Reorientation-effect measurement of the $\langle 2_1^{+}\parallel \widehat{E2}\parallel 2_1^{+}\rangle$ matrix element in ${}^{10}$Be

The highly-efficient and segmented TIGRESS $\gamma$-ray spectrometer at TRIUMF has been used to perform a reorientation-effect Coulomb-excitation study of the 2${}_1^{+}$ state at 3.368 MeV in ${}^{10}$Be. This is the first Coulomb-excitation measurement that enables one to obtain information on diagonal matrix elements for such a high-lying first excited state from $\gamma$-ray data. With the availability of accurate lifetime data, a value of $-0.110\pm 0.087$ $e$b is determined for the $\langle 2_1^{+}\parallel \widehat{E2}\parallel 2_1^{+}\rangle$ diagonal matrix element, which assuming the rotor model, leads to a negative spectroscopic quadrupole moment of $Q_{{}_S}\left(2_1^{+}\right)=-0.083\pm 0.066$ $e$b. This result is in agreement with both no-core shell-model calculations performed in this work with the CD-Bonn 2000 two-nucleon potential and large shell-model spaces, and Green's function Monte Carlo predictions with two- plus three-nucleon potentials.

Figure 1

Particle-energy spectrum for rings at average angles of $\theta =$ 35.6${}^{\circ }$ (top panel) and 60.0${}^{\circ }$ (bottom panel). The application of $|E_{\mathrm{ring}}-E_{\mathrm{sector}}|\le$ 350 keV permits a good identification of the 2${}_1^{+}$ inelastic peak in ${}^{10}$Be. The larger background at $\theta =35.6$ arises from the higher density of dead layers for innermost rings in the silicon detector.

Figure 2

Gamma-ray energy spectra for the ${}^{194}$Pt(${}^{10}$Be,${}^{10}$Be${}^{*}$)${}^{194}$Pt${}^{*}$ reaction at 41 MeV. The main panel shows Doppler-corrected energy spectra with (black) and without (brown) a ${}^{10}$Be inelastic-particle-coincidence condition. The former results in a cleaner background, yet conserves the number of counts for the 3.368 MeV transition depopulating the 2${}_1^{+}$ state in ${}^{10}$Be. The inset shows a Doppler-uncorrected $\gamma$-ray spectrum with the 2${}_1^{+}$ peak at 328 keV in ${}^{194}$Pt.

Figure 3

Angular distributions showing experimental and calculated $\gamma$-ray yields as a function of particle angle (in the laboratory frame) for the de-excitation of the 2${}_1^{+}$ states in ${}^{10}$Be (top) and ${}^{194}$Pt (bottom).

Figure 4

The parabolic lines indicate the central value (dashed) and 1-$\sigma$ limits (solid) of $\langle 2_1^{+}\parallel \widehat{E2}\parallel 0_1^{+}\rangle$ versus $\langle 2_1^{+}\parallel \widehat{E2}\parallel 2_1^{+}\rangle$ matrix elements in ${}^{10}$Be for $k\left(2_1^{+}\right)=0.81$. The horizontal band represents the 1-$\sigma$ boundary for $\langle 2_1^{+}\parallel \widehat{E2}\parallel 0_1^{+}\rangle =0.0690\left(15\right)$ [15, 42, 43]. The extrapolated NCSM results are given by the square data point; see the text for details.

Figure 5

Calculated transitional (top panel) and diagonal (bottom panel) matrix elements involving the first two $J^{\pi }=2^{+}$ states using NCSM with the CD-Bonn 2000 $2N$ potential at different $\hslash \Omega$ and $N_{\max }$.