# source:sasmodels/sasmodels/models/lamellarPC.py@d18f8a8

core_shell_microgelscostrafo411magnetic_modelrelease_v0.94release_v0.95ticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since d18f8a8 was d18f8a8, checked in by Paul Kienzle <pkienzle@…>, 8 years ago

fix multiline equation alignment

• Property mode set to 100644
File size: 4.2 KB
Line
1# Note: model title and parameter table are inserted automatically
2r"""
3This model calculates the scattering from a stack of repeating lamellar
4structures. The stacks of lamellae (infinite in lateral dimension) are
5treated as a paracrystal to account for the repeating spacing. The repeat
6distance is further characterized by a Gaussian polydispersity. **This model
7can be used for large multilamellar vesicles.**
8
9Definition
10----------
11
12In the equations below,
13
14- *scale* is used instead of the mass per area of the bilayer $\Gamma_m$
15  (this corresponds to the volume fraction of the material in the bilayer,
16  *not* the total excluded volume of the paracrystal),
17
18- *sld* $-$ *solvent_sld* is the contrast $\Delta \rho$,
19
20- *thickness* is the layer thickness $t$,
21
22- *Nlayers* is the number of layers $N$,
23
24- *spacing* is the average distance between adjacent layers
25  $\langle D \rangle$, and
26
27- *spacing_polydisp* is the relative standard deviation of the Gaussian
28  layer distance distribution $\sigma_D / \langle D \rangle$.
29
30
31The scattering intensity $I(q)$ is calculated as
32
33.. math::
34
35    I(q) = 2\pi\Delta\rho^2\Gamma_m\frac{P_\text{bil}(q)}{q^2} Z_N(q)
36
37The form factor of the bilayer is approximated as the cross section of an
38infinite, planar bilayer of thickness $t$
39
40.. math::
41
42    P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2
43
44$Z_N(q)$ describes the interference effects for aggregates
45consisting of more than one bilayer. The equations used are (3-5)
46from the Bergstrom reference:
47
48.. math::
49
50
51    Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)}
52        + x_N S_N + (1 - x_N) S_{N+1}
53
54where
55
56.. math::
57
58    S_N(q) = \frac{a_N}{N}[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2
59
60and
61
62.. math::
63
64    a_N &= 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) \\
65        &\quad - 4w^{N+2}\cos(Nq \langle D \rangle)
66        + 2 w^{N+3}\cos[(N-1)q \langle D \rangle]
67        + 2w^{N+1}\cos[(N+1)q \langle D \rangle]
68
69for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$.
70
71Non-integer numbers of stacks are calculated as a linear combination of
72the lower and higher values
73
74.. math::
75
76    N_L = x_N N + (1 - x_N)(N+1)
77
78The 2D scattering intensity is the same as 1D, regardless of the orientation
79of the $q$ vector which is defined as
80
81.. math::
82
83    q = \sqrt{q_x^2 + q_y^2}
84
85
86.. figure:: img/lamellarPC_1d.jpg
87
88    1D plot using the default values above (w/20000 data point).
89
90Reference
91---------
92
93M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf,
94*J. Phys. Chem. B*, 103 (1999) 9888-9897
95
96"""
97
98from numpy import inf
99
100name = "lamellarPC"
101title = "Random lamellar sheet with paracrystal structure factor"
102description = """\
103    [Random lamellar phase with paracrystal structure factor]
104        randomly oriented stacks of infinite sheets
105        with paracrytal S(Q), having polydisperse spacing.
106        sld = sheet scattering length density
107        sld_solvent = solvent scattering length density
108        background = incoherent background
109        scale = scale factor
110"""
111category = "shape:lamellae"
112
113#             ["name", "units", default, [lower, upper], "type","description"],
114parameters = [["thickness", "Ang", 33.0, [0, inf], "volume",
115               "sheet thickness"],
116              ["Nlayers", "", 20, [0, inf], "",
117               "Number of layers"],
118              ["spacing", "Ang", 250., [0.0, inf], "",
119               "d-spacing of paracrystal stack"],
120              ["spacing_polydisp", "Ang", 0.0, [0.0, inf], "",
121               "d-spacing polydispersity"],
122              ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "",
123               "layer scattering length density"],
124              ["solvent_sld", "1e-6/Ang^2", 6.34, [-inf, inf], "",
125               "Solvent scattering length density"],
126             ]
127
128
129source = ["lamellarPC_kernel.c"]
130
131form_volume = """
132    return 1.0;
133    """
134
135Iqxy = """
136    return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS);
137    """
138
139# ER defaults to 0.0
140# VR defaults to 1.0
141
142demo = dict(scale=1, background=0,
143            thickness=33, Nlayers=20, spacing=250, spacing_polydisp=0.2,
144            sld=1.0, solvent_sld=6.34,
145            thickness_pd=0.2, thickness_pd_n=40)
146
147oldname = 'LamellarPCrystalModel'
148oldpars = dict(spacing_polydisp='pd_spacing', sld='sld_layer',
149               solvent_sld='sld_solvent')
Note: See TracBrowser for help on using the repository browser.