Four α correlations in nuclear fragmentation: a game of resonances


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M. Huang, A. Bonasera, S. Zhang, H. Zheng, D. X. Wang, J. C. Wang, N. Song, X. Tang, L. Lu, G. Zhang, Z. Kohley, M. R. D. Rodrigues, Y. G. Ma and S. J. Yennello. Four α correlations in nuclear fragmentation: a game of resonances[J]. Chinese Physics C.
M. Huang, A. Bonasera, S. Zhang, H. Zheng, D. X. Wang, J. C. Wang, N. Song, X. Tang, L. Lu, G. Zhang, Z. Kohley, M. R. D. Rodrigues, Y. G. Ma and S. J. Yennello. Four α correlations in nuclear fragmentation: a game of resonances[J]. Chinese Physics C. shu
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Four α correlations in nuclear fragmentation: a game of resonances

  • 1. College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao, 028000, China
  • 2. Institute of Nuclear Physics, Inner Mongolia University for Nationalities, Tongliao 028000, China
  • 3. Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA
  • 4. Laboratori Nazionali del Sud, INFN, via Santa Sofia, 62, 95123 Catania, Italy
  • 5. School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710119, China
  • 6. Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China
  • 7. Chemistry Department, Texas A&M University, College Station, TX 77843, USA
  • 8. Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
  • 9. Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China

Abstract: Heavy ion collisions near the Fermi energy produce a ‘freezout’ region where fragments appear and later decay emitting mainly neutrons, protons, alphas and gamma rays. These products carry information on the decaying nuclei in the medium. Fragmentation events might result in high yields of boson particles, especially alpha particles, and carry important information on the nuclear Bose Einstein Condensate (BEC). We study ‘in medium’ 4α correlations and link them to the ‘fission’ of 16O in two 8Be in the ground state or 12C*(Hoyle State)+α. Using novel techniques for the correlation functions we confirm a resonance of 16O at 15.2 MeV excitation energy and the possibility of a lower resonance close to 14.72 MeV. The latter resonance is the result of all α particles having 92 keV relative kinetic energies.


    • Long ago, Fred Hoyle [1] proposed the possibility of a 0+ resonance near the 3$ \alpha $ threshold in 12C: the Hoyle state (HS). Today, clustering is still a hot topic in nuclear structure for C and more complex nuclei [2-25]. This is true in nuclear dynamics as well, for instance in fragmentation reactions [26-33], together with the possibility of observing a Bose Einstein Condensate (BEC) and Efimov states [22, 34]. In Refs. [35, 36] we have extended the study of 12C to ‘in medium’ decays by studying 3$ \alpha $ correlations in 70Zn+70Zn, 64Zn+64Zn and 64Ni+64Ni heavy ion collisions at 35 MeV/nucleon, an experiment performed at the Cyclotron Institute, Texas A&M University [32]. Following the works from Refs. [37, 38] we searched events with all the $ \alpha $ particles in mutual resonance through the ground state (g.s.) of 8Be. If this mechanism is at an excited level of 12C, it should be located at 7.46 MeV, just below the HS. Because of the lower excitation energy with respect to the HS, its decay probability is strongly hindered [38]. The situation might change ‘in medium’ if in heavy ion collisions an $ \alpha $-BEC is strongly resonating through the g.s. of 8Be. This implies that all the relative energies of the 3$ \alpha $ (two by two) are equal to 92 keV and it is dubbed as Thomas state (TS) in Refs. [39, 40]. Following Ref. [39], we can identify the HS as an Efimov State (ES) [34, 41, 42] because of its 100% decay into 8Be+$ \alpha $, i.e, with the lowest relative energy of two $ \alpha $s equal to 92 keV. The mechanism is that an interaction between two $ \alpha $ particles is mediated by a third $ \alpha $ particle. The Boson mediation produces an effective field that binds the 3 particles system [40].

      This mechanism might be extended to 4 or more Bosons [43] and provides new insight into many body interactions. In particular, if 16O can be described as 4$ \alpha $ clusters, we can study the relative energy distributions of all six 2$ \alpha $ possible combinations.

      On one hand, if 16O decays into 4$ \alpha $ at once then its Q-value will be -14.44 MeV, and the six relative energies combinations will equal to 92 keV (the 8Be ground state). Therefore we expect an excited level at E* = 14.72 MeV [43]. The Coulomb and the finite $ \alpha $-$ \alpha $ interaction range might deny its occurrence [34, 41, 43]. This mechanism has been observed in atoms [44] for three particles and a possible extension to N$ > $3 Boson systems is currently investigated [43]. On the other hand, similarly to the 12C [35, 36], we might expect an excited level of 16O, which decays sequentially to 12C(HS)+$ \alpha $ with the 12C(HS) decaying into 8Be+$ \alpha $ [45-48]. In this case the available excitation energy is not divided ‘democratically’ (equally) among all the $ \alpha $s. Only the lowest relative energy of two $ \alpha $s is equal to 92 keV (the last 8Be decay). In recent papers [18, 21], this level has been observed experimentally around 15.2 MeV excitation energy of 16O but not confirmed in Ref. [22], probably because of low statistics in the energy region of interest. We notice that in the region near 14.72 MeV excitation energy a large number of energy levels [49] has been observed close to the threshold for neutron decay and a few MeV above the proton decay. Thus, the decay into 4$ \alpha $ might be missed in those experiments because of the low tunneling probability.

      We carefully investigate the decay of 16O into 4$ \alpha $ from the fragmentation data. The paper is organized as follows: section 2 describes the experimental setup briefly and summarizes analyzing method of selecting and reconstructing events. In Sec. 3, we analyze the important ingredients from relative-kinetic-energy distributions of 2$ \alpha $s and energy correlation function of the 16O from 4$ \alpha $s. We conclude and summarize our work in Sec. 4.

    • The beams of 70Zn, 64Zn, and 64Ni at 35 MeV/u were produced by the K500 Superconducting Cyclotron at the Texas A&M University Cyclotron Institute and collided with 70Zn, 64Zn, and 64Ni self-supporting targets, respectively. Each fragment on an event-by-event basis was measured on a 4$ \pi $ array Neutron Ion Multidetector for Reaction Oriented Dynamics with the Indiana Silicon Sphere (NIMROD-ISiS), which consisted of 14 concentric rings covering from 3.6° to 167° in the laboratory frame [50, 51]. In the forward rings of the 3.6° to 45°, two modules were set. The supertelescopes modules had two Si detectors (150 and 500 $ \mu $m) placed in front of a CsI(Tl) detector(3–10 cm). The telescopes modules in the forward and backward rings had one Si detector (one of 150,300, or 500 $ \mu $m) followed by a CsI(Tl) detector. The light charged particles with Z = 1-3 were identified by the pulse shape discrimination method in the CsI(Tl) detectors. Intermediate mass fragments (IMFs), were identified with the telescopes and supertelescopes using the ‘$ \Delta $E-E’ method. In the forward rings an isotopic resolution up to Z = 12 and an elemental identification up to Z = 20 were achieved. In the backward rings only Z = 1–2 particles were identified because of the detector energy thresholds. Further experimental details can be found in the Refs. [25, 29, 30, 32, 50, 51].

      We emphasize that only the events with $ \alpha $ multiplicity equal to four are analyzed and the results of the 3 nuclear collision systems are combined to increase the statistics. This is the same idea of the 12C analysis [35, 36]. The momenta of the $ \alpha $s can be measured very well. The only major problem is that when the relative kinetic energy of the 2$ \alpha $s could be as small as tens of keV. These particles are detected in two nearby detectors (or in the same one). Due to the limited size(finite granularity) of the detector, the angle and relative momentum errors arise. This is a problem for all types of detectors, and the resulting error or minimum measurable relative kinetic energy is about 40 keV [9, 10, 25]; The better the granularity, the smaller the error. Our method has an obvious advantage because the beam energy we are working at is close to the Fermi energy, that is, each ion has a high kinetic energy greater than a few MeV/nucleon [32, 36]. This is ideal for our detector. In order to reveal the low energy excited levels of 8Be, 12C and 16O, the relative kinetic energy needs to be very small.

      Let us start by recalling the relation that calculates the excitation energy E* of 16O decaying into 4$ \alpha $s with Q-value, Q = −14.44 MeV. This relationship is given by Eq.(1):

      $ {E^*} = \frac{1}{2}\sum\limits_{i = 1,j > i}^4 {{E_{ij}}} - Q, $


      where ${\rm{E}}_{ij} $ is the two $ \alpha $s relative energies and we have classified the (undistinguishable) $ \alpha $ particles according to their relative energies in such a way that ${\rm{E}}_{ij}^{1} \leqslant{\rm{E}}_{ij}^{2} \leqslant \dots \leqslant {\rm{E}}_{ij}^{6}$.

      Using Eq.(1) we can easily estimate the value of an excited level in 16O assuming that we have not one but two HS in 16O. In the rest frame of the 8Be(g.s.), it assumes that the first $ \alpha $ is at an energy corresponding to the HS of 12C. Now it also assumes that the second $ \alpha $ is at an energy corresponding to the HS of 12C. Thus the excitation energy of 16O in this configuration is given by E*(16O) = 2*0.235+0.092+14.44 = 15.0 (MeV), i.e. very close to the suggested level [18, 21, 22], where 0.235 MeV is the relative energy of the 2$ \alpha $s from the decay of 12C in the HS, 0.092 MeV is the energy of 8Be(g.s.) and 14.44 MeV is the Q-value [35, 36]. We would like to stress that to calculate the tunneling probability for such a configuration at this stage might be tricky since the intermediate state involves the 12C which is not in the ground state. Thus reactions on the excited level of 12C might occur with higher probability as discussed in [52] and we will investigate this scenario in a following paper. We would like to point out that the occurrence of an excited level of 16O at 15.0 MeV as discussed above involves multiple resonances in the 8Be+$ \alpha $+$ \alpha $ system similar to the Efimov mechanism [34, 40-44], thus a careful investigation is needed.

      The proposed analysis relies heavily on the detector performance. The finite granularity of the detector (like all detectors) demands some choices which become crucial when searching for fine details. Because of the finite granularity, the first important choice is to assign a position to the fragment in the single detector. Commonly, two possible avenues are followed [10, 32, 53]. One is to assign the fragment position at the center of the single detector (CD), the second is to assign a random position on the surface of the single detector (RD). While the first choice is sufficient to determine resonances within the reach of the detector granularity, the second spreads the resonance but it would be sensitive to resonances located at even lower excitation energies. Thus, the two approaches are somehow complementary to each other and we will analyze the data with both. The RD approach has an advantage with respect to the CD. In fact, we can randomly choose the position of the real events N$ > >1 $ times (RDN). In this way we can uniformly explore the surface of the detector and, if we normalize the number of events to one, it becomes the probability of finding a fragment at a certain angle and energy. This procedure (not to be confused with the mixing method discussed below) smooths the fluctuations due to statistics or resonances and if two resonances are too close in energy, we might not be able to discriminate them. Of course, for large relative angles between the two fragments and large statistics, the RD(N) and CD methods give similar results.

      Another detector feature to consider is double hits (DH). Because of the finite granularity it is possible that two fragments hit the same detector in the same event. The signals induced by DH of $ \alpha $-particles are quite unique and the DH events can be distinguished clearly from other fragments such as 6,7Li [53]. However, only the total energy of the two $ \alpha $s is determined. The authors of Ref. [53] adopted the ‘democratic’ assumption that the total energy is equally divided between the two fragments. This together with the CD method automatically produces a ‘resonance’ for relative energies ${\rm{E}}_{ij} $(DH) = 0 MeV. The RD(N) method, on the other hand, might give non-zero relative energy since the positions of the two particles in the single detector are randomly chosen. As we have discussed in Ref. [35] for the 12C case (where the RD method was adopted) the lowest relative energy of two $ \alpha $ is peaked at 92 keV after correcting for the detector acceptance. This result suggests to modify the double hits choice ${\rm{E}}_{ij} $(DH) to 92 keV, of course this assumption especially in the CD case will improve on the 8Be resonance decay and we will show that it is amply justified.

      Notice that the real events carry strong quantum correlations since the $ \alpha $ particles are not distinguishable Bosons, thus at low excitation energies or temperatures they might be in a BEC. Therefore, the next step in the data analysis is to generate mixing events for each assumption discussed above. This is achieved by choosing four different $ \alpha $-particles from four different events. This procedure can be repeated many times (more than the number of real events) in order to get a smooth benchmark of the available phase space. When the events are mixed, the quantum correlations are lost and the distribution can be described as a classical one or a Maxwell-Boltzmann distribution. In the absence of resonances we expect the correlation function to be given by the ratio of the Bose-Einstein distribution divided by the Maxwell-Boltzmann distribution opportunely corrected by Coulomb interactions [54]. Strong resonances, especially close to the threshold, might dominate the correlation function and this could be observed in the data as we show below. As for the real events, we normalize the total number of mixing events to 1. A four-body correlation function can be defined as:

      $ 1+R_4 = \frac{Y_R}{Y_M}. $


      Where ${\rm{Y}}_{R} $ is the yield of real events and ${\rm{Y}}_{M} $ is the yield of mixing events. Similarly, the three-body (1+${\rm{R}}_{3} $) or the two-body (1+${\rm{R}}_{2} $) correlation functions can be obtained. The ratio can be performed as function of the 16O excitation energy defined in Eq. (1) or other relevant physical quantities. In Ref. [35] we have defined an alternative way of deriving correlation functions using the transverse energy distribution ($ E_T $) instead of the mixing events technique. This is based on the same assumption that for equilibrated systems, the transverse and total energy distributions are the same apart a trivial 3/2 scaling factor [35].

      A major problem in our analysis is the detector acceptance. Most of our results provide circumstantial evidence for the low energy resonances and a repetition of the methods discussed here with more performing detectors would be crucial to shed some light on the discussed mechanisms. Of course, a perfect 4$ \pi $ detector is quite impossible to realize, thus it is very important to optimize it according to the phenomena to be studied. For $ \alpha $ coming from heavy ion collisions it is important to select the events which produce the largest amount of correlated particles. In our experiment this can be easily visualized by deriving the average angle of the $ \alpha $s (i.e. average of the four $ \alpha $s angles for each event) as function of the E* of 16O. In the Fig. 1 we plot such distribution and different color codes gives the value of 1+${\rm{R}}_4 $ as reported in the figure. As we can see the correlation function is larger than 1 for energies below 30 MeV where many resonances of 16O are found. These events are located at small angles which suggest peripheral collisions with low excitation energy deposited. In particular the excitation energy region near 15 MeV corresponds to angles less than 15 degrees. Thus in order to increase the results sensitivity, a larger granularity is required especially for those forward angles.

      Figure 1.  4$\alpha$s emitting angles Vs. Excitation energy for RD case. (a) is with DH events and (b) is without DH events.

    • Following Ref. [35], we plot the relative two body energy distributions for the RDN case in Fig. 2 with similar results to the 12C case. In the top panels, the bumps obtained for the real events(the solid black circles) are due to the decays of 8Be, 12C or 16O into $ \alpha $s, and completely non-correlated processes, for instance, a heavy fragment can emit $ \alpha $s as well [32, 36]. To distinguish the correlated events from the non-correlated events, we plot the distributions from mixing events showed in Fig. 2 (red open circles). The total number of real and mixing events are normalized to one, respectively. The two distributions look similar on the logarithmic scale, but show significant differences when the relative kinetic energy is low. When the relative energy is small, the distribution of real events in Fig. 2 (a) shows a downward trend, while in (b-f) shows an upward trend. As mentioned earlier, when the relative kinetic energy becomes very small, it is difficult to allocate the detection angle due to the granularity of the detector. Of course, this is less important for the situation in the Fig. 2(b-f), since the relative kinetic energy of the event obtained in the first panel is the smallest. In order to correct this feature, we use an exponential function to fit the highest point in Figure 2(a). This allows us to deduce the instrument error $ \Delta $E = 1/22 MeV = 0.045 MeV, which is slightly larger than that found in the Refs. [9-17] but small enough to let us derive the results discussed below. It can be clearly seen from Fig. 2(b-f) that the experimental error is less important, and the slope change can be seen around 0.1 MeV(the 8Be$ _{g.s.} $). In order to find resonances if any, we can use the same slope (or experimental error) as before for the exponential fitting, and reproduce the experimental point at 0.045 MeV. By subtracting the fits from the real events, we get the open squares in Fig. 2, which can be regarded as the real events corrected by the detector acceptance. As we can see, all six cases show a bump at approximately 0.08 MeV (very close to 0.092 MeV), corresponding to decay of 8Be$ _{g.s.} $. According to our ranking of these 2$ \alpha $'s relative kinetic energies, we can infer that if the maximum relative kinetic energy is 0.092 MeV, then the other five kinetic energies must also be 0.092 MeV (because they are smaller). Therefore, we have 4$ \alpha $s in a mutual resonance event, which is a mechanism similar to the Efimov states of exchanging a Boson between the other two [34, 37, 41]. If this cannot be related to the characteristics of the strong resonant boson gas [6], then it may be an unexplored 16O state at ${\rm{E}} ^* $ = 14.72 MeV (as given by Eq.(1)). Ref. [38] calculations suggested that when 12C was formed in a vacuum, its observation probability is 8-orders of magnitude smaller than the HS. When in a medium, because of the presence of other fragments, the mechanism of mutual resonances might be enhanced. After subtracting the exponential fit, the ratio of the real (green squares) and the mixing events (red open circles) gives the correlation functions 1+${\rm{R}} _2 $ displayed in the bottom panels of Fig. 2. The ratio shows that peaks present around 0.08 MeV for all cases.

      Figure 2.  (color online) Selected events from $^{70(64)}$Zn(64Ni) + $^{70(64)}$Zn(64Ni) at E/A = 35 MeV/nucleon with $\alpha$ multiplicity equal to four with RDN method. Relative kinetic energy distribution as a function of the relative kinetic energy of 2$\alpha$s with the order of ${\rm{E}}_{ij}^{1}\leqslant {\rm{E}}_{ij}^{2}\leqslant\dots\leqslant{\rm{E}}_{ij}^{6}$ (a-f). The solid black circles represent data from real events, red open circles are from mixing events, and the green open squares represent the difference between the real events and the exponential function (solid line), which takes into account the experimental error. The ratios of the real (pink open triangles) data and the real data minus the fitting function (blue crosses) divided by the mixing events are, respectively, a function of the relative kinetic energy of 2$\alpha$s classified as ${\rm{E}}_{ij}^{1}\leqslant{\rm{E}}_{ij}^{2}\leqslant \dots \leqslant{\rm{E}}_{ij}^{6}$ (g-l).

      In order to see the double hit effects on the relative kinetic energy, we add ${\rm{E}} _{ij} $(DH) = 92 keV (middle panels) and no DH case (bottom panel) for the RDN case in Fig. 3. Some important features are worth stressing from Fig. 3. The first one is that for higher relative energies the result with and without the exponential subtraction are the same (top panels). The second is that all the relative energies display a peak around 92 keV, thus confirm the occurrence of strong resonances in the BEC (top and middle panels). The ${\rm{E}} _{ij} $(DH) = 92 keV (middle panels) does not need any exponential correction indicating the RDN method plus the choice ${\rm{E}} _{ij} $(DH) = 92 keV well enough to extract the energy level. The ${\rm{E}} _{ij} $ original produces the increase of the correlation function at low energies and needs a correction to be consistent with the ${\rm{E}} _{ij} $(DH) = 92 keV case. Finally, in the bottom panel we show the results without DH. This shows that the RDN method is able to determine the 92 keV resonance but it needs DH in order to reveal the 8Be resonance for all relative energy combinations.

      Figure 3.  The 1+R2 correlation function vs the $\alpha$-$\alpha$ relative energy for all possible combinations. The events are classified such that ${\rm{E}}_{ij}^{1} \leqslant{\rm{E}}_{ij}^{2} \leqslant \dots \leqslant{\rm{E}}_{ij}^{6}$. The RDN method with ${\rm{E}}_{ij}$ original (top), ${\rm{E}}_{ij}$(DH) = 92 keV (middle) and no DH (bottom) are plotted as function of the relative energy. In the top panels, the exponential correction to the detector acceptance case is given by the open circles [35].

      For completeness, in Fig. 4 we have repeated the analysis for the CD case. There are some differences in the choice of the relative energy for double hits, but we notice that this choice is complementary to the RDN one since the peak near 92 keV is evident already for the smallest ${\rm{E}} _{ij} $1 without any correction for the experimental acceptance and no DH.

      Figure 4.  Same as for Fig. 3 for the CD case. The ${\rm{E}}_{ij}$ original (top), ${\rm{E}}_{ij}$(DH) = 92 keV (middle) and no DH (bottom) cases display multiple resonances at 92 keV. For large relative energies the 92 keV shifts to higher energies because of low statistics and the finite detector granularity.

      In Fig. 5, we plot the excitation energy distributions (a-c) and the energy correlation function (d-f) of the 16O from 4$ \alpha $s. In the panels (a,d), the ${\rm{E}} _{ij} $(DH) = 92 keV is adopted while the ${\rm{E}} _{ij} $ original(i.e. ${\rm{E}} _{ij} $ = 0 MeV) is given in the panels (b,e). The results without DH are displayed in the panels (c,f). As we can see, in the panels (d-f), the CD (black full circles) choice gives a positive correlation function around 15 MeV. A peak at 15.2 MeV is clearly seen when the ${\rm{E}} _{ij} $(DH) = 92 keV assumption is adopted. The RDN events (red open circles) show a smoothing of the CD case (see also the insets). The interesting feature is that two clear peaks appear at 14.85 and 15.2 MeV in the panel (d), while the two peaks are smoothed in the panel (e). The CD case is smoothen into one large peak for the RDN case. With the original choice for the DH, we observe a decreasing trend of the correlation function. The panel(f) (no DH) shows the positive correlation function above 15 MeV but no data points are found for lower energies, thus it suggests that the DH relevance for the decay of 8Be is crucial to determine the exact position of the resonance(s). Another important ingredient in the data analysis displayed in Fig. 5 is the bin-width chosen to be 60 keV. Larger bin-widths smoothen the fluctuations and may increase the resonance widths.

      Figure 5.  (color online) The energy correlation function of the 16O from 4$\alpha$s: (a) relative energy for DH is equal to 92 keV and (b) original. No double hits in panel (c). In all cases, the bin-width is 60 keV. In the insets we display the results for the RDN cases only.

      In Fig. 6, we have repeated the analysis of the previous Fig. 5 for different bin-widths ((a),(b),(c),(d) corresponding to the bin-widths of 60 keV, 80 keV, 120 keV, 200 keV, and similarly for the middle and bottom panels and for the CD case only. The 15.2 MeV is clearly visible in the two top panels with some hints in the bottom panel where DH is not included. Some data points are also present near the 14.85 MeV excitation energy but the error bars are too large. In the inset of Fig. 6(d) we have indicated the positions and widths of some observed excited levels coming from the decays into 8Be+8Be (black full lines) and $ \alpha $+12C*(HS) (red dashed lines) [45-48]. We notice the large bump below 16 MeV which might be dominated by the suggested resonances plus the detector acceptance. For larger excitation energies, resonances are embedded into the BEC, thus are not clearly distinguishable [26, 30, 31]. Of course repeating the experiment with an improved detector granularity might shed more light on these in medium levels. In such a scenario the RDN method proposed in this work might be crucial.

      Figure 6.  Excitation energy distribution for the CD choice: (top) ${\rm{E}}_{ij}$(DH) = 92 keV, (middle) ${\rm{E}}_{ij}$ original and (bottom) no double hits. The (a), (b), (c), (d) correspond to the bin-widths of 60 keV, 80 keV, 120 keV, 200 keV, and the middle and bottom panels are similar. In the inset of (d), the positions and widths of known resonances are indicated as well, see text.

      For completeness we have repeated the CD case using the transverse energy method to derive the correlation function as in Ref.[35]. In Fig. 7 we plot the correlation function for different bin widths, compare to Fig. 6. This method is in agreement with the mixing method and actually shows sharper resonance at the lowest excitation energies.

      Figure 7.  Transverse excitation energy distribution for the CD choice: (top) ${\rm{E}}_{ij}$(DH) = 92 keV, (middle) ${\rm{E}}_{ij}$ original and (bottom) no double hits. The (a), (b), (c), (d) correspond to the bin-widths of 60 keV, 80 keV, 120 keV, 200 keV, and the middle and bottom panels are similar. In the inset of (d), the positions and widths of known resonances are indicated as well, see text.

    • In conclusion, in this paper we have shown that strong resonances among $ \alpha $-particles occur in fragmentation reactions and they could be a signature of a BEC. Furthermore, the strong correlations at relative energies ${\rm{E}} _{ij}^{1} \leqslant{\rm{E}}_{ij}^{2} \leqslant \dots \leqslant{\rm{E}}_{ij}^{6} $ confirm the mechanism of Boson exchanges [43] and the possibility of an excited level of 16O at 14.72 MeV. Experimental search for this resonance in reactions with no medium effects (for instance p+$ ^{15}{\rm{N}}\rightarrow $8Be+8Be$ \rightarrow $4$ \alpha $, $ {\rm{J}}^{\pi} $ = 0+) would be extremely interesting but we anticipate a very low cross section for the decay because of the Coulomb barrier and the competition with other open channels, nevertheless the presence of multiple resonances might modify the probabilities substantially [52]. We have also confirmed the resonance in 16O at 15.2 MeV [21, 22].

Reference (54)



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