Began writing documentation for free gas scattering.

docs/source/methods/index.rst

@@ -26,3 +26,93 @@ where :math:`\hat{\mathbf{n}}` is a unit vector normal to the surface at the
 point of the surface crossing. The direction of the surface normal will be the
 gradient to the surface at the point of crossing, i.e. :math:`\mathbf{n} =
 \nabla f(x,y,z)`.
+
+------------------------------
+Free Gas Scattering Kinematics
+------------------------------
+
+When a neutron scatters off of a nucleus, many times it is assumed that the
+target nucleus is at rest. However, if the material is at a temperature greater
+than 0 K, it will have motion associated with the thermal vibration. Thus, the
+velocity of the neutrno relative to the target nucleus is in general not the
+same as the velocity of the neutron entering the collision.
+
+The affect of the thermal motion on the interaction probability can be written
+as
+
+.. math::
+    :label: freegas1
+
+    v_n \sigma (v_n, T) = \int_0^\infty d\mathbf{v}_T \sigma(v_r, 0)
+    \mathbf{v}_r p(\mathbf{v}_T)
+    
+One assumption we can make here is that the velocity distribution for the
+thermal motion is isotropic, i.e.
+
+.. math::
+    :label: freegas2
+
+    p(\mathbf{v}_T) d\mathbf{v}_T = \frac{1}{4\pi} p(v_T) dv_T d\mu d\phi
+
+With this assumption, we can now rewrite equation :eq:`freegas1` as
+
+.. math::
+    :label: freegas3
+
+    v_n \sigma (v_n, T) = \frac{1}{2} \int_{-1}^1 d\mu \int\limits_{v_r > 0}
+    v_r \sigma (v_r, 0) p(v_T) dv_T
+
+To change the outer variable of integration from :math:`\mu` to :math:`v_r`, we
+can establish a relation between these variables based on the law of cosines.
+
+.. math::
+    :label: lawcosine
+
+    2 v_n v_T \mu = v_n^2 + v_T^2 - v_r^2
+
+The probability distribution for the magnitude of the velocity of the target
+nucleus and the angle between the neutron and target velocity is
+
+.. math::
+    :label: freegas4
+
+    P(v_T, \mu) = \frac{\sigma (v_r, 0) v_r P(v_T)}{2 \sigma (v_n, T) v_n}
+
+It is normally assumed that :math:`\sigma (v_r, 0)` is constant over the range
+of relative velocities of interest. This is a good assumption for almost all
+cases since the elastic scattering cross section varies slowly with velocity for
+light nuclei, and for heavy nuclei where large variations can occur due to
+resonance scattering, the moderating effect is rather small. Nonetheless, this
+assumption can cause incorrect answers in systems with U-238 where the low-lying
+resonances can cause a significant amount of upscatter that would be ignored by
+this assumption.
+
+With this (sometimes incorrect) assumption, we see that the probability
+distribution is proportional to
+
+.. math::
+    :label: freegas5
+
+    P(v_T, \mu) \propto v_r P(v_T) = | v_n - v_T | P(v_T)
+
+We can divide this probability distribution into two parts as such:
+
+.. math::
+    :label: freegas6
+
+    P(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\
+    f_1(v_T, \mu) &= \frac{| v_n - v_T |}{\hat{f_1} (v_n + v_T)} \\
+    f_2(v_T) &= (v_n + v_T) P(v_T)
+
+In general, any probability distribution function of the form :math:`p(x) =
+f_1(x) f_2(x)` with :math:`f_1(x)` bounded can be sampled by sampling
+:math:`x_s` from the distribution
+
+.. math:: \frac{f_2(x)}{\int f_2(x) dx}
+
+and accepting it with probability
+
+.. math:: \frac{f_1(x_s)}{\max f_1(x)}
+
+It is normally assumed that the velocity distribution of the target nucleus
+assumes a Maxwellian distribution in velocity.