A new, alternate test distribution, REVERSE, has been created by P.R. Jemian to test several key questions:
#1. How close can the program get to a known volume fraction?
Note that there is no specification for the exact answer of total volume fraction for BIMODAL.SAS, only a normalized distribution [Culverwell, 1986].
#2. Does MaxSas handle data in the size range of a double-crystal intrument? #3. Do the solved distributions always look the same?
The distribution is (once again) two Gaussians in f(D) space where the Gaussian at lower diameter (1100 A, sigma = 300) is 25% the height of the other Gaussian (3400 A, sigma = 680). REVERSE.DIS is the starting distribution, from which is calculated the scattering (REVERSE.SAS) using the exact form factor for spheres. An artificial volume fraction of 1.5%, artificial scattering contrast of 10.0E28 1/m**4, an artificial background of 5.0 1/cm, and artificial random noise of 4% were added to the data. A summary of the analysis of the REVERSE.SAS dataset follows:
| term | analysis | actual |
|---|---|---|
| qMin | 0.0005 | — |
| qMax | 0.025 | — |
| NumPts | 24 | — |
| ChiSquared | 23.972 | — |
| Dmin | 80 | — |
| Dmax | 8000 | — |
| NumBins | 100 | — |
| flat entropy | 4.605 | — |
| entropy | 3.925 | — |
| Total vol. frac. | 1.436% | 1.5% |
| suggested background | 4.78 | 5.0 |
| vol-mean diameter | 3231 | 3172 |
| number-mean diameter | 1181 | 905 |
| error scaling factor | 1.0 | 1.0 |
It appears that the spheres model can deliver the character of the correct distribution and volume fraction.
While the oscillations in the distribution suggest that there is statistical evidence for such irregular features, these cannot be believed as we know, a priori, the starting distribution and that distribution is smooth. We must conclude therefore that the entropy is not adequately maximized, subject to the constraint that ChiSquared equals the number of intensity points. While decreasing the maximum value allowed for TEST (currently set at 0.05) might seem to produce a better alignment between the entropy and ChiSquared gradients, a value as low as 0.0001 does not seem to alter the final entropy more than about 0.5%. A discussion with G.J. Daniell might bring us to resolve this point. Probably the oscillations have an origin in the introduction of the baseline “b” into the definition of the entropy as done by [Skilling, 1984]. This simplifies the math when calculating the entropy gradients but that probably makes the algorithm of [Skilling, 1984] very sensitive to gradients in the form factor.
One method to circumvent this unsightly “noise” in the solved distributions has been to replace the form factors that are defined with trig terms by ones defined by algebra. These approximations are only as good as the algebraic form can model the scattering and can render truly fictional volume fractions in the worst possible cases.
To answer, then, the three questions above, the volume fraction of the solution was very close to the actual volume fraction. The mean diameter was also very close, with the volume-weighted mean being the closest. The solved distribution was very close to the input distribution which differed dramatically in shape to the distribution of BIMODAL.SAS, hence the solved distributions do not always look alike. The range of diameters in the distribution for REVERSE.SAS was in the range of the double-crystal instrument and so that question can be answered affirmatively. The answers are also believable and so MaxSas is not limited by the experimental range of a particular type of scattering camera.