@@ -25,19 +25,38 @@ KERMA (Kinetic Energy Release in Materials) [Mack97]_ coefficients for reaction
2525:math: `\times ` cross-section (e.g., eV-barn) and can be used much like a reaction
2626cross section for the purpose of tallying energy deposition.
2727
28- KERMA coefficients can be computed using the energy-balance method with
29- a nuclear data processing code like NJOY, which performs the following
30- iteration over all reactions :math: `r` for all isotopes :math: `i`
31- requested
28+ KERMA coefficients can be computed using the energy-balance method with a
29+ nuclear data processing code like NJOY, which estimates the KERMA coefficients
30+ using the following equation:
3231
3332.. math ::
3433
35- k_{i, r}(E) = \left (E + Q_{i, r} - \bar {E}_{i, r, n}
34+ k_{i, r}(E) = \left (E + Q_{i, r} - \sum \limits _x \bar {E}_{i, r, x}
35+ \right )\sigma _{i, r}(E),
36+
37+ where the summation is over each secondary particle type :math: `x`. This
38+ equation states that the energy deposited is equal to the energy of the incident
39+ particle plus the reaction :math: `Q` value less the energy of secondary
40+ particles that are transported away from the reaction site. For neutron
41+ interactions, the energy-balance KERMA coefficient is
42+
43+ .. math ::
44+
45+ k_{i, r}(E) = \left (E + Q_{i, r} - \sum \limits _x \bar {E}_{i, r, n}
3646 - \bar {E}_{i, r, \gamma }\right )\sigma _{i, r}(E),
3747
38- removing the energy of neutral particles (neutrons and photons) that are
39- transported away from the reaction site :math: `\bar {E}`, and the reaction
40- :math: `Q` value.
48+ where :math: `\bar {E}_{i, r, n}` is the average energy of secondary neutrons and
49+ :math: `\bar {E}_{i, r, \gamma }` is the average energy of secondary photons. For
50+ photon and charged particle interactions the KERMA coefficient is
51+
52+ .. math ::
53+ :label: energy-balance-photon
54+
55+ k_{i, r}(E) = \left (E + Q_{i, r} - \sum \limits _x \bar {E}_{i, r, x}
56+ \right )\sigma _{i, r}(E).
57+
58+ where the :math: `Q` value is zero for all interactions except for pair
59+ production and positron annihilation.
4160
4261-------
4362Fission
@@ -120,7 +139,7 @@ run with :math:`N918` reflecting fission heating computed from NJOY.
120139 This modified heating data is stored as the MT=901 reaction and will be scored
121140if ``heating-local `` is included in :attr: `openmc.Tally.scores `.
122141
123- Coupled neutron-photon transport
142+ Coupled Neutron-Photon Transport
124143--------------------------------
125144
126145Here, OpenMC instructs ``heatr `` to assume that energy from photons is not
@@ -138,6 +157,50 @@ Let :math:`N301` represent the total heating number returned from this
138157 This modified heating data is stored as the MT=301 reaction and will be scored
139158if ``heating `` is included in :attr: `openmc.Tally.scores `.
140159
160+ Photons and Charged Particles
161+ -----------------------------
162+
163+ In OpenMC, energy deposition from photons or charged particles is scored using
164+ the energy balance method based on Equation :eq: `energy-balance-photon `. Special
165+ consideration is given to electrons and positrons as described below.
166+
167+ +++++++++++++++++
168+ Charged Particles
169+ +++++++++++++++++
170+
171+ OpenMC tracks photons interaction by interaction so the energy deposited in each
172+ collision is easily attributed back to the nuclide and reaction for which the
173+ photon interacted with. Charged particles (electrons and photons) aren't tracked
174+ in the same way. For charged particles, OpenMC assumes that all their energy
175+ (less the energy of bremsstrahlung radiation) is deposited in the material in
176+ which they were born. In this way it is harder to trace how much energy should
177+ be attributed in each nuclide.
178+
179+ According to the CSDA approximation (see :ref: `ttb `) the energy deposited by a
180+ charged particle with kinetic energy :math: `T` in the :math: `i`-th element can
181+ be calculated as:
182+
183+ .. math ::
184+
185+ E_{i} = \int _{0 }^{R(T)} w_{i}S_{\text {col,i}} dx
186+
187+ where :math: `R(T)` is the CSDA range of the charged particle,
188+ :math: `S_{\text {col},i}` is the collision stopping power of the charged particle
189+ in the :math: `i`-th element and :math: `w_i` is the mass fraction of the
190+ :math: `i`-th element. According to the Bethe formula the collision stopping
191+ power of the :math: `i`-th element is proportional to :math: `Z_i/A_i`, so the
192+ fractional collision stopping power from the :math: `i`-th element is:
193+
194+ .. math ::
195+
196+ \frac {w_{i}S_{\text {col},i}(T)}{S_{\text {col}}(T)} =
197+ \frac {\frac {w_{i}Z_{i}}{A_{i}}}{\sum _{i}\frac {w_{i}Z_{i}}{A_{i}}} =
198+ \frac {\gamma _i Z_{i}}{\sum _{i}\gamma _i Z_{i}}.
199+
200+ where :math: `\gamma _i` is the atomic fraction of the :math: `i`-th element.
201+ Therefore, the energy deposited by charged particles should be attributed to
202+ a given element according to its fractional charge density.
203+
141204----------
142205References
143206----------
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