OpenMC/docs/source/methods/neutron_physics.rst

.. _methods_neutron_physics:

===============
Neutron Physics
===============

There are limited differences between physics treatments used in the
continuous-energy and multi-group modes.  If distinctions are necessary, each
of the following sections will provide an explanation of the differences.
Otherwise, replacing any references of the particle's energy (`E`) with
references to the particle's energy group (`g`) will suffice.

-----------------------------------
Sampling Distance to Next Collision
-----------------------------------

As a particle travels through a homogeneous material, the probability
distribution function for the distance to its next collision :math:`\ell` is

.. math::
    :label: distance-pdf

    p(\ell) d\ell = \Sigma_t e^{-\Sigma_t \ell} d\ell

where :math:`\Sigma_t` is the total macroscopic cross section of the
material. Equation :eq:`distance-pdf` tells us that the further the distance is
to the next collision, the less likely the particle will travel that
distance. In order to sample the probability distribution function, we first
need to convert it to a cumulative distribution function

.. math::
    :label: distance-cdf

    \int_0^{\ell} d\ell' p(\ell') = \int_0^{\ell} d\ell' \Sigma_t e^{-\Sigma_t
    \ell'} = 1 - e^{-\Sigma_t \ell}.

By setting the cumulative distribution function equal to :math:`\xi`, a random
number on the unit interval, and solving for the distance :math:`\ell`, we
obtain a formula for sampling the distance to next collision:

.. math::
    :label: sample-distance-1

    \ell = -\frac{\ln (1 - \xi)}{\Sigma_t}.

Since :math:`\xi` is uniformly distributed on :math:`[0,1)`, this implies that
:math:`1 - \xi` is also uniformly distributed on :math:`[0,1)` as well. Thus,
the formula usually used to calculate the distance to next collision is

.. math::
    :label: sample-distance-2

    \ell = -\frac{\ln \xi}{\Sigma_t}

----------------------------------------------------
:math:`(n,\gamma)` and Other Disappearance Reactions
----------------------------------------------------

All absorption reactions other than fission do not produce any secondary
neutrons. As a result, these are the easiest type of reactions to handle. When a
collision occurs, the first step is to sample a nuclide within a material. Once
the nuclide has been sampled, then a specific reaction for that nuclide is
sampled. Since the total absorption cross section is pre-calculated at the
beginning of a simulation, the first step in sampling a reaction is to determine
whether a "disappearance" reaction occurs where no secondary neutrons are
produced. This is done by sampling a random number :math:`\xi` on the interval
:math:`[0,1)` and checking whether

.. math::
    :label: disappearance

    \xi \sigma_t (E) < \sigma_a (E) - \sigma_f (E)

where :math:`\sigma_t` is the total cross section, :math:`\sigma_a` is the
absorption cross section (this includes fission), and :math:`\sigma_f` is the
total fission cross section. If this condition is met, then the neutron is
killed and we proceed to simulate the next neutron from the source bank.

Note that photons arising from :math:`(n,\gamma)` and other neutron reactions
are not produced in a microscopically correct manner. Instead, photons are
sampled probabilistically at each neutron collision, regardless of what reaction
actually takes place. This is described in more detail in
:ref:`photon_production`.

------------------
Elastic Scattering
------------------

Note that the multi-group mode makes no distinction between elastic or
inelastic scattering reactions. The specific multi-group scattering
implementation is discussed in the :ref:`multi-group-scatter` section.

Elastic scattering refers to the process by which a neutron scatters off a
nucleus and does not leave it in an excited state. It is referred to as "elastic"
because in the center-of-mass system, the neutron does not actually lose
energy. However, in lab coordinates, the neutron does indeed lose
energy. Elastic scattering can be treated exactly in a Monte Carlo code thanks
to its simplicity.

Let us discuss how OpenMC handles two-body elastic scattering kinematics. The
first step is to determine whether the target nucleus has any associated
motion. Above a certain energy threshold (400 kT by default), all scattering is
assumed to take place with the target at rest. Below this threshold though, we
must account for the thermal motion of the target nucleus. Methods to sample the
velocity of the target nucleus are described later in section
:ref:`freegas`. For the time being, let us assume that we have sampled the
target velocity :math:`\mathbf{v}_t`. The velocity of the center-of-mass system
is calculated as

.. math::
    :label: velocity-com

    \mathbf{v}_{cm} = \frac{\mathbf{v}_n + A \mathbf{v}_t}{A + 1}

where :math:`\mathbf{v}_n` is the velocity of the neutron and :math:`A` is the
atomic mass of the target nucleus measured in neutron masses (commonly referred
to as the *atomic weight ratio*). With the velocity of the center-of-mass
calculated, we can then determine the neutron's velocity in the center-of-mass
system:

.. math::
    :label: velocity-neutron-com

    \mathbf{V}_n = \mathbf{v}_n - \mathbf{v}_{cm}

where we have used uppercase :math:`\mathbf{V}` to denote the center-of-mass
system. The direction of the neutron in the center-of-mass system is

.. math::
    :label: angle-neutron-com

    \mathbf{\Omega}_n = \frac{\mathbf{V}_n}{|| \mathbf{V}_n ||}.

At low energies, elastic scattering will be isotropic in the center-of-mass
system, but for higher energies, there may be p-wave and higher order scattering
that leads to anisotropic scattering. Thus, in general, we need to sample a
cosine of the scattering angle which we will refer to as :math:`\mu`. For
elastic scattering, the secondary angle distribution is always given in the
center-of-mass system and is sampled according to the procedure outlined in
:ref:`sample-angle`. After the cosine of the angle of scattering has been
sampled, we need to determine the neutron's new direction
:math:`\mathbf{\Omega}'_n` in the center-of-mass system. This is done with the
procedure in :ref:`transform-coordinates`. The new direction is multiplied by
the speed of the neutron in the center-of-mass system to obtain the new velocity
vector in the center-of-mass:

.. math::
    :label: velocity-neutron-com-2

    \mathbf{V}'_n = || \mathbf{V}_n || \mathbf{\Omega}'_n.

Finally, we transform the velocity in the center-of-mass system back to lab
coordinates:

.. math::
    :label: velocity-neutron-lab

    \mathbf{v}'_n = \mathbf{V}'_n + \mathbf{v}_{cm}

In OpenMC, the angle and energy of the neutron are stored rather than the
velocity vector itself, so the post-collision angle and energy can be inferred
from the post-collision velocity of the neutron in the lab system.

For tallies that require the scattering cosine, it is important to store the
scattering cosine in the lab system. If we know the scattering cosine in the
center-of-mass, the scattering cosine in the lab system can be calculated as

.. math::
    :label: cosine-lab

    \mu_{lab} = \frac{1 + A\mu}{\sqrt{A^2 + 2A\mu + 1}}.

However, equation :eq:`cosine-lab` is only valid if the target was at rest. When
the target nucleus does have thermal motion, the cosine of the scattering angle
can be determined by simply taking the dot product of the neutron's initial and
final direction in the lab system.

.. _inelastic-scatter:

--------------------
Inelastic Scattering
--------------------

Note that the multi-group mode makes no distinction between elastic or
inelastic scattering reactions. The specific multi-group scattering
implementation is discussed in the :ref:`multi-group-scatter` section.

The major algorithms for inelastic scattering were described in previous
sections. First, a scattering cosine is sampled using the algorithms in
:ref:`sample-angle`. Then an outgoing energy is sampled using the algorithms in
:ref:`sample-energy`. If the outgoing energy and scattering cosine were given in
the center-of-mass system, they are transformed to laboratory coordinates using
the algorithm described in :ref:`transform-coordinates`. Finally, the direction
of the particle is changed also using the procedure in
:ref:`transform-coordinates`.

Although inelastic scattering leaves the target nucleus in an excited state, no
secondary photons from nuclear de-excitation are tracked in OpenMC.

------------------------
:math:`(n,xn)` Reactions
------------------------

Note that the multi-group mode makes no distinction between elastic or
inelastic scattering reactions. The specific multi-group scattering
implementation is discussed in the :ref:`multi-group-scatter` section.

These types of reactions are just treated as inelastic scattering and as such
are subject to the same procedure as described in :ref:`inelastic-scatter`. For
reactions with integral multiplicity, e.g., :math:`(n,2n)`, an appropriate
number of secondary neutrons are created. For reactions that have a multiplicity
given as a function of the incoming neutron energy (which occasionally occurs
for MT=5), the weight of the outgoing neutron is multiplied by the multiplicity.

.. _multi-group-scatter:

----------------------
Multi-Group Scattering
----------------------

In multi-group mode, a scattering collision requires that the outgoing energy
group of the simulated particle be selected from a probability distribution,
the change-in-angle selected from a probability distribution according to
the outgoing energy group, and finally the particle's weight adjusted again
according to the outgoing energy group.

The first step in selecting an outgoing energy group for a particle in a given
incoming energy group is to select a random number (:math:`\xi`) between 0 and
1.  This number is then compared to the cumulative distribution function
produced from the outgoing group (`g'`) data for the given incoming group (`g`):

.. math::
    CDF = \sum_{g'=1}^{h}\Sigma_{s,g \rightarrow g'}

If the scattering data is represented as a Legendre expansion, then the
value of :math:`\Sigma_{s,g \rightarrow g'}` above is the 0th order for the
given group transfer. If the data is provided as tabular or histogram data, then
:math:`\Sigma_{s,g \rightarrow g'}` is the sum of all bins of data for a given
`g` and `g'` pair.

Now that the outgoing energy is known the change-in-angle, :math:`\mu` can be
determined. If the data is provided as a Legendre expansion, this is done by
rejection sampling of the probability distribution represented by the Legendre
series. For efficiency, the selected values of the PDF (:math:`f(\mu)`) are
chosen to be between 0 and the maximum value of :math:`f(\mu)` in the domain of
-1 to 1. Note that this sampling scheme automatically forces negative values of
the :math:`f(\mu)` probability distribution function to be treated as zero
probabilities.

If the angular data is instead provided as a tabular representation, then the
value of :math:`\mu` is selected as described in the :ref:`angle-tabular`
section with a linear-linear interpolation scheme.

If the angular data is provided as a histogram representation, then
the value of :math:`\mu` is selected in a similar fashion to that described for
the selection of the outgoing energy (since the energy group representation is
simply a histogram representation) except the CDF is composed of the angular
bins and not the energy groups.  However, since we are interested in a specific
value of :math:`\mu` instead of a group, then an angle is selected from a uniform
distribution within from the chosen angular bin.

The final step in the scattering treatment is to adjust the weight of the
neutron to account for any production of neutrons due to :math:`(n,xn)`
reactions. This data is obtained from the multiplicity data provided in the
multi-group cross section library for the material of interest.
The scaled value will default to 1.0 if no value is provided in the library.

.. _fission:

-------
Fission
-------

While fission is normally considered an absorption reaction, as far as it
concerns a Monte Carlo simulation it actually bears more similarities to
inelastic scattering since fission results in secondary neutrons in the exit
channel. Other absorption reactions like :math:`(n,\gamma)` or
:math:`(n,\alpha)`, on the contrary, produce no neutrons. There are a few other
idiosyncrasies in treating fission. In an eigenvalue calculation, secondary
neutrons from fission are only "banked" for use in the next generation rather
than being tracked as secondary neutrons from elastic and inelastic scattering
would be. On top of this, fission is sometimes broken into first-chance fission,
second-chance fission, etc. The nuclear data file either lists the partial
fission reactions with secondary energy distributions for each one, or a total
fission reaction with a single secondary energy distribution.

When a fission reaction is sampled in OpenMC (either total fission or, if data
exists, first- or second-chance fission), the following algorithm is used to
create and store fission sites for the following generation. First, the average
number of prompt and delayed neutrons must be determined to decide whether the
secondary neutrons will be prompt or delayed. This is important because delayed
neutrons have a markedly different spectrum from prompt neutrons, one that has a
lower average energy of emission. The total number of neutrons emitted
:math:`\nu_t` is given as a function of incident energy in the ENDF format. Two
representations exist for :math:`\nu_t`. The first is a polynomial of order
:math:`N` with coefficients :math:`c_0,c_1,\dots,c_N`. If :math:`\nu_t` has this
format, we can evaluate it at incoming energy :math:`E` by using the equation

.. math::
    :label: nu-polynomial

    \nu_t (E) = \sum_{i = 0}^N c_i E^i.

The other representation is just a tabulated function with a specified
interpolation law. The number of prompt neutrons released per fission event
:math:`\nu_p` is also given as a function of incident energy and can be
specified in a polynomial or tabular format. The number of delayed neutrons
released per fission event :math:`\nu_d` can only be specified in a tabular
format. In practice, we only need to determine :math:`nu_t` and
:math:`nu_d`. Once these have been determined, we can calculated the delayed
neutron fraction

.. math::
    :label: beta

    \beta = \frac{\nu_d}{\nu_t}.

We then need to determine how many total neutrons should be emitted from
fission. If no survival biasing is being used, then the number of neutrons
emitted is

.. math::
    :label: fission-neutrons

    \nu = \frac{w \nu_t}{k_{eff}}

where :math:`w` is the statistical weight and :math:`k_{eff}` is the effective
multiplication factor from the previous generation. The number of neutrons
produced is biased in this manner so that the expected number of fission
neutrons produced is the number of source particles that we started with in the
generation. Since :math:`\nu` is not an integer, we use the following procedure
to obtain an integral number of fission neutrons to produce. If :math:`\xi >
\nu - \lfloor \nu \rfloor`, then we produce :math:`\lfloor \nu \rfloor`
neutrons. Otherwise, we produce :math:`\lfloor \nu \rfloor + 1` neutrons. Then,
for each fission site produced, we sample the outgoing angle and energy
according to the algorithms given in :ref:`sample-angle` and
:ref:`sample-energy` respectively. If the neutron is to be born delayed, then
there is an extra step of sampling a delayed neutron precursor group since they
each have an associated secondary energy distribution.

The sampled outgoing angle and energy of fission neutrons along with the
position of the collision site are stored in an array called the fission
bank. In a subsequent generation, these fission bank sites are used as starting
source sites.

The above description is similar for the multi-group mode except the data are
provided as group-wise data instead of in a continuous-energy format. In this
case, the outgoing energy of the fission neutrons are represented as histograms
by way of either the nu-fission matrix or chi vector.

------------------------------------
Secondary Angle-Energy Distributions
------------------------------------

Note that this section is specific to continuous-energy mode since the
multi-group scattering process has already been described including the
secondary energy and angle sampling.

For a reaction with secondary products, it is necessary to determine the
outgoing angle and energy of the products. For any reaction other than elastic
and level inelastic scattering, the outgoing energy must be determined based on
tabulated or parameterized data. The `ENDF-6 Format`_ specifies a
variety of ways that the secondary energy distribution can be represented. ENDF
File 5 contains uncorrelated energy distribution whereas ENDF File 6 contains
correlated energy-angle distributions. The ACE format specifies its own
representations based loosely on the formats given in ENDF-6. OpenMC's HDF5
nuclear data files use a combination of ENDF and ACE distributions; in this
section, we will describe how the outgoing angle and energy of secondary
particles are sampled.

One of the subtleties in the nuclear data format is the fact that a single
reaction product can have multiple angle-energy distributions. This is mainly
useful for reactions with multiple products of the same type in the exit channel
such as :math:`(n,2n)` or :math:`(n,3n)`. In these types of reactions, each
neutron is emitted corresponding to a different excitation level of the compound
nucleus, and thus in general the neutrons will originate from different energy
distributions. If multiple angle-energy distributions are present, they are
assigned incoming-energy-dependent probabilities that can then be used to
randomly select one.

Once a distribution has been selected, the procedure for determining the
outgoing angle and energy will depend on the type of the distribution.

Uncorrelated Angle-Energy Distributions
---------------------------------------

The first set of distributions we will look at are uncorrelated angle-energy
distributions, where angle and energy are specified separately. For these
distributions, OpenMC first samples the angular distribution as described
:ref:`sample-angle` and then samples an energy as described in
:ref:`sample-energy`.

.. _sample-angle:

Sampling Angular Distributions
++++++++++++++++++++++++++++++

For elastic scattering, it is only necessary to specific a secondary angle
distribution since the outgoing energy can be determined analytically. Other
reactions may also have separate secondary angle and secondary energy
distributions that are uncorrelated. In these cases, the secondary angle
distribution is represented as either

- An isotropic angular distribution,
- A tabular distribution.

Isotropic Angular Distribution
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In the first case, no data is stored in the nuclear data file, and the cosine of
the scattering angle is simply calculated as

.. math::
    :label: isotropic-angle

    \mu = 2\xi - 1

where :math:`\mu` is the cosine of the scattering angle and :math:`\xi` is a
random number sampled uniformly on :math:`[0,1)`.

.. _angle-tabular:

Tabular Angular Distribution
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In this case, we have a table of cosines and their corresponding values for a
probability distribution 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 cumulative distribution function. We first find the interpolation factor on
the incoming energy grid:

.. math::
    :label: interpolation-factor

    f = \frac{E - E_i}{E_{i+1} - E_i}

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_1 > f` and
:math:`\ell = i + 1` otherwise, where :math:`\xi_1` is a random number. Another
random number :math:`\xi_2` is used to sample a scattering cosine bin :math:`j`
using the cumulative distribution function:

.. math::
    :label: sample-cdf

    c_{\ell,j} < \xi_2 < 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
function as

.. math::
    :label: cdf

    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_{\ell,j}) = c_{\ell,j}`, this implies that for :math:`\mu >
\mu_{\ell,j}`,

.. math::
    :label: cdf-2

    c(\mu) = c_{\ell,j} + \int_{\mu_{\ell,j}}^{\mu} p(\mu') d\mu'

For histogram interpolation, we have that :math:`p(\mu') = p_{\ell,j}` for
:math:`\mu_{\ell,j} \le \mu' < \mu_{\ell,j+1}`. Thus, after integrating
:eq:`cdf-2` we have that

.. math::
    :label: cumulative-dist-histogram

    c(\mu) = c_{\ell,j} + (\mu - \mu_{\ell,j}) p_{\ell,j} = \xi_2

Solving for the scattering cosine, we obtain the final form for histogram
interpolation:

.. math::
    :label: cosine-histogram

    \mu = \mu_{\ell,j} + \frac{\xi_2 - 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
values on the probability distribution function, we know that

.. math::
    :label: pdf-interpolation

    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

.. math::
    :label: cdf-linlin

    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_{\ell,j+1} - p_{\ell,j}}{\mu_{\ell,j+1} - \mu_{\ell,j}}
    (\mu' - \mu_{\ell,j}) + p_{\ell,j}.

Equation :eq:`cdf-linlin` then becomes

.. math::
    :label: cdf-linlin-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_{\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_{\ell,j} + \frac{1}{2m} \left ( \left [ m (\mu - \mu_{\ell,j} ) +
    p_{\ell,j} \right ]^2 - p_{\ell,j}^2 \right ) = \xi_2

Solving for :math:`\mu`, we have the final form for the scattering cosine using
linear-linear interpolation:

.. math::
    :label: cosine-linlin

    \mu = \mu_{\ell,j} + \frac{1}{m} \left ( \sqrt{p_{\ell,j}^2 + 2 m (\xi_2 -
    c_{\ell,j} )} - p_{\ell,j} \right )

.. _sample-energy:

Sampling Energy Distributions
+++++++++++++++++++++++++++++

Inelastic Level Scattering
^^^^^^^^^^^^^^^^^^^^^^^^^^

It can be shown (see Foderaro_) that in inelastic level scattering, the outgoing
energy of the neutron :math:`E'` can be related to the Q-value of the reaction
and the incoming energy:

.. math::
    :label: level-scattering

    E' = \left ( \frac{A}{A+1} \right )^2 \left ( E - \frac{A + 1}{A} Q \right )

where :math:`A` is the mass of the target nucleus measured in neutron masses.

.. _continuous-tabular:

Continuous Tabular Distribution
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In a continuous tabular distribution, a tabulated energy distribution is
provided for each of a set of incoming energies. While the representation itself
is simple, the complexity lies in how one interpolates between incident as well
as outgoing energies on such a table. If one performs simple interpolation
between tables for neighboring incident energies, it is possible that the
resulting energies would violate laws governing the kinematics, i.e., the
outgoing energy may be outside the range of available energy in the reaction.

To avoid this situation, the accepted practice is to use a process known as
`scaled interpolation`_. First, we find the tabulated incident energies which
bound the actual incoming energy of the particle, i.e., find :math:`i` such that
:math:`E_i < E < E_{i+1}` and calculate the interpolation factor :math:`f` via
:eq:`interpolation-factor`. Then, we interpolate between the minimum and maximum
energies of the outgoing energy distributions corresponding to :math:`E_i` and
:math:`E_{i+1}`:

.. math::
    :label: continuous-minmax

    E_{min} = E_{i,1} + f ( E_{i+1,1} - E_{i,1} ) \\
    E_{max} = E_{i,M} + f ( E_{i+1,M} - E_{i,M} )

where :math:`E_{min}` and :math:`E_{max}` are the minimum and maximum outgoing
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` 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, and :math:`\xi_1` is a
random number. 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. We then sample an outgoing energy bin
:math:`j` using the cumulative distribution function:

.. math::
    :label: continuous-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 interpolate 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 calculated in equation :eq:`continuous-minmax` to
get the final outgoing energy:

.. math::
    :label: continuous-eout

    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:`continuous-minmax`.

.. _maxwell:

Maxwell Fission Spectrum
^^^^^^^^^^^^^^^^^^^^^^^^

One representation of the secondary energies for neutrons from fission is the
so-called Maxwell spectrum. A probability distribution for the Maxwell spectrum
can be written in the form

.. math::
    :label: maxwell-spectrum

    p(E') dE' = c E'^{1/2} e^{-E'/T(E)} dE'

where :math:`E` is the incoming energy of the neutron and :math:`T` is the
so-called nuclear temperature, which is a function of the incoming energy of the
neutron. The ENDF format contains a list of nuclear temperatures versus incoming
energies. The nuclear temperature is interpolated between neighboring incoming
energies using a specified interpolation law. Once the temperature :math:`T` is
determined, we then calculate a candidate outgoing energy based on rule C64 in
the `Monte Carlo Sampler`_:

.. math::
    :label: maxwell-E-candidate

    E' = -T \left [ \log (\xi_1) + \log (\xi_2) \cos^2 \left ( \frac{\pi
    \xi_3}{2} \right ) \right ]

where :math:`\xi_1, \xi_2, \xi_3` are random numbers sampled on the unit
interval. The outgoing energy is only accepted if

.. math::
    :label: maxwell-restriction

    0 \le E' \le E - U

where :math:`U` is called the restriction energy and is specified in the ENDF
data. If the outgoing energy is rejected, it is resampled using equation
:eq:`maxwell-E-candidate`.

Evaporation Spectrum
^^^^^^^^^^^^^^^^^^^^

Evaporation spectra are primarily used in compound nucleus processes where a
secondary particle can "evaporate" from the compound nucleus if it has
sufficient energy. The probability distribution for an evaporation spectrum can
be written in the form

.. math::
    :label: evaporation-spectrum

    p(E') dE' = c E' e^{-E'/T(E)} dE'

where :math:`E` is the incoming energy of the neutron and :math:`T` is the
nuclear temperature, which is a function of the incoming energy of the
neutron. The ENDF format contains a list of nuclear temperatures versus incoming
energies. The nuclear temperature is interpolated between neighboring incoming
energies using a specified interpolation law. Once the temperature :math:`T` is
determined, we then calculate a candidate outgoing energy based on the algorithm
given in LA-UR-14-27694_:

.. math::
    :label: evaporation-E

    E' = -T \log ((1 - g\xi_1)(1 - g\xi_2))

where :math:`g = 1 - e^{-w}`, :math:`w = (E - U)/T`, :math:`U` is the
restriction energy, and :math:`\xi_1, \xi_2` are random numbers sampled on the
unit interval. The outgoing energy is only accepted according to the restriction
energy as in equation :eq:`maxwell-restriction`. This algorithm has a much
higher rejection efficiency than the standard technique, i.e. rule C45 in the
`Monte Carlo Sampler`_.

Energy-Dependent Watt Spectrum
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The probability distribution for a `Watt fission spectrum`_ can be written in
the form

.. math::
    :label: watt-spectrum

    p(E') dE' = c e^{-E'/a(E)} \sinh \sqrt{b(E) \, E'} dE'

where :math:`a` and :math:`b` are parameters for the distribution and are given
as tabulated functions of the incoming energy of the neutron. These two
parameters are interpolated on the incoming energy grid using a specified
interpolation law. Once the parameters have been determined, we sample a
Maxwellian spectrum with nuclear temperature :math:`a` using the algorithm
described in :ref:`maxwell` to get an energy :math:`W`. Then, the outgoing
energy is calculated as

.. math::
    :label: watt-E

    E' = W + \frac{a^2 b}{4} + (2\xi - 1) \sqrt{a^2 b W}

where :math:`\xi` is a random number sampled on the interval :math:`[0,1)`. The
outgoing energy is only accepted according to a specified restriction energy
:math:`U` as defined in equation :eq:`maxwell-restriction`.

A derivation of the algorithm described here can be found in a paper by Romano_.

Product Angle-Energy Distributions
----------------------------------

If the secondary distribution for a product was given in file 6 in ENDF, the
angle and energy are correlated with one another and cannot be sampled
separately. Several representations exist in ENDF/ACE for correlated
angle-energy distributions.

Kalbach-Mann Correlated Scattering
++++++++++++++++++++++++++++++++++

This law is very similar to the uncorrelated continuous tabular energy
distribution except now the outgoing angle of the neutron is correlated to the
outgoing energy and is not sampled from a separate distribution. For each
incident neutron energy :math:`E_i` tabulated, there is an array of precompound
factors :math:`R_{i,j}` and angular distribution slopes :math:`A_{i,j}`
corresponding to each outgoing energy bin :math:`j` in addition to the outgoing
energies and distribution functions as in :ref:`continuous-tabular`.

The calculation of the outgoing energy of the neutron proceeds exactly the same
as in the algorithm described in :ref:`continuous-tabular`. In that algorithm,
we found an interpolation factor :math:`f`, statistically sampled an incoming
energy bin :math:`\ell`, and sampled an outgoing energy bin :math:`j` based on
the tabulated cumulative distribution function. Once the outgoing energy has
been determined with equation :eq:`continuous-eout`, we then need to calculate
the outgoing angle based on the tabulated Kalbach-Mann parameters. These
parameters themselves are subject to either histogram or linear-linear
interpolation on the outgoing energy grid. For histogram interpolation, the
parameters are

.. math::
    :label: KM-parameters-histogram

    R = R_{\ell,j} \\
    A = A_{\ell,j}.

If linear-linear interpolation is specified, the parameters are

.. math::
    :label: KM-parameters-linlin

    R = R_{\ell,j} + \frac{\hat{E} - E_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} (
    R_{\ell,j+1} - R_{\ell,j} ) \\
    A = A_{\ell,j} + \frac{\hat{E} - E_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} (
    A_{\ell,j+1} - A_{\ell,j} )

where :math:`\hat{E}` is defined in equation :eq:`energy-linlin`. With the
parameters determined, the probability distribution function for the cosine of
the scattering angle is

.. math::
    :label: KM-pdf-angle

    p(\mu) d\mu = \frac{A}{2 \sinh (A)} \left [ \cosh (A\mu) + R \sinh (A\mu)
    \right ] d\mu.

The rules for sampling this probability distribution function can be derived
based on rules C39 and C40 in the `Monte Carlo Sampler`_. First, we sample two
random numbers :math:`\xi_3, \xi_4` on the unit interval. If :math:`\xi_3 > R`
then the outgoing angle is

.. math::
    :label: KM-angle-1

    \mu = \frac{1}{A} \ln \left ( T + \sqrt{T^2 + 1} \right )

where :math:`T = (2 \xi_4 - 1) \sinh (A)`. If :math:`\xi_3 \le R`, then the
outgoing angle is

.. math::
    :label: KM-angle-2

    \mu = \frac{1}{A} \ln \left ( \xi_4 e^A + (1 - \xi_4) e^{-A} \right ).

.. _correlated-energy-angle:

Correlated Energy and Angle Distribution
++++++++++++++++++++++++++++++++++++++++

This distribution is very similar to a Kalbach-Mann distribution in the sense
that the outgoing angle of the neutron is correlated to the outgoing energy and
is not sampled from a separate distribution. In this case though, rather than
being determined from an analytical distribution function, the cosine of the
scattering angle is determined from a tabulated distribution. For each incident
energy :math:`i` and outgoing energy :math:`j`, there is a tabulated angular
distribution.

The calculation of the outgoing energy of the neutron proceeds exactly the same
as in the algorithm described in :ref:`continuous-tabular`. In that algorithm,
we found an interpolation factor :math:`f`, statistically sampled an incoming
energy bin :math:`\ell`, and sampled an outgoing energy bin :math:`j` based on
the tabulated cumulative distribution function. Once the outgoing energy has
been determined with equation :eq:`continuous-eout`, we then need to decide
which angular distribution to use. If histogram interpolation was used on the
outgoing energy bins, then we use the angular distribution corresponding to
incoming energy bin :math:`\ell` and outgoing energy bin :math:`j`. If
linear-linear interpolation was used on the outgoing energy bins, then we use
the whichever angular distribution was closer to the sampled value of the
cumulative distribution function for the outgoing energy. The actual algorithm
used to sample the chosen tabular angular distribution has been previously
described in :ref:`angle-tabular`.

N-Body Phase Space Distribution
+++++++++++++++++++++++++++++++

Reactions in which there are more than two products of similar masses are
sometimes best treated by using what's known as an N-body phase
distribution. This distribution has the following probability density function
for outgoing energy and angle of the :math:`i`-th particle in the center-of-mass
system:

.. math::
    :label: n-body-pdf

    p_i(\mu, E') dE' d\mu = C_n \sqrt{E'} (E_i^{max} - E')^{(3n/2) - 4} dE' d\mu

where :math:`n` is the number of outgoing particles, :math:`C_n` is a
normalization constant, :math:`E_i^{max}` is the maximum center-of-mass energy
for particle :math:`i`, and :math:`E'` is the outgoing energy. We see in
equation :eq:`n-body-pdf` that the angle is simply isotropic in the
center-of-mass system. The algorithm for sampling the outgoing energy is based
on algorithms R28, C45, and C64 in the `Monte Carlo Sampler`_. First we
calculate the maximum energy in the center-of-mass using the following equation:

.. math::
    :label: n-body-emax

    E_i^{max} = \frac{A_p - 1}{A_p} \left ( \frac{A}{A+1} E + Q \right )

where :math:`A_p` is the total mass of the outgoing particles in neutron masses,
:math:`A` is the mass of the original target nucleus in neutron masses, and
:math:`Q` is the Q-value of the reaction. Next we sample a value :math:`x` from
a Maxwell distribution with a nuclear temperature of one using the algorithm
outlined in :ref:`maxwell`. We then need to determine a value :math:`y` that
will depend on how many outgoing particles there are. For :math:`n = 3`, we
simply sample another Maxwell distribution with unity nuclear temperature. For
:math:`n = 4`, we use the equation

.. math::
    :label: n-body-y4

    y = -\ln ( \xi_1 \xi_2 \xi_3 )

where :math:`\xi_i` are random numbers sampled on the interval
:math:`[0,1)`. For :math:`n = 5`, we use the equation

.. math::
    :label: n-body-y5

    y = -\ln ( \xi_1 \xi_2 \xi_3 \xi_4 ) - \ln ( \xi_5 ) \cos^2 \left (
    \frac{\pi}{2} \xi_6 \right )

After :math:`x` and :math:`y` have been determined, the outgoing energy is then
calculated as

.. math::
    :label: n-body-energy

    E' = \frac{x}{x + y} E_i^{max}

There are two important notes to make regarding the N-body phase space
distribution. First, the documentation (and code) for MCNP5-1.60 has a mistake
in the algorithm for :math:`n = 4`. That being said, there are no existing
nuclear data evaluations which use an N-body phase space distribution with
:math:`n = 4`, so the error would not affect any calculations. In the
ENDF/B-VII.1 nuclear data evaluation, only one reaction uses an N-body phase
space distribution at all, the :math:`(n,2n)` reaction with H-2.

.. _transform-coordinates:

-------------------------------------
Transforming a Particle's Coordinates
-------------------------------------

Since all the multi-group data exists in the laboratory frame of reference, this
section does not apply to the multi-group mode.

Once the cosine of the scattering angle :math:`\mu` has been sampled either from
a angle distribution or a correlated angle-energy distribution, we are still
left with the task of transforming the particle's coordinates. If the outgoing
energy and scattering cosine were given in the center-of-mass system, then we
first need to transform these into the laboratory system. The relationship
between the outgoing energy in center-of-mass and laboratory is

.. math::
    :label: energy-com-to-lab

    E' = E'_{cm} + \frac{E + 2\mu_{cm} (A + 1) \sqrt{EE'_{cm}}}{(A+1)^2}.

where :math:`E'_{cm}` is the outgoing energy in the center-of-mass system,
:math:`\mu_{cm}` is the scattering cosine in the center-of-mass system,
:math:`E'` is the outgoing energy in the laboratory system, and :math:`E` is the
incident neutron energy. The relationship between the scattering cosine in
center-of-mass and laboratory is

.. math::
    :label: angle-com-to-lab

    \mu = \mu_{cm} \sqrt{\frac{E'_{cm}}{E'}} + \frac{1}{A + 1}
    \sqrt{\frac{E}{E'}}

where :math:`\mu` is the scattering cosine in the laboratory system. The
scattering cosine still only tells us the cosine of the angle between the
original direction of the particle and the new direction of the particle. If we
express the pre-collision direction of the particle as :math:`\mathbf{\Omega} =
(u,v,w)` and the post-collision direction of the particle as
:math:`\mathbf{\Omega}' = (u',v',w')`, it is possible to relate the pre- and
post-collision components. We first need to uniformly sample an azimuthal angle
:math:`\phi` in :math:`[0, 2\pi)`. After the azimuthal angle has been sampled,
the post-collision direction is calculated as

.. math::
    :label: post-collision-angle

    u' = \mu u + \frac{\sqrt{1 - \mu^2} ( uw \cos\phi - v \sin\phi )}{\sqrt{1 -
    w^2}} \\

    v' = \mu v + \frac{\sqrt{1 - \mu^2} ( vw \cos\phi + u \sin\phi )}{\sqrt{1 -
    w^2}} \\

    w' = \mu w - \sqrt{1 - \mu^2} \sqrt{1 - w^2} \cos\phi.

.. _freegas:

------------------------------------------
Effect of Thermal Motion on Cross Sections
------------------------------------------

Since all the multi-group data should be generated with thermal scattering
treatments already, this section does not apply to the multi-group mode.

When a neutron scatters off of a nucleus, it may often be assumed that the
target nucleus is at rest. However, the target nucleus will have motion
associated with its thermal vibration, even at absolute zero (This is due to the
zero-point energy arising from quantum mechanical considerations). Thus, the
velocity of the neutron relative to the target nucleus is in general not the
same as the velocity of the neutron entering the collision.

The effect of the thermal motion on the interaction probability can be written
as

.. math::
    :label: doppler-broaden

    v_n \bar{\sigma} (v_n, T) = \int d\mathbf{v}_T v_r \sigma(v_r)
    M (\mathbf{v}_T)

where :math:`v_n` is the magnitude of the velocity of the neutron,
:math:`\bar{\sigma}` is an effective cross section, :math:`T` is the temperature
of the target material, :math:`\mathbf{v}_T` is the velocity of the target
nucleus, :math:`v_r = || \mathbf{v}_n - \mathbf{v}_T ||` is the magnitude of the
relative velocity, :math:`\sigma` is the cross section at 0 K, and :math:`M
(\mathbf{v}_T)` is the probability distribution for the target nucleus velocity
at temperature :math:`T` (a Maxwellian). In a Monte Carlo code, one must account
for the effect of the thermal motion on both the integrated cross section as
well as secondary angle and energy distributions. For integrated cross sections,
it is possible to calculate thermally-averaged cross sections by applying a
kernel Doppler broadening algorithm to data at 0 K (or some temperature lower
than the desired temperature). The most ubiquitous algorithm for this purpose is
the `SIGMA1 method`_ developed by Red Cullen and subsequently refined by
others. This method is used in the NJOY_ and PREPRO_ data processing codes.

The effect of thermal motion on secondary angle and energy distributions can be
accounted for on-the-fly in a Monte Carlo simulation. We must first qualify
where it is actually used however. All threshold reactions are treated as being
independent of temperature, and therefore they are not Doppler broadened in NJOY
and no special procedure is used to adjust the secondary angle and energy
distributions. The only non-threshold reactions with secondary neutrons are
elastic scattering and fission. For fission, it is assumed that the neutrons are
emitted isotropically (this is not strictly true, but is nevertheless a good
approximation). This leaves only elastic scattering that needs a special thermal
treatment for secondary distributions.

Fortunately, it is possible to directly sample the velocity of the target
nuclide and then use it directly in the kinematic calculations. However, this
calculation is a bit more nuanced than it might seem at first glance. One might
be tempted to simply sample a Maxwellian distribution for the velocity of the
target nuclide.  Careful inspection of equation :eq:`doppler-broaden` however
tells us that target velocities that produce relative velocities which
correspond to high cross sections will have a greater contribution to the
effective reaction rate. This is most important when the velocity of the
incoming neutron is close to a resonance. For example, if the neutron's velocity
corresponds to a trough in a resonance elastic scattering cross section, a very
small target velocity can cause the relative velocity to correspond to the peak
of the resonance, thus making a disproportionate contribution to the reaction
rate. The conclusion is that if we are to sample a target velocity in the Monte
Carlo code, it must be done in such a way that preserves the thermally-averaged
reaction rate as per equation :eq:`doppler-broaden`.

The method by which most Monte Carlo codes sample the target velocity for use in
elastic scattering kinematics is outlined in detail by [Gelbard]_. The
derivation here largely follows that of Gelbard. Let us first write the reaction
rate as a function of the velocity of the target nucleus:

.. math::
    :label: reaction-rate

    R(\mathbf{v}_T) = || \mathbf{v}_n - \mathbf{v}_T || \sigma ( ||
    \mathbf{v}_n - \mathbf{v}_T || ) M ( \mathbf{v}_T )

where :math:`R` is the reaction rate. Note that this is just the right-hand side
of equation :eq:`doppler-broaden`. Based on the discussion above, we want to
construct a probability distribution function for sampling the target velocity
to preserve the reaction rate -- this is different from the overall probability
distribution function for the target velocity, :math:`M ( \mathbf{v}_T )`. This
probability distribution function can be found by integrating equation
:eq:`reaction-rate` to obtain a normalization factor:

.. math::
    :label: target-pdf-1

    p( \mathbf{v}_T ) d\mathbf{v}_T = \frac{R(\mathbf{v}_T) d\mathbf{v}_T}{\int
    d\mathbf{v}_T \, R(\mathbf{v}_T)}

Let us call the normalization factor in the denominator of equation
:eq:`target-pdf-1` :math:`C`.


Constant Cross Section Model
----------------------------

It is often assumed that :math:`\sigma (v_r)` 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
may cause incorrect answers in systems with low-lying resonances that can cause
a significant amount of up-scatter that would be ignored by this assumption
(e.g. U-238 in commercial light-water reactors). We will revisit this assumption
later in :ref:`energy_dependent_xs_model`. For now, continuing with the
assumption, we write :math:`\sigma (v_r) = \sigma_s` which simplifies
:eq:`target-pdf-1` to

.. math::
    :label: target-pdf-2

    p( \mathbf{v}_T ) d\mathbf{v}_T = \frac{\sigma_s}{C} || \mathbf{v}_n -
    \mathbf{v}_T || M ( \mathbf{v}_T ) d\mathbf{v}_T

The Maxwellian distribution in velocity is

.. math::
    :label: maxwellian-velocity

    M (\mathbf{v}_T) = \left ( \frac{m}{2\pi kT} \right )^{3/2} \exp \left (
    \frac{-m || \mathbf{v}_T^2 ||}{2kT} \right )

where :math:`m` is the mass of the target nucleus and :math:`k` is Boltzmann's
constant. Notice here that the term in the exponential is dependent only on the
speed of the target, not on the actual direction. Thus, we can change the
Maxwellian into a distribution for speed rather than velocity. The differential
element of velocity is

.. math::
    :label: differential-velocity

    d\mathbf{v}_T = v_T^2 dv_T d\mu d\phi

Let us define the Maxwellian distribution in speed as

.. math::
    :label: maxwellian-speed

    M (v_T) dv_T = \int_{-1}^1 d\mu \int_{0}^{2\pi} d\phi \, dv_T \, v_T^2
    M(\mathbf{v}_T) = \sqrt{ \frac{2}{\pi} \left ( \frac{m}{kT} \right )^3}
    v_T^2 \exp \left ( \frac{-m v_T}{2kT} \right ) dv_T.

To simplify things a bit, we'll define a parameter

.. math::
    :label: maxwellian-beta

    \beta = \sqrt{\frac{m}{2kT}}.

Substituting equation :eq:`maxwellian-beta` into equation
:eq:`maxwellian-speed`, we obtain

.. math::
    :label: maxwellian-speed2

    M (v_T) dv_T = \frac{4}{\sqrt{\pi}} \beta^3 v_T^2 \exp \left ( -\beta^2
    v_T^2 \right ) dv_T.

Now, changing variables in equation :eq:`target-pdf-2` by using the result from
equation :eq:`maxwellian-speed`, our new probability distribution function is

.. math::
    :label: target-pdf-3

    p( v_T, \mu ) dv_T d\mu = \frac{4\sigma_s}{\sqrt{\pi}C'} || \mathbf{v}_n -
    \mathbf{v}_T || \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right ) dv_T d\mu

Again, the Maxwellian distribution for the speed of the target nucleus has no
dependence on the angle between the neutron and target velocity vectors. Thus,
only the term :math:`|| \mathbf{v}_n - \mathbf{v}_T ||` imposes any constraint
on the allowed angle. Our last task is to take that term and write it in terms
of magnitudes of the velocity vectors and the angle rather than the vectors
themselves. We can establish this relation based on the law of cosines which
tells us that

.. math::
    :label: lawcosine

    2 v_n v_T \mu = v_n^2 + v_T^2 - v_r^2.

Thus, we can infer that

.. math::
    :label: change-terms

    || \mathbf{v}_n - \mathbf{v}_T || = || \mathbf{v}_r || = v_r = \sqrt{v_n^2 +
       v_T^2 - 2v_n v_T \mu}.

Inserting equation :eq:`change-terms` into :eq:`target-pdf-3`, we obtain

.. math::
    :label: target-pdf-4

    p( v_T, \mu ) dv_T d\mu = \frac{4\sigma_s}{\sqrt{\pi}C'} \sqrt{v_n^2 +
       v_T^2 - 2v_n v_T \mu} \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right )
       dv_T d\mu

This expression is still quite formidable and does not lend itself to any
natural sampling scheme. We can divide this probability distribution into two
parts as such:

.. math::
    :label: divide-pdf

    \begin{aligned}
    p(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\
    f_1(v_T, \mu) &= \frac{4\sigma_s}{\sqrt{\pi} C'} \frac{ \sqrt{v_n^2 +
       v_T^2 - 2v_n v_T \mu}}{v_n + v_T} \\
    f_2(v_T) &= (v_n + v_T) \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2 \right ).
    \end{aligned}

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'` from the distribution

.. math::
    :label: freegas-f2

    q(x) dx = \frac{f_2(x) dx}{\int f_2(x) dx}

and accepting it with probability

.. math::
    :label: freegas-accept

    p_{accept} = \frac{f_1(x')}{\max f_1(x)}

The reason for dividing and multiplying the terms by :math:`v_n + v_T` is to
ensure that the first term is bounded. In general, :math:`|| \mathbf{v}_n -
\mathbf{v}_T ||` can take on arbitrarily large values, but if we divide it by
its maximum value :math:`v_n + v_T`, then it ensures that the function will be
bounded. We now must come up with a sampling scheme for equation
:eq:`freegas-f2`. To determine :math:`q(v_T)`, we need to integrate :math:`f_2`
in equation :eq:`divide-pdf`. Doing so we find that

.. math::
    :label: integrate-f2

    \int_0^{\infty} dv_T (v_n + v_T) \beta^3 v_T^2 \exp \left ( -\beta^2 v_T^2
    \right ) = \frac{1}{4\beta} \left ( \sqrt{\pi} \beta v_n + 2 \right ).

Thus, we need to sample the probability distribution function

.. math::
    :label: freegas-f2-2

    q(v_T) dv_T = \left ( \frac{4\beta^2 v_n v_T^2}{\sqrt{\pi} \beta v_n + 2} +
    \frac{4\beta^4 v_T^3}{\sqrt{\pi} \beta v_n + 2} \right ) exp \left (
    -\beta^2 v_T^2 \right ).

Now, let us do a change of variables with the following definitions

.. math::
    :label: beta-to-x

    x = \beta v_T \\
    y = \beta v_n.

Substituting equation :eq:`beta-to-x` into equation :eq:`freegas-f2-2` along
with :math:`dx = \beta dv_T` and doing some crafty rearranging of terms yields

.. math::
    :label: freegas-f2-3

    q(x) dx = \left [ \left ( \frac{\sqrt{\pi} y}{\sqrt{\pi} y + 2} \right )
    \frac{4}{\sqrt{\pi}} x^2 e^{-x^2} + \left ( \frac{2}{\sqrt{\pi} y + 2}
    \right ) 2x^3 e^{-x^2} \right ] dx.

It's important to make note of the following two facts. First, the terms outside
the parentheses are properly normalized probability distribution functions that
can be sampled directly. Secondly, the terms inside the parentheses are always
less than unity. Thus, the sampling scheme for :math:`q(x)` is as follows. We
sample a random number :math:`\xi_1` on the interval :math:`[0,1)` and if

.. math::
    :label: freegas-alpha

    \xi_1 < \frac{2}{\sqrt{\pi} y + 2}

then we sample the probability distribution :math:`2x^3 e^{-x^2}` for :math:`x`
using rule C49 in the `Monte Carlo Sampler`_ which we can then use to determine
the speed of the target nucleus :math:`v_T` from equation
:eq:`beta-to-x`. Otherwise, we sample the probability distribution
:math:`\frac{4}{\sqrt{\pi}} x^2 e^{-x^2}` for :math:`x` using rule C61 in the
`Monte Carlo Sampler`_.

With a target speed sampled, we must then decide whether to accept it based on
the probability in equation :eq:`freegas-accept`. The cosine can be sampled
isotropically as :math:`\mu = 2\xi_2 - 1` where :math:`\xi_2` is a random number
on the unit interval. Since the maximum value of :math:`f_1(v_T, \mu)` is
:math:`4\sigma_s / \sqrt{\pi} C'`, we then sample another random number
:math:`\xi_3` and accept the sampled target speed and cosine if

.. math::
    :label: freegas-accept-2

    \xi_3 < \frac{\sqrt{v_n^2 + v_T^2 - 2 v_n v_T \mu}}{v_n + v_T}.

If is not accepted, then we repeat the process and resample a target speed and
cosine until a combination is found that satisfies equation
:eq:`freegas-accept-2`.

.. _energy_dependent_xs_model:

Energy-Dependent Cross Section Model
------------------------------------

As was noted earlier, assuming that the elastic scattering cross section is
constant in :eq:`reaction-rate` is not strictly correct, especially when
low-lying resonances are present in the cross sections for heavy nuclides. To
correctly account for energy dependence of the scattering cross section entails
performing another rejection step. The most common method is to sample
:math:`\mu` and :math:`v_T` as in the constant cross section approximation and
then perform a rejection on the ratio of the 0 K elastic scattering cross
section at the relative velocity to the maximum 0 K elastic scattering cross
section over the range of velocities considered:

.. math::
    :label: dbrc

    p_{dbrc} = \frac{\sigma_s(v_r)}{\sigma_{s,max}}

where it should be noted that the maximum is taken over the range :math:`[v_n -
4/\beta, 4_n + 4\beta]`. This method is known as Doppler broadening rejection
correction (DBRC) and was first introduced by `Becker et al.`_. OpenMC has an
implementation of DBRC as well as an accelerated sampling method that samples the `relative velocity`_ directly.

.. _Becker et al.: https://doi.org/10.1016/j.anucene.2008.12.001
.. _relative velocity: https://doi.org/10.1016/j.anucene.2017.12.044

.. _sab_tables:

------------
|sab| Tables
------------

Note that |sab| tables are only applicable to continuous-energy transport.

For neutrons with thermal energies, generally less than 4 eV, the kinematics of
scattering can be affected by chemical binding and crystalline effects of the
target molecule. If these effects are not accounted for in a simulation, the
reported results may be highly inaccurate. There is no general analytic
treatment for the scattering kinematics at low energies, and thus when nuclear
data is processed for use in a Monte Carlo code, special tables are created that
give cross sections and secondary angle/energy distributions for thermal
scattering that account for thermal binding effects. These tables are mainly
used for moderating materials such as light or heavy water, graphite, hydrogen
in ZrH, beryllium, etc.

The theory behind |sab| is rooted in quantum mechanics and is quite
complex. Those interested in first principles derivations for formulae relating
to |sab| tables should be referred to the excellent books by [Williams]_ and
[Squires]_. For our purposes here, we will focus only on the use of already
processed data as it appears in the ACE format.

Each |sab| table can contain the following:

- Thermal inelastic scattering cross section;
- Thermal elastic scattering cross section;
- Correlated energy-angle distributions for thermal inelastic and elastic
  scattering.

Note that when we refer to "inelastic" and "elastic" scattering now, we are
actually using these terms with respect to the *scattering system*. Thermal
inelastic scattering means that the scattering system is left in an excited
state; no particular nucleus is left in an excited state as would be the case
for inelastic level scattering. In a crystalline material, the excitation of the
scattering could correspond to the production of phonons. In a molecule, it
could correspond to the excitation of rotational or vibrational modes.

Both thermal elastic and thermal inelastic scattering are generally divided into
incoherent and coherent parts. Coherent elastic scattering refers to scattering
in crystalline solids like graphite or beryllium. These cross sections are
characterized by the presence of *Bragg edges* that relate to the crystal
structure of the scattering material. Incoherent elastic scattering refers to
scattering in hydrogenous solids such as polyethylene. As it occurs in ACE data,
thermal inelastic scattering includes both coherent and incoherent effects and
is dominant for most other materials including hydrogen in water.

Calculating Integrated Cross Sections
-------------------------------------

The first aspect of using |sab| tables is calculating cross sections to replace
the data that would normally appear on the incident neutron data, which do not
account for thermal binding effects. For incoherent inelastic scattering, the
cross section is stored as a linearly interpolable function on a specified
energy grid. For coherent elastic data, the cross section can be expressed as

.. math::
    :label: coherent-elastic-xs

    \sigma(E) = \frac{1}{E} \sum_{E_i < E} s_i

where :math:`E_i` are the energies of the Bragg edges and :math:`s_i` are
related to crystallographic structure factors. Since the functional form of the
cross section is just 1/E and the proportionality constant changes only at Bragg
edges, the proportionality constants are stored and then the cross section can
be calculated analytically based on equation :eq:`coherent-elastic-xs`. For
incoherent elastic data, the cross section can be expressed as

.. math::
    :label: incoherent-elastic-xs

    \sigma(E) = \frac{\sigma_b}{2} \left( \frac{1 - e^{-4EW'}}{2EW'} \right)

where :math:`\sigma_b` is the characteristic bound cross section and :math:`W'`
is the Debye-Waller integral divided by the atomic mass.

Outgoing Angle for Coherent Elastic Scattering
----------------------------------------------

Another aspect of using |sab| tables is determining the outgoing energy and
angle of the neutron after scattering. For incoherent and coherent elastic
scattering, the energy of the neutron does not actually change, but the angle
does change. For coherent elastic scattering, the angle will depend on which
Bragg edge scattered the neutron. The probability that edge :math:`i` will
scatter then neutron is given by

.. math::
    :label: coherent-elastic-probability

    \frac{s_i}{\sum_j s_j}.

After a Bragg edge has been sampled, the cosine of the angle of scattering is
given analytically by

.. math::
    :label: coherent-elastic-angle

    \mu = 1 - \frac{2E_i}{E}

where :math:`E_i` is the energy of the Bragg edge that scattered the neutron.

.. _incoherent elastic angle:

Outgoing Angle for Incoherent Elastic Scattering
------------------------------------------------

For incoherent elastic scattering, OpenMC has two methods for calculating the
cosine of the angle of scattering. The first method uses the Debye-Waller
integral, :math:`W'`, and the characteristic bound cross section as given
directly in an ENDF-6 formatted file. In this case, the cosine of the angle of
scattering can be sampled by inverting equation 7.4 from the `ENDF-6 Format`_:

.. math::
    :label: incoherent-elastic-mu-exact

    \mu = \frac{1}{c} \log \left( 1 + \xi \left( e^{2c} - 1 \right) \right) - 1

where :math:`\xi` is a random number sampled on unit interval and :math:`c =
2EW'`. In the second method, the probability distribution for the cosine of the
angle of scattering is represented as a series of equally-likely discrete
cosines :math:`\mu_{i,j}` for each incoming energy :math:`E_i` on the thermal
elastic energy grid. First the outgoing angle bin :math:`j` is sampled. Then, if
the incoming energy of the neutron satisfies :math:`E_i < E < E_{i+1}` the
cosine of the angle of scattering is

.. math::
    :label: incoherent-elastic-angle

    \mu' = \mu_{i,j} + f (\mu_{i+1,j} - \mu_{i,j})

where the interpolation factor is defined as

.. math::
    :label: sab-interpolation-factor

    f = \frac{E - E_i}{E_{i+1} - E_i}.

To better represent the true, continuous nature of the cosine distribution, the
sampled value of :math:`mu'` is then "smeared" based on the neighboring values.
First, values of :math:`\mu` are calculated for outgoing angle bins :math:`j-1`
and :math:`j+1`:

.. math::
    :label: incoherent-elastic-smear1

    \mu_\text{left} = \mu_{i,j-1} + f (\mu_{i+1,j-1} - \mu_{i,j-1}) \\

    \mu_\text{right} = \mu_{i,j+1} + f (\mu_{i+1,j+1} - \mu_{i,j+1}).

Then, a final cosine is calculated as:

.. math::
    :label: incoherent-elastic-smear2

    \mu = \mu' + \min (\mu - \mu_\text{left}, \mu + \mu_\text{right} ) \cdot
    \left( \xi - \frac{1}{2} \right)

where :math:`\xi` is again a random number sampled on the unit interval. Care
must be taken to ensure that :math:`\mu` does not fall outside the interval
:math:`[-1,1]`.

Outgoing Energy and Angle for Inelastic Scattering
--------------------------------------------------

Each |sab| table provides a correlated angle-energy secondary distribution for
neutron thermal inelastic scattering.  There are three representations used
in the ACE thermal scattering data: equiprobable discrete outgoing
energies, non-uniform yet still discrete outgoing energies, and continuous
outgoing energies with corresponding probability and cumulative distribution
functions provided in tabular format.  These three representations all
represent the angular distribution in a common format, using a series of
discrete equiprobable outgoing cosines.

Equi-Probable Outgoing Energies
+++++++++++++++++++++++++++++++

If the thermal data was processed with :math:`iwt = 1` in NJOY, then the
outgoing energy spectra is represented in the ACE data as a set of discrete and
equiprobable outgoing energies.  The procedure to determine the outgoing energy
and angle is as such. First, the interpolation factor is determined from
equation :eq:`sab-interpolation-factor`.  Then, an outgoing energy bin is
sampled from a uniform distribution and then interpolated between values
corresponding to neighboring incoming energies:

.. math::
    :label: inelastic-energy

    E = E_{i,j} + f (E_{i+1,j} - E_{i,j})

where :math:`E_{i,j}` is the j-th outgoing energy corresponding to the i-th
incoming energy. For each combination of incoming and outgoing energies, there
is a series equiprobable outgoing cosines. An outgoing cosine bin is sampled
uniformly and then the final cosine is interpolated on the incoming energy grid:

.. math::
    :label: inelastic-angle

    \mu = \mu_{i,j,k} + f (\mu_{i+1,j,k} - \mu_{i,j,k})

where :math:`\mu_{i,j,k}` is the k-th outgoing cosine corresponding to the j-th
outgoing energy and the i-th incoming energy.

Skewed Equi-Probable Outgoing Energies
++++++++++++++++++++++++++++++++++++++

If the thermal data was processed with :math:`iwt=0` in NJOY, then the
outgoing energy spectra is represented in the ACE data according to the
following: the first and last outgoing energies have a relative probability of
1, the second and second-to-last energies have a relative probability of 4, and
all other energies have a relative probability of 10.  The procedure to
determine the outgoing energy and angle is similar to the method discussed
above, except that the sampled probability distribution is now skewed
accordingly.

Continuous Outgoing Energies
++++++++++++++++++++++++++++

If the thermal data was processed with :math:`iwt=2` in NJOY, then the outgoing
energy spectra is represented by a continuous outgoing energy spectra in tabular
form with linear-linear interpolation.  The sampling of the outgoing energy
portion of this format is very similar to :ref:`correlated-energy-angle`, but
the sampling of the correlated angle is performed as it was in the other two
representations discussed in this sub-section.  In the Law 61 algorithm, we
found an interpolation factor :math:`f`, statistically sampled an incoming
energy bin :math:`\ell`, and sampled an outgoing energy bin :math:`j` based on
the tabulated cumulative distribution function. Once the outgoing energy has
been determined with equation :eq:`continuous-eout`, we then need to decide
which angular distribution data to use.  Like the linear-linear interpolation
case in Law 61, the angular distribution closest to the sampled value of the
cumulative distribution function for the outgoing energy is utilized.  The
actual algorithm utilized to sample the outgoing angle is shown in equation
:eq:`inelastic-angle`. As in the case of incoherent elastic scattering with
discrete cosine bins, the sampled cosine is :ref:`smeared <incoherent elastic
angle>` over neighboring angle bins to better approximate a continuous
distribution.

.. _probability_tables:

----------------------------------------------
Unresolved Resonance Region Probability Tables
----------------------------------------------

Note that unresolved resonance treatments are only applicable to
continuous-energy transport.

In the unresolved resonance energy range, resonances may be so closely spaced
that it is not possible for experimental measurements to resolve all
resonances. To properly account for self-shielding in this energy range, OpenMC
uses the `probability table method`_. For most thermal reactors, the use
of probability tables will not significantly affect problem results. However,
for some fast reactors and other problems with an appreciable flux spectrum in
the unresolved resonance range, not using probability tables may lead to
incorrect results.

Probability tables in the ACE format are generated from the UNRESR module in
NJOY following the method of Levitt. A similar method employed for the RACER and
MC21_ Monte Carlo codes is described in a paper by `Sutton and Brown`_. For the
discussion here, we will focus only on use of the probability table table as it
appears in the ACE format.

Each probability table for a nuclide contains the following information at a
number of incoming energies within the unresolved resonance range:

- Cumulative probabilities for cross section bands;
- Total cross section (or factor) in each band;
- Elastic scattering cross section (or factor) in each band;
- Fission cross section (or factor) in each band;
- :math:`(n,\gamma)` cross section (or factor) in each band; and
- Neutron heating number (or factor) in each band.

It should be noted that unresolved resonance probability tables affect only
integrated cross sections and no extra data need be given for secondary
angle/energy distributions. Secondary distributions for elastic and inelastic
scattering would be specified whether or not probability tables were present.

The procedure for determining cross sections in the unresolved range using
probability tables is as follows. First, the bounding incoming energies are
determined, i.e. find :math:`i` such that :math:`E_i < E < E_{i+1}`. We then
sample a cross section band :math:`j` using the cumulative probabilities for
table :math:`i`. This allows us to then calculate the elastic, fission, and
capture cross sections from the probability tables interpolating between
neighboring incoming energies. If interpolation is specified, then
the cross sections are calculated as

.. math::
    :label: ptables-linlin

    \sigma = \sigma_{i,j} + f (\sigma_{i+1,j} - \sigma{i,j})

where :math:`\sigma_{i,j}` is the j-th band cross section corresponding to the
i-th incoming neutron energy and :math:`f` is the interpolation factor defined
in the same manner as :eq:`sab-interpolation-factor`. If logarithmic
interpolation is specified, the cross sections are calculated as

.. math::
    :label: ptables-loglog

    \sigma = \exp \left ( \log \sigma_{i,j} + f \log
    \frac{\sigma_{i+1,j}}{\sigma_{i,j}} \right )

where the interpolation factor is now defined as

.. math::
    :label: log-interpolation-factor

    f = \frac{\log \frac{E}{E_i}}{\log \frac{E_{i+1}}{E_i}}.

A flag is also present in the probability table that specifies whether an
inelastic cross section should be calculated. If so, this is done from a normal
reaction cross section (either MT=51 or a special MT). Finally, if the
cross sections defined are above are specified to be factors and not true
cross sections, they are multiplied by the underlying smooth cross section in
the unresolved range to get the actual cross sections. Lastly, the total cross
section is calculated as the sum of the elastic, fission, capture, and inelastic
cross sections.

-----------------------------
Variance Reduction Techniques
-----------------------------

.. _survival_biasing:

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
scoring events. 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, absorption reactions are prohibited from occurring and
instead, 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 serve as source sites for the next generation in the method of
successive generations. 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.

Weight Windows
--------------

In fixed source problems, it can often be difficult to obtain sufficiently low
variance on tallies in regions that are far from the source. The `weight window
method <https://doi.org/10.13182/FST84-A23082>`_ was developed to increase the
population of particles in important spatial regions and energy ranges by
controlling particle weights. Each spatial region and particle energy range is
assigned upper and lower weight bounds, :math:`w_u` and :math:`w_\ell`,
respectively. When a particle is in a given spatial region / energy range, its
weight, :math:`w`, is compared to the lower and upper bounds. If the weight of
the particle is above the upper weight bound, the particle is split into
:math:`N` particles, where

.. math::
    :label: ww-split

    N = \min(N_{max}, \lceil w/w_u \rceil)

and :math:`N_{max}` is a user-defined maximum number of splits. To ensure a
fair game, each of the :math:`N` particles is assigned a weight :math:`w/N`. If
the weight is below :math:`w_\ell`, it is Russian rouletted as described in
:ref:`survival_biasing` with a survival weight :math:`w_s` that is set equal to

.. math::
    :label: ww-survival-weight

    w_s = \min(N_{max} w, f_s w_l)

where :math:`f_s` is a user-defined survival weight ratio greater than one.

On top of the standard weight window method described above, OpenMC implements
two additional checks intended to mitigate problems with long histories. First,
particles with a weight that falls below some very small cutoff (defaults to
:math:`10^{-38}`) are killed with no Russian rouletting. Additionally, the total
number of splits experienced by a particle is tracked and if it reaches some
maximum value, it is prohibited from splitting further.

At present, OpenMC allows weight windows to be defined on all supported mesh
types.

.. only:: html

   .. rubric:: References

.. [Gelbard] Ely M. Gelbard, "Epithermal Scattering in VIM," FRA-TM-123, Argonne
   National Laboratory (1979).

.. [Squires] G. L. Squires, *Introduction to the Theory of Thermal Neutron
   Scattering*, Cambridge University Press (1978).

.. [Williams] M. M. R. Williams, *The Slowing Down and Thermalization of
   Neutrons*, North-Holland Publishing Co., Amsterdam (1966). **Note:** This
   book can be obtained for free from the OECD_.

.. |sab| replace:: S(:math:`\alpha,\beta,T`)

.. _SIGMA1 method: https://doi.org/10.13182/NSE76-1

.. _scaled interpolation: https://doi.org/10.13182/NSE73-A26575

.. _probability table method: https://doi.org/10.13182/NSE72-3

.. _Watt fission spectrum: https://doi.org/10.1103/PhysRev.87.1037

.. _Foderaro: https://dspace.mit.edu/handle/1721.1/1716

.. _OECD: https://www.oecd-nea.org/tools/abstract/detail/NEA-1792

.. _NJOY: https://www.njoy21.io/NJOY2016/

.. _PREPRO: https://www-nds.iaea.org/public/endf/prepro/

.. _ENDF-6 Format: https://www.oecd-nea.org/dbdata/data/manual-endf/endf102.pdf

.. _Monte Carlo Sampler: https://mcnp.lanl.gov/pdf_files/TechReport_1983_LANL_LA-9721-MS_EverettCashwell.pdf

.. _LA-UR-14-27694: https://www.osti.gov/biblio/1159204

.. _MC21: https://www.osti.gov/biblio/903083

.. _Romano: https://doi.org/10.1016/j.cpc.2014.11.001

.. _Sutton and Brown: https://www.osti.gov/biblio/307911

.. _lectures: https://mcnp.lanl.gov/pdf_files/TechReport_2005_LANL_LA-UR-05-4983_Brown.pdf