Changeset e46dde6 in sasmodels

Timestamp:

Jun 12, 2018 5:39:34 AM (6 years ago)

Author:

Adam Washington <adam.washington@…>

Branches:

master

Children:

02e3c54

Parents:

38935ec

Message:

Start writing up log_sesans theory

File:

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

doc/guide/sesans/sans_to_sesans.rst

@@ -31 +31 @@
 
 in which :math:`t` is the thickness of the sample and :math:`\lambda` is the wavelength of the neutrons.
+
+Log Spaced SESANS
+-----------------
+
+For computational efficiency, the integral in the Hankel transform is
+converted into a Reimann sum
+
+
+.. math:: G(\delta) \approx
+          2 \pi
+          \sum_{Q=q_{min}}^{q_{max}} J_0(Q \delta)
+          \frac{d \Sigma}{d \Omega} (Q)
+          Q \Delta Q \!
+
+However, this model approximates more than is strictly necessary.
+Specifically, it is approximating the entire integral, when it is only
+the scattering function that cannot be handled analytically.  A better
+approximation might be
+
+.. math:: G(\delta) \approx
+          \sum_{n=0} 2 \pi \frac{d \Sigma}{d \Omega} (q_n)
+          \int_{q_n}^{q_{n+1}} J_0(Q \delta) Q dQ
+          =
+          \sum_{n=0} \frac{2 \pi}{\delta} \frac{d \Sigma}{d \Omega} (q_n)
+          (q_{n+1}J_1(q_{n+1} \delta) - q_{n}J_1(q_{n} \delta))\!,
+
+Assume that vectors :math:`q_n` and :math:`I_n` represent the q points
+and corresponding intensity data, respectively.  Further assume that
+:math:`\delta_m` and :math:`G_m` are the spin echo lengths and
+corresponding Hankel transform value.
+
+.. math:: G_m = H_{nm} I_n
+
+where
+
+.. math:: H_{nm} = \frac{2 \pi}{\delta_m}
+          (q_{n+1} J_1(q_{n+1} \delta_m) - q_n J_1(q_n \delta_m))