1 | # Note: model title and parameter table are inserted automatically |
---|
2 | r""" |
---|
3 | This model provides the scattering intensity, $I(q)$, for a lyotropic lamellar |
---|
4 | phase where a random distribution in solution are assumed. The SLD of the head |
---|
5 | region is taken to be different from the SLD of the tail region. |
---|
6 | |
---|
7 | Definition |
---|
8 | ---------- |
---|
9 | |
---|
10 | The scattering intensity $I(q)$ is |
---|
11 | |
---|
12 | .. math:: |
---|
13 | |
---|
14 | I(q) = 2\pi\frac{\text{scale}}{2(\delta_H + \delta_T)} P(q) \frac{1}{q^2} |
---|
15 | |
---|
16 | The form factor $P(q)$ is |
---|
17 | |
---|
18 | .. math:: |
---|
19 | |
---|
20 | P(q) = \frac{4}{q^2} |
---|
21 | \left\lbrace |
---|
22 | \Delta \rho_H |
---|
23 | \left[\sin[q(\delta_H + \delta_T)\ - \sin(q\delta_T)\right] |
---|
24 | + \Delta\rho_T\sin(q\delta_T) |
---|
25 | \right\rbrace^2 |
---|
26 | |
---|
27 | where $\delta_T$ is *tail_length*, $\delta_H$ is *head_length*, |
---|
28 | $\Delta\rho_H$ is the head contrast (*head_sld* $-$ *solvent_sld*), |
---|
29 | and $\Delta\rho_T$ is tail contrast (*sld* $-$ *solvent_sld*). |
---|
30 | |
---|
31 | The 2D scattering intensity is calculated in the same way as 1D, where |
---|
32 | the $q$ vector is defined as |
---|
33 | |
---|
34 | .. math:: |
---|
35 | |
---|
36 | q = \sqrt{q_x^2 + q_y^2} |
---|
37 | |
---|
38 | |
---|
39 | .. figure:: img/lamellarFFHG_1d.jpg |
---|
40 | |
---|
41 | 1D plot using the default values (w/1000 data point). |
---|
42 | |
---|
43 | References |
---|
44 | ---------- |
---|
45 | |
---|
46 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
---|
47 | |
---|
48 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
---|
49 | |
---|
50 | *2014/04/17 - Description reviewed by S King and P Butler.* |
---|
51 | """ |
---|
52 | |
---|
53 | from numpy import inf |
---|
54 | |
---|
55 | name = "lamellar_FFHG" |
---|
56 | title = "Random lamellar phase with Head Groups" |
---|
57 | description = """\ |
---|
58 | [Random lamellar phase with Head Groups] |
---|
59 | I(q)= 2*pi*P(q)/(2(H+T)*q^(2)), where |
---|
60 | P(q)= see manual |
---|
61 | layer thickness =(H+T+T+H) = 2(Head+Tail) |
---|
62 | sld = Tail scattering length density |
---|
63 | head_sld = Head scattering length density |
---|
64 | solvent_sld = solvent scattering length density |
---|
65 | background = incoherent background |
---|
66 | scale = scale factor |
---|
67 | """ |
---|
68 | category = "shape:lamellae" |
---|
69 | |
---|
70 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
71 | parameters = [["tail_length", "Ang", 15, [0, inf], "volume", |
---|
72 | "Tail thickness"], |
---|
73 | ["head_length", "Ang", 10, [0, inf], "volume", |
---|
74 | "head thickness"], |
---|
75 | ["sld", "1e-6/Ang^2", 0.4, [-inf,inf], "", |
---|
76 | "Tail scattering length density"], |
---|
77 | ["head_sld", "1e-6/Ang^2", 3.0, [-inf,inf], "", |
---|
78 | "Head scattering length density"], |
---|
79 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf,inf], "", |
---|
80 | "Solvent scattering length density"], |
---|
81 | ] |
---|
82 | |
---|
83 | # No volume normalization despite having a volume parameter |
---|
84 | # This should perhaps be volume normalized? |
---|
85 | form_volume = """ |
---|
86 | return 1.0; |
---|
87 | """ |
---|
88 | |
---|
89 | Iq = """ |
---|
90 | const double qsq = q*q; |
---|
91 | const double drh = head_sld - solvent_sld; |
---|
92 | const double drt = sld - solvent_sld; //correction 13FEB06 by L.Porcar |
---|
93 | const double qT = q*tail_length; |
---|
94 | double Pq, inten; |
---|
95 | Pq = drh*(sin(q*(head_length+tail_length))-sin(qT)) + drt*sin(qT); |
---|
96 | Pq *= Pq; |
---|
97 | Pq *= 4.0/(qsq); |
---|
98 | |
---|
99 | inten = 2.0e-4*M_PI*Pq/qsq; |
---|
100 | |
---|
101 | // normalize by the bilayer thickness |
---|
102 | inten /= 2.0*(head_length+tail_length); |
---|
103 | |
---|
104 | return inten; |
---|
105 | """ |
---|
106 | |
---|
107 | Iqxy = """ |
---|
108 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
---|
109 | """ |
---|
110 | |
---|
111 | # ER defaults to 0.0 |
---|
112 | # VR defaults to 1.0 |
---|
113 | |
---|
114 | demo = dict(scale=1, background=0, |
---|
115 | tail_length=15,head_length=10, |
---|
116 | sld=0.4, head_sld=3.0, solvent_sld=6.0, |
---|
117 | tail_length_pd= 0.2, tail_length_pd_n=40, |
---|
118 | head_length_pd= 0.01, head_length_pd_n=40) |
---|
119 | |
---|
120 | oldname = 'LamellarFFHGModel' |
---|
121 | oldpars = dict(head_length='h_thickness', tail_length='t_length', |
---|
122 | sld='sld_tail', head_sld='sld_head', solvent_sld='sld_solvent') |
---|
123 | # |
---|
124 | tests = [ |
---|
125 | [ {'scale': 1.0, 'background' : 0.0, 'tail_length' : 15.0, 'head_length' : 10.0,'sld' : 0.4, |
---|
126 | 'head_sld' : 3.0, 'solvent_sld' : 6.0, }, [0.001], [653143.9209]] |
---|
127 | ] |
---|
128 | |
---|
129 | |
---|