point spacing of polydispersity should not always be linear.

[butlerpd]

A NIST user recently noticed that they were getting incorrect answers when using the lognormal distribution with a very large dispersity. After a lot of work to track down they realized it is because all polydispersity distributions use a linear point spacing for the integration when in fact something like lognormal (or the Schulz-Zim) distribution should be using a log spacing of points, particularly with large dispersity values.

It may be that for low values of the dispersity the spacing should still be linear. This needs a bit of investigation. In the meantime the user fixed their problem by altering the sasmodels code. I include the code snippets here as they look good to me and could make for a quick patch?

Below are the distribution points for a lognormal distribution (using the default sphere radius to be polydisperse) for a pd of 2, 1 and 0.5 respectively

PD 2.0

PD 1.0

PD 0.5

CODE

First add a new _logspace method to the Dispersion class in weights.py immediately following the existing _linspace method (and mostly copied form it)

def _logspace(self, center, sigma, lb, ub):
"""helper function to provide log(e) spaced weight points within range"""
npts, nsigmas = self.npts, self.nsigmas
# sigma in the lognormal function is in ln(R) space, thus needs converting
sig=np.fabs(sigma/center)
x = np.logspace(np.log(center)-nsigmas*sig, np.log(center)+nsigmas*sig, npts,base=np.e)
x = x[(x >= lb) & (x <= ub)]
return x

Next, replace the call to _linspace with a call to _logspace in the LogNormalDispersion class. The same should probably be done for the SchulzDispersion class. Note that he also added 4 new lines at the end just before the return to transform x and px which should probably be folded directly into the way x and px are calculated?

class LogNormalDispersion(Dispersion):
r"""
log Gaussian dispersion, with 1-$\sigma$ width.

.. math::

w = \frac{\exp\left(-\tfrac12 (\ln x - c)^2/\sigma^2\right)}{x\sigma}
"""
type = "lognormal"
default = dict(npts=80, width=0, nsigmas=8)
def _weights(self, center, sigma, lb, ub):
x = self._logspace(center, sigma, max(lb, 1e-8), max(ub, 1e-8))
# sigma in the lognormal function is in ln(R) space, thus needs converting
sig = np.fabs(sigma/center)
px = np.exp(-0.5*((np.log(x)-np.log(center))/sig)**2)/(x*sig)
x1 = (x[1:] + x[:-1])*0.5
px1 = 0.5*(px[1:]+px[:-1])*(x[1:]-x[:-1])
norm = sum(px1)
px1 = px1/norm
return x1, px1

[pkienzle]

Which models were they looking at and how did they determine the correct values?

For the polydispersity integral ∫p(r) F²(q;r) dr with lognormal p and sphere F we can plot the integrand p(r) F²(q;r) to see how it behaves. The result suggests we do want linear spacing, and further that we want to restrict PD to be less than about 0.5.

plot ( exp( -((ln(r)-ln(100))/σ)^2/2 )/(σr*sqrt(2π)) (r^2/10 j1(r/10))^2 ) from 1 to 600 for σ = ΔR/R = 0.5, 1.0 and 2.0

With R=100, ΔR/R = 0.5, nsigma = 8, the integral ranges from 0 to 500, which covers most of the function. Unfortunately the integrand is positive everywhere so we will always underestimate the true result.

See also the discussion in #585