From 4f69e3474e9366993f02e894d05cfb80015713dd Mon Sep 17 00:00:00 2001 From: amandalund Date: Tue, 5 Mar 2019 19:16:38 -0600 Subject: [PATCH] Update photon physics documentation --- docs/source/methods/photon_physics.rst | 170 +++++++++++++++++++++++-- 1 file changed, 158 insertions(+), 12 deletions(-) diff --git a/docs/source/methods/photon_physics.rst b/docs/source/methods/photon_physics.rst index 028a71de7..bde6d2a99 100644 --- a/docs/source/methods/photon_physics.rst +++ b/docs/source/methods/photon_physics.rst @@ -728,10 +728,18 @@ the cross section differential in energy loss. The total stopping power 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. To obtain -the radiative stopping power for positrons, the radiative stopping power for -electrons is multiplied by :eq:`positron-factor`. Currently, the collision -stopping power for electrons is also used for positrons. +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 @@ -754,14 +762,97 @@ power is calculated using Bragg's additivity rule as 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. 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. Currently, we use Bragg's additivity rule to -calculate the collision stopping power as well, but this is not a good -approximation and should be fixed in the future. +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 = \nu_i \rho / \nu_p. + +The plasma energy :math:`h\nu_p` of the medium is given by + +.. math:: + :label: plasma-frequency + + h\nu_p = \sqrt{(hc)^2/\pi r_e \rho_m N_A Z / 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 + + l_i &= (\bar{v}_i^2 + 2/3f_i)^{1/2} ~~~~&\text{for}~~ \bar{v}_i > 0 \\ + l_n &= f_n^{1/2} ~~~~&\text{for}~~ \bar{v}_n = 0, + +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: @@ -891,6 +982,55 @@ direction of the incident charged particle, which is a reasonable approximation at higher energies when the bremsstrahlung radiation is emitted at small angles. +----------------- +Photon Production +----------------- + +In coupled neutron-photon transport, a source neutron is tracked, and photons +produced from neutron reactions are transported after the neutron's history has +terminated. Since these secondary photons form the photon source for the +problem, it is important to correctly describe their energy and angular +distributions as the accuracy of the calculation relies on the accuracy of this +source. The photon production cross section for a particular reaction :math:`i` +and incident neutron energy :math:`E` is defined as + +.. math:: + :label: photon-production-xs + + \sigma_{\gamma, i}(E) = y_i(E)\sigma_i(E), + +where :math:`y_i(E)` is the photon yield corresponding to an incident neutron +reaction having cross section :math:`\sigma_i(E)`. + +The yield of photons during neutron transport is determined as the sum of the +photon yields from each individual reaction. In OpenMC, production of photons +is treated in an average sense. That is, the total photon production cross +section is used at a collision site to determine how many photons to produce +rather than the photon production from the reaction that actually took place. +This is partly done for convenience but also because the use of variance +reduction techniques such as implicit capture make it difficult in practice to +directly sample photon production from individual reactions. + +In OpenMC, secondary photons are created after a nuclide has been sampled in a +neutron collision. The expected number of photons produced is + +.. math:: + :label: expected-number-photons + + n = w\frac{\sigma_{\gamma}(E)}{\sigma_T(E)}, + +where :math:`w` is the weight of the neutron, :math:`\sigma_{\gamma}` is the +photon production cross section for the sampled nuclide, and :math:`\sigma_T` +is the total cross section for the nuclide. :math:`\lfloor n \rfloor` photons +are created with an additional photon produced with probability :math:`n - +\lfloor n \rfloor`. Next, a reaction is sampled for each secondary photon. The +probability of sampling the :math:`i`-th reaction is given by +:math:`\sigma_{\gamma, i}(E)/\sum_j\sigma_{\gamma, j}(E)`, where +:math:`\sum_j\sigma_{\gamma, j} = \sigma_{\gamma}` is the total photon +production cross section. The secondary angle and energy distributions +associated with the reaction are used to sample the angle and energy of the +emitted photon. + .. _Koblinger: https://doi.org/10.13182/NSE75-A26663 .. _anomalous scattering: http://pd.chem.ucl.ac.uk/pdnn/diff1/anomscat.htm @@ -906,3 +1046,9 @@ angles. .. _Kaltiaisenaho: https://aaltodoc.aalto.fi/bitstream/handle/123456789/21004/master_Kaltiaisenaho_Toni_2016.pdf .. _Salvat: http://www.oecd-nea.org/globalsearch/download.php?doc=77434 + +.. _Sternheimer: https://journals.aps.org/prb/pdf/10.1103/PhysRevB.26.6067 + +.. [Sternheimer] R. M. Sternheimer, S.M.Seltzer, and M.J.Berger, *Density + effect for the ionization loss of charged particles in various substances*, + Phys. Rev. B, 26, 6067–6076 (1982).