Interaction of Solar Neutrinos with ${}^{98}\mathrm{Mo}$ and ${}^{100}\mathrm{Mo}$ Nuclei

Abstract

The process of neutrino interaction with ${}^{98}\mathrm{Mo}$ and ${}^{100}\mathrm{Mo}$ nuclei is studied with allowance for the effect of charge-exchange resonances. The results obtained by calculating the cross section for solar-neutrino capture by the isotopes ${}^{98}$Mo and ${}^{100}$Mo, $\sigma(E_{\nu})$, 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 finite Fermi systems. The effect 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 $\sigma(E_{\nu})$ 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.

1 INTRODUCTION

In calculating the cross sections for neutrino interaction with atomic nuclei, $\sigma(E_{\nu})$, it is necessary to determine, for the nuclei involved, the charge-exchange strength function $S(E)$, which has a resonance character. For solar neutrinos, the upper boundary of the spectrum is determined by the $hep$ reaction ${}^{3}\textrm{He}+p\to^{4}\textrm{He}+e^{+}+\nu_{e}$, in which case $E_{x}\leqslant 18.77$ 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 solar-neutrino capture, $\sigma(E_{\nu})$, 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 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 $\sigma(E_{\nu})$ for neutrino capture by atomic nuclei [13, 15].

Fig. 1

Scheme of excited levels of ${}^{98,100}$Mo nuclei.

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 $Q_{1}=Q_{\beta}$ is $1684\pm 3$ keV for the isotope ${}^{98}$Tc, while $Q_{2}=172.1\pm 1.4$ 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 $p+p\to^{2}\textrm{H}+e^{+}+\nu_{e}$), for which $E_{x}\leqslant 420$ keV [1] and whose number is several orders of magnitude greater. This is the reason why the cross sections $\sigma(E_{\nu})$ for neutrino capture by these nuclei differ strongly (see below).

2 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 ${}^{100}\mathrm{Mo}(^{3}\mathrm{He},t)^{100}$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 $\sigma(E_{\nu})$ for the ${}^{100}$Mo nucleus, but this is not so for ${}^{98}$Mo, in which case $Q_{\beta}=1684$ keV and where a dominant contribution comes from harder boron and $hep$ neutrinos (see Fig. 2c).

Fig. 2

Charge-exchange strength function $S(E)$ for GT excitations of the isotopes (a) ${}^{98}$Tc and (b) ${}^{100}$Tc: (thin curves) experimental data from [2] for ${}^{98}$Tc and from [4] for ${}^{100}$Tc, (thick curves) results of the present calculation based on the theory of finite Fermi systems, and (dashed curves) resonances (GTR, PR1, PR2, and PR3). (c) Fluxes of solar neutrinos, where various contributions are indicated.

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 ${f}^{\prime}_{0}$ and ${g}^{\prime}_{0}$ of local isospin–isospin 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 $28\pm 5$ [2], which is $0.67\pm 0.08$ 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 $\Sigma B$(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]

$$\Sigma M_{i}^{2}=\Sigma B_{i}(\textrm{GT})=q[3(N-Z)]$$

(1)

$${}=e_{q}^{2}[3(N-Z)]\approx\int\limits_{0}^{E_{\max}}{S(E)}dE=I(E_{\max}).$$

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 $E_{\max}\approx 19$ MeV [4]. The parameter $q$, $q<1$, in Eq. (1) determines the quenching effect (deficit in the sum rule); at $q=1$, $\Sigma M_{i}^{2}=\Sigma B_{i}(\mathrm{GT})=3(N-Z)$, which corresponds to the maximum value. Within the TFFS framework, $q=e_{q}^{2}$, 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 $e_{q}(\mathrm{F})=1$, while, for GT transitions, $e_{q}(\mathrm{GT})=1$–$2\zeta_{S}$ (see [18, p. 223]), where $\zeta_{S}$, $0<\zeta_{S}<1$, is an empirical parameter. Thus, we see that, in the case of Mo $\to$ Tc transitions considered here, the effective charge $e_{q}=e_{q}$(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.

3 CROSS SECTIONSFOR SOLAR-NEUTRINO CAPTURE BY ${}^{98,100}$Mo NUCLEI

The ($\nu_{e},e^{-}$) cross section, which depends on the incident-neutrino energy $E_{\nu}$, is given by [21]

$$\sigma(E_{\nu})=\frac{(G_{\textrm{F}}g_{A})^{2}}{\pi c^{3}\hbar^{4}}\!\!\int\limits_{0}^{E_{\nu}-Q}\!\!{E_{e}p_{e}F(Z,A,E_{e})S(x)dx,}$$

(2)

$$E_{e}=E_{\nu}-Q-x+m_{e}c^{2},$$

$$cp_{e}=\sqrt{E_{e}^{2}-(mc^{2})^{2}},$$

where $F(Z,A,E_{e})$ is the Fermi function, $S(E)$ is the strength function, $G_{\mathrm{F}}/(\hbar c)^{3}=1.1663787(6)\times 10^{-5}$ GeV${}^{-2}$ is the Fermi weak coupling constant, and $g_{A}={-}1.2723(23)$ is the axial-vector coupling constant [25].

The neutrino-capture cross section $\sigma(E)$ is shown in Fig. 3 for the reaction ${}^{98}$Mo$(\nu_{e},e^{-})^{98}$Tc and in Fig. 4 for the reaction ${}^{100}$Mo$(\nu_{e},e^{-})^{100}$Tc. The cross sections $\sigma(E)$ are given both according to the calculations with the experimental strength function $S(E)$ (see Fig. 2) and according to the calculations with the strength function $S(E)$ obtained within the TFFS approach. Also presented here are the results of the calculations performed without allowing for GT and pygmy resonances. From the figures in question, one can see that the calculations with the strength functions $S(E)$ obtained within the TFFS approach describe fairly well the cross sections $\sigma(E)$ calculated with the experimental strength functions, the average discrepancies for the total cross section not exceeding 10${\%}$ both for ${}^{98}$Mo and for ${}^{100}$Mo.

Fig. 3

Neutrino-capture cross section $\sigma(E)$ in the reaction ${}^{98}$Mo$(\nu_{e},e^{-})^{98}$Tc. The points on display stand for the results of the calculation based on the experimental strength function $S(E)$ (see Fig. 2). The solid and dashed curves represent the results of the calculations performed with the strength function $S(E)$ obtained within the TFFS approach: (1) total cross section and (2, 3, 4, and 5) results of the calculations not including, respectively, GTR; GTR and PR1; GTR, PR1, and PR2; and GTR, PR1, PR2, and PR3, where GTR is the GT resonance.

Fig. 4

Neutrino-capture cross section $\sigma(E)$ in the reaction ${}^{100}$Mo$(\nu_{e},e^{-})^{100}$Tc. The points on display stand for the results of the calculation based on the experimental strength function $S(E)$ (see Fig. 2). The solid and dashed curves represent the results of the calculations performed with the strength function $S(E)$ obtained within the TFFS approach: (1) total cross section, (2) results obtained without including GTR, and (3) results obtained without including GTR and PR1.

From Figs. 3 and 4, one can see that the effect of charge-exchange resonances on the cross section $\sigma(E)$ is quite significant. The disregard of only two resonances, the GT resonance and PR1, reduces the cross section $\sigma(E)$ for ${}^{98}$Mo by a value of about 10${\%}$ to a value of about 60${\%}$ for neutrino energies between 4 and 14 MeV; for ${}^{100}$Mo, the respective reduction is about 5${\%}$ to 40${\%}$. Thus, the effect of the resonances on the cross section $\sigma(E)$ is smaller for the ${}^{100}$Mo nucleus than for the ${}^{98}$Mo nucleus. We can see this in Fig. 5, which shows the ratios of the cross sections $\sigma_{i}(E)$ calculated for the reactions ${}^{98}$Mo$(\nu_{e},e^{-})^{98}$Tc and ${}^{100}$Mo$(\nu_{e},e^{-})^{100}$Tc and normalized to the total cross section $\sigma_{\mathrm{tot}}(E)$ determined by employing the strength functions $S(E)$ calculated within the TFFS framework. The reason behind the reduction of the effect of charge-exchange resonances on the cross section $\sigma(E)$ for neutrino capture by the ${}^{100}$Mo nucleus in relation to the respective results for ${}^{98}$Mo is that the cross section for ${}^{100}$Mo receives a contribution primarily from low-energy solar neutrinos, whose number is several orders of magnitude smaller than the number of neutrinos having energies in the region of $E_{\nu}>2$ MeV and making a dominant contribution to the region of resonances in ${}^{98}$Mo.

Fig. 5

Ratios of the cross sections $\sigma_{i}(E)$ calculated for the reactions (curves 3 and 5) ${}^{98}$Mo$(\nu_{e},e^{-})^{98}$Tc and (curves 2 and 4) ${}^{100}$Mo$(\nu_{e},e^{-})^{100}$Tc to the total cross section $\sigma_{\mathrm{tot}}(E)$ based on the TFFS approach (curve 1). Curves 2 and 3 were calculated without taking into account GTR, while curves 4 and 5 were calculated without taking into account GTR and PR1.

4 RATE OF SOLAR-NEUTRINO CAPTURE BY ${}^{98,100}$Mo NUCLEI

The solar-neutrino capture rate $R$ (number of neutrinos absorbed per unit time) is related to the solar-neutrino flux and the capture cross section by the equation

$$R=\int\limits_{0}^{E_{\max}}{\rho_{\mathrm{solar}}(E_{\nu})\sigma_{\mathrm{total}}(E_{\nu})}dE_{\nu},$$

(3)

where, for the energy $E_{\max}$, we can restrict ourselves (see [26]) to $hep$ neutrinos (that is, those from the reaction ${}^{3}\textrm{He}+p\to^{4}\textrm{He}+e^{+}+\nu_{e}$), in which case $E_{\max}\leqslant 18.79$ MeV, or to boron neutrinos (that is, those from the reaction ${}^{8}\textrm{B}\to^{8}\textrm{Be}+e^{+}+\nu_{e}$), in which case $E_{\max}\leqslant 16.36$ 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 ${}^{100}\mathrm{Mo}(^{3}\mathrm{He},t)^{100}$Tc [3, 4] (see Fig. 2).

Table 1 Solar-neutrino capture rate $R$ (in SNU) calculated for the isotope ${}^{98}$Mo with the strength function obtained from experimental data reported in [2] (the reduction (in percent) of the capture rates upon the disregard of GTR and GTR $+$ PR1 is indicated parenthetically)

Table 2 Solar-neutrino capture rate $R$ (in SNU) calculated for the isotope ${}^{98}$Mo with the strength function obtained within the TFFS framework [17] (the reduction (in percent) of the capture rates upon the disregard of GTR and GTR $+$ PR1 is indicated parenthetically)

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)

Table 4 Solar-neutrino capture rate $R$ (in SNU) calculated for the isotope ${}^{100}$Mo with the strength function obtained within the TFFS framework [17] (the reduction (in percent) of the capture rates upon the disregard of GTR and GTR $+$ PR1 is indicated parenthetically)

The capture rate obtained for the isotope ${}^{98}$Mo is $R_{\mathrm{Total}}=18.52$ SNU (Table 1), which is close to the value of $17.4_{-11}^{+18.5}$ SNU from [1] and to the result found by employing the calculated strength functions; our result is $R_{\mathrm{Total}}=19.02$8 SNU (Table 2), while the respective result reported earlier in [27] is $28_{-8}^{+15}$ SNU.

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 $E\leqslant 4$ 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 $R_{\mathrm{Total}}$ are approximately 0.4${\%}$. In the article published in 2017 [29], the same authors presented the value of $R_{\mathrm{Total}}=975$ SNU, which differs from their earlier value and from our estimate by about 1${\%}$. The discrepancies in question stem from special features of experimental-data 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 $R_{\mathrm{Total}}$ 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 $R_{\mathrm{Total}}$ 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 $R_{\mathrm{Total}}$ 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 neutrino-capture cross section $\sigma(E_{\nu})$, and, for ${}^{100}$Mo, to the inaccuracies in describing low-lying states, where the calculated value of $R$ depends greatly on the changes in $E_{x}$ and $B$(GT). For example, the change in the ground-state position from 0 to 100 keV with a step of $\Delta E=50$ keV causes a sequential change of about 150 SNU in $R_{\mathrm{Total}}$ at each step $\Delta E$ (in all, about 300 SNU). Almost everything is associated here with the $pp$-neutrino channel. For neutrinos from ${}^{7}$Be, the decrease is about 10 SNU at each step $\Delta E$.

The effect of charge-exchange resonances on the rates $R$ of solar-neutrino capture by the isotopes ${}^{98}$Mo and ${}^{100}$Mo is also illustrated in Tables 1–4. One can see that the values of $R_{\mathrm{Total}}$ for ${}^{100}$Mo undergo virtually no change in the case of the calculation without GTR (decrease of about 1${\%}$) and in the case of the calculation without GTR and PR1 (decrease of about 2${\%}$), but that, for ${}^{98}$Mo, these changes are significant: ${-}34{\%}$ and ${-}43{\%}$, respectively. As was indicated above, the reason is that a dominant contribution to $R_{\mathrm{Total}}$ comes from low-energy neutrinos (about 70${\%}$)—mostly from $pp$ solar neutrinos—for ${}^{100}$Mo and from boron neutrinos (about 99${\%}$) for ${}^{98}$Mo. As a result, the calculations without GTR and PR1 make nearly identical contributions to $R_{\mathrm{Total}}$ and $R(^{8}\mathrm{B})$ for ${}^{98}$Mo. The situation is similar for the iodine isotope ${}^{127}\mathrm{I}$ [30], in which case $R_{\mathrm{Total}}=37.904$ SNU and $R(^{8}\mathrm{B})=33.232$ SNU differ by a value as small as 12.3${\%}$, whereas the GTR and PR1 contributions reduce $R_{\mathrm{Total}}$ by 72.7${\%}$ to 10.345 SNU (27.3${\%}$) owing primarily to boron neutrinos.

The analog resonances at the energies of $E(\mathrm{AR})_{\textrm{exp}}=9.7$ MeV [2] and $E(\mathrm{AR})_{\textrm{calc}}=9.78$ MeV [17] in ${}^{98}$Mo and at the energies of $E(\mathrm{AR})_{\textrm{exp}}=11.085$ MeV [4] and $E(\mathrm{AR})_{\textrm{calc}}=10.99$ MeV [17] in ${}^{100}$Mo have but a slight effect on the cross sections $\sigma(E)$ and on the solar-neutrino capture rates $R$. For example, the inclusion of the analog resonances increases $R$ by $\Delta R\leqslant 5{\%}$ for ${}^{98}$Mo and by $\Delta R\leqslant 1{\%}$ for ${}^{100}$Mo.

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 ${}^{\mathrm{99}}$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].

5 CONCLUSIONS

We have explored the interaction of solar neutrinos with ${}^{98}\mathrm{Mo}$ and ${}^{100}\mathrm{Mo}$ 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 [2–4] 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 resonance-peak 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 $\sigma(E)$ 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 $\sigma(E)$ 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 $\sigma(E)$ for the ${}^{98}$Mo and ${}^{100}$Mo nuclei are also different. Thus, $\sigma(E)$ 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 $\sigma(E)$ 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 $R_{\mathrm{Total}}$ 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 $R_{\mathrm{Total}}$ 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, $\sigma(E)$, and in solar-neutrino capture rate.