The electric dipole response of exotic nuclei

T Aumann; T Nakamura

doi:10.1088/0031-8949/2013/T152/014012

1. Introduction

When we add neutrons one by one to a β-stable nucleus, how does the nucleus change its structure and dynamics? This is one of the fundamental questions that have been raised in nuclear physics. Indeed, a large excess of the neutron number (N) over the proton number (Z) in a nucleus induces anomalous and intriguing features. Figure 1 illustrates such a drastic change in the structure toward the neutron-drip line. A β stable nucleus, as shown in figure 1(a), has almost identical S_n (one neutron separation energy) and S_p (one proton separation energy). The stable nucleus also has a similar neutron and proton density profile, i.e. the root mean square (rms) radius of the proton distribution is almost identical to that of the neutron's. This situation drastically changes as the N/Z ratio becomes larger. As shown schematically in figure 1(b), a larger N/Z ratio causes the difference of Fermi energies between protons and neutrons, S_p − S_n, and accordingly a thick neutron skin of 0.2–0.9 fm develops [1, 2]. Note that the difference of the Fermi energies reaches almost 20 MeV near the drip line, which is almost half the depth of the mean field potential of a normal nucleus. At the neutron-drip line (c), when S_n itself approaches zero, the neutron wave function extends far outside of the nuclear mean field as a result of quantum tunneling and a neutron halo is formed [3–5]. In this case, the value of S_n is typically less than 1 MeV, much smaller than for normal nuclei of about 8 MeV. Further interesting features of neutron halo and skin nuclei are the resultant dynamical aspects as revealed, for instance, in the electric dipole response as discussed here.

**Figure 1.** The change of the mean field potential (top) and density profiles (bottom) from β stable (a) to neutron-rich nuclei (b) and to much more neutron-rich nuclei near the neutron-drip line (c) is schematically shown. The upper panels indicate how the change of the ratio of proton/neutron numbers can induce a significant difference of Fermi energies between protons and neutrons as measured by S_p − S_n. Such a difference causes the formation of a thick neutron skin as in the bottom panel of (b). When S_n approaches zero, the neutron halo structure appears as a consequence of the quantum tunneling.
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The study of the physics of halo nuclei was initiated in the mid-1980s by Tanihata et al [3, 6, 7] when they started to apply high-energy fragmentation reactions to produce very exotic nuclei by using heavy ions at about 800 MeV per nucleon at the Bevalac at LBNL (Lawlence Berkeley National Laboratory). They measured systematically the interaction cross sections of light neutron-rich nuclei up to the neutron-drip line from He to Be isotopes and found that the most neutron-rich Li isotope, ¹¹Li, exhibits by far a much larger matter radius as compared to the conventional r₀A^1/3 systematics [3]. While it was not evident a priori if the origin would be a strong deformation or a long tail in the matter distribution [6], additional experimental information on the magnetic moment obtained at ISOLDE [8] (and later also of the quadrupole moment [9]) suggested an explanation as detailed by Hansen and Jonson [4] that the large interaction radius is related to a long tail of the neutron wave function due to the small binding, which they named 'Halo'. In the same paper, Hansen and Jonson also predicted that, as a consequence, the Coulomb breakup cross section of such halo nuclei should be extremely large. Direct experimental evidence of the halo structure of ¹¹Li, where two valence neutrons are extended over the densely packed ⁹Li core, then came from the observation of the narrow momentum distribution of ⁹Li, following the breakup of a ¹¹Li projectile with a carbon target, by Kobayashi et al [10] (transverse component of the momentum), and later confirmed by Orr et al [11] (longitudinal component of the momentum). These observations directly reflect the small internal momentum equivalent to the extended density distribution of the two valence neutrons, which are removed in the reaction. The third important observation on ¹¹Li was the anomalously large electromagnetic dissociation (EMD) cross section of close to one barn by Kobayashi et al [5]. They bombarded the heavy Pb target with a ¹¹Li beam and observed inclusively the ⁹Li fragment [5]. This was interpreted as evidence of an enhanced electric dipole (E1) response at low excitation energies, called 'Soft E1 excitation' as predicted by Hansen and Jonson [4]. These three observations, large interaction cross sections, narrow momentum distribution of the fragment and the enhanced electric dipole response at low excitation energies, have been the major signals of any halo system. The pioneering experiments by Tanihata et al [3], which led to the discovery of the halo nuclei, constitute the birth of physics using reactions with radioactive beams. Reaction experiments with radioactive ion (RI) beams have been extremely successful over the last two decades; in conjunction with increased efforts in nuclear theory, they have allowed to unravel and understand the properties of these exotic nuclear systems.

In this article, we focus on one observable to investigate exotic nuclei, that is, the electric dipole (E1) response. Considering the E1 response of stable nuclei, where the giant dipole resonance (GDR) around E_x ∼ 80A^−1/3 MeV (∼13–20 MeV) exhausts most of the E1 strength, the significant E1 strength well below the GDR energy for neutron halo and neutron-skin nuclei is very unique. Hence, the E1 response of exotic nuclei, in particular that of neutron-rich nuclei exhibiting neutron-halo or neutron-skin structure, has been a major subject of physics of exotic nuclei. Experimentally, the E1 strength function of exotic nuclei can be accessed by using heavy-ion-induced electromagnetic excitation. Most of the experiments discussed here study the neutron or particle decay after excitation. In this case, the process is called the Coulomb breakup⁴.

The mechanism of the enhanced E1 response for halo nuclei has been a major issue of the physics of halo nuclei. Hansen and Jonson [4] discussed the possibility of 'soft electric dipole mode' for ¹¹Li, which is considered as having a non-resonant character, inferred from the analogy of deuteron. Meanwhile, as shown in figure 2, Ikeda attributed the enhanced low-energy E1 strength to the soft dipole resonance (SDR), caused by the low-frequency motion of the charged core nucleus against the low-density neutron halo [12].

**Figure 2.** The upper panel shows the vibration of the 'neutron fluid' relative to the 'saturated core'. The bottom panel shows the transition probability as a function of excitation energy. The lower-energy bump corresponds to a 'new soft-dipole giant resonance', whereas the higher-energy bump corresponds to 'GDR analogous to the conventional one'. The figure is reprinted with kind permission from Ikeda [12].
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As shown in detail in section 3, the primary mechanism is now established as having a non-resonant character, called the direct breakup mechanism, as has been investigated for one-neutron halo nuclei such as ¹¹Be [13–15], ¹⁵C [16, 17] and ¹⁹C [18]. The cases of two-neutron halo systems, such as ⁶He [19] and ¹¹Li [20], are much more complicated. The current understanding for two-neutron halo cases is also non-resonant origin, but the initial two-neutron correlation and the final state interaction (FSI) affect largely the E1 spectra. From the non-resonant feature of this excitation, we may call this phenomena 'soft E1 excitation'. Soft E1 excitation has become one of the sensitive tools to probe the microscopic structure of halo states, such as the single particle state for the one-neutron halo nuclei. In the case of two-neutron halo nuclei, the dipole strength at low energy reflects the intrinsic correlations of the halo neutrons, as well. This has been studied quantitatively so far only for the cases of ⁶He [19] and ¹¹Li [20], which will be discussed in section 3.2.

For neutron-rich systems not exhibiting a halo structure, the dipole response is expected to change as well. In contrast to stable nuclei for which the dipole response is dominated by the GDR, new collective modes have been suggested on the basis of macroscopic models. In 1971, Mohan et al [21] discussed the possibility of a new excitation mode related to excess neutrons in a three-fluid model. Experiments with stable nuclei, in particular neutron capture measurements, reported low-lying dipole strength below the threshold at around 7 MeV, which has been discussed in this context and has often been called pygmy resonances (for an overview on the experimental status of pygmy resonances in stable nuclei, see the recent review article by Savran et al [22]). The soft dipole mode for halo nuclei discussed by Hansen and Jonson [4] has triggered new interest and the hope that such a new mode could be observable in neutron-rich nuclei in general. Suzuki et al [23] have studied this possibility in the framework of the Steinwedel and Jensen hydrodynamical model and predicted the strength and position of a such a mode as a function of neutron excess and mass. The situation of a vibration of excess neutrons against a core is illustrated schematically in figure 2, which was originally proposed as SDR for neutron halo nuclei.

While the dominant mode of excitation in stable nuclei is the collective out-phase vibration of neutrons against protons as the GDR located at around 13–20 MeV, the vibration of less bound excess neutrons against the core is expected at a lower frequency due to the weaker restoring force.

Such a redistribution of the dipole strength for neutron–proton asymmetric nuclei and the appearance of a soft vibration mode has been predicted also by microscopic calculations. Figure 3 shows as an example a result from the work of Vretenar et al [24] for nickel isotopes.

**Figure 3.** Left frames: dipole strength distribution calculated in relativistic RPA for nickel isotopes [24]. Right frames: transition densities for the high-energy peak (GDR) and the low-energy peak (PDR). Figure reprinted from Vretenar *et al* [24]. Copyright (2001), with permission from Elsevier.
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The left frames show the strength functions calculated in a relativistic random phase approximation (RPA) approach. For the neutron-rich isotopes ^68,78Ni a peak-like structure at around 9 MeV excitation energy is visible, well below the GDR region around 17 MeV. The transition densities for the high-energy peaks and the low-energy peak show pronounced differences. While the high-energy transitions exhibit the typical out-of-phase oscillation of neutrons versus protons, a pure neutron oscillation is visible at the surface region for the low-energy peak, which corresponds to the vibration of the neutrons against an N = Z core as discussed in the macroscopic picture. A comprehensive discussion from a theoretical point of view can be found in the review article by Paar et al [25].

The first experimental study of the full E1 response of neutron-rich nuclei has been carried out at GSI for the oxygen isotopic chain [26]. It was found that the dipole response indeed changes dramatically with adding more and more neutrons. A clear separation into two energy domains representing the two modes of vibration discussed above has not been observed. A later measurement for the neutron-rich tin isotopes ^130,132Sn, however, gave clear evidence for a low-lying pygmy dipole resonance (PDR) located around 10 MeV excitation energy [27].

Since the change in dipole response for neutron-rich nuclei is caused by the less bound excess neutrons, which are responsible for the development of a neutron skin as well, it is clear that the low-lying dipole strength is related to the neutron-skin thickness. It has been noted that the predicted dipole strength function depends sensitively on the parameters of the effective interactions used, which describe the isospin-asymmetric part of the nuclear equation of state. In turn, it was suggested that such parameters may be constrained from measurements of the pygmy dipole mode in exotic nuclei [28] or from the dipole strength or polarizability in general [29]. First attempts in this direction have been made by using data on neutron-rich nuclei [30, 31] and also for the stable nucleus ²⁰⁸Pb [32].

In this article, we first give in section 2 a brief introduction to the method of Coulomb excitation at intermediate and high energies. We then discuss the soft E1 excitation of halo nuclei and its spectroscopic relevance in section 3. Here, we will emphasize also the case of two-neutron haloes, where the dipole response gives insights into correlations among the halo neutrons. Section 4 is devoted to the dipole response of non-halo neutron-rich nuclei and the PDR, including a discussion of its relevance for constraining the density dependence of the symmetry energy and its relation to the neutron-skin thickness. Finally, section 5 gives a brief summary and points out the open questions and future perspectives of this field in the era of the new-generation high-energy RI beam facilities such as FAIR at GSI, FRIB at MSU and RIBF (RI Beam Factory) at RIKEN.

2. Experimental method

2.1. Coulomb excitation and breakup at intermediate and high energies

Coulomb excitation is a process where a fast projectile at relativistic energies is excited by an impulse of the Lorentz-contracted Coulomb field of a high-Z target. When the final state is above the particle decay threshold and thus breaks up, this specific channel is called Coulomb breakup or EMD, as schematically shown in figure 4.

**Figure 4.** The Coulomb breakup process is shown schematically for a projectile with ^AZ on a Pb target, decaying into ^A−1Z + neutron. The excitation process can be equivalent to the absorption of a virtual photon with E_γ(=E_x).
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Coulomb excitation at low incident energies below the Coulomb barrier has long been used successfully in spectroscopy of low-lying excited states of stable nuclei, where the nuclear excitation does not contribute. More recently, however, it has been proven that the Coulomb excitation at relativistic energies has advantages over low-energy Coulomb excitation because of the large experimental yield due to the kinematic focusing and the availability of a thick target of the order 0.1–1 g cm⁻². It should be noted that the nuclear contribution can be estimated by using a less electromagnetic sensitive target (such as ⁹Be and ¹²C). The nuclear breakup contribution can also be controlled by the selection of the scattering angle, or equivalently of the impact parameter. Since the intensity of the RI beam is generally low, the Coulomb excitation at intermediate/high energies becomes a powerful spectroscopic tool. This also matches the intermediate-/high-energy RI beams available at the heavy-ion facilities based on the fragmentation reactions, such as GANIL, GSI, MSU and RIKEN.

The Coulomb excitation at relativistic energies can be expressed as a photo-absorption process induced by a virtual photon. The way of treating Coulomb excitation/breakup in terms of the virtual photon is called equivalent photon method. In this method, the Coulomb excitation/breakup cross section for the electric excitation with a multipolarity λ can be described as

$\begin{equation} \frac{{\rm d}\sigma(E\lambda)}{{\rm d}E_{{\rm x}}} =\frac{(2\pi)^3(\lambda{+}1)}{\lambda \left[(2\lambda{+}1)!!\right]^2} \left(\frac{E_{{\rm x}}}{\hbar c}\right)^{2\lambda{-}1} N_{E\lambda}(E_{{\rm x}}) \frac{{\rm d}B(E\lambda)}{{\rm d}E_{{\rm x}}} \frac{{\rm d}E_{{\rm x}}}{E_{{\rm x}}}, \end{equation} \tag{ 1 }$

where N_Eλ(E_x) is the number of virtual photons of the electric multipole λ with photon energy E_x [33]. Hence, the reduced transition probability B(Eλ), which contains the physics essence of Eλ response, can be directly extracted from the excitation (breakup) cross section spectrum. The simplicity with which the cross section can be factorized in this way is one of the advantages of using the Coulomb excitation/breakup. The magnetic excitation is treated similarly, and the corresponding photon number can be seen, for instance, in [33].

The spectrum of the virtual photon can be calculated since it is an electromagnetic process [33]. In figure 5, the virtual photon spectrum for E1 excitation is shown for a projectile at 70, 250 and 800 MeV per nucleon on a Pb target. The impact parameter cut is fixed at 12.3 fm here, corresponding to the halo nucleus ¹¹Be on Pb [13]. The characteristic feature of this photon spectrum is that it is a nearly exponentially falling function, so that the low-energy excitation is, in particular, amplified. It is interesting to note that the curves are crossing around E_γ ∼ 7 MeV. The yield at the low excitation energies is thus larger for the lower incident energies, while it is the inverse for the higher incident energies. Hence, the soft E1 excitation has a larger cross section for the lower incident energies, while it is almost similar or a little larger for the higher incident energies in exciting PDR at around E_x = 8–10 MeV. The GDR is excited more favorably for the higher incident energies. The incident energy dependences of the cross sections for the different multipolarities and for different excitation energies are shown in figure 6.

**Figure 6.** Coulomb breakup (excitation) cross sections estimated for the E1, E2 and M1 transitions, with one Weisskopf unit (W.u.) for a nucleus with A = 11, are shown as a function of the incident energy. A strong dependence on the multipolarity and the incident energy is seen, which could be useful in distinguishing the multipolarity.
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2.2. Inclusive measurement and kinematically complete measurement

There are two major methods of measuring the Coulomb breakup, the inclusive method and the kinematically complete measurement (final-state exclusive measurement). The inclusive measurement extracts the 1n (2n) removal cross section (inclusive cross section) for the aimed nucleus. For instance, in the case of the ¹¹Li breakup, the ⁹Li produced in the reaction of a ¹¹Li projectile with the target is counted as in [5]. This corresponds to measure

$\begin{equation} \sigma(E{1}) = \int_{E_{{\rm th}}}^{\infty} \frac{16\pi^3}{9\hbar c} N_{E{1}}(E_{{\rm x}})\frac{{\rm d}B(E{1})}{{\rm d}E_{{\rm x}}} {\rm d}E_{{\rm x}}. \end{equation} \tag{ 2 }$

The cross section is now an integration of the product of N_E1(E_x) and dB(E1)/dE_x in terms of E_x from the decay threshold energy E_th to infinity. Since the photon spectrum falls nearly exponentially with E_x, as shown in figure 5, σ(E1) becomes large when the dipole strength B(E1) is concentrated at low excitation energies (soft E1 excitation) as in halo nuclei.

The advantage of the inclusive measurement is that the experiment is simpler, and the yield is higher since one does not need to measure neutron(s), compared with the kinematically complete measurement. The experiment is feasible even for secondary beam intensities of the order of 1 count per second or even less. However, as one cannot map the B(Eλ) distribution, the analysis covered in this method is restricted. Nevertheless, one can obtain evidence of the halo state as demonstrated for ³¹Ne [34], which is discussed in section 3.

The kinematically complete measurement maps the B(Eλ) distribution as a function of the relative energy E_rel. It reconstructs the invariant mass from the four-momentum vectors of all the particles following the breakup of the projectile. As an example, we consider the case of ^AZ + Pb → ^A−1Z + n + X, namely the one neutron removal of ^AZ with a Pb target. One can then measure exclusively the final states of the excited ^AZ. In this case, the invariant mass M(^AZ*) can be related to the four-momentum vectors of the outgoing particles as in

$\begin{eqnarray} M(^{A}Z^{*}) &=& \sqrt{ \left(\sum_i{E_i}\right)^2- \left(\sum_i{\vec{P}_i}\right)^2 }\nonumber\\ &=& \sqrt{(E(^{A-1}Z) + E(n))^2 - (\vec{P}(^{A-1}Z) + \vec{P}(n))^2},\quad\quad \end{eqnarray} \tag{ 3 }$

where E(^A−1Z) and E(n) are the total energy of the fragment (^A−1Z) and that of the neutron, respectively. The relative energy E_rel between the fragment and the neutron is then related to M(^AZ*) as

$\begin{equation} E_{{\rm rel}} = M(^{A}Z^{*}) - M(^{A-1}Z) - m_{{\rm n}}, \end{equation} \tag{ 4 }$

where M(^A−1Z) and m_n denote the mass of the fragment and of the neutron, respectively. The excitation energy E_x is related to the relative energy by E_x = E_rel + S_n. Note that when the fragment is produced as an excited bound state, the mass of the fragment should be replaced by

$\begin{equation} M(^{A-1}Z^*) = M(^{A-1}Z) + \sum_i{E_\gamma^{(i)}}, \end{equation} \tag{ 5 }$

where the second term is the summation of the γ-ray energies associated with the daughter nucleus ^AZ when it is produced in a bound excited state. The excitation energy then becomes

$\begin{equation} E_{{\rm x}}= E_{{\rm rel}} + S_{{\rm n}} + \sum_i{E_\gamma^{(i)}}. \end{equation} \tag{ 6 }$

For the soft E1 excitation of halo nuclei, the contribution of the bound excited states is generally weak due to the rapidly falling function of the virtual photons. On the other hand, higher excitation as in PDR and GDR involves more excited daughter state contributions, where γ-ray energy measurements are essential. Typical setups for the kinematical complete measurement are shown in figures 8 and 17 for the RIKEN setup and in figure 21 for the GSI setup, which are explained in later sections.

3. Soft E1 excitation of halo nuclei

The E1 response of halo nuclei has a unique feature that the excitation also occurs strongly at low excitation energies at E_x ∼ 1 MeV, much below the GDR peak. In this section, we first clarify the underlying mechanism. We then discuss how this excitation can be related to the halo property and we show how this can be applied to the spectroscopy. Figure 7 shows the location of neutron halo nuclei, whose E1 response is studied by either the kinematically complete measurement or inclusive Coulomb breakup cross section. At RIBF at RIKEN, other halo nuclei such as ¹⁹B and ²²C have been studied by kinematically complete measurements, and some heavier neutron-rich nuclei such as ³⁷Mg have been studied inclusively, whose analysis is in progress. Hence, in the near future, the E1 response will be clarified in those nuclei as well.

**Figure 7.** The halo nuclei and the current status of the Coulomb breakup experiments of those nuclei are shown in the nuclear chart up to Z = 20 (Ca). The known neutron halo nuclei (published) are shown as indicated. The neutron halo nuclei studied by a kinematically complete measurement (published) are shown by filled circles, and the ones studied only by inclusive Coulomb breakup (published) are shown by the gradation of filled circles.
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A one-neutron halo nucleus has a simpler structure due to its two-body nature. Unlike a two-neutron halo nucleus, which has a three-body Borromean structure, it does not suffer from strong FSIs in the breakup reactions. Hence, we first describe the case of one-neutron halo nuclei and discuss the mechanism causing the soft E1 excitation.

3.1. Soft E1 excitation of a one-neutron halo nucleus

First we clarify the underlying mechanism of soft E1 excitation and its spectroscopic significance by taking the example of ¹¹Be. Then we show the application to clarify the structure of ¹⁹C, which had been controversial before the Coulomb breakup experiment [18], and to the case of ¹⁵C, which is important in the stellar reaction.

3.1.1. Soft E1 excitation of ¹¹Be.

¹¹Be has been a benchmark halo nucleus since it has a simple single-neutron halo structure ¹⁰Be + n with a well-determined separation energy S_n = 504(4) keV. The ground-state structure is known to be dominated by the following two configurations:

$\begin{eqnarray} |^{11}{{\rm Be}}(1/2^+;{{\rm gs}})\rangle&=&\alpha|^{10}{{\rm Be(0}}^+)\otimes \nu 1{{\rm s}}_{1/2}\rangle\nonumber\\ && + \beta|^{10}{{\rm Be(2}}^+)\otimes \nu 0{{\rm d}}_{5/2}\rangle, \end{eqnarray} \tag{ 7 }$

where the first term corresponds to the halo state. There were spectroscopic studies of ¹¹Be by using the transfer reaction,¹⁰Be(d,p)¹¹Be, from which the dominance of 1s_1/2 neutron in the ground state (α² = 0.73 [35] and 0.77 [36]) was extracted. More recently, the knockout reaction of ¹¹Be [37] and the ¹⁰Be(d,p) reaction in the inverse kinematics [38] also showed similar values close to 0.7. We also show here that the Coulomb breakup can be used to extract the spectroscopic factor for the halo state.

The earlier Coulomb breakup experiments were devoted, on the other hand, primarily to the two-neutron halo nucleus ¹¹Li [39–41]. However, they failed to clarify the mechanism of the soft E1 excitation, where the obtained B(E1) strength distributions resulted in inconsistent spectra as shown later in figure 17 (right). Theoretical interpretation of the soft E1 excitation of ¹¹Li requires an understanding of the correlations of the two neutrons and other two-body FSIs. Hence, the Coulomb breakup of the simpler 1n halo nucleus ¹¹Be played a significant role in understanding the underlying mechanism of the soft E1 excitation.

The first full kinematical complete measurement of ¹¹Be was made at RIKEN at 72 MeV per nucleon [13], where the E1 spectrum was obtained, and evidence for the direct breakup mechanism of the Coulomb breakup is shown. There the spectroscopic factor α² was also extracted and the spectroscopic significance was suggested for the first time. In GSI, at much higher energies of 520 MeV per nucleon [14], the Coulomb breakup of ¹¹Be was measured to support the result of [13] with higher statistics. This experiment also measured γ-ray in coincidence to pin down the component of the second term of equation (7). It favors comparatively higher excitations due to the incident energy dependence of the virtual photons shown in figure 5. The third experiment performed at RIKEN [15], which has further higher statistics and better resolutions, utilized the angular distribution to refine the E1 spectrum. Here, we explain the essence of the Coulomb breakup and the soft E1 excitation of ¹¹Be by showing the latter experiment of RIKEN [15].

The experimental setup used in this experiment [15] is shown in figure 8. The ¹¹Be secondary beam, produced by fragmentation of an ¹⁸O beam at 100 MeV per nucleon at RIPS at RIKEN, bombarded a Pb target at an average energy of 68.7 MeV per nucleon. The momentum vectors of the beam ( $\vec {P}(^{11}{\mathrm { Be}})$ ) as well as those of the outgoing ¹⁰Be fragment ( $\vec {P}(^{10}{\mathrm { Be}})$ ) and the neutron ( $\vec {P}(n)$ ) were measured in coincidence.

The invariant mass of ¹¹Be was then extracted to deduce the Coulomb breakup spectrum as a function of ¹⁰Be − n relative energy, E_rel. Spectra of the breakup cross sections with the Pb and C targets are shown in figure 9 (left) (a) and figure 9 (left) (b), respectively. The breakup data on the carbon target, where nuclear breakup is dominant, are used to investigate the characteristic features of the target dependence. The cross sections are plotted for different selection of scattering angle θ in the center of mass of the projectile and the target.

**Figure 9.** Left: breakup cross sections as a function of E_rel of ¹⁰Be and the neutron in the breakup of ¹¹Be + Pb at 69 MeV per nucleon (a) and ¹¹Be + C at 67 MeV per nucleon (b). See text and [15] for details. Right: angular distribution of the ¹¹Be in the center of mass of ¹¹Be + Pb for the two E_rel ranges shown. The grazing angles are indicated by θ_gr(=3.8°). Solid curves are calculations of the ECIS code, whereas the dotted ones are calculations of the equivalent photon method. Both calculations are E1 only, and no nuclear excitation is included. Figure reprinted with permission from Fukuda *et al* [15]. Copyright (2004) by the American Physical Society.
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The striking contrast is evident between the spectra for the two targets. The total one-neutron removal breakup cross section for the Pb target is 1790 ± 20(stat.) ± 110(syst.) mb, whereas that for the C target is 93.3 ± 0.8(stat.) ^+5.6_−10.3(syst.) mb, about 1/20 of the former⁵. The large difference in cross sections can be attributed to the different reaction mechanisms between Pb and C targets. The former is dominated by Coulomb breakup, whereas the latter is dominated by the nuclear breakup. It should be noted that the ratio of the 'nuclear' breakup cross sections for these two targets would be about 2 [15], much smaller than this observed ratio.

The spectral shape exhibits a remarkable difference as well according to the target. The spectra for Pb for the two angular ranges have an asymmetric peak at very low E_rel, whereas that for C for the whole angular range shows two prominent resonance peaks (E_x = 1.78 MeV, 5/2⁺ and E_x = 3.41 MeV, 3/2⁺) upon the structureless continuum. This also suggests the Coulomb breakup dominance for the Pb target data, and the nuclear breakup dominance for the C target data.

In this experiment, the scattering angle was measured and used effectively to control the reaction channel. The scattering angle is obtained by the difference between the incoming momentum vector of ¹¹Be and the outgoing vector (sum of momentum vectors of ¹⁰Be and the neutron) in the center of mass (c.m.) frame of the ¹¹Be + Pb(C). In the Coulomb breakup, the scattering particle (c.m. of ¹⁰Be + n) largely follows the Rutherford trajectory, the impact parameter b can be determined event by event from θ as b = a cot(θ/2), where a represents half the distance of closest approach in the classical Coulomb head-on collision. Although the nuclear breakup does not follow this rule, the selection of forward angles can be a good way of selecting the Coulomb breakup component. This can be seen in the forward-angle-selected spectra for Pb and C targets. There, the spectra have almost identical asymmetric shapes for both targets, which follow perfectly the calculation of the direct breakup mechanism of the Coulomb breakup of the one-neutron halo nucleus, as shown by the solid curves. The result shows that the direct breakup mechanism is indeed the underlying mechanism of the soft E1 excitation. It is also noted that for the Pb target, 6° data (whole acceptance) can also be explained by the direct breakup calculation for the most part. This again proves the dominance of the Coulomb breakup for the Pb target.

The selection of the most forward angles, well within the grazing angle (θ_gr = 3.8°), has been found useful in extracting almost purely the 'Coulomb' component of the breakup, which was confirmed by detailed reaction theories [42–44]. For the whole acceptance data, we find a slight deviation from the direct breakup curve, which can be attributed to nuclear breakup and higher order Coulomb breakup effects.

The B(E1) distribution was extracted from the forward-angle-selected data on Pb. In this case, it is useful to use the double differential cross section formula for the E1 excitation in the equivalent photon method, written as

$\begin{equation} \frac{{\rm d}^2\sigma}{{\rm d}\Omega\, {\rm d}E_{{\rm x}}} = \frac{16\pi^3}{9\hbar c}\frac{{\rm d}N_{E{1}}(E_{{\rm x}},\theta)}{{\rm d}\Omega}\frac{{\rm d}B(E{1})}{{\rm d}E_{{\rm x}}}. \end{equation}\noindent \tag{ 8 }$

The virtual photon number is extracted by integrating over the selected angles in this case. The B(E1) spectrum was obtained by dividing the dσ/dE_rel by this integrated photon number. The result is shown in figure 10, in agreement with the direct breakup mechanism shown in the solid curve. The integrated B(E1) obtained from the data selected for the forward angles amounts to 1.05 ± 0.06 e² fm² corresponding to 3.29 ± 0.19 W.u. (Weisskopf unit) for E_x ⩽4 MeV, which is huge, considering that the E1 strength in this energy region for normal nuclei is negligible (below 0.1 W.u.).

**Figure 10.** B(E1) strength distribution for ¹¹Be as a function of E_x, obtained from the angle-selected Coulomb breakup data on a Pb target (θ < 1.3°). The solid curve is the result of a calculation with the direct breakup mechanism with α² = 0.72. Figure reprinted from Nakamura *et al* [13]. Copyright (1994), with permission from Elsevier.
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In the soft E1 excitation, the non-energy weighted E1 cluster sum rule is useful in extracting the geometrical information. For the one-neutron halo nucleus, the sum rule can be expressed as

$\begin{equation} B(E{1}) = \int_{-\infty}^{+\infty}\frac{{\rm d}B(E{1})}{{\rm d}E} {\rm d}E = \frac{3}{4\pi}\left(\frac{Ze}{A}\right)^2 \langle\, r^2\,\rangle, \end{equation}\noindent \tag{ 9 }$

which shows that the integrated B(E1) is proportional to 〈r²〉, where r represents the distance of the halo neutron from the c.m. of the core. For ¹¹Be, Fukuda et al [15] estimated $\sqrt {\langle \, r^2\,\rangle } = 5.77\pm 0.16\,{\mathrm { fm}}$ by summing the E1 strength up to E_x = 4 MeV, which is 1.05 ± 0.06 e² fm². Note, however, that the sum should be taken up to a sufficiently high excitation energy, and also take into account the transitions to bound final states [45, 46], which resulted in $\sqrt {\langle \, r^2\,\rangle } = 6.4 \pm 0.5\,{\mathrm { fm}}$ [47].

3.1.2. Direct breakup mechanism.

It was clearly shown above that the direct breakup mechanism explains the B(E1) spectrum of the one-neutron halo nucleus ¹¹Be. The direct breakup itself has been known for long, for instance, for the photodisintegration of the deuteron [48].

In the direct breakup mechanism, the 1n-halo nucleus, in this case ¹¹Be, breaks up into ¹⁰Be and a neutron without forming any intermediate resonances. The possibility of the SDR being a main mechanism of this large E1 strength is unlikely [13–15].

The matrix element of the direct breakup is described as

$\begin{equation} \frac{{\rm d}B(E{1})}{{\rm d}E_{{\rm rel}}} = \mid \langle \Phi_{{\rm f}}(\vec{r},\vec{q}) \mid e_{{\rm eff}}^{E{1}} \hat{T}(E{1}) \mid \Phi_{{\rm i}}(\vec{r})\rangle \mid^2 , \end{equation} \tag{ 10 }$

where $\Phi _{\mathrm {i}}(\vec {r})$ and $\Phi _{\mathrm {f}}(\vec {r},\vec {q})$ represent the wave function of the ground state and the final state in the continuum of the neutron relative to the core, respectively. $\vec {r}$ is the relative coordinate of the neutron relative to the c.m. of the core. $\Phi _{\mathrm {f}}(\vec {r},\vec {q})$ is also a function of the relative momentum $q=\sqrt {2\mu E_{\mathrm { rel}}/\hbar }$ . The electric dipole operator is $\hat {T}(E{\mathrm { 1}})(=rY^{(1)}(\Omega)$ ).

It is worth considering a simple formula for the case where the ground state is a pure single particle state and a spinless core, as in |¹⁰Be(0⁺)⊗ν1s_1/2〉. Equation (10) can then be expressed in a simpler form as

$\begin{equation} \frac{{\rm d}B(E{1})}{{\rm d}E_{{\rm rel}}}= \frac{3}{4\pi}(e_{{\rm eff}}^{E{1}})^2\langle \ell_{{\rm i}} 010 \mid \ell_{{\rm f}}0\rangle^2 \left| \int {\rm d}r r^2 \phi_{{\rm f}}(r,q) r\phi_{{\rm i}}(r) \right|^2,\!\! \end{equation} \tag{ 11 }$

where ℓ_i,ℓ_f are the orbital angular momenta of the valence neutron in the initial state, and the scattered neutron in the final state, respectively [49–51]. ϕ_i is the radial wave function involved in Φ_i, and ϕ_f is the radial component of the final scattering state of the neutron. The E1 effective charge e^E1_eff for the one-neutron halo nucleus is Ze/A. Although the analysis is to be performed with a distorted wave for ϕ_f due to the FSI between the core and the neutron, it is instructive to consider the plane wave approximation

$\begin{equation} \phi_{{\rm f}}(r,q)= \sqrt{\frac{2\mu q}{\hbar^2 \pi}}j_{\ell_{{\rm f}}}(qr), \end{equation} \tag{ 12 }$

where j_{ℓ_f}(qr) is the spherical Bessel function. This implies that the above equations (equations (10) and (11)) have the form of the Fourier transform of rϕ_i(r). Hence, the B(E1) distribution at low q (low E_rel) is an amplified image of the density profile of the valence neutron, i.e. the halo state. The amplification is attributed to the factor 'r' in the E1 operator, as well as to the phase space factor 4πr² in the integrand. The theoretical calculation in [49] shows that B(E1) at low E_rel is almost solely determined by the density distribution outside of the range of the mean-field potential.

This interpretation in terms of a Fourier transform also implies that the spatial decoupling of the halo and the core is important. The spatially decoupled halo state overlaps very well with the E1 operator which further emphasizes large separations of the valence neutron and the core. Such an intermediate state has a large overlap with the final scattering state. It is then natural that for stable nuclei, which have no such spatial decoupling, only a high-energy E1 photon can separate the whole neutron fluid from the proton fluid which is realized as GDR.

3.1.3. Spectroscopic significance of soft E1 excitation of one-neutron halo nuclei.

The direct breakup of the one-neutron halo nucleus has led to the idea that the Coulomb breakup can now be used as a spectroscopic tool. For ¹¹Be, Φ_i is represented by equation (7), and the strong sensitivity of the low-energy E1 to the tail part of the radial wave function (the first term of equation (7)) restricts the contribution to the soft E1 excitation only to the first term (halo term) of equation (7). This allows us to extract the spectroscopic factor α². The spectroscopic factor extracted in [13–15] was in the range α² = 0.6–0.8. For instance, Fukuda et al [15] extracted the value to be 0.72(4), which is consistent with the values from the transfer reactions [35, 36, 38] and the knockout reaction [37].

Not only the amplitude, but also the spectral shape of B(E1) is useful, as demonstrated below for ¹⁹C. Table 1 shows the dependence of the spectral shape on S_n and the angular orbital momentum ℓ in the plane wave approximation [50–52]. These two parameters are important since the halo is formed as a result of the quantum tunneling. The ℓ value determines the height of the centrifugal barrier, and it is known that only s and p neutrons can form the halo states [53]. Note that the spectral shape is dependent also on the angular momentum of the final state, ℓ_f. Another important feature is that the peak is very sensitive to S_n. The lower the S_n value, the lower the energy the peak shifts. This is nothing but the revelation of the soft E1 excitation.

Table 1. Characteristic features of the spectral shapes of dB(E1)/dE_rel depending on S_n, the orbital angular momentum of the valence neutron (ℓ) in the initial state, and that of the scattered neutron (ℓ_f) in the final state. The calculation is performed in the plane wave approximation [50–52].

ℓ→ℓ_f	dB(E1)/dE_rel (E_rel∼0 MeV)	Peak of dB(E1)/dE_rel
	∝E^ℓ_f+1/2_rel
s→p	∝E^3/2_rel	$E_{\mathrm { rel}}=\frac {3}{5}S_{\mathrm {n}}$
p→s	∝E^1/2_rel	E_rel≃0.18S_n
p→d	∝E^5/2_rel	$E_{\mathrm { rel}}=\frac {5}{3}S_{\mathrm {n}}$
d→p	∝E^3/2_rel	$E_{\mathrm { rel}}=\frac {5}{3}S_{\mathrm {n}}$

3.1.4. E1 versus E2 excitation for halo nuclei.

As shown in figure 6, the photon numbers vary according to the excitation energy, incident energy and the multipolarity of the excitation. The E1 dominance of the Coulomb breakup of the neutron-drip-line nuclei can be attributed not only to this photon number but also to the effective charge involved in the matrix element. For the direct breakup of a one-neutron halo nucleus, the matrix element is shown in equation (10), which involves the effective charge Ze/A. In general, the same effective charge should be involved in any transition matrix, when the E1 transition is associated with the halo state since the transition occurs for the halo neutron relative to the c.m. of the core. In other words, the cluster-like structure of the nucleus is the cause of the electromagnetic transition to separate the halo part from the core part. Hence, the cross section is roughly proportional to the square of the effective charge.

The effective charge for the Eλ transition for a projectile with a cluster-like structure ^AZ = (^A₁Z₁ + ^A₂Z₂) is written as involved in equation (3.4.19) of [33],

$\begin{equation} e_{{\rm eff}}^{E\lambda} = e \left[ Z_1 \beta_1^{\lambda} + (-)^\lambda Z_2 \beta_2^{\lambda} \right], \end{equation} \tag{ 13 }$

where the β_i corresponds to the parameters describing the position and orientation of the cluster i from the center of mass of the projectile, i.e. $\vec {r_1}= \beta _1 \vec {r}$ and $\vec {r_2}= -\beta _2 \vec {r}$ , where $\vec {r}$ is the relative vector directing from 2 to 1. Here, β₁ = A₂/A and β₂ = A₁/A.

For the one-neutron halo nucleus, A₁ = 1, Z₁ = 0, A₂ = A − 1 and Z₂ = Z, then, the E1 and E2 effective charges squared are, respectively, written as

$\begin{eqnarray} \left (e_{{\rm eff}}^{E{1}} \right)^2 &=& e^2 \left(\frac{Z}{A} \right)^2, \end{eqnarray}\noindent \tag{ 14 }$

$\begin{eqnarray} \left (e_{{\rm eff}}^{E2} \right)^2 &=& e^2 \left(\frac{Z}{A^2} \right)^2. \end{eqnarray} \tag{ 15 }$

The formulae explain why the E1 transition is dominant over the E2 transition for neutron halo nuclei. For instance, for ¹¹Be, the E1 effective charge squared is larger by two orders of magnitude (∼A²) than that of E2. The dominance of E1 transition is confirmed for Coulomb breakup of ¹¹Be [13] and ¹⁹C [18] by the angular correlation of the fragment in the c.m. of the fragment + neutron. The one obtained for ¹¹Be is shown in figure 11, where the squared sine function expected for the E1 transition is clearly seen. The theoretical curve shown is based on the first-order perturbation theory of the Coulomb breakup [54, 55].

**Figure 11.** Azimuthal angular distribution of ¹⁰Be in the ¹¹Be rest frame. The solid curve is the result for the pure E1 transition based on the first-order perturbation theory [54, 55].
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On the other hand, for the proton halo (proton cluster) and α-cluster nucleus, E2 transitions are comparatively important. For instance, in the case of ¹⁶O + γ → ¹²C + α, which is one of the key stellar reactions, the e^E1_eff = 0, so that the E2 transition becomes dominant. For the proton + core system, such as the one-proton halo nucleus, the effective charges for E1 and E2 transitions are

$\begin{eqnarray} \left(e_{{\rm eff}}^{E{1}} \right)^2 &=& e^2 \left(\frac{A-Z}{A} \right)^2, \end{eqnarray} \tag{ 16 }$

$\begin{eqnarray} \left(e_{{\rm eff}}^{E2} \right)^2 &=& e^2 \left[ \left(\frac{A-1}{A} \right)^2 + \frac{Z-1}{A^2} \right]^2. \end{eqnarray} \tag{ 17 }$

For ⁸B + γ → ⁷Be + p, the key reaction of the pp chain in the sun and thus important in estimating the solar neutrino flux, the effective charge squared of E2 (0.68 e²) is larger than that of E1(0.14e²). Due to the photon number spectrum for the low excitation energies, the E2 transition in this reaction is still not significant compared to the E1, but not negligible anymore. The point is that the proton-halo-like case needs to take into consideration the E2 mixture with the E1, but for neutron-halo nuclei the E1 transition is by far dominant, so the E2 transitions can thus be neglected.

3.1.5. Application of Coulomb breakup to the structure study—soft E1 excitation of ¹⁹C.

Here we show two applications of Coulomb breakup, which are on ¹⁹C and ¹⁵C. In this subsection, we discuss the case of ¹⁹C. Before the Coulomb breakup experiment [18], the structure of ¹⁹C had been controversial partially because of a large uncertainty in the direct mass measurements ranging from almost 0 to 700 keV [56–59]. The mass evaluation in 1993 based on these direct mass measurements was 0.16 ± 0.11 MeV [60]. Such a small S_n value for ¹⁹C suggested a possible halo state. However, one should note that the condition to form a neutron halo is not only the smallness of the S_n value, but also the smallness of the orbital angular momentum of the least bound neutron. For ¹⁹C, it was not clear if its valence neutron satisfies that condition. Since the 13th neutron in ¹⁹C occupies an orbital in the sd shell, the following two possibilities for the ground state configuration should be considered:

$\begin{eqnarray} &&|^{19}{{\rm C}(1/2}^+)\rangle = \alpha|^{18}{{\rm C}(0}^+)\otimes \nu 1{{\rm s}}_{1/2}\rangle + \beta|^{18}{{\rm C}(2}^+)\otimes \nu 0{{\rm d}}_{5/2}\rangle, \nonumber\\ \end{eqnarray}\noindent \tag{ 18 }$

$\begin{eqnarray} &&|^{19}{{\rm C}(5/2}^+)\rangle = \gamma|^{18}{{\rm C}(2}^+)\otimes \nu 1{{\rm s}}_{1/2}\rangle + \delta|^{18}{{\rm C}(0}^+)\otimes \nu 0{{\rm d}}_{5/2}\rangle, \nonumber\\ \end{eqnarray} \tag{ 19 }$

where α, β, γ and δ denote the spectroscopic amplitude for each configuration. It is interesting to note that the shell model calculations [61, 62] suggest that these two states are almost degenerate in energy.

A kinematically complete measurement of the Coulomb breakup of ¹⁹C at 67 MeV per nucleon was carried out at RIKEN [18]. Figure 12 (left) shows the Coulomb breakup cross section as a function of E_rel. Here, the nuclear breakup component was subtracted using the breakup spectra for a C target, assuming that the breakup cross section with the light target arises from the nuclear breakup origin, and the nuclear breakup cross section is scaled by the target + projectile radii. The method of selecting the forward angles, used in more recent ¹¹Be data [15] shown above was not adopted in these data due to lower statistics. The Coulomb breakup spectrum in figure 12 (left) shows an asymmetric shape with a large cross section of 1.19 ± 0.11 barn (b), whose large amplitude and the shape clearly show the halo characteristics.

The data are compared with the direct-breakup calculations for the possible four configurations, as shown in figure 12 (left). It is shown that good agreement is obtained for a higher S_n value of 530 keV for |¹⁸C(0⁺)⊗ν1s_1/2〉 with J^π = 1/2⁺. This S_n value 530 keV adopted here was obtained from an independent analysis using the angular distribution, as shown below. The large spectroscopic factor of α² = 0.67 was also deduced. The obtained configuration matches the prediction by a shell model calculation with the WBP interaction [61, 63]. The dominance of a weakly bound s-wave valence neutron shows that ¹⁹C is another well-established case of the 1n halo nucleus as in ¹¹Be.

The mass of ¹⁹C (more precisely, S_n) was extracted from the Coulomb breakup for the first time in this experiment [18]. The result has been indeed incorporated into the newer mass evaluation [64]. Here, a feature of Coulomb breakup, namely that the angular distribution of ¹⁹C (¹⁸C + n system) is sensitive to S_n, was used. As in equation (8), the angular distribution of the cross section is proportional to that of the E1 virtual photon (dN_E1/dΩ). The point is that the angular distribution of the photon is a function of E_x, which is related to S_n via E_x = E_rel + S_n. Note that the determination of S_n from the angular distribution is less model dependent compared to the determination of S_n from the peak position of the B(E1) spectrum which needs to assume the spectral shape to be the pure direct breakup.

Figure 12 (right) shows the measured angular distribution in comparison with the calculated spectra, where an angular resolution of 8.4 mrad (in 1σ) is folded. The value S_n = 530 ± 130 keV gives a best fit to the distribution, whereas the previously adopted value of S_n = 160 keV did not reproduce the angular distribution. This higher S_n value was confirmed by the measurement of the momentum distribution of ¹⁸C in coincidence with the de-excitation γ-ray from the first excited state of ¹⁸C at MSU [65].

3.1.6. Application of the Coulomb breakup to the radiative capture reaction ¹⁴C(n,γ)¹⁵C of astrophysical interest.

Coulomb breakup is also useful in extracting the radiative neutron capture reaction (n,γ), which is the inverse of the (γ,n) reaction. Coulomb breakup has been used in the application to nuclear astrophysics [66, 67], not only for the cases relevant to the neutron halo, but also and more widely for the proton-rich nuclei, i.e. proton radiative capture.

Here, we show the case of the Coulomb breakup of ¹⁵C whose inverse process is ¹⁴C(n,γ)¹⁵C.

The following equation based on the principle of the detailed balance relates the photo-absorption cross section (σ_γn) to the radiative neutron capture cross section (σ_nγ), i.e.

$\begin{equation} \sigma_{{{\rm n}}\gamma}(E_{{\rm c.m.}}) = \frac{2I_A + 1}{2I_{A-1}+1} \frac{E_\gamma^2}{2\mu c^2 E_{{\rm c.m.}}} \sigma_{\gamma {{\rm n}}}(E_\gamma), \end{equation} \tag{ 20 }$

where I_A = 1/2 and I_A−1 = 0 for the present case, and μ denotes the reduced mass of ¹⁴C + n, and E_c.m. is the c.m. energy of n + ¹⁴C which is equivalent to E_rel.

Due to the phase-space factor, the photo-absorption cross section is significantly larger than the inverse process. For the current case with E_c.m. = 0.5 MeV, this ratio is very large, as σ_γn(E_γ)/σ_nγ(E_c.m.) ≃ 150. The Coulomb breakup has an additional advantage that the photon numbers are further multiplied, which is of 2–3 orders of magnitude. Furthermore, the kinematic focusing and the availability of a thick target of the order of 100 mg cm⁻²–1 g cm⁻², are also advantages, which can compensate for a weak intensity of the RI beam. Note also that the Coulomb breakup of ¹⁵C can avoid the difficulty of using the radioactive ¹⁴C target as well.

The neutron capture reaction on ¹⁴C is considered to be important in the nucleosynthesis possibilities, such as the neutron-induced CNO cycle [68], revised r-process scenario initiating from the lightest elements [69, 70] and the inhomogeneous big-bang nucleosynthesis [71]. The Coulomb breakup of ¹⁵C is also important in terms of the spectroscopic significance of the ¹⁵C nucleus itself. ¹⁵C has a moderate-sized neutron halo with a neutron separation energy S_n of only 1.218 MeV. Its main configuration is |¹⁴C(0⁺)⊗ν1s_1/2〉 upon the N = 8 core of ¹⁴C. Coulomb breakup of ¹⁵C is expected to proceed via soft E1 excitation due to the direct breakup mechanism.

In such a situation, the direct neutron capture process for a p-wave neutron [50] is considered to be a dominant process in the inverse reaction. This particular capture process is called direct neutron capture since the neutron in the continuum is captured onto the ¹⁴C nucleus, as the inverse process of direct Coulomb breakup. The p-wave capture process, as a stellar reaction, is an exceptional case, considering that the neutron capture of a low-energy s-wave neutron which follows the 1/v law is usually dominant.

Another important feature of this Coulomb breakup is that the inverse reaction of the neutron capture of ¹⁴C has been measured [72, 73], and one can thus evaluate the validity of the Coulomb breakup (or vice versa). There were three Coulomb breakup measurements [16, 17, 74]. The latter two [16, 17] are consistent with the newer neutron capture measurement [73]. Here we introduce the most recent Coulomb breakup result [17], where the experiment was done at 68 MeV per nucleon at RIKEN.

Figure 13 (left) shows the Coulomb breakup cross sections of ¹⁴C + n as a function of E_rel obtained in the breakup of ¹⁵C on a Pb target at 68 MeV per nucleon. For extracting the Coulomb breakup component, we adopt the same procedure as in the Coulomb breakup of ¹¹Be [15], using the angular selection at forward angles. The solid squares show the breakup cross section integrated over the scattering angular range 0° ⩽ θ ⩽ 6.0° which nearly corresponds to the whole acceptance. Open circles show the cross section for a selected angular range, 0° ⩽ θ ⩽ 2.1°, which corresponds to the impact parameter range b > 20 fm.

The breakup cross sections integrated up to E_rel = 4 MeV are 670 ± 14(stat.) ± 40(syst.) mb and 294 ± 12(stat.) ± 18(syst.) mb for the whole acceptance (0–6°) and the selected angular range (0–2.1°), respectively. Although the spectral shape is attributed to Coulomb breakup of halo nuclei, the cross section is about 1/5–1/3 of those for the conventional halo nuclei such as ¹¹Be [15], and ¹⁹C [18]. This indicates that the size of the halo in ¹⁵C is not as large as in the more weakly bound halo nuclei. The B(E1) spectrum has been extracted for the angular selected data, and is shown in figure 13 (right), showing excellent agreement with the direct breakup calculation, as shown in the solid curve. The dot-dashed curve shows the calculation before folding the experimental resolutions.

Figure 14 shows the neutron capture cross section σ_nγ extracted from the B(E1) spectrum by applying the principle of detailed balance (equation (20)). Excellent agreement is obtained with the p-wave direct radiative capture model calculation [50]. This result is consistent with the consideration that the final state of the ¹⁴C capture reaction (namely ¹⁵C g.s.) is a halo state with a dominant s-wave component. The result is also found to be consistent with the neutron capture measurement in [73]. The overall agreement of the Coulomb breakup result with the direct capture measurement suggests that Coulomb breakup can be a good alternative for obtaining the neutron capture cross sections involving radioactive nuclei. One should note, however, that in the inverse reaction of Coulomb breakup, one can restrict the reaction to the ground state. For ¹⁵C, there is only one excited state at 0.74 MeV, and the neutron capture to this state is estimated to be negligible [46, 71, 75]. Although such fortunate situations may be scarce for heavier neutron-rich nuclei, we may expect lower level densities near closed shells, such as N = 50 and 82, where Coulomb breakup can be applicable. These nuclei may be relevant to the r-process, and thus may also be of importance.

3.1.7. Inclusive Coulomb breakup of ³¹Ne.

The experimental efforts of Coulomb breakup on halo nuclei are now directing toward the heavier nuclei. As shown in the nuclear chart in figure 7, nuclear species investigated by the Coulomb breakup have been limited to the very light neutron-rich nuclei as far as the halo physics is concerned. However, the halo could be formed in the heavier nuclei. The questions are how the halo structure appear in heavier neutron-drip-line nuclei and whether halo states can appear in many regions of the nuclear chart or not. These are not trivial since in heavier regions configuration mixing is more complex, higher angular momentum is involved in the structure and a variety of deformed shapes is expected. Here, we show the first experiment of Coulomb breakup of nuclei in the heavier region over Z = 8 or over the sd neutron shell. Before this experiment, the Coulomb breakup has been measured only up to ¹⁹C, and has for long been the heaviest halo nucleus experimentally established.

The beam intensity of the neutron-drip-line nuclei in this region has been limited. One breakthrough was the start of the new-generation RI beam facility RIBF at RIKEN, whose layout is shown in figure 15. The final goal of this facility is to obtain all the heavy ion beams up to ²³⁸U at the intensity over 1 pμA (particle micro-ampere), corresponding to ∼6 × 10¹² particles per second. So far, for instance, a rather high maximum intensity of 250 pnA has already been achieved for ⁴⁸Ca. With this intense ⁴⁸C beam, one can now have access to neutron-rich nuclei nearly up to the neutron-drip line for B–Si.

**Figure 15.** The layout of the RIBF at RIKEN as of 2012, which represents the world's largest RI beam facility in the world at this moment. The sequence of the accelerators, for instance, RILAC, RRC, FRC, IRC and SRC, is used for the acceleration up to 345 MeV per nucleon.
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As demonstrated above, Coulomb breakup reaction is a powerful tool for investigating 1n halo nuclei in mapping the dB(E1)/dE_rel function by a kinematically complete measurement. However, as a first step, the inclusive Coulomb breakup may also be important, due to the simplicity and the feasibility with a smaller beam intensity. For ³¹Ne, we measured the inclusive Coulomb breakup cross section at 230 MeV per nucleon [34] at RIBF at RIKEN. The beam intensity was only about 5 counts per second with the ⁴⁸Ca beam at 345 MeV per nucleon with typical 60 pnA at that time.

Since the first observation of ³¹Ne in the late 1990s [76], ³¹Ne has been the next heavier candidate for a 1n halo nucleus due to the small 1n separation energy. According to the mass evaluation in 2003 [64], S_n was theoretically estimated as 0.332 ± 1.069 MeV, while the recent direct mass measurement showed S_n to be 0.29 ± 1.64 MeV [77]. Experimental studies of ³¹Ne are thus important as to how and in what form heavier halo nuclei appear. Once the halo structure of ³¹Ne is established, we may obtain a key to understand how a single-particle configuration plays a role in halo formation. ³¹Ne (N = 21) is also interesting in that it may be inside the island of inversion where shell model calculations showed that N = 20 magicity is lost and significant 2p–2h configurations are mixed [78–80]. Note that the neighboring nuclei ³²Mg [81], ³²Na [82] and ^30,32Ne [83, 84] were found within this island experimentally.

It should be noted that for the conventional shell order, where the valence neutron resides in the 1f_7/2 orbital upon the ³⁰Ne core, a halo never develops since the tunneling effect is blocked by the high centrifugal barrier. The halo formation of ³¹Ne is only possible when a strong shell modification occurs, such that the 1p_3/2 orbital lowers below the 0f_7/2 orbital. As mentioned in section 2.2, the halo signature can be obtained by the inclusive cross section. For ³¹Ne, the significant cross section of about 0.5 b or more can be obtained only when the soft E1 excitation occurs.

Figure 16 (left) shows the 1n removal cross sections of ^19,20C and ³¹Ne on Pb and C targets obtained in this experiment. We readily find that the cross sections of ¹⁹C and ³¹Ne are both significantly larger than that of ²⁰C. Secondly, the ratio of the cross section for Pb to that for C is 9.0 ± 1.1 for ³¹Ne and 7.4 ± 0.4 for ¹⁹C, much larger than the ratio for nuclear breakup only, which is estimated to be about 1.7–2.6. This demonstrates that the cross section for a Pb target is dominated by Coulomb breakup for ¹⁹C and ³¹Ne.

The Coulomb breakup component of the 1n removal cross section on Pb was deduced by subtracting the nuclear component estimated from σ_−1n(C). The Coulomb breakup cross section for ³¹Ne was thus obtained to be σ_−1n(E1) = 540 ± 70 mb. The dominance of the Coulomb breakup for the reaction of ³¹Ne on Pb and the deduced σ_−1n(E1) of some 0.5 b, nearly as high as the established halo nucleus ¹⁹C, indicates the occurrence of soft E1 excitation for ³¹Ne, which provides evidence for 1n halo structure of ³¹Ne.

According to the direct breakup mechanism, the single-particle structure of the ground state of ³¹Ne can be examined. Figure 16 (right) compares the experimentally deduced σ_−1n(E1) with calculations for possible valence–neutron configurations as a function of S_n. Since there is a large experimental uncertainty in the S_n value of ³¹Ne, the calculations were shown as a function of S_n. The results of the direct-breakup calculations for the valence neutron either in the 1s_1/2, 0d_3/2, 0f_7/2 or 1p_3/2 orbital, being coupled to the ground state of ³⁰Ne with C²S = 1. A more detailed analysis including possible configurations of the valence neutron coupled to the ³⁰Ne(2⁺₁) state is presented in [34]. The comparison in figure 16 (right) shows that the data are reproduced by the configuration of |³⁰Ne(0⁺₁)⊗ν1p_3/2〉 (J^π = 3/2⁻) or |³⁰Ne(0⁺₁)⊗ν1s_1/2〉 (J^π = 1/2⁺), while not by |³⁰Ne(0⁺₁)⊗ν0d_3/2〉 or |³⁰Ne(0⁺₁)⊗ν0f_7/2〉. The significant contribution of such a low-ℓ valence neutron is again consistent with the 1n halo structure in ³¹Ne. The other important implication of this result is that the conventional shell model configuration of |³⁰Ne(0⁺₁)⊗ν0f_7/2〉 for the N = 21 nucleus does not represent a primary configuration of the ³¹Ne ground state. Large-scale Monte-Carlo shell model (MCSM) calculations employing the SDPF-M effective interactions [80] support the assignment of J^π = 3/2⁻ having a |³⁰Ne(0⁺₁)⊗ν1p_3/2〉 contribution, which is consistent with the current findings.

The shell model calculations (MCSM) also showed the large configuration mixing, where |³⁰Ne(0⁺₁)⊗ν1p_3/2〉 could be mixed with |³⁰Ne(2⁺₁)⊗ν1p_3/2〉 and |³⁰Ne(2⁺₁)⊗ν0f_7/2〉. Such a large configuration mixing may be described in terms of deformation. Recently, the current data were interpreted by a deformed mean-field model [85]. There, the 21st neutron (1p_3/2) can be orbiting in a deformed mean-field potential, namely in the Nilsson levels [330]1/2⁻ or [321]3/2⁻, although the possibility of an s-wave valence neutron ([200]1/2⁺) in a strongly deformed mean field (β > 0.59) was not fully excluded. More recently, the ground state properties of ³¹Ne were also discussed using the particle-rotor model, which takes into account the rotational excitation of the ³⁰Ne core [86]. It is interesting to note that halo properties, such as direct breakup dynamics and soft E1 excitations, are kept for the wave functions obtained in a deformed mean-field potential.

One neutron halo structure of ³¹Ne has recently been confirmed also by the reaction cross section measurement of this nucleus at RIBF [87]. Further experimental work on determining the content of the halo structure of ³¹Ne is highly called for. In particular, it is important to determine S_n as well as the configuration mixing experimentally. For that, a kinematically complete measurement of ³¹Ne would be one of the key experiments, where C²S and S_n can be extracted unambiguously for this 1n halo nucleus. At RIBF at RIKEN, such experiments are expected to be performed using the SAMURAI (Superconducting Analyser for MUlti-particles from Radio-Isotope beam) facility (see figure 20 in section 3.3).

3.2. Soft E1 excitation of 2n halo nuclei

The soft E1 excitation of 2n halo nuclei has a different aspect from that of 1n halo nuclei, i.e. it has two neutrons involved in the initial state and excitation. The three-body nature makes the interpretation of the experiment difficult. As will be mentioned, the two neutron correlation in the initial state and the FSIs between n–core and n–n will play important roles.

The first Coulomb breakup measurement for halo nuclei was made for ¹¹Li in an inclusive manner, which showed significant enhancement of low-energy dipole strength as mentioned in [5]. Later on, final-state exclusive, kinematically complete measurements of Coulomb breakup were made for ⁶He [19, 88], ¹¹Li [20, 39–41] and ¹⁴Be [89]. Here, the kinematical complete measurements of ⁶He and ¹¹Li are primarily described in order to clarify the characteristic features.

3.2.1. Coulomb breakup and soft E1 excitation of ¹¹Li.

In the 1990s, kinematical complete measurements for ¹¹Li were made at 28 MeV per nucleon at MSU [39], at 43 MeV per nucleon at RIKEN [40] and at 280 MeV per nucleon at GSI [41] (see figure 17 (right)). However, these results are controversial with each other. More recently, new data on ¹¹Li were obtained [20] with much higher statistics and with higher sensitivity for two-neutron detections.

The controversial results on ¹¹Li are considered to arise from the difficulty of the multi-neutron measurements. The fast-neutron measurement made above uses plastic or liquid scintillators, where the recoiled protons or knocked-out protons in the detector ingredients are a major source of the detection. In this process, this neutron that induced this recoil particle is scattered, and if it induces another signal at a different part of the detector, then the hit signals detected become double. Such an event is called cross talk, which mimics the true two-neutron hit.

The Coulomb breakup measurement of ¹¹Li on Pb at 70 MeV per nucleon at RIKEN [20] used the setup shown in figure 17 (left), which took care of the cross talk events as follows. The neutron detectors composed of 54 rods of plastic scintillators (214(H) × 6.1(V) × 6.1(D) cm³ each) were arranged into two walls (12 × 2 rods for the front (NEUT-A) and 15 × 2 rods for the rear (NEUT-B)), separated by 1.09 m. This separation is essential to distinguish the two neutrons by imposing the cross-talk-cut condition. When two hits are detected in NEUT-A and NEUT-B independently, one can reconstruct the velocity v_A for the hit in A, and the apparent velocity v_AB between the two walls from the hit timing of each wall. Then we discard all the events with v_A ⩾ v_AB, thereby excluding cross talk events since the scattered neutron has less velocity than that of the incoming neutron. This measurement also uses the two hits in the same wall (see [20, 47] for details). This revised measurement enhanced significantly the sensitivity down to E_rel ∼ 0 MeV in the B(E1) spectrum.

Figure 17 (right) shows the B(E1) distribution in this experiment, which is compared with the previous three data sets. The result shows substantial E1 strength at very low E_rel around 0.3 MeV. This feature is in contrast to the previous data, which showed much reduced strength toward low relative energies. This is considered due to the lower neutron detection efficiency near E_rel ∼ 0 MeV in those experiments.

The spectrum amounts to an energy-integrated B(E1) strength of 1.42 ± 0.18 e² fm² (4.5(6) W.u.) for E_rel ⩽ 3 MeV, which is the largest soft E1 strength ever observed. In fact, this E1 strengths is about 40% larger than that for the one-neutron halo nucleus ¹¹Be [14, 15]. The extraction using the equivalent photon method depends on the S_2n value since the virtual photon spectrum depends on E_x(=E_rel + S_2n). Nakamura et al [20] adopted S_2n = 0.3 MeV from the 2003 mass evaluation [64]. When one uses the most recent precise mass measurement result of ¹¹Li (S_2n = 0.36915(65) MeV) [90] using the Penning trap technique, the E1 strength for ¹¹Li is reevaluated as 1.49(19) (e² fm²) for E_rel ⩽ 3 MeV, or about 5% larger. Such a slight change within the experimental uncertainty does not change the conclusion, however.

Figure 17 (right) also compares the data with the three-body model calculation of Esbensen and Bertsch [45], which includes the two-neutron correlations in both the initial and final states in the E1 excitation. This calculation reproduces the data very well with no adjustment of normalization. Recently, a revised three-body calculation by Esbensen et al [91], which took into account the recoil effect, also showed overall agreement with the data, although some deviation occurs around E_rel ∼ 0.5 MeV (see figure 2 of [91]). This deviation may indicate the effect of the core excitation or other effects. The recent calculation by Myo et al [92], which incorporated the effect of the core polarization using the tensor optimized shell model, reproduced better the peak region. These overall agreements of both the spectral shape and absolute strength in these calculations indicate the presence of a strong two-neutron correlation in ¹¹Li, in both the ground and final states (FSIs).

3.2.2. Coulomb breakup and soft E1 excitation of ⁶He.

⁶He exhibits a Borromean structure composed of α + n + n. The S_2n value is 972.44(66) keV [64], larger than that of ¹¹Li by a factor of about 3. The shell configuration of the two halo neutrons in ⁶He is p dominant, while that of ¹¹Li has a mixture of s and p. The other difference is that the core of ⁶He is a rigid ⁴He with J^π = 0⁺, while that of ¹¹Li is a soft ⁹Li with an odd spin (J^π = 3/2⁻). Hence, it is expected that the ⁶He Coulomb breakup could provide a more rigorous test of the three-body- and reaction models of the soft E1 excitation of 2n halo nuclei.

For the Coulomb breakup of ⁶He, two kinematically complete measurements at 25 MeV per nucleon on U at MSU [88] and at 240 MeV per nucleon on Pb at GSI [19] were made. The B(E1) spectra from these two data show apparent different behavior in strength and in shape. This difference could be due to the different sensitivity of the two data sets. Since the experiment at MSU [88] had a much smaller acceptance which cut the spectra at higher excitation energies, we hereafter discuss the experiment at GSI [19].

The result of GSI is shown in figure 18 for the cross section and the extracted B(E1) spectrum using the equivalent photon method. The integrated B(E1) sum is 0.59(12) e² fm² (2.8(6) W.u.) for E_x ⩽ 5 MeV and 1.2(2) e² fm² (5.7(10) W.u.) for E_x ⩽ 10 MeV. Compared to ¹¹Li, the absolute value of E1 strength is smaller, and the spectrum shows a broader peak, extending over 6 MeV in E_rel.

Qualitatively, such a difference may be attributed to the fact that ⁶He is more bound than ¹¹Li, and is p-wave dominant which has a finite centrifugal barrier even though it is small. Theoretically, there are quite a few theoretical publications for the electromagnetic excitation of ⁶He [93–103]. Theoretical interpretations of the data are still not fully established, and in this review we cannot fully trace those aspects. We introduce here one recent theoretical study by Kikuchi et al [98], which investigated separately the effects of direct breakup into non-interacting three-body particles and those of initial and FSIs.

Figure 19 shows the comparison in [98] with the data, where the left figure shows the comparison with the cross section dσ/dE_x, whereas the right two figures show the cross sections in terms of the two-body relative energies α + n and n + n. We find overall agreement for the three-body decay spectrum, although a slight enhancement around E_rel ∼ 1 MeV and hindrance over 2 MeV in the theoretical curve is seen. On the other hand, better agreement is achieved for the two-body decay spectra. In particular, the α + n data are in excellent agreement showing a strong effect of the ⁵He(3/2⁻) ground state, i.e. the effect of the FSI between α and neutron. Such an effect was already pointed out by the data analysis in [19]. Kikuchi et al [98] also found the effect of the strong nn FSI, while the nn correlation in the initial ground state affects negligibly the spectrum. In the next section, we discuss the non-energy weighted cluster sum rule, where the ground state nn correlation can be revealed in the summed E1 strength.

3.2.3. Non-energy weighted cluster sum rule for 2n halo nuclei and dineutron correlation.

The spatial two-neutron correlation in the ground state (initial state) of two-neutron halo nuclei can be quantitatively estimated by the non-energy weighted E1 cluster sum rule [45, 104]. Such a sum rule was discussed for 1n halo nuclei as in equation (9). For the case of a two-neutron halo nucleus, the E1 operator $\vec {r}$ is replaced by $\vec {r_1}+\vec {r_2}$ , where $\vec {r_1}$ and $\vec {r_2}$ are the position vectors of the two valence neutrons relative to the c.m. of the core. Namely, the sum rule is written as

$\begin{equation} B(E{1}) = \frac{3}{4\pi} \left(\frac{Ze}{A}\right)^2 \langle r_1^2 + r_2^2 + 2 \vec{r_1} \cdot \vec{r_2} \rangle = \frac{3}{\pi} \left(\frac{Ze}{A}\right)^2 \langle r^2_{c,2{\rm n}} \rangle. \end{equation} \tag{ 21 }$

Here, $\vec {r_1}+\vec {r_2}$ can be related to the rms of r_c,2n, which is the distance from the center of the core to the c.m. of the two halo neutrons. Note that the term of $\vec {r_1}\cdot \vec {r_2}$ involves the opening angle θ₁₂ between the two position vectors of the two valence neutrons, i.e. the information on the two-neutron correlation. The value of 〈r_c,2n〉, and hence B(E1), becomes larger for smaller spatial separation of the two neutrons when θ₁₂ approaches 0°. The integrated B(E1) thus provides a good measure of the two-neutron spatial correlation.

For the ¹¹Li data in [20], the integral of the E1-response calculation up to E_rel = 3 MeV accounts for about 80% of the total E1 cluster sum-rule strength when one assumes the extrapolation according to the calculation in [45]. The estimated B(E1) strength of 1.78 ± 0.22 e² fm² for E_x ⩾ S_2n is thus deduced, which corresponds to $\sqrt {\langle r^2_{c,2\mathrm {n}}\rangle } = 5.01 \pm 0.32\,{\mathrm { fm}}$ , which is consistent with other theoretical estimations of 5–6 fm [91, 92, 107]. The charge radius measurement of ¹¹Li extracted $\sqrt {\langle r^2_{c,2\mathrm {n}}\rangle }=5.97(22)$ [91, 105, 106]. The slight deviation between the value from the Coulomb breakup and the charge radius could be due to a possible core polarization effect [91].

Assuming the sum rule value of 1.07 e² fm² for the two non-correlated neutrons as calculated in [45], the total experimental strength of 1.78 ± 0.22 e² fm² can be translated to 〈θ₁₂〉 = 48⁺¹⁴₋₁₈ degrees. This angle is significantly smaller than the mean opening angle of 90° expected for the two non-correlated neutrons. Hence, an appreciable two-neutron spatial correlation is suggested for the two halo neutrons. An important feature of the soft E1 excitation of 2n halo nucleus is that the larger polarization of the charge due to a stronger dineutron-like correlation enhances the soft E1 excitation.

It can be shown that the spatial nn correlation is related to a mixture of different parity states in the valence neutrons of ¹¹Li. In a simple shell model picture where the inert ⁹Li core is assumed, the ¹¹Li could be described as

$\begin{eqnarray} \Psi(^{11}{{\rm Li}}({{\rm g.s.}})) &=& \alpha|^{9}{{\rm Li(g.s.)}}\otimes\nu (1{{\rm s}}_{1/2})^2\rangle \nonumber\\ &&+ \beta|^{9}{{\rm Li(g.s.)}}\otimes\nu (0{{\rm p}}_{1/2})^2\rangle. \end{eqnarray} \tag{ 22 }$

Here, we are interested in the mean opening angle between $\vec {r_1}$ and $\vec {r_2}$ . The expectation value of cos θ₁₂ can be written as

$\begin{eqnarray} \langle \cos{\theta_{12}}\rangle &=& \alpha^2 \langle(1{{\rm s}})^2| \cos{\theta_{12}} |(1{{\rm s}})^2\rangle + \beta^2 \langle(0{{\rm p}})^2| \cos{\theta_{12}} |(0{{\rm p}})^2\rangle \nonumber \\ & &+ 2\alpha\beta \langle(0{{\rm p}})^2| \cos{\theta_{12}} |(1{{\rm s}})^2\rangle \end{eqnarray}\noindent \tag{ 23 }$

$\begin{eqnarray} &&= 2\alpha\beta \langle(0{{\rm p}})^2| \cos{\theta_{12}} |(1{{\rm s}})^2\rangle. \end{eqnarray} \tag{ 24 }$

The terms for α² and β² are null since these are odd functions of cos θ₁₂. Hence, a non-zero expectation value implies a mixture of these different parity configurations. Namely, no mixture of different parity states (either α² = 0 or β² = 0) gives rise to 〈θ₁₂〉 = 90° and no 2n correlation, while the mixture leads to the 2n spatial correlation.

A similar discussion can be applied also to ⁶He. The authors of [19] showed that the B(E1) strength up to E_x = 10 MeV in ⁶He is translated into $\sqrt {\langle r^2_{c,2\mathrm {n}}\rangle }=3.36(39)\,{\mathrm { fm}}$ . The opening angle estimated in [107] using this value is 83⁺²⁰₋₁₀ degrees, which is certainly larger than the case of ¹¹Li. The closeness to 90° shows that the two-neutron halo may have a smaller mixture of different parities from p states.

3.3. Perspectives on the soft E1 excitation

We find that the Coulomb breakup of 1n halo nuclei and the revealed enhanced low-energy strength (soft E1 excitation) can be well understood by the direct breakup mechanism, thereby providing a good experimental tool to probe the single-particle halo state. The Coulomb breakup of 2n halo nuclei provides, in addition, information on dineutron correlation for 2n halo nuclei. However, we note that full understanding of Coulomb breakup of 2n halo nuclei is still on the way. There are quite a few theoretical works on the Coulomb breakup on ¹¹Li (see, for instance, [91, 102, 104, 108–117]), which are still controversial. For the two-neutron halo case, the non-resonant character seems to persist for 2n halo nuclei except for the strong core-n and nn FSIs. However, for instance, a more recent theory [117] relates the enhanced strength with a strong dipole resonance nature of ¹¹Li. In this sense, the issue whether it is resonance or non-resonance remains to be solved.

Experimentally, it is important to explore other 2n halo cases with different shell properties. The most neutron-rich carbon isotope, ²²C, provides such a case, which was recently confirmed as a two-neutron halo nucleus from the reaction cross section measurement [118] and the 2n-removal reaction and the momentum distribution of ²⁰C fragment in the ²²C + C reaction at RIBF [119]. The two neutrons in ²²C should reside in s and d [118, 119], although the detailed mixture is still controversial and should be investigated further. At RIBF at RIKEN the inclusive Coulomb breakup of ¹⁹B and ²²C on Pb around 240 MeV per nucleon has recently been measured, and the results show the enhanced cross section of about 1 b, consistent with the revelation of soft E1 excitation. More recently, kinematical complete measurements for ²²C have also been made in 2012 at the newly commissioned large acceptance spectrometer SAMURAI as shown in figure 20. We also note that further high-precision measurement for ⁶He and ¹¹Li is necessary to pin down the nature of Coulomb breakup and soft E1 excitation of 2n halo nuclei. This is also important for obtaining clearer evidence of the dineutron correlation, and its understanding. In the near future, we also note that the 4n halo could be in the scope.

**Figure 20.** The schematic view of the SAMURAI facility at RIBF, which was commissioned in 2012. The secondary beam, for instance ²²C, comes from the left-hand side of this figure, impinges on the target, located in the inner side of the DALI2 detector array. The breakup particle, for instance ²⁰C, is bent about 60° in the superconducting magnet at about 3 T, detected with the heavy-ion detectors (the two multi-wire drift chambers FDC1 and FDC2, and the plastic scintillator hodoscope HODF). The fast neutrons are detected in NEBULA neutron detectors, which have 120 modules of plastics scintillators, each of which has 12(H) × 12(D) × 180(V) cm³.
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4. Giant and pygmy resonances in neutron-rich nuclei

4.1. Experimental investigations on the multipole response of exotic nuclei

The excitation spectra of nuclei are dominated by collective giant resonances of various different character. The question of how the multipole response of nuclei evolves when moving from stable to more and more neutron–proton asymmetric systems has been studied in many theoretical works and also experimentally. For an overview of theoretical studies, see the review article by Paar et al [25]. From the experimental point of view, it is mainly the electric dipole response which has been studied so far due to experimental difficulties. Isoscalar monopole and quadrupole giant resonances, for instance, are studied usually in α scattering experiments at low momentum transfer. In inverse kinematics, this results in very-low-energy recoiling α particles. In order to be able to measure the angular distribution and energy of the recoils, very thin targets, i.e. gas targets, are required implying a low luminosity. An attempt has been made by using an active target at GANIL [120] for an investigation of the GMR and GQR in ⁵⁶Ni in deuteron scattering. At GSI, a method is being developed to study elastic and inelastic hadron and light-ion scattering in the storage ring by using a gas-jet target [121, 122]. The luminosity that can be reached with this method is much higher and thus will be very well suited for giant-resonance studies in the future.

So far, mainly the process of heavy-ion-induced electromagnetic excitation as discussed above has been employed in the study of radioactive nuclei. In order to excite the nucleus up to energies in the giant-resonance domain, however, rather high beam energies are needed due to the adiabatic cutoff, see above. Typical beam energies used are around 500 MeV per nucleon. The cross section is dominated by electric dipole transitions. Typically, the cross section for exciting the giant quadrupole resonance is about one order of magnitude smaller than the one for exciting the isovector GDR at energies around 500 MeV per nucleon. For high-precision measurements, however, different multipoles have to be considered. Measurements at different beam energies as well as a precise measurement of the scattering-angle distribution can be used to decompose the different multipolarities. The results from the experiments with radioactive beams discussed below have not been corrected for possible quadrupole contributions. In the analysis, however, contributions from E2 excitations have been taken into account by assuming standard parameters for the giant quadrupole resonance.

In addition to possible admixtures of excitations other than E1, nuclear excitations have to be considered. For impact parameters significantly larger than the sum of the two radii of the target and the projectile, only the electromagnetic interaction is important, while for smaller impact parameters, nuclear processes dominate. For the calculation of the electromagnetic cross section, either a minimum impact parameter is considered, or the nuclear processes are explicitly taken into account [123]. Most of the nuclear reaction channels, however, populate final states other than considered here. Only inelastic scattering, i.e. excitation of the projectile to the excitation-energy region of interest with subsequent decay (excluding individual nucleon–nucleon collision with large momentum transfer), contribute to the cross section of interest. In the small impact parameter range, where nuclear inelastic excitation is important, also electromagnetic excitation is competing and the two amplitudes might interfere. Since the integrated cross section for electromagnetic excitation is much larger in the cases considered here, and only a small fraction of the cross sections stems from an impact parameter range where both processes are competing, the effects due to interference are usually small, and in particular for scattering-angle integrated cross sections negligible. In extracting the cross section for electromagnetic excitation, the nuclear contribution is determined usually by a measurement with a carbon target and then subtracted from the total cross section after proper scaling. Typically, the nuclear contribution in the cases discussed here is of the order of 10–20%.

Differential cross sections with respect to excitation energies can be deduced using the invariant-mass method and a measurement of the four momenta of all outgoing particles, i.e. a kinematically complete measurement of the final state. The experimental setup used at GSI is shown schematically in figure 21.

**Figure 21.** Schematic view of the LAND experimental setup at GSI. The incoming ion beam is tracked and identified event by event with projectile-tracking detectors. The target is surrounded by a modular NaI detector array (160 crystals) for photon detection and calorimetry (Crystal Ball). Charged fragments are deflected by the large-gap dipole magnet ALADIN and tracked by fragment-tracking detectors which serve in conjunction with energy-loss and time-of-flight measurements (TOF) for fragment identification. Neutrons are detected with an acceptance of typically ±80 mrad around 0° with the LAND detector placed 12 m downstream of the target. From the combined information the excitation energy can be reconstructed by using the invariant-mass method. The inset shows a typical example of the identification of the incoming beam projectiles. Figure reprinted with permission from Adrich *et al* [27]. Copyright (2005) by the American Physical Society.
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The energy resolution typically achieved by this method ranges from 0.2 MeV close to the threshold to around 1 MeV around the giant-resonance region of about 12–15 MeV. The resolution is dominated by the determination of the excitation energy of the fragment, which is obtained by a measurement of the photon decays, i.e. the γ sum energy, and by the determination of the neutron kinetic energy. The instrumental response, however, might be rather complex due to deficiencies of the photon detection, or due to misidentification of the number of outgoing neutrons (or the fragment mass). This might lead to a distortion of the reconstructed spectrum which is usually accounted for by a complete simulation of the response when comparing data to theory. This procedure includes the simulation of the neutron decay of continuum states to (excited) fragment states with subsequent γ decay. Next-generation experiments aim at a calorimetric γ detection, a high-resolution tracking for heavy fragments, as well as high-efficient tracking of multi-neutron events in order to achieve a less complex energy response of the system.

4.2. Dipole strength distribution in neutron-rich oxygen isotopes and ²⁶Ne

The first experiment measuring the dipole strength of unstable nuclei including the GDR region has been performed at GSI for neutron-rich oxygen isotopes ¹⁸⁻²²O [26] and for the 1n channel for ²³O [124]. From the measurement of the differential cross section dσ/dE* the photo-neutron cross section σ_(γ,xn) has been deduced for ^18,20,22O [26]. The deduced cross section for the stable isotope ¹⁸O is in good agreement with previous measurements using real photons. Figure 22 shows the results for the neutron-rich isotopes.

**Figure 22.** Left panels: photo-neutron cross sections σ_(γ,xn) for the unstable isotopes ²⁰O (upper panel) and ²²O (lower panel) as extracted from the measured electromagnetic excitation cross section (symbols) [26]. For comparison, the photo-neutron cross section for the stable ¹⁶O [127] is displayed as the solid curve in the upper panel. Right panels: evolution with neutron excess N − Z of integrated (up to 10 MeV excitation energy) strength $S_{\exp }$ in units of the Thomas–Reiche–Kuhn (TRK) sum rule S_TRK (upper panel) and of the cluster-sum-rule S_clus (lower panel) for oxygen isotopes. For the stable isotope ¹⁸O, experimental values from real-photon absorption experiments are given by the circle [125] and open square [126]. The values extracted from the electromagnetic-excitation experiment [26] are given by the filled squares. The data are compared with shell-model calculations of Sagawa and Suzuki [128] (solid curve in the lower left frame and stars in the right two frames) and the quasi-particle RPA calculation of Colò and Bortignon [129] (filled circles). Figure reprinted from Aumann [130]. With kind permission from Springer Science and Business Media.
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**Figure 22.** Left panels: photo-neutron cross sections σ_(γ,xn) for the unstable isotopes ²⁰O (upper panel) and ²²O (lower panel) as extracted from the measured electromagnetic excitation cross section (symbols) [26]. For comparison, the photo-neutron cross section for the stable ¹⁶O [127] is displayed as the solid curve in the upper panel. Right panels: evolution with neutron excess N − Z of integrated (up to 10 MeV excitation energy) strength $S_{\exp }$ in units of the Thomas–Reiche–Kuhn (TRK) sum rule S_TRK (upper panel) and of the cluster-sum-rule S_clus (lower panel) for oxygen isotopes. For the stable isotope ¹⁸O, experimental values from real-photon absorption experiments are given by the circle [125] and open square [126]. The values extracted from the electromagnetic-excitation experiment [26] are given by the filled squares. The data are compared with shell-model calculations of Sagawa and Suzuki [128] (solid curve in the lower left frame and stars in the right two frames) and the quasi-particle RPA calculation of Colò and Bortignon [129] (filled circles). Figure reprinted from Aumann [130]. With kind permission from Springer Science and Business Media.
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For comparison, the photo-absorption cross section for ¹⁶O measured with real photons [127] is overlaid on the spectrum for ²⁰O in the upper left frame. In contrast to ¹⁶O, where the dipole strength is concentrated in the giant-resonance region around 24 MeV, the strength is much more fragmented for the neutron-rich isotopes. The low-lying dipole strength extends down to the neutron separation threshold around 8 MeV. The redistribution of the strength is reproduced by the shell-model calculation of Sagawa and Suzuki [128], as shown by the solid curve for ²²O in the lower left frame. The experimental finding is also reproduced in relativistic RPA calculations by Vretenar et al [24]. In particular, the relativistic RPA not only reproduces the strong fragmentation, but also exhibits a low-lying peak in the dipole response of ²²O at around 9 MeV, very close to the experimental finding, see figure 22. Vretenar et al [24] discuss the collectivity of the low-lying strength and conclude that the dipole transitions in the neutron-rich oxygen isotopes mainly result from single-particle transitions. Although the transition strength of the low-lying peak shows a characteristic shape for a neutron-skin oscillation as anticipated for the pygmy resonance. From the same theoretical calculations, the authors also concluded that the low-lying strength in heavier neutron-rich nuclei like Ni or Sn isotopes develops a more collective character. We will come back to the experimental results for the heavier nuclei later in this section.

The right panels of figure 22 show the integrated strength up to 10 MeV as a function of neutron excess in units of the energy-weighted sum rule (upper frame) and the energy-weighted cluster sum rule (lower frame) assuming a ¹⁶O core. About 5–10% of the TRK sum rule exhaustion is observed below 10 MeV corresponding to a large fraction of the cluster sum rule of 40–80% with a decreasing trend for the heavier isotopes. The magnitude and trend are well reproduced by the shell model [128] (stars) as well as the quasi-particle RPA calculation by Colò and Bortignon [129] (filled circles). It is interesting to note that if the strength is integrated up to 15 MeV, still in a region where no strength is found in ¹⁶O, the experimentally observed strength exceeds the cluster sum rule implying that also core excitations play a role in that energy region for the neutron-rich isotopes [26].

The experiments just discussed have measured the strength only above the neutron separation threshold. In an experiment at MSU, Tryggestad et al [131, 132] have measured the strength also below threshold for ¹⁸O and ²⁰O by detecting the γ decays after Coulomb excitation at 100 MeV per nucleon. The measured γ spectrum is shown in the left part of figure 23, showing significant E1 strength in the region between 5 and 8 MeV, not present in the stable isotopes. The right part of the figure shows the extracted B(E1) and B(E2) strengths. The integrated value for the dipole strength below threshold is, however, much smaller than the one in the low-energy part above the threshold as discussed above.

**Figure 23.** Left panel: Doppler-corrected γ energy spectrum measured after electromagnetic excitation of 100 MeV per nucleon ²⁰O projectiles on a lead target [131, 132]. Right panels: dipole and quadrupole strength distributions below the neutron separation threshold as extracted from the γ spectrum (left frame). Figures reprinted with permission from Tryggestad *et al* [131]. Copyright (2002) by the American Physical Society.
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The low-energy dipole response has been studied in RIKEN for the neutron-rich isotope ²⁶Ne by Gibelin et al [133] using Coulomb excitation at 58 MeV per nucleon on a Pb target. One neutron and γ-rays from the decay of the excited nucleus were measured in coincidence for invariant-mass analysis. The excitation-energy spectrum for the 1n channel is shown in the upper part of figure 24.

**Figure 24.** Excitation-energy spectrum for the 1n-decay channel of ²⁶Ne excited on a Pb target (upper frame) and an Al target (lower frame) [133]. The shaded area indicates a Lorentzian fit to the data. The one- and two-neutron separation thresholds are indicated. Figure reprinted with permission from Gibelin *et al* [133]. Copyright (2008) by the American Physical Society.
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The lower frame shows the cross section obtained with the Al target representing mainly nuclear excitations. The spectra show a peak-like structure at an excitation energy of around 9 MeV, rather close to the low-lying dipole structure predicted by Cao and Ma [134] at 8.5 MeV. Also the extracted integrated E1 strength, which exhausts 4.9(1.6)% of the Thomas–Reiche–Kuhn (TRK) sum rule, is in agreement with theoretical predictions. The analysis of microscopical calculations suggests that the low-lying structure mainly stems from single-particle transitions of neutrons occupying the least bound orbits in ²⁶Ne (see discussion in [133] and references therein), similar to the interpretation discussed for the oxygen isotopes above. In the same experiment, the decay to the final fragment states has been analyzed as well. An interesting and surprising finding is that the excited ²⁶Ne decays exclusively to excited states of ²⁵Ne rather than to the ground state as predicted by the theoretical studies mentioned above.

4.3. Giant and PDRs in medium-heavy neutron-rich Ni and Sn isotopes

The low-energy dipole strength in light neutron-rich nuclei, as discussed above, has been interpreted theoretically as mainly arising from single-particle transitions rather than showing a collective nature as suggested by a macroscopic picture of the pygmy resonance of a collective vibration of excess neutrons versus a core nucleus. The onset of collectivity in the low-energy part of the dipole excitation spectrum for heavier nuclei has been discussed by Vretenar et al [24] in a relativistic RPA framework. They have shown for neutron-rich Ni and Sn isotopes that the low-energy peaks in the spectrum have a very different and more complicated character compared to the light nuclei involving many neutron p–h configurations. Vretenar et al concluded that the low-energy response in these nuclei indeed show the characteristics of a collective neutron vibration against the inert core composed of an equal number of neutrons and protons [24].

The first experimental evidence for a resonance-like structure in the low-energy dipole response was found for the neutron-rich isotopes ^130,132Sn at GSI [27]. The experiment used secondary beams produced by fission of a primary uranium beam. The secondary beams impinged on a Pb target with energies around 500 MeV per nucleon. The differential cross section dσ/dE* for one-to-three neutron decay after electromagnetic excitation has been extracted and is shown for ^130,132Sn in the left frames of the upper part of figure 25.

**Figure 25.** Upper panel: cross section for electromagnetic excitation of Sn isotopes on Pb at around 500 MeV per nucleon (left frames) and corresponding photo-neutron cross section (right frames) [27]. The solid curve is a fit to the data containing a Lorentzian plus a Gaussian for the GDR (dash-dotted) and pygmy (dotted) contributions, respectively. Lower panel: pygmy dipole strength distribution dB(E1)/dE obtained for unstable Sn and Sb isotopes [30] with odd neutron number (upper row) and with even neutron number (bottom row). The solid lines show the results for ^130,132Sn from the RQRPA calculation with a particular choice of the DD–ME interaction. Figures reprinted with permission from Adrich *et al* [27] and Klimkiewicz *et al* [30]. Copyright (2005) and (2007), respectively, by the American Physical Society.
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The right frames show the corresponding photo-neutron cross section, which can be directly compared with the photo-absorption cross section measured for stable nuclei. Evidently, the Lorentzian shape of the cross section for the GDR cannot account for the additional strength observed at low energy. The observed two-hump structure in the cross section is fitted by a Lorentzian for the GDR (dash-dotted curve) and a Gaussian accounting for the low-energy bump (dotted curve) which gives in total a good description of the measured spectrum (solid curve). The fit takes into account the experimental response including the distortions of the energy response due to gamma detection, misidentification of the number of neutrons or fragment mass, as well as all resolutions. The resulting parameters for the GDR for position width and sum rule exhaustion are in agreement with the expectation of the systematics of giant resonances collected for stable nuclei. Thus, although within the still large error bars, no systematic change of the frequency of the GDR mode in neutron-rich nuclei was observed. For the low-energy peak located at around 10 MeV, which exhausts around 5% of the energy-weighted sum rule, a lower limit on the width of 3.4 (2.5) MeV has been extracted for ¹³⁰Sn (¹³²Sn). The extracted strength agrees well with several theoretical predictions. It should be noted, however, that the predictions depend strongly on the effective interactions used, in particular on the symmetry energy as will be discussed below. The position of the pygmy mode is predicted in relativistic RPA calculations around 8–9 MeV, for some Skyrme forces around 10 MeV, i.e. closer to the experimental finding. For a comparison and discussion of the different theoretical results and models, see the review article of Paar et al [25].

The same collaboration has published data on low-lying strength in a couple of nuclei around ^130,132Sn [30] including nuclei with odd neutron number. The results after subtracting the GDR contribution and after conversion into dipole strength B(E1) is displayed in the lower part of figure 25. In all nuclei, strength of a similar magnitude is observed located between 6 and 12 MeV. It is interesting to note that the odd nuclei have significant lower neutron separation thresholds compared to even ones extending the sensitivity of the experiment to lower excitation energies (the experiment relied on the measurement in coincidence with at least one neutron). A clear odd–even staggering in the centroid is visible. The interpretation of this effect is not clear. It could be related to an odd–even staggering of the position of the pygmy strength, or it could be explained by missing strength for the even isotopes due to the higher threshold. It is evidently mandatory, that experiments with an extended detection scheme including the measurement of γ decays below the threshold have to be carried out in order to understand this behavior. The integrated strength in excess to the GDR tail for neutron-rich tin isotopes is significantly larger than the one found in stable nuclei in the same mass range. It has to be noted, however, that the strength extracted from (γ,γ') measurements at stable nuclei provides often a lower limit only since the measurement is sensitive only to the direct γ decay back to the ground state, which might be a small fraction depending on structure and level density. For the measurements with neutron-rich tin isotopes discussed above, on the other hand, additional non-observed strength below the threshold cannot be excluded. New more complete experiments are needed for both stable and unstable nuclei for clarification and for a proper comparison of the measurements for stable and unstable nuclei. For a more complete discussion including a comparison to the experimental findings with stable nuclei, see the recent review by Savran et al [22].

Another method based on heavy-ion-induced electromagnetic excitation has been employed by the RISING collaboration. In an experiment performed at GSI, the γ decay of ⁶⁸Ni projectiles at 600 MeV per nucleon has been measured after excitation on gold target nuclei [135]. As can be seen in the right part of figure 26, a peak-like structure can be observed in the Doppler corrected γ spectrum above a background stemming mainly from statistical decay of the GDR excited in the target nuclei or in the projectile.

**Figure 26.** Left: cross section for electromagnetic excitation and photon decay for ⁶⁸Ni at 600 MeV per nucleon on an Au target [135]. The Doppler corrected γ spectrum has been measured with BaF₂ detectors at GSI at the FRS and is compared to statistical-model calculations for the decay after target excitation (dotted line) and projectile excitation (dashed line). The inset shows a comparison of a GEANT simulation for a γ-transition at 11 MeV. The right panel shows the contributions of the GDR and the PDR to the background.subtracted γ spectrum. Figures reprinted with permission from Wieland *et al* [135]. Copyright (2009) by the American Physical Society.
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The excess cross section is peaked at around 11 MeV, i.e. above the neutron separation threshold at around 7.8 MeV. The apparent width of the peak mainly reflects the experimental resolution. The inset shows a GEANT simulation assuming a γ transition at 11 MeV with a very narrow width. The position of the peak is close to the predicted value by the relativistic RPA calculations of Vretenar et al [24] of 9 MeV, and the relativistic quasiparticle RPA (RQRPA) of Cao and Ma [134]. The result of an extended RQRPA calculation including coupling to more complicated states (two-phonon relativistic quasiparticle time blocking approximation) of Litvinova et al [136] shows a centroid of the pygmy strength at slightly higher energy peaking at 10.3 MeV rather close to the experimental finding.

The integrated cross section of the peak cannot be directly translated into a B(E1) value for the excitation since the transition lies above the neutron threshold resulting in a rather small probability for a direct photon decay. Wieland et al [135] estimated this branching ratio assuming a level density to be of the order of 4%, which yields a dipole strength for the 11 MeV peak corresponding to about 5% of the energy-weighted sum rule. The level density used was significantly smaller than the systematic one for stable nuclei. If using the latter would result in a smaller decay branch and thus in a larger B(E1) value corresponding to 9% of the sum rule. A new experiment measuring the neutron decay after electromagnetic excitation of ⁶⁸Ni has been carried out at the R3B-LAND setup at GSI [137]. Preliminary results shown on conferences show that low-lying dipole strength has been observed [137], which is located between 10 and 11 MeV and exhausting 4.1(1.9)% of the energy-weighted sum rule, which is in agreement with the RISING experiment. The final analysis will also allow for extracting the γ decay branching ratio for the pygmy resonance.

4.4. The dipole response of nuclei and the neutron skin

Since the low-lying dipole strength is related to an excitation of the less-well bound excess neutrons, it is clear that the low-lying dipole response should be related to the development of a neutron skin in medium–heavy neutron-rich nuclei. A quantitative analysis of this was first presented by Piekarewicz [28] who proposed that the systematics of the pygmy dipole mode may be used to constrain the density dependence of the symmetry energy. It has been pointed out by many authors, see for instance [138, 139], that the neutron-skin thickness calculated in mean-field models depends strongly on the density dependence of the symmetry energy (close to saturation density) incorporated into the effective interactions used. A similar dependence is found for the low-lying dipole strength. Klimkiewicz et al [30] used a relativistic interaction with a systematically changed symmetry-energy parameter a₄ and found an almost linear correlation with the low-lying dipole strength and the neutron-skin thickness, see figure 27.

**Figure 27.** Left panel: summed B(E1) strength exhausted by the low-energy peak relative to the strength exhausted by the GDR (upper frames) and neutron-skin thickness (lower frames) calculated in relativistic RPA as a function of the symmetry-energy parameter a₄ used in the calculation for ¹³⁰Sn (left frames) and ¹³²Sn (right frames). The dashed lines in the upper frames indicate the experimental value for the strength of the 10 MeV peak resulting in an average symmetry parameter a₄ = 32.0(1.8) MeV; this result corresponds to neutron-skin thicknesses as indicated in the lower frames. Right panel: parameter L describing the density dependence of the symmetry energy as deduced by various approaches (see [31]). The values deduced from the PDR by Carbone *et al* [31] are based on RPA calculations with a variety of different Skyrme forces. Figures reprinted with permission from Klimkiewicz *et al* [30] and Carbone *et al* [31]. Copyright (2007) and (2010), respectively, by the American Physical Society.
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In this procedure, all other parameters of the interaction have been refitted to properties of stable nuclei in order to ensure that always a well-calibrated force is used. From the comparison with the data on ^130,132Sn discussed above, they deduced a value a₄ = 32.0(1.8) MeV for the symmetry energy (upper panels of figure 27) and a corresponding neutron-skin thickness for ¹³²Sn of 0.24(4) fm (see the lower panels of figure 27).

A similar analysis based on RPA calculations with different Skyrme effective forces has been performed by Carbone et al [31]. Comparing the theoretical calculations with the experimental results on ¹³²Sn and ⁶⁸Ni discussed above, they deduced a value for the slope parameter L, which is related to the density dependence of the symmetry energy at saturation density. The results are compared in figure 27 to the results obtained by other methods. The results are in good agreement with the values deduced using other approaches, as can be seen in figure 27. It should be noted, however, that the procedure is not model independent. It should also be noted that the experimental value for ⁶⁸Ni assumed a γ decay branch to extract the B(E1) strength, see the discussion above. Another difficulty in the method is defining the PDR strength in theory and experiment which have to be compared. Depending on the theoretical calculation, the strength observed in addition to the GDR tail does not peak at the same excitation energy as in the calculation. It has been pointed out by Reinhard and Nazarewicz [29] that a better correlated and more robust observable would be the polarizability of the nucleus. Since the polarizability is the inversely energy-weighted integral of the E1 strength, the need of defining an energy window for the low-energy dipole strength is not necessary, while the value of the polarizability is still very sensitive to a redistribution of the dipole strength. Based on such calculations, Tamii et al [32] deduced a value for the neutron-skin thickness of ²⁰⁸Pb of 0.156(25) fm from a precise measurement of the polarizability of ²⁰⁸Pb deduced from the measured Coulomb excitation cross section in (p,p) scattering close to zero degree [32]. A value of 0.168(22) fm has been deduced by Piekarewicz et al [141] from the same experimental measurement.

Although the relative effect of the density dependence of the symmetry energy, or correspondingly the thickness of the neutron skin, is larger for the strength of the low-energy part of dipole response as compared to the polarizability, a less strong model dependence can be a big advantage. An advantage is here to study exotic nuclei as neutron-rich as possible to maximize the effect. The influence of different symmetry-energy parameters in the mean-field model on the calculated dipole response can clearly be seen in figure 28, where the dipole response calculated in relativistic RPA is shown for ⁶⁸Ni [141].

The left panel of figure 28 shows the energy-weighted response for ⁶⁸Ni calculated with effective forces using different values for the symmetry energy resulting in different calculated neutron-skin thicknesses as well as in a different shape of the response [141]. The corresponding value of the neutron-skin thickness calculated for ²⁰⁸Pb is indicated in the figure. The larger the neutron-skin thickness, the more the strength is shifted from high excitation energies to lower energy. The inset shows the running sum of the energy-weighted strength, i.e. the exhaustion of the energy-weighted dipole sum rule as a function of excitation energy. It is clearly seen that while the shape of the spectrum changes significantly, the curves converge if integrated above the GDR region, i.e. each calculation respects the sum rule. The change in the shape of the response is more drastically seen in the right frame, where the inversely weighted strength is plotted, i.e. the polarizability as a function of excitation energy. The running sum keeps diverging until a flat region is reached if integrated above the GDR region. Most noticeably, the polarizability calculated with the different effective forces differs by about 20% for a range of parameters shown. This effect is much bigger than in ²⁰⁸Pb, thus demonstrating the large sensitivity one might reach with radioactive beams. Even a measurement with moderate accuracy could yield a rather precise value for the neutron-skin thickness or the slope parameter of the symmetry energy.

5. Conclusion and outlook

The characteristic changes of the properties of nuclei when going from stable to neutron–proton asymmetric nuclei, such as the asymmetric Fermi energies for protons and neutrons and the development of neutron skins and haloes, manifest themselves in the electric dipole response of these nuclei. From an experimental point of view, such effects have been studied so far mainly for neutron-rich nuclei. An extreme case is the halo nuclei at the neutron-drip line for which a huge dipole-transition strength is found just above the neutron separation threshold. The decoupling of the loosely bound neutrons exhibiting an extended density distribution from the well-bound core nucleons with a normal density distribution is reflected in the decoupling of the dipole response of the halo, which is observed as low-energy threshold strength, and the response of the core, which we believe corresponds to the usual GDR. It has to be said, however, that the experimental determination of the full strength function including the GDR has not been possible so far for a halo nucleus. The characteristic features of the soft E1 excitation related to the halo are meanwhile understood quantitatively for the one-neutron halo nuclei and used as a tool to study the single-particle structure of the halo wave function, such as its spatial extension, its single-particle configurations, and the associated spectroscopic factors. For two- or more-neutron haloes, the dipole response contains additional information on the spatial correlations among the halo neutrons, which in turn can be investigated by Coulomb-breakup experiments. The method of Coulomb breakup is thus an ideal tool to study heavier drip-line nuclei in the future, as they will become accessible at the new fragmentation-beam facilities such as RIBF at RIKEN, FRIB at MSU and FAIR at Darmstadt.

Measurements of the electric dipole response of neutron-rich non-halo nuclei, and in particular of heavier nuclei, are still rather scarce. For the oxygen isotopic chain, a strong fragmentation of the dipole strength has been found for the neutron-rich isotopes, but a clear separation of giant-dipole excitations and low-lying excitations of excess neutrons is not evident. For the heavier neutron-rich nuclei around ¹³²Sn and ⁶⁸Ni, evidence for a low-lying resonance has been observed, which is understood by microscopical models as a vibration of excess neutrons versus a core. A systematic investigation is still lacking, and even a comparison with the results from stable nuclei is hampered by different experimental sensitivities. Next-generation experiments are already on the way both for stable as well as for radioactive beams at the different facilities. In particular, the newly developed instrumentation such as SAMURAI at RIKEN and R3B at GSI and later FAIR will provide the basis for improved and more complete measurements. The new facilities will not only allow to study nuclei with larger neutron excess, but also in a wider mass range including neutron-rich lead isotopes which could be studied at R3B at FAIR. In addition, other experimental techniques have now become available that will allow investigation of the dipole response of exotic nuclei using different probes, such as the α scattering to study the isoscalar dipole response. Finally, with a measurement of the full strength function below and above the threshold including the GDR it will be possible to extract the polarizability of neutron-rich nuclei yielding important information on the neutron-skin thickness and the isospin-asymmetric part of the equation of state of nuclear matter.

In this article, we have concentrated on experimental results and developments. Progress in understanding the dipole response of neutron-rich nuclei, however, has been made in conjunction with theoretical efforts in this field over the last two decades. The fact that both theory and experiment are intensively developing in this field of research makes us confident that progress on answering the many still open questions related to the dipole response of exotic nuclei will be made in the near future.

Acknowledgments

We would like to thank all colleagues who contributed to the experiments discussed here. In particular, we thank our colleagues who have allowed us to use the figures from experiments we were not involved in. TA is much indebted to the members of the LAND-, GSI-Halo-, and R3B collaborations and especially to his long-term GSI colleagues K Boretzky, L Chulkov, H Emling, D Rossi and H Simon. TN would like to thank the collaborators in the RIPS collaboration for the breakup experiments of halo nuclei, and the ZDS- and SAMURAI collaborations at RIBF at RIKEN, and in particular, Y Kondo, H Satou, T Kobayashi, T Motobayashi and K Yoneda. Both authors would like to thank the late P G Hansen for giving us invaluable inspiration and insights about physics of exotic nuclei. We gratefully acknowledge many valuable discussions with B Jonson, M Ishihara and I Tanihata. TA acknowledges support by the Helmholtz International Center for FAIR, GSI and the BMBF project 06DA7047I. TN has been supported in part by a Grant-in-Aid for Scientific Research (B) (no. 22340053), and that on Innovative Areas (no. 24115005) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT, Japan).

The electric dipole response of exotic nuclei

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Abstract

1. Introduction

2. Experimental method

2.1. Coulomb excitation and breakup at intermediate and high energies

2.2. Inclusive measurement and kinematically complete measurement

3. Soft E1 excitation of halo nuclei

3.1. Soft E1 excitation of a one-neutron halo nucleus

3.1.1. Soft E1 excitation of ¹¹Be.

3.1.2. Direct breakup mechanism.

3.1.3. Spectroscopic significance of soft E1 excitation of one-neutron halo nuclei.

3.1.4. E1 versus E2 excitation for halo nuclei.

3.1.5. Application of Coulomb breakup to the structure study—soft E1 excitation of ¹⁹C.

3.1.6. Application of the Coulomb breakup to the radiative capture reaction ¹⁴C(n,γ)¹⁵C of astrophysical interest.

3.1.7. Inclusive Coulomb breakup of ³¹Ne.

3.2. Soft E1 excitation of 2n halo nuclei

3.2.1. Coulomb breakup and soft E1 excitation of ¹¹Li.

3.2.2. Coulomb breakup and soft E1 excitation of ⁶He.

3.2.3. Non-energy weighted cluster sum rule for 2n halo nuclei and dineutron correlation.

3.3. Perspectives on the soft E1 excitation

4. Giant and pygmy resonances in neutron-rich nuclei

4.1. Experimental investigations on the multipole response of exotic nuclei

4.2. Dipole strength distribution in neutron-rich oxygen isotopes and ²⁶Ne

4.3. Giant and PDRs in medium-heavy neutron-rich Ni and Sn isotopes

4.4. The dipole response of nuclei and the neutron skin

5. Conclusion and outlook

Acknowledgments

Footnotes

The electric dipole response of exotic nuclei

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Author e-mails

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Dates

Abstract

1. Introduction

2. Experimental method

2.1. Coulomb excitation and breakup at intermediate and high energies

2.2. Inclusive measurement and kinematically complete measurement

3. Soft E1 excitation of halo nuclei

3.1. Soft E1 excitation of a one-neutron halo nucleus

3.1.1. Soft E1 excitation of 11Be.

3.1.2. Direct breakup mechanism.

3.1.3. Spectroscopic significance of soft E1 excitation of one-neutron halo nuclei.

3.1.4. E1 versus E2 excitation for halo nuclei.

3.1.5. Application of Coulomb breakup to the structure study—soft E1 excitation of 19C.

3.1.6. Application of the Coulomb breakup to the radiative capture reaction 14C(n,γ)15C of astrophysical interest.

3.1.7. Inclusive Coulomb breakup of 31Ne.

3.2. Soft E1 excitation of 2n halo nuclei

3.2.1. Coulomb breakup and soft E1 excitation of 11Li.

3.2.2. Coulomb breakup and soft E1 excitation of 6He.

3.2.3. Non-energy weighted cluster sum rule for 2n halo nuclei and dineutron correlation.

3.3. Perspectives on the soft E1 excitation

4. Giant and pygmy resonances in neutron-rich nuclei

4.1. Experimental investigations on the multipole response of exotic nuclei

4.2. Dipole strength distribution in neutron-rich oxygen isotopes and 26Ne

4.3. Giant and PDRs in medium-heavy neutron-rich Ni and Sn isotopes

4.4. The dipole response of nuclei and the neutron skin

5. Conclusion and outlook

Acknowledgments

Footnotes

3.1.1. Soft E1 excitation of ¹¹Be.

3.1.5. Application of Coulomb breakup to the structure study—soft E1 excitation of ¹⁹C.

3.1.6. Application of the Coulomb breakup to the radiative capture reaction ¹⁴C(n,γ)¹⁵C of astrophysical interest.

3.1.7. Inclusive Coulomb breakup of ³¹Ne.

3.2.1. Coulomb breakup and soft E1 excitation of ¹¹Li.

3.2.2. Coulomb breakup and soft E1 excitation of ⁶He.

4.2. Dipole strength distribution in neutron-rich oxygen isotopes and ²⁶Ne