First-excited state $g$ factors in the stable, even Ge and Se isotopes

Transient-field $g$-factor measurements in inverse kinematics were performed for the first-excited states of the stable, even isotopes of Ge and Se. The $g$ factors of ${}^{74}\mathrm{Ge}$ and ${}^{74}\mathrm{Se}$ were measured simultaneously using a cocktail beam, which eliminates most possible sources of systematic error in a relative $g$-factor measurement. The results are $g\left({}^{74}\mathrm{Se}\right)/g\left({}^{74}\mathrm{Ge}\right)=1.34\left(7\right), g\left({}^{70}\mathrm{Ge}\right)/g\left({}^{74}\mathrm{Ge}\right)=1.16\left(15\right), g\left({}^{72}\mathrm{Ge}\right)/g\left({}^{74}\mathrm{Ge}\right)=0.92\left(13\right), g\left({}^{76}\mathrm{Ge}\right)/g\left({}^{74}\mathrm{Ge}\right)=0.88\left(5\right), g\left({}^{76}\mathrm{Se}\right)/g\left({}^{74}\mathrm{Se}\right)=0.96\left(7\right), g\left({}^{78}\mathrm{Se}\right)/g\left({}^{74}\mathrm{Se}\right)=0.82\left(5\right), g\left({}^{80}\mathrm{Se}\right)/g\left({}^{74}\mathrm{Se}\right)=0.99\left(7\right)$, and $g\left({}^{82}\mathrm{Se}\right)/g\left({}^{74}\mathrm{Se}\right)=1.19\left(6\right)$. The measured $g$-factor ratios are in agreement with ratios from previous measurements, despite considerable variation in previous reported absolute values. The absolute values of the $g$ factors remain uncertain, however, the Rutgers parametrization was used to set the transient-field strength and then compare the experimental $g$ factors with shell-model calculations based on the JUN45 and jj44b interactions. Modest agreement was found between experiment and theory for both interactions. The shell-model calculations indicate that the $g\left(2_1^{+}\right)$ values and trends are determined largely by the balance of the spin carried by orbital motion of the protons.

Figure 1

Experimental geometry for run 1 and run 2 showing the beam axis and the particle detector dimensions (run 1/ run 2). The average scattering angles for the ${}^{12}\mathrm{C}$ target ions are indicated by $\langle {\theta }_p\rangle$.

Figure 2

(a) Experimental geometry for runs $3–5$ showing the beam axis, a $\gamma$-ray detector and the particle detectors. (b) A schematic of the particle detector array indicating dimensions and joined detectors.

Figure 3

Illustration of the particle-$\gamma$ coincidence geometry, drawn from a top-down view. Symmetry is maintained when particle-$\gamma$ coincidence pairs are formed with ${\gamma }_i$ at $+\theta$ paired with the particle detector at ${\varphi }_p$ and ${\gamma }_j$ at $-\theta$ paired with the particle detector at ${180}^{\circ }-{\varphi }_p$, shown by the equivalent dotted vs dashed angles. In short, ${\gamma }_i\left({\varphi }_p\right)$ must be paired with ${\gamma }_j({180}^{\circ }-{\varphi }_p)$. Figure 2 shows the definition of the angle ${\varphi }_p$.

Figure 4

Spectrum of $\gamma$ rays at ${65}^{\circ }$ to the beam in coincidence with ${}^{12}\mathrm{C}$ recoils measured during run 1 with the ${}^{74}\mathrm{Ge}/{}^{74}\mathrm{Se}$ cocktail beam. Random coincidences have been subtracted. The resulting spectrum has almost no background. This spectrum represents $\approx 25\%$ of the data taken for the simultaneous measurement of ${}^{74}\mathrm{Ge}$ and ${}^{74}\mathrm{Se}$ during run 1.

Figure 5

Angular correlations for the ${}^{74}\mathrm{Ge}$ and ${}^{74}\mathrm{Se}{2}_1^{+}\to 0_1^{+}$ transitions from run 1 for the two unique particle-detector angles. (a) ${}^{74}\mathrm{Ge}$, top and bottom particle detectors. (b) ${}^{74}\mathrm{Ge}$, center particle detector. (c) ${}^{74}\mathrm{Se}$, top and bottom particle detectors. (d) ${}^{74}\mathrm{Se}$, center particle detector. Statistical errors are smaller than or similar in size to the data points. Dark red (light blue) points correspond to the $\gamma$-ray detector that moves to positive (negative) angles. The measured data are normalized to the theoretical angular correlations shown as continuous lines. Although there is a distinct difference between the correlations for the top/bottom and center particle detectors near $0^{\circ }$, they are similar near $\pm {65}^{\circ }$, meaning the two sets of particle detectors have similar sensitivity for the $g$-factor measurement. An offset of $+3^{\circ }$ was required on the negative angle data (all taken with the same $\gamma$-ray detector) to optimize the fit.

Figure 6

Angular correlations for the ${}^{76}\mathrm{Ge}{2}_1^{+}\to 0_1^{+}$ transition from run 2 for the two unique particle-detector angles. (a) Top and bottom particle detectors. (b) Center particle detector. See also the caption of Fig. 5.

Figure 7

Angular correlations for the ${}^{74}\mathrm{Ge}{2}_1^{+}\to 0_1^{+}$ transition from run 3, with measurements taken at varying $\gamma$-ray detector angles for the five particle detector angles. (a) Particle detector at ${\varphi }_p=0^{\circ }$. (b) Particle detectors at ${\varphi }_p=\pm {45}^{\circ }$. (c) Particle detectors at ${\varphi }_p=\pm {90}^{\circ }$. (d) Particle detectors at ${\varphi }_p=\pm {135}^{\circ }$. (e) Particle detector at ${\varphi }_p={180}^{\circ }$. The measured data are normalized to the theoretical angular correlations. Dark red (light blue) points correspond to the $\gamma$-ray detector that moves to positive (negative) angles. Note the reflection symmetry between the correlations for ${\varphi }_p=0$ and ${\varphi }_p={180}^{\circ }, {\varphi }_p={45}^{\circ }$ and ${\varphi }_p={135}^{\circ }$.

Figure 8

As for Fig. 7, for the ${}^{74}\mathrm{Se}{2}_1^{+}\to 0_1^{+}$ transition in run 3.

Figure 9

Ratios of $g\left(2_1^{+}\right)$ in the Ge isotopes relative to ${}^{74}\mathrm{Ge}$, from present and previous work: 1984Pa20 [30], 1987La20 [31], and 2013Gu23 [7]. Values from Ref. [7] are taken from their target II measurements only.

Figure 10

Ratios of $g\left(2_1^{+}\right)$ in the Se isotopes relative to ${}^{74}\mathrm{Se}$, from present and previous work: 1998Sp03 [6].

Figure 11

Comparison of the adopted Ge and Se $g$-factor values (Table 4). There is a shift in magnitude between the two isotope chains but the trend with neutron number appears to be similar. The independently measured IPAC result for ${}^{76}\mathrm{Se}$ [44] is shown for reference (slightly displaced from $N=42$ for clarity).

Figure 12

Measured $2_1^{+}$-level $g$ factors for (a) Ge and (b) Se isotopes from the present work (red squares) and the literature (blue circles). Previous work is designated by the Nuclear Science Reference code, namely 2013Gu23 [7] and 1998Sp03 [6]. Shell-model calculations using the JUN45 and jj44b interactions are shown as solid lines and dashed lines, respectively. Experimental $g$ factors from Ref. [7] shown in (a) are their reported weighted-averages that include the previous measurements [30, 31] along with their own data. (See also Table 4.)