Role of momentum transfer in the quenching of Gamow-Teller strength

Background: Differential cross sections for the $(p,n)$ and $(n,p)$ reactions on ${}^{90}$Zr over the interval of $0–50$-MeV excitation energy were used to determine the corresponding Gamow-Teller (GT) strengths, and the resulting quenching factor is $\approx$0.9 with respect to the Ikeda sum rule. In this procedure the contribution of the isovector spin monopole (IVSM) strength was subtracted from the total strength without taking into account the interference between the GT and the IVSM modes.

Purpose: To determine the quantitative effect of the IVSM excitation mode on the $L=0$ strength in charge-exchange reactions on several closed-shell nuclei and the Sn isotopic chain.

Method: The fully consistent relativistic Hartree-Bogoliubov model plus proton-neutron relativistic quasiparticle random-phase approximation ($pn$-RQRPA) are employed in the calculation of transition strength in the ${\beta }^{-}$ and ${\beta }^{+}$ channels.

Results: The inclusion of the higher-order terms, that include the effect of finite momentum transfer, in the transition operator shifts a portion of the strength to the high-energy region above the GT resonance. The total strength is slightly enhanced in nuclei with small neutron-to-proton ratios but remains unchanged with increasing neutron excess.

Conclusions: Terms that include momentum transfer in the transition operator act mostly to shift the strength to high excitation energies but hardly affect the total strength. Based on the strength obtained using the full $L=0$ transition operator in the $pn$-RQRPA calculation, we have estimated the impact of the IVSM on the strength measured in the charge-exchange reactions on ${}^{90}$Zr and found that the data are consistent with the Ikeda sum rule.

Figure 1

The $pn$-RQRPA strength distribution of isovector spin monopole states in ${}^{90}$Zr. The dashed curve denotes the total strength with $0\hslash \omega$ transitions included, whereas the solid curve corresponds to the strength for which only $2\hslash \omega$ and higher excitations are included in the configuration space, for the (a) ${\beta }^{-}$ and (b) ${\beta }^{+}$ channels.

Figure 2

Energy centroids of the isovector spin monopole strength in tin isotopes (excluding the $0\hslash \omega$ transitions), in comparison with the corresponding Gamow-Teller centroids.

Figure 3

Momentum transfer for a $(p,n)$ reaction, calculated using Eqs. (17, 18, 19, 20). (a) $q$ plotted for a constant kinetic energy of the incoming proton equal to $T=300$ MeV. (b) $q$ plotted for constant excitation energy of the nucleus of $E_x=50$ MeV.

Figure 4

Comparison of the $pn$-RQRPA strengths obtained with the Gamow-Teller operator (dashed), the GT $+$ IVSM operator Eq. (16) (full), and the full $L=0$ operator Eq. (14) (dash-dotted) in ${}^{90}$Zr. The upper panels display the strength in the ${\beta }^{-}$ channel, whereas the strength in the ${\beta }^{+}$ channel is shown in the lower panels. Different scales are used for the region of low excitation energy below 30 MeV (left) and excitation energies in the interval 20 to 70 MeV (right).

Figure 5

Ratios $\eta$ and $\zeta$ defined in Eqs. (23) and (24), respectively, for the Sn isotopes. Open symbols denote ratios calculated with the GT $+$ IVSM operator Eq. (16), and filled symbols are for those obtained using the full $L=0$ operator Eq. (14).

Figure 6

Comparison of the ${\beta }^{-}$ strength distributions in ${}^{100}$Sn (upper panels) and ${}^{144}$Sn (lower panels), calculated with the Gamow-Teller operator, the GT $+$ IVSM operator Eq. (16), and the full $L=0$ operator Eq. (14). Note the different scales used for the resonance region and the region of high-excitation energies.

Figure 7

${\beta }^{-}$ $pn$-RQRPA strength located at energies above $E_{95\%}$ in Sn isotopes with $100\le A\le 150$. The GT operator (circles), the GT $+$ IVSM operator defined in Eq. (16) (squares), and the full $L=0$ operator Eq. (14) (diamonds) have been used in the calculation of the strength distributions.