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Added description of survival biasing in documentation.

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Paul Romano 2012-07-26 13:08:46 -04:00
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docs/source/methods/physics.rst
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@ -778,6 +778,8 @@ weight of the outgoing neutron is multiplied by the number of secondary
neutrons, e.g. for (n,2n), only one outgoing neutron is tracked but its weight
is doubled.
.. _fission:
-------
Fission
-------
@ -868,7 +870,7 @@ produced. This is done by sampling a random number :math:`\xi` on the interval
:math:`[0,1)` and checking whether
.. math::
:label: absorption-condition
:label: disappearance
\xi \sigma_t (E) < \sigma_a (E) - \sigma_f (E)
@ -882,6 +884,56 @@ alpha-particles are produced or tracked. To truly capture the affects of gamma
heating in a problem, it would be necessary to explicitly track photons
originating from :math:`(n,\gamma)` and other reactions.
----------------
Survival Biasing
----------------
In problems with highly absorbing materials, a large fraction of neutrons may be
killed through absorption reactions thus leading to tallies with very few events
scoring in them. To remedy this situation, an algorithm known as survival
biasing or implicit absorption (or sometimes implicit capture, even though this
is a misnomer) is commonly used.
In survival biasing, rather than ever sample absorption reactions, at every
collision, the weight of neutron is reduced by probability of absorption
occurring, i.e.
.. math::
:label: survival-biasing-weight
w' = w \left ( 1 - \frac{\sigma_a (E)}{\sigma_t (E)} \right )
where :math:`w'` is the weight of the neutron after adjustment and :math:`w` is
the weight of the neutron before adjustment. A few other things need to be
handled differently if survival biasing is turned on. Although fission reactions
never actually occur with survival biasing, we still need to create fission
sites to preserve the basic criticality algorithm. The algorithm for sampling
fission sites is the same as that described in :ref:`fission`. The only
difference is in equation :eq:`fission-neutrons`. We now need to produce
.. math::
:label: fission-neutrons-survival
\nu = \frac{w}{k} \frac{\nu_t \sigma_f(E)}{\sigma_t (E)}
fission sites, where :math:`w` is the weight of the neutron before being
adjusted. One should note this is just the expected number of neutrons produced
*per collision* rather than the expected number of neutrons produced given that
fission has already occurred.
Additionally, since survival biasing can reduce the weight of the neutron to
very low values, it is always used in conjunction with a weight cutoff and
Russian rouletting. Two user adjustable parameters :math:`w_c` and :math:`w_s`
are given which are the weight below which neutrons should undergo Russian
roulette and the weight should they survive Russian roulette. The algorithm for
Russian rouletting is as follows. After a collision if :math:`w < w_c`, then the
neutron is killed with probability :math:`1 - w/w_s`. If it survives, the weight
is set equal to :math:`w_s`. One can confirm that the average weight following
Russian roulette is simply :math:`w`, so the game can be considered "fair". By
default, the cutoff weight in OpenMC is :math:`w_c = 0.25` and the survival
weight is :math:`w_s = 1.0`. These parameters vary from one Monte Carlo code to
another.
.. _freegas:
------------------------------------------