Doppler-shift attenuation lifetime measurement of the ${}^{36}\mathbf{Ar}{2}_1^{+}$ level

At TRIUMF, the TIGRESS Integrated Plunger device and its suite of ancillary detector systems have been implemented for charged-particle tagging and light-ion identification in coincidence with $\gamma$-ray spectroscopy for Doppler-shift lifetime studies and low-energy Coulomb excitation measurements. As a test of the device, the lifetime of the first $2^{+}$ excited state in ${}^{36}\mathrm{Ar}$ was measured from the $\gamma$-ray line shape of the $2_1^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+}$ transition using the Doppler-shift attenuation technique following Coulomb excitation. The line-shape signatures, vital for precision lifetime measurements, were significantly improved by enhanced reaction-channel selectivity using a complementary approach of kinematic gating and digital rise-time discrimination of recoiling charged particles in a silicon PIN diode array. The lifetime was determined by comparisons between the data and simulated line shapes generated using our TIGRESS Coulomb excitation code as an input to the Lindhard method, which was then extended and included as a class in geant4. The model-independent lifetime result of $490\pm 50$ fs corresponds to a reduced quadrupole transition strength of $B(E2;2_1^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+})=56\pm 6{\mathrm{e}}^2{\mathrm{fm}}^4$ and agrees well with previous intermediate energy Coulomb excitation measurements, thereby resolving reported discrepancies in the $2_1^{+}$ level lifetime in this self-conjugate nucleus.

Figure 1

$\gamma$-ray energy spectra detected with the ${45}^{\circ }$ TIGRESS clovers in coincidence with all recoiling charged particles in the TIP silicon PIN diode array in panel (a). The ${}^{36}\mathrm{Ar}{2}_1^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+}$ transition at 1970.3 keV exhibits well-developed slowing and stopped components. The signal to noise was improved with kinematic suppression as demonstrated in panel (b)—like panel (a), but only $\gamma$ rays in coincidence with recoiling charged particles in rings 1 and 2 of the silicon array—and silicon rise-time discrimination as demonstrated in panel (c)—like panel (b), but with the additional requirement that particles have carbon-like rise times in the silicon array (cf. Fig. 2).

Figure 2

Two representative $1\text{−}\mu \mathrm{s}$ wave-form traces (black) from the TIP silicon PIN diode array, normalized with baselines aligned. The rise times of the traces corresponding to carbon and $\alpha$ particles are 98 and 212 ns, respectively, and are extracted from event-by-event fits; the best-fit function (red) to the carbon wave form is overlaid atop the data. The resulting particle identification from silicon DRD is demonstrated in the inset, where the incident ion energy is plotted against the extracted rise time for recoils detected in ring 2 of the silicon array. The three distinct regions $A$, $B$, and $C$ correspond to the detection of carbon, one $\alpha$ particle, and two $\alpha$ particles in a single photodiode, respectively. Only $\gamma$ rays in coincidence with particles in region $A$ from the silicon array rings 1 and 2 were used in the line-shape analysis to extract the ${}^{36}\mathrm{Ar}{2}_1^{+}$ lifetime.

Figure 3

Distributions of reduced-${\chi }^2$ values as a goodness-of-fit test between simulations and experimental data for backward TIGRESS rings of ${145}^{\circ }$ (blue squares) and ${125}^{\circ }$ (red triangles) in panel (a) and forward TIGRESS rings of ${55}^{\circ }$ (blue squares) and ${35}^{\circ }$ (red triangles) in panel (b).

Figure 4

$\mathcal{R}={\mathcal{R}}_{\mathrm{stopped}}/{\mathcal{R}}_{\mathrm{shifted}}$ as a function of simulated lifetime for backward TIGRESS rings of ${145}^{\circ }$ (blue squares) and ${125}^{\circ }$ (red triangles) in panel (a) and forward TIGRESS rings of ${55}^{\circ }$ (blue squares) and ${35}^{\circ }$ (red triangles) in panel (b).

Figure 5

Experimental (blue) and method I best-fit simulated (red) $\gamma$-ray line shapes at four TIGRESS rings: ${145}^{\circ }$ in panel (a), ${125}^{\circ }$ in panel (b), ${55}^{\circ }$ in panel (c), and ${35}^{\circ }$ in panel (d). The simulations have an enforced coincident detection of $\gamma$ rays and carbon recoils detected in rings 1 and 2 of the TIP silicon PIN diode array. The dashed vertical lines in panels (b) and (d) illustrate the normalization intervals for the simulated line shapes while those in panels (a) and (c) indicate the stopped and shifted intervals for the investigation of $\mathcal{R}$.

Figure 6

The normalized experimental $\gamma$-ray yields as a function of the TIP silicon PIN diode array ring laboratory polar angle (orange circles) were obtained by summing all TIGRESS clovers in coincidence with carbon recoils in each ring. Simulated TIGRESS yields are plotted in comparison for gold backing thicknesses of infinitely thin (dot-dot-dashed black), 4.5 $\mu \mathrm{m}$ (dashed blue), 9.5 $\mu \mathrm{m}$ (dot-dashed orange), 11.0 $\mu \mathrm{m}$ (solid black), and 13 $\mu \mathrm{m}$ (solid red). The shapes of the simulated yields best agree with the data for gold backing thicknesses between 10 and 12 $\mu \mathrm{m}$.

Figure 7

Experimental (blue) and method II simulated (red) carbon recoil energy spectra from ring 1 of the TIP silicon PIN diode array. An $11.5-\mu \mathrm{m}$ gold backing thickness was used in the simulations, which also employed a step-function low-energy threshold cutoff as an approximation of the steep experimental energy threshold. For comparison, the inset shows the same simulations with a $9.5-\mu \mathrm{m}$ gold backing thickness. Notice the significant peak shift to higher energies, away from the experimental result.

Figure 8

Distributions of reduced-${\chi }^2$ values as a goodness-of-fit test between simulations and experimental data for the forward ${45}^{\circ }$ (red squares) and backward ${135}^{\circ }$ (blue squares) TIGRESS clovers. The distributions have been fit with cubic polynomials to extract the lifetime values.

Figure 9

Experimental (blue) and method II best-fit simulated (red) $\gamma$-ray line shapes for the $2_1^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+}$ transition in ${}^{36}\mathrm{Ar}$ for TIGRESS clover detectors at ${135}^{\circ }$ in panel (a), ${90}^{\circ }$ in panel (b), and ${45}^{\circ }$ in panels (c) and (d). The simulations in panels (a)–(c) have an enforced coincident detection of $\gamma$ rays and carbon recoils detected in rings 1 and 2 of the TIP silicon PIN diode array. For comparison, no such kinematic gate was applied for the simulated line shape in panel (d). Note the contaminant peak near 1915 keV that is not simulated; this is also visible in Fig. 1. Panels (e) and (f) are like panel (c), but the comparison between experimental and simulated TIGRESS clover data is split into the two forward TIGRESS rings at ${55}^{\circ }$ and ${35}^{\circ }$, respectively.

Figure 10

$\mathcal{R}={\mathcal{R}}_{\mathrm{stopped}}/{\mathcal{R}}_{\mathrm{shifted}}$ as a function of simulated lifetime for the ${45}^{\circ }$ TIGRESS clovers in panel (a) and the ${135}^{\circ }$ TIGRESS clovers in panel (b). The blue squares and red triangles represent $\mathcal{R}$ values calculated using different stopped peak and shifted plateau energy intervals.

Figure 11

The $B\left(E2\right)$ derived from the lifetime result reported in this work and the associated shell-model calculation (filled black squares) are compared to the adopted model-independent literature value (open black square) of Ref. [54] (“Pritychenko 2016”), the two intermediate energy Coulomb excitation results (filled blue squares) of Ref. [25] (“Pritychenko 1999”) and [26] (“Cottle 1999”), and the DSAM result and associated shell-model calculation (filled red squares) of Ref. [27] (“Speidel 2006”).

Figure 12

$\mathcal{R}={\mathcal{R}}_{\mathrm{stopped}}/{\mathcal{R}}_{\mathrm{shifted}}$ as a function of simulated lifetime for the ${45}^{\circ }$ (red squares) and ${135}^{\circ }$ (blue squares) TIGRESS clovers. The data are the same as the blue data of Fig. 10, but the error bars are enlarged due to 20- and 100-fold beam-intensity reductions in panels (a) and (b), respectively. The spread in $\mathcal{R}=1$ intersections of the best-fit lines for the upper and lower $1\sigma$ limits yields a measure of the loss of precision with decreasing experimental statistics.