Added discussion of analog, collision, and track-length estimators for tallies in documentation.

docs/source/methods/tallies.rst

@@ -4,14 +4,156 @@
 Tallies
 =======
 
+------------------
+Filters and Scores
+------------------
+
+------------------------------
+Using Maps for Filter-Matching
+------------------------------
+
+
+-----------------------------------------
+Volume-Integrated Flux and Reaction Rates
+-----------------------------------------
+
+One quantity we may wish to compute during the course of a Monte Carlo
+simulation is the flux or a reaction rate integrated over a finite volume. The
+volume may be a particular cell, a collection of cells, or the entire
+geometry. There are various methods by which we can estimate reaction rates
+
 ----------------
 Analog Estimator
 ----------------
 
+The analog estimator is the simplest type of estimator for reaction rates. The
+basic idea is that we simply count the number of actual reactions that take
+place and use that as our estimate for the reaction rate. This can be written
+mathematically as
+
+.. math::
+    :label: analog-estimator
+
+    R_x = \frac{1}{W} \sum_{i \in A} w_i
+
+where :math:`R_x` is the reaction rate for reaction :math:`x`, :math:`i` denotes
+an index for each event, :math:`A` is the set of all events resulting in
+reaction :math:`x`, and :math:`W` is the total starting weight of the particles,
+and :math:`w_i` is the pre-collision weight of the particle as it enters event
+:math:`i`. One should note that equation :eq:`analog-estimator` is
+volume-integrated so if we want a volume-averaged quantity, we need to divided
+by the volume of the region of integration.
+
+-------------------
+Collision Estimator
+-------------------
+
+While the analog estimator is conceptually very simple and easy to implement, it
+can suffer higher variance due to the fact low probability events will not occur
+often enough to get good statistics if they are being tallied. Thus, it is
+desirable to use a different estimator that allows us to score to the tally more
+often. One such estimator is the collision estimator. Instead of tallying a
+reaction only when it happens, the idea is to make a contribution to the tally
+at every collision.
+
+We can start by writing a formula for the collision estimate of the flux. Since
+:math:`R = \Sigma_t \phi` where :math:`R` is the total reaction rate,
+:math:`\Sigma_t` is the total macroscopic cross section, and :math:`\phi` is the
+scalar flux, it stands to reason that we can estimate the flux by taking an
+estimate of the total reaction rate and dividing it by the total macroscopic
+cross section. This gives us the following formula:
+
+.. math::
+    :label: collision-estimator-flux
+
+    \phi = \frac{1}{W} \sum_{i \in C} \frac{w_i}{\Sigma_t (E_i)}
+
+where :math:`W` is again the total starting weight of the particles, :math:`C`
+is the set of all events resulting in a collision with a nucleus, and
+:math:`\Sigma_t (E)` is the total macroscopic cross section of the target
+material at the incoming energy of the particle :math:`E_i`.
+
+If we multiply both sides of equation :eq:`collision-estimator-flux` by the
+macroscopic cross section for some reaction :math:`x`, then we get the collision
+estimate for the reaction rate for that reaction:
+
+.. math::
+    :label: collision-estimator
+
+    R_x = \frac{1}{W} \sum_{i \in C} \frac{w_i \Sigma_x (E_i)}{\Sigma_t (E_i)}
+
+where :math:`\Sigma_x (E_i)` is the macroscopic cross section for reaction
+:math:`x` at the incoming energy of the particle :math:`E_i`. In comparison to
+equation :eq:`analog-estimator`, we see that the collision estimate will result
+in a tally with a larger number of events that score to it with smaller
+contributions (since we have multiplied it by :math:`\Sigma_x / \Sigma_t`).
+
 ----------------------
 Track-length Estimator
 ----------------------
 
+One other method we can use to increase the number of events that scores to
+tallies is to use an estimator the scores contributions to a tally at every
+track for the particle rather than every collision. This is known as a
+track-length estimator, sometimes also called a path-length estimator. We first
+start with an expression for the volume integrated flux, which can be written as
+
+.. math::
+    :label: flux-integrated
+
+    V \phi = \int d\mathbf{r} \int dE \int d\mathbf{\Omega} \int dt \,
+    \psi(\mathbf{r}, \mathbf{\hat{\Omega}}, E, t).
+
+where :math:`V` is the volume, :math:`\psi` is the angular flux,
+:math:`\mathbf{r}` is the position of the particle, :math:`\mathbf{\hat{\Omega}}`
+is the direction of the particle, :math:`E` is the energy of the particle, and
+:math:`t` is the time. By noting that :math:`\psi(\mathbf{r},
+\mathbf{\hat{\Omega}}, E, t) = v n(\mathbf{r}, \mathbf{\hat{\Omega}}, E, t)`
+where :math:`n` is the angular neutron density, we can rewrite equation
+:eq:`flux-integrated` as
+
+.. math::
+    :label: flux-integrated-2
+
+    V \phi = \int d\mathbf{r} \int dE \int dt v \int d\mathbf{\Omega} \, n(\mathbf{r},
+    \mathbf{\hat{\Omega}}, E, t))
+
+Using the relations :math:`N(\mathbf{r}, E, t) = \int d\mathbf{\Omega}
+n(\mathbf{r}, \mathbf{\hat{\Omega}}, E, t)` and :math:`d\ell = v \, dt` where
+:math:`d\ell` is the differential unit of track length, we then obtain
+
+.. math::
+    :label: track-length-integral
+
+    V \phi = \int d\mathbf{r} \int dE \int d\ell N(\mathbf{r}, E, t)
+
+Equation :eq:`track-length-integral` indicates that we can use the length of a
+particle's trajectory as an estimate for the flux, i.e. the track-length
+estimator of the flux would be
+
+.. math::
+    :label: track-length-flux
+
+    \phi = \frac{1}{W} \sum_{i \in T} w_i \ell_i
+
+where :math:`T` is the set of all the particle's trajectories within the desired
+volume and :math:`\ell_i` is the length of the :math:`i`-th trajectory. In the
+same vein as equation :eq:`collision-estimator`, the track-length estimate of a
+reaction rate is found by multiplying equation :eq:`track-length-flux` by a
+macroscopic reaction cross section:
+
+.. math::
+    :label: track-length-estimator
+
+    R_x = \frac{1}{W} \sum_{i \in T} w_i \ell_i \Sigma_x (E_i)
+
+One important fact to take into consideration is that the use of a track-length
+estimator precludes us from using any filter that requires knowledge of the
+particle's state following a collision because by definition, it will not have
+had a collision at every event. Thus, for tallies with outgoing-energy filters
+(which require the post-collision energy) or for tallies of scattering moments
+(which require the scattering cosine), we must use an analog estimator.
+
 ---------------
 Surface Current
 ---------------