Interaction of Solar Neutrinos with 98 Mo and 100 Mo Nuclei

— The process of neutrino interaction with 98 Mo and 100 Mo nuclei is studied with allowance for the e ﬀ ect of charge-exchange resonances. The results obtained by calculating the cross section for solar-neutrino capture by the isotopes 98 Mo and 100 Mo, σ ( E ν ) , are presented. Use is made of both experimental data obtained for the strength functions S ( E ) in ( p, n ) and ( 3 He, t ) charge-exchange reactions and the strength functions S ( E ) calculated within the theory of ﬁ nite Fermi systems. The e ﬀ ect of the resonance structure of S ( E ) on the calculated cross sections for solar-neutrino capture is studied, and the contribution of each high-lying resonance to the capture cross section σ ( E ν ) is isolated. The contributions of all components of the solar neutrino spectrum are calculated. The contribution of background solar neutrinos to the double-beta decay of 100 Mo nuclei is estimated.


INTRODUCTION
In calculating the cross sections for neutrino interaction with atomic nuclei, σ(E ν ), it is necessary to determine, for the nuclei involved, the chargeexchange strength function S(E), which has a resonance character. For solar neutrinos, the upper boundary of the spectrum is determined by the hep reaction 3 He + p → 4 He + e + + ν e , in which case E x 18.77 MeV [1]. For the isotopes 98 Mo and 100 Mo, which are considered here, the strength functions S(E) were measured up to E x = 18 MeV for 98 Mo [2] and in the region of E x > 20 MeV for 100 Mo [3,4]. The isotopes 98 Mo and 100 Mo differ in structure only by two neutrons, but, in the cross section for solarneutrino capture, σ(E ν ), they differ many times, and this is what we will discuss in the present article.
Yet another reason why we have chosen these nuclei is that large-scale international projects aimed at studying double-beta decay employ the isotope 100 Mo, and it is of paramount importance to explore the effect of background solar neutrinos. In the NEMO-3 experiment, where use was made of 6.914 kg of the isotope 100 Mo and 0.932 kg of 1) National Research Center Kurchatov Institute, Moscow, Russia. 2) Moscow Institute of Physics and Technology (National Research University), Moscow, Russia. 3) Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia. * E-mail: lutostansky@yandex.ru the isotope 82 Se, the half-life of 100 Mo was measured with respect to decay to the ground state of 100 Ru [5]. In planning experiments that would involve substantially longer exposures, it is illegitimate to disregard the background of solar neutrinos. These backgrounds will be taken into account for the SuperNEMO project involving a higher mass and a greater number of isotopes [6]. The situation around backgrounds is similar in the CUPID-Mo experiment, which is being performed at the Laboratoire Souterrain de Modane (LSM, France) [7], and at the initial stage of the AMoRE experiment [8]. Figure 1 shows the scheme of charge-exchange excitations of 98,100 Mo nuclei upon neutrino capture followed by the decay of arising 98,100 Tc nuclei. One can see that the excited states of technetium isotopes have a resonance structure. The giant Gamow-Teller (GT) resonance is the most intense [9]. An isobaric analog resonance (AR) lies below GTR [10], while so-called pygmy resonances (PR) [11], which are of importance in charge-exchange reactions [12,13] and in beta-decay processes [14], lie still lower. Accordingly, these charge-exchange resonances manifest themselves in the strength function S(E) and change substantially the results of the calculation of cross sections for charge-exchange reactions, including the cross sections σ(E ν ) for neutrino capture by atomic nuclei [13,15].
In addition, Fig. 1 gives the energy thresholds Q 1 and Q 2 for the neighboring isobaric nuclei 98 Tc and 100 Tc, respectively. They also differ markedly-for example, the energy Q 1 = Q β is 1684 ± 3 keV for the isotope 98 Tc, while Q 2 = 172.1 ± 1.4 keV for the isotope 100 Tc [16]. As a result, a dominant role in the process of solar-neutrino capture is played by hard solar neutrinos in the case of the 98 Mo nucleus and by neutrinos of lower energy in the case of the 100 Mo nucleus. In the latter case, these are primarily pp solar neutrinos (that is, those from the reaction p + p → 2 H + e + + ν e ), for which E x 420 keV [1] and whose number is several orders of magnitude greater. This is the reason why the cross sections σ(E ν ) for neutrino capture by these nuclei differ strongly (see below).

CHARGE-EXCHANGE EXCITATIONS OF THE ISOTOPES 98,100 Mo
The resonance structure of charge-exchange excitations of the 98,100 Mo nuclei is illustrated in Fig. 2, where the experimental data obtained for the strength functions in the reactions 98 Mo(p, n) 98 Tc [2] and 100 Mo( 3 He, t) 100 Tc [3,4] are shown along with the respective data calculated in [17] within the theory of finite Fermi systems (TFFS) [18]. The data in Fig. 2 are given in the form of a graph that represents the dependence of the strength function S(E) on the excitation energy E reckoned from the ground state of the isotope 100 Mo(see Fig. 1). Reckoned with respect to this reference value, the energies of the isobaric resonances have close values, since the isotopes 98 Mo and 100 Mo differ by only two neutrons. The same reference system permits determining the solar-neutrino sorts in the graph (see Fig. 2c) that make contributions in various regions of energies of the isotopes 98,100 Mo considered here. One can see that low-energy solar neutrinos (see Fig. 2c) make a dominant contribution, which is several orders of magnitudes larger than the contribution of other neutrinos of the solar spectrum, to the capture cross section σ(E ν ) for the 100 Mo nucleus, but this is not so for 98 Mo, in which case Q β = 1684 keV and where a dominant contribution comes from harder boron and hep neutrinos (see Fig. 2c).
The charge-exchange strength functions S(E) given in Fig. 2 for the isotopes 98,100 Mo were calculated within the theory of finite Fermi systems (TFFS) [18], as was done earlier for other nuclei [13,19]. For the excited states of the daughter nucleus, the energies and matrix elements were determined according to [18] from the set of secular equations for the effective TFFS field. In the calculations, the values obtained in [20] from an analysis of experimental data on the energies of analog resonances (in 38 nuclei) and GT resonances (in 20 nuclei) were used for the parameters f 0 and g 0 of local isospinisospin and spin-isospin quasiparticle interactions. In just the same way as in [13], the continuum part of the spectrum of the strength function S(E) was calculated with a Breit-Wigner broadening (see [21,22]). In describing both the experimental and the calculated data on the strength function S(E) for the isotopes 98,100 Mo in Fig. 2

, the normalization of S(E)
is an issue of importance. For example, the experimental data for 98 Mo were obtained in the reaction 98 Mo(p, n) 98 Tc [2], whereupon the charge-exchange strength function S(E) was determined up to the excitation energy of E max = 18 MeV. It was found that the total sum of the GT matrix elements B(GT) up to the energy of 18 MeV is 28 ± 5 [2], which is 0.67 ± 0.08 of the maximum value of 3(N − Z) = 42, which is given by the sum rule for GT excitations of the 98 Mo nucleus. This means that there is a deficit in the sum rule for GT excitations. In [4], the results of processing B(GT) for 100 Mo are given over the energy range extending up to 4 MeV. For other energy values, the authors of [4] do not present the dependence of B(GT) on the energy E, nor do they give the sum ΣB(GT). In the earlier study reported in [3], it was found, however, that the sum of GT matrix elements up to the energy of 18.8 MeV is 34.56 or 0.72 (72%) of the maximum possible value of 3(N − Z) = 48. This is greater by 7.5% than the respective result for 98 Mo [2]. The observed deficit in the sum rule for GT excitations is due to the quenching effect [23] or to a violation of the normalization of GT matrix elements. According to the sum rule, the normalization for GT transitions has the form [16] where E max is the maximum energy taken into account in the calculation or in the experiment and S(E) is the charge-exchange strength function. In the present calculations, the value of E max = 20 MeV was employed for the isotopes 98 Mo and 100 Mo, while, in the experiments, this energy was set to, respectively, E max = 18 MeV [2] and E max ≈ 19 MeV [4]. The parameter q, q < 1, in Eq. (1) determines the quenching effect (deficit in the sum rule); at q = 1, , which corresponds to the maximum value. Within the TFFS framework, q = e 2 q , where e q is an effective charge [18]. As was shown by A.B. Migdal [24], the effective charge should not exceed unity; for Fermi transitions, we have e q (F) = 1, while, for GT transitions, e q (GT) = 1-2ζ S (see [18, p. 223]), where ζ S , 0 < ζ S < 1, is an empirical parameter. Thus, we see that, in the case of Mo → Tc transitions considered here, the effective charge e q = е q (GT) is a parameter that is extracted from experimental data. A detailed analysis of the quenching effect was performed in [17], where it was found that e q = 0.90 (q = 0.81) for the isotope 98 Mo and e q = 0.8 (q = 0.64) for the isotope 100 Mo. This confirms the presence of the quenching effect.

CROSS SECTIONS FOR SOLAR-NEUTRINO CAPTURE
BY 98,100 Mo NUCLEI The (ν e , e − ) cross section, which depends on the incident-neutrino energy E ν , is given by [21] σ( (23) is the axial-vector coupling constant [25].
The neutrino-capture cross section σ(E) is shown in Fig. 3 where, for the energy E max , we can restrict ourselves (see [26]) to hep neutrinos (that is, those from the reaction 3 He + p → 4 He + e + + ν e ), in which case E max 18.79 MeV, or to boron neutrinos (that is, those from the reaction 8 B → 8 Be + e + + ν e ), in which case E max 16.36 MeV. The solar-neutrino capture rate is given in SNU (SNU is a solar neutrino unit that corresponds to one detection event per second per 10 36 target nuclei). In calculating the cross sections for solar-neutrino capture, it is important to simulate correctly the flux of solar neutrinos. Several solar models have been vigorously developed in recent years. They include the BS05(OP), BS05(AGS, OP), and BS05(AGS, OPAL) models evolved by the group headed by J.N. Bahcall [26]. The helium concentration and metallicity (the specific number of atoms heavier than helium), as well as their distributions over the star volume, are the most important simulated parameters, along with the medium opaqueness parameter and the dimensions of the convection zone. A description of neutrino fluxes requires a detailed knowledge of the cross section for neutrino interaction with a detector material and, as a consequence, knowledge of the strength function and its resonance structure for nuclei of this material. In this article, we present calculations based on the BS05(OP) model, which is the most convenient for a comparison with experimental data. The rescaling to other solar models reduces to a normalization of the fluxes.
The numerical values of the solar-neutrino capture rates R calculated for the isotopes 98 Mo and 100 Mo are listed in Tables 1-4 (in SNU). The tables give the results obtained by calculating R with the experimental and theoretical strength functions S(E) and with and without allowance for GT and pygmy resonances. The calculations with the experimental strength functions S(E) (see Table 1 and 3) were performed by employing data obtained in the reactions 98 Mo(p, n) 98 Tc [2] and 100 Mo( 3 He, t) 100 Tc [3,4] (see Fig. 2).
The capture rate obtained for the isotope 98 Mo is R Total = 18.52 SNU (Table 1), which is close to the value of 17.4 +18.5

−11
SNU from [1] and to the result found by employing the calculated strength functions; our result is R Total = 19.028 SNU (Table 2), while the respective result reported earlier in [27] is In the calculations for 100 Mo (Table 3), use was made of two sets of experimental data-one from [3] and the other from [4]. The point is that the table of the data presented in [4] for the energies E and matrix elements B(GT) covers the energy range of E 4 MeV, whereas, in the article of H. Akimune and his coauthors [3], which was published earlier, there are tabulated data on high-lying excitations of the daughter nucleus 100 Tc. In addition to the R values calculated for 100 Mo with the experimental strength functions, Table 3 gives the results obtained by H. Ejiri and S.R. Elliott [28] on the basis of the data from [4] extending to 4 MeV. In our results, this corresponds to the calculations that disregard GTR, and the discrepancies are insignificant, whereas the discrepancies between the values of R Total are approximately 0.4%. In the article published in 2017 [29], the same authors presented the value of R Total = 975 SNU, which differs from their earlier value and from our estimate by about 1%. The discrepancies in question stem from special features of experimentaldata processing and are irrelevant to the present analysis.
Comparing the results of the calculations for 98 Mo and 100 Mo (those in Tables 1 and 2 versus those in  Tables 3 and 4), we note, first of all, a large difference, larger than by a factor of 45, between the values of R Total for these isotopes. This is explained by a large difference between the energy of Q 1 = 1684 keV for the isotope 98 Tc and the energy of Q 2 = 172.1 keV for the isotope 100 Tc (see Fig. 1). As a result, a dominant contribution to the process of solar-neutrino capture comes from hard solar neutrinos in the case of the 98 Mo nucleus and from lower energy neutrinos, mostly pp solar neutrinos, whose number is several orders of magnitude greater (see Fig. 2), in the case of the 100 Mo nucleus. For example, the contribution of hard boron neutrinos to R Total in the case of 98 Mo  is 99%, whereas their contribution in the case of 100 Mo is as small as 2.6%; at the same time, soft pp neutrinos make a 70% contribution in the latter case (see Fig. 2).
The discrepancies between the R values obtained from the experimental and calculated data on the strength functions S(E) are more significant, amounting, for R Total to about 3% for 98 Mo and to about 14% for 100 Mo. This is due, for 98 Mo, to discrepancies in the description of resonance states [17], which make a dominant contribution to the neutrinocapture cross section σ(E ν ), and, for 100 Mo, to the    Table 3. Solar-neutrino capture rate R (in SNU) calculated for the isotope 100 Mo with the strength function obtained from the experimental data reported in [3,4] (also given here are the results of the calculations performed in [28] and based on the use of the data from [4]; the reduction (in percent) of the capture rates upon the disregard of GTR and GTR + PR1 is indicated parenthetically) Without taking into account neutrino oscillations, our calculations of the number of neutrino events for 100 Mo up to the neutron-separation energy with allowance for both GT and analog resonances yield 188.3 events per ton per year. The calculations were performed up to the energy of neutron separation from the 100 Tc nucleus, since higher lying excitations undergo de-excitation via neutron emission and transitions to excited states of the 99 Tc nucleus. This process will not contribute to backgrounds to 100 Mo double-beta decay involving solar neutrinos. The articles published in recent years present the following values: for example, the estimate obtained by the authors of [28] is R = 989 SNU (R = 975 SNU in [29]); upon rescaling to one ton of matter, this gives 187.6 (184.9 [29]) events per ton per year.
In real ground-based experiments, the flux of electron neutrinos from the Sun is approximately one-half as large as that in the model not featuring oscillations. Moreover, there are detection schemes making it possible to suppress the background partly [31].

CONCLUSIONS
We have explored the interaction of solar neutrinos with 98 Mo and 100 Mo nuclei, taking into account the effect of charge-exchange resonances. We have studied the effect of high-lying charge-exchange resonances in the strength function S(E) on the cross sections for solar-neutrino capture by 98 Mo and 100 Mo nuclei. We have employed both experimental data obtained for the strength functions S(E) in (p, n) and ( 3 He, t) charge-exchange reactions [2][3][4] and the strength functions S(E) calculated within the theory of finite Fermi systems [17].
A comparison of the calculated strength function S(E) with experimental data shows good agreement both in resonance-peak energy and in resonancepeak amplitude. There is a deficit in the sum rule for GT excitations. It is due either to the quenching effect [23] or to a violation of the normalization of GT matrix elements. Within the TFFS framework [18], this deficit is compensated by the introduction of an effective charge-e q = 0.90 (q = 0.81) for the isotope 98 Mo and e q = 0.8 (q = 0.64) for the isotope 100 Mo [17].
We have calculated the cross sections σ(E) for solar-neutrino capture and have analyzed the contribution of all charge-exchange resonances. We have found that, in all energy ranges, the cross section σ(E) is substantially larger for the 100 Mo nucleus than for the 98 Mo nucleus. This is because the energy thresholds Q 1 and Q 2 are markedly different for the neighboring isobaric nuclei 98 Tc and 100 Tc (see Fig. 1). As a result, the cross sections σ(E) for the 98 Mo and 100 Mo nuclei are also different. Thus, σ(E) for 100 Mo receives a dominant contribution from soft neutrinos, whose number is several orders of magnitude greater (see Fig. 2), the resonance energy region having no effect here. Accordingly, the contribution of high-energy nuclear resonances to σ(E) is smaller for 100 Mo than for 98 Mo.
We have also calculated solar-neutrino capture rates R for the isotopes 98 Mo and 100 Mo, taking into account all components of the solar neutrino spectrum. The calculations have been performed both with experimental and with theoretical strength functions S(E) and with and without allowance for the GT and pygmy resonances.
Comparing the results of our calculations for 98 Mo and 100 Mo nuclei, we note that the values of R Total for these isotopes differ by a factor greater than 45. The reason for this is that, as was already indicated, hard solar neutrinos play a major role in solar-neutrino capture by the 98 Mo nucleus, but, in the case of the 100 Mo nucleus, neutrinos of lower energy, which are more copious by several orders of magnitude, contribute predominantly.
The problem of the change in the background values of R Total because of neutrino oscillations was not addressed in the present article, since this would require taking into account changes in all components of the solar flux, which have different energies.
Thus, the two isotopes 98 Mo and 100 Mo of the same element, which differ only slightly in structure and in charge-exchange strength function, differ sharply in solar-neutrino capture cross section, σ(E), and in solar-neutrino capture rate.