SESANS q min and q max shouldn't depend on the density of correlation points

[pkienzle]

Looking at the calculation of sesans $G(ξ) = \frac{1}{2\pi} \int I(q) J_0(qξ) q dq$, the value of G is going to be driven primarily by the rapid decrease in I(q), with higher q values contributing minimally to the integral.

The proposed qmin, qmax range for single point ξ in #563 will not work. Instead q min should be based on the optional parameter Rmax, which defaults to ξ max if it is not provided. q max can be set to a number of decades above q min, assuming scattering intensity follows a power law, or perhaps from an additional parameter Rmin whose default value comes from ξ min, with a minimal number of decades in q max/q min.

In principle the calculation should be independent of the particular ξ points, with a single point calculation giving the same result as that point in a vector of ξ values. That is, we should only use Rmin and Rmax to determine q, not the ξ density.

Regarding structure factors, there will be correlations beyond maximum particle dimension but these show up as structure factor peaks scaling the I(q) values. Again, since high q values contribute minimally to the integral, we may not need to increase Rmax to capture the correlation structure in ξ above Rmax. This is another reason the Rmax parameter should not be ignored.

[pkienzle]

To evaluate the Hankel transform we are using $$G(ξ) = \int_0^\infty I(q) J_0(qξ)q dq = \int_0^{q_{min}} I(q) J_0(qξ)q dq + \int_{q_{min}}^{q_{max}} I(q) J_0(qξ)q dq + \int_{q_{max}}^\infty I(q) J_0(qξ)q dq = ε_{min} + ε_{mid} + ε_{max}$$ where we approximate
$$ε_{mid} ≈ \sum I(q_k) J_0(q_k ξ) q_k$$

The goal is to find $q_{min}$ and $q_{max}$ so that we achieve a desired relative error $ε_{max}/G(ξ) < Δ$ for some small constant $Δ$. What follows are some musings on the topic.

Consider a power law $I(q) \propto q^{-4}$ as the limiting case for $I(q)$. Since $ε_{min}$ diverges, we need to introduce a correlation length cutoff $R_{max}$ where $I(q) = q_{min}^{-4}$ is constant for $q < q_{min}$. The cutoff will be determined by the coherence length of the neutron, and may be above the largest correlation length probed in the measurement. Analytically this gives $$ε_{min}(ξ) = I(q_{min}) J_1(q_{min} ξ)q_{min}/ξ $$

For the $ε_{max}$ term we can use the approximation $$J_0(z) ≈ \sqrt\frac{2}{πz}\cos(z-π/4)$$ for $z = qξ > 1$. Ignoring the oscillation in the cosine we can get a weak upper bound $$ε_{max} < \int_{q_{max}}^\infty I(q)\sqrt\frac{2}{πqξ} q dq = \sqrt\frac{2}{πξ} \frac{q_{max}^{-2.5}}{2.5}$$
For $q_{max} = 10$ / nm and $ξ_{min} = 1$ nm this gives $ε_{max} ≈ 0.0010$, whereas the value from the Hankel integral ≈-0.00012.

Even if we had a good bound for $ε_{max}$ we can't use it to estimate error since we don't have a bound on $ε_{mid}$. Let's assume that oscillations in $I(q)$ are small relative to $I(q)$ and we might be able to get away with using the envelope function for $J_0$, which gives $$ε_{mid}+ε_{max} \sim \int_{q_{min}}^\infty I(q)\sqrt\frac{2}{πqξ} q dq = \sqrt\frac{2}{πξ} \frac{q_{min}^{-2.5}}{2.5}$$
The hope is that the overestimate in this term will cancel the overestimate in the $ε_{max}$ term. Then we can set $ε_{max} / (ε_{mid} + ε_{max}) < Δ$. Solving this gives $$q_{max} = q_{min} Δ^{-1/2.5}$$ For $Δ=10^{-5}$ this gives $q_{max} = 100 q_{min}$. Testing for a variety $q_{min}$ and $ξ$ in the SANS/USANS range using the Hankel integral rather than the approximation the relative error is sometimes 10x too high.

This scheme only works when $q_{min}$ is at the start of the power law region. For the sphere model this is at $q_{min} = 1/R$. Given an $R_{max}$ parameter, then, we would need $q_{min} = 0.1 / R_{max}$ and $q_{max} = 100 / R_{max}$ for better than 0.1% accuracy. Checking $\int_0^y J_0(ξ q) (3 j_1(q R)/(q R))^2 q dq$ for $y = 10^n/R$ the range from $n=-1$ to $n=2$ gives good accuracy. I don't expect structure factor will have a big influence: the $J_0$ is oscillating so rapidly that the integral will be dominated by the $J_0$ envelope, especially at high $ξ$. Lower values of $ξ$ require a larger $q$ range to achieve the same accuracy.

Our $q_k$ are sampled far to sparsely to hit every oscillation in $J_0$. We should compute a Hankel integrals to high precision for a few select (model, ξ) sets to verify that our sampling is high enough.