SetupMGXSAction

Total Cross Sections

At a minimum, this action always computes total MGXS, which is computed with the following formula:

$$\Sigma_{t,i,g} = \frac{\langle\Sigma_{t}\psi\rangle_{i,g}}{\langle\psi\rangle_{i,g}}$$(1)

where

$$\langle\Sigma_{t}\psi\rangle_{i,g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega\int_{E_{g}}^{E_{g-1}}\,dE\,\Sigma_{t}(\vec{r}, E)\psi(\vec{r}, E, \hat{\Omega})$$(2)

and

$$\langle\phi\rangle_{i,g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega\int_{E_{g}}^{E_{g-1}}\,dE\,\psi(\vec{r}, E, \hat{\Omega})$$(3)

$\Sigma_{t,i,g}$ is the homogenized total MGXS, $V_{i}$ is the homogenization volume (determined by tally_type), $\Sigma_{t}(\vec{r}, E)$ is the continuous-energy total macroscopic cross section, and $\psi(\vec{r}, E, \hat{\Omega})$ is the continuous energy angular flux.

Absorption Cross Sections

Absorption cross sections can be added by setting add_absorption = true, and are calculated with the following:

$$\Sigma_{a,i,g} = \frac{\langle\Sigma_{a}\psi\rangle_{i,g}}{\langle\psi\rangle_{i,g}}$$(4)

where

$$\langle\Sigma_{a}\psi\rangle_{i,g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega\int_{E_{g}}^{E_{g-1}}\,dE\,\Sigma_{a}(\vec{r}, E)\psi(\vec{r}, E, \hat{\Omega})$$(5)

$\Sigma_{a,i,g}$ is the homogenized absorption MGXS and $\Sigma_{a}(\vec{r}, E)$ is the equivalent continuous-energy macroscopic cross section.

Nu-Scattering Matrices

By default this action adds MGXS scattering matrices with particle multiplication, though this behaviour can be disabled by setting add_scattering = false. Scattering cross sections are expanded in Legendre moments of $\mu = \hat{\Omega}'\cdot\hat{\Omega}$, where $\hat{\Omega}'$ is the direction entering a scattering reaction and $\hat{\Omega}$ is the direction exiting a scattering reaction. The maximum order of the Legendre expansion can be specified by setting legendre_order (where the default of 0 indicates isotropic scattering). At present, SetupMGXSAction implements a 'simple' formulation of the scattering matrix where the elements are computed with analog estimators:

$$\Sigma_{s,i,\ell,g'\rightarrow g} = \frac{\langle\Sigma_{s}\psi\rangle_{i,\ell,g\rightarrow g'}}{\langle\psi\rangle_{i,g}}$$(6)

where

$$\langle\nu\Sigma_{s}\psi\rangle_{i,\ell,g'\rightarrow g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega'\int_{E'_{g}}^{E'_{g-1}}\,dE'\int_{4\pi}\,d\Omega\int_{E_{g}}^{E_{g-1}}\,dE\,\nu\Sigma_{s}(\vec{r}, \mu, E'\rightarrow E)P_{\ell}(\mu)\psi(\vec{r}, E, \hat{\Omega}')$$(7)

$\nu\Sigma_{s,i,\ell,g'\rightarrow g}$ is the homogenized nu-scatter MGXS, $\nu\Sigma_{s}(\vec{r}, \mu, E'\rightarrow E)$ is the equivalent continuous-energy macroscopic cross section, $P_{\ell}(\mu)$ are the Legendre polynomials, and $E'$ indicates the energy entering a scattering reaction. SetupMGXSAction supports transport corrected P0 ($\ell = 0$) scattering cross sections, which are computed with the following:

$$\Sigma_{s,i,0,g'\rightarrow g} = \frac{\langle\Sigma_{s}\psi\rangle_{i,0,g\rightarrow g'} - \delta_{gg'}\sum_{g''}\langle\Sigma_{s}\psi\rangle_{i,1,g''\rightarrow g}}{\langle\psi\rangle_{i,g}}$$(8)

where $\delta_{gg'}$ is the Khronecker delta function. The transport correction can be applied by setting transport_correction = true.

Nu-Fission Cross Sections

Nu-fission (neutron production) cross sections can be added by setting add_fission = true, and are calculated with the following:

$$\nu\Sigma_{f,i,g} = \frac{\langle\nu\Sigma_{f}\psi\rangle_{i,g}}{\langle\psi\rangle_{i,g}}$$(9)

where

$$\langle\Sigma_{a}\psi\rangle_{i,g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega\int_{E_{g}}^{E_{g-1}}\,dE\,\nu\Sigma_{f}(\vec{r}, E)\psi(\vec{r}, E, \hat{\Omega})$$(10)

$\nu\Sigma_{f,i,g}$ is the homogenized neutron production MGXS and $\nu\Sigma_{f}(\vec{r}, E)$ is the equivalent continuous-energy macroscopic cross section. The tallies used to compute $\nu\Sigma_{f,i,g}$ are also used to compute $\chi_{i,g}$, necessitating the use of an analog estimator for $\nu\Sigma_{f,i,g}$. This cross section is not available for photons.

Discrete Chi Spectra

Discrete chi spectra can also be added by setting add_fission = true, and are computed with the following:

$$\chi_{i,g} = \frac{\langle\chi\nu\Sigma_{f}\psi\rangle_{i,g}}{\langle\nu\Sigma_{f}\psi\rangle_{i}}$$(11)

where

$$\langle\chi\nu\Sigma_{f}\psi\rangle_{i,g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega'\int_{0}^{\infty}\,dE'\int_{E_{g}}^{E_{g-1}}\,dE\,\chi(\vec{r}, E)\nu\Sigma_{f}(\vec{r}, E')\psi(\vec{r}, E', \hat{\Omega}')$$(12)

and

$$\langle\nu\Sigma_{f}\psi\rangle_{i} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega'\int_{0}^{\infty}\,dE'\int_{0}^{\infty}\,dE\,\chi(\vec{r}, E)\nu\Sigma_{f}(\vec{r}, E')\psi(\vec{r}, E', \hat{\Omega}')$$(13)

$\chi_{i,g}$ is the homogenized MG chi spectra and $\chi(\vec{r}, E)$ is the equivalent continuous-energy spectra. The need to know the entering and exiting energies for the nu-fission reaction rate necessitates the use of an analog estimator for $\chi_{i,g}$. This property is not available for photons.

Fission Heating Values

Fission heating values can be added by setting add_fission_heating = true, and are computed with the following:

$$\kappa\Sigma_{f,i,g} = \frac{\langle\kappa\Sigma_{f}\psi\rangle_{i,g}}{\langle\psi\rangle_{i,g}}$$(14)

where

$$\langle\kappa\Sigma_{f}\psi\rangle_{i,g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega\int_{E_{g}}^{E_{g-1}}\,dE\,\kappa\Sigma_{f}(\vec{r}, E)\psi(\vec{r}, E, \hat{\Omega})$$(15)

$\kappa\Sigma_{f,i,g}$ is the homogenized MG fission heating cross section and $\kappa\Sigma_{f}(\vec{r}, E)$ is the equivalent continuous-energy cross section. This property is not available for photons.

Inverse Velocity Values

Inverse velocity values can be added by setting add_inverse_velocity = true, and are computed with the following:

$$\Big(\frac{1}{v}\Big)_{i,g} = \frac{\langle\frac{\psi}{v}\rangle_{i,g}}{\langle\psi\rangle_{i,g}}$$(16)

where

$$\Big\langle\frac{\psi}{v}\Big\rangle_{i,g} = \int_{V_i}\,dr^3\int_{4\pi}\,d\Omega\int_{E_{g}}^{E_{g-1}}\,dE\,\frac{\psi(\vec{r}, E, \hat{\Omega})}{v(E)}$$(17)

$(\frac{1}{v})_{i,g}$ is the homogenized MG inverse velocity and $v(E)$ is the associated continuous energy velocity.

Particle Diffusion Coefficients

Particle diffusion coefficients can be added by setting add_diffusion_coefficient = true, and are computed with the following:

$$D_{i,g} = \frac{1}{3\Sigma_{tr,i,g}}$$(18)

$$\Sigma_{tr,i,g} = \frac{\langle\Sigma_{t}\psi\rangle_{i,g} - \sum_{g'}\langle\nu\Sigma_{s}\psi\rangle_{i,1,g'\rightarrow g}}{\langle\psi\rangle_{i,g}}$$(19)

$D_{i,g}$ is the homogenized MG diffusion coefficient and $\Sigma_{tr,i,g}$ is the homogenized transport MGXS.