Probing isospin symmetry in the $({}^{50}\mathrm{Fe}, {}^{50}\mathrm{Mn}, {}^{50}\mathrm{Cr})$ isobaric triplet via electromagnetic transition rates

Lifetimes of the $T=1$ isobaric analog $I^{\pi }=2^{+}$ states in ${}^{50}\mathrm{Mn}$ and ${}^{50}\mathrm{Cr}$ were measured simultaneously by employing the recoil distance Doppler-shift (RDDS) technique. The states were populated in a fusion-evaporation reaction with a ${}^{12}\mathrm{C}$ beam on a ${}^{40}\mathrm{Ca}$ target. An analysis of the data and the calculations from the present work, together with the available $B(E2:2^{+}\to 0^{+})$ data of the isobaric analog states in ${}^{50}\mathrm{Fe}$ and ${}^{50}\mathrm{Cr}$, were used to study isospin symmetry in the three $A=50$ isobaric nuclei. Shell-model calculations reproduce the magnitudes as well as the increasing trend of the $B\left(E2\right)$ data with increasing $Z$. To draw a firm conclusion on the level of isospin mixing in the triplet, a new precision measurement of the $2^{+}\to 0^{+}$ transition rate in ${}^{50}\mathrm{Fe}$ will be required.

Figure 1

A schematic representation of the $T=1$, $2^{+}\to 0^{+}\gamma$-ray transition energies (in keV) in the $A=50$ triplet. In ${}^{50}\mathrm{Mn}$, the known second $2^{+}$ state at an excitation energy of 1873 keV is also shown with a tentative assignment of $T=0$ along with the observed 1073-keV $\gamma$ decay to the first $T=1$, $2^{+}$ state [6].

Figure 2

$\gamma$-ray energy spectra of Ring(1), showing the total statistics collected for all of the distances given in Table 1. The histograms were obtained by utilizing the Ring(1) versus Ring(2) $\gamma \gamma$ energy matrix and (a) taking the total projection, (b) gating with the 343-keV line corresponding to the $3^{+}\to 2^{+}\gamma$ decay in ${}^{50}\mathrm{Mn}$, and (c) gating with the 1098-keV line corresponding to the $4^{+}\to 2^{+}\gamma$ decay in ${}^{50}\mathrm{Cr}$. The 610-keV transition is known to be much weaker compared to the 662-keV transition, therefore, was not prominent in the spectra [18].

Figure 3

Partial level schemes of (a) ${}^{50}\mathrm{Mn}$ and (b) ${}^{50}\mathrm{Cr}$ obtained from the present data. Only decays from low-lying states with spins $\le 5$ can be seen in ${}^{50}\mathrm{Mn}$ due to much lower cross sections compared to ${}^{50}\mathrm{Cr}$.

Figure 4

$\gamma$-ray energy spectra of Ring(1) showing the fully shifted and the unshifted peaks corresponding to the 783-keV $\gamma$-ray transition in ${}^{50}\mathrm{Cr}$. The histograms were obtained by utilizing the Ring(1) versus Ring(2) $\gamma \gamma$ energy matrix and gating with the fully-shifted peak of the $4^{+}\to 2^{+}$ 1098-keV transition on the Ring(2) axis. Each of the spectra [from (a) to (j)] is labeled with the effective plunger distance $x_{\mathrm{eff}}^{\mathrm{Cr}}$. The dotted lines corresponds to the centroids of fully shifted and unshifted peaks of the 783-keV $\gamma$-ray at $x_{\mathrm{eff}}^{\mathrm{Cr}}=20.8\mu \mathrm{m}$.

Figure 5

An analysis of the intensities of the 783-keV $\gamma$-ray given in Table 1 to obtain the lifetime of the $2^{+}$ state in ${}^{50}\mathrm{Cr}$. (a) The individual lifetimes obtained at eight distances in the “region of sensitivity” are shown. The final result from a weighted average is also given. The intensities, $I^s$ and $I^{\mathrm{us}}$, are shown as a function of distance in (b) and (c), respectively, along with the best fits (solid lines) obtained using napatau [29].

Figure 6

Same as Fig. 5 for the $4^{+}$ state in ${}^{50}\mathrm{Cr}$.

Figure 7

Similar to Fig. 4, $\gamma$-ray energy spectra gated by the fully shifted peak of the $3^{+}\to 2^{+}$ 343-keV transition in ${}^{50}\mathrm{Mn}$, showing the fully shifted and unshifted peaks of the 800-keV $\gamma$-ray.

Figure 8

Same as Fig. 5 for the $2^{+}$ state in ${}^{50}\mathrm{Mn}$.

Figure 9

The matrix element $M\left(E2\right)$ as a function of $T_z$ for the three $T=1,A=50$ isobaric nuclei. Datum for ${}^{50}\mathrm{Mn}$ was obtained for the first time in the present work. In the case of ${}^{50}\mathrm{Cr}$, the data point is a weighted average of our result and that from Refs. [16, 17, 19] while the result from Ref. [22] has been used for the case of ${}^{50}\mathrm{Fe}$ with $T_z=-1$. The predictions from our calculations are also shown (see the text for details).