First-excited state $g$ factor of ${}^{136}\mathrm{Te}$ by the recoil in vacuum method

The $g$ factor of the first $2^{+}$ state of radioactive ${}^{136}\mathrm{Te}$ with two valence protons and two valence neutrons beyond double-magic ${}^{132}\mathrm{Sn}$ has been measured by the recoil in vacuum (RIV) method. The lifetime of this state is an order of magnitude longer than the lifetimes of excited states recently measured by the RIV method in Sn and Te isotopes, requiring a new evaluation of the free-ion hyperfine interactions and methodology used to determine the $g$ factor. The calibration data are reported and the analysis procedures are described in detail. The resultant $g$ factor has a similar magnitude to the $g$ factors of other nuclei with an equal number of valence protons and neutrons in the major shell. However, an unexpected trend is found in the $g$ factors of the $N=84$ isotones, which decrease from ${}^{136}\mathrm{Te}$ to ${}^{144}\mathrm{Nd}$. Shell model calculations with interactions derived from the CD Bonn potential show good agreement with the $g$ factors and $E2$ transition rates of $2^{+}$ states around ${}^{132}\mathrm{Sn}$, confirming earlier indications that ${}^{132}\mathrm{Sn}$ is a good doubly magic core.

Figure 1

Examples of $\gamma$-ray spectra in coincidence with Ti ions detected in rings 2, 3, and 4 of BareBall from (a) 390 MeV ${}^{125}\mathrm{Te}$ and (b) 400 MeV ${}^{130}\mathrm{Te}$.

Figure 2

Angular correlations of ${}^{122}\mathrm{Te}$ excited on Ti. (a)–(c) correspond to ${\theta }_{\gamma }={90}^{\circ }$, (d)–(f) to ${\theta }_{\gamma }={132}^{\circ }$. The unperturbed correlations are indicated by the dotted curves; solid red lines show the fit to the attenuated angular correlations. See text for details of the normalization procedure.

Figure 3

Angular correlations of ${}^{125}\mathrm{Te}$ excited on Ti with Ti recoils detected in BareBall ring 3. (a)–(d) correspond to ${\theta }_{\gamma }={90}^{\circ }$, (e)–(h) to ${\theta }_{\gamma }={132}^{\circ }$. The unperturbed correlations are indicated by the dotted curves; solid red lines show the fit to the attenuated angular correlations. The transitions are 444 keV $\left(3/2_2^{+}\to 1/2_1^{+}\right)$, 408 keV $\left(3/2_2^{+}\to 3/2_1^{+}\right)$, 463 keV $\left(5/2_1^{+}\to 1/2_1^{+}\right)$, and 428 keV $\left(5/2_1^{+}\to 3/2_1^{+}\right)$. See text for details of the normalization procedure.

Figure 4

Angular correlations of ${}^{136}\mathrm{Te}$ excited on Ti. (a), (b) correspond to ${\theta }_{\gamma }={90}^{\circ }$, (c), (d) to ${\theta }_{\gamma }={132}^{\circ }$, and (e), (f) to ${\theta }_{\gamma }={154}^{\circ }$. To give a visual representation of the overall statistical quality of the angular correlation data, individual data points have been combined and projected onto the interval $0\le \mathrm{\Delta }\varphi \le {180}^{\circ }$. The red solid line indicates the best fit to the data (not combined). The black dotted line indicates the unperturbed angular correlation and the blue dashed line indicates the angular correlation that would be associated with $g=0.1$.

Figure 5

Experimental data and $G_4$ parametrized as a function of $G_2$ for each BareBall ring: (a) ring 2, (b) ring 3, and (c) ring 4. Fits to Eq. (7) are shown as the black solid line. The parametrizations of $G_k$ versus $g^2\tau$ used in the $g$-factor analysis give the red dot-dashed lines, which are almost indistinguishable from the solid black lines. The static model used previously to analyze RIV data on Sn and Te isotopes [14, 15, 16] is shown for reference (blue dashed line), as is the Abragam-Pound formula, Eq. (6), (green dotted line).

Figure 6

(a) Total ${\chi }^2$ versus $g^2\tau$. (b)–(d) $G_k$ versus $g^2\tau$ calibration curves (solid lines) for BareBall rings 2–4. The best fit $g^2\tau$ value for ${}^{136}\mathrm{Te}$, and its uncertainty, is projected onto the curves (red filled). Also shown are the calibration data for the stable Te isotopes that define the $G_k$ curves [13]. Results for ${}^{125}\mathrm{Te}$ are blue filled. Note that there is no $G_4$ term for $I=3/2$ states and that the differences in $G_k$ values for $I=3/2,2,5/2$ are negligible within the experimental uncertainty (Sec. 4a). The dotted lines indicate the calibration curves that result if the ${}^{125}\mathrm{Te}3/2^{+}$ state (extreme right data point) is ignored, which effectively sets the hard core parameter ${\alpha }_k=0$.

Figure 7

Toy model Monte Carlo calculations of (a) $G_4(I=5/2)/G_4(I=2)$, (b) $G_2(I=5/2)/G_2(I=2)$, and (c) $G_2(I=3/2)/G_2(I=2)$. The nuclear $g$ factor was set to $g=0.35$ and the nuclear lifetime varied. The thick black line represents the model parameters that best fit the RIV data for Te ions in Ref. [13]. The other lines examine sensitivity to the average atomic spin and the rate and duration of atomic transitions.

Figure 8

Atomic spin $\left(J\right)$ dependence of the hard-core vacuum attenuation coefficients for nuclear spins $I=3/2$, 2, $5/2$.

Figure 9

$G_k$ versus $g\tau$ calibration curves for BareBall ring 3 ignoring data for the $5/2^{+}$ state in ${}^{125}\mathrm{Te}$. The best fit $g\tau$ value for ${}^{136}\mathrm{Te}$, and its uncertainty, is projected onto the curves (red filled). See also Fig. 6.

Figure 10

Dependence of the parameters (a) ${\alpha }_k$ and (b) $C_k$, Eqs. (10) and (11), on ion velocity. Lines indicate linear fits.

Figure 11

First-excited $2^{+}$ state $g$ factors as a function of the valence proton fraction, $N_p/N_t$, for nuclei with $50<Z<82$ and $82\le N<126$. The broken line indicates the trend for $N_p/N_t<0.7$. Data points (from Refs. [14, 41] and present work) are colored (shaded) to indicate $R_{42}=E_x\left(4_1^{+}\right)/E_x\left(2_1^{+}\right)$. Note that only the $N=82$ and $N=84$ isotones have $R_{42}<2$.

Figure 12

Reduced transition rates and $g$ factors of Te and Xe isotopes spanning $N=82$. The dotted lines in (a) and (b) correspond to an increased neutron effective charge of $e_n=0.8$ for $N<82$. Experimental data from Refs. [1, 7, 12, 14, 18, 27, 48,55, 56, 57].