crp/OpenMC
1
0
Fork 0
mirror of https://github.com/openmc-dev/openmc.git synced 2026-07-26 13:15:39 -04:00
Code Issues Projects Releases Packages Wiki Activity Actions

Changed beginning of derivation for free gas scattering kinematics.

Browse source
This commit is contained in:
Paul Romano 2012-07-26 14:30:55 -04:00
parent 1f66f140a4
commit 2b024bb7aa
1 changed files with 108 additions and 58 deletions
Download patch file Download diff file Expand all files Collapse all files

166
docs/source/methods/physics.rst
View file

@ -947,21 +947,25 @@ than 0 K, it will have motion associated with the thermal vibration. 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 affect of the thermal motion on the interaction probability can be written
The effect of the thermal motion on the interaction probability can be written
as
.. math::
:label: freegas1
:label: doppler-broaden
v_n \sigma (v_n, T) = \int_0^\infty d\mathbf{v}_T \sigma(v_r, 0)
\mathbf{v}_r p(\mathbf{v}_T)
v_n \bar{\sigma} (v_n, T) = \int d\mathbf{v}_T v_r \sigma(v_r)
\phi (\mathbf{v}_T, T)
where :math:`v_n` is the magnitude of the velocity of the neutron,
:math:`\mathbf{v}_T` is the velocity of the target nucleus, :math:`\mathbf{v}_r`
is the relative velocity, and :math:`T` is the temperature of the target
material. 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
: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:`\phi
(\mathbf{v}_T, T)` is the probability distribution for the target nucleus
velocity at temperature :math:`T` (normally 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]_
@ -983,83 +987,129 @@ 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:`freegas1` 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:`freegas`.
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. The first assumption we can
make is that the velocity distribution for the thermal motion is isotropic, i.e.
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: freegas2
:label: reaction-rate
p(\mathbf{v}_T) d\mathbf{v}_T = \frac{1}{4\pi} p(v_T) dv_T d\mu d\phi
R(\mathbf{v}_T) = || \mathbf{v}_n - \mathbf{v}_T || \sigma ( ||
\mathbf{v}_n - \mathbf{v}_T || ) \phi ( \mathbf{v}_T, T )
With this assumption, we can now rewrite equation :eq:`freegas1` as
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:`\phi (\mathbf{v}_T,
T)`. This probability distribution function can be found by integrating equation
:eq:`reaction-rate` to obtain a normalization factor:
.. math::
:label: freegas3
:label: target-pdf-1
v_n \sigma (v_n, T) = \frac{1}{2} \int_{-1}^1 d\mu \int\limits_{v_r > 0}
dv_T v_r \sigma (v_r, 0) p(v_T)
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)}
after integrating over :math:`d\phi`. To change the outer variable of
integration from :math:`\mu` to :math:`v_r`, we can establish a relation between
these variables based on the law of cosines.
Let us call the normalization factor in the denominator of equation
:eq:`target-pdf-1` :math:`C`.
It is normally 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
can cause incorrect answers in systems with U-238 where the low-lying resonances
can cause a significant amount of up-scatter that would be ignored by this
assumption. Nevertheless, with this 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 || \phi ( \mathbf{v}_T, T ) d\mathbf{v}_T
At this point, we'd like to go from a distribution in :math:`\mathbf{v}_T` to
one in :math:`v_T` and :math:`\mu`, the angle between the neutron and target
velocity vectors. This can be done in the following manner. First we note that
the velocity distribution for the thermal motion (a Maxwellian) is isotropic and
therefore we can write
.. math::
:label: velocity-isotropic
\phi (\mathbf{v}_T, T) d\mathbf{v}_T = \frac{1}{4\pi} \phi (v_T, T) dv_T
d\mu d\phi
Integrating out the azimuthal angle, we can rewrite our desired probability
distribution function as
.. math::
:label: target-pdf-3
p( v_T, \mu ) dv_T d\mu = \frac{\sigma_s}{C'} || \mathbf{v}_n - \mathbf{v}_T
|| \phi ( v_T, T ) dv_T d\mu
Note that 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
The probability distribution for the magnitude of the velocity of the target
nucleus and the angle between the neutron and target velocity is
Thus, we can infer that
.. math::
:label: freegas4
:label: change-terms
P(v_T, \mu) = \frac{\sigma (v_r, 0) v_r P(v_T)}{2 \sigma (v_n, T) v_n}
|| \mathbf{v}_n - \mathbf{v}_T || = || \mathbf{v}_r || = v_r = \sqrt{v_n^2 +
v_T^2 - 2v_n v_T \mu}
It is normally assumed that :math:`\sigma (v_r, 0)` is constant over the range
of relative velocities of interest. This is a good assumption for almost all
cases since the elastic scattering cross section varies slowly with velocity for
light nuclei, and for heavy nuclei where large variations can occur due to
resonance scattering, the moderating effect is rather small. Nonetheless, this
assumption can cause incorrect answers in systems with U-238 where the low-lying
resonances can cause a significant amount of up-scatter that would be ignored by
this assumption.
With this (sometimes incorrect) assumption, we see that the probability
distribution is proportional to
Inserting equation :eq:`change-terms` into :eq:`target-pdf-3`, we obtain
.. math::
:label: freegas5
:label: target-pdf-4
P(v_T, \mu) \propto v_r P(v_T) = | v_n - v_T | P(v_T)
p( v_T, \mu ) dv_T d\mu = \frac{\sigma_s}{C'} \sqrt{v_n^2 + v_T^2 - 2v_n v_T
\mu} \, \phi ( v_T, T ) dv_T d\mu
We can divide this probability distribution into two parts as such:
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: freegas6
:label: divide-pdf
P(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\
f_1(v_T, \mu) &= \frac{| v_n - v_T |}{C (v_n + v_T)} \\
f_2(v_T) &= (v_n + v_T) P(v_T)
p(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\
where :math:`C = \int dv_T \sigma v_r P(v_T)`. In general, any probability
distribution function of the form :math:`p(x) = f_1(x) f_2(x)` with
:math:`f_1(x)` bounded can be sampled by sampling :math:`x_s` from the
distribution
f_1(v_T, \mu) &= \frac{\sigma_s \sqrt{v_n^2 + v_T^2 - 2v_n v_T \mu}}{C'
(v_n + v_T)} \\
f_2(v_T) &= (v_n + v_T) \phi(v_T, T)
In general, any probability distribution function of the form :math:`p(x) =
f_1(x) f_2(x)` with :math:`f_1(x)` bounded can be sampled by sampling
:math:`x_s` from the distribution
.. math::
:label: freegas7
@ -1309,7 +1359,7 @@ References
.. [Foderaro] Anthony Foderaro, *The Elements of Neutron Interaction Theory*,
MIT Press, Cambridge, Massachusetts (1971). **Note:** Students, faculty, and
staff at MIT can obtian a PDF copy of this book for free from the `MIT
staff at MIT can obtain a PDF copy of this book for free from the `MIT
Press`_.
.. [Gelbard] Ely M. Gelbard, "Epithermal Scattering in VIM," FRA-TM-123, Argonne