Changeset c9d052d in sasmodels for doc

Timestamp:

Jun 17, 2018 1:45:45 PM (6 years ago)

Author:

awashington

Branches:

master

Children:

298d2d4

Parents:

befe905 (diff), b032200 (diff)
Note: this is a merge changeset, the changes displayed below correspond to the merge itself.
Use the (diff) links above to see all the changes relative to each parent.

Message:

Merge branch 'log_sesans' of github.com:rprospero/sasmodels into log_sesans

Location:

doc/guide

Files:

• sesans/sans_to_sesans.rst (modified) (1 diff)
• magnetism/magnetism.rst (modified) (2 diffs)

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-1}}^{q_n} J_0(Q \delta) Q dQ
+          =
+          \sum_{n=0} \frac{2 \pi}{\delta} \frac{d \Sigma}{d \Omega} (q_n)
+          (q_n J_1(q_n \delta) - q_{n-1}J_1(q_{n-1} \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 J_1(q_n \delta_m) - q_{n-1} J_1(q_{n-1} \delta_m))
+
+Also not that, for the limit as :math:`\delta_m` approaches zero,
+
+.. math:: G(0)
+          =
+          \sum_{n=0} \pi \frac{d \Sigma}{d \Omega} (q_n) (q_n^2 - q_{n-1}^2)

doc/guide/magnetism/magnetism.rst

@@ -39 +39 @@
 
 .. math::
-    -- &= ((1-u_i)(1-u_f))^{1/4} \\
-    -+ &= ((1-u_i)(u_f))^{1/4} \\
-    +- &= ((u_i)(1-u_f))^{1/4} \\
-    ++ &= ((u_i)(u_f))^{1/4}
+    -- &= (1-u_i)(1-u_f) \\
+    -+ &= (1-u_i)(u_f) \\
+    +- &= (u_i)(1-u_f) \\
+    ++ &= (u_i)(u_f)
 
 Ideally the experiment would measure the pure spin states independently and
@@ -104 +104 @@
 | 2015-05-02 Steve King
 | 2017-11-15 Paul Kienzle
+| 2018-06-02 Adam Washington