[9cb1415] | 1 | # Note: model title and parameter table are inserted automatically |
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[3c56da87] | 2 | r""" |
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[c1e44e5] | 3 | Calculates the interparticle structure factor for a hard sphere fluid |
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| 4 | with a narrow, attractive, potential well. Unlike the :ref:`squarewell` |
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| 5 | model, here a perturbative solution of the Percus-Yevick closure |
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| 6 | relationship is used. The strength of the attractive well is described |
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[5f3c534] | 7 | in terms of "stickiness" as defined below. |
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| 8 | |
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| 9 | The perturbation parameter (perturb), $\tau$, should be fixed between 0.01 |
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[c1e44e5] | 10 | and 0.1 and the "stickiness", $\epsilon$, allowed to vary to adjust the |
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| 11 | interaction strength. The "stickiness" is defined in the equation below and is |
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| 12 | a function of both the perturbation parameter and the interaction strength. |
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| 13 | $\epsilon$ and $\tau$ are defined in terms of the hard sphere diameter $(\sigma = 2 R)$, |
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| 14 | the width of the square well, $\Delta$ (having the same units as $R$\ ), |
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| 15 | and the depth of the well, $U_o$, in units of $kT$. From the definition, it |
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[5f3c534] | 16 | is clear that smaller $\epsilon$ means a stronger attraction. |
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[9cb1415] | 17 | |
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[eb69cce] | 18 | .. math:: |
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| 19 | |
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[5f3c534] | 20 | \epsilon &= \frac{1}{12\tau} \exp(u_o / kT) \\ |
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| 21 | \tau &= \Delta / (\sigma + \Delta) |
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[9cb1415] | 22 | |
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| 23 | where the interaction potential is |
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| 24 | |
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[eb69cce] | 25 | .. math:: |
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| 26 | |
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| 27 | U(r) = \begin{cases} |
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| 28 | \infty & r < \sigma \\ |
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| 29 | -U_o & \sigma \leq r \leq \sigma + \Delta \\ |
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| 30 | 0 & r > \sigma + \Delta |
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| 31 | \end{cases} |
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[9cb1415] | 32 | |
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[5f3c534] | 33 | The Percus-Yevick (PY) closure is used for this calculation, and is an |
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| 34 | adequate closure for an attractive interparticle potential. The solution |
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[3c56da87] | 35 | has been compared to Monte Carlo simulations for a square well fluid, with |
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| 36 | good agreement. |
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| 37 | |
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[5f3c534] | 38 | The true particle volume fraction, $\phi$, is not equal to $h$ which appears |
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[c1e44e5] | 39 | in most of reference [1]. The two are related in equation (24). Reference |
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| 40 | [1] also describes the relationship between this perturbative solution and |
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[5f3c534] | 41 | the original sticky hard sphere (or "adhesive sphere") model of Baxter [2]. |
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| 42 | |
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| 43 | .. note:: |
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| 44 | |
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| 45 | The calculation can go haywire for certain combinations of the input |
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| 46 | parameters, producing unphysical solutions. In this case errors are |
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| 47 | reported to the command window and $S(q)$ is set to -1 (so it will |
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| 48 | disappear on a log-log plot!). |
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[c1e44e5] | 49 | |
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| 50 | Use tight bounds to keep the parameters to values that you know are |
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| 51 | physical (test them), and keep nudging them until the optimization |
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[5f3c534] | 52 | does not hit the constraints. |
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| 53 | |
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| 54 | .. note:: |
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| 55 | |
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[c1e44e5] | 56 | Earlier versions of SasView did not incorporate the so-called |
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| 57 | $\beta(q)$ ("beta") correction [3] for polydispersity and non-sphericity. |
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[4d00de6] | 58 | This is only available in SasView versions 5.0 and higher. |
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[c1e44e5] | 59 | |
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[5f3c534] | 60 | In SasView the effective radius may be calculated from the parameters |
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[eb69cce] | 61 | used in the form factor $P(q)$ that this $S(q)$ is combined with. |
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[9cb1415] | 62 | |
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[eb69cce] | 63 | For 2D data the scattering intensity is calculated in the same way |
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| 64 | as 1D, where the $q$ vector is defined as |
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[9cb1415] | 65 | |
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| 66 | .. math:: |
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| 67 | |
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[eb69cce] | 68 | q = \sqrt{q_x^2 + q_y^2} |
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[9cb1415] | 69 | |
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| 70 | |
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[eb69cce] | 71 | References |
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| 72 | ---------- |
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[9cb1415] | 73 | |
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[0507e09] | 74 | .. [#] S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190 |
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| 75 | |
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[5f3c534] | 76 | .. [#] R J Baxter, *J. Chem. Phys.*, 49 (1968), 2770-2774 |
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| 77 | |
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| 78 | .. [#] M Kotlarchyk and S-H Chen, *J. Chem. Phys.*, 79 (1983) 2461-2469 |
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| 79 | |
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[0507e09] | 80 | Authorship and Verification |
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| 81 | ---------------------------- |
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| 82 | |
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[c1e44e5] | 83 | * **Author:** |
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| 84 | * **Last Modified by:** |
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[5f3c534] | 85 | * **Last Reviewed by:** Steve King **Date:** March 27, 2019 |
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[9cb1415] | 86 | """ |
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| 87 | |
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| 88 | # TODO: refactor so that we pull in the old sansmodels.c_extensions |
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[7e224c2] | 89 | |
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[2d81cfe] | 90 | import numpy as np |
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[7e224c2] | 91 | from numpy import inf |
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[9cb1415] | 92 | |
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| 93 | name = "stickyhardsphere" |
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[5f3c534] | 94 | title = "'Sticky' hard sphere structure factor with Percus-Yevick closure" |
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[9cb1415] | 95 | description = """\ |
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[3e428ec] | 96 | [Sticky hard sphere structure factor, with Percus-Yevick closure] |
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[c1e44e5] | 97 | Interparticle structure factor S(Q) for a hard sphere fluid |
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[5f3c534] | 98 | with a narrow attractive well. Fits are prone to deliver non- |
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[c1e44e5] | 99 | physical parameters; use with care and read the references in |
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| 100 | the model documentation.The "beta(q)" correction is available |
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[5f3c534] | 101 | in versions 4.2.2 and higher. |
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[9cb1415] | 102 | """ |
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[a5d0d00] | 103 | category = "structure-factor" |
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[8e45182] | 104 | structure_factor = True |
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[9cb1415] | 105 | |
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[13ed84c] | 106 | single = False |
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[3e428ec] | 107 | # ["name", "units", default, [lower, upper], "type","description"], |
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[9cb1415] | 108 | parameters = [ |
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[7e224c2] | 109 | # [ "name", "units", default, [lower, upper], "type", |
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| 110 | # "description" ], |
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[54954e1] | 111 | ["radius_effective", "Ang", 50.0, [0, inf], "volume", |
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[7e224c2] | 112 | "effective radius of hard sphere"], |
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| 113 | ["volfraction", "", 0.2, [0, 0.74], "", |
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| 114 | "volume fraction of hard spheres"], |
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| 115 | ["perturb", "", 0.05, [0.01, 0.1], "", |
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[5f3c534] | 116 | "perturbation parameter, tau"], |
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[7e224c2] | 117 | ["stickiness", "", 0.20, [-inf, inf], "", |
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[5f3c534] | 118 | "stickiness, epsilon"], |
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[9cb1415] | 119 | ] |
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[7e224c2] | 120 | |
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[8f04da4] | 121 | def random(): |
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[b297ba9] | 122 | """Return a random parameter set for the model.""" |
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[8f04da4] | 123 | pars = dict( |
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| 124 | scale=1, background=0, |
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| 125 | radius_effective=10**np.random.uniform(1, 4.7), |
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| 126 | volfraction=np.random.uniform(0.00001, 0.74), |
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| 127 | perturb=10**np.random.uniform(-2, -1), |
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| 128 | stickiness=np.random.uniform(0, 1), |
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| 129 | ) |
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| 130 | return pars |
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| 131 | |
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[9cb1415] | 132 | # No volume normalization despite having a volume parameter |
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| 133 | # This should perhaps be volume normalized? |
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| 134 | form_volume = """ |
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| 135 | return 1.0; |
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| 136 | """ |
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| 137 | |
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| 138 | Iq = """ |
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[3c56da87] | 139 | double onemineps,eta; |
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| 140 | double sig,aa,etam1,etam1sq,qa,qb,qc,radic; |
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| 141 | double lam,lam2,test,mu,alpha,beta; |
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| 142 | double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq; |
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| 143 | |
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| 144 | onemineps = 1.0-perturb; |
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| 145 | eta = volfraction/onemineps/onemineps/onemineps; |
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| 146 | |
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[54954e1] | 147 | sig = 2.0 * radius_effective; |
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[3c56da87] | 148 | aa = sig/onemineps; |
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| 149 | etam1 = 1.0 - eta; |
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| 150 | etam1sq=etam1*etam1; |
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| 151 | //C |
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| 152 | //C SOLVE QUADRATIC FOR LAMBDA |
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| 153 | //C |
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[034e19a] | 154 | qa = eta/6.0; |
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| 155 | qb = stickiness + eta/etam1; |
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[3c56da87] | 156 | qc = (1.0 + eta/2.0)/etam1sq; |
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[034e19a] | 157 | radic = qb*qb - 2.0*qa*qc; |
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[3c56da87] | 158 | if(radic<0) { |
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| 159 | //if(x>0.01 && x<0.015) |
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[3e428ec] | 160 | // Print "Lambda unphysical - both roots imaginary" |
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[3c56da87] | 161 | //endif |
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| 162 | return(-1.0); |
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| 163 | } |
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| 164 | //C KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL |
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[034e19a] | 165 | radic = sqrt(radic); |
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| 166 | lam = (qb-radic)/qa; |
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| 167 | lam2 = (qb+radic)/qa; |
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[3c56da87] | 168 | if(lam2<lam) { |
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| 169 | lam = lam2; |
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| 170 | } |
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| 171 | test = 1.0 + 2.0*eta; |
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| 172 | mu = lam*eta*etam1; |
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| 173 | if(mu>test) { |
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| 174 | //if(x>0.01 && x<0.015) |
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| 175 | // Print "Lambda unphysical mu>test" |
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| 176 | //endif |
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| 177 | return(-1.0); |
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| 178 | } |
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| 179 | alpha = (1.0 + 2.0*eta - mu)/etam1sq; |
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| 180 | beta = (mu - 3.0*eta)/(2.0*etam1sq); |
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| 181 | //C |
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| 182 | //C CALCULATE THE STRUCTURE FACTOR |
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| 183 | //C |
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| 184 | kk = q*aa; |
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| 185 | k2 = kk*kk; |
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| 186 | k3 = kk*k2; |
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| 187 | SINCOS(kk,ds,dc); |
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| 188 | //ds = sin(kk); |
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| 189 | //dc = cos(kk); |
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| 190 | aq1 = ((ds - kk*dc)*alpha)/k3; |
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| 191 | aq2 = (beta*(1.0-dc))/k2; |
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| 192 | aq3 = (lam*ds)/(12.0*kk); |
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| 193 | aq = 1.0 + 12.0*eta*(aq1+aq2-aq3); |
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| 194 | // |
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| 195 | bq1 = alpha*(0.5/kk - ds/k2 + (1.0 - dc)/k3); |
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| 196 | bq2 = beta*(1.0/kk - ds/k2); |
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| 197 | bq3 = (lam/12.0)*((1.0 - dc)/kk); |
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| 198 | bq = 12.0*eta*(bq1+bq2-bq3); |
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| 199 | // |
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| 200 | sq = 1.0/(aq*aq +bq*bq); |
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| 201 | |
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| 202 | return(sq); |
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[9cb1415] | 203 | """ |
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[bfb195e] | 204 | |
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[54954e1] | 205 | demo = dict(radius_effective=200, volfraction=0.2, perturb=0.05, |
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| 206 | stickiness=0.2, radius_effective_pd=0.1, radius_effective_pd_n=40) |
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[7f47777] | 207 | # |
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| 208 | tests = [ |
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[40a87fa] | 209 | [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0, |
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| 210 | 'perturb': 0.05, 'stickiness': 0.2, 'volfraction': 0.1, |
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| 211 | 'radius_effective_pd': 0}, |
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| 212 | [0.001, 0.003], [1.09718, 1.087830]], |
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| 213 | ] |
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