source: sasmodels/sasmodels/models/hardsphere.py @ e98c1e0

# Note: model title and parameter table are inserted automatically
r"""Calculate the interparticle structure factor for monodisperse
spherical particles interacting through hard sphere (excluded volume)
interactions.
May be a reasonable approximation for other shapes of particles that 
freely rotate, and for moderately polydisperse systems. Though strictly 
the maths needs to be modified (no \Beta(Q) correction yet in sasview).

radius_effective is the effective hard sphere radius.
volfraction is the volume fraction occupied by the spheres.

In sasview the effective radius may be calculated from the parameters
used in the form factor $P(q)$ that this $S(q)$ is combined with.

For numerical stability the computation uses a Taylor series expansion 
at very small $qR$, there may be a very minor glitch at the transition point
in some circumstances.

The S(Q) uses the Percus-Yevick closure where the interparticle
potential is

.. math::

    U(r) = \begin{cases}
    \infty & r < 2R \\
    0 & r \geq 2R
    \end{cases}

where $r$ is the distance from the center of the sphere of a radius $R$.

For a 2D plot, the wave transfer is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}


References
----------

J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1
"""

from numpy import inf

name = "hardsphere"
title = "Hard sphere structure factor, with Percus-Yevick closure"
description = """\
    [Hard sphere structure factor, with Percus-Yevick closure]
        Interparticle S(Q) for random, non-interacting spheres.
    May be a reasonable approximation for other shapes of
    particles that freely rotate, and for moderately polydisperse
        systems. Though strictly the maths needs to be modified -
    which sasview does not do yet.
    radius_effective is the hard sphere radius
    volfraction is the volume fraction occupied by the spheres.
"""
category = "structure-factor"
structure_factor = True
single = False

#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["radius_effective", "Ang", 50.0, [0, inf], "volume",
               "effective radius of hard sphere"],
              ["volfraction", "", 0.2, [0, 0.74], "",
               "volume fraction of hard spheres"],
             ]

# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
form_volume = """
    return 1.0;
    """

Iq = """
      double D,A,B,G,X,X2,X4,S,C,FF,HARDSPH;

      if(fabs(radius_effective) < 1.E-12) {
               HARDSPH=1.0;
               return(HARDSPH);
      }
      // removing use of pow(xxx,2) and rearranging the calcs of A, B & G cut ~40% off execution time ( 0.5 to 0.3 msec)
      X = 1.0/( 1.0 -volfraction);
      D= X*X;
      A= (1.+2.*volfraction)*D;
      A *=A;
      X=fabs(q*radius_effective*2.0);

      if(X < 5.E-06) {
                 HARDSPH=1./A;
                 return(HARDSPH);
      }
      X2 =X*X;
      B = (1.0 +0.5*volfraction)*D;
      B *= B;
      B *= -6.*volfraction;
      G=0.5*volfraction*A;

      if(X < 0.2) {
      // RKH Feb 2016, use Taylor series expansion for small X, IT IS VERY PICKY ABOUT THE X CUT OFF VALUE, ought to be lower in double. 
      // else no obvious way to rearrange the equations to avoid needing a very high number of significant figures. 
      // Series expansion found using Mathematica software. Numerical test in .xls showed terms to X^2 are sufficient 
      // for 5 or 6 significant figures, but I put the X^4 one in anyway 
            //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;
            // refactoring the polynomial makes it very slightly faster (0.5 not 0.6 msec)
            //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;

            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;

            // combining the terms makes things worse at smallest Q in single precision
            //FF = (8-0.8*X2)*A +(3.0-X2/3.)*2*B + (4+0.5*X2)*G +(A/35. +B/40. +G/50.)*X4;
            // note that G = -volfraction*A/2, combining this makes no further difference at smallest Q
            //FF = (8 +2.*volfraction + ( volfraction/4. -0.8 +(volfraction/100. -1./35.)*X2 )*X2 )*A  + (3.0 -X2/3. +X4/40.)*2.*B;
            HARDSPH= 1./(1. + volfraction*FF );
            return(HARDSPH);
      }
      X4=X2*X2;
      SINCOS(X,S,C);

// RKH Feb 2016, use version FISH code as is better than original sasview one at small Q in single precision, and more than twice as fast in double.
      //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;
      // refactoring the polynomial here & above makes it slightly faster

      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 ;
      HARDSPH= 1./(1. + 24.*volfraction*FF/X2 );

      // changing /X and /X2 to *MX1 and *MX2, no significantg difference?
      //MX=1.0/X;
      //MX2=MX*MX;
      //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)) ;
      //HARDSPH= 1./(1. + 24.*volfraction*FF*MX2*MX );

// grouping the terms, was about same as sasmodels for single precision issues
//     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 );
//     HARDSPH= 1./(1. + 24.*volfraction*FF/X2 );
// remove 1/X2 from final line, take more powers of X inside the brackets, stil bad
//      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;
//      HARDSPH= 1./(1. + 24.*volfraction*FF );
      return(HARDSPH);
   """

Iqxy = """
    // never called since no orientation or magnetic parameters.
    return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS);
    """

# ER defaults to 0.0
# VR defaults to 1.0

demo = dict(radius_effective=200, volfraction=0.2, radius_effective_pd=0.1, radius_effective_pd_n=40)
oldname = 'HardsphereStructure'
oldpars = dict(radius_effective="effect_radius",radius_effective_pd="effect_radius_pd",radius_effective_pd_n="effect_radius_pd_n")
# Q=0.001 is in the Taylor series, low Q part, so add Q=0.1, assuming double precision sasview is correct
tests = [
        [ {'scale': 1.0, 'background' : 0.0, 'radius_effective' : 50.0, 'volfraction' : 0.2,
           'radius_effective_pd' : 0}, [0.001,0.1], [0.209128,0.930587]]
        ]
# ADDED by: RKH  ON: 16Mar2016  using equations from FISH as better than orig sasview, see notes above. Added Taylor expansions at small Q,