11. Heating and Energy Deposition

As particles traverse a problem, some portion of their energy is deposited at collision sites. This energy is deposited when charged particles, including electrons and recoil nuclei, undergo electromagnetic interactions with surrounding electrons and ions. The information describing how much energy is deposited for a specific reaction is referred to as “heating numbers” and can be computed using a program like NJOY with the heatr module.

The heating rate is the product of reaction-specific coefficients and a reaction cross section

\[H(E) = \phi(E)\sum_i\rho_i\sum_rk_{i, r}(E),\]

and has units energy per time, typically eV/s. Here, \(k_{i, r}\) are the KERMA (Kinetic Energy Release in Materials) [Mack97] coefficients for reaction \(r\) of isotope \(i\). The KERMA coefficients have units of energy \(\times\) cross-section (e.g., eV-barn) and can be used much like a reaction cross section for the purpose of tallying energy deposition.

KERMA coefficients can be computed using the energy-balance method with a nuclear data processing code like NJOY, which estimates the KERMA coefficients using the following equation:

\[k_{i, r}(E) = \left(E + Q_{i, r} - \sum\limits_x \bar{E}_{i, r, x} \right)\sigma_{i, r}(E),\]

where the summation is over each secondary particle type \(x\). This equation states that the energy deposited is equal to the energy of the incident particle plus the reaction \(Q\) value less the energy of secondary particles that are transported away from the reaction site. For neutron interactions, the energy-balance KERMA coefficient is

\[k_{i, r}(E) = \left(E + Q_{i, r} - \sum\limits_x \bar{E}_{i, r, n} - \bar{E}_{i, r, \gamma}\right)\sigma_{i, r}(E),\]

where \(\bar{E}_{i, r, n}\) is the average energy of secondary neutrons and \(\bar{E}_{i, r, \gamma}\) is the average energy of secondary photons. For photon and charged particle interactions the KERMA coefficient is

(1)\[k_{i, r}(E) = \left(E + Q_{i, r} - \sum\limits_x \bar{E}_{i, r, x} \right)\sigma_{i, r}(E).\]

where the \(Q\) value is zero for all interactions except for pair production and positron annihilation.

11.1. Fission

During a fission event, there are potentially many secondary particles, and all must be considered. The total energy released in a fission event is typically broken up into the following categories:

• \(E_{fr}\) - kinetic energy of fission fragments

• \(E_{n,p}\) - energy of prompt fission neutrons

• \(E_{n,d}\) - energy of delayed fission neutrons

• \(E_{\gamma,p}\) - energy of prompt fission photons

• \(E_{\gamma,d}\) - energy of delayed fission photons

• \(E_{\beta}\) - energy of released \(\beta\) particles

• \(E_{\nu}\) - energy of neutrinos

These components are defined in MF=1, MT=458 data in a standard ENDF-6 formatted file. All these quantities may depend upon incident neutron energy, but this dependence is not shown to make the following demonstrations cleaner. As neutrinos scarcely interact with matter, the recoverable energy from fission is defined as

\[E_r\equiv E_{fr} + E_{n,p} + E_{n, d} + E_{\gamma, p} + E_{\gamma, d} + E_{\beta}\]

Furthermore, the energy of the secondary neutrons and photons is given as \(E_{n, p}\) and \(E_{\gamma, p}\), respectively.

NJOY computes the fission KERMA coefficient using this energy-balance method to be

\[k_{i, f}(E) = \left[E + Q(E) - \bar{E}(E)\right]\sigma_{i, f}(E) = \left[E_{fr} + E_{\gamma, p}\right]\sigma_{i, j}(E)\]

Note

The energy from delayed neutrons and photons and beta particles is intentionally left out from the NJOY calculations.

11.2. OpenMC Implementation

For fissile isotopes, OpenMC makes modifications to the heating reaction to include all relevant components of fission energy release. These modifications are made to the total heating reaction, MT=301. Breaking the total heating KERMA into a fission and non-fission section, one can write

\[k_i(E) = k_{i, nf}(E) + \left[E_{fr}(E) + E_{\gamma, p}\right]\sigma_{i, f}(E)\]

OpenMC seeks to modify the total heating data to include energy from \(\beta\) particles and, conditionally, delayed photons. This conditional inclusion depends on the simulation mode: neutron transport, or coupled neutron-photon transport. The heating due to fission is removed using MT=318 data, and then re-built using the desired components of fission energy release from MF=1,MT=458 data.

11.2.1. Neutron Transport

For this case, OpenMC instructs heatr to produce heating coefficients assuming that energy from photons, \(E_{\gamma, p}\) and \(E_{\gamma, d}\), is deposited at the fission site. Let \(N901\) represent the total heating number returned from this heatr run with \(N918\) reflecting fission heating computed from NJOY. \(M901\) represent the following modification

\[M901_{i}(E)\equiv N901_{i}(E) - N918_{i}(E) + \left[E_{i, fr} + E_{i, \beta} + E_{i, \gamma, p} + E_{i, \gamma, d}\right]\sigma_{i, f}(E).\]

This modified heating data is stored as the MT=901 reaction and will be scored if heating-local is included in openmc.Tally.scores.

11.2.2. Coupled Neutron-Photon Transport

Here, OpenMC instructs heatr to assume that energy from photons is not deposited locally. However, the definitions provided in the NJOY manual indicate that, regardless of this mode, the prompt photon energy is still included in \(k_{i, f}\), and therefore must be manually removed. Let \(N301\) represent the total heating number returned from this heatr run and \(M301\) be

\[M301_{i}(E)\equiv N301_{i}(E) - N318_{i}(E) + \left[E_{i, fr}(E) + E_{i, \beta}(E)\right]\sigma_{i, f}(E).\]

This modified heating data is stored as the MT=301 reaction and will be scored if heating is included in openmc.Tally.scores.

11.2.3. Photons and Charged Particles

In OpenMC, energy deposition from photons or charged particles is scored using the energy balance method based on Equation (1). Special consideration is given to electrons and positrons as described below.

11.2.3.1. Charged Particles

OpenMC tracks photons interaction by interaction so the energy deposited in each collision is easily attributed back to the nuclide and reaction for which the photon interacted with. Charged particles (electrons and photons) aren’t tracked in the same way. For charged particles, OpenMC assumes that all their energy (less the energy of bremsstrahlung radiation) is deposited in the material in which they were born. In this way it is harder to trace how much energy should be attributed in each nuclide.

According to the CSDA approximation (see Thick-Target Bremsstrahlung Approximation) the energy deposited by a charged particle with kinetic energy \(T\) in the \(i\)-th element can be calculated as:

\[E_{i} = \int_{0}^{R(T)} w_{i}S_{\text{col,i}} dx\]

where \(R(T)\) is the CSDA range of the charged particle, \(S_{\text{col},i}\) is the collision stopping power of the charged particle in the \(i\)-th element and \(w_i\) is the mass fraction of the \(i\)-th element. According to the Bethe formula the collision stopping power of the \(i\)-th element is proportional to \(Z_i/A_i\), so the fractional collision stopping power from the \(i\)-th element is:

\[\frac{w_{i}S_{\text{col},i}(T)}{S_{\text{col}}(T)} = \frac{\frac{w_{i}Z_{i}}{A_{i}}}{\sum_{i}\frac{w_{i}Z_{i}}{A_{i}}} = \frac{\gamma_i Z_{i}}{\sum_{i}\gamma_i Z_{i}}.\]

where \(\gamma_i\) is the atomic fraction of the \(i\)-th element. Therefore, the energy deposited by charged particles should be attributed to a given element according to its fractional charge density.

11.3. References

[Mack97]

Abdou, M.A., Maynard, C.W., and Wright, R.Q. MACK: computer program to calculate neutron energy release parameters (fluence-to-kerma factors) and multigroup neutron reaction cross sections from nuclear data in ENDF Format. Oak Ridge National Laboratory report ORNL-TM-3994.