Review
Recent experimental progress in nuclear halo structure studies

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Abstract

Recent developments (since the last review in J. of Physics G by I. Tanihata in 1996 [1]) at RIB facilities opened possibilities of detailed studies of halo nuclei. New facilities have been constructed to provide higher intensity beams of radioactive nuclei in a wide range of energies. At the time of the last review, only secondary beams by projectile fragmentation were the production source of halo nuclei for use in reaction studies. Since then, re-acceleration facilities have been developed and thus high-quality low-energy beams become available for the reaction studies. The wide variety of new data are thus available on halo nuclei and nuclei on and outside of proton and neutron drip lines.

Low energy beams provided a means to determine the masses and charge radii of halo nuclei (6,8He, 11Li). Also transfer reactions have been measured in many nuclei far from the stability line. In fragmentation facilities, new experimental methods such as gamma ray detection in coincidence with breakup fragments of halo nuclei have been developed. Also the reaction cross sections have been measured in a wide range of beam energies. In addition, proton elastic scattering of halo nuclei has been measured at high energies. All together, studies of density distribution, identification of shell orbitals and spectroscopic factors of halo wave function became possible. Such studies reveal many new important information such as the change of magic numbers in nuclei far from the stability line.

In this article, we would like to review the experimental developments on halo nuclei and other related drip line nuclei. Also the new view of the nuclear structure learned from such studies will be discussed. Development of selected theories on related nuclear structure problems will be mentioned briefly.

Introduction

A neutron halo was discovered in 11Li nucleus from the series of experiments including the interaction cross section, the momentum distribution of the 9Li fragment from 11Li, and enhancement of the Electro-Magnetic Dissociation (EMD) cross section.

The main concept of the halo is a long tail in the density distribution of a nucleus. In stable nuclei with separation energy of about 6–8 MeV, the density distributions ρ(r) are usually described by a Woods–Saxon type distribution as, (for a spherical nucleus) ρws(r)=ρ0[1+exp(rRa)]1, with diffuseness parameter a0.53fm. Here ρ00.17fm1 is the density at the center and the radius parameter R is parameterized by mass number A as, R1.10A1/3(fm).

The density distributions of protons and neutrons were considered to be similar with same R and a in a nucleus. The distribution at large r is essentially exponential shape but the slope factor a is independent of mass number of stable nuclei. However this slope is directly related to the wave function of outer nucleon. As an example, let us see the s-wave wave function of a neutron in square well potential. The wave function of the neutron outside the potential is, Ψ(r)=(2πk)(ekrr)[ekR(1+kR)], where R is the width of the potential. Using this wave function, the density distribution of the neutron is written as, ρ(r)=|Ψ(r)|2e2krr2.

The parameter k, that determines the slope of the density tail, is related to a separation energy of the neutron (Es) by, (ħk)2=2μEs where μ is the reduced mass of the system. Since Es68MeV for most of the stable nuclei, the slope factor does not change from a nucleus to another nucleus.

An important concept of a halo is the decoupling of the halo wave function from the core of the nucleus. Cluster models, that assume a halo nucleus to be a core+halo nucleon(s), have been successful in most cases to describe halo nuclei. Therefore it is simple but relevant to know the behavior of the wave function of a valence nucleon in a certain orbital.

The density distribution observed in 11Li is shown in a later Section 3.2 shows the tail that extends much longer than those observed in stable nuclei. It can be understood by the extremely small neutron separation energy of the 11Li compared with that of stable nuclei. In this case, the two-neutron separation energy is about 360 keV that is only a few % of the separation energy of a stable nucleus and thus the k become much smaller giving a long extended tail.

More realistic behavior of the density distributions in different single-particle orbitals in Wood–Saxon potential is shown in Fig. 1.1. Density distributions of s-, p-, d-orbitals are presented for several selected separation energies. As can be seen, distributions with longer tail are formed for smaller separation energies. The slope also depends on the angular momentum of the orbital. The lower the orbital angular momentum, the longer the tail. It is due to the centrifugal barrier of the specific orbital. It is also seen that the proton distributions show a shorter tail than the neutron for the same separation energy. It is due to the Coulomb barrier, that does not exist for neutron.

Therefore the constant diffuseness (or tail length) in stable nuclei just originate from the constant separation energies of stable nuclei. However the separation energy of nucleons changes as a nucleus moves out from the stability line. The diffuseness of the nuclear surface therefore changes in unstable nuclei. Therefore the cut line of a halo and non-halo distribution is not defined clearly. But there are a special behavior of the density distribution of s- and p-wave.

Riisager et al. [2] calculated the radius of the two-body system for different separation energies and orbitals. As shown in Fig. 1.2, the radius of the system become larger and larger as separation energy decreases. In the figure the separation energy is normalized by the size of the square well potential R0 as ER0 and the root-mean-square radius is also normalized as Rrms/R0 in the figure. The radius diverges when the separation energy approaches to zero for s- (l=0) and p- (l=1) waves. On the other hand the radius converges to certain value for d-wave (l=2). This is true also for larger l orbitals. Therefore s-, and p-orbitals behave differently from other orbitals of larger l. That is one of the reason why the halo phenomena associated with a low orbital angular momentum orbital is interesting.

A similar analysis for a two-neutron halo system was made by Fedorov et al. [3], [4]. A three-body wave function is an eigenfunction of an angular momentum operator K with eigenvalue K(K+4) where the quantum number K (usually called as hypermoment) is a non-negative integer, even or odd depending on the parity of the system. The value of K defines the effective centrifugal barrier. The calculations are done for a Borromean system. (A three body bound system in which no two-body sub-system is bound. 6He, 11Li are examples of Borromean system.) For K=0 system the radius shows a logarithmic divergence. But a divergence is not seen for larger K system. It is therefore a large halo would probably made only for K=0 state (see Fig. 1.3).

The density distributions or the radii are determined by several methods but most of the nucleon distributions (radii) have been determined by the interaction or reaction cross sections.

Other important ingredient is the amplitude of the density distribution. A tail may have smaller slope parameter and thus shows a long tail when the separation energy is small, but the density itself may change due to the different normalization amplitude of the wave function. In addition to the slope of the density distribution, the magnitude of a density also affect the observed variables such as the interaction cross section. As a halo wave function it is related to the amplitude of the s- and p-waves in the wave function.

Another important behavior of the loosely bound orbital should be noted. Fig. 7.8 shown in the later section shows the binding energy of single-particle orbitals of a neutron in an Woods–Saxon potential. The values of the Woods–Saxon potential parameters are taken from the text book of Bohr and Mottelson [5]. It is easily seen that orbital energies of s-wave come down compared with other orbitals when the orbital become loosely bound. This is because of the formation of a halo. The s-wave does not have centrifugal barrier so that a long tail leaks out from the potential and then the kinetic energy become smaller and thus gain the energy. The higher angular momentum orbital has higher centrifugal barrier so that the wave function does not extend much thus the kinetic energy remains high. Therefore neutron would gain energy when it occupies the s-orbital when it becomes loosely bound. Near the drip line, a lower angular momentum, that forms a halo, is also favored by the binding energy.

The narrow momentum distribution of a nucleon is another indication of a halo. As can be seen in a simple square well model, the momentum distribution of halo neutrons in s-orbital is described by a Lorentzian shape, F(p)=C(p2+k2) where p is the momentum of the neutron. The width of the distribution is characterized by k as 2k being the FWHM of the distribution. The smaller the k, the narrower the width of the momentum distribution. Therefore a nucleon with wider density distribution appears as a narrower momentum distribution. It is a concept of Heisenberg’s uncertainty principle.

The remaining part of nucleus (so called core part of the halo nucleus) has exactly the same momentum (but with opposite direction). Thus in most of the experiments, the momentum distribution of the fragment after removal of the halo nucleon is observed to obtain momentum distribution of a halo nucleon.

Although Eq. (1.6) is shown only for s-wave neutron to show the dependence on the separation energy, the shape of the momentum distribution also strongly depends on the angular momentum (l) of the orbital and also the reaction mechanism. Therefore the momentum distribution is a sensitive tool to see he details of the configuration of a halo nucleon.

Although it was mentioned that a long tail of the density and a narrow momentum width is a concept of uncertainty principle, it has to be noted that the sensitivity to the configuration of the halo nucleon differs between the density distribution and the momentum distribution. If a halo nucleon is in a pure single particle state in a potential the relation between those two quantities are unique. However, in a nucleus, a nucleon wave function usually is a mixture of different configurations. When different angular momentum orbitals are mixed, a smaller angular momentum component extend in a space and thus affects the reaction (or interaction cross section) strongly, but a larger angular momentum component does not extend much and thus is difficult to be detected in the reaction cross section. Instead, the larger angular momentum component appears as a wider momentum distribution and thus is easily detected. The width and the shape of a fragment momentum distribution are then actively used to identify orbitals of the removed nucleon as described in Section 5.

Another important feature of a halo nucleus is the soft mode of excitation. Firstly, it has been predicted by Hansen and Jonson that a soft mode of electric dipole excitation may occur in a neutron halo nucleus [6], [7]. It is also predicted by Ikeda that a soft dipole resonance may exist at a very low excitation energy [8]. Searches for such phenomena have been made by electromagnetic dissociation (EMD) measurements. The first evidence was reported as an enhancement of the EMD cross sections in 11Li nucleus [9]. The enhancement of the EMD cross sections of halo nuclei were reported to be consistent with the view of soft excitation followed by neutron emission. However the excitation spectrum was not observed in the first experiment. Since the last review, many more detailed studies have been reported showing the excitation spectra or B (E1) strength. Those will be discussed in Section 5 of this review. It is noted that EMD measurements are also sensitive to the correlation of neutrons in a two-neutron halo.

So far halo are observed either in forms of one-nucleon halos or two-nucleon halos. The confirmed and suggested halo nuclei are so far only in lightest part of the nuclear chart as shown in Fig. 1.4. It can be seen from the figure that neutron halos are not rare phenomena but exists in most of the nuclei near the drip line. It is suggestive to see it in the beginning and the middle of p–sd shell. A standard ordering of the single particle orbitals does not necessary favor s-wave in all the region of p–sd shell. It appears like s-wave contribute to form halo for gaining the energy to expand the region of bound nuclei as discussed before. The halo with more than two neutrons (giant halo) was predicted by Terasaki et al. [10] at heavier mass region of nuclei but no experimental search has been done yet. It is natural to expect four-neutron halos when pure p3/2 orbital is loosely bound.

For a Borromean halo (most of the two neutron halos) the correlation between two-halo neutrons are important to be understood. One is the spatial correlation so called di-neutron correlation. The other is the mixing of the orbitals with different parity. Recent studies on these directions are presented in Section 8.

The reaction, interaction, and charge changing cross sections have been determined mostly by the transmission type measurements. The reaction cross section (σR) and the interaction cross section (σI) are defined as, σR=σtσel,σI=σRσinel, where σt is the total cross section, σel is the elastic scattering cross section, and σinel is the inelastic scattering cross section of the reaction. The inelastic scattering here is defined as the excitation of the nucleus to its bound excited states without emission of nucleons. The interaction cross section is measured by the transmission measurement by detecting the same nuclide as the incident nucleus in the final state. At incident beam energies lower than 100A MeV, the reaction cross sections are also measured. At such incident energies, the energy of un-interacted (a nucleus with the same A and Z with the incident nucleus) nucleus can be measured with good resolution so that any excitation of a part of the system can be detected and thus the inelastic scattering cross section can also be measured by the transmission measurement. It is this interaction cross section measurement that gave the first evidence of the neutron halo [11], [12]. Examples of the transmission measurements of interaction cross sections are shown in [13] and recent reaction cross section measurements can be found in [14]. Those measurements are extensively used for the determination of radii and also nucleon density distributions of unstable nuclei. The details of such measurements can be seen in previous review papers [1], [13]. Recent developments with these methods are presented in Section 3.1.

The charge-changing cross section is defined as the total cross section of changing the number of protons from a projectile nucleus. This is also determined by the transmission type measurement. Recently, the relation between proton distribution radii and the charge-changing cross sections is one of the interesting possibilities.

The first measurement of the momentum distribution of the unstable nuclei was reported by Kobayashi et al. [15]. The narrow momentum spread of the fragments were observed in this experiment and confirmed the neutron halo in 11Li. The first EMD enhancement was also discovered by the target Z-dependence of the fragmentation cross section [9].

Those experiments used high-energy beams of radioactive nuclei, however many new experiments were made later by the development of low energy radioactive beams from ISOL type (or re-acceleration type) facilities. Such development together with the future perspective of radioactive beam facilities can be seen in [16].

In the following, firstly new developments in experimental techniques are reviewed in Section 2. In Section 3, the radii and density distribution studies made after the last review [1] are presented. Not only the nucleon distribution but also the recent development in charge radii of halo nuclei are reviewed. The knowledge that can be obtained by combining nucleon radii and charge radii is also presented. In Section 4, the studies with the lowest energy beams, nuclear beta decay and moments are reviewed. Recent studies with fragmentation are reviewed in Section 5 and studies with transfer reactions are reviewed in Section 6. The details of the wave function of the halo nucleon are the subject of these sections. One of the most important subject, changes of orbitals are discussed in Section 7. In Section 8, the correlations of two-neutrons in halos are discussed with various methods. At the Section 9, unbound nuclei that sit next to halo nuclei are reviewed. Their structure is deeply related to the structure of halo nuclei.

In the present review, we concentrated to present the halo nuclei from the view point of nuclear structure. Studies of reaction mechanism using halo nuclei were not included in the present review. There are many reaction studies with halo nuclei in particular the fusion reactions. The recent paper by Signorini et al. [17] and the references therein present such studies.

Section snippets

Reaching the neutron- and proton-drip lines

The drip line is defined by the last isotopes that are bound. In neutron rich side (the neutron drip line), it is usually even neutron nuclei and the nuclei with one more neutron (odd neutron) are unbound, namely the neutron separation energy is negative. Often a neutron drip line nucleus has a two-neutron separation energy smaller than the one-neutron separation energy. Beyond the neutron drip line, the lifetime of a nucleus is very short and they exist only in the form of a resonance. Under a

Interaction cross section and the density distribution

Measurements of interaction cross sections and reaction cross sections have been the main method for determining the matter radii of short lived nuclei. In fact, the first observation of the halo is made by the measurements of the interaction cross sections of He and Li isotopes [52], [11], [12]. This has been possible because of the success of the Glauber model for calculating the cross sections [53]. Measurements of the interaction and reaction cross sections are summarized by, Ozawa et al.

Beta decay of 11Li

The beta decay of a halo nucleus 11Li is complex. Due to the large decay Q value to the daughter 11Be (Q=20.6MeV), many decay channels (1n, 2n, 4He, 6He, d, t delayed emissions) have been observed. Summary of the decay modes is shown in Table 4.1. Among them two halo-related modes will be discussed here. One is the decay to a halo excited state in 11Be and the other is the decay of the halo neutrons into deuteron. The delayed deuteron emission is considered to be related to a β decay of the

Nuclear breakup

The removal of one or two nucleons from a loosely bound halo nucleus can provide us information on the halo neutrons. In the statistical models of the fragmentation, the width of the momentum distribution is expected to follow the relation [116], σ2=σ02F(AF)/(A1), where F is the mass number of a fragment and A is the mass number of a projectile nucleus. Experiments on stable nuclei have shown that σ090MeV/c agrees within 10% for a wide range of nuclei. However, the first experiments on

Spectroscopy of halo nuclei by transfer reactions

Nucleon transfer reactions have served as one of the most precise tools for spectroscopy of nuclei to determine spin of levels, energies of excited states, spectroscopic factors and pairing correlation. In a one-nucleon transfer reaction a neutron (or proton) is transferred from the projectile to the target or vice versa. An example of such a reaction is: d+Ap+B, where one neutron from the nucleus d(=p+n) is transferred to ‘A’ forming ‘B’. The transition amplitude for this process in a

Changes of shell structures and halo

In neutron rich nuclei, it was first suggested that the neutron shell closure at N=20 may be disappearing for Na isotopes. Atomic spectroscopy of the Na isotope chain (from 21Na to 31Na) showed anomalies in spins, magnetic moments as well as the charge radii across the N=20. Those characters were not as predicted by the spherical shell model that has the magic number N=20 [234]. They were consistent with prolate deformation for 28 - 31Na in addition to 21 - 24Na, while 25 - 27Na are spherical.

Correlation of neutrons in two-neutron halo

Since the observation of the 11Li halo, special attention has been paid to nuclei showing a halo state consisting of two neutrons. These two-neutron halo nuclei are three-body systems with no bound binary sub-system and are referred to as Borromean nuclei [79].

The lightest Borromean nucleus is 6He. Since its two-neutron separation is 0.96 MeV, the other components such as t+t configuration is much smaller than expected [215] and the Borromean component is the main component of the 6He ground

Unbound nuclei around halo nuclei

The Borromean nuclei with a neutron halo may be described as a three body bound system consisting of a core surrounded by valence neutrons. Its pairwise subsystems is unbound but may still be produced and observed as a resonance. If we look at the sequence of He isotopes, as an example, we find that 4He cannot bind one neutron to form 5He but two, to form 6He (T1/2=806.7ms). The next bound isotope is then 8He (T1/2=119ms) since 7He is unbound. The investigation of such resonances is similar to

Halo in excited states

Suzuki and Yabana [423] studied the possibility of analog halo states that are excited states in more stable nuclei. They have estimated the excitation energy of such isobaric analog states (IAS) based on the Coulomb energy difference. They predicted the T=5/2 IAS of 11Li ground state in 11Be to be at 21 MeV excitation. They also discuss the decay width of such analog states.

The experiment was made by Teranishi et al. [424], [425] for 11Li(p,n)11Be reaction using 64AMeV beam of 11Li. They

Future perspective

The halo nuclei have been providing a new test ground for the models of nuclei. It also opened a new view to the nuclear models, (i) necessity of the treatment of bound state and continuum under the same footing, (ii) system of decoupled proton and neutrons, (iii) effects of weak binding, (iv) behavior of single-particle orbitals in asymmetric system, (v) Borromean system. After decades of studies we still need more studies to understand the structure of halo nuclei. The study of halo nuclei

Acknowledgments

The part of this work is supported by the grant-in-aid program of Japanese government. One of the authors (IT) gratefully acknowledges the support of MEXT of Japan. The support of the PR China government and Beihang university under the Thousand Peoples Plan is also gratefully acknowledged. He also expresses sincere thanks to Dr. A. Tosaki for his support for the cosmonuclear physics group in Osaka university. One of the authors (RK) gratefully acknowledge the research support from NSERC,

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