[5d4777d] | 1 | r""" |
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[40a87fa] | 2 | For information about polarised and magnetic scattering, see |
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[9a4811a] | 3 | the :ref:`magnetism` documentation. |
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[19dcb933] | 4 | |
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| 5 | Definition |
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| 6 | ---------- |
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| 7 | |
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| 8 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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| 9 | |
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| 10 | .. math:: |
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| 11 | |
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[eb69cce] | 12 | I(q) = \frac{\text{scale}}{V} \cdot \left[ |
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| 13 | 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} |
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[19dcb933] | 14 | \right]^2 + \text{background} |
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| 15 | |
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| 16 | where *scale* is a volume fraction, $V$ is the volume of the scatterer, |
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[7e6bea81] | 17 | $r$ is the radius of the sphere and *background* is the background level. |
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[49da079] | 18 | *sld* and *sld_solvent* are the scattering length densities (SLDs) of the |
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[7e6bea81] | 19 | scatterer and the solvent respectively, whose difference is $\Delta\rho$. |
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[19dcb933] | 20 | |
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| 21 | Note that if your data is in absolute scale, the *scale* should represent |
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| 22 | the volume fraction (which is unitless) if you have a good fit. If not, |
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| 23 | it should represent the volume fraction times a factor (by which your data |
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| 24 | might need to be rescaled). |
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| 25 | |
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| 26 | The 2D scattering intensity is the same as above, regardless of the |
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| 27 | orientation of $\vec q$. |
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| 28 | |
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| 29 | Validation |
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| 30 | ---------- |
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| 31 | |
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| 32 | Validation of our code was done by comparing the output of the 1D model |
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| 33 | to the output of the software provided by the NIST (Kline, 2006). |
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| 34 | |
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| 35 | |
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[eb69cce] | 36 | References |
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| 37 | ---------- |
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[934a001] | 38 | |
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| 39 | .. [#] A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
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| 40 | John Wiley and Sons, New York, (1955) |
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[19dcb933] | 41 | |
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[0507e09] | 42 | Source |
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| 43 | ------ |
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| 44 | |
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| 45 | `sphere.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/sphere.py>`_ |
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| 46 | |
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| 47 | `sphere.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/sphere.c>`_ |
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| 48 | |
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| 49 | Authorship and Verification |
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| 50 | ---------------------------- |
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| 51 | |
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[b39bf3b] | 52 | * **Author:** |
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| 53 | * **Last Modified by:** |
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[ef07e95] | 54 | * **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06 |
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[0507e09] | 55 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[5d4777d] | 56 | """ |
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| 57 | |
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[2d81cfe] | 58 | import numpy as np |
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[3c56da87] | 59 | from numpy import inf |
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[5d4777d] | 60 | |
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| 61 | name = "sphere" |
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[19dcb933] | 62 | title = "Spheres with uniform scattering length density" |
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[5d4777d] | 63 | description = """\ |
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[49da079] | 64 | P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) |
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[eb69cce] | 65 | /(qr)^3]^2 + background |
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| 66 | r: radius of sphere |
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[19dcb933] | 67 | V: The volume of the scatter |
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| 68 | sld: the SLD of the sphere |
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[49da079] | 69 | sld_solvent: the SLD of the solvent |
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[5d4777d] | 70 | """ |
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[a5d0d00] | 71 | category = "shape:sphere" |
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[5d4777d] | 72 | |
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[3e428ec] | 73 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 74 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[3e428ec] | 75 | "Layer scattering length density"], |
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[42356c8] | 76 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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[3e428ec] | 77 | "Solvent scattering length density"], |
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[6140894] | 78 | ["radius", "Ang", 50, [0, inf], "volume", |
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[3e428ec] | 79 | "Sphere radius"], |
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| 80 | ] |
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[5d4777d] | 81 | |
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[b297ba9] | 82 | source = ["lib/sas_3j1x_x.c", "sphere.c"] |
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[71b751d] | 83 | have_Fq = True |
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[a34b811] | 84 | radius_effective_modes = ["radius"] |
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[c036ddb] | 85 | |
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[404ebbd] | 86 | def random(): |
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[b297ba9] | 87 | """Return a random parameter set for the model.""" |
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[404ebbd] | 88 | radius = 10**np.random.uniform(1.3, 4) |
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| 89 | pars = dict( |
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| 90 | radius=radius, |
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| 91 | ) |
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| 92 | return pars |
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[934a001] | 93 | #2345678901234567890123456789012345678901234567890123456789012345678901234567890 |
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[7e6bea81] | 94 | tests = [ |
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[934a001] | 95 | [{}, 0.2, 0.726362], # each test starts with default parameter values |
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| 96 | # inside { }, unless modified. Then Q and expected value of I(Q) |
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| 97 | # putting None for an expected result will pass the test if there are no |
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| 98 | # errors from the routine, but without any check on the value of the result |
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| 99 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
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| 100 | "radius": 120.}, [0.01,0.1,0.2], |
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| 101 | [1.34836265e+04, 6.20114062e+00, 1.04733914e-01]], |
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[6140894] | 102 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
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[934a001] | 103 | # careful tests here R=120 Pd=.2, then with S(Q) at default Reff=50 |
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| 104 | # (but this gets changed to 120) phi=0,2 |
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[6140894] | 105 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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[934a001] | 106 | [0.01,0.1,0.2], [1.74395295e+04, 3.68016987e+00, 2.28843099e-01]], |
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| 107 | # a list of Q values and list of expected results is also possible |
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| 108 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
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| 109 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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| 110 | 0.01, 335839.88055473, 1.41045057e+11, 120.0, 8087664.122641933, 1.0], |
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| 111 | # the longer list here checks F1, F2, R_eff, volume, volume_ratio |
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[db3947c] | 112 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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[934a001] | 113 | 0.1, 482.93824329, 29763977.79867414, 120.0, 8087664.122641933, 1.0], |
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[db3947c] | 114 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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[934a001] | 115 | 0.2, 1.23330406, 1850806.1197361, 120.0, 8087664.122641933, 1.0], |
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[6140894] | 116 | # But note P(Q) = F2/volume |
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[934a001] | 117 | # F and F^2 are "unscaled", with for n <F F*>S(q) or for beta approx |
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| 118 | # I(q) = n [<F F*> + <F><F*> (S(q) - 1)] |
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| 119 | # for n the number density and <.> the orientation average, and |
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| 120 | # F = integral rho(r) exp(i q . r) dr. |
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[a34b811] | 121 | # The number density is volume fraction divided by particle volume. |
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[934a001] | 122 | # Effectively, this leaves F = V drho form, where form is the usual |
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| 123 | # 3 j1(qr)/(qr) or whatever depending on the shape. |
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| 124 | # @S RESULTS using F1 and F2 from the longer test strng above: |
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| 125 | # |
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| 126 | # I(Q) = (F2 + F1^2*(S(Q) -1))*volfraction*scale/Volume + background |
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| 127 | # |
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| 128 | # with by default scale=1.0, background=0.001 |
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| 129 | # NOTE currently S(Q) volfraction is also included in scaling |
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| 130 | # structure_factor_mode 0 = normal decoupling approx, |
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| 131 | # 1 = beta(Q) approx |
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| 132 | # radius_effective_mode 0 is for free choice, |
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| 133 | # 1 is use radius from F2(Q) |
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| 134 | # (sphere only has two choices, other models may have more) |
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[b39bf3b] | 135 | [{"@S": "hardsphere", |
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[934a001] | 136 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45,"volfraction":0.2, |
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| 137 | #"radius_effective":50.0, # hard sphere structure factor |
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| 138 | "structure_factor_mode": 1, # mode 0 = normal decoupling approx, |
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| 139 | # 1 = beta(Q) approx |
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| 140 | "radius_effective_mode": 0 # this used default hardsphere Reff=50 |
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| 141 | }, [0.01,0.1,0.2], [1.32473756e+03, 7.36633631e-01, 4.67686201e-02] ], |
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[6140894] | 142 | [{"@S": "hardsphere", |
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| 143 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45, |
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| 144 | "volfraction":0.2, |
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[934a001] | 145 | "radius_effective":45.0, # explicit Reff over rides either 50 or 120 |
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| 146 | "structure_factor_mode": 1, # beta approx |
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| 147 | "radius_effective_mode": 0 # |
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| 148 | }, 0.01, 1316.2990966463444 ], |
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[6140894] | 149 | [{"@S": "hardsphere", |
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| 150 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45, |
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| 151 | "volfraction":0.2, |
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[934a001] | 152 | "radius_effective":120.0, # over ride Reff |
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| 153 | "structure_factor_mode": 1, # beta approx |
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| 154 | "radius_effective_mode": 0 # (mode=1 here also uses 120) |
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[db3947c] | 155 | }, [0.01,0.1,0.2], [1.57928589e+03, 7.37067923e-01, 4.67686197e-02 ]], |
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| 156 | [{"@S": "hardsphere", |
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| 157 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45, |
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| 158 | "volfraction":0.2, |
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[934a001] | 159 | #"radius_effective":120.0, # hard sphere structure factor |
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| 160 | "structure_factor_mode": 0, # normal decoupling approximation |
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| 161 | "radius_effective_mode": 1 # this uses 120 from the form factor |
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[db3947c] | 162 | }, [0.01,0.1,0.2], [1.10112335e+03, 7.41366536e-01, 4.66630207e-02]], |
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| 163 | [{"@S": "hardsphere", |
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| 164 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45, |
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| 165 | "volfraction":0.2, |
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[934a001] | 166 | #"radius_effective":50.0, # hard sphere structure factor |
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| 167 | "structure_factor_mode": 0, # normal decoupling approximation |
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[db3947c] | 168 | "radius_effective_mode": 0 # this used 50 the default for hardsphere |
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| 169 | }, [0.01,0.1,0.2], [7.82803598e+02, 6.85943611e-01, 4.71586457e-02 ]] |
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[934a001] | 170 | ] |
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| 171 | # |
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