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\textbf{Search for the four-neutron quasi-resonance state populated in the $^2$H($^8$He,$^6$Li)4n reaction studied with the detection of the $^6$Li recoil nuclei.} \\

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\section{Introduction}

The search for the the multineutron systems is one of the most attractive fields of modern nuclear physics.
The first suggestion about the stability of such systems was made in works \cite{Zeldovich:1960,Goldansky:1960}, but multiple experimental attempts of search for bound states of the neutron clusters (e.g. 2n in Ref.\ \cite{WILLARD1964339}, 3n in Ref.\ \cite{belozyorov:1988}, 4n in Ref.\ \cite{Marques_PhysRev:2002,Kisamori:2016}) were unsuccessful.
Nevertheless, the issue of bound neutron nuclei existence is still addressed in the modern theoretical works, see Ref.\ \cite{Pieper:2003,Timofeyuk:2003b,Higgins:2021}.
The recently published work \cite{Duer:2022} reported the observation of the resonance at 2.37\,MeV  with $\Gamma=1.75$\,MeV, which was interpreted as a tetraneutron state produced in a high-energy knockout of the alpha core from the $^8$He beam nuclei.
The ensuing theoretical work \cite{Lazauskas:2023} provides the possible realistic explanation of the observed phenomenon using the model based on a transition between initial and final state of four studied neutrons.
The published in Ref.\ \cite{Duer:2022} results are undoubtedly convincing but the used reaction of knockout of the alpha core from the $^8$He was studied only at very backward angles.
Moreover, authors summarized that the obtained results are limited by the single approach of four-neutron system production and do not describe the correlation of the component neutrons.
This work is dedicated to the results on the $^2$H($^8$He,$^6$Li) reaction at the ACCULINNA-2 fragment separator in the experiment dedicated to the study of the $^7$H resonant states populated in the $^2$H($^8$He,$^7$H) reaction. 


\section{Experiment}


ACCULINNA-2 facility, FLNR, JINR, produced the 26\,AMeV $^{8}$He beam with intensity $\approx10^{5}$\,pps.
This beam was focused on the cryogenic gaseous deuterium 27\,K target with a temperature of 27\,K equipped with the stainless-steel and mylar windows.
The target thickness was 3.7$\times$10$^{20}$ cm$^{-2}$. 
Initially, the experiment was dedicated to the $^{7}$H studies, see Ref.\ \cite{Muzalevskii:2021}, but the detection system allowed to measure the product of the $^2$H($^8$He,$^6$Li)4n reaction and the further multibody decay of the produced unbound system.
The employed detector system is widely described in Ref.\ \cite{Muzalevskii:2021}.

The beam particles were identified by two plastic scintillators, which allowed to measure the energy of the projectile from its time-of-flight (ToF). 
%The particle identification (PID) of the beam was performed by the dE-ToF 
%For the beam particle identification and its energy determination two plastic scintillators were used. 
The motion vector of were reconstructed by two pairs of multi-wire proportional chambers. 
The special run with the empty target cell was conducted to estimate the background conditions, which had $\approx$16\% of the total $^8$He beam time.

%-------------------------------------------------------------------------------
\begin{figure}
	\centering
	\begin{tabular}{cc}			
		\includegraphics[width=0.51\linewidth]{figures/deE_side} 
		&
		\includegraphics[width=0.49\linewidth]{figures/tof} \\
		(a) & (b)
	\end{tabular}
	\caption{
		(a) Identification of $^{6}$Li recoil nuclei by $\Delta E$-$E$ method in side telescopes.
		(b) The ToF distribution obtained for the stilbene-array signals triggered by the $^6$Li recoils.
		The set of three peaks below ToF = 1-ns corresponds to the gamma rays produced in the diaphragm installed 20 cm upstream the target plane, target frame, and CsI(Tl) array which was in use in experiments \cite{Bezbakh:2018}.
		The green and red-line histograms are formed by the events, identified as gamma/neutron by the dE-TAC method.
	}
	\label{fig:ND_id}
\end{figure}
%-------------------------------------------------------------------------------

The detection of the charged reaction products was realized by the $\Delta E$-$E-E$ telescopes.
The recoil $^6$Li nuclei, appearing in the $^2$H($^8$He,$^6$Li) reaction hit the array of four identical $\Delta E$-$E-E$ telescopes.
The telescope array was located 179 mm downstream the target.
Each telescope consisted of three layers of silicon strip detectors (SSDs).
The 20-$\mu$m-thick SSD with a sensitive area of 50$\times$50 mm$^2$ was divided into 16 strips, the second and the third layers were created by the two identical 1 mm-thick SSDs (60$\times$60 mm$^2$ with 16 strips). 
The $^6$Li recoiles emitted from the deuterium gas target in the $^2$H($^8$He,$^6$Li) reaction in a range 6-24 degrees in the laboratory system were detected by this telescope array with probability rising from about 10\% obtained at the small $\theta_{\text{lab}}$ to about 50\% at the larger angles. 
The signals obtained from these telescope detectors allowed to identify $^6$Li with clear separation from other registered lithium isotopes.
Shown in Fig.\,\ref{fig:exp-deltaee-3he} is the typical identification plot obtained for Z = 1, 2, and 3 recoil nuclei in one of the four side silicon telescopes.
%The side assembly of three (20\,$\mu$m, 1\,mm and 1\,mm thick) silicon strip detector (SSD) telescope, and the front telescope made of the 1.5\,mm double side SSD coupled to the CsI(Tl) scintillator array.
%The thin, 20\,$\mu$m detectors in the side telescopes allowed one to reliably identify and reconstruct low-energy particles (the recoil $^4$He nuclei with energy $\ge$5\,MeV), see Ref.\ \cite{Muzalevskii:2020}, emitted from the target in the laboratory angular range between $8^{\circ}$ and $26^{\circ}$.
%The first one covered the laboratory angular range between $8^{\circ}$ and $26^{\circ}$ and allowed to identify and reconstruct low-energy particles, see Ref.\ \cite{Muzalevskii:2020}.
%The front telescope covered angles $\leq9^{\circ}$ and was used to measure the high-energy particles (tritons with energy up to 160\,MeV), stopping them in the CsI(Tl) crystal.



%-------------------------------------------------------------------------------
%\begin{figure}
%	\begin{center}
%		\includegraphics[width=0.49\textwidth]{figures/deE_side}
%	\end{center}
	%
%	\caption{
%		Identification of $^{6}$Li recoil nuclei by $\Delta E$-$E$ method in side telescopes.
%	}
	%
%	\label{fig:exp-deltaee-3he}
%\end{figure}
%-------------------------------------------------------------------------------

The group of four neutrons remaining free as a result of the $\alpha$-core removal from the $^8$He projectile.
Neutron were detected by the time-of-flight (ToF) stilbene modules \cite{Bezbakh:2018}%, which provides clear neutron-gamma separation and allow to calculate the particle energy from its ToF.
The setup included 48 stilbene scintilator crystals placed on a 0.7$\times$1.1 mm$^2$ area moved 2\,m back from the target at zero angle to the $^8$He beam axis. 
The distance between the 50-mm thick and 80\,mm diameter stilbene crystals was approximately
12\,cm which resulted in a $\sim$30\% probability for neutrons to hit a stilbene detector. 
The stilbene array provided 4.5\% energy resolution and the single neutron registration efficiency
of $\approx$15\%. 
The probability of a neutron registration in coincidence with the $^{6}$Li recoil  was around 10\%, taking into account that four neutrons are flying forward, towards the stilbene array, in each case when this recoil is detected.



%The neutron identification was realized 

\section{Results}

Important part of the $^2$H($^8$He,$^6$Li)4n reaction analysis was neutron identification and reconstruction.
%The dE-TAC correlation presented Fig.\,\ref{fig:ND_id} (a) shows that the signals produced by gamma and neutron interaction with detector material are well separated.
The neutron-gamma separation was made by means of the Pulse Shape Discrimination (PSD) \cite{MOSZYNSKI1994226}. 
The selection each of them on the ToF distribution, see Fig.\,\ref{fig:tof} leads to the suggestion that some gamma-type signals correspond to neutron-like ToF.
These gammas are produced by the interaction of the neutrons with the stilbene housing and should be considered as neutron events.


%-------------------------------------------------------------------------------
%\begin{figure}
%	\begin{center}
%		\includegraphics[width=0.49\textwidth]{figures/tof}
%	\end{center}
	%
%	\caption{
%		The ToF distribution obtained for the stilbene-array signals triggered by the $^6$Li recoils.
%		The set of three peaks below ToF = 1-ns corresponds to the gamma rays produced in the %diaphragm installed 20 cm upstream the target plane, target frame, and CsI(Tl) array which was in use in experiments \cite{Bezbakh:2018}.
%		The green and red-line histograms are formed by the events, identified as gamma/neutron by the dE-TAC method.
%	}
	%
%	\label{fig:tof}
%\end{figure}
%-------------------------------------------------------------------------------

Tetranuetron was reconstructed from the recoil $^6$Li as a missing component in the $^2$H($^8$He,$^6$Li) reaction.
The total number of $^6$Li-neutron coincidences found in the recorded data was 136.
%But, due to the kinematic selection, only 108 of these events were identified as the population of 4n.
%The correlation we used for this selection is shown in Fig.\ \ref{fig:mm_4n} (b).
The data points of these events are presented in Fig.\ \ref{fig:mm_4n} (a), where the neutron kinetic energy $E_n$(4n c.m.s.) in the 4n center-of-mass frame is compared with the reconstructed missing-mass (MM) energy $E_T$ of the 4n group. .
The fact that the magority of these events (108 data points out of 136) are located below the kinematic border proves the good channel identification of the studied reaction.
Another evidence for the 4n population is that the data collected with the empty target has zero events satisfying the used selections.

The MM spectrum of 4n group, occuring in the $^2$H($^8$He,$^6$Li)4n reaction, was reconstracted from the measured $^6$Li recoil energy and emission angle, taking the 108 events with the data points lying below the kinematical border $E_{n} < $3/4$E_T$(4n) reconstructed after kinematic selection.
This spectrum is shown in Fig.\ \ref{fig:mm_4n} (b).
Obviously, the obtained spectrum shape can not be described just as a contribution of the 4-body phase space volume, see the orange dotted curve in Fig.\ \ref{fig:mm_4n} (b).
Apparently, one can assume the presence of some resonance-like state (or states) lying at 3-4\,MeV above the 4n decay threshold.
It is notable that the observed peak energy is consistent with the reported in Ref.\ \cite{Duer:2022}  observation of the correlated free four-neutron system.
%Its important evidence for the tetraneutron resonance at low energies above the 4n decay threshold obtained at forward angles in the reaction of alpha core knockout from the $^8$He beam.

%-------------------------------------------------------------------------------
\begin{figure}
	\centering
	\begin{tabular}{cc}			
		\includegraphics[width=0.518\linewidth]{figures/triangle} 
		&
		\includegraphics[width=0.482\linewidth]{figures/mm_spec} \\
		(a) & (b)
	\end{tabular}
	\caption{
		(a) Correlation between the neutron energy in the 4n frame and the 4n decay energy.
		The kinematical border showing the $E_{n} < 3/4 E_T (4$n) relation is shown with the red line.	
		(b) The 4n MM spectrum projected from red events in (a) by using the kinematical condition.	
		The orange dotted curve illustrates the 4-body phase volume $\sim E_T^{7/2}$.
	}
	\label{fig:mm_4n}
\end{figure}
%-------------------------------------------------------------------------------


\section{Conclusion}

The $^2$H($^8$He,$^6$Li)4n reaction was studied at the center-of-mass forward angles  $\theta_{\text{cm}}$ = 10-50 degrees.
The observed peak at 3-4 MeV above the 4n threshold is in agreement with the correlated free four-neutron system observation made in Ref.\ \cite{Duer:2022}.
Cross-section value $\sim$10$^{-28}$ cm$^2$/sr is estimated for the reaction resulting in the population of this peak.
%The obtained results are pushing the investigation of the neutron clusters.
The choice of the detector setup made specifically for the study of such reaction and is demanded in the future experiments.
In particular, the neutron detection efficiency will be increased drastically by the use of the detector array presented in Ref.\ \cite{Bezbakh:2023}.
The latter will increase statistics and will allow to study neutron cluster decays in full kinematics reconstructed by the invariant mass methods. 


\section{Acknowledgement}

%Institute of Physics of Silesian University in Opava 
We acknowledge the principal support of this work by the Russian Science
Foundation grant No.\ 22-12-00054. The authors are grateful to
Profs.\ Yu.Ts.\ Oganessian and S.N.\ Dmitriev for the long-term support and
development of this activity. We acknowledge important contribution of Prof.\
M.S.\ Golovkov to the development of the experimental methods and useful
discussions. Also, authors express their gratitude to the
acceleration team for the stable work of U-400M cyclotron during all runs.

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