Changeset f0fc507 in sasmodels for doc/guide/sesans

Timestamp:

Aug 17, 2017 4:59:19 AM (7 years ago)

Author:

GitHub <noreply@…>

Branches:

master, core_shell_microgels, costrafo411, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests

Children:

f8a2baa

Parents:

cdf7dac

git-author:

Adam Washington <rprospero@…> (08/17/17 04:59:19)

git-committer:

GitHub <noreply@…> (08/17/17 04:59:19)

Message:

Clarification of the Hankel transform

I've tried to tidy up the documentation and get it more in line with what's going on with in the actual code.

File:

• doc/guide/sesans/sans_to_sesans.rst (modified) (1 diff)

doc/guide/sesans/sans_to_sesans.rst

@@ -15 +15 @@
 in which :math:`J_0` is the zeroth order Bessel function, :math:`\delta`
 the spin-echo length, :math:`Q` the wave vector transfer and :math:`\frac{d \Sigma}{d \Omega} (Q)`
-the scattering cross section in absolute units. This is a 1-dimensional
-integral, which can be rather fast. In the numerical calculation we integrate
-from :math:`Q_{min} = 0.1 \times 2 \pi / R_{max}` in which :math:`R_{max}`
-will be model dependent. We determined the factor 0.1 by varying its value
-until the value of the integral was stable. This happened at a value of 0.3.
-The have a safety margin of a factor of three we have choosen the value 0.1.
-For the solid sphere we took 3 times the radius for :math:`R_{max}`. The real
-integration is performed to :math:`Q_{max}` which is an instrumental parameter
-that is read in from the measurement file. From the equation above we can
-calculate the polarisation that we measure in a SESANS experiment:
+the scattering cross section in absolute units.
+
+Out of necessity, a 1-dimensional numerical integral approximates the exact Hankel transform.
+The upper bound of the numerical integral is :math:`Q_{max}`, which is calculated from the wavelength and the instrument's maximum acceptance angle, both of which are included in the file.
+While the true Hankel transform has a lower bound of zero, most scattering models are undefined at :math: `Q=0`, so the integral requires an effective lower bound.
+The lower bound of the integral is :math:`Q_{min} = 0.1 \times 2 \pi / R_{max}`, in which :math:`R_{max}` is the maximum length scale probed by the instrument multiplied by the number of data points.
+This lower bound is the minimum expected Q value for the given length scale multiplied by a constant.
+The constant, 0.1, was chosen empirically by integrating multiple curves and finding where the value at which the integral was stable.
+A constant value of 0.3 gave numerical stability to the integral, so a factor of three safety margin was included to give the final value of 0.1.
+
+
+From the equation above we can calculate the polarisation that we measure in a SESANS experiment:
 
 .. math:: P(\delta) = e^{t \left( \frac{ \lambda}{2 \pi} \right)^2 \left(G(\delta) - G(0) \right)} \!,
 
-in which :math:`t` is the thickness of the sample and :math:`\lambda` is
-the wavelength of the neutrons.
+in which :math:`t` is the thickness of the sample and :math:`\lambda` is the wavelength of the neutrons.