First direct mass measurement of the two-neutron halo nucleus {}^{6}Heand improved mass for the four-neutron halo {}^{8}He

# First direct mass-measurement of the two-neutron halo nucleus 6He and improved mass for the four-neutron halo 8He

## Abstract

The first direct mass-measurement of He has been performed with the TITAN Penning trap mass spectrometer at the ISAC facility. In addition, the mass of He was determined with improved precision over our previous measurement. The obtained masses are (He) = 6.018 885 883(57) u and (He) = 8.033 934 44(11) u. The He value shows a deviation from the literature of 4. With these new mass values and the previously measured atomic isotope shifts we obtain charge radii of 2.060(8) fm and 1.959(16) fm for He and He respectively. We present a detailed comparison to nuclear theory for He, including new hyperspherical harmonics results. A correlation plot of the point-proton radius with the two-neutron separation energy demonstrates clearly the importance of three-nucleon forces.

###### pacs:
21.10.Dr 27.20.+n 21.45.-v

Nuclei with exceptionally weak binding lie at the limits of stability and exhibit fascinating phenomena. One of them is the formation of a halo structure of one or more loosely-bound nucleons surrounding a tightly bound core, similar to electrons in atoms. The experimentally best studied cases are the two-neutron halo nuclei He and Li (1). These nuclei are of Borromean nature, where all two-body (two-neutron and neutron-core) subsystems are unbound, but the three-body system is loosely bound (2). Due to a lack of pairing correlations, the neighbouring isotopes of He (He and He) are unbound, while He is again bound with a four-neutron halo. This heaviest helium isotope also marks the nucleus with the most extreme neutron-to-proton ratio ( = 3). Neutron halo nuclei are distinguished by their extended matter radius and a small neutron separation energy compared to other nuclei. The size of their core can be associated with the (root-mean-square) charge radius (its deviation from the halo-free core results from polarization effects due to strong interactions), while the halo extension depends exponentially on the separation energy (3).

To date, charge radii of halo nuclei can be determined only from the measurement of the change in energy of an atomic transition between isotopes and . This so-called isotopic shift is linked to the mean-square charge radius difference by:

 δνA,A′=δνA,A′MS+KFS⋅((r2c)A−(r2c)A′), (1)

where the mass shift and the field shift constant are obtained using atomic structure calculations (4). Both terms need to be known with the same absolute precision. Because of their larger fractional change in mass and their smaller volume, light nuclei have a mass shift term typically times larger than the field shift . Also, the mass shift depends sensitively on the nuclear mass and in order for the mass uncertainty to make a negligible contribution to the charge radius determination of halo nuclei, reliable atomic masses with relative uncertainty on the order of are needed, as for example achieved in (5).

The nuclear charge radii of He have been measured by laser spectroscopy (6); (7). However, to date, the mass of He (8) is determined only from the -value comparison of two nuclear reactions (9) and has never been measured directly. Over the past years, direct Penning-trap mass measurements have uncovered large deviations with indirectly measured masses, while yielding consistent results with other direct mass measurement methods (e.g., the 5 deviation of the Li mass (10)). Hence, a precise and accurate mass measurement of He is highly desirable to update the charge radius analysis.

Understanding and predicting the properties of halo nuclei also presents a theoretical challenge. He and He are the lightest known halo nuclei and, due to their few-nucleon () structure, are amenable to different ab-initio calculations based on microscopic nuclear forces (11); (12); (13); (14); (15). Therefore, they represent an ideal testing ground for nuclear structure theory, leading to a deeper understanding of the strong interaction in neutron-rich systems.

In this Letter, we present the first direct mass measurement of He, together with a more precise value for He, using the TRIUMF Ion Trap for Atomic and Nuclear science (TITAN) (16) Penning trap mass spectrometer. The TITAN facility is situated in the low-energy section of the TRIUMF’s Isotope Separator and ACcelerator (ISAC) experimental hall (17). The He mass was first directly measured in an earlier TITAN experiment (18). Based on the new masses presented here, we determine reliable binding energies, and the resulting values for the charge radii of He and He. These observables provide key tests for nuclear theory. We make a detailed comparison to theory for He, where ab-initio calculations based on different nucleon-nucleon (NN) and three-nucleon (3N) forces are available. To date, no calculation exists based on chiral effective field theory interactions. This approach has the advantage that the corresponding 3N and 4N forces are largely predicted. As a first step towards this goal, we present new ab-initio hyperspherical harmonics results based on chiral low-momentum interactions. A natural correlation between separation energy and radii is found when only NN interactions are included. The results and the precise experimental data clearly illustrate the importance of including 3N forces.

Both radioactive helium isotopes were produced via spallation reaction using 500 MeV protons from the TRIUMF cyclotron at a current of 80 A impinging a high power silicon-carbide target. The beam was ionized using the Forced Electron Beam Ion Arc Discharge (FEBIAD) source (19) and transported at an energy of 20 keV to the TITAN facility. Contamination in both beams was removed using a two-stage high resolving power dipole-magnet mass separator. Upon reaching the TITAN facility, the purified continuous ion beam was thermalized, accumulated and bunched using a hydrogen-filled Radio Frequency Quadrupolar (RFQ) ion trap (20). After their extraction from the RFQ, the ions were transported at an energy of approximately one keV to the Penning trap, where the mass measurement was performed.

The basic principle behind Penning trap mass spectrometry consists of measuring the cyclotron frequency of an ion of mass and charge in a magnetic field . TITAN, like most on-line Penning trap mass spectrometers, uses the Time Of Flight Ion-Cyclotron Resonance (TOF-ICR) technique (21); (22) to determine the ion’s cyclotron frequency (we refer the reader to (23) for more details about mass measurements using the TOF-ICR technique at TITAN).

Typical He and He time-of-flight ion-cyclotron resonances are shown in Fig. 1. These measurements took 27 minutes each and comprised 1656 and 1171 detected ions yielding statistical relative uncertainties on the cyclotron frequencies of 9 and 14 parts per billion (ppb), respectively. For both isotopes, the magnetic field was calibrated by measuring the cyclotron frequency of stable Li produced by the TITAN off-line ion source between the He (or He) cyclotron frequency measurements. From these measurements, one calculates the frequency ratio , which yields the ratio of the masses of the two ions.

A total of 12 He and 17 He frequency ratios where measured and for these measurements, the different sources of systematic errors such as magnetic field inhomogeneities, misalignment with the magnetic field, harmonic distortion of the trap potential, non-harmonic terms in the trapping potential, interaction of multiple ions in the trap, magnetic field time-fluctuations and error due to relativistic effects were considered (see (24) for a detailed analysis and treatment of these effects for the He measurements). The main systematic errors on the He and He cyclotron frequency ratios arise from the interaction of multiple ions in the trap and are found to be 8 and 13 ppb for He and He, respectively. The contributions from the other effects are all below the ppb level and therefore have a negligible contribution to the final uncertainty. The weighted averages of the cyclotron frequency ratios are 0.857 868 442 9(42)82 and 1.145 098 361(7)16 for He and He, respectively (where the statistical uncertainty is given in parenthesis and the total uncertainty in curly brackets).

In mass spectrometry, the quantity of interest is the atomic mass, which is given by , where and are the last electron binding energies of the calibrant ion and of the ion of interest, is the electron mass, and is the calibrant atomic mass.

Using the more precise mass measurement of the calibrant Li from (25), the He mass reported in (18) becomes 8.033 935 67(72) u. The He measurement presented here yields a mass of 8.033 934 40(12) u, which agrees with the previous result within 1.7, with a factor of 12 improvement in precision. Combining these two results, the mass and mass excess of He become 8.033 934 44(11) u and 31 609.72(11) keV. This is within 1.7 of the atomic mass evaluation (AME03) value (8). On the other hand, for the He mass and mass excess we obtain 6.018 885 883(57) u and 17 592.087(54) keV, which deviate from the AME03 by 4 while improving the precision by a factor of 14.

Following the TITAN measurements, the new He and He two-neutron separation energies are 975.46(23) keV and 2125.00(33) keV, respectively. Using the new masses we also computed the charge radii of He following the procedure presented in (7). The TITAN masses enters in the mass shift evaluation obtained from atomic structure calculations (4). These new mass shifts, together with the corresponding isotopic shifts from (7); (6) and updated field shifts are presented in Table 1. The total field shift for He was taken as the weighted average of all transitions and the various systematic uncertainties presented in (7) were added in quadrature yielding a field shift of -1.020(64) MHz. For He, (6) and (7) were treated as independent measurements. Consequently, we took the weighted average of the two final field shifts, except for the Zeeman systematic uncertainty (0.03 MHz) which was present in both measurements. We also applied the nuclear polarization correction (-0.014(3) MHz) to the measurement (6) as done in (7). The total field shift for He is then -1.430(31) MHz. The resulting mean-square charge radii of He are computed using Eq. (1), where = 1.681(4) fm (26) is the mean-square charge radius of He, and = 1.008 MHz/fm (4). The updated values for the He charge radii are 2.060(8) fm and 1.959(16) fm, respectively. The new mass measurements lead to a decrease in the He charge radius by 0.011 fm and an increase in He by 0.025 fm compared to the values of (7) with the He charge radius from (26), which significantly reduces the difference between the two isotopes.

In order to compare the experimental charge radii with theory, we also calculate the corresponding point-proton radii given by (27):

 r2pp=r2c−R2p−(N/Z)⋅R2n−3/(4M2p)−r2so, (2)

where and fm (28) are the proton and neutron mean-square charge radii, respectively, fm is a first-order relativistic (Darwin-Foldy) correction (29) and is a spin-orbit nuclear charge-density correction. The latter is estimated to be fm and fm in the extreme case of pure -wave halo neutrons (27) for He and He, respectively (see also (30) for an improved estimate). Realistic values should be somewhere between zero and these extremes, so we conservatively took 0.08 and 0.017 fm as the corresponding error.

For the Review of Particle Physics (28) value is fm. Recently, has been also precisely measured from spectroscopy of muonic hydrogen (31) leading to fm. Using these two values for with the above mentioned spin-orbit corrections in Eq. (2) we obtain () and () fm for He (He), respectively. The experimental range in Fig. 2 includes both cases within the errors shown for He.

In Fig. 2, we compare the point-proton radius and the two-neutron separation energy of He to ab-initio calculations based on different NN and 3N interactions. The Green’s Function Monte Carlo (GFMC) results (11) are the only existing converged calculations that include 3N forces, which are constrained to reproduce the properties of light nuclei, including He and He. The scatter in Fig. 2 gives some measure of the numerical uncertainty in the GFMC method as well as an uncertainty in the 3N force models used (the IL2 and IL6 three-body forces were used with the AV18 NN potential) (11). The comparison of the experimental range to theory clearly demonstrates the importance of including and advancing 3N forces. The theoretical results shown in Fig. 2 based on NN interactions only are consistently at lower and smaller values. The NN-only calculations include the Fermionic Molecular Dynamics (FMD) results based on the UCOM NN potential and a phenomenological term (to account for three-body physics) (13), the No-Core Shell Model (NCSM) results based on the CD Bonn and INOY NN potentials (12), and variational Microscopic Cluster Model (MCM) results based on the Minnesota (MN) and MN without spin-orbit (MN-LS) NN potentials (14). Figure 2 also shows the importance of comparing theoretical predictions to more than one observable. To illustrate this, both NCSM (using CD Bonn) and the GFMC results show a good agreement for the point-proton radius, while the NCSM result has a large error for and tends to underpredict the two-neutron separation energy.

In addition, we present new Effective Interaction Hyperspherical Harmonics (EIHH) results (15) based on chiral low-momentum NN interactions  (32). In the EIHH approach the wave function falls off exponentially by construction, making it ideally suited for the study of halo nuclei (for calculational details see (15)). The obtained energies and radii are converged within the few-body calculational uncertainty given by the error bars. The three EIHH results shown in Fig. 2 are for different NN cutoff scales , and . The running of observables with is due to neglected many-body forces. The EIHH results lie on a line indicated in Fig. 2, leading to a decreasing and increasing , that does not go through the experimental range. Such a correlation is expected, because a smaller stretches out the core (7). This correlation is also similar to the Phillips and Tjon lines in few-body systems (33), which arises from strong NN interactions (large scattering lengths). Three-body physics manifests itself as a breaking from this line/band. The correlation is also supported by the variational MCM results. A key future step will be to include chiral 3N forces in the EIHH calculations.

We have presented the first direct mass-measurement of the two-neutron halo nucleus He and a more precise mass value for the four-neutron halo He. Both measurements where performed using the TITAN Penning trap mass spectrometer. While the He mass value is 1.7 within the AME03 (8), the He mass deviates by 4. The new masses lead to improved values of the charge (and point-proton) radii and the two-neutron separation energies, which combined provide stringent tests for three-body forces at neutron-rich extremes.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the National Research Council of Canada (NRC). We would like to thank the TRIUMF technical staff, especially Melvin Good. S.E. acknowledges support from the Vanier CGS program, T.B. from the Evangelisches Studienwerk e.V. Villigst, D.L. from TRIUMF during his sabbatical, and A.S. from the Helmholtz Alliance HA216/EMMI.

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