Charge-exchange isobaric resonances

Three types of the charge-exchange isobaric resonances — giant Gamow–Teller (GTR), the analog (AR) and pygmy (PR) ones are investigated using the microscopic theory of finite Fermi systems and its approximated version. The calculated energies of GTR, AR and three PR’s are in good agreement with the experimental data. Calculated differences ∆EG−A=EGTR−EAR go to zero in heavier nuclei indicating the restoration of Wigner SU(4)-symmetry. The average deviation for ∆EG−A is 0.30 MeV for the 33 considered nuclei where experimental data are available. The comparison of calculations with experimental data on the energies of charge-exchange pygmy resonances gives the standard deviation δE<0.40 MeV. Strength functions for the 118Sn, 71Ga, 98Mo and 127I isotopes are calculated and the calculated resonance energies and amplitudes of the resonance peaks are close to the experimental values. Strong influence of the charge-exchange resonances on neutrino capturing cross sections is demonstrated.


Introduction
Charge-exchange isobaric states are manifested in the corresponding charge-exchange reactions such as (ν, e), (p, n), (n, p), ( 3 He, t), (t, 3 He), ( 6 Li, 6 He) and others, or in β-transitions in nuclei. Among these states, collective resonance excitations are of the most interest. The theoretical studies of these collective states have began with the first calculations of the giant Gamow-Teller resonance (GTR) [1] and other collective states [2] long before their experimental studies in charge-exchange reactions [3,4]. These collective states lying below the giant GTR [5] were called "pygmy" resonances (PR).
The most complete experimental information on the charge-exchange excitations in 9 tin isotopes with A=112-124 were obtained in [6], where the Sn( 3 He, t)Sb chargeexchange reaction at the energy E( 3 He)=200 MeV was used. The excitation energies (E x ), widths (Γ), and cross sections dσ/dΩ (mb/sr) were measured for the analog, Gamow-Teller, and three pygmy resonances. Also some charge-exchange strength-functions S (E) were measured earlier in (p, n) and ( 3 He, t) reactions. The influence of the charge-exchange resonances on neutrino capturing cross sections was investigated also.

Method of calculation
Charge-exchange excitations of nuclei are described in the microscopic theory of finite Fermi systems (TFFS) by the system of equations for the effective field [7] V pn = e q V ω pn + p n F ω np,n p ρ p n , where V pn and V h pn are the effective fields of quasi-particles and holes in a nucleus whereas V ω pn is the external chargeexchange field. The system of secular equations (1) is solved for allowed transitions with local nucleon-nucleon interaction in the Landau-Migdal form [7] F with the parameters f 0 =1.35 and g 0 =1.22 as in [8].
The energies of charge-exchange excitations were calculated both in the self-consistent TFFS and in its approximate version [5], which allowed to obtain the analytical solutions for the most collective states. For the energies E GTR and E PR , the solutions ω k (k=0 for GTR and k=1, 2, 3 for PR1, PR2, and PR3, correspondingly) divided by the average value E ls of the spin-orbit splitting [8] at ∆E>E ls , has the form Parameters a k , b k and g k in (3) are defined as follows: and for k > 0 The energies of analog resonances (AR) were calculated within TFFS and also from the simple formulas for the difference ∆E C between the Coulomb energies of neighboring isobar nuclei where δQ=Q β +∆ np , and ∆ np =0.78234 MeV is the dif- ference between the neutron mass and the mass of hydrogen atom. For all nuclei with 5≤A≤ 244 we have obtained a C =1416 keV and b C =−698 keV [9] with the standard deviation δE<100 keV. Deformation was taken into account as in [9], by introducing the correction to ∆E C = ∆E sph C −δE def C with the deformation parameters β 2 and β 4 from [10].  169 Tm, and 208 Pb for which experimental data are available [6,11]. The results of theoretical calculations and experimental data are presented in Fig. 1. Strictly speaking, in Fig. 1 we display the calculated and extracted from experimental data the dimensionless ratio y(x)=∆E G−A /E ls as a function of parameter x=∆E/E ls . Moreover, the values of ∆y 0 (x) calculated by the following equation:

Differences of energies between Gamow-Teller and analog resonances ∆E
is also depicted in Fig. 1. Equation (6) is related with Eq. (3) where k=0 for the GTR energy. The standard r.m.s. deviation of the values calculated within TFFS Eq. (2) from the experimental data for the listed above nuclei is δ(∆ε)≤0.30 MeV which is comparable to the accuracy of the E GTR experimental data [6]. Calculated differences ∆E G−A go to zero in heavier nuclei (for 208 Pb x=2. 15) indicating the restoration of Wigner SU(4)-symmetry [8]. Thus the analog and Gamow-Teller resonances belong to the same supermultiplet. AR energies were analyzed using both TFFS and simple Eqs. (4) and (5). Based on the results of our analysis of the energies ∆E C for more than 400 nuclei [8], one can state that the observed functional dependence corresponds to the SU(4)theory.
The most complete experimental studies of the entire spectrum of the charge-exchange excitations were performed in [6] for nine tin isotopes 112,114,116,117,118,119,120,122,124 Sn exploiting the chargeexchange reaction ( 3 He, t) at beam energy 200 MeV. In particular, three charge-exchange pygmy resonances were identified in every Sn isotopes. Our results for the energies of five charge-exchange resonances AR, GTR, PR1, PR2, and PR3 for nine tin isotopes calculated within microscopic TFFS presented in Table 1 together with the experimental data from [6]. As it can be seen from Table 1, the standard r.m.s. deviations of the calculated from experimental ones for the energies are small enough δE<0.40 MeV. These values of δE are comparable with the experimental errors ∆E exp =±0.25 MeV. Our results are close to that of other calculations of high-lying excitations, e.g., performed in the framework of self-consistent QRPA with Skyrme forces [12]. Figure 2 demonstrates the difference between the energies of GTR and lying below pygmy resonances PR for Sn isotopes as functions of the mass number A. The calculations by Eq. (3) produces standard deviations δE PR1 =0.53  Table 1). Thus, on average the calculations by two methods describe the experimental data satisfactorily. Charge-exchange strength functions S (E x ) were calculated within TFFS for more than 100 isotopes in the middle-mass region. At present only few of these nuclides are investigated experimentally in charge-exchange nuclear reactions such as (p, n) or ( 3 He, t). We present in Figure 3a) the experimental data on excitation spectra obtained in 118 Sn( 3 He, t) 118 Sb reaction [6]. The calculated charge-exchange strength function S (E x ) for 118 Sn is depicted in Figure 3b). Unfortunately, direct measurements of the strength function S (E x ) were not performed, but the data on counts shown in Figure 3a) are proportional to the partial data on the function S (E x ). One more example of the experimental and theoretical strength functions are given in Figure 4 for the 71 Ga nucleus. Here, the GTR resonance as well as three pygmy resonances located at lower excitation energy are seen as well. Theoretical and experimental energies of the resonances are in reasonable agreement. These resonances play significant role in the neutrino capturing reactions as will be discussed in the next section 4.

Charge-exchange resonances in the neutrino capturing reactions
The charge-exchange resonances strongly influence neutrino capturing cross sections by nuclei σ(E ν ). We calculated within TFFS charge-exchange strength functions S (E x ) for 71 Ga, 98 Mo and 127 I nuclei. Then, knowing S (E x ) we calculated cross sections σ(E ν ) applying the following equation (see [14]): In Eq. (7), E ν is an incident neutrino energy, F(Z, A, E e )the Fermi function, G F /( c) 3 -the Fermi weak interaction constant, g A -the axial-vector constant from [15]. Cal- culations were carried out for 71 Ga, 98 Mo and 127 I nuclei which are of interest for neutrino physics. The results of calculations σ(E ν ) are shown partly in Figure 5. It is seen that the effect of charge-exchange resonances on the cross section is quite sizable. Even at E ν =4 MeV, the exclusion of the Gamow-Teller resonance alone leads to reduction in the cross section by 25% for the reaction 71 Ga(ν e , e − ) 71 Ge. For the 98 Mo(ν e , e − ) 98 Tc reaction exclusion of GTR reduces the cross-section by 16% at E ν =6 MeV and by 45% at E ν =14 MeV. For the 127 I(ν e , e − ) 127 Xe reaction the crosssection appears to be reduced by 19% at E ν =2 MeV and by 69% at E ν =14 MeV. Certainly, the effect becomes stronger

Conclusions
Observed in all charge-exchange reactions three types of the isobaric resonances -the giant Gamow-Teller (GTR), the analog (AR) and pygmy (PR) ones -are investigated exploiting the microscopic theory of finite Fermi systems and its approximate version. The calculated energies of GTR, AR and three PRs are in good agreement with the experimental data. The root-mean-square deviation for the value ∆E G−A is 0.30 MeV for the 33 considered nuclei where experimental data are available. The comparison of calculations with experimental data on the energies of charge-exchange pygmy resonances gives the standard deviation δE<0.40 MeV. Calculated difference of energies of GTR and AR resonances ∆E G−A =E GTR −E AR go to zero in heavier nuclei indicating the restoration of Wigner SU(4)symmetry. This allows to describe the properties of heavy nuclei more confidently using SU(4)-theory, especially for mass relations [16]. Also the analysis of the Coulomb displacement energies using the SU(4)-approach allows to describe AR energies and the masses of superheavy nuclei with a good accuracy. Strength functions for the 118 Sn, 71 Ga, 98 Mo and 127 I nuclides were calculated and analyzed. The calculated resonance energies and amplitudes of resonance peaks appear to be close to the experimental values. Strong influence of the charge-exchange resonances on neutrino capturing cross sections is demonstrated.
Further investigation of the resonant structure of the strength functions S (E) will allow us to analyze the data of the charge-exchange reactions and β-delayed processes. So the appearance of PR in the energy window of the emission of β-delayed neutrons leads to a sharp increase in the probability of this process [17]. Similar effect can be observed for the β-delayed fission process [18].