Nuclear track emulsion in search for the Hoyle-state in dissociation of relativistic ${}^{12}$C nuclei

TL;DR

This study uses nuclear track emulsion to identify Hoyle state decays in relativistic ${}^{12}$C dissociation, providing insights into nuclear structure and decay processes with high angular measurement precision.

Contribution

It demonstrates a method to detect Hoyle state contributions in ${}^{12}$C dissociation using nuclear track emulsion with precise angular measurements.

Findings

Estimated 10-15% contribution of Hoyle state to ${}^{12}$C dissociation.

Reconstructed Hoyle state decays via alpha particle emission angles.

Identified ${}^{8}$Be nucleus in decay processes.

Abstract

Production of ensembles of $\alpha$-particle triples associated with the Hoyle state (the second excited state of the ${}^{12}$C nucleus) in peripheral dissociation of relativistic ${}^{12}$C nuclei is studied. Stacks of nuclear track emulsion pellicles exposed to ${}^{12}$C with an energy from hundreds MeV to a few GeV per nucleon serve as the material for studies. The Hoyle state decays are reconstructed via measurement of emission angles of $\alpha$ particles with the precision sufficient for identification of the unstable ${}^{8}$Be nucleus. The estimate of the contribution of Hoyle's state to the ${}^{12}$C $\to$ 3$\alpha$ dissociation is 10-15\%.

Figures

Tables

Table 1. Table 1: The average values of ⟨ Θ 2 ​ α ⟩ delimited-⟨⟩ subscript Θ 2 𝛼 \langle\Theta_{2\alpha}\rangle and ⟨ Q 2 ​ α ⟩ delimited-⟨⟩ subscript Q 2 𝛼 \langle\textit{Q}_{2\alpha}\rangle ( Q 2 ​ α subscript 𝑄 2 𝛼 Q_{2\alpha} < < 300 keV) in the angular regions of 8 Be decays.

Nucleus ($P_0$, $A$ GeV/$c$)	$⟨{\Theta }_{2\alpha }⟩$ (RMS), ${10}^{-3}$ rad ($Q_{2\alpha }$ 300 keV)	$⟨{\text{Q}}_{2\alpha }⟩$ (RMS), keV
12C (4.5)	2.1 $\pm$ 0.1 (0.8)	109 $\pm$ 11 (83)
14N (2.9)	2.9 $\pm$ 0.2 (1.9)	119.6 $\pm$ 9.5 (72)
9Be (2.0)	4.4 $\pm$ 0.2 (2.1)	86 $\pm$ 4 (48)
10C (2.0)	4.6 $\pm$ 0.2 (1.9)	63 $\pm$ 7 (83)
11C (2.0)	4.7 $\pm$ 0.3 (1.9)	77 $\pm$ 7 (40)
10B (1.6)	5.9 $\pm$ 0.2 (1.6)	101 $\pm$ 6 (46)
12C (1.0)	10.4 $\pm$ 0.6 (3.7)	117 $\pm$ 12 (75)

Full text

Introduction

Dissociation of relativistic nuclei in a nuclear track emulsion (NTE) is a well-established phenomenon allowing holistic exploration of relativistic ensembles of lightest nuclei. The NTE technique remains the only source of such observations staying unconquerable in sensitivity and angular resolution. Individual features of investigated nuclei manifest themselves in their projectile fragmentation cones. Events of coherent dissociation which does not feature either slow fragments or charged mesons (“white” stars) are clearly observed in NTE. Since distortions of projectile initial states are minima in them they are especially evident in studies of nuclear structure (example in Fig 1) (see Table 1).

The cluster structure of light nuclei and the role of the unstable 8Be and 9B nuclei in them is a subject of the $\href http://becquerel.jinr.ru/$ project (reviewed in 1 and 2 ). The studies are performed on a basis of NTE layers longitudinally exposed to relativistic Be, B, C and N isotopes, including radioactive ones. To set new limits for NTE technique it is suggested to search for the Hoyle-state (HS) in dissociation relativistic 12C nuclei using the invariant mass approach.

Despite capabilities of the NTE technique its history seemed to be completed in the early 2000s. However, since 2012, the company “Slavich” (Pereslavl Zalessky, Russia) has resumed production of NTE layers of a thickness from 50 to 200 $\mu$m on a glass base. NTE samples were tested in state-of-art experiments (reviewed in 3 ). On the basis of photography on microscopes, the experience of computer recognition of short nuclear tracks in NTE was obtained. At the present time, production of baseless 500 $\mu$m thick layers of is being mastered. Reproduction of NTE allows one to put forward new proposals grounded on this classical technique.

The status of the experimental and theoretical studies of the second excited state of the 12C nucleus is reviewed in 4 . This excitation is named after the astrophysicist F. Hoyle who postulated its existence to explain the prevalence of the 12C isotope. Following an accurate prediction of the HS energy it was experimentally confirmed that the 12C nucleus has the excited state located at only 378 keV above the mass threshold of the three $\alpha$ particles. Although it is unstable, its width is only 8.5 eV. Such a value indicates that the HS lifetime is comparable with the values for 8Be or $\pi^{0}$-meson. Observation of HS at a contrast of relativistic energy and the minimum possible energy stored by 3$\alpha$-ensembles can demonstrate HS as a nuclear-molecular object similar to 8Be. First of all it is necessary to establish the very possibility of HS appearance in the relativistic fragmentation cone that is the purpose of the present study.

Reconstruction of invariant mass

In general, energy of a few-particle system Q   can be defined as difference between the invariant mass of the system $\textit{M}^{*}$   and a primary nucleus mass or a sum of masses of the particles M , that is, $\textit{Q}=\textit{M}^{*}-M$. M is defined as the sum of all products of 4-momenta $\textit{P}_{i,k}$ fragments $\textit{M}^{*2}=(\sum{P}_{j})^{2}=\sum(P_{i}P_{k})$. Subtraction of $M$ is a matter of convenience and $Q$ is also named an invariant mass. Reconstruction of $Q$ makes possible to identify decays unstable particles and nuclei.

For the most part, fragments of a relativistic nucleus are contained in a narrow cone of the polar angle $\theta$, which is estimated as $\theta$ = $0.2/P_{0}$, where the factor 0.2 GeV/$c$ is determined by the spectator-nucleon transverse momentum, while $P_{0}$ is the momentum of the accelerated projectile nucleon. The fragment 4-momenta $P_{i,k}$ in the cone can be determined in assumption of conservation of momentum per nucleon by fragments of a projectile (or its velocity). This approximation is well grounded when primary energy above 1 $A$ GeV 5 . Then, $Q$ is functionally related with opening angles $\theta$ between fragments. In the 12C context the assumption about the correspondence of a doubly charged fragment to the 4He isotope is well justified also.

The unstable 8Be nucleus is an imminent participant of HS decay and its reconstruction is the precondition of HS identification. The ground state 8Be is sufficiently separated from the first excited state $2^{+}$ 6 to be identified in a spectrum over the invariant mass $Q_{2\alpha}$ calculated by $\alpha$-pair opening angles $\Theta_{2\alpha}$ 1 ; 2 . Like 8Be, HS is well separated from the higher 12C excitations 6 . Therefore, the same approach can be extended to the identification of HS with respect to the invariant mass of $\alpha$-triples $Q_{3\alpha}$, according to formula

$Q_{3\alpha}=\sqrt{\sum\limits_{i\neq{j}}E_{\alpha_{i}}E_{\alpha_{j}}-P_{\alpha_{i}}P_{\alpha_{j}}cos\Theta_{2\alpha}}-3m_{\alpha}$

,where $E_{\alpha}$ and $P_{\alpha}$ are energy and momentum values of the $\alpha$-particles $i$ and $j$, $\Theta_{2\alpha}$ is the angle of separation between them, $m_{\alpha}$ is the mass of the $\alpha$-particle; $P_{\alpha}=4P_{0}$, where $P_{0}$ is the momentum per nucleon of incident nuclei.

The foundations of required methods of measurements on microscopes in exposure NTE layers were laid at the beginning of studies on the physics of cosmic rays 7 and, then, used widely beams of relativistic nuclei became available. For these purposes microscopes KSM-1 manufactured by $\mathsf{Carl}$ $\mathsf{Zeiss}$ (Jena) about half of century and still functioning well are applied in JINR. Each microscope is equipped with apochromatic and achromatic lenses providing an increase of 15x and 50x, two eyepieces 12.5x and a tube lens 2x which together give an image magnification 375$-$1250x. The maximum error due to manufacturing tolerances does not exceed 0.05 $\mu$m. Further, samples of 99 events 10B $\to$ 2He + H at 1.6 $A$ GeV/$c$ and 212 events 11C $\to$ 2He + 2H at 2.0 $A$ GeV/$c$ are used to describe in brief the procedure of coordinate measurements as a key aspect of the ongoing HS search.

When an event in an emulsion plate is found it is fixed on a microscope stage in such a way that direction of a beam track coincides with direction of lengthwise movement of a microscope stage with an accuracy of 0.1 $-$ 0.2 $\mu$m over a 1mm length. All measurements are carried out in 3D-geometry in a Cartesian coordinate system associated with a microscope. Coordinates x, y, z of 10 points at 1mm paths are measured over the primary and fragment tracks. NTE shrinkage due to development has to be taken into account. The found coordinates are linearly approximated to derive the dip and planar angles ($\alpha$ and $\varphi$) for the beam and secondary tracks. The primary track angles are used to transit for the fragment tracks to a coordinate system associated with this track.

Fig 2 shows an outline a reconstructed event 10B $\to$ 2He + H. The distributions of the He and H tracks over the angles $\alpha$ and $\varphi$ are shown in Fig. 3. They are characterized by mean values (RMS) $\langle\alpha_{He}\rangle$ = $-$0.7 $\pm$ 0.9 (24) mrad and $\langle\varphi_{He}\rangle$ = 0.2 $\pm$ 0.8 (21) mrad, for the nuclei H – $\langle\alpha_{H}\rangle$ = $-$0.1 $\pm$ 2.0 (37) mrad and $\langle\varphi_{H}\rangle$ = $-$0.7 $\pm$ 2.1 (36) mrad. Mean values (RMS) of angular errors are $\langle\Delta\alpha\rangle$ = 0.08 $\pm$ 0.02 (0.8) mrad and $\langle\Delta\varphi\rangle$ = 0.06 $\pm$ 0.01 (0.4) mrad (Fig. 4). Scattering of the coordinate fitting residuals $\Delta$y and $\Delta$z (Fig. 5) differs about 4 times and doesn’t exceed more than 2 $-$ 3 times a track thickness. A dip coordinate accuracy is less than planar one due to noise generated by vertical displacements of a microscope tube, shrinkage coefficient and vertical NTE distortions during development.

Resulting reconstruction of values $Q_{2\alpha p}$ and $Q_{2\alpha}$ for the 10B and 11C fragmentation is presented in Fig. 6 in the range which is relevant for 9B ($Q_{2\alpha p}$ $<$ 400 keV). In these cases the 9B decays serves as source of 8Be 2 . Both distributions are similar. Their mean values (at RMS) $\langle Q_{2\alpha p}\rangle$ = 265 $\pm$ 14 (100) keV and $\langle Q_{2\alpha}\rangle$ = 91 $\pm$ 7 (53) keV match the accepted values 6 and expected resolution. Thus, condition $Q_{2\alpha}$ $<$ 200 keV is a practical cut-off for 8Be identification.

Angular measurements in 12C exposure

The current material for the HS search is a set 200 $\mu$m NTE pellicles on 2mm glass of size 9$-$12 cm which is irradiated longitudinally 12C nuclei at initial momentum $P_{0}$=1 $A$ GeV/$c$. This exposure was performed recently in the medical-biological beam of the Institute of High Energy Physics (Protvino). This 12C beam has energy of about 400 $A$ MeV and used for medical and biological studies. 2% irradiation homogeneity is provided by application of two rotating electrostatic wobblers. The steps taken in December 2016 and April 2017 resulted in the controllable irradiation with a particle density at the area of irradiation of 2000$-$4500 nuclei/cm2. Accelerated search for 3$\alpha$-events the developed pellicles is carried out by scanning along bands that are transverse to the beam direction. By May 2018, 86 events of 12C $\to$ 3$\alpha$ at 1 $A$ GeV/$c$, including 36 “white” stars, are founded and measured following the described procedure.

Besides, measurements made in the 90s in NTE layers exposed to 12C beam at momentum $P_{0}$ = 4.5 $A$ GeV/$c$ at the JINR Synchrophasotron are available for 72 (G.M. Chernov’s group, Tashkent) 8 and 114 “white” stars 12C $\to$ 3$\alpha$ (A.Sh. Gaitinov’s group, Alma-Ata) as a legacy of the emulsion community. At that time, the HS problem was not set. Fig. 4 shows jointly distributions of $\alpha$-particles at both momentum values over the polar emission angle $\theta_{\alpha}$. They are described by the Rayleigh distribution with the parameters $\sigma_{\theta_{\alpha}}$ equal to 27 $\pm$ 3 (1.0 $A$ GeV/$c$) and 6.5 $\pm$ 0.6 (4.5 $A$ GeV/$c$) corresponding to a simple inverse relationship between $P_{0}$ and $\sigma_{\theta_{\alpha}}$. In addition, Fig. 7 shows data on He fragments for the 2.0 $A$ GeV/$c$ 11C dissociation where the 4He isotope dominates.

The $Q_{2\alpha}$ distributions obtained on a basis of angular measurements of events 12C $\to$ 3$\alpha$ at two values $P_{0}$ are presented jointly in Fig. 8. Both are distributions do not differ within statistics. The region $Q_{2\alpha}$ $<$ 200 keV contains a peak pressed to the origin which corresponds to decays of 8Be. Although the 8Be signal is present the $Q_{2\alpha}$ distribution appears to be significantly wider than in Fig. 6(b).

In the $Q_{3\alpha}$ distribution over the invariant mass of the $\alpha$-triples (Fig. 9) there is a peak in the region $Q_{3\alpha}$ $<$ 1 MeV where HS decays could be reflected. For events at 4.5 $A$ GeV/$c$ the mean value for the events at the peak $\langle Q_{3\alpha}\rangle$ (at RMS) is 441 $\pm$ 34 (190) keV, and at 1 $A$ GeV/$c$, respectively, 346 $\pm$ 28 (85) keV. According to the “soft” condition $Q_{3\alpha}$ $<$ 1 MeV in the 4.5 $A$ GeV/$c$ exposure 30 (of 186) events can be attributed to HS and 9 (of 86) including 5 “white” stars (of 36) in 1 $A$ GeV/$c$ exposure.

When selecting $\alpha$-pairs from $\alpha$-triples that correspond to the HS criterion $Q_{3\alpha}$ $<$ 1 MeV the $Q_{2\alpha}$ distribution acquires the form shown in Fig. 10. The average value $\langle Q_{2\alpha}\rangle$ (RMS) is 210 $\pm$ 15 (156) keV. The distribution form becomes wider and separation of the 8Be peak in the region $Q_{2\alpha}$ $<$ 200 keV is impossible. This change is caused by the increased contribution of non-8Be-resonance $\alpha$ pairs of HS decays masking the 8Be signal. In turn, this circumstance makes unattainable a more detailed analysis of the HS inner structure. It characterizes a limitation of our approach to penetrate in the HS structure. Nevertheless, it is concluded that HS is observed in a relativistic dissociation 12C $\to$ 3$\alpha$ with probability about 10-15%.

The angular measurements make it possible to conclude about the dynamics of the HS appearance according to the distribution of $\alpha$-particle triples over their total transverse momentum $P_{Tsum}$ (Fig. 11). Its average value $\langle P_{Tsum}\rangle$ (RMS) is equal to 190 $\pm$ 19 (118) MeV/$c$ corresponding to the nuclear-diffraction mechanism. In the case of electromagnetic dissociation on Ag and Br nuclei composing NTE the limitation is expected to be $P_{Tsum}$ $<$ 100 MeV/$c$ 9 . It is surprising that such a “fragile” formation of three $\alpha$-particles as HS can arise in relativistic collisions as an ensemble which is “bouncing off” with the transverse momentum $P_{Tsum}$ characteristic for strong interactions rather than electromagnetic ones. It can be assumed that increased statistics allow registration of the HS formation outside the angular cone of fragmentation of the parent nucleus. Events of this kind were observed in the cases 9Be $\to$ 8Be and 10C $\to$ 9B. Such observations would clearly demonstrate HS as a holistic and long-lived nuclear-molecular state.

Conclusion

In dissociation 12C $\to$ 3$\alpha$ at 4.5 and 1 $A$ GeV/$c$ in nuclear track emulsion production of the Hoyle’s-state is identified by using approximate invariant mass representation. Contribution of HS is estimated to be about 10-15%. This conclusion is grounded on the basis of the most precise angular measurements performed by three research groups in two exposures at two momentum values that are separated in time by two decades. By itself, this finding demonstrates the thoroughness of the NTE technique.

However, the NTE grounded approach doesn’t allow one to address the features of the HS decay. Nevertheless reconstruction of HS in NTE by the invariant mass of relativistic $\alpha$-triples can be applied to study processes with the HS formation as a wholesome relativistic object at large moment transfers. It is possible that a HS wouldn’t be limited only as the 12C excitation but can manifest itself similarly to 8Be as a universal object in fragmentation of heavier nuclei. In this respect, the closest source to verify such a assumption is the 14N nucleus. The 13N and 13C nuclei whose beams can be formed in the 14N fragmentation are even more convenient in this respect.

Reference