INTRODUCTION

One of the key aspects of nuclear structure is the presence of degrees of freedom in which quartets of spin-paired protons and neutrons behave as constituent clusters, manifested in the intense production of α particles in a wide variety of nuclear reactions and decays. The transition to the study of ensembles of α particles immediately above the binding thresholds makes it possible to identify the role of unstable 8Be and 9B nuclei and the 3α Hoyle state (HS) and to search for their analogs.

The use of a technically simple and inexpensive nuclear emulsion (NE) technique in relativistic nuclear beams provides flexibility and uniformity at the search stage and, in the theoretical aspect, transparency of interpretation. It would be particularly good to demonstrate the similarity of conclusions based on relativistic invariance. During the dissociation of relativistic nuclei in a narrow fragmentation solid angle, ensembles of He and H nuclei are intensively generated (Fig. 1). Particularly valuable are the so-called “white” stars, in the region of the interaction vertex of which no tracks of the target nucleus and produced mesons are observed. In addition, when tracking in the direction of the heavy nuclei fragmentation cone, one can observe stars that do not have an incoming track from the event vertex, which arise in the interactions of relativistic neutron fragments and nuclei in the NE material [1].

Fig. 1.
figure 1

Macrophotograph of the peripheral interaction of the 84Kr nucleus with an energy of 950 MeV per nucleon with multiple formation of secondary He and H fragments.

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According to the widths, the decays of 8Be, 9B, and HS occur over ranges from several thousand (8Be and HS) to several tens (9B) of atomic sizes and should be identified by a minimum invariant mass. Due to the minimum energy, the decays of 8Be, 9B, and HS should appear as pairs and triplets with relativistic He and H fragments with the smallest separation angles. The invariant mass of a system of relativistic fragments is defined as the sum of all scalar products of 4‑momenta Pi,k of the fragments M*2 = Σ(Pi · Pk). For convenience of presentation, we introduce a variable Q defined as the difference between the invariant mass and the sum of the fragment masses Q = M* − Σm. The components Pi,k are determined in the approximation of conservation of initial momentum per nucleon by fragments.

Current interest in nuclear α-clustering is largely motivated by the concept of α-particle Bose–Einstein condensate (αBEC). The unstable 8Be and HS nuclei are described as 2- and 3αBEC states, and their decays can serve as signatures of the decays of more complex nαBEC states. The existence of the latter can expand the picture of the nucleosynthesis of heavy nuclei. Recently, the statistics of dozens of 8Be decays revealed an increase in the probability of detecting 8Be with an increase in the number of associated α particles nα. A preliminary conclusion is drawn that the contributions from 9B and HS decays are also increasing.

IDENTIFICATION OF DECAYS OF 8Be AND 9B NUCLEI IN THE DISSOCIATION OF LIGHT RELATIVISTIC NUCLEI

Analysis of the irradiation of NE layers in a beam of 10B nuclei with an energy of 1A GeV made it possible to reveal the effect of dominance of the multiple fragmentation channel [2, 3]. In the distribution of fragments over the charge state, the fraction of the 10B → 2He + H channel was 77%. Based on measurements of the outgoing angles of He and H fragments, it was established that the unstable nucleus 8Beg.s. (Fig. 2, right) appears with a probability of (25 ± 5)%, of which (13 ± 3)% are due to decays of the unstable 9B nucleus (Fig. 2, left). What is unexpected is the fact that the number of white stars 9B + n is 10 times greater than 9Be + p. This observation may indicate a wider spatial distribution of neutrons in the 10B nucleus compared to protons, resulting in a larger cross section for the 9B + n channel compared to the mirror channel. In addition, with a probability of 8%, stars are observed in the 10B → 6Li + α channel. It is possible that the Li nucleus, weakly manifested in the dissociation of 10B, is also present in 10B mainly in a “dissolved” form, giving a nonresonant contribution to the Θ2He distribution [2–4].

Fig. 2.
figure 2

Distributions of Q2αp of 2αp triplets by excitation energy Q2αp for fragmentation (solid line) 10B → 2He + H at 1.6 A GeV/c and 11C → 2He + 2H at (added, dotted line) 2.0 A GeV/c and (right) of Q2α of α pairs in 9B decays identified in these events [7].

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The charge topology of the dissociation channels of the 11C nuclei in NE with an energy of 1.2 A GeV has been studied. Among 11C stars, events with observations of only relativistic fragments He and H, especially 2He + 2H, are dominant; their contribution was 77% [5]. A channel that is characteristic only of the 11C nucleus with the formation of Li + He + H fragments has been established. Based on the measured outgoing angles of He and H fragments, in the representation of the invariant variable Q, it was shown that decays of the 8Beg.s. nuclei (Fig. 2, right) of all found white 11C stars are represented in 21% of the 11C → 2He + 2H events and 19% for the 11C→3He channel. 9B decays (Fig. 2, left) were detected in white stars 11C → 2He + 2H, constituting 14% of white stars 11C → 2He + 2H. It was found that, as in the case of 10C, 8Beg.s. decays of white 11C stars almost always occur due to 9B decays of [5, 6]. It is worth noting the lowest-energy peak in the Q2α2p distribution of the 18 found stars 11C → 2He + 2H, characterized by an average Q2α2p value of (2.7 ± 0.4) MeV with an rms value of 2.0 MeV [6].

The constraint established during the analysis of data on the dissociation of 10B and 11C nuclei on the identification of decays of 8Be nuclei (Q2α < 0.2 MeV) made it possible to estimate the contribution of such decays to the dissociation of relativistic nuclei 12C → 3α and 16O → 4α in nuclear emulsion at the level of (45 ± 4)% and (62 ± 3)%, respectively (Fig. 3) [7–9].

Fig. 3.
figure 3

Distribution of the number of 2α pairs N2α over the excitation energy Q2α in the coherent dissociation (solid line) 12C → 3α and (dashed line) 16O → 4α at 3.65 A GeV. (Inset) Enlarged region Q2α < 1 MeV (step 40 keV). The histograms are normalized to the number of white stars Nws [9].

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OBSERVATION OF EVENTS WITH DECAYS OF THE HOYLE STATE

The certainty in the identification of 8Be and 9B became the basis for the search for HS decays in the 12C → 3α dissociation (Fig. 4), where the limit on the Q variable of α triples to 0.7 MeV was established [7]. A comprehensive analysis of the 12C → 3α and 16O → 4α stars made it possible to establish that the fraction of events containing HS decays is (11 ± 3)% for 12C and (22 ± 2)% for 16O (Fig. 4) [7–11]; 33 16O → 28Be events have been identified, accounting for 5 ± 1% of the 16O → 4α white stars. The Q distribution of a system of 4α particles in the 16O → 28Be events [12] indicates two candidates 16O(0+6) → 28Be in the region Q < 1.0 MeV. The dissociation statistics for the 16O → 28Be and 16O → αHS channels have the ratio (0.22 ± 0.02) [9].

Fig. 4.
figure 4

Distribution of the number of 3α triples N3α over excitation energy Q3α for (solid lines) 316 white stars 12C → 3α and (dashed lines) 641 white stars 16O → 4α at an energy of 3.65 A GeV. (Inset) Enlarged part Q3α < 2 MeV, normalized to the number of white stars Nws [9].

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It should be noted that, with an increase in 2α- and 3α-combinations in the event, the manifestation of unstable 8Be and HS increases. Thus, HS identified in the relativistic dissociation of 12C also appears in the case of 16O. This result shows that HS does not reduce to the usual excitation of the 12C nucleus, but, like 8Be, is a more universal object of a nuclear-molecular nature. The closest confirmation of this assumption can be the observations of HS in the relativistic fragmentation 14N → 3α [13]. Nevertheless, such an observation deserves verification for heavier nuclei, where α combinatorics increases rapidly with mass number.

ALPHA-PARTICLE DISSOCIATION OF HEAVY RELATIVISTIC NUCLEI

Having been assessed in the study of light nuclei, a similar search was applied to the study of dissociation events of medium and heavy nuclei to identify 8Be and HS and search for more complex nαBEC states. The analysis made it possible to trace the contribution of unstable states with a higher multiplicity of He and H fragments using the method of transverse scanning of NE layers. Starting with the fragmentation of 16O nuclei in the energy range 3.65–200 GeV/nucleon, the analysis showed a relative increase in the contribution of 8Be nuclei with an increase in the number of relativistic α particles per event [14]. The results of measurements of 4301 interactions of 22Ne nuclei at an energy of 3.22 A GeV have been analyzed [15]. This data set includes precision measurements of the outgoing angles of 2 to 5 relativistic α particles, which allowed analysis in the Q(2–5)α variables. It has been established that the probability of identifying 8Be, 9B, and HS increases with the multiplicity of α particles in the event [15]. A similar result was obtained by analyzing 1093 events of nα fragmentation of 28Si nuclei with an energy of 14.6 GeV/nucleon in a nuclear emulsion up to 6 α particles in the event [14]. The distribution of events with identified 8Be nuclei by the Q3α variable for the data set on 12C, 22N, and 28S nuclei is shown in Fig. 5.

Fig. 5.
figure 5

Distributions of 3α systems with identified 8Be N(3α)(8Be) decays over excitation energy Q3α (≤2 MeV) in events of fragmentation of (solid line) 22Ne nuclei with an energy of 3.22 GeV/nucleon and (dashed line) 28Si nuclei with an energy of 14.6 GeV/nucleon; (dots) the distribution of N(3α)(8Be) in the 12C → 3α dissociation, normalized to the statistics for 22Ne and 28Si [14].

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Also, the nα events were studied during transverse scanning of NE layers longitudinally irradiated in a beam of 84Kr nuclei with an energy of 950 MeV/nucleon [12, 16]. In this analysis, the fragment momentum was corrected for the ionization loss in nuclear emulsion and multiplied by 0.8 to approximately calculate the drop in the initial momentum value upon interaction [16]. Being inessential for the selection of Q2α(8Be) ≤ 0.4 MeV, it further allows one to maintain the selection condition Q3α(HS) < 0.7 MeV, focusing on the Q3α(HS) peak (Fig. 6). The Q4α distributions up to 10 MeV indicate an α quartet at nα = 6 with an isolated value of Q4α = 0.6 MeV, corresponding to both the αHS and 28Be variants [12]. Without contradicting the 16O(0+6) decay, this single observation serves as the starting point for further accumulation of statistics on the problem of the 4αBEC state.

Fig. 6.
figure 6

Distribution in the region of low excitation energies of nα-systems Q of (dots, factor 0.1) pairs, (solid line) triplets, and (shaded) quadruples of α particles formed in the fragmentation of Kr nuclei. [12].

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Wide coverage of nα is provided by measurements of 1316 inelastic interactions of 197Au at 10.7 GeV/nucleon [14]. The proportion of events with nα > 3 among those measured was 16%. Due to the increasing complexity of measurements, the selection condition for Q2α(8Be) was relaxed to ≤ 0.4 MeV. It turned out that the ratio of the number of Nnα(8Be) events with at least one identified 8Be decay to their number Nnα demonstrates a strong increase with nα [14, 17]. In general, the correlation picture of the dependence of the multiplicity of α particles in an event and on the number of events with identified 8Be decays (at least one) is shown in Fig. 7.

Fig. 7.
figure 7

Dependence of the relative contribution of decays of 8Be nuclei Nnα(8Be) to the statistics of Nnα events with α-particle multiplicity nα in the relativistic fragmentation of C, O, Ne, Si, and Au nuclei; white stars 12C → 3α and 16O → 4α (WS) are marked; for convenience, the points are slightly shifted around the nα values and connected by lines.

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