The He and He spectra studied in the , reaction
The low-lying spectra of He and He nuclei were studied in the H(He,)He and H(He,)He transfer reactions. The ground state (g.s.) of He and excited states, at MeV and at MeV, were populated with cross sections of 200, , and b/sr, respectively. Some evidence for He state at about 7.5 MeV is obtained. We discuss a possible nature of the near-threshold anomaly above 2.14 MeV in He and relate it to the population of a continuum (soft dipole excitation) with peak value at about 3 MeV. The lowest energy group of events in the He spectrum was observed at MeV with a cross section of b/sr. We argue that this result is possibly consistent with the previously reported observation of He, in that case providing a new g.s. position for He at about 3 MeV.
keywords:He, He beams, tritium gas target, resonance states, hyperspherical harmonic method, soft dipole mode, neutron halo.
Pacs:25.10.+s – Nuclear reactions involving few-nucleon systems, 24.50.+g – Direct reactions, 25.55.Hp – H, He, He induced reactions; transfer reactions, 25.60.Ge – Reactions induced by unstable nuclei; transfer reactions, 21.60.Gx – Cluster models.
To study drip-line nuclei with large neutron excess one should either transfer neutrons or remove protons or make multi-nucleon charge-exchange. Two-neutron transfer from tritium provides here important opportunities connected with the simplicity of reaction mechanism and simplicity of recoil particle (proton) registration. This class of reactions remains practically not exploited in the radioactive beam research. Availability of the unique cryogenic tritium target  in the Flerov Laboratory of Nuclear Reactions (JINR, Dubna) makes possible systematic studies of these reactions. The effectiveness of such an approach in the investigation of exotic nuclei was demonstrated in the recent studies of the H system [2, 3].
Although He has been discovered more than a decade ago , very limited information on this system is available. The ground state properties were found in the H(Li,He) reaction as , MeV , and in the Be(C,O)He reaction as , MeV . Here and below denotes the energy relative to the lowest breakup threshold for the systems, while denotes the excitation energy.
The He g.s. was theoretically predicted  to be a narrow three-body He++ resonance with MeV, MeV and the valence neutrons populating mainly the configuration. A widely discussed shell inversion phenomenon in the nuclei became the source of new interest to He. Possible existence of a virtual state in He was demonstrated in Ref.  and an upper limit fm was established for the scattering length. Following this finding, the existence of a narrow near-threshold state in He (, MeV) with a structure was predicted in Ref.  in addition to the state. It was suggested in  that the ground state of He had not been observed so far and the resonance at MeV is actually the first excited state. The low-lying spectrum of He was revised in the recent experiment  resulting in a higher, than in the previous studies, position of the state (experiment  provided unique spin-parity identification for the He states below 5 MeV). The presence of the contribution is evident in the data , but the exact nature of this contribution (virtual state or nonresonant -wave continuum) was not clarified and only a lower limit fm was set in this work. This work triggered further theoretical research: problems with the interpretation of the He spectrum and controversy between the He and He data were demonstrated in Ref. .
This intriguing situation inspired us to revisit the He issue. The study of the H(He,)He reaction was accompanied by the study of the H(He,)He reaction providing a reference case of the relatively well investigated He system.
2 Experimental setup
Experiments were performed using a 34 MeV/amu primary beam of B delivered by the JINR U-400M cyclotron. The secondary beams of He and He nuclei were produced by the separator ACCULINNA  and focused in a 20 mm spot on the target cell. For safety reasons, the main target cell, filled with 900 mPa tritium gas and cooled down to 28 K, was inserted into an evacuated protective box. Thus, the target had twin entrance and exit windows sealed with 12.7 m stainless steel foils. For 4 mm distance between the inner entrance and exit windows the thickness of the tritium target was cm. Typical beam intensities incident on the target were s for the He and s for the He projectile nuclei. The admixtures of other particles in the beams were no more than and the beam diagnostics completely eliminated them.
Experimental setup and kinematical diagram for the H(He,)He and H(He,)He reactions are shown in Fig. 1. For the small centre-of-mass system (cms) angles, where the maximal cross section is expected, the protons fly in backward direction in the lab system. The residuals (He and He) and their decay products (He and He) are moving in a relatively narrow angular cone in forward direction. Protons escaping back from the target hit a telescope consisting of one 300 m and one 1 mm thick annular Si detectors. The active areas of these detectors had the outer and inner diameters of 82 mm and 32 mm, respectively. The proton telescope was installed 100 mm upstream of the target and covered an angular range of in lab system. The first detector was segmented in 16 rings on one side and 16 sectors on the other side and the second, 1 mm detector was not segmented. A veto detector was installed upstream of the proton telescope to alert to the signals generated by the beam halo.
Zero angle telescope for the He and He detection was installed on the beam axis at a distance of 36.5 cm in the case of the He beam and at 28.8 cm in the experiment with the He beam. The telescope included six squared ( mm) 1 mm thick detectors. The first two detectors of the telescope were segmented in 16 strips each in vertical and horizontal directions. All other detectors in the telescope were segmented in 4 strips in the He run and in 16 strips in the He run.
A set of beam detectors was installed upstream of the veto detector (not shown in Fig. 1). Two 0.5 mm plastic scintillators placed on a 8 m base provided the particle identification and projectile energy measurement. The overall time resolution was 0.5 ns. Beam tracking, giving a 1.5 mm resolution for the target hit position, was made by two multiwire chambers installed 26 and 80 cm upstream of the target.
Particle identification in the proton telescope was not imperative because, due to kinematical constraints, nothing but protons could be emitted in the backward direction in these reactions. The main background source were protons originating from the interactions of beam nuclei with the target windows. Test irradiations done with empty target showed that this background was almost completely eliminated when -He or/and -He coincidences were considered. In the case of the H(He,)He reaction the detection of the -He coincidence events granted a selection for the reaction channel populating the He g.s. For the decays of He and excited He nuclei the respective -He and -He coincidence information was used to clean the missing mass spectra and reconstruct the charged fragment energy in the cms of He or He.
Array of 48 detector modules of the neutron time-of-flight spectrometer DEMON  was installed in the forward direction at a distance of 3.1 m from the target. In more rare events where triple -He- coincidences were detected the complete reaction kinematics was reconstructed.
For the He and He beams the projectile energies in the middle of the tritium target were on average about 25 MeV/amu and 27.4 MeV/amu, respectively; integral fluxes and were collected. The missing mass spectra of He and He were measured up to 14 MeV and 16 MeV, respectively. The upper limits were set by the low-energy proton detection threshold. Monte Carlo (MC) simulations taking into account the details of these experiments showed that a 450 keV (FWHM) resolution was inherent to the He and He missing mass energy spectra obtained from the data. The precision of the beam energy measurement made the most important contribution to the error of the missing mass.
3 H(He,)He reaction
Missing mass spectra of He from the H(He,)He reaction are presented in Fig. 2. The peak corresponding to the He g.s. is well seen in the -He coincidence data. The tail visible in Fig. 2 (a) on the right side of the g.s. peak was caused by the pile-ups in the second (non-segmented) detector. Protons emitted from the target with energy MeV correspond to the g.s. peak of He. They passed through the 300 m Si detector and were stopped in the second (1 mm) detector of the proton telescope. The background signals arose here from the beam halo particles [count rate of s]. The veto detector allowed taking away these events in the data analysis but the energy resolution of the second detector was deteriorated. Operation conditions were much better for the segmented 300 m detector. The count rate per any of its sectors was at least 10 times lower. Consequently, the background signals did not cause the resolution deterioration when the -He coincidences were detected. In that case protons with energy MeV were emitted from the target and practically all of them were stopped in the 300 m detector. Therefore, for the He excited states the stated 450 keV resolution is valid.
There are two peaks apparent in the He excitation spectrum. We assign 2 to the He resonance at excitation energy MeV. The 2 resonance with energy 3.570.12 MeV and width =0.50.35 MeV was for the first time unambiguously, and with that good precision, obtained in Ref. . Later on, this resonance was reported in a number of papers with energies close to 3.6 MeV and widths MeV (see, e.g., [14, 15, and Refs. therein]). We assume that the MeV peak seen in Fig. 2 is the resonance of He. The ground for this assumption comes from various theoretical results (e.g. [16, 17, 18]) stably predicting that in the He excitation spectrum the next state after the should be the state. We note that evidence for the peak at MeV was found in Ref. . The He excited state at 5.4 MeV was recently reported also in Ref. . A rapid rise of the He spectrum at the He++ decay threshold is seen in Fig. 2. This rise cannot be explained by the left “wing” of the resonance. The peculiar threshold behaviour is discussed in Section 6. We note also that the spectra in Fig. 2 show some evidence for a He state at E7.5 MeV.
In the H(He,)He reaction the population cross section for the He g.s., averaged in a range of of the reaction cms, is found to be b/sr. The observed threshold anomaly makes the cross section derivation for the excited states of He more complicated (and model dependent). The cross sections for the excited states are further discussed in Sections 5 and 6.
4 H(He,)He reaction
Data obtained for the H(He,)He reaction are shown in Fig. 3 (a) as a scatter plot He) vs. , where He) is the energy of He in the He cms. Condition He should be valid for the He decay. Therefore, He events should be below the boundary shown by the dashed line in the scatter plot of Fig. 3 (a). The shaded area in Fig. 3 (a) extends this boundary accounting for the experimental resolution. One can see that practically all the events presented in Fig. 3 (a) fall into the He locus indicating very clean background conditions. The missing mass spectrum in Fig. 3 (b) was obtained projecting the events confined in the He locus.
Not a single event was detected in the He spectrum below 2.5 MeV. This imposes a stringent (one count corresponds to 14 b/sr) limit on the population cross section in the expected He ground state region at about 1.2 MeV . The lowest energy feature in the He spectrum is a group of 10 events in between 2.5 and 5.5 MeV [see Fig. 3]. This MeV group is well isolated from the rest of the spectrum and has a typical resonant cross section ( b/sr averaged for cms angles ), see estimates in Section 5. Also, this group has a distinct feature: the energy distribution of the He fragments obtained in the He cms appears to be different from that in the rest of events in the He spectrum. One can see in Fig. 3 (a) that within this group the He) energies are around the maximal possible value. This means that the relative energy of the decay neutrons for such events tends to zero. This could be evidence for some strong specific momentum correlations or/and strong - final state interaction in this part of the He spectrum. We think that the MeV group of events represents a resonant state for He; the possible nature of this state is discussed in Section 7.
5 Cross section estimates
Both the He and He states were populated in our experiments by the same “dineutron” transfer in the same kinematical conditions and, presumably, by the same direct reaction mechanism. This fact makes it very probable that spectroscopic information can be extracted from the cross sections in a straightforward way. For theoretical estimates of the spectroscopic factors we used somewhat extended phenomenological Cluster Oscillator Shell Model Approximation (COSMA) of Ref. . Within this model the g.s. wave functions (WF) of the He isotopes can be written as
The schematic notation denotes the Slater determinant of neutrons occupying orbital projected on the total spin and normalized. The -particle is considered to be an inert core and it is omitted in the notation. In the original paper  only the configuration in Eq. (1) was considered.
The model looks very schematic. However, it lists all the possible -shell configurations, representing the dominant part of the WF. Particularly, for the He g.s. coefficients , can be inferred from the three-cluster model calculations 
exhausting of the WF normalization (the corresponding of and of components are considered). The simplified He WF can also be used with only configuration (, ) to test the sensitivity to the He structure. Assuming the He WF (1) is normalized, we end up with only one unknown parameter in the model.
The cluster overlaps for the He WFs within this model are:
Using spin algebra and Talmi coefficients, the overlaps of the shell model configurations with the “dineutron” being in the -wave motion relative to the core are obtained as
Dineutron here is the the two neutrons with angular momentum and total spin equal to zero represented by minimal oscillator. The spectroscopic weight of the He g.s. configuration in the He WF is obtained by Eq. (2) as
For the reactions studied in this work a reasonable estimate of the cross section ratio for the He and He g.s. population is the ratio of the dineutron spectroscopic factors. They are found as
The spectroscopic information obtained in the model is illustrated by Fig. 4. In the region the cross section ratio is changing dramatically [Fig. 4 (c)]. However, this region is presumably unphysical. In this region the weight of the dineutron configuration in He is minimal [Fig. 4 (b)] and the weight of the He g.s. configuration in He is minimal as well [Fig. 4 (a)]. These configurations are expected to be maximized by the variational procedure as they are energetically highly preferable. Simple heuristic considerations show that the coefficient should be confined by condition [to maximize attractive interaction] and sign [to maximize pairing].
For a reasonable weight of coefficient (for example, sign) the population of the state in He is expected to be larger than the state.
Population cross section for the He state can not differ strongly from that obtained for the He g.s. For the realistic structure of He the values lying in a range of are expected.
Population rate for the 3 MeV group of events in He is found consistent with the resonant cross section estimated for the population of the -wave state. Coefficient can be obtained from the experimentally measured cross sections for the population of He and He ground states: . In this work such a derivation is methodologically clean as both cross sections are obtained in the same experimental conditions.
Note that the model proposed here (with neutrons situated only in the -shell) shows that the basic dynamics of the He system strongly limits the possible range of the He g.s. configuration weight in the He structure [see Fig. 4 (a)]. This implies that the weights corresponding to the He g.s. and He configurations in the structure of He have only a weak dependence on the and configuration mixing.
The spectroscopic factor for processes with the disintegration of He in He(g.s)+2n continuum is connected with the weight in Eq. (3) by relation
A discrepancy can be seen in Ref.  between the experimentally obtained and theoretical “shell model” value given as 1/6 (see Table 1 in ). The values obtained in our model vary between 0.8 and 1.1 (depending on the value) in a good agreement with the experiment of Ref. .
6 Possible nature of the threshold state in He
In the missing mass spectrum of He (see Fig. 2) attention is attracted by a steep rise ensuing straight from the three-body He++ threshold. The lowest known resonant state of He is at MeV , MeV. It decays sequentially via the He ground state resonance ( at MeV, MeV) by a -wave neutron emission. This guarantees negligible population of the continuum below MeV where decay takes place in a “three-body regime”, . Above that energy, population probability transfers to the “two-body -wave regime”, . Consequently, the low-energy tail of the state can not be responsible for the near threshold events.
The only plausible source of the low-energy events, we have found, is the population of the E1 (means ) continuum. Theoretical studies of such continuum populated in reactions [22, 23, 24] show that the profile of the cross section typically well resembles the profile of the electromagnetic strength function . Such functions for spatially extended halo systems could provide very low-energy peak — the so called soft dipole mode — even without the formation of any resonant state.
We estimate the E1 strength function for the HeHe+ dissociation using the model developed in . For the WF with outgoing asymptotic
generated by the dipole operator , acting on the g.s. WF , the E1 strength function is found as
Vectors and are Jacobi coordinates for the He- and (He-)- subsystems, respectively. Estimating the dipole strength for the light -shell nuclei we can well take into account only the transitions and neglect the interactions and -wave interaction between the core and neutron (unless the latter is not strongly attractive). In this approximation the three-body Green’s function (GF) has a simple analytical form
where is a free motion GF in the subsystem, and the GF in the subsystem corresponds to the -wave continuum with the He g.s. resonance at MeV.
The results of the model calculations, including the He test, are shown in Fig. 5. The estimated He strength function giving peak at about 1.1 MeV is in a reasonable agreement with the complete three-body calculations  giving peak at about 1.3 MeV. It can be seen that the strength function profile in He is sensitive to two main aspects of the dynamics. (i) Energy of the resonance ground state in the -wave subsystem: dashed curve shows that the strength function peak is shifting to the lower energy if the He state is artificially shifted from the experimental MeV position to the lower 0.445 MeV. (ii) “Size” of the ground state WF: dotted curve shows the strength function peak shifting to higher energy if we artificially overbound the He g.s. WF to MeV instead of 0.9 MeV decreasing its radial extent. When we turn from He to He these dynamical trends work in the opposite directions and largely compensate each other (the He g.s. is more “compact” than the He g.s., but the He g.s. resonance is lower than the He g.s. resonance). As a result we find the strength function peak position in He to be somewhat lower than respective position in He. This indicates that in He, where the state is significantly higher than in He, the lowest-energy feature in the continuum could be the excitation.
The behaviour of the cross section with the estimated E1 component taken into account is shown in Fig. 6. The state profile is given here by the standard R-matrix expression for the -wave decay via the He g.s. providing the widths MeV for excitation energies MeV (the reduced width is taken as Wigner limit). Without E1 contribution the data are in agreement with the standard position ( MeV) of the state, but the near threshold behaviour of the cross section can not be reproduced. The population cross section in this case can be estimated as b/sr. The addition of the contribution allows to reproduce the low-energy part of the spectrum much better. In that case we can allow up to feeding to the continuum. Then we get b/sr for the population and have to shift to about MeV the position of this state.
The proposed significant contribution of the cross section is not absolutely unexpected and never seen phenomenon. For example, the experimental spectrum from paper  is shown in the inset to Fig. 6. Inspected around the He++ threshold “on a large scale” it shows the same presence of the low-energy intensity which can not be attributed to the tail of the state. Strong population of the E1 continuum in He by nuclear processes has been demonstrated in a comparison made for the nuclear and Coulomb dissociation data [27, 28]. However, in the interpretation of the data presented in [27, 28] the idea was accepted that the E1 cross section in He should peak at higher energy than in He (maximum at about MeV above the threshold). This idea is based on the argument (ii) discussed above (smaller size of He compared to He); actual situation appears to be more complicated. As a result the authors of [27, 28] have had to position the state below the E1 peak. Consequently, they had to ascribe to it a very low excitation energy 2.9 MeV (compared to about 3.6 MeV in the other recent works). The assumption of the very low-energy soft E1 peak in He would probably allow to explain in a more natural way the data from [27, 28]. Also, there exists a large uncertainty in the definition of the “standard” position of the state in He ( MeV, see Ref. ). We think that a significant component of the disagreement among different experimental works could be connected with the possibility that the state is typically observed in a mixture with the contribution. Correlation measurements could clarify this situation.
7 Interpretation of the He spectrum
There is an evident discrepancy between the group of events at about 3 MeV observed in this experiment and the recognized position of He g.s. at about 1.2 MeV. A possible explanation is that an excited state of He was observed in our work and the ground state was not populated for some reason. We, however, find a different explanation preferable.
There are two important problems, pointed by theoreticians, in the interpretation of the He spectrum. (i) Possible existence of a near-threshold state with the structure, due to the shell inversion phenomenon . In this case we would have two states in the low-energy continuum of He, nearby each other. The state is predicted in  to have very specific properties (tentatively assigned as “three-body virtual state”) and it distorts strongly the higher-lying spectrum associated with the state. At first blush it is not impossible that the state is not populated in our experiment. (ii) Reaction mechanism issue was pointed in Ref. . The most clear observation of the He g.s. was made so far in the experiment with the Li beam . It was shown in  that, in contrast to the typical transfer reactions, the experiments with the Li beam can provide very specific signal for the state: in the Li case the spectrum is shifted downwards due to the abnormal size of the halo component of the Li WF.
Let us consider the second issue first. The measured missing mass spectrum of He is shown in Fig. 7 in comparison with the experimental data  and calculations  taking into account the reaction mechanisms in both cases. It is clear that the calculations are somewhat overbound ( MeV), but otherwise consistent with the data in both cases. It has also been shown in Sec. 5 that the absolute cross section value for the 3 MeV group of events is quantitatively consistent with the population of a -wave state. We can conclude here that it is very probable that the 1.2 MeV peak observed in Ref.  and the 3 MeV peak in our work represent the same state. It should be emphasized that the calculated peak energy for the , reaction cross section is consistent with the resonance properties inferred from the -matrix in : the eigenphase for scattering is passing at about the peak energy. Therefore, the peak energy observed in the transfer reaction could provide a better access to the He properties.
Now we return to the first issue. Is it possible that the theoretically predicted in  low-lying state with the structure exists, but it is not populated in our reaction? It was shown in  that the expected states with the and structures would interfere strongly. The momentum distributions for the state were predicted to be strongly different in the cases when there is a state below it and when there is no such state. This point is illustrated in Fig. 8 (a) for different interactions in the He- -wave channel (the positive values of scattering length indicated for two curves in Fig. 8 (a) imply that repulsive interaction takes place in the -wave state). In Ref.  the cases of fm in He correspond to the formation of extremely sharp near threshold He states. Otherwise, there is only the state at in the He continuum. It can be seen in Fig. 8 (b) that only scattering lengths fm (and hence no state) are qualitatively consistent with our data. Thus the data favour the situation of the ground state of He. In this way our data also indirectly lead to contradiction with the He- scattering length limit fm claimed in Ref. .
The interpretation proposed above is very nonorthodox and is based, at the moment, on the limited statistics data. However, alternatively we face a problem to explain why the “real” ground state was not observed in our experiment despite the very low cross section limit achieved ( b/sr) and the estimates of Section 5 indicating large population probability for possible state.
In this work we studied the He and He spectra in the same transfer reaction. This allowed us, when interpreting the data, to be free in our speculations of the reaction mechanism peculiarities. We think that our results are not in contradiction with the previous works done on these nuclei in the sense of the data, however, making various theoretical estimates we arrived at different conclusions on several issues.
The ground and the excited , states of He are populated with cross sections 200, , and b/sr. The presence of near-threshold events at about MeV can be an evidence for the formation of the soft dipole mode in the He continuum. The generation of such a mode with the very low peak energy ( MeV, MeV) in nuclear reactions could possibly be an explanation to the respective controversial features of the other He data as well.
The population cross section of the 3 MeV peak in He b/sr is consistent with the estimated resonance cross section for the population of the He state with the structure. The weight of the configuration in He was inferred from the ratio.
According to the calculations of Ref.  the 3 MeV peak position obtained here for the He g.s. is in agreement with the 1.2 MeV position found in Ref. , if one takes into account the peculiar reaction mechanism for the Li beam used in . If this interpretation is valid, a new ground state energy of about 3 MeV should be established for He since the peak position obtained in the transfer reaction corresponds to the -matrix pole position, while for reactions with Li there is a strong difference.
Further measurements of a similar style are desirable. This would allow to reveal the potential of correlation measurements for such complicated systems and to resolve the interesting problems outlined in this work.
We are grateful to Profs. B.V. Danilin, S.N. Ershov, and M.V. Zhukov for illuminating discussions. The authors acknowledge the financial support from the INTAS Grants No. 03-51-4496 and No. 05-100000-8272, Russian RFBR Grants Nos. 05-02-16404, 08-02-00089 and 05-02-17535 and Russian Ministry of Industry and Science grant NS-1885.2003.2. Support provided for this work by the Department of Science and Technology of South Africa is acknowledged.
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