     39of radius $r$, $r_c$ is the radius of the core, $t_s$ is the thickness of 
     40the shell, $t_w$ is the thickness of the solvent layer between the shells, 
     41$\rho_\text{shell}$ is the scattering length density of a shell, and 
     42$\rho_\text{solv}$ is the scattering length density of the solvent. 
     43 
     44USAGE NOTES 
     45 
     46* The outer-most shell radius $R_N$ is used as the effective radius 
     47  for $P(Q)$ when $P(Q) * S(Q)$ is applied. 
     48  calculations rather slow. 
     49* The number of shells is always rounded to an integer value as a non interger 
     50  number of layers is not physical. 
     51* Thus Polydispersity should only be applied to number of shells **VERY 
     52  CAREFULLY**.  A possible legitimate use would be for mixed systems in which 
     53  some vesicles have 1 shell, some have 2, etc. A polydispersity on $N$ can be 
     54  used to model the data by using the "array distriubtion" feature. First 
     55  create a file such as *shell_dist.txt* containing the relative portion 
     56  of each vesicle size:: 
     57 
     58    1 20 
     59    2  4 
     60    3  1 
     61 
     62  Turn on polydispersity and select an array distribution for the *n_shells* 
     63  parameter.  Choose the above *shell_dist.txt* file, and the model will be 
     64  computed with 80% 1-shell vesicles, 16% 2-shell vesicles and 4% 
     65  3-shell vesicles. 
     66* This is a highly non-linear, highly oscillatory (especially around the 
     67  q-values that correspond to the repeat distance of the layers), model 
     68  function complicated by the fact that the number of water/shell pairs must 
     69  physically be an integer value, although the optimization treats it as a 
     70  floating point value. Thus it may be that the resolution interpolation is not 
     71  sufficiently fine grained in certain cases. Please report any such occurences 
     72  to the SasView team. Generally, for the best possible experience: 
     73 * Start with the best possible guess 
     74 * Using a priori knowledge, hold as many parameters fixed as possible 
     75 * if N=1, tw (water thickness) must by definition be zero. Both N and tw should 
     76   be fixed during fitting. 
     77 * If N>1, use constraints to keep N > 1 
     78 * Because N only really moves in integer steps, it may get "stuck" if the 
     79   optimizer step size is too small so care should be taken 
     80   If you experience problems with this please contact the SasView team and let 
     81   them know the issue preferably with example data and model which fail to 
     82   converge. 
    4083 
    4184The 2D scattering intensity is the same as 1D, regardless of the orientation 
    … …  
    4689    q = \sqrt{q_x^2 + q_y^2} 
    4790 
    48  
    49 The outer most radius 
    50  
    51 $radius + n\_pairs * thick\_shell + (n\_pairs- 1) * thick\_solvent$ 
    52  
    53 is used for both the volume fraction normalization and for the  
    54 effective radius for *S(Q)* when $P(Q) * S(Q)$ is applied. 
    55  
    5691For information about polarised and magnetic scattering, see 
    5792the :ref:`magnetism` documentation. 
    58  
    59 This code is based on the form factor calculations implemented in the NIST 
    60 Center for Neutron Research provided c-library (Kline, 2006). 
    6193 
    6294References 
    6395---------- 
    6496 
    65 B Cabane, *Small Angle Scattering Methods*, 
    66 in *Surfactant Solutions: New Methods of Investigation*, 
    67 Ch.2, Surfactant Science Series Vol. 22, Ed. R Zana and M Dekker, 
    68 New York, (1987). 
     97.. [#] B Cabane, *Small Angle Scattering Methods*, in *Surfactant Solutions: 
     98   New Methods of Investigation*, Ch.2, Surfactant Science Series Vol. 22, Ed. 
     99   R Zana and M Dekker, New York, (1987). 
    69100 
    70 **Author:** NIST IGOR/DANSE **on:** pre 2010 
     101Authorship and Verification 
     102---------------------------- 
    71103 
    72 **Last Modified by:** Piotr Rozyczko**on:** Feb 24, 2016 
    73  
    74 **Last Reviewed by:** Paul Butler **on:** March 20, 2016 
     104* **Author:** NIST IGOR/DANSE **Date:** pre 2010 
     105* **Converted to sasmodels by:** Piotr Rozyczko **Date:** Feb 24, 2016 
     106* **Last Modified by:** Paul Kienzle **Date:** Feb 7, 2017 
     107* **Last Reviewed by:** Paul Butler **Date:** March 12, 2017 
    75108 
    76109""" 
    … …  
    89122    sld_solvent: solvent scattering length density 
    90123    sld: shell scattering length density 
    91     n_pairs:number of "shell plus solvent" layer pairs 
     124    n_shells:number of "shell plus solvent" layer pairs 
    92125    background: incoherent background 
    93126        """ 
    … …  
    98131parameters = [ 
    99132    ["volfraction", "",  0.05, [0.0, 1],  "", "volume fraction of vesicles"], 
    100     ["radius", "Ang", 60.0, [0.0, inf],  "", "radius of solvent filled core"], 
    101     ["thick_shell", "Ang",        10.0, [0.0, inf],  "", "thickness of one shell"], 
    102     ["thick_solvent", "Ang",        10.0, [0.0, inf],  "", "solvent thickness between shells"], 
     133    ["radius", "Ang", 60.0, [0.0, inf],  "volume", "radius of solvent filled core"], 
     134    ["thick_shell", "Ang",        10.0, [0.0, inf],  "volume", "thickness of one shell"], 
     135    ["thick_solvent", "Ang",        10.0, [0.0, inf],  "volume", "solvent thickness between shells"], 
    103136    ["sld_solvent",    "1e-6/Ang^2",  6.4, [-inf, inf], "sld", "solvent scattering length density"], 
    104137    ["sld",   "1e-6/Ang^2",  0.4, [-inf, inf], "sld", "Shell scattering length density"], 
    105     ["n_pairs",     "",            2.0, [1.0, inf],  "", "Number of shell plus solvent layer pairs"], 
     138    ["n_shells",     "",            2.0, [1.0, inf],  "volume", "Number of shell plus solvent layer pairs"], 
    106139    ] 
    107140# pylint: enable=bad-whitespace, line-too-long 
    108141 
     142# TODO: proposed syntax for specifying which parameters can be polydisperse 
     143#polydispersity = ["radius", "thick_shell"] 
     144 
    109145source = ["lib/sas_3j1x_x.c", "multilayer_vesicle.c"] 
    110146 
    111 polydispersity = ["radius", "n_pairs"] 
     147def ER(radius, thick_shell, thick_solvent, n_shells): 
     148    n_shells = int(n_shells+0.5) 
     149    return radius + n_shells * (thick_shell + thick_solvent) - thick_solvent 
    112150 
    113151demo = dict(scale=1, background=0, 
    … …  
    118156            sld_solvent=6.4, 
    119157            sld=0.4, 
    120             n_pairs=2.0) 
     158            n_shells=2.0) 
    121159 
    122160tests = [ 
    … …  
    127165      'sld_solvent': 6.4, 
    128166      'sld': 0.4, 
    129       'n_pairs': 2.0, 
     167      'n_shells': 2.0, 
    130168      'scale': 1.0, 
    131169      'background': 0.001, 
    … …  
    138176      'sld_solvent': 6.4, 
    139177      'sld': 0.4, 
    140       'n_pairs': 2.0, 
     178      'n_shells': 2.0, 
    141179      'scale': 1.0, 
    142180      'background': 0.001, 
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