Spectroscopy of proton-unbound nuclei by tracking their decay products in-flight: One- and two- proton decays of ${}^{15}\mathrm{F}$, ${}^{16}\mathrm{Ne}$, and ${}^{19}\mathrm{Na}$

A powerful method of investigating proton-unbound nuclear states by tracking their decay products in flight is discussed in detail. To verify the method, four known levels in ${}^{15}\mathrm{F}$, ${}^{16}\mathrm{Ne}$, and ${}^{19}\mathrm{Na}$ were investigated by measuring the angular correlations between protons and the respective heavy-ion fragments stemming from the precursor decays in flight. The parent nuclei of interest were produced in nuclear reactions of one-neutron removal from ${}^{17}\mathrm{Ne}$ and ${}^{20}\mathrm{Mg}$ projectiles at energies of 410–450 $A$ MeV. The trajectories of the respective decay products, ${}^{14}\mathrm{O}$ $+$ p $+$ p and ${}^{18}\mathrm{Ne}$ $+$ p $+$ p, were measured by applying a tracking technique with microstrip detectors. These data were used to reconstruct the angular correlations of the fragments, which provided information on energies and widths of the parent states. In addition for reproducing properties of known states, evidence for hitherto unknown excited states in ${}^{15}\mathrm{F}$ and ${}^{16}\mathrm{Ne}$ was found. This tracking technique has an advantage in studies of exotic nuclei beyond the proton drip line measuring the resonance energies and widths with a high precision although by using low-intensity beams and very thick targets.

Figure 1

Upper panel: Sketch of the experimental setup located at the FRS midplane. The secondary beams were tracked by a DSSD and a multiwire chamber (MW). The reaction products were formed in fragmentation reactions in the reaction target located at the FRS focus F2. Outgoing protons and heavy ions were tracked in triple coincidences HI $+$ p $+$ p by three planes of silicon micro-strip detectors (D1–D3). The heavy ions were identified at the foci F3 and F4 by magnetic-rigidity, time-of-flight (TOF), and energy loss measurements with the scintillator detector Sc4 at F4. Lower panel: Illustration of tracking 2p decays of ${}^{19}\mathrm{Mg}$ with the three planes of microstrip detectors.

Figure 2

Transverse spatial profile of ions versus their energy loss of a secondary beam detected in front of the FRS reaction target by the DSSD detector. The ion-optical settings of the first half of FRS are optimized for a 410 $A$ MeV beam of ${}^{20}\mathrm{Mg}$.

Figure 3

Identification plot for ions detected at the FRS focal planes F2 and F4. The ion-optical setting was optimized for the transmission of ${}^{17}\mathrm{Ne}$ ions produced in the reaction target by a 410 $A$ MeV beam of ${}^{20}\mathrm{Mg}$. The $Y$ axis represents the element number, $Z$, the $X$ axis displays the mass-to-charge ratio, AoQ.

Figure 4

Typical numbers of strips fired in microstrip detectors by 400 $A$ MeV protons (left) and ${}^{17}\mathrm{Ne}$ ions (right).

Figure 5

Typical energy-loss spectrum of different ions derived from the hit-cluster areas on the $X$ side of the microstrip detector [23]. The inset shows the proton energy-loss spectrum derived from coincident triple ${}^{17}\mathrm{Ne}$ $+$ p $+$ p events.

Figure 6

Distribution of uncorrected cluster areas from the $Y$ side of the microstrip detector derived from triple ${}^{15}\mathrm{O}$ $+$ p $+$ p coincidences.

Figure 7

Uncertainties of positions of ${}^{20}\mathrm{Mg}$ ions at the reaction target in the ($X,Y$)-transverse directions. The uncertainties were evaluated from the differences between two alternative ($X,Y$)-coordinate values of the ${}^{20}\mathrm{Mg}$ positions at the target derived from two alternative ${}^{20}\mathrm{Mg}$ trajectories defined by the MSD pairs D1–D3 and D2–D3, see Fig. 1.

Figure 8

Distances of the closest approach $d$ of two trajectories of a proton and an ${}^{17}\mathrm{Ne}$ ion derived from tracked ${}^{17}\mathrm{Ne}$ $+$ p $+$ p events.

Figure 9

Spatial differences between two decay vertices along the $Z$ (beam) direction defined for the $\mathrm{p}{}_1$-${}^{17}\mathrm{Ne}$ and $\mathrm{p}{}_2$-${}^{17}\mathrm{Ne}$ trajectories which were taken from the same ${}^{17}\mathrm{Ne}$ $+$ p${}_1$ $+$ p${}_2$ event. (a) Distribution for events with large p-${}^{17}\mathrm{Ne}$ angles ($>$50 mrad). (b) Distribution for events with small p-${}^{17}\mathrm{Ne}$ angles ($<$50 mrad).

Figure 10

Cartoons of transverse momentum correlations $k_{p1-\mathrm{HI}}-k_{p2-\mathrm{HI}}$ (lower panels) expected for three different mechanisms of 2p emission (illustrated in the upper panels) from a parent nucleus with mass number $A$ to a daughter nucleus with A-2: (a) direct three-body decay, (b) sequential emission of protons via a narrow intermediate state in nucleus A-1, and (c) deexcitation of broad continuum states in the nucleus $A$ with final-state interaction due to a narrow resonance in the intermediate nucleus with A-1.

Figure 11

(a) Kinematics of isotropic mono-energetic proton emission from a high-energy heavy ion. (b) The corresponding angular p–HI distribution exhibiting the peak corresponding to the $Q$-value of the proton decay.

Figure 12

The level schemes of known states in ${}^{14}\mathrm{O}$, ${}^{15}\mathrm{F}$, and ${}^{16}\mathrm{Ne}$ (Refs. [12, 13, 14, 31, 32]). 1p decays of states in ${}^{16}\mathrm{Ne}$ and ${}^{15}\mathrm{F}$ into the g.s. of ${}^{14}\mathrm{O}$ are indicated by the dash-dotted arrows. The 2p decay of the ${}^{16}\mathrm{Ne}$ g.s. is illustrated by the dashed arrow. The hatched area indicates the unspecified continuum states in ${}^{16}\mathrm{Ne}$. Decay energies and level widths are given in MeV; the decay energies are relative to the respective 1p and 2p thresholds.

Figure 13

(a) Simulated angular correlations ${\theta }_{p1-{}^{14}\mathrm{O}}-{\theta }_{p2-{}^{14}\mathrm{O}}$ from Monte-Carlo simulations of the experimental response to the direct 2p decay of the ${}^{16}\mathrm{Ne}$ g.s. with a 2p-decay energy of 1.4 MeV. (b) Similar simulation for the 2p decay of an assumed excited state in ${}^{16}\mathrm{Ne}$ with a 2p-decay energy of 7.6 MeV decaying sequentially by 2p emission through the g.s. of ${}^{15}\mathrm{F}$. (c) Experimental angular correlations (p${}_1$–${}^{14}\mathrm{O}$)–(p${}_2$–${}^{14}\mathrm{O}$) obtained from measured ${}^{14}\mathrm{O}$ $+$ p $+$ p events (color boxes with scale shown on the right-hand side). The shadowed arc areas indicate the locations of simultaneous or sequential 2p decays of the most intensively populated states in ${}^{16}\mathrm{Ne}$ (the ground and the 7.6-MeV excited state).

Figure 14

(a) Opening angles ${\theta }_{p-{}^{}O}$ (full circles with statistical uncertainties) projected from the measured ${}^{14}\mathrm{O}$ $+$ p $+$ p events shown in Fig. 13c. The upper axis shows the transverse momenta $k$ of protons with respect to ${}^{14}\mathrm{O}$. The apparent peaks are labeled (1–4). (b) Angular $\rho =\sqrt{{\theta }_{p1-\mathrm{O}}^2+{\theta }_{p2-\mathrm{O}}^2}$ distribution (full circles with statistical uncertainties) obtained from the data shown in Fig. 13a. The dashed and dash-dotted curves show the results of Monte Carlo simulations of the experimental response to 2p decays of the known ground 0${}^{+}$ and first-excited 2${}^{+}$ states in ${}^{16}\mathrm{Ne}$ with $Q_{2p}$ values of 1.35 and 3.2 MeV, respectively. For illustration purpose, the dash-dot-dotted curve shows a hypothetical contribution from the excited second 0${}^{+}$ state at 3.5 MeV. The dotted curve is the contribution of a previously unknown excited state with $Q_{2p}~7.6$ MeV (the properties of the excited states will be considered in Sec. 5). The solid curve results from the sum fit.

Figure 15

(a) Angular p-${}^{14}\mathrm{O}$ correlations (full circles with statistical uncertainties) obtained from the measured p $+$ p $+$ ${}^{14}\mathrm{O}$ events by selecting the other proton angle ${\theta }_{p2-\mathrm{O}}$ within the range from 0 to 45 mrad, which corresponds to the ${}^{16}\mathrm{Ne}$ g.s. region. The solid curve represents the Monte Carlo simulation of the detector response for ${}^{16}\mathrm{Ne}$${}_{\mathrm{g}.\mathrm{s}.}\to {}^{14}$O $+$ p $+$ p with the fitted 2p-decay energy of 1.35(8) MeV. The dashed line is a sum fit to the data. The dotted curve indicates the assumed background contribution. (b) Calculated probability that the simulation matches the data as a function of the assumed decay energy (see text). The curve represents a Gaussian function fitted to the histogram.

Figure 16

Angular p-${}^{14}\mathrm{O}$ correlations (full circles with statistical uncertainties) from the ${}^{16}\mathrm{Ne}$ g.s. obtained from the measured p $+$ p $+$ ${}^{14}\mathrm{O}$ events by selecting their arc-area $\rho =\sqrt{{\theta }_{p1-\mathrm{O}}^2+{\theta }_{p2-\mathrm{O}}^2}$ within the range from 35 to 50 mrad [see Fig. 14b]. The solid curve is the corresponding calculation of the decay ${}^{16}\mathrm{Ne}$${}_{\mathrm{g}.\mathrm{s}.}\to {}^{14}$O $+$ p $+$ p with a 2p-decay energy of 1.35(8) MeV cut by the same condition as the data. The similar but un-truncated calculation is presented by the dashed curve whose shape practically coincides with the calculation in Fig. 15a.

Figure 17

Angular p-${}^{14}\mathrm{O}$ distribution obtained from the data shown in Fig. 13c by selecting the other proton angle from 120 to 150 mrad, which corresponds to p-${}^{14}\mathrm{O}$ final-state interactions due to the ground and excited states of ${}^{15}\mathrm{F}$. The dashed and dotted curves result from Monte Carlo simulations of the known 1p decays of the ground and first-excited states of ${}^{15}\mathrm{F}$ with 1p-decay energies of 1.56(13) and 2.80(4) MeV, respectively [12]. The dash-dotted and dash-dot-dotted curves show the fit of two other peaks by suggesting two unknown excited states in ${}^{15}\mathrm{F}$ with 1p-decay energies of 4.9(2) and 6.4(2) MeV, respectively. Their derived properties are given in Sec. 5. The solid line is the sum fit.

Figure 18

(a) Angular (p${}_1$-${}^{18}\mathrm{Ne}$)–(p${}_2$-${}^{18}\mathrm{Ne}$) correlations obtained from the measured ${}^{18}\mathrm{Ne}$ $+$ p $+$ p events (color boxes with scale shown on the right-hand side). The grey slices indicate the areas where final-state interaction in the p-${}^{18}\mathrm{Ne}$ pairs due to the ${}^{18}\mathrm{Na}$ g.s. is expected. (b) Angular correlations ${\theta }_{p1-{}^{18}\mathrm{Ne}}-{\theta }_{p2-{}^{18}\mathrm{Ne}}$ obtained by Monte Carlo simulations of the experimental response for 2p emission from broad continuum states in ${}^{20}\mathrm{Mg}$ through the g.s. of ${}^{19}\mathrm{Na}$ with a known 1p-decay energy of 0.32(2) MeV [36].

Figure 19

(a) Angular p-${}^{18}\mathrm{Ne}$ correlations (full circles with statistical uncertainties) projected from Fig. 18a. The peak reflects the final-state interaction due to the ${}^{19}\mathrm{Na}$ g.s. The dashed curve represents the Monte Carlo simulation of the detector response for ${}^{19}\mathrm{Na}$${}_{\mathrm{g}.\mathrm{s}.}\to {}^{18}$Ne $+$ p with the fitted 1p-decay energy of 0.328(22) MeV. The assumed background (dotted line) is interpolated from the data outside the ${}^{19}\mathrm{Na}$ peak. The solid line is a sum fit to the data. The lower panels show the probability that the simulations match the data as functions of (b) the assumed decay energy and (c) width of the ${}^{19}\mathrm{Na}$ g.s. The probability histograms are fitted by Gaussian and sigmoidal functions (solid lines).

Figure 20

(a) The ${\theta }_{p-{}^{}\mathrm{O}}$ correlations shown in Fig. 14b in comparison with the simulations of different 1p decays of ${}^{15}\mathrm{F}$ states. The simulation curves are similar to those described in (b). (b) One-dimensional ${\theta }_{p-{}^{}\mathrm{O}}$ projection (full circles with statistical uncertainties) of the two-dimensional data shown in Fig. 13c, obtained by gating on ${\theta }_{p_2-\mathrm{O}}>120$ mrad, which corresponds to ${}^{15}\mathrm{F}$ resonances due to final-state interactions in $p_1$-${}^{14}\mathrm{O}$ pairs. The dashed and dotted curves result from simulations of the response of our setup to the known 1p decays of the ground and first-excited states in ${}^{15}\mathrm{F}$, see Sec. 4b. The dash-dotted and dash-dot-dotted curves indicate two new states in ${}^{15}\mathrm{F}$ with fitted $Q_p$ values of 4.9(2) and 6.4(2) MeV, respectively. The solid line is the sum fit. The short-dash curved shows the sum fit with all level widths set to 1 keV, which illustrates the experimental resolution. The short-dash-dotted curve is the 1p-decay estimate of the 7.8 MeV state in ${}^{15}\mathrm{F}$. (c) The ${\theta }_{p-\mathrm{O}}$ distribution selected within the arc-area $\sqrt{{\theta }_{p1-\mathrm{O}}^2+{\theta }_{p2-\mathrm{O}}^2}$ around 115 mrad, which corresponds to the 7.6 MeV state in ${}^{16}\mathrm{Ne}$. The solid curve is a fit obtained by simulating the sequential 2p decay of the ${}^{16}\mathrm{Ne}$${}^{*}$ state via the g.s. (dashed curve) and the first-excited state (dash-dotted curve) in ${}^{15}\mathrm{F}$. The dotted curve shows a similar fit with the ${}^{16}\mathrm{Ne}$${}^{*}$ width set to 1 keV. The insets in (b) and (c) show the probability (as a function of the assumed resonance width) that the simulations match the data.

Figure 21

(a) Angular correlations ${\theta }_{p1-{}^{}\mathrm{N}}$-${\theta }_{p2-{}^{}\mathrm{N}}$ from the ${}^{13}\mathrm{N}$ $+$ p $+$ p events (color boxes with scale shown on right). The arc area indicates 2p emission from an unknown ${}^{15}\mathrm{F}$${}^{*}$ resonance. The bands show the p $+$ ${}^{13}\mathrm{N}$ final-state interaction due to the 3${}^{-}$ state in ${}^{14}\mathrm{O}$. (b) Inclusive projection ${\theta }_{p-{}^{}\mathrm{N}}$ of the data shown in panel (a). (c) Distribution ${\theta }_{p-{}^{}\mathrm{N}}$ (full circles with statistical errors) gated by 78 $\mathrm{mrad}⩽\sqrt{{\theta }_{p1-\mathrm{N}}^2+{\theta }_{p2-\mathrm{N}}^2}⩽88$ mrad, which corresponds to the 2p decay of a resonance in ${}^{15}\mathrm{F}$${}^{*}$. The solid curve is the simulation of the sequential 2p decay of ${}^{15}\mathrm{F}$${}^{*}$ via the 2${}^{+}$ state in ${}^{14}\mathrm{O}$ at $E^{*}=6.59$ MeV [39]. The fitted parameters of the ${}^{15}\mathrm{F}$${}^{*}$ state are $Q_p=7.8(2)$ MeV and $\Gamma =0.4(4)$ MeV. The dotted and dash-dotted curves show similar calculations with assumed ${}^{15}\mathrm{F}$${}^{*}$ widths of 1 and 800 keV, respectively. The inset shows the probability (as a function of the assumed resonance width) that the simulations match the data. (d) The ${\theta }_{p1-{}^{}\mathrm{N}}$ histogram obtained for ${\theta }_{p2-{}^{}\mathrm{N}}>$80 mrad, which corresponds to p $+$ ${}^{13}\mathrm{N}$ final-state interactions. The solid, dotted, dashed, dash-dotted and dash-dot-dotted curves are simulations of the 1p resonances in ${}^{14}\mathrm{O}$${}^{*}$ at $E^{*}$ of 5.173, 5.92, 6.272, 6.59, and 7.768 MeV, respectively [39].

Figure 22

Angular p-${}^{14}\mathrm{O}$ correlations (full circles with statistical uncertainties) obtained from the measured p $+$ p $+$ ${}^{14}\mathrm{O}$ events by selecting the $\rho$ angle within the range from 65 to 80 mrad [see Fig. 14b], which corresponds to the first-excited state in ${}^{16}\mathrm{Ne}$. The solid curve represents the Monte Carlo simulation of the detector response for ${}^{16}\mathrm{Ne}$${}^{*}(2^{+})\to {}^{14}$O $+$ p $+$ p direct 2p decay with the fitted values $Q_{2p}=3.2(2)$ MeV and $\Gamma =0.2(2)$ MeV. The dotted line presents the same calculation but assuming a different 2p-decay mechanism, namely a sequential proton emission via ${}^{15}\mathrm{F}$ g.s. The dashed line is a simulation of a hypothetical direct 2p decay of the $0^{+}$ state with $Q_{2p}=3.5$ MeV and $\Gamma =1$ MeV reported in Ref. [41]. The dash-dotted curve indicates the background contribution scaled from the 7.6 MeV state data.

Figure 23

Level scheme of all states observed in our experiment. Single-proton 1p decays of the states in ${}^{16}\mathrm{Ne}$, ${}^{15}\mathrm{F}$, and ${}^{14}\mathrm{O}$ are shown by dash-dotted arrows. The corresponding decay energies are shown relative to the respective ${}^{14}\mathrm{O}$ $+$ 1p(2p) thresholds. A 2p decay of the ${}^{16}\mathrm{Ne}$ g.s. is illustrated by the dashed arrow. The hatched area indicates the high-energy continuum in ${}^{16}\mathrm{Ne}$.