[dc02af0] | 1 | # Note: model title and parameter table are inserted automatically |
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| 2 | r""" |
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[eb69cce] | 3 | This model provides the scattering intensity, $I(q)$, for a lyotropic lamellar |
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[3c56da87] | 4 | phase where a random distribution in solution are assumed. The SLD of the head |
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| 5 | region is taken to be different from the SLD of the tail region. |
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[dc02af0] | 6 | |
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[d138d43] | 7 | Definition |
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| 8 | ---------- |
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[dc02af0] | 9 | |
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[d138d43] | 10 | The scattering intensity $I(q)$ is |
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[dc02af0] | 11 | |
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| 12 | .. math:: |
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| 13 | |
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[d138d43] | 14 | I(q) = 2\pi\frac{\text{scale}}{2(\delta_H + \delta_T)} P(q) \frac{1}{q^2} |
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[dc02af0] | 15 | |
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[d138d43] | 16 | The form factor $P(q)$ is |
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[dc02af0] | 17 | |
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[d138d43] | 18 | .. math:: |
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| 19 | |
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| 20 | P(q) = \frac{4}{q^2} |
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| 21 | \left\lbrace |
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| 22 | \Delta \rho_H |
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| 23 | \left[\sin[q(\delta_H + \delta_T)\ - \sin(q\delta_T)\right] |
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| 24 | + \Delta\rho_T\sin(q\delta_T) |
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| 25 | \right\rbrace^2 |
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[dc02af0] | 26 | |
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[eb69cce] | 27 | where $\delta_T$ is *tail_length*, $\delta_H$ is *head_length*, |
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[6ab4ed8] | 28 | $\Delta\rho_H$ is the head contrast (*sld_head* $-$ *sld_solvent*), |
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| 29 | and $\Delta\rho_T$ is tail contrast (*sld* $-$ *sld_solvent*). |
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[dc02af0] | 30 | |
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[52a3db3] | 31 | The total thickness of the lamellar sheet is $\delta_H$ + $\delta_T$ + $\delta_T$ + $\delta_H$. |
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| 32 | Note that in a non aqueous solvent the chemical "head" group may be the |
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| 33 | "Tail region" and vice-versa. |
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| 34 | |
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[3c56da87] | 35 | The 2D scattering intensity is calculated in the same way as 1D, where |
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[eb69cce] | 36 | the $q$ vector is defined as |
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[dc02af0] | 37 | |
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| 38 | .. math:: |
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| 39 | |
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[d138d43] | 40 | q = \sqrt{q_x^2 + q_y^2} |
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[dc02af0] | 41 | |
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| 42 | |
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[d138d43] | 43 | References |
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| 44 | ---------- |
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[dc02af0] | 45 | |
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| 46 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
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| 47 | |
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| 48 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
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| 49 | |
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| 50 | *2014/04/17 - Description reviewed by S King and P Butler.* |
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| 51 | """ |
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| 52 | |
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[3c56da87] | 53 | from numpy import inf |
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[dc02af0] | 54 | |
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[6ab4ed8] | 55 | name = "lamellar_hg" |
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| 56 | title = "Random lamellar phase with Head and Tail Groups" |
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[dc02af0] | 57 | description = """\ |
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[6ab4ed8] | 58 | [Random lamellar phase with Head and Tail Groups] |
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[3e428ec] | 59 | I(q)= 2*pi*P(q)/(2(H+T)*q^(2)), where |
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| 60 | P(q)= see manual |
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| 61 | layer thickness =(H+T+T+H) = 2(Head+Tail) |
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| 62 | sld = Tail scattering length density |
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[6ab4ed8] | 63 | sld_head = Head scattering length density |
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| 64 | sld_solvent = solvent scattering length density |
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[3e428ec] | 65 | background = incoherent background |
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| 66 | scale = scale factor |
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[dc02af0] | 67 | """ |
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[a5d0d00] | 68 | category = "shape:lamellae" |
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[dc02af0] | 69 | |
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[3eb6b90] | 70 | # pylint: disable=bad-whitespace, line-too-long |
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[3e428ec] | 71 | # ["name", "units", default, [lower, upper], "type","description"], |
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[6ab4ed8] | 72 | parameters = [["tail_length", "Ang", 15, [0, inf], "volume", "Tail thickness ( total = H+T+T+H)"], |
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| 73 | ["head_length", "Ang", 10, [0, inf], "volume", "Head thickness"], |
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[3eb6b90] | 74 | ["sld", "1e-6/Ang^2", 0.4, [-inf,inf], "", "Tail scattering length density"], |
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[6ab4ed8] | 75 | ["sld_head", "1e-6/Ang^2", 3.0, [-inf,inf], "", "Head scattering length density"], |
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| 76 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf,inf], "", "Solvent scattering length density"]] |
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[3eb6b90] | 77 | # pylint: enable=bad-whitespace, line-too-long |
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[dc02af0] | 78 | |
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| 79 | # No volume normalization despite having a volume parameter |
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| 80 | # This should perhaps be volume normalized? |
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| 81 | form_volume = """ |
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| 82 | return 1.0; |
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| 83 | """ |
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| 84 | |
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| 85 | Iq = """ |
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| 86 | const double qsq = q*q; |
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[6ab4ed8] | 87 | const double drh = sld_head - sld_solvent; |
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| 88 | const double drt = sld - sld_solvent; //correction 13FEB06 by L.Porcar |
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[3c56da87] | 89 | const double qT = q*tail_length; |
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| 90 | double Pq, inten; |
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| 91 | Pq = drh*(sin(q*(head_length+tail_length))-sin(qT)) + drt*sin(qT); |
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| 92 | Pq *= Pq; |
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| 93 | Pq *= 4.0/(qsq); |
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| 94 | |
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| 95 | inten = 2.0e-4*M_PI*Pq/qsq; |
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| 96 | |
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| 97 | // normalize by the bilayer thickness |
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| 98 | inten /= 2.0*(head_length+tail_length); |
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| 99 | |
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| 100 | return inten; |
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[dc02af0] | 101 | """ |
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[3c56da87] | 102 | |
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[dc02af0] | 103 | Iqxy = """ |
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[bfb195e] | 104 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[dc02af0] | 105 | """ |
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| 106 | |
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| 107 | # ER defaults to 0.0 |
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| 108 | # VR defaults to 1.0 |
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| 109 | |
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[3e428ec] | 110 | demo = dict(scale=1, background=0, |
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[3eb6b90] | 111 | tail_length=15, head_length=10, |
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[6ab4ed8] | 112 | sld=0.4, sld_head=3.0, sld_solvent=6.0, |
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[3eb6b90] | 113 | tail_length_pd=0.2, tail_length_pd_n=40, |
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| 114 | head_length_pd=0.01, head_length_pd_n=40) |
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[3c56da87] | 115 | |
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[7f47777] | 116 | # |
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[3eb6b90] | 117 | tests = [[{'scale': 1.0, 'background': 0.0, 'tail_length': 15.0, 'head_length': 10.0, |
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[6ab4ed8] | 118 | 'sld': 0.4, 'sld_head': 3.0, 'sld_solvent': 6.0}, [0.001], [653143.9209]]] |
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| 119 | # ADDED by: RKH ON: 18Mar2016 converted from sasview previously, now renaming everything & sorting the docs |
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