[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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[d138d43] | 28 | $\Delta\rho_H$ is the head contrast (*head_sld* $-$ *solvent_sld*), |
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| 29 | and $\Delta\rho_T$ is tail contrast (*sld* $-$ *solvent_sld*). |
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[dc02af0] | 30 | |
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[3c56da87] | 31 | The 2D scattering intensity is calculated in the same way as 1D, where |
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[eb69cce] | 32 | the $q$ vector is defined as |
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[dc02af0] | 33 | |
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| 34 | .. math:: |
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| 35 | |
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[d138d43] | 36 | q = \sqrt{q_x^2 + q_y^2} |
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[dc02af0] | 37 | |
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| 38 | |
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[d138d43] | 39 | .. figure:: img/lamellarFFHG_1d.jpg |
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[dc02af0] | 40 | |
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[d138d43] | 41 | 1D plot using the default values (w/1000 data point). |
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[dc02af0] | 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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| 55 | name = "lamellar_FFHG" |
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[d138d43] | 56 | title = "Random lamellar phase with Head Groups" |
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[dc02af0] | 57 | description = """\ |
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[3e428ec] | 58 | [Random lamellar phase with Head Groups] |
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| 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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[22eac46] | 63 | head_sld = Head scattering length density |
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| 64 | solvent_sld = 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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[3e428ec] | 70 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 71 | parameters = [["tail_length", "Ang", 15, [0, inf], "volume", |
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| 72 | "Tail thickness"], |
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| 73 | ["head_length", "Ang", 10, [0, inf], "volume", |
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| 74 | "head thickness"], |
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| 75 | ["sld", "1e-6/Ang^2", 0.4, [-inf,inf], "", |
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| 76 | "Tail scattering length density"], |
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| 77 | ["head_sld", "1e-6/Ang^2", 3.0, [-inf,inf], "", |
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| 78 | "Head scattering length density"], |
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| 79 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf,inf], "", |
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| 80 | "Solvent scattering length density"], |
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| 81 | ] |
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[dc02af0] | 82 | |
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| 83 | # No volume normalization despite having a volume parameter |
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| 84 | # This should perhaps be volume normalized? |
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| 85 | form_volume = """ |
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| 86 | return 1.0; |
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| 87 | """ |
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| 88 | |
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| 89 | Iq = """ |
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| 90 | const double qsq = q*q; |
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[3c56da87] | 91 | const double drh = head_sld - solvent_sld; |
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| 92 | const double drt = sld - solvent_sld; //correction 13FEB06 by L.Porcar |
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| 93 | const double qT = q*tail_length; |
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| 94 | double Pq, inten; |
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| 95 | Pq = drh*(sin(q*(head_length+tail_length))-sin(qT)) + drt*sin(qT); |
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| 96 | Pq *= Pq; |
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| 97 | Pq *= 4.0/(qsq); |
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| 98 | |
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| 99 | inten = 2.0e-4*M_PI*Pq/qsq; |
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| 100 | |
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| 101 | // normalize by the bilayer thickness |
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| 102 | inten /= 2.0*(head_length+tail_length); |
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| 103 | |
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| 104 | return inten; |
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[dc02af0] | 105 | """ |
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[3c56da87] | 106 | |
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[dc02af0] | 107 | Iqxy = """ |
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[bfb195e] | 108 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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[dc02af0] | 109 | """ |
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| 110 | |
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| 111 | # ER defaults to 0.0 |
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| 112 | # VR defaults to 1.0 |
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| 113 | |
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[3e428ec] | 114 | demo = dict(scale=1, background=0, |
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| 115 | tail_length=15,head_length=10, |
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| 116 | sld=0.4, head_sld=3.0, solvent_sld=6.0, |
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| 117 | tail_length_pd= 0.2, tail_length_pd_n=40, |
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| 118 | head_length_pd= 0.01, head_length_pd_n=40) |
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[3c56da87] | 119 | |
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[dc02af0] | 120 | oldname = 'LamellarFFHGModel' |
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[22eac46] | 121 | oldpars = dict(head_length='h_thickness', tail_length='t_length', |
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| 122 | sld='sld_tail', head_sld='sld_head', solvent_sld='sld_solvent') |
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[7f47777] | 123 | # |
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| 124 | tests = [ |
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| 125 | [ {'scale': 1.0, 'background' : 0.0, 'tail_length' : 15.0, 'head_length' : 10.0,'sld' : 0.4, |
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| 126 | 'head_sld' : 3.0, 'solvent_sld' : 6.0, }, [0.001], [653143.9209]] |
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| 127 | ] |
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| 128 | |
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[dc02af0] | 129 | |
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