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Added description of ACE Law 4 in documentation.

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docs/source/methods/physics.rst
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@ -44,8 +44,8 @@ of the scattering angle is simply calculated as
where :math:`\xi` is a random number sampled uniformly on :math:`[0,1)`.
Equiprobable Bin Distribution
+++++++++++++++++++++++++++++
Equiprobable Angle Bin Distribution
+++++++++++++++++++++++++++++++++++
For a 32 equiprobable bin distribution, the procedure to determine the
scattering cosine is as follows. First, we select a random number :math:`\xi` to
@ -64,8 +64,10 @@ The same random number can then also be used to interpolate between neighboring
\mu = \mu_i + (32\xi - i) (\mu_{i+1} - \mu_i)
Tabular Distribution
++++++++++++++++++++
.. _angle-tabular:
Tabular Angular Distribution
++++++++++++++++++++++++++++
As the MCNP Manual points out, using an equiprobable bin distribution works well
for high-probability regions of the scattering cosine probability, but for
@ -74,7 +76,7 @@ is to represent the scattering cosine with a tabular distribution. In this case,
we have a table of cosines and their corresponding values for a probability
distributions function and cumulative distribution function. For each incoming
neutron energy :math:`E_i`, let us call :math:`p_{i,j}` the j-th value in the
probability distribution function and :math:`c_i,j` the j-th value in the
probability distribution function and :math:`c_{i,j}` the j-th value in the
cumulative distribution function. We first find the interpolation factor on the
incoming energy grid:
@ -83,17 +85,18 @@ incoming energy grid:
f = \frac{E - E_i}{E_{i+1} - E_i}
Then, statistical interpolation is performed to choose between using the cosines
and distribution functions corresponding to energy :math:`E_i` or
:math:`E_{i+1}`. Let :math:`\ell` be the chosen table where :math:`\ell = i` if
:math:`\xi > f` and :math:`\ell = i + 1` otherwise where :math:`\xi` is a random
number. A different random number is used to sample a scattering cosine bin
:math:`j` using the cumulative distribution function:
where :math:`E` is the incoming energy of the particle. Then, statistical
interpolation is performed to choose between using the cosines and distribution
functions corresponding to energy :math:`E_i` and :math:`E_{i+1}`. Let
:math:`\ell` be the chosen table where :math:`\ell = i` if :math:`\xi > f` and
:math:`\ell = i + 1` otherwise where :math:`\xi` is a random number. A different
random number is used to sample a scattering cosine bin :math:`j` using the
cumulative distribution function:
.. math::
:label: sample-cdf
c_{l,j} < \xi < c_{l,j+1}
c_{\ell,j} < \xi < c_{\ell,j+1}
The final scattering cosine will depend on whether histogram or linear-linear
interpolation is used. In general, we can write the cumulative distribution
@ -105,21 +108,22 @@ function as
c(\mu) = \int_{-1}^\mu p(\mu') d\mu'
where :math:`c(\mu)` is the cumulative distribution function and :math:`p(\mu)`
is the probability distribution function. Since we know that :math:`c(\mu_{l,k})
= c_{l,k}`, this implies that for :math:`\mu > \mu_{l,k}`,
is the probability distribution function. Since we know that
:math:`c(\mu_{\ell,j}) = c_{\ell,j}`, this implies that for :math:`\mu >
\mu_{\ell,j}`,
.. math::
:label: cdf-2
c(\mu) = c_{l,k} + \int_{\mu_{l,k}}^{\mu} p(\mu') d\mu'
c(\mu) = c_{\ell,j} + \int_{\mu_{\ell,j}}^{\mu} p(\mu') d\mu'
For histogram interpolation, we have that :math:`p(\mu') = p_{l,k}`. Thus,
For histogram interpolation, we have that :math:`p(\mu') = p_{\ell,j}`. Thus,
after integration we have that
.. math::
:label: cumulative-dist-histogram
c(\mu) = c_{l,k} + (\mu - \mu_{l,k}) p_{l,k} = \xi
c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi
Solving for the scattering cosine, we obtain the final form for histogram
interpolation:
@ -127,7 +131,7 @@ interpolation:
.. math::
:label: cosine-histogram
\mu = \mu_{l,k} + \frac{\xi - c_{l,k}}{p_{l,k}}
\mu = \mu_{\ell,j} + \frac{\xi - c_{\ell,j}}{p_{\ell,j}}
For linear-linear interpolation, we represent the function :math:`p(\mu')` as a
first-order polynomial in :math:`\mu'`. If we interpolate between successive
@ -136,8 +140,8 @@ values on the probability distribution function, we know that
.. math::
:label: pdf-interpolation
p(\mu') - p_{l,k} = \frac{p_{l,k+1} - p_{l,k}}{\mu_{l,k+1} - \mu_{l,k}}
(\mu' - \mu_{l,k})
p(\mu') - p_{\ell,j} = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} -
\mu_{\ell,j}} (\mu' - \mu_{\ell,j})
Solving for :math:`p(\mu')` in equation :eq:`pdf-interpolation` and inserting it
into equation :eq:`cdf-2`, we obtain
@ -145,40 +149,40 @@ into equation :eq:`cdf-2`, we obtain
.. math::
:label: cdf-linlin
c(\mu) = c_{l,k} + \int_{\mu_{l,k}}^{\mu} \left [ \frac{p_{l,k+1} -
p_{l,k}}{\mu_{l,k+1} - \mu_{l,k}} (\mu' - \mu_{l,k}) + p_{l,k} \right ]
d\mu'
c(\mu) = c_{\ell,j} + \int_{\mu_{\ell,j}}^{\mu} \left [ \frac{p_{\ell,j+1} -
p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}} (\mu' - \mu_{\ell,j}) +
p_{\ell,j} \right ] d\mu'
Let us now make a change of variables using
.. math::
:label: introduce-eta
\eta = \frac{p_{l,k+1} - p_{l,k}}{\mu_{l,k+1} - \mu_{l,k}}
(\mu' - \mu_{l,k})
\eta = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}}
(\mu' - \mu_{\ell,j})
Equation :eq:`cdf-linlin` then becomes
.. math::
:label: cdf-linlin-eta
c(\mu) = c_{l,k} + \frac{1}{m} \int_{p_{l,k}}^{m(\mu - \mu_{l,k}) + p_{l,k}}
\eta \, d\eta
c(\mu) = c_{\ell,j} + \frac{1}{m} \int_{p_{\ell,j}}^{m(\mu - \mu_{\ell,j}) +
p_{\ell,j}} \eta \, d\eta
where we have used
.. math::
:label: slope
m = \frac{p_{l,k+1} - p_{l,k}}{\mu_{l,k+1} - \mu_{l,k}}
m = \frac{p_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}}
Integrating equation :eq:`cdf-linlin-eta`, we have
.. math::
:label: cdf-linlin-integrated
c(\mu) = c_{l,k} + \frac{1}{2m} \left ( \left [ m (\mu - \mu_{l,k} ) +
p_{l,k} \right ]^2 - p_{l,k}^2 \right ) = \xi
c(\mu) = c_{\ell,j} + \frac{1}{2m} \left ( \left [ m (\mu - \mu_{\ell,j} ) +
p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi
Solving for :math:`\mu`, we have the final form for the scattering cosine using
linear-linear interpolation:
@ -186,8 +190,8 @@ linear-linear interpolation:
.. math::
:label: cosine-linlin
\mu = \mu_{l,k} + \frac{1}{m} \left ( \sqrt{p_{l,k}^2 + 2 m (\xi - c_{l,k}
)} - p_{l,k} \right )
\mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi -
c_{\ell,j} )} - p_{\ell,j} \right )
.. _sample-energy:
@ -217,6 +221,8 @@ Once a secondary energy distribution has been sampled, the procedure for
determining the outgoing energy will depend on which ACE law has been specified
for the data.
.. _ace-law-1:
ACE Law 1 - Tabular Equiprobable Energy Bins
++++++++++++++++++++++++++++++++++++++++++++
@ -248,7 +254,7 @@ energies of a scaled distribution, :math:`E_{i,j}` is the j-th outgoing energy
corresponding to the incoming energy :math:`E_i`, and :math:`M` is the number of
outgoing energy bins. Next, statistical interpolation is performed to choose
between using the outgoing energy distributions corresponding to energy
:math:`E_i` or :math:`E_{i+1}`. Let :math:`\ell` be the chosen table where
:math:`E_i` and :math:`E_{i+1}`. Let :math:`\ell` be the chosen table where
:math:`\ell = i` if :math:`\xi_1 > f` and :math:`\ell = i + 1` otherwise where
:math:`\xi_1` is a random number. Now, we randomly sample an equiprobable
outgoing energy bin :math:`j` and interpolate between successive values on the
@ -287,6 +293,65 @@ where :math:`A` is the mass of the target nucleus measured in neutron masses.
ACE Law 4 - Continuous Tabular Distribution
+++++++++++++++++++++++++++++++++++++++++++
This representation is very similar to :ref:`ace-law-1` except that instead of
equiprobable outgoing energy bins, the outgoing energy distribution for each
incoming energy is represented with a probability distribution function. For
each incoming neutron energy :math:`E_i`, let us call :math:`p_{i,j}` the j-th
value in the probability distribution function, :math:`c_{i,j}` the j-th value
in the cumulative distribution function, and :math:`E_{i,j}` the j-th outgoing
energy.
Weproceed first as we did for ACE Law 1, determining the bounding energies of
the particle's incoming energy such that :math:`E_i < E < E_{i+1}` and
calculating an interpolationg factor :math:`f` with equation
:eq:`interpolation-factor`. Next, statistical interpolation is performed to
choose between using the outgoing energy distributions corresponding to energy
:math:`E_i` and :math:`E_{i+1}`. Let :math:`\ell` be the chosen table where
:math:`\ell = i` if :math:`\xi_1 > f` and :math:`\ell = i + 1` otherwise where
:math:`\xi_1` is a random number. Then, we sample an outgoing energy bin
:math:`j` using the cumulative distribution function:
.. math::
:label: ace-law-4-sample-cdf
c_{\ell,j} < \xi_2 < c_{\ell,j+1}
where :math:`\xi_2` is a random number sampled uniformly on :math:`[0,1)`. At
this point, we need to inteporlate between the successive values on the outgoing
energy distribution using either histogram or linear-linear interpolation. The
formulas for these can be derived along the same lines as those found in
:ref:`angle-tabular`. For histogram interpolation, the interpolated outgoing
energy on the :math:`\ell`-th distribution is
.. math::
:label: energy-histogram
\hat{E} = E_{\ell,j} + \frac{\xi_2 - c_{\ell,j}}{p_{\ell,j}}
If linear-linear interpolation is to be used, the outgoing energy on the
:math:`\ell`-th distribution is
.. math::
:label: energy-linlin
\hat{E} = E_{\ell,j} + \frac{E_{\ell,j+1} - E_{\ell,j}}{p_{\ell,j+1} -
p_{\ell,j}} \left ( \sqrt{p_{\ell,j}^2 + 2 \frac{p_{\ell,j+1} -
p_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} ( \xi_2 - c_{\ell,j} )} - p_{\ell,j}
\right )
Since this outgoing energy may violate reaction kinematics, we then scale it to
minimum and maximum energies interpolated between the neighboring outgoing
energy distributions to get the final outgoing energy:
.. math::
:label: ace-law-4-energy
E' = E_{min} + \frac{\hat{E} - E_{\ell,1}}{E_{\ell,M} - E_{\ell,1}}
(E_{max} - E_{min})
where :math:`E_{min}` and :math:`E_{max}` are defined the same as in equation
:eq:`ace-law-1-minmax`.
ACE Law 7 - Maxwell Fission Spectrum
++++++++++++++++++++++++++++++++++++