[cf404cb] | 1 | r""" |
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[841753c] | 2 | This model provides the form factor for a pearl necklace composed of two |
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| 3 | elements: *N* pearls (homogeneous spheres of radius *R*) freely jointed by *M* |
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[cf404cb] | 4 | rods (like strings - with a total mass *Mw* = *M* \* *m*\ :sub:`r` + *N* \* *m*\ |
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[841753c] | 5 | :sub:`s`, and the string segment length (or edge separation) *l* |
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[cf404cb] | 6 | (= *A* - 2\ *R*)). *A* is the center-to-center pearl separation distance. |
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| 7 | |
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[2f0c07d] | 8 | .. figure:: img/pearl_necklace_geometry.jpg |
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[cf404cb] | 9 | |
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| 10 | Pearl Necklace schematic |
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| 11 | |
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| 12 | Definition |
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| 13 | ---------- |
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| 14 | |
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[a2f9aa2] | 15 | The output of the scattering intensity function for the pearl_necklace is |
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[cf404cb] | 16 | given by (Schweins, 2004) |
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| 17 | |
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| 18 | .. math:: |
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| 19 | |
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| 20 | I(q)=\frac{ \text{scale} }{V} \cdot \frac{(S_{ss}(q)+S_{ff}(q)+S_{fs}(q))} |
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| 21 | {(M \cdot m_f + N \cdot m_s)^2} + \text{bkg} |
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| 22 | |
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| 23 | where |
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| 24 | |
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| 25 | .. math:: |
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| 26 | |
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[708bc7a] | 27 | S_{ss}(q) &= 2m_s^2\psi^2(q)\left[\frac{N}{1-sin(qA)/qA}-\frac{N}{2}- |
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| 28 | \frac{1-(sin(qA)/qA)^N}{(1-sin(qA)/qA)^2}\cdot\frac{sin(qA)}{qA}\right] \\ |
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| 29 | S_{ff}(q) &= m_r^2\left[M\left\{2\Lambda(q)-\left(\frac{sin(ql/2)}{ql/2}\right)\right\}+ |
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[cf404cb] | 30 | \frac{2M\beta^2(q)}{1-sin(qA)/qA}-2\beta^2(q)\cdot |
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[708bc7a] | 31 | \frac{1-(sin(qA)/qA)^M}{(1-sin(qA)/qA)^2}\right] \\ |
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| 32 | S_{fs}(q) &= m_r \beta (q) \cdot m_s \psi (q) \cdot 4\left[ |
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[cf404cb] | 33 | \frac{N-1}{1-sin(qA)/qA}-\frac{1-(sin(qA)/qA)^{N-1}}{(1-sin(qA)/qA)^2} |
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[708bc7a] | 34 | \cdot \frac{sin(qA)}{qA}\right] \\ |
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[cf404cb] | 35 | \psi(q) &= 3 \cdot \frac{sin(qR)-(qR)\cdot cos(qR)}{(qR)^3} \\ |
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| 36 | \Lambda(q) &= \frac{\int_0^{ql}\frac{sin(t)}{t}dt}{ql} \\ |
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| 37 | \beta(q) &= \frac{\int_{qR}^{q(A-R)}\frac{sin(t)}{t}dt}{ql} |
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| 38 | |
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[841753c] | 39 | where the mass *m*\ :sub:`i` is (SLD\ :sub:`i` - SLD\ :sub:`solvent`) \* |
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[cf404cb] | 40 | (volume of the *N* pearls/rods). *V* is the total volume of the necklace. |
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| 41 | |
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[841753c] | 42 | The 2D scattering intensity is the same as $P(q)$ above, regardless of the |
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[cf404cb] | 43 | orientation of the *q* vector. |
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| 44 | |
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| 45 | The returned value is scaled to units of |cm^-1| and the parameters of the |
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| 46 | pearl_necklace model are the following |
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| 47 | |
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[a807206] | 48 | NB: *num_pearls* must be an integer. |
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[cf404cb] | 49 | |
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[2f0c07d] | 50 | References |
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| 51 | ---------- |
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[cf404cb] | 52 | |
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[0507e09] | 53 | .. [#] R Schweins and K Huber, *Particle Scattering Factor of Pearl Necklace Chains*, |
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[cf404cb] | 54 | *Macromol. Symp.* 211 (2004) 25-42 2004 |
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[0507e09] | 55 | .. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659 |
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| 56 | |
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| 57 | Source |
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| 58 | ------ |
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| 59 | |
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| 60 | `pearl_necklace.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/pearl_necklace.py>`_ |
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| 61 | |
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| 62 | `pearl_necklace.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/pearl_necklace.c>`_ |
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| 63 | |
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| 64 | Authorship and Verification |
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| 65 | ---------------------------- |
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| 66 | |
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[708bc7a] | 67 | * **Author:** |
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| 68 | * **Last Modified by:** |
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| 69 | * **Last Reviewed by:** |
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[0507e09] | 70 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[cf404cb] | 71 | """ |
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| 72 | |
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[2d81cfe] | 73 | import numpy as np |
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[b297ba9] | 74 | from numpy import inf |
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[cf404cb] | 75 | |
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| 76 | name = "pearl_necklace" |
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[a2f9aa2] | 77 | title = "Colloidal spheres chained together with no preferential orientation" |
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[cf404cb] | 78 | description = """ |
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| 79 | Calculate form factor for Pearl Necklace Model |
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| 80 | [Macromol. Symp. 2004, 211, 25-42] |
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| 81 | Parameters: |
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| 82 | background:background |
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| 83 | scale: scale factor |
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| 84 | sld: the SLD of the pearl spheres |
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| 85 | sld_string: the SLD of the strings |
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| 86 | sld_solvent: the SLD of the solvent |
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[a807206] | 87 | num_pearls: number of the pearls |
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[cf404cb] | 88 | radius: the radius of a pearl |
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[a807206] | 89 | edge_sep: the length of string segment; surface to surface |
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| 90 | thick_string: thickness (ie, diameter) of the string |
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[cf404cb] | 91 | """ |
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| 92 | category = "shape:cylinder" |
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| 93 | |
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| 94 | # ["name", "units", default, [lower, upper], "type","description"], |
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[a2f9aa2] | 95 | parameters = [["radius", "Ang", 80.0, [0, inf], "volume", |
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[cf404cb] | 96 | "Mean radius of the chained spheres"], |
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[a807206] | 97 | ["edge_sep", "Ang", 350.0, [0, inf], "volume", |
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[cf404cb] | 98 | "Mean separation of chained particles"], |
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[a807206] | 99 | ["thick_string", "Ang", 2.5, [0, inf], "volume", |
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[cf404cb] | 100 | "Thickness of the chain linkage"], |
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[1bd1ea2] | 101 | ["num_pearls", "none", 3, [1, inf], "volume", |
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[a2f9aa2] | 102 | "Number of pearls in the necklace (must be integer)"], |
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[42356c8] | 103 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", |
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[cf404cb] | 104 | "Scattering length density of the chained spheres"], |
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[42356c8] | 105 | ["sld_string", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", |
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[cf404cb] | 106 | "Scattering length density of the chain linkage"], |
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[42356c8] | 107 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", |
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[cf404cb] | 108 | "Scattering length density of the solvent"], |
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[841753c] | 109 | ] |
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[cf404cb] | 110 | |
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[4b541ac] | 111 | source = ["lib/sas_Si.c", "lib/sas_3j1x_x.c", "pearl_necklace.c"] |
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[d18582e] | 112 | single = False # use double precision unless told otherwise |
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[b297ba9] | 113 | effective_radius_type = ["equivalent volume sphere"] |
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| 114 | |
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[8f04da4] | 115 | def random(): |
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[b297ba9] | 116 | """Return a random parameter set for the model.""" |
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[8f04da4] | 117 | radius = 10**np.random.uniform(1, 3) # 1 - 1000 |
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| 118 | thick_string = 10**np.random.uniform(0, np.log10(radius)-1) # 1 - radius/10 |
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| 119 | edge_sep = 10**np.random.uniform(0, 3) # 1 - 1000 |
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| 120 | num_pearls = np.round(10**np.random.uniform(0.3, 3)) # 2 - 1000 |
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| 121 | pars = dict( |
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| 122 | radius=radius, |
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| 123 | edge_sep=edge_sep, |
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| 124 | thick_string=thick_string, |
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| 125 | num_pearls=num_pearls, |
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| 126 | ) |
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| 127 | return pars |
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| 128 | |
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[cf404cb] | 129 | # parameters for demo |
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[a807206] | 130 | demo = dict(scale=1, background=0, radius=80.0, edge_sep=350.0, |
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| 131 | num_pearls=3, sld=1, sld_solvent=6.3, sld_string=1, |
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| 132 | thick_string=2.5, |
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[cf404cb] | 133 | radius_pd=.2, radius_pd_n=5, |
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[a807206] | 134 | edge_sep_pd=25.0, edge_sep_pd_n=5, |
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| 135 | num_pearls_pd=0, num_pearls_pd_n=0, |
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| 136 | thick_string_pd=0.2, thick_string_pd_n=5, |
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[cf404cb] | 137 | ) |
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[b297ba9] | 138 | # ER function is not being used here, not that it is likely very sensible to |
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| 139 | # include an S(Q) with this model, the default in sasview 5.0 would be to the |
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[2cc8aa2] | 140 | # "unconstrained" radius_effective. |
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| 141 | #tests = [[{}, 0.001, 17380.245], [{}, 'ER', 115.39502]] |
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| 142 | tests = [[{}, 0.001, 17380.245]] |
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