Invariant-mass spectroscopy in projectile fragmentation reactions

The fragmentation of a projectile into a number of pieces can lead to the creation of many resonances in different nuclei. We discuss application of the invariant-mass method to the products from such reactions to find some of the most exotic resonances located furthest beyond the proton drip line. We show examples from fragmentation of a fast ${}^{13}\mathrm{O}$ beam including the production of the newly identified ${}^9\mathrm{N}$ resonance. In extracting resonance parameters from invariant-mass spectra, accurate estimates of the background from nonresonant prompt protons are needed. These prompt protons are correlated with the resonances due to long-range Coulomb interactions and will suppress events when resonance decay products and the prompt protons have small invariant masses. The shape of this background is especially important in determining the widths of wide resonances typically found at the edge of the chart of nuclides. An event-mixing recipe is proposed to describe this background, where the mixed events have reduced weighting for the smaller invariant masses to account for the effect of the Coulomb final-state interactions. The weighting is based on the measured correlations of heavier hydrogen isotopes with the resonances or the projectile residues. We also show that the relative magnitude of the background can be reduced in some cases by selecting events where the resonance decay products are accompanied by a deuteron or triton cluster. The deuteron and triton clusters are largely not products of resonant decay and thus events with an accompanying cluster are associated with fewer background protons than events without them. The width of the $5/2^{-}$ state in ${}^9\mathrm{C}$ with the new background prescription was found to be consistent with the value obtained using proton elastic scattering on ${}^8\mathrm{B}$. Improved values for the widths of ${}^{10}\mathrm{C}, {}^{10,11}\mathrm{N}$, and ${}^{12}\mathrm{O}$ resonances were also obtained using the new background prescription.

Figure 1

Distribution of reconstructed center-of-mass velocity for some of the resonant-decay channels considered in this work. Panel (c) shows the center-of-mass velocity distribution of the $p+{}^9\mathrm{C}$ subevents of the detected $d+p+{}^9\mathrm{C}$ events as analysed in Sec. 6. In panel (a), the arrows indicate the low and high-velocity thresholds for proton identification. The dotted vertical lines in all panels give the beam velocity in the center of the target.

Figure 2

Distribution of ${}^9\mathrm{C}$ excitation energy derived from the invariant masses of detected $p+{}^8\mathrm{B}$ events. Experimental data points were obtained with (a) the ${}^{13}\mathrm{O}$ beam and [(b), (c)] the ${}^9\mathrm{C}$ beam. The latter are subdivided into events where (b) the target is not excited and (c) all other events. Solid red curves show fits to these distributions with backgrounds from the prompt protons indicated with the dashed blue curves and the contribution from the wide $5/2^{-}$ state indicated by the green dotted curves. The results for the ${}^9\mathrm{C}$ beam in [(b), (c)] are scaled up by a factor of 3 above $E^{*}=2.7$ MeV.

Figure 3

Distribution of ${}^9\mathrm{C}$ excitation energy obtained from the invariant masses of detected $p+{}^8\mathrm{B}$ events using the ${}^{13}\mathrm{O}$-beam data (black histogram). The red histogram shows the contribution from events where two protons and a ${}^8\mathrm{B}$ fragment are emitted in the angular region probed with the HiRA array, but where only one of these protons is detected.

Figure 4

Comparison of the distributions of ${}^9\mathrm{C}$ excitation energy obtained from the invariant masses of detected $p+{}^8\mathrm{B}$ events (black data points) to those where a cluster was also detected in coincidence (colored histograms). All results were obtained with the ${}^{13}\mathrm{O}$-beam data.

Figure 5

Correlations functions obtained for the $\text{cluster}+p+{}^8\mathrm{B}$ events as a function of [(a)–(d)] the relative energy between the cluster and the reconstructed ${}^9\mathrm{C}$ fragment and [(e)–(h)] the relative energy between the cluster and the ${}^8\mathrm{B}$ residual. For comparison, the solid red curves in [(e), (f)] are the correlation functions for prompt protons obtained in the fit to the excitation-energy distribution in Fig. 2.

Figure 6

Distributions of total decay energy for $\text{cluster}+p+{}^8\mathrm{B}$ events. The data points show the experimental results, the dotted blue curves are from the mixed events, and the solid red curves were obtained when the latter events are weighted by the fitted correlations functions.

Figure 7

Distributions of total decay energy for (a) $2p+{}^8\mathrm{B}$ and (b) $3p+{}^8\mathrm{B}$ events. Data points are the experimental results. The solid red curves show the expected background from $p+{}^8\mathrm{B}$ and $2p+{}^8\mathrm{B}$ events, where an extra prompt proton is included.

Figure 8

[(a), (b)] Distributions of ${}^9\mathrm{C}$ excitation energy gated on the indicated regions of proton decay angle ${\theta }_p$. The solid red curves are fits to these distributions including the prompt proton background distributions (dashed blue curves) obtained from event mixing with the same ${\theta }_p$ gates. The dotted green curves give the fitted contribution from the $5/2^{-}$ state. (c) The angular distribution obtained from these fits where the error bars are smaller than the size of the data points. The solid green curve shows a fit using Eq. (3).

Figure 9

Distributions of ${}^{10}\mathrm{C}$ excitation energy obtained from the detected $p+{}^9\mathrm{B}$(g.s.)$\to 2p+2\alpha$ events. The background from $2p+{}^9\mathrm{B}$(g.s.) and $3p+{}^9\mathrm{B}$(g.s.) events where the protons are emitted in the angular range probed by the HiRA array, but only one is detected, is shown as the red histogram. Arrows show the location of ${}^{10}\mathrm{C}$ states known to decay into this channel.

Figure 10

Correlation function obtained from the $d+p+{}^9\mathrm{B}$(g.s.)$\to d+2p+2\alpha$ events as a function of the relative energy between the $d$ and the reconstructed ${}^9\mathrm{B}$ fragment. The solid green curve shows a fit with Eq. (2).

Figure 11

Data points show the ${}^{10}\mathrm{C}$ excitation-energy distribution for the $p+{}^9\mathrm{B}$(g.s.)$\to 2p+2\alpha$ events corrected for the background from $2p+{}^9\mathrm{B}$(g.s.) and $3p+{}^9\mathrm{B}$(g.s.) events. The solid red curve shows the fit to these data with three ${}^{10}\mathrm{C}$ resonances and the background (dashed blue curve) associated with detecting prompt protons. The dotted green curves show the contributions for each resonance. The dotted blue curve, which was obtained from the mixed events without weighting and normalized to reproduce the high-energy tail, cannot reproduce the data at low values of $E^{*}$.

Figure 12

Histograms show the decay-energy spectra for (a) ${}^{11}\mathrm{N}\to 2p+{}^9\mathrm{B}$(g.s.)$\to 3p+2\alpha$ events and (b) ${}^{12}\mathrm{O}\to 3p+{}^9\mathrm{B}$(g.s.)$\to 4p+2\alpha$ events. The distribution in (a) has been corrected for the contribution $3p+{}^9\mathrm{B}$(g.s.) events where one of the protons flies in between HiRA telescopes and is not detected. The dashed blue curves show the predicted backgrounds from coincidences with prompt protons estimated using weighted mixed events.

Figure 13

Data obtained from $d+p+{}^{10}\mathrm{C}$ events. The $d\text{−}{}^{10}\mathrm{C}$ correlation function is shown in (a) plotted versus their relative energy. The distribution of ${}^{11}\mathrm{N}$ decay energy obtained from the $p+{}^{10}\mathrm{C}$ subevents is shown in (b) with the results of the fit shown by the solid red curve. The fitted background from the prompt protons is indicated by the dashed blue curve, while the contributions from the different ${}^{11}\mathrm{N}$ states are indicated by the dotted curves. Solid arrows indicate the location of previously known states, and the dashed arrows locate peaks associated with proton decay of the $2^{+}$ first excited state of ${}^{10}\mathrm{C}$ as assumed in the fit. The simulated invariant-mass resolution (FWHM) increases from 274 keV at $E_{{\text{p-}}^{10}\text{C}}=1$ MeV to 519 keV at $E_{{\text{p-}}^{10}\text{C}}=3$ MeV.

Figure 14

Experimental distribution of ${}^{10}\mathrm{N}$ decay energy obtained from the $p+{}^9\mathrm{C}$ subevents of the detected $t+p+{}^9\mathrm{C}$ events. The distributions have been gated on (a) longitudinal decay $|cos{\theta }_p|>0.5$ and (b) transverse decay $|cos{\theta }_p|<0.5$. The solid red curves show a joint fit to these data with two peaks (dotted curves) and the background from prompt protons (dashed blue curves).

Figure 15

Histogram shows the experimental distribution of $5p+\alpha$ total decay energy obtained from $p+{}^8\mathrm{C}$(g.s.) events in Ref. [34]. The two blues curves show the background from promptly emitted protons as determined from the weighted mixed events. The upper dotted curve has been normalized to a similar maximum value as the experimental data to emphasis its difference in shape to the experimental result. The lower dashed curve is normalized to reproduce just the high-energy and lowest-energy tail regions, which appear to be dominated from this background.

Figure 16

Data points show the distribution of ${}^{12}\mathrm{O}$ decay energy obtained from the detected $2p+{}^{10}\mathrm{C}$ events where the emission angle ${\theta }_{10}$ of the ${}^{10}\mathrm{C}$ fragment in the parent's reference frame is constrained to $|cos{\theta }_{10}|<$ 0.2. The solid red curve is a fit to this distribution with four peaks (dotted green curves) and the prompt-proton background (dashed blue curve). Arrows indicate the energies of levels with known spin-parity.

Figure 17

Distribution of the ${}^{10}\mathrm{C}$ decay angle in the parent ${}^{12}\mathrm{O}$ fragment's reference frame for the two $2^{+}$ states in ${}^{12}\mathrm{O}$.