# source:sasmodels/sasmodels/models/hardsphere.py@71b751d

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 71b751d was 71b751d, checked in by Paul Kienzle <pkienzle@…>, 23 months ago

update remaining form factors to use Fq interface

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1# Note: model title and parameter table are inserted automatically
2r"""Calculate the interparticle structure factor for monodisperse
3spherical particles interacting through hard sphere (excluded volume)
4interactions.
5May be a reasonable approximation for other shapes of particles that
6freely rotate, and for moderately polydisperse systems. Though strictly
7the maths needs to be modified (no \Beta(Q) correction yet in sasview).
8
10volfraction is the volume fraction occupied by the spheres.
11
12In sasview the effective radius may be calculated from the parameters
13used in the form factor $P(q)$ that this $S(q)$ is combined with.
14
15For numerical stability the computation uses a Taylor series expansion
16at very small $qR$, there may be a very minor glitch at the transition point
17in some circumstances.
18
19The S(Q) uses the Percus-Yevick closure where the interparticle
20potential is
21
22.. math::
23
24    U(r) = \begin{cases}
25    \infty & r < 2R \\
26    0 & r \geq 2R
27    \end{cases}
28
29where $r$ is the distance from the center of the sphere of a radius $R$.
30
31For a 2D plot, the wave transfer is defined as
32
33.. math::
34
35    q = \sqrt{q_x^2 + q_y^2}
36
37
38References
39----------
40
41J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1
42"""
43
44import numpy as np
45from numpy import inf
46
47name = "hardsphere"
48title = "Hard sphere structure factor, with Percus-Yevick closure"
49description = """\
50    [Hard sphere structure factor, with Percus-Yevick closure]
51        Interparticle S(Q) for random, non-interacting spheres.
52    May be a reasonable approximation for other shapes of
53    particles that freely rotate, and for moderately polydisperse
54        systems. Though strictly the maths needs to be modified -
55    which sasview does not do yet.
57    volfraction is the volume fraction occupied by the spheres.
58"""
59category = "structure-factor"
60structure_factor = True
61single = False # TODO: check
62
63#             ["name", "units", default, [lower, upper], "type","description"],
64parameters = [["radius_effective", "Ang", 50.0, [0, inf], "",
65               "effective radius of hard sphere"],
66              ["volfraction", "", 0.2, [0, 0.74], "",
67               "volume fraction of hard spheres"],
68             ]
69
70Iq = r"""
71      double D,A,B,G,X,X2,X4,S,C,FF,HARDSPH;
72      // these are c compiler instructions, can also put normal code inside the "if else" structure
73      #if FLOAT_SIZE > 4
74      // double precision
75      // orig had 0.2, don't call the variable cutoff as PAK already has one called that!
76      // Must use UPPERCASE name please.
77      // 0.05 better, 0.1 OK
78      #define CUTOFFHS 0.05
79      #else
80      // 0.1 bad, 0.2 OK, 0.3 good, 0.4 better, 0.8 no good
81      #define CUTOFFHS 0.4
82      #endif
83
85               HARDSPH=1.0;
86//printf("HS1 %g: %g\n",q,HARDSPH);
87               return(HARDSPH);
88      }
89      // removing use of pow(xxx,2) and rearranging the calcs
90      // of A, B & G cut ~40% off execution time ( 0.5 to 0.3 msec)
91      X = 1.0/( 1.0 -volfraction);
92      D= X*X;
93      A= (1.+2.*volfraction)*D;
94      A *=A;
96
97      if(X < 5.E-06) {
98                 HARDSPH=1./A;
99//printf("HS2 %g: %g\n",q,HARDSPH);
100                 return(HARDSPH);
101      }
102      X2 =X*X;
103      B = (1.0 +0.5*volfraction)*D;
104      B *= B;
105      B *= -6.*volfraction;
106      G=0.5*volfraction*A;
107
108      if(X < CUTOFFHS) {
109      // RKH Feb 2016, use Taylor series expansion for small X
110      // else no obvious way to rearrange the equations to avoid
111      // needing a very high number of significant figures.
112      // Series expansion found using Mathematica software. Numerical test
113      // in .xls showed terms to X^2 are sufficient
114      // for 5 or 6 significant figures, but I put the X^4 one in anyway
115            //FF = 8*A +6*B + 4*G - (0.8*A +2.0*B/3.0 +0.5*G)*X2 +(A/35. +B/40. +G/50.)*X4;
116            // refactoring the polynomial makes it very slightly faster (0.5 not 0.6 msec)
117            //FF = 8*A +6*B + 4*G + ( -0.8*A -2.0*B/3.0 -0.5*G +(A/35. +B/40. +G/50.)*X2)*X2;
118
119            FF = 8.0*A +6.0*B + 4.0*G + ( -0.8*A -B/1.5 -0.5*G +(A/35. +0.0125*B +0.02*G)*X2)*X2;
120
121            // combining the terms makes things worse at smallest Q in single precision
122            //FF = (8-0.8*X2)*A +(3.0-X2/3.)*2*B + (4+0.5*X2)*G +(A/35. +B/40. +G/50.)*X4;
123            // note that G = -volfraction*A/2, combining this makes no further difference at smallest Q
124            //FF = (8 +2.*volfraction + ( volfraction/4. -0.8 +(volfraction/100. -1./35.)*X2 )*X2 )*A  + (3.0 -X2/3. +X4/40.)*2.*B;
125            HARDSPH= 1./(1. + volfraction*FF );
126//printf("HS3 %g: %g\n",q,HARDSPH);
127            return(HARDSPH);
128      }
129      X4=X2*X2;
130      SINCOS(X,S,C);
131
132// RKH Feb 2016, use version FISH code as is better than original sasview one
133// at small Q in single precision, and more than twice as fast in double.
134      //FF=A*(S-X*C)/X + B*(2.*X*S -(X2-2.)*C -2.)/X2 + G*( (4.*X2*X -24.*X)*S -(X4 -12.*X2 +24.)*C +24. )/X4;
135      // refactoring the polynomial here & above makes it slightly faster
136
137      FF=  (( G*( (4.*X2 -24.)*X*S -(X4 -12.*X2 +24.)*C +24. )/X2 + B*(2.*X*S -(X2-2.)*C -2.) )/X + A*(S-X*C))/X ;
138      HARDSPH= 1./(1. + 24.*volfraction*FF/X2 );
139
140      // changing /X and /X2 to *MX1 and *MX2, no significantg difference?
141      //MX=1.0/X;
142      //MX2=MX*MX;
143      //FF=  (( G*( (4.*X2 -24.)*X*S -(X4 -12.*X2 +24.)*C +24. )*MX2 + B*(2.*X*S -(X2-2.)*C -2.) )*MX + A*(S-X*C)) ;
144      //HARDSPH= 1./(1. + 24.*volfraction*FF*MX2*MX );
145
146// grouping the terms, was about same as sasmodels for single precision issues
147//     FF=A*(S/X-C) + B*(2.*S/X - C +2.0*(C-1.0)/X2) + G*( (4./X -24./X3)*S -(1.0 -12./X2 +24./X4)*C +24./X4 );
148//     HARDSPH= 1./(1. + 24.*volfraction*FF/X2 );
149// remove 1/X2 from final line, take more powers of X inside the brackets, stil bad
150//      FF=A*(S/X3-C/X2) + B*(2.*S/X3 - C/X2 +2.0*(C-1.0)/X4) + G*( (4./X -24./X3)*S -(1.0 -12./X2 +24./X4)*C +24./X4 )/X2;
151//      HARDSPH= 1./(1. + 24.*volfraction*FF );
152//printf("HS4 %g: %g\n",q,HARDSPH);
153      return(HARDSPH);
154   """
155
156def random():
157    pars = dict(
158        scale=1, background=0,
160        volfraction=10**np.random.uniform(-2, 0),  # high volume fraction
161    )
162    return pars
163
164# ER defaults to 0.0
165# VR defaults to 1.0
166