New Feature: electron/positron independent source.

docs/source/methods/charged_particles_physics.rst

@@ -0,0 +1,358 @@
+.. _methods_charged_particles_physics:
+
+=========================
+Charged Particles Physics
+=========================
+
+OpenMC neglects the spatial transport of charged particles (electrons and positrons), assuming they deposit all their energy locally and produce bremsstrahlung photons at their birth location.
+This approximation, called thick-target bremsstrahlung (TTB) approximation is justified by the fact that charged particles have much shorter stopping ranges compared to neutrons and photons, especially in high-density materials.
+
+------------------------------
+Charged Particles Interactions
+------------------------------
+
+Bremsstrahlung
+--------------
+
+When a charged particle is decelerated in the field of an atom, some of its
+kinetic energy is converted into electromagnetic radiation known as
+bremsstrahlung, or 'braking radiation'. In each event, an electron or positron
+with kinetic energy :math:`T` generates a photon with an energy :math:`E`
+between :math:`0` and :math:`T`. Bremsstrahlung is described by a cross section
+that is differential in photon energy, in the direction of the emitted photon,
+and in the final direction of the charged particle. However, in Monte Carlo
+simulations it is typical to integrate over the angular variables to obtain a
+single differential cross section with respect to photon energy, which is often
+expressed in the form
+
+.. math::
+    :label: bremsstrahlung-dcs
+
+    \frac{d\sigma_{\text{br}}}{dE} = \frac{Z^2}{\beta^2} \frac{1}{E}
+    \chi(Z, T, \kappa),
+
+where :math:`\kappa = E/T` is the reduced photon energy and :math:`\chi(Z, T,
+\kappa)` is the scaled bremsstrahlung cross section, which is experimentally
+measured.
+
+Because electrons are attracted to atomic nuclei whereas positrons are
+repulsed, the cross section for positrons is smaller, though it approaches that
+of electrons in the high energy limit. To obtain the positron cross section, we
+multiply :eq:`bremsstrahlung-dcs` by the :math:`\kappa`-independent factor used
+in Salvat_,
+
+.. math::
+    :label: positron-factor
+
+    \begin{aligned}
+    F_{\text{p}}(Z,T) =
+    & 1 - \text{exp}(-1.2359\times 10^{-1}t + 6.1274\times 10^{-2}t^2 - 3.1516\times 10^{-2}t^3 \\
+    & + 7.7446\times 10^{-3}t^4 - 1.0595\times 10^{-3}t^5 + 7.0568\times 10^{-5}t^6 \\
+    & - 1.8080\times 10^{-6}t^7),
+    \end{aligned}
+
+where
+
+.. math::
+    :label: positron-factor-t
+
+    t = \ln\left(1 + \frac{10^6}{Z^2}\frac{T}{\text{m}_\text{e}c^2} \right).
+
+:math:`F_{\text{p}}(Z,T)` is the ratio of the radiative stopping powers for
+positrons and electrons. Stopping power describes the average energy loss per
+unit path length of a charged particle as it passes through matter:
+
+.. math::
+    :label: stopping-power
+
+    -\frac{dT}{ds} = n \int E \frac{d\sigma}{dE} dE \equiv S(T),
+
+where :math:`n` is the number density of the material and :math:`d\sigma/dE` is
+the cross section differential in energy loss. The total stopping power
+:math:`S(T)` can be separated into two components: the radiative stopping
+power :math:`S_{\text{rad}}(T)`, which refers to energy loss due to
+bremsstrahlung, and the collision stopping power :math:`S_{\text{col}}(T)`,
+which refers to the energy loss due to inelastic collisions with bound
+electrons in the material that result in ionization and excitation. The
+radiative stopping power for electrons is given by
+
+.. math::
+    :label: radiative-stopping-power
+
+    S_{\text{rad}}(T) = n \frac{Z^2}{\beta^2} T \int_0^1 \chi(Z,T,\kappa)
+    d\kappa.
+
+
+To obtain the radiative stopping power for positrons,
+:eq:`radiative-stopping-power`  is multiplied by :eq:`positron-factor`.
+
+While the models for photon interactions with matter described above can safely
+assume interactions occur with free atoms, sampling the target atom based on
+the macroscopic cross sections, molecular effects cannot necessarily be
+disregarded for charged particle treatment. For compounds and mixtures, the
+bremsstrahlung cross section is calculated using Bragg's additivity rule as
+
+.. math::
+    :label: material-bremsstrahlung-dcs
+
+    \frac{d\sigma_{\text{br}}}{dE} = \frac{1}{\beta^2 E} \sum_i \gamma_i Z^2_i
+    \chi(Z_i, T, \kappa),
+
+where the sum is over the constituent elements and :math:`\gamma_i` is the
+atomic fraction of the :math:`i`-th element. Similarly, the radiative stopping
+power is calculated using Bragg's additivity rule as
+
+.. math::
+    :label: material-radiative-stopping-power
+
+    S_{\text{rad}}(T) = \sum_i w_i S_{\text{rad},i}(T),
+
+where :math:`w_i` is the mass fraction of the :math:`i`-th element and
+:math:`S_{\text{rad},i}(T)` is found for element :math:`i` using
+:eq:`radiative-stopping-power`. The collision stopping power, however, is a
+function of certain quantities such as the mean excitation energy :math:`I` and
+the density effect correction :math:`\delta_F` that depend on molecular
+properties. These quantities cannot simply be summed over constituent elements
+in a compound, but should instead be calculated for the material. The Bethe
+formula can be used to find the collision stopping power of the material:
+
+.. math::
+    :label: material-collision-stopping-power
+
+    S_{\text{col}}(T) = \frac{2 \pi r_e^2 m_e c^2}{\beta^2} N_A \frac{Z}{A_M}
+    [\ln(T^2/I^2) + \ln(1 + \tau/2) + F(\tau) - \delta_F(T)],
+
+where :math:`N_A` is Avogadro's number, :math:`A_M` is the molar mass,
+:math:`\tau = T/m_e`, and :math:`F(\tau)` depends on the particle type. For
+electrons,
+
+.. math::
+    :label: F-electron
+
+    F_{-}(\tau) = (1 - \beta^2)[1 + \tau^2/8 - (2\tau + 1) \ln2],
+
+while for positrons
+
+.. math::
+    :label: F-positron
+
+    F_{+}(\tau) = 2\ln2 - (\beta^2/12)[23 + 14/(\tau + 2) + 10/(\tau + 2)^2 +
+    4/(\tau + 2)^3].
+
+The density effect correction :math:`\delta_F` takes into account the reduction
+of the collision stopping power due to the polarization of the material the
+charged particle is passing through by the electric field of the particle.
+It can be evaluated using the method described by Sternheimer_, where the
+equation for :math:`\delta_F` is
+
+.. math::
+    :label: density-effect-correction
+
+    \delta_F(\beta) = \sum_{i=1}^n f_i \ln[(l_i^2 + l^2)/l_i^2] -
+    l^2(1-\beta^2).
+
+Here, :math:`f_i` is the oscillator strength of the :math:`i`-th transition,
+given by :math:`f_i = n_i/Z`, where :math:`n_i` is the number of electrons in
+the :math:`i`-th subshell. The frequency :math:`l` is the solution of the
+equation
+
+.. math::
+    :label: density-effect-l
+
+    \frac{1}{\beta^2} - 1 = \sum_{i=1}^{n} \frac{f_i}{\bar{\nu}_i^2 + l^2},
+
+where :math:`\bar{v}_i` is defined as
+
+.. math::
+    :label: density-effect-nubar
+
+    \bar{\nu}_i = h\nu_i \rho / h\nu_p.
+
+The plasma energy :math:`h\nu_p` of the medium is given by
+
+.. math::
+    :label: plasma-frequency
+
+    h\nu_p = \sqrt{\frac{(hc)^2 r_e \rho_m N_A Z}{\pi A}},
+
+where :math:`A` is the atomic weight and :math:`\rho_m` is the density of the
+material. In :eq:`density-effect-nubar`, :math:`h\nu_i` is the oscillator
+energy, and :math:`\rho` is an adjustment factor introduced to give agreement
+between the experimental values of the oscillator energies and the mean
+excitation energy. The :math:`l_i` in :eq:`density-effect-correction` are
+defined as
+
+.. math::
+    :label: density-effect-li
+
+    \begin{aligned}
+    l_i &= (\bar{\nu}_i^2 + 2/3f_i)^{1/2} ~~~~&\text{for}~~ \bar{\nu}_i > 0 \\
+    l_n &= f_n^{1/2} ~~~~&\text{for}~~ \bar{\nu}_n = 0,
+    \end{aligned}
+
+where the second case applies to conduction electrons. For a conductor,
+:math:`f_n` is given by :math:`n_c/Z`, where :math:`n_c` is the effective
+number of conduction electrons, and :math:`v_n = 0`. The adjustment factor
+:math:`\rho` is determined using the equation for the mean excitation energy:
+
+.. math::
+    :label: mean-excitation-energy
+
+    \ln I = \sum_{i=1}^{n-1} f_i \ln[(h\nu_i\rho)^2 + 2/3f_i(h\nu_p)^2]^{1/2} +
+    f_n \ln (h\nu_pf_n^{1/2}).
+
+.. _ttb:
+
+
+Thick-Target Bremsstrahlung Approximation
++++++++++++++++++++++++++++++++++++++++++
+
+Since charged particles lose their energy on a much shorter distance scale than
+neutral particles, not much error should be introduced by neglecting to
+transport electrons. However, the bremsstrahlung emitted from high energy
+electrons and positrons can travel far from the interaction site. Thus, even
+without a full electron transport mode it is necessary to model bremsstrahlung.
+We use a thick-target bremsstrahlung (TTB) approximation based on the models in
+Salvat_ and Kaltiaisenaho_ for generating bremsstrahlung photons, which assumes
+the charged particle loses all its energy in a single homogeneous material
+region.
+
+To model bremsstrahlung using the TTB approximation, we need to know the number
+of photons emitted by the charged particle and the energy distribution of the
+photons. These quantities can be calculated using the continuous slowing down
+approximation (CSDA). The CSDA assumes charged particles lose energy
+continuously along their trajectory with a rate of energy loss equal to the
+total stopping power, ignoring fluctuations in the energy loss. The
+approximation is useful for expressing average quantities that describe how
+charged particles slow down in matter. For example, the CSDA range approximates
+the average path length a charged particle travels as it slows to rest:
+
+.. math::
+    :label: csda-range
+
+    R(T) = \int^T_0 \frac{dT'}{S(T')}.
+
+Actual path lengths will fluctuate around :math:`R(T)`. The average number of
+photons emitted per unit path length is given by the inverse bremsstrahlung
+mean free path:
+
+.. math::
+    :label: inverse-bremsstrahlung-mfp
+
+    \lambda_{\text{br}}^{-1}(T,E_{\text{cut}})
+    = n\int_{E_{\text{cut}}}^T\frac{d\sigma_{\text{br}}}{dE}dE
+    = n\frac{Z^2}{\beta^2}\int_{\kappa_{\text{cut}}}^1\frac{1}{\kappa}
+    \chi(Z,T,\kappa)d\kappa.
+
+The lower limit of the integral in :eq:`inverse-bremsstrahlung-mfp` is non-zero
+because the bremsstrahlung differential cross section diverges for small photon
+energies but is finite for photon energies above some cutoff energy
+:math:`E_{\text{cut}}`. The mean free path
+:math:`\lambda_{\text{br}}^{-1}(T,E_{\text{cut}})` is used to calculate the
+photon number yield, defined as the average number of photons emitted with
+energy greater than :math:`E_{\text{cut}}` as the charged particle slows down
+from energy :math:`T` to :math:`E_{\text{cut}}`. The photon number yield is
+given by
+
+.. math::
+    :label: photon-number-yield
+
+    Y(T,E_{\text{cut}}) = \int^{R(T)}_{R(E_{\text{cut}})}
+    \lambda_{\text{br}}^{-1}(T',E_{\text{cut}})ds = \int_{E_{\text{cut}}}^T
+    \frac{\lambda_{\text{br}}^{-1}(T',E_{\text{cut}})}{S(T')}dT'.
+
+:math:`Y(T,E_{\text{cut}})` can be used to construct the energy spectrum of
+bremsstrahlung photons: the number of photons created with energy between
+:math:`E_1` and :math:`E_2` by a charged particle with initial kinetic energy
+:math:`T` as it comes to rest is given by :math:`Y(T,E_1) - Y(T,E_2)`.
+
+To simulate the emission of bremsstrahlung photons, the total stopping power
+and bremsstrahlung differential cross section for positrons and electrons must
+be calculated for a given material using :eq:`material-bremsstrahlung-dcs` and
+:eq:`material-radiative-stopping-power`. These quantities are used to build the
+tabulated bremsstrahlung energy PDF and CDF for that material for each incident
+energy :math:`T_k` on the energy grid. The following algorithm is then applied
+to sample the photon energies:
+
+1. For an incident charged particle with energy :math:`T`, sample the number of
+   emitted photons as
+
+   .. math::
+
+       N = \lfloor Y(T,E_{\text{cut}}) + \xi_1 \rfloor.
+
+2. Rather than interpolate the PDF between indices :math:`k` and :math:`k+1`
+   for which :math:`T_k < T < T_{k+1}`, which is computationally expensive, use
+   the composition method and sample from the PDF at either :math:`k` or
+   :math:`k+1`. Using linear interpolation on a logarithmic scale, the PDF can
+   be expressed as
+
+   .. math::
+
+       p_{\text{br}}(T,E) = \pi_k p_{\text{br}}(T_k,E) + \pi_{k+1}
+       p_{\text{br}}(T_{k+1},E),
+
+   where the interpolation weights are
+
+   .. math::
+
+       \pi_k = \frac{\ln T_{k+1} - \ln T}{\ln T_{k+1} - \ln T_k},~~~
+       \pi_{k+1} = \frac{\ln T - \ln T_k}{\ln T_{k+1} - \ln T_k}.
+
+   Sample either the index :math:`i = k` or :math:`i = k+1` according to the
+   point probabilities :math:`\pi_{k}` and :math:`\pi_{k+1}`.
+
+3. Determine the maximum value of the CDF :math:`P_{\text{br,max}}`.
+
+3. Sample the photon energies using the inverse transform method with the
+   tabulated CDF :math:`P_{\text{br}}(T_i, E)` i.e.,
+
+   .. math::
+
+       E = E_j \left[ (1 + a_j) \frac{\xi_2 P_{\text{br,max}} -
+       P_{\text{br}}(T_i, E_j)} {E_j p_{\text{br}}(T_i, E_j)} + 1
+       \right]^{\frac{1}{1 + a_j}}
+
+   where the interpolation factor :math:`a_j` is given by
+
+   .. math::
+
+       a_j = \frac{\ln p_{\text{br}}(T_i,E_{j+1}) - \ln p_{\text{br}}(T_i,E_j)}
+       {\ln E_{j+1} - \ln E_j}
+
+   and :math:`P_{\text{br}}(T_i, E_j) \le \xi_2 P_{\text{br,max}} \le
+   P_{\text{br}}(T_i, E_{j+1})`.
+
+We ignore the range of the electron or positron, i.e., the bremsstrahlung
+photons are produced in the same location that the charged particle was
+created. The direction of the photons is assumed to be the same as the
+direction of the incident charged particle, which is a reasonable approximation
+at higher energies when the bremsstrahlung radiation is emitted at small
+angles.
+
+
+Electron-Positron Annihilation
+------------------------------
+
+When a positron collides with an electron, both particles are annihilated and
+generally two photons with equal energy are created. If the kinetic energy of
+the positron is high enough, the two photons can have different energies, and
+the higher-energy photon is emitted preferentially in the direction of flight
+of the positron. It is also possible to produce a single photon if the
+interaction occurs with a bound electron, and in some cases three (or, rarely,
+even more) photons can be emitted. However, the annihilation cross section is
+largest for low-energy positrons, and as the positron energy decreases, the
+angular distribution of the emitted photons becomes isotropic.
+
+In OpenMC, we assume the most likely case in which a low-energy positron (which
+has already lost most of its energy to bremsstrahlung radiation) interacts with
+an electron which is free and at rest. Two photons with energy equal to the
+electron rest mass energy :math:`m_e c^2 = 0.511` MeV are emitted isotropically
+in opposite directions.
+
+
+.. _Kaltiaisenaho: https://aaltodoc.aalto.fi/bitstream/handle/123456789/21004/master_Kaltiaisenaho_Toni_2016.pdf
+
+.. _Salvat: https://doi.org/10.1787/32da5043-en
+
+.. _Sternheimer: https://doi.org/10.1103/PhysRevB.26.6067

docs/source/methods/index.rst

@@ -14,11 +14,12 @@ Theory and Methodology
     random_numbers
     neutron_physics
     photon_physics
+    charged_particles_physics
     tallies
     eigenvalue
     depletion
     energy_deposition
     parallelization
     cmfd
     variance_reduction
-    random_ray
+    random_ray