Reactor antineutrino spectra

To derive a number of emitted antineutrinos, N_em, from the number of fis-sions, additional information on the isotopes’ neutron-rich fission products, namely theirβ-decay spectra, is needed. For a givenν_eenergyEwe can write

N_em(E) =

Z

tn_f(t)

∑

k

α_k(t)S_k(E) (3.9) Here, S_k(E) is the antineutrino yield or spectrum per fission of the isotope k. These spectra are weighed with the fraction of fissions undergone by each isotope,α_k(t), and summed over all four isotopes. The combined antineutrino spectrum can be written as

S_tot(E,t) =

∑

k

α_k(t)S_k(E) (3.10)

3.1. REACTORν_E FLUX PREDICTION 65

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time (days)

1019

1020

1021

1022

1023

number of fissions (/day)

235U

238U

239Pu

241Pu

Figure 3.4: The fission rates of the four fissile isotopes: ²³⁵U, ²³⁸U, ²³⁹Pu and ²⁴¹Pu during cycle 3 of the BR2 reactor in 2018. Given the log scale, it is clear that ²³⁵U contributes more than 99.9% of the fissions.

TheS_k’s need to include all possible outcomes of the fission reaction and depend on the variety of configurations for the β-decays of the resulting daughter nuclides. In practice, the β-spectra emitted by the reactor are cal-culated first and are then translated to ν_e-spectra. To obtain these combined spectra two main methods exist: the summation method and the conversion method.

The summation method is a so-called ab initio approach, where the β-spectrum from fissile isotopekis computed by summing the contributions of each of the possible fission productsFk [104]:

S_k(E_β) =

N_Fk

∑

A_Fk(t)×^S_Fk(E_β) (3.11) where A_Fk(t)is the activity of fission productFk at timetfor fissions of iso-topek.S_Fk(E_β)is theβ-spectrum of the fission productFkand is an aggregate of the spectra of all possible β-decay channels, weighed by their branching ratio. The fission product activity and information on the involved β-spectra can be retrieved from nuclear databases. More details on the calculation of theS_Fk(E_β)can e.g. be found in reference [104].

A single β-branch spectrum with energy E_β can be translated to an ν_e -spectrum in E_νe by using the simple energy relation E_νe = E₀^b_Fk−^Eβ, where

66 CHAPTER 3. SIGNAL PREDICTION E^b_0Fk is the end-point energy of the branch b of the fission productFk. This implies that for the summation method theν_e-spectra can be determined with the same accuracy as theβ-spectra.

The conversion method relies on measured reference β-spectra of the four fissile isotopes. The electron spectra from ²³⁵U, ²³⁹Pu and²⁴¹Pu fissions were measured with high precision in the 80’s at the ILL reactor in Grenoble, France [105]. The fissions were induced by the irradiation of²³⁵U/²³⁹Pu/²⁴¹Pu target foils with reactor neutrons and the β-spectra were acquired after an ir-radiation time between a few hours and 2 days. The β-spectrum of²³⁸U was measured much later at the FRMII neutron source in Garching, Germany [106].

A method to convert these to ν_e-spectra exists in fitting the β-spectrum shape with a large number of virtualβ-decay branches of an assumed shape.

The fitted spectra of those branches are then converted to antineutrino spec-tra via the relation E_νe = E^v₀−^Eβ, where E^v₀ is now the end-point energy of the virtual branchi.

Both methods for the calculation of reactor antineutrino spectra have their advantages and disadvantages. Spectra calculated with the conversion method need to be corrected for so-called off-equilibrium effects. This follows from the fact that β-decays have finite lifetimes and as a consequence the spectrum requires some time from the beginning of the fission process to reach a steady equilibrium. Due to the limited measurement time for the ILL reference spectra, compared to that of antineutrino experiments, the flux con-tributions of long lived fission products are not included. This is taken into account with off-equilibrium corrections.

The summation method, on the other hand, depends on the knowledge of a large number of variables, such as the fission yields, the lifetimes of the daughter nuclei, the individual beta branching ratios, etc.. Systematic effects and missing information in nuclear databases lead to relatively large uncertainties on the predicted spectra.

However, by combining the methods and using information from nuclear databases for the conversion of the reference spectra, very precise predictions of antineutrino spectra can be made.

For the SoLid experiment, spectra calculated with both methods are pro-vided by a team of SCK•CEN and Subatech staff. For each of the calculation methods, they use, in addition to the MCNPX/CINDER90 simulation of the reactor, the MURE (MCNP Utility for Reactor Evolution) code to track the

3.1. REACTORν_E FLUX PREDICTION 67 burn-up of the fissile products.

The conversion method is then implemented by coupling the simulation for the complete reactor model with converted neutrino spectra from a refer-ence of choice; Huber [107], Mueller [104], ... . The off-equilibrium effects are implemented as explained in reference [104].

The evaluation of the antineutrino spectra with the summation method is based on a more extensive use of MURE. In addition to the simulation of the reactor core evolution, the MURE code can compute the associated antineu-trino energy spectrum. For this purpose, it is coupled to nuclear databases containing all beta decay branches of the fission products for each fissile iso-tope.

In figure 3.5, examples of the spectra resulting from the conversion model from Huber for²³⁵U,²³⁹Pu and²⁴¹Pu and the Mueller prediction for²³⁸U and for a recent summation calculation by Estienne et al. [108] are shown. The ratio of the summation spectra to the conversion results is shown in figure 3.6.

The uncertainties on the predicted spectra are derived from the errors on the input models and/or the variables from nuclear databases. The uncer-tainty on the ²³⁵U yield determined with the conversion method is shown in figure 3.7. The errors increase with increasing energy, but remain below 2%

(4%) for the lower (higher) halve of the E_νe range of interest. For the spec-tra calculated with the summation method, an overall uncertainty of 2% is generally quoted. However, the actual errors are expected to be larger, as the complex and often unknown correlations between fission yields are not taken into account in these quoted values [109].