Search for weak $M1$ transitions in ${}^{48}\mathrm{Ca}$ with inelastic proton scattering

Background: The quenching of spin-isospin modes in nuclei is an important field of research in nuclear structure. It has an impact on astrophysical reaction rates and on fundamental processes like neutrinoless double-$\beta$ decay. Gamow–Teller (GT) and spin-flip $M1$ strengths are quenched. Concerning the latter, the $J^{\pi }=1^{+}$ resonance in the doubly magic nucleus ${}^{48}\mathrm{Ca}$, dominated by a single transition, serves as a reference case.

Purpose: The aim of the present work is to search for weak $M1$ transitions in ${}^{48}\mathrm{Ca}$ with a high-resolution $(p,p^{\prime })$ experiment at 295 MeV and forward angles including $0^{\circ }$ and a comparison with results from a similar study using backward-angle electron scattering at low momentum transfers in order to estimate their contribution to the total $B\left(M1\right)$ strength in ${}^{48}\mathrm{Ca}$.

Methods: The spin-$M1$ cross sections of individual peaks in the spectra are deduced with a multipole decomposition analysis (MDA) and converted to reduced spin-$M1$ transition strengths by using the unit cross-section method. For a comparison with electron-scattering results, corresponding reduced $B\left(M1\right)$ transition strengths are extracted following the approach outlined in Birkhan et al. [Phys. Rev. C 93, 041302(R) (2016)].

Results: In total, 30 peaks containing a $M1$ contribution are found in the excitation energy region 7–13 MeV. The resulting $B\left(M1\right)$ strength distribution compares well to the electron-scattering results considering different factors limiting the sensitivity in both experiments and the enhanced importance of mechanisms breaking the proportionality of nuclear cross sections and electromagnetic matrix elements for weak transitions as studied here. The total strength of 1.14(7) ${\mu }_{\mathrm{N}}^2$ deduced assuming a nonquenched isoscalar part of the $(p,p^{\prime })$ cross sections agrees with the $(e,e^{\prime })$ result of 1.21(13) ${\mu }_{\mathrm{N}}^2$. A bin-wise analysis above 10 MeV provides an upper limit of 1.51(17) ${\mu }_{\mathrm{N}}^2$.

Conclusions: The present results confirm the previous electron-scattering work that weak transitions contribute about 25% to the total $B\left(M1\right)$ strength in ${}^{48}\mathrm{Ca}$ and the quenching factors of GT and spin-$M1$ strength are then comparable in $fp$-shell nuclei. Thus, the role of meson-exchange currents seems to be negligible in ${}^{48}\mathrm{Ca}$, in contrast to $sd$-shell nuclei.

Figure 1

(a) Spectrum of the ${}^{48}\mathrm{Ca}(p,p^{\prime })$ reaction at $E_0=295$ MeV and ${\theta }_{\mathrm{c}.\mathrm{m}.}=0.4^{\circ }$. The strong $M1$ transition at $E_{\mathrm{x}}=10.23$ MeV extends the scale by a factor of 14 and has a maximum cross section of about 350 mb/(sr MeV). (b) Spectra in the excitation energy range $E_{\mathrm{x}}=10.1–10.6$ MeV for scattering angles ${\theta }_{\mathrm{c}.\mathrm{m}.}=0.4^{\circ }$ (solid line), $1.8^{\circ }$ (dotted line), and $4.5^{\circ }$ (dashed line).

Figure 2

Calculated angular distributions used in the MDA for $E1,M1,E2$, and $E3$ transitions populating (a) $J^{\pi }=1^{-}$, (b) $1^{+}$, (c) $2^{+}$, and (d) $3^{-}$ states in ${}^{48}\mathrm{Ca}$, respectively, normalized at ${\theta }_{\mathrm{c}.\mathrm{m}.}=0^{\circ }$.

Figure 3

Examples of the MDA for the peak at $E_{\mathrm{x}}=12.275$ MeV using different combinations of theoretical angular distributions from Fig. 2.

Figure 4

Extended view of the spectrum of Fig. 1 between 7 and 13 MeV. The arrows indicate the energies of structures investigated in the single-peak analysis.

Figure 5

Examples of the MDA with the smallest $S_{\mathrm{red}}^2$ values for the transitions to the states at (a) 7.648 MeV, (b) 9.383 MeV, (c) 9.475 MeV, and (d) 9.973 MeV.

Figure 6

Angular distribution of the transition to the state at 10.138 MeV. The MDA clearly favors a pure $M1$ character.

Figure 7

$B\left(M1\right)\uparrow$ strength extracted from the ${}^{48}\mathrm{Ca}(p,p^{\prime })$ data. Orange (light gray) histogram: bin-wise analysis above the prominent peak at 10.23 MeV. Blue (dark gray) histogram: single-peak analysis (Table 2) summed in the same energy bins.

Figure 8

Comparison between the $B\left(M1\right)$ strength distributions in ${}^{48}\mathrm{Ca}$ deduced from (a) the $(p,p^{\prime })$ reaction (present work) and (b) the $(e,e^{\prime })$ reaction [16]. Arrows indicate upper limits. The prominent transition to the state at $E_{\mathrm{x}}=10.23$ MeV is excluded.

Figure 9

Running sums of the $B\left(M1\right)\uparrow$ strengths in ${}^{48}\mathrm{Ca}$ between 7 and 13 MeV (excluding the prominent transition to the state at $E_{\mathrm{x}}=10.23$ MeV) from the $(p,p^{\prime })$ reaction [present work, red (light grey) histogram] and the $(e,e^{\prime })$ reaction [Ref. [16], blue (dark gray) histogram]. The bands indicate the experimental uncertainties.