Neutron capture and the antineutrino yield from nuclear reactors
Abstract
We identify a new, fluxdependent correction to the antineutrino spectrum as produced in nuclear reactors. The abundance of certain nuclides, whose decay chains produce antineutrinos above the threshold for inverse beta decay, has a nonlinear dependence on the neutron flux, unlike the vast majority of antineutrino producing nuclides, whose decay rate is directly related to the fission rate. We have identified four of these socalled nonlinear nuclides and determined that they result in an antineutrino excess at lowenergies below , dependent on the reactor thermal neutron flux. We develop an analytic model for the size of the correction and compare it to the results of detailed reactor simulations for various real existing reactors, spanning 3 orders of magnitude in neutron flux. In a typical pressurized water reactor the resulting correction can reach % of the low energy flux which is comparable in size to other, known lowenergy corrections from spent nuclear fuel and the nonequilibrium correction. For naval reactors the nonlinear correction may reach the 10% level.
Science with antineutrinos and nuclear reactors and have been intimately connected since the discovery of the antineutrino by Cowan and Reines Cowan et al. (1956). Reactors are the largest terrestrial source of antineutrinos through the production of unstable fission fragments. These fission fragments are neutronrich and, thus, will beta decay to stability producing antineutrinos. An average of six beta decays occurs per fission, thus a reactor will produce . The vast majority of reactor nuclides lighter than uranium are generated directly as a fission product or by the beta decays of fission products, for instance
(1) 
where we have truncated the chain at Tc as its halflife of allows us to consider it stable. Here, the daughter nuclides are produced from decays of their parents, which are dominantly produced via fissions. Thus, the decay rates of both the daughters and parents in the chain are linearly dependent on the fission rates. Equivalently, these nuclides are said to be linear in the neutron flux as the fission rate goes as for a macroscopic fission crosssection . A second mechanism for antineutrino production is from neutron captures on certain isotopes, such as:
(2) 
where the neutrons are the prompt neutrons from fission, thermalized by the moderator. Nuclides that are primarily produced via neutron captures require two neutrons: one to initiate the fission whose fission products result in a beta decay chain yielding the capture isotope, Tc in above example, and a second neutron for the actual neutron capture. Thus naively, one would conclude that the production of Tc is quadratic in the neutron flux. Thus, these nuclides will be produced in different quantities for different values of even if is kept constant; we therefore name these nonlinear nuclides. This letter examines how many such nonlinear nuclides with a relevant antineutrino yield exist and how large the resulting correction to the antineutrino spectrum can become.
To be a relevant nonlinear nuclide , several conditions have to be met:

Large cumulative fission yield, of the capture isotope .

Large neutron capture cross section .

The nonlinear nuclide must decay sufficiently quickly, that is the decay constant must be large enough.

The beta decay of the nonlinear nuclide has to have an endpoint above the inverse beta decay threshold of 1.8 MeV.
There are approximately candidate nuclides fulfilling these conditions. An example of a relevant nonlinear nuclide, Tc, is given in Fig. 1.
Here, Tc is the beta decaying nonlinear nuclide and it is fissionblocked from the beta decay chain by its stable isobar Mo. Thus, Tc is practically absent from fission products, i.e. its fission yield is negligible. Being fission blocked by a doublebeta decay isotope, like Mo, is characteristic for all candidates. The production is then primarily governed by its precursor nuclide , in this case Tc. Furthermore, Tc is relatively stable and linear as it is fed through its own decay chain meaning that, with a large enough cumulative yield and neutron capture crosssection, the production of Tc may be nonnegligible. To simplify our discussion we consider only that have stable precursors (including Tc), are significantly blocked (), and have a significant feeder cumulative yield , where we used the JEFF3.1 Koning et al. (2006) fission yields. With these criteria we are able to reduce our list to four major nonlinear nuclides listed in Tab. 1. Note, that Tc, Rh, Ag show predominantly () allowed GamowTeller decays, whereas Pr exhibits a nonunique forbidden decay, which as we will see later, contributes less than 10% to the total nonlinear correction.
Tc  Rh  Ag  Pr  
3.2  2.4  2.9  2.2  
15.5  42.3  24.6  68830  

U  0.061  0.031  0.00029  0.059  
Pu  0.062  0.069  0.017  0.052  
Pu  0.056  0.065  0.030  0.049  
17.0  127  80.9  6.53  
2.75  39.3  0.57  32.5  
1.57  7.08  18.2  26.7 
From Fig. 1 it is apparent we must solve a set of three linearly coupled nonhomogeneous differential equations, the Bateman equations Bateman (1910), in order to express the abundance of N in terms of the thermal neutron flux and the irradiation time . Similar sets have been solved, without the neutron component, as an eigenvalue problem Moral and Pacheco (2003) and recursively Cetnar (2006).
The limiting cases of the solutions can be identified from the information provided in Tab. 1. For irradiation times larger than the longest involved halflives, which generally occur for the longlived precursor parent , and the relevant halflives range from 0.57 d to 39.3 d, we can assume that the isotope is in equilibrium.
Once has reached equilibrium, the next concern is that capture directly from the longlived nuclide to a stable isotope can prevent the production of the neutron capture isotope. The decay rate of the longlived nuclide equals the capture rate for a neutron flux density of
(3) 
using the values given in Tab. 1 this yields a range of , which is nearly an order of magnitude above the values found for any of the reactors considered here. Thus we conclude, that this mechanism can practically be neglected. Note, that for Cs, the neutron capture on Xe with a cross section of prevents any significant production.
The decay rate of the antineutrino producing nonlinear nuclide is large with halflives in seconds to hours range and thus will be always in equilibrium with its much slower production rate. Therefore, for irradation times which are long compared to the halflives of , the production rate of the nonlinear nuclide, which is the same as its decay rate is given by
(4) 
hence the name nonlinear nuclide. The decay rate of a fission product in equilibrium is given proportional to and thus the relative contribution of a nonlinear nuclide scales as . From Eq. 4 and Tab. 1 we also can conclude that Rh will have the largest contribution for the fissile isotopes investigated, for fission of U the second most important nonlinear nuclide is Tc, whereas for the fission of both plutonium isotopes the second largest contribution stems from Ag. We note, that reactors with a high neutron flux density and very long core lifetimes, in principle, can exhibit corrections in the 510% range. In one example assuming a neutron flux density of and after an irradiation of 5 years, we find, based on our analytic calculation, a 4% correction. Clearly, naval reactors fulfill these characteristics and a precise measurement of the nonlinear correction to their antineutrino emissions may allow to draw conclusions about some of the design characteristics, like core size, and operational history.
IR40  PWR  IRT  ILL  HFIR  
1batch  3batch  U  Pu  Pu  
Fuel/Moderator  NU+C  NU+DO  LEU+HO  HEU+HO  HEU+DO  HEU+HO  
Burnup [MWd/t]  
Max[] [%] 
We can formulate an explicit solution for the nonlinear nuclides by solving the corresponding set of Bateman equations. The longlived nuclide is dominantly produced via fission and destroyed through its neutron captures and decays. The precursor is produced via fission and decays from . It is destroyed by neutron captures. Finally, the nonlinear nuclide is produced solely through captures on and is destroyed via its own decays. Therefore, our nonlinear set is given by:
(5) 
where , is the fission rate vector, and () is the cumulative (individual) fission yields. All nuclear parameters are denoted by their subscript. It is straightforward to solve Eq. 5 analytically, but the salient features are contained in above description of the limiting cases. Many of the reactor physics effects neglected in the simplified reaction network used result in nonnegligible corrections and we find that the analytic result generally is within a factor of two the solution derived from using a full reaction network. The full reaction network is evaluated using the Standardized Computer Analyses for Licensing and Evaluation (SCALE6.1) ORNL (2011) reactor simulation suite, developed by Oak Ridge National Laboratory.
Now that we have an expression for the activity of these four nonlinear beta decaying nuclides we can apply a neutrino spectrum to each decay to generate a neutrino rate. The neutrino spectra are applied to our four nuclides following Ref. Huber (2011), which generates the neutrino spectra for each nonlinear nuclide. The neutrino spectra were then summed to determine the total nonlinear correction.
Solving Eq. 5 will lead to an expression for the activity of the nonlinear nuclides, which can be combined with the spectra of each nonlinear isotope to produce the total nonlinear spectral contribution. Comparing this with the total reactor spectra, shows that the nonlinear spectra falls off steeply at . This nonlinear spectral contribution is important as it interferes with other lowenergy corrections, such as the spent fuel signal Zhou et al. (2012), the nonequilibrium correction for inverse beta decay experiments^{1}^{1}1For elastic antineutrinoelectron scattering experiments at very low energies a detailed discussion of the nonequilibrium correction including some neutron captures (different than those considered here) can be found in Refs. Mikaelyan (2002); Kopeikin (2003) . which has been evaluated in Ref. Mueller et al. (2011), where neutron capture was specifically neglected. All of these corrections, including the nonlinear correction, will directly impact geoneutrino searches Sramek et al. (2012); Domogatski et al. (2005) wherever a sizable reactor signal needs to be subtracted, like for instance in JUNO Han et al. (2015). .
Using SCALE, we are able to model nine different reactor configurations, spanning three orders of magnitude in their thermal neutron flux. The first is a natural uranium loaded and graphitemoderated reactor, similar in design to the British CalderHall reactor. This reactor is referred to as the reactor and has been previously modeled Christensen et al. (2015). The next reactor uses natural uranium as fuel and heavy water as a moderator, similar in design to the CANDU reactors. This reactor, referred to as the IR40, has also been previously modeled Christensen et al. (2014). The third reactor is fueled with lowenriched uranium (LEU) with a water moderator. These reactors are pressurized water reactors (PWR) similar in design to the Daya Bay cores. The Daya Bay reactors have also been previously modeled to estimate the spent fuel contribution An et al. (2012). The PWR cores are simulated using a 3batch method, where a full core consists of 3 parts: a third each of fresh, onceirradiated, and twiceirradiated fuel. We also include a singlebatch calculation for comparison. Next, we simulate a research reactor, named the IRT reactor, which is a pooltype reactor using highlyenriched uranium (HEU) fuel elements, natural uranium target elements, and water as a moderator. It was previously simulated, also in Ref. Christensen et al. (2015). We have also recreated the measurements conducted at the ILL reactor, irradiating a fissile mass with a specific neutron flux according to Ref. Schreckenbach et al. (1985); Haag et al. (2014); Hahn et al. (1989). Finally, we simulate the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory, which represents the highest steadystate neutron flux commercially available. Our simulation closely follows that of Ref. Ilas (2012). This reactor database spans over orders of magnitude for the neutron fluxes and we aim to find a nonlinear correction trend as a function of . Each reactor is irradiated with its own typical power history using the SCALE simulation suite. The burnup and reactor specifications are given in Tab. 2.
We use the Origen depletion subroutine to compute the fission rates and nuclide activities as a function of irradiation time. We use the linear antineutrino yields for U, Pu and Pu from Ref. Huber (2011) and for U from Ref. Mueller et al. (2011) to convert the fission rates to a total neutrino spectrum for each reactor during its power cycle. The nonlinear correction is isolated by selecting the activities of our four nonlinear nuclides and converting these into a neutrino spectrum using the beta decay description in Ref. Huber (2011). Each spectrum is binned into bins and a nonlinear correction is determined from the ratio of the nonlinear contribution to the total reactor spectrum at all irradiation times. This result is then used to calculate a timeaveraged nonlinear correction.
With this nonlinear lowenergy neutrino correction, we can see that commercial reactors can be very sensitive to the resulting effects, where it becomes comparable with spent fuel () Zhou et al. (2012) and the nonequilibrium correction () Mikaelyan (2002); Kopeikin (2003). Therefore, neutrino experiments will need to consider the nonlinear correction when predicting the total reactor neutrino spectrum, especially in the low energy region where detailed reactor simulations are necessary.
A final item of note is that the widely used measurements of the cumulative beta spectra from fissions of U Schreckenbach et al. (1985), Pu and Pu Hahn et al. (1989), and now U Haag et al. (2014) have utilized research reactors with fluxes of . Our analysis has been conducted to reproduce the measurements conducted by Schreckenbach et al. to determine if nonlinear effects appear in these measurements. A flux of was used with irradiation times of , and for U, Pu, Pu, and U, respectively in accordance with Ref. Schreckenbach et al. (1981); Von Feilitzsch et al. (1982); Schreckenbach et al. (1985); Hahn et al. (1989); Haag et al. (2014). The results for these calculations, shown in Tab. 2 illustrate that these measurements are not contaminated by nonlinear corrections and thus the extracted neutrino fluxes Huber (2011); Mueller et al. (2011) are unaffected.
The lack of nonlinear corrections in the Schreckenbach measurements is due to the short irradiation times , which are all less than two days. As we have noted earlier, such as in Fig. 1, most of our nonlinear nuclides are fed via a precursor nuclide with a longlived parent . The large halflives, relative to , of the nuclides prevents the buildup of the feeder nuclides, which then prevents the buildup of the nonlinear nuclides, thus preserving the Schreckenbach measurements. This same effect is seen in the diminished nonlinear correction for the HFIR reactor, which involves irradiation cycles less than . Two nonlinear nuclides (Rh and Pr) are fed through an with , so their contribution to the HFIR correction is lower than would be expected for longer irradiation times.
In our note we have introduced a new lowenergy correction to the reactor antineutrino spectrum. This correction is due to nonlinear nuclides in the reactor, which are dominantly produced via neutron captures. Demanding that our nuclides of interest meet several criteria, we have limited the list of these nonlinear nuclides to four that can impact neutrino studies: Tc, Rh, Ag, and Pr. We derived an analytic solution for the abundance of these nuclides in a reactor environment, which depends on the neutron flux in a nonlinear fashion. We calculated the nonlinear corrections for several reactor designs spanning thermal neutron fluxes from to , discovering a nonlinear neutrino excess as large as . The resulting nonlinear nuclide production is negligible for short irradiation times less than 30 d, but much larger for multibatch commercial reactors, which can reach large burnup values. This result indicates that special attention should be given to the nonlinear correction in future neutrino experiments, requiring detailed reactor simulations to correctly predict this excess.
Acknowledgements.
We would like to thank A. Hayes for the many discussions on nonlinear nuclide production in a reactor as well as D. Ilas and the ORNL team behind the maintenance and verification of SCALE. This work was supported by the U.S. Department of Energy under award DESC0013632.References
 Cowan et al. (1956) C. Cowan, F. Reines, F. Harrison, H. Kruse, and A. McGuire, Science 124, 103 (1956).
 (2) J. Tuli, Evaluated Nuclear Structure Data File, http://www.nndc.bnl.gov/ensdf/.
 Koning et al. (2006) A. Koning, R. Forrest, M. Kellett, R. Mills, H. Henriksson, and Y. Rugama, Joint Evaluated Fission and Fusion File, Incidentneutron data, http://wwwnds.iaea.org/exfor/endf00.htm (2006).
 Department of Energy (1993) Department of Energy, Nuclear Physics and Reactor Theory Module 2: Reactor Theory (Neutron Characteristics) (National Technical Information Services, 1993).
 Wilson et al. (2000) W. Wilson, T. England, and M. Brady, CinderLibrary, https://rsicc.ornl.gov/codes/ccc/ccc7/ccc755.html (2000).
 Bateman (1910) H. Bateman, Proc. Cambridge Philos. Soc. 15, 423 (1910).
 Moral and Pacheco (2003) L. Moral and A. F. Pacheco, American Journal of Physics 71, 684 (2003).
 Cetnar (2006) J. Cetnar, Annals of Nuclear Energy 33, 640 (2006), ISSN 03064549.
 ORNL (2011) ORNL, Standardized Computation and Licensing Evaluation, http://scale.ornl.gov/ (2011).
 Huber (2011) P. Huber, Phys. Rev. C84, 024617 (2011), eprint 1106.0687.
 Zhou et al. (2012) B. Zhou, X.C. Ruan, Y.B. Nie, Z.Y. Zhou, F.P. An, et al., Chin. Phys. C36, 1 (2012).
 Mikaelyan (2002) L. Mikaelyan, Phys. Atom. Nucl. 65, 1173 (2002), eprint hepph/0210047.
 Kopeikin (2003) V. Kopeikin, Phys. Atom. Nucl. 66, 472 (2003), eprint hepph/0110030.
 Mueller et al. (2011) T. A. Mueller et al., Phys. Rev. C 83, 054615 (2011), eprint 1101.2663.
 Sramek et al. (2012) O. Sramek, W. F. McDonough, and J. G. Learned, Adv. High Energy Phys. 2012, 235686 (2012).
 Domogatski et al. (2005) G. Domogatski, V. Kopeikin, L. Mikaelyan, and V. Sinev, Phys. Atom. Nucl. 68, 69 (2005), eprint hepph/0401221.
 Han et al. (2015) R. Han, Y.F. Li, L. Zhan, W. F. McDonough, and J. Cao (2015), eprint 1510.01523.
 Christensen et al. (2015) E. Christensen, P. Huber, and P. Jaffke, Science and Global Security 23, 20 (2015), eprint 1312.1959.
 Christensen et al. (2014) E. Christensen, P. Huber, P. Jaffke, and T. E. Shea, Phys. Rev. Lett. 113, 042503 (2014).
 An et al. (2012) F. An et al. (Daya Bay), Phys. Rev. Lett. 108, 171803 (2012), eprint 1203.1669.
 Schreckenbach et al. (1985) K. Schreckenbach, G. Colvin, W. Gelletly, and F. Von Feilitzsch, Phys. Lett. B160, 325 (1985).
 Haag et al. (2014) N. Haag, A. Gütlein, M. Hofmann, L. Oberauer, W. Potzel, et al., Phys. Rev. Lett. 112, 122501 (2014), eprint 1312.5601.
 Hahn et al. (1989) A. Hahn, K. Schreckenbach, G. Colvin, B. Krusche, W. Gelletly, et al., Phys. Lett. B218, 365 (1989).
 Ilas (2012) D. Ilas, ORNL/TM2011/367, Oak Ridge National Laboratory (2012).
 Schreckenbach et al. (1981) K. Schreckenbach, H. Faust, F. von Feilitzsch, A. Hahn, K. Hawerkamp, and J. Vuilleumier, Phys. Lett. B 99, 251 (1981), ISSN 03702693.
 Von Feilitzsch et al. (1982) F. Von Feilitzsch, A. Hahn, and K. Schreckenbach, Phys. Lett. B118, 162 (1982).