Lifetime measurements of $N\simeq 20$ phosphorus isotopes using the AGATA $\gamma$-ray tracking spectrometer

Lifetimes of excited states of the phosphorus isotopes ${}_{15}^{33,34,35,36}\mathrm{P}$ have been measured by using the differential recoil-distance method. The isotopes of phosphorus were populated in binary grazing reactions initiated by a beam of ${}^{36}\mathrm{S}$ ions of energy 225 MeV incident on a thin ${}^{208}\mathrm{Pb}$ target mounted in the Cologne plunger apparatus. The combination of the PRISMA magnetic spectrometer and an early implementation of the AGATA $\gamma$-ray tracking array was used to detect $\gamma$ rays in coincidence with projectile-like nuclear species. Lifetime measurements of populated states were made within the range from about 1 to 100 ps. The number of states for which lifetime measurements were possible was limited by statistics. For ${}^{33}\mathrm{P}$, lifetime limits were determined for the first $3/2^{+}$ and $5/2^{+}$ states at 1431 and 1848 keV, respectively; the results are compared with previous published lifetime values. The lifetime of the first $2^{+}$ state of ${}^{34}\mathrm{P}$ at 429 keV was determined and compared with earlier measurements. For ${}^{35}\mathrm{P}$, the states for which lifetimes, or lifetime limits, were determined were those at 2386, 3860, 4101, and 4493 keV, with $J^{\pi }$ values of $3/2^{+}$, $5/2^{+}$, $7/2_1^{-}$, and $7/2_2^{-}$, respectively. There have been no previous published lifetimes for states in this nucleus. A lifetime was measured for the stretched $\pi \left(1f_{7/2}\right)\otimes \nu \left(1f_{7/2}\right)J^{\pi }=\left(7^{+}\right)$ state of ${}^{36}\mathrm{P}$ at 5212 keV and a lifetime limit was established for the stretched $\pi \left(1{\mathrm{d}}_{3/2}\right)\otimes \nu \left(1f_{7/2}\right)J^{\pi }=\left(5^{-}\right)$ state at 2030 keV. There are no previously published lifetimes for states of ${}^{36}\mathrm{P}$. Measured lifetime values were compared with the results of state-of-the-art shell-model calculations based on the PSDPF effective interaction. In addition, measured branching ratios, published mixing ratios, and electromagnetic transition rates, where available, have been compared with shell-model values. In general, there is good agreement between experiment and the shell model; however there is evidence that the shell-model values of the $M1$ transition rates for the $3/2_1^{+}\to 1/2^{+}$ (ground state) and $5/2_1^{+}\to 3/2_1^{+}$ transitions in ${}^{33}\mathrm{P}$ underestimate the experimental values by a factor between 5 and 10. In ${}^{35}\mathrm{P}$ there are some disagreements between experimental and shell-model values of branching ratios for the first and second excited $7/2^{-}$ states. In particular, there is a serious disagreement for the decay characteristics of the second $7/2^{-}$ state at 4493 keV, for which the shell-model counterpart lies at 4754 keV. In this case, the shell-model competing electromagnetic decay branches are dominated by $E1$ and $M1$ transitions.

Figure 1

The mass spectrum of phosphorus ions detected at the focal plane of the PRISMA spectrometer in coincidence with $\gamma$ rays.

Figure 2

$\gamma$-ray-singles energy spectrum observed in coincidence with ${}^{33}\mathrm{P}$ ions detected at the focal plane of PRISMA. The target-like photopeaks correspond to $\gamma$ rays emitted from ${}^{209\text{–}211}\mathrm{Bi}$. The photopeaks separately labeled with the symbol $\blacksquare$ correspond to $\gamma$ rays from the complementary unobserved fragment, ${}^{209}\mathrm{Bi}$.

Figure 3

Level scheme of ${}^{33}\mathrm{P}$ from the earlier ${}^{36}\mathrm{S}+{}^{208}\mathrm{Pb}$ experiment [17].

Figure 4

Section of the measured $\gamma$-ray spectra for target-degrader distances of 7 and $120\mu \mathrm{m}$ which shows the two components of the 1431-keV photopeak corresponding to the $3/2_1^{+}$ to $1/2^{+}$ (ground-state) transition in ${}^{33}\mathrm{P}$. The black curve corresponds to a fit to the data (Gaussian peaks plus linear background). Background-subtracted photopeaks are shown in green (corresponding to $\gamma$-ray emission between the target and absorber) and in blue (corresponding to $\gamma$-ray emission after the decaying nucleus has passed through the degrader foil).

Figure 5

Section of a ${}^{33}\mathrm{P}\gamma$-ray energy spectrum. The spectrum in blue corresponds to no $Q$ gate while that in red corresponds to a $Q$ gate ($Q>-35$ MeV) which results in the suppression of the 2378-keV $\gamma$-ray photopeak.

Figure 6

Section of the measured $\gamma$-ray spectra for target-degrader distances of 7 and $120\mu \mathrm{m}$ corresponding to the 1848-keV $5/2^{+}$ to $1/2^{+}$ (ground-state) transition of ${}^{33}\mathrm{P}$. A $Q$ gate has been applied to the $\gamma$-ray spectra (see text).

Figure 7

γ-ray singles energy spectrum observed in coincidence with ${}^{34}\mathrm{P}$ ions detected at the focal plane of PRISMA. The photopeaks labeled with $▾$ and $•$ correspond to $\gamma$-ray emission from the associated target-like fragments, ${}^{208}\mathrm{Bi}$ and ${}^{209}\mathrm{Bi}$, respectively.

Figure 8

Level scheme of ${}^{34}\mathrm{P}$ from the earlier ${}^{36}\mathrm{S}+{}^{208}\mathrm{Pb}$ experiment [9].

Figure 9

The decay curve corresponding to the $2^{+}\to 1^{+}$(ground-state) transition in ${}^{34}\mathrm{P}$ with no correction for feeding of the state from higher-lying states. See text for details.

Figure 10

(a) $Q$-value spectrum with the $Q$ gate shown in red, and (b) a section of the measured $\gamma$-ray spectrum, which includes the 1876- and 1891-keV photopeaks, corresponding to no $Q$ gate (spectrum in blue) and to a $Q$ gate (spectrum in red). See text for details.

Figure 11

Section of the measured $\gamma$-ray spectra for target-degrader distances of 7 and $120\mu \mathrm{m}$ corresponding to the 429-keV $2^{+}$ to $1^{+}$ (ground-state) transition in ${}^{34}\mathrm{P}$.

Figure 12

The decay curve corresponding to the $2^{+}\to 1^{+}$(ground-state) transition in ${}^{34}\mathrm{P}$ with the removal of feeding through the application of a $Q$ gate. See text for details.

Figure 13

$\gamma$-ray singles energy spectrum observed in coincidence with ${}^{35}\mathrm{P}$ ions detected at the focal plane of PRISMA. Photopeaks labeled with the symbol $\circ$ correspond to $\gamma$ rays emitted from the associated target-like nucleus, ${}^{209}\mathrm{Bi}$, while those labeled with the symbol $▴$ correspond to $\gamma$-ray emission from ${}^{208}\mathrm{Bi}$.

Figure 14

Level scheme of ${}^{35}\mathrm{P}$ observed in the earlier ${}^{36}\mathrm{S}+{}^{208}\mathrm{Pb}$ experiment [9]. See text for details of the $J^{\pi }$ values of the 4766-, 4962-, and 5089-keV states.

Figure 15

Section of the measured $\gamma$-ray spectra for target-degrader distances of 7, 65, and $120\mu \mathrm{m}$ corresponding to the 2386-keV $3/2_1^{+}$ to $1/2^{+}$ (ground-state) transition of ${}^{35}\mathrm{P}$.

Figure 16

Sections of $\gamma$-ray spectra which show the two components of the 3860-keV $\gamma$-ray photopeak with a $Q$ gate ($Q>-17$ MeV) (spectrum in red) and without a gate (in blue). See text for details.

Figure 17

Section of a $\gamma$-ray spectrum which show the 4101-keV $\gamma$-ray photopeak. The spectrum corresponds to the sum of $\gamma$-ray spectra at all five target-degrader distances. See text for details.

Figure 18

Section of the measured $\gamma$-ray spectra for target-degrader distances of 7, 65, and $120\mu \mathrm{m}$ corresponding to the 4101-keV ($7/2_1^{-}$) $\to 1/2^{+}$ (ground-state) transition of ${}^{35}\mathrm{P}$.

Figure 19

Sections of $\gamma$-ray spectra which show the two components of the 392-keV $\gamma$-ray photopeak and the feeding transitions at 273 and 469 keV with (spectrum in red) a $Q$ gate and (in blue) without a $Q$ gate. See text for details.

Figure 20

Sections of the measured $\gamma$-ray spectra for target-degrader distances of 7 and $120\mu \mathrm{m}$ corresponding to the 392-keV $(7/2_2^{-})$ to $(7/2_1^{-})$ transition of ${}^{35}\mathrm{P}$.

Figure 21

The decay curve corresponding to the $(7/2_2^{-})\to (7/2_1^{-})$ transition in ${}^{35}\mathrm{P}$.

Figure 22

γ-ray singles energy spectrum observed in coincidence with ${}^{36}\mathrm{P}$ ions detected at the focal plane of PRISMA. Photopeaks labeled with the symbol $•$ correspond to $\gamma$-ray emission from ${}^{207}\mathrm{Bi}$, while those labeled with the symbol $\blacksquare$ correspond to emission from ${}^{208}\mathrm{Bi}$.

Figure 23

Level scheme of ${}^{36}\mathrm{P}$ observed in the earlier ${}^{36}\mathrm{S}+{}^{208}\mathrm{Pb}$ experiment [9, 17].

Figure 24

Section of the measured ${}^{36}\mathrm{P}\gamma$-ray spectrum which shows the two components of the 2030-keV photopeak, labeled with the symbol $•$, and corresponding to the ($5^{-}$) to $4^{-}$ transition. Spectra at all measured target-degrader distances have been summed to produce the $\gamma$-ray spectrum.

Figure 25

Section of the measured $\gamma$-ray spectra for target-degrader distances of 7, 65, and $120\mu \mathrm{m}$ corresponding to the 2030-keV ($5^{-}$) to $4^{-}$ transition of ${}^{36}\mathrm{P}$.

Figure 26

The decay curve corresponding to the $\left(5^{-}\right)\to 4^{-}$ transition in ${}^{36}\mathrm{P}$.

Figure 27

Section of the measured ${}^{36}\mathrm{P}\gamma$-ray spectrum which shows the two components of the 3182-keV ($7^{+}$) to ($5_1^{-}$) transition, indicated with the symbol $•$. Spectra at all measured target-degrader distances have been summed to produce the $\gamma$-ray spectrum.

Figure 28

Sections of the measured $\gamma$-ray spectra for target-degrader distances of 7, 65, and $120\mu \mathrm{m}$ corresponding to the 3182-keV ($7^{+}$) to ($5_1^{-}$) transition of ${}^{36}\mathrm{P}$.

Figure 29

The decay curve corresponding to the $\left(7^{+}\right)\to \left(5^{-}\right)$ transition in ${}^{36}\mathrm{P}$.