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Greatly expanded documentation.
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Paul Romano
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24
docs/source/methods/geometry.rst
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24
docs/source/methods/geometry.rst
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.. _methods_geometry:
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========
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Geometry
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========
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-------------------
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Reflective Surfaces
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-------------------
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In general, a surface can be written in the form :math:`f(x,y,z) = 0`. If a
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neutron is traveling in direction :math:`\vec{v}` and crosses a reflective
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surface of the above form, it can be shown that the velocity vector will then
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become
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.. math::
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\mathbf{v'} = \mathbf{v} - 2 (\mathbf{v} \cdot \hat{\mathbf{n}})
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\hat{\mathbf{n}}
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where :math:`\hat{\mathbf{n}}` is a unit vector normal to the surface at the
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point of the surface crossing. The direction of the surface normal will be the
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gradient to the surface at the point of crossing, i.e. :math:`\mathbf{n} =
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\nabla f(x,y,z)`.
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@ -1,118 +1,18 @@
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.. _methods:
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===========
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Methodology
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===========
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======================
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Theory and Methodology
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======================
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The OpenMC code solves the neutron transport equation using the Monte Carlo
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method whereby particles are tracked as they randomly move through a geometry,
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undergoing collisions, and creating secondary particles.
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-------------------
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Reflective Surfaces
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-------------------
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.. toctree::
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:numbered:
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:maxdepth: 2
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In general, a surface can be written in the form :math:`f(x,y,z) = 0`. If a
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neutron is traveling in direction :math:`\vec{v}` and crosses a reflective
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surface of the above form, it can be shown that the velocity vector will then
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become
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.. math::
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\mathbf{v'} = \mathbf{v} - 2 (\mathbf{v} \cdot \hat{\mathbf{n}})
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\hat{\mathbf{n}}
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where :math:`\hat{\mathbf{n}}` is a unit vector normal to the surface at the
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point of the surface crossing. The direction of the surface normal will be the
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gradient to the surface at the point of crossing, i.e. :math:`\mathbf{n} =
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\nabla f(x,y,z)`.
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------------------------------
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Free Gas Scattering Kinematics
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------------------------------
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When a neutron scatters off of a nucleus, many times it is assumed that the
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target nucleus is at rest. However, if the material is at a temperature greater
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than 0 K, it will have motion associated with the thermal vibration. Thus, the
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velocity of the neutrno relative to the target nucleus is in general not the
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same as the velocity of the neutron entering the collision.
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The affect of the thermal motion on the interaction probability can be written
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as
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.. math::
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:label: freegas1
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v_n \sigma (v_n, T) = \int_0^\infty d\mathbf{v}_T \sigma(v_r, 0)
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\mathbf{v}_r p(\mathbf{v}_T)
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One assumption we can make here is that the velocity distribution for the
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thermal motion is isotropic, i.e.
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.. math::
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:label: freegas2
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p(\mathbf{v}_T) d\mathbf{v}_T = \frac{1}{4\pi} p(v_T) dv_T d\mu d\phi
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With this assumption, we can now rewrite equation :eq:`freegas1` as
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.. math::
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:label: freegas3
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v_n \sigma (v_n, T) = \frac{1}{2} \int_{-1}^1 d\mu \int\limits_{v_r > 0}
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v_r \sigma (v_r, 0) p(v_T) dv_T
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To change the outer variable of integration from :math:`\mu` to :math:`v_r`, we
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can establish a relation between these variables based on the law of cosines.
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.. math::
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:label: lawcosine
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2 v_n v_T \mu = v_n^2 + v_T^2 - v_r^2
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The probability distribution for the magnitude of the velocity of the target
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nucleus and the angle between the neutron and target velocity is
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.. math::
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:label: freegas4
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P(v_T, \mu) = \frac{\sigma (v_r, 0) v_r P(v_T)}{2 \sigma (v_n, T) v_n}
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It is normally assumed that :math:`\sigma (v_r, 0)` is constant over the range
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of relative velocities of interest. This is a good assumption for almost all
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cases since the elastic scattering cross section varies slowly with velocity for
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light nuclei, and for heavy nuclei where large variations can occur due to
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resonance scattering, the moderating effect is rather small. Nonetheless, this
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assumption can cause incorrect answers in systems with U-238 where the low-lying
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resonances can cause a significant amount of upscatter that would be ignored by
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this assumption.
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With this (sometimes incorrect) assumption, we see that the probability
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distribution is proportional to
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.. math::
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:label: freegas5
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P(v_T, \mu) \propto v_r P(v_T) = | v_n - v_T | P(v_T)
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We can divide this probability distribution into two parts as such:
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.. math::
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:label: freegas6
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P(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\
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f_1(v_T, \mu) &= \frac{| v_n - v_T |}{\hat{f_1} (v_n + v_T)} \\
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f_2(v_T) &= (v_n + v_T) P(v_T)
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In general, any probability distribution function of the form :math:`p(x) =
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f_1(x) f_2(x)` with :math:`f_1(x)` bounded can be sampled by sampling
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:math:`x_s` from the distribution
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.. math:: \frac{f_2(x)}{\int f_2(x) dx}
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and accepting it with probability
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.. math:: \frac{f_1(x_s)}{\max f_1(x)}
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It is normally assumed that the velocity distribution of the target nucleus
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assumes a Maxwellian distribution in velocity.
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introduction
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geometry
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physics
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tallies
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10
docs/source/methods/introduction.rst
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10
docs/source/methods/introduction.rst
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.. _methods_introduction:
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============
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Introduction
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============
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---------------------
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Criticality Algorithm
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---------------------
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95
docs/source/methods/physics.rst
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95
docs/source/methods/physics.rst
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.. _methods_physics:
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=======
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Physics
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=======
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------------------------------
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Free Gas Scattering Kinematics
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------------------------------
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When a neutron scatters off of a nucleus, many times it is assumed that the
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target nucleus is at rest. However, if the material is at a temperature greater
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than 0 K, it will have motion associated with the thermal vibration. Thus, the
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velocity of the neutrno relative to the target nucleus is in general not the
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same as the velocity of the neutron entering the collision.
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The affect of the thermal motion on the interaction probability can be written
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as
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.. math::
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:label: freegas1
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v_n \sigma (v_n, T) = \int_0^\infty d\mathbf{v}_T \sigma(v_r, 0)
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\mathbf{v}_r p(\mathbf{v}_T)
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One assumption we can make here is that the velocity distribution for the
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thermal motion is isotropic, i.e.
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.. math::
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:label: freegas2
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p(\mathbf{v}_T) d\mathbf{v}_T = \frac{1}{4\pi} p(v_T) dv_T d\mu d\phi
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With this assumption, we can now rewrite equation :eq:`freegas1` as
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.. math::
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:label: freegas3
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v_n \sigma (v_n, T) = \frac{1}{2} \int_{-1}^1 d\mu \int\limits_{v_r > 0}
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v_r \sigma (v_r, 0) p(v_T) dv_T
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To change the outer variable of integration from :math:`\mu` to :math:`v_r`, we
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can establish a relation between these variables based on the law of cosines.
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.. math::
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:label: lawcosine
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2 v_n v_T \mu = v_n^2 + v_T^2 - v_r^2
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The probability distribution for the magnitude of the velocity of the target
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nucleus and the angle between the neutron and target velocity is
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.. math::
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:label: freegas4
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P(v_T, \mu) = \frac{\sigma (v_r, 0) v_r P(v_T)}{2 \sigma (v_n, T) v_n}
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It is normally assumed that :math:`\sigma (v_r, 0)` is constant over the range
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of relative velocities of interest. This is a good assumption for almost all
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cases since the elastic scattering cross section varies slowly with velocity for
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light nuclei, and for heavy nuclei where large variations can occur due to
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resonance scattering, the moderating effect is rather small. Nonetheless, this
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assumption can cause incorrect answers in systems with U-238 where the low-lying
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resonances can cause a significant amount of upscatter that would be ignored by
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this assumption.
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With this (sometimes incorrect) assumption, we see that the probability
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distribution is proportional to
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.. math::
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:label: freegas5
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P(v_T, \mu) \propto v_r P(v_T) = | v_n - v_T | P(v_T)
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We can divide this probability distribution into two parts as such:
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.. math::
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:label: freegas6
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P(v_T, \mu) &= f_1(v_T, \mu) f_2(v_T) \\
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f_1(v_T, \mu) &= \frac{| v_n - v_T |}{\hat{f_1} (v_n + v_T)} \\
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f_2(v_T) &= (v_n + v_T) P(v_T)
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In general, any probability distribution function of the form :math:`p(x) =
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f_1(x) f_2(x)` with :math:`f_1(x)` bounded can be sampled by sampling
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:math:`x_s` from the distribution
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.. math:: \frac{f_2(x)}{\int f_2(x) dx}
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and accepting it with probability
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.. math:: \frac{f_1(x_s)}{\max f_1(x)}
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It is normally assumed that the velocity distribution of the target nucleus
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assumes a Maxwellian distribution in velocity.
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17
docs/source/methods/tallies.rst
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17
docs/source/methods/tallies.rst
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@ -0,0 +1,17 @@
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.. _methods_tallies:
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=======
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Tallies
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=======
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----------------
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Analog Estimator
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----------------
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----------------------
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Track-length Estimator
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----------------------
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---------------
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Surface Current
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---------------
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