Nuclear size and related topics

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1 Nuclear Physics A 693 (2001) Nuclear size and related topics A. Ozawa a,, T. Suzuki a,b,i.tanihata a a RIKEN, Hirosawa 2-1, Wako-shi, Saitama , Japan b Department of Physics, Niigata University, Niigata , Japan Abstract Experimental studies on nuclear sizes and related topics are reviewed. The recent development of radioactive nuclear beams has enabled us to study the nuclear sizes of unstable nuclei. The nuclear sizes for unstable nuclei, which are deduced by the interaction cross sections and reaction cross sections, are mainly reviewed. From a theoretical view point, a Glauber-model analysis is important to deduce nuclear sizes. Other related topics, such as halo and skin from a nuclear size point of view are also discussed Elsevier Science B.V. All rights reserved. PACS: Gv; Dz; Lg; n; t Keywords: Measured interaction, reaction, charge-changing σ ; Deduced r.m.s. matter radii; Deduced matter-density distribution; Glauber-model analysis 1. Introduction The nuclear size and density distribution are important bulk properties of nuclei that determine the nuclear potential, single-particle orbitals, and wave function. Historically, the nuclear size has been studied by electron scattering and muonic atoms (see reviews in Refs. [1 3]). Also, the density parameters are summarized in Ref. [4]. Those are studies of charge distributions. Nucleon distributions have been studied by several nuclear reactions with strong interacting probes. Among those, proton elastic scattering provides the best information [1,5]. From these studies, three basic properties of nuclear density have been established: (1) the half-density radius of the matter distribution is expressed as r 0 A 1/3,wherer 0 is the radius constant; (2) protons and neutrons are uniformly mixed in the nucleus, namely ρ p (r) ρ n (r); (3) the surface thickness is constant. * Corresponding author. address: ozawa@rarfaxp.riken.go.jp (A. Ozawa) /01/$ see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (01)

2 A. Ozawa et al. / Nuclear Physics A 693 (2001) However, these properties were obtained mainly from studies of stable nuclei. Isotope-shift measurements have been a unique technology to determine the nuclear size of unstable nuclei (reviewed by Ref. [6]). However, due to the limitation of accessible species in ion sources, the measurements have been made for isotopes of a limited number of elements. Also, only the radii of the charge distribution can be studied by this method. In the mid-80s, radioactive nuclear beam (RNB, also called RI Beam, which stands for Radioactive Ion Beam) technology was developed drastically (reviewed by Ref. [7]). Using such technology, it has become possible to determine the nuclear size for unstable nuclei, mainly based on the interaction and reaction cross sections. The measurements have no limitation of accessible species. As one of the successful applications of RNB, exotic structures of nuclei, such as neutron halos and neutron skins, have been discovered through the determination of the matter radii of nuclei near to the drip lines. It was also shown that the common properties of the nuclear density just mentioned above do not hold in unstable nuclei. Some pioneering experimental studies on neutron halo nuclei are reviewed in Ref. [8]. In the present paper, recent experimental studies of the nuclear sizes for unstable nuclei, mainly measured by the interaction and reaction cross sections, and some related topics are reviewed. The part discussed in a 1996 Review [8] is not discussed, unless necessary. In the following section, brief descriptions of the nuclear densities are presented for an introduction. In Section 3, recent measurements are summarized. The method to deduce the nuclear radii and the density distributions is described in Section 4. Information obtained by those studies is presented in Section 5, followed by a summary in Section Overview of the nuclear density distribution 2.1. Density distribution of stable nuclei The density distributions for stable nuclei, mainly charge distributions, have been wellstudied by electron scattering, isotope-shift measurements and studies of muonic atoms. Good fits of the experimental data can be obtained if one assumes a distribution of the form [ ρ(r)= ρ exp r R ] 1. (2.1) a The electron scattering and muonic atoms data are consistent with the parameters ρ 0 = 0.17 nucleon fm 3, a = 0.54 fm and R = 1.1A 1/3 fm for all stable nuclei with A>16 [3]. 1 On the other hand, for light stable nuclei (A <20), a harmonic-oscillator (HO)-type density distribution is sufficient to describe the experimental data. The HO-type density distributions including the contributions up to the sd-shell are given by the following equations: 1 Here, we do not discuss the detail concerning, the density distribution of stable nuclei, but show only the global properties of stable nuclei that are related to the present discussion on unstable nuclei.

3 34 A. Ozawa et al. / Nuclear Physics A 693 (2001) for2<z<8, 2 <N<8: ρ n (r) = 2π 3/2 λ 3 ( 1 1 A ρ p (r) = 2π 3/2 λ 3 ( 1 1 A forz>8, N>8: ( ρ n (r) = 4π 3/2 λ A ρ p (r) = 4π 3/2 λ 3 ( 1 1 A ) 3/2 exp ( x 2)( 1 + N 2 3x 2 ) 3/2 exp ( x 2)( 1 + Z 2 3x 2 ), ) ; (2.2) ) 3/2 N N + 8 exp( x 2) ( 1 + 2x 2 + N 8 15x 4 ) 3/2 Z Z + 8 exp( x 2) ( 1 + 2x 2 + Z 8 15x 4 ), ), (2.3) where A, N and Z are the mass, neutron, and proton numbers, x 2 = (r/λ) 2 and λ denotes the size parameter [9]. The shapes for protons and neutrons are quite similar for stable nuclei. The differences in the proton and neutron distributions have been studied in detail for stable nuclei including 48 Ca and 208 Pb, which have many more neutrons than protons. However, no large differences in the radii between the proton and neutron distributions have been observed in stable nuclei [10 12] Density distributions of unstable nuclei The density distributions for unstable nuclei are quite different from those for stable nuclei. A significant character is the existence of skins and halos [8]. Although no large difference in the radii between the proton and neutron distributions has been observed in stable nuclei, most of the mean-field nuclear models predict a large difference in the radii for neutron (or proton) -rich nuclei. Such neutron skins have been observed in neutron-rich He isotopes [13] and neutron-rich Na isotopes [14]. The details concerning the neutronskin observation are discussed in Section 5.1. In nuclei near the drip line, the separation energy of the last nucleon(s) becomes extremely small. The density distribution in such a loosely bound nucleus shows an extremely long tail, called a halo. Although the density of a halo is very low, it strongly affects the reaction cross sections, and leads to new properties for the nuclei. Because a detailed discussion on halo nuclei has been reviewed in several articles [8,15 17] and also discussed in a chapter of this issue, no details are discussed, except in Section 5.4, where a determination of the density distributions and mixings in halo neutrons is presented. 3. Recent measurements for nuclear sizes 3.1. Interaction cross-section (σ I )measurements Interaction cross-section (σ I ) measurements, which were the first experiments using RI beams, started at Bevalac at LBL in the early 80s. There, σ I were measured by

4 A. Ozawa et al. / Nuclear Physics A 693 (2001) Fig. 1. Schematic view of the experimental setup at the fragment separator FRS. TPC and MUSIC mean Time-Projection Chambers and Multi-Sampling Ion-Chamber (MUSIC), respectively. The measured quantities (TOF, Bρ and E)areshownbyboldletters. transmission-type experiments, as follows. The isotopes produced in the production target were separated by rigidity (Bρ) with a beam line magnet system. The rigidity separated isotopes were further identified before incidence on a reaction target by the time-of-flight (TOP) and by energy loss ( E). After the reaction targets, Bρ, TOF,and E for the isotopes were measured by a large acceptance spectrometer (HISS) [18]. The acceptance and the separation power of the beam-line limited σ I measurements to only light nuclei A<20 in the setup. The FRagment Separator (FRS) at GSI is the only RI beam-facility for relativistic energies after the shutdown of Bevalac at LBL. FRS is a projectile-fragment separator with four focusing points. The new setup is shown in Fig. 1. Secondary beams were produced through projectile fragmentation of a primary beam accelerated to around 1 A GeV by the heavy-ion synchrotron SIS. The first half of the FRS, down to the intermediate focal plane (F2), was used to separate and identify the incident secondary beams. A reaction target of carbon was placed at F2. The second half of the FRS was used as a spectrometer to transport the noninteracting secondary nuclei. Those nuclei were then identified and counted at the final focal plane of the FRS (F4). Because this new setup enables measurements with small detectors, the analysis of the data became much simpler than that of the LBL measurements. For future facilities, such as the RI Beam Factory in RIKEN, a setup with two separators in series would be the best tool for the measurements. At the FRS, σ I, for Na and Mg isotopes [14,19] and light neutron-rich nuclei [20] have been measured. The experimental data for σ I are summarized in Table 1.

5 36 A. Ozawa et al. / Nuclear Physics A 693 (2001) Table 1 Summary for interaction cross-section (σ I ) measurements (σ I ) (mb) Nuclei Energy (A MeV) Be-target C-target Al-target Reference 3 He ± ± ± 9 [18] 4 He ± ± ± 13 [18] 6 He ± ± ± 8 [18] 8 He ± ± ± 9 [18] 6 Li ± ± ± 11 [9] 7 Li ± ± ± 7 [9] 8 Li ± ± ± 14 [9] 9 Li ± ± ± 7 [9] 11 Li ± 60 [9] ± 40 [21] 7 Be ± ± ± 17 [9] 9 Be ± ± ± 11 [9] 10 Be ± ± ± 16 [9] 11 Be ± ± ± 25 [21] 12 Be ± ± ± 31 [21] 14 Be ± 90 [21] ± 34 [22] 8 B ± ± ± 32 [21] ± 6 [23] 10 B ± 16 [20] 11 B ± 30 [20] 12 B ± ± ± 15 [21] 13 B ± ± ± 28 [21] 14 B ± ± ± 43 [21] 15 B ± ± ± 180 [21] ± ± ± 30 [24] ± 15 [24] 17 B ± 200 [25] ± 22 [22] 19 B ± 83 [22] 9 C ± ± ± 29 [24] ± 13 [24] 10 C ± ± ± 20 [24] 11 C ± ± ± 63 [26] ± 23 [26] 12 C ± 9 [25] ± 6 [20] 13 C ± 12 [20] 14 C ± 19 [20] 15 C ± ± ± 20 [24] 16 C ± 11 [20]

6 A. Ozawa et al. / Nuclear Physics A 693 (2001) Table 1 continued (σ I ) (mb) Nuclei Energy (A MeV) Be-target C-target Al-target Reference 17 C ± 10 [20] 18 C ± 15 [20] 19 C ± 28 [20] 20 C ± 20 [20] 12 N ± ± ± 128 [26] ± 27 [26] 13 N ± ± ± 30 [26] ± 15 [26] 14 N ± 9 [20] 15 N ± 30 [20] 16 N ± 32 [20] 17 N ± ± ± 61 [27] ± 24 [27] 18 N ± 8 [20] 19 N ± 9 [20] 20 N ± 17 [28] 21 N ± 9 [20] 22 N ± 49 [20] 23 N ± 98 [20] 13 O ± ± ± 54 [24] ± 24 [24] 14 O ± ± ± 54 [24] ± 23 [24] 15 O ± ± ± 28 [24] ± 13 [24] 16 O ± 6 [20] 17 O ± 15 [20] 18 O ± 26 [20] 19 O ± 9 [20] 20 O ± 10 [28] 21 O ± 11 [20] 22 O ± 22 [20] 23 O ± 16 [20] 24 O ± 52 [20] 17 F ± ± ± 93 [27] ± 32 [27] 18 F ± 50 [20] 19 F ± 24 [20] 20 F ± 11 [28] 21 F ± 12 [20] 23 F ± 16 [20] 24 F ± 23 [20]

7 38 A. Ozawa et al. / Nuclear Physics A 693 (2001) Table 1 continued (σ I ) (mb) Nuclei Energy (A MeV) Be-target C-target Al-target Reference 25 F ± 31 [20] 26 F ± 54 [20] 17 Ne ± ± ± 224 [27] ± 31 [27] 18 Ne ± 25 [29] 19 Ne ± 15 [29] 20 Ne ± 10 [28] 21 Ne ± 25 [29] 23 Ne ± 15 [29] 24 Ne ± 48 [29] 25 Ne ± 16 [29] 26 Ne ± 20 [29] 28 Ne ± 40 [29] 29 Ne ± 32 [29] 20 Na ± 11 [19] 21 Na ± 9 [14] 22 Na ± 6 [14] 23 Na ± 12 [14] 25 Na ± 9 [14] 26 Na ± 16 [14] 27 Na ± 18 [14] 28 Na ± 10 [14] 29 Na ± 22 [14] 30 Na ± 15 [14] 31 Na ± 41 [14] 32 Na ± 61 [14] 20 Mg ± 12 [28] 22 Mg ± 33 [19] 23 Mg ± 68 [19] 24 Mg ± 72 [19] 25 Mg ± 113 [19] 27 Mg ± 16 [19] 29 Mg ± 16 [19] 30 Mg ± 8 [19] 31 Mg ± 35 [19] 32 Mg ± 24 [19] 3.2. Reaction cross-section (σ R )measurements The reaction cross sections (σ R ) have been measured extensively for stable nuclei (proton to Ca) with various targets (proton to Pb) and for various energies (10 A to

8 A. Ozawa et al. / Nuclear Physics A 693 (2001) A MeV) (reviewed by Ref. [30]). The development of RI-beam technology for intermediate energies (< 100 A MeV) allows one to measure σ R for unstable nuclei. Measurements of σ R started at GANIL in the late 80s. And now, almost all RI-beam facilities, such as GANIL [31 34], MSU [35 37], RIKEN [38,39], RIBLL [40], GSI [41], have programs for the measurements of σ R for unstable nuclei. There are two major methods to measure σ R at intermediate energies: one is the associated γ -ray detection method [42]; the other is the transmission method using Bρ E TOF identification or E E identification. The principle of the associated-γ method is very simple: the basic assumption is that a nuclear reaction is associated by the emission of at least one γ -ray. Supposing an ideal γ -detector with 100% detection efficiency, it is necessary only to count the number of γ -events normalized to the number of incident particles and to the thickness of the reaction target. However, two uncertainties exist in this method. One is the efficiency of γ -ray detection; the other is the validity of the basic assumption, the emission of γ -rays in all reaction events. The second uncertainty is particularly large for the reaction of a nucleus near to the drip lines. Because the nucleus is bound only by an extremely small energy, the removal of nucleons may occur with very soft collisions, and thus a γ -ray may not be emitted at all. Therefore, σ R of near drip-line nuclei measured by this method may not be used to determine the nuclear sizes. Instead, the difference in σ R from the transmission method and the γ -ray method provides support for the existence of loosely bound halo neutrons. On the other hand, the transmission method allows a direct measurement of σ R without any interpretation by a theoretical model. However, complete particle identification is necessary after the reaction target. Such identification can be done by E E measurements at intermediate energies (< 100 A MeV) for light nuclei (A<20). A difficulty in this measurement is to identify an inelastic scattering and a reaction of particle removal only from the target. (It has to be noted that σ I is the sum of the particle removal cross section from the projectile. Therefore no experimental ambiguity exists for the determination of σ I.) These events have to be identified by the change of the energy and momentum of the projectile. It therefore becomes difficult as the beam energy increases. The experimental data of σ R measured by the transmission method are summarized in Table 2. Because of the easier setup, measurements by the associated-γ method cover a much wider region of nuclides than those by the transmission method. Since the reaction targets and reaction energies are widely spread, a summary of experimental data for the associated-γ method is omitted (see references in Refs. [32,33,36,37]) Charge-changing cross-section (σ CC )measurements Measurements of σ CC for stable nuclei have been made with the objective to interpreting the production of secondary nuclei during cosmic-ray propagation in the galaxy to better estimate the source elemental and isotopic composition of cosmic rays. At the same time, they provide an understanding of peripheral interactions of energetic heavy nuclei.

9 40 A. Ozawa et al. / Nuclear Physics A 693 (2001) Table 2 Summary of reaction cross sections (σ R ) measured by the transmission method; the data with only light targets (Be to Al) are summarized (σ R ) (mb) Nuclei Energy (A MeV) Be-target C-target Al-target Reference 6 Li ± ± ± 56 [39] 7 Be ± ± ± 96 [39] ± ± ± 79 [39] ± ± 7 [41] 11 Be ± ± 50 [38] 8 B ± ± ± 21 [39] ± ± ± 35 [39] ± 3 [41] ± ± 7 [41] 9 C ± ± 50 [41] 12 C ± 56 [40] ± 40 [30] ± 30 [30] ± 45 [30] ± 60 [30] ± 60 [30] 13 C ± 40 [40] ± ± 80 [38] 14 C ± 75 [40] 15 C ± 130 [40] 16 C ± 44 [40] 14 N ± 66 [40] 15 N ± 55 [40] 16 N ± 69 [40] 17 N ± 34 [40] 16 O ± 74 [40] 17 O ± 42 [40] 18 O ± 59 [40] Data concerning σ CC for stable nuclei (C to Ni) with energies ranging from 300 A to 1700 A MeV were tabulated at the end of the 80s [43]. The development of RI beam technology also allows us to measure σ CC for unstable nuclei. Measurements of σ CC are relatively easier than those of σ R and σ I, since only Z identification is necessary after the reaction target. σ CC at relativistic energies can provide a method to estimate the charge radii using a Glauber-model analysis. σ CC can be larger than the Glauber (geometrical) model estimation due to possible proton emission at the ablation stage of the reaction. Therefore, σ CC provides an upper limit of the charge radius. A combination of σ I and σ CC, therefore, allows us to deduce the minimum estimation of the neutron skin thickness [44]. On the

10 A. Ozawa et al. / Nuclear Physics A 693 (2001) Fig. 2. Charge-changing cross sections (σ CC ) as a function of the mass number A. The experimental data are shown by filled circles (boron isotopes), open circles (carbon isotopes), filled triangles (nitrogen isotopes), open triangles (oxygen isotopes), and filled rhombuses (fluorine isotopes). The solid curve corresponds to a power function, σ CC Z 0.55 [43]. other hand, it is difficult to estimate the proton skin by this method, because it would likely be over-estimated. After the pioneering work at SATURNE [45], σ CC for light unstable nuclei have been extensively measured, mainly, at GSI [28,46]. It is generally accepted, on the basis of measurements of σ CC for stable nuclei at relativistic energies that σ CC can be parameterized, similar to σ I, as a power function of the mass number (σ CC A 0.55±0.02 ) [44]. However, the σ CC values for drip-line nuclei do not follow this dependence. For light nuclei (3 <Z<10), the charge radii strongly depend on the atomic number, but not on the neutron number. The charge radii stay nearly constant for a given isotope sequence until the very edge of the neutron stability, as shown in Fig Related experiments Elastic-scattering measurements As known from studies on stable nuclei, the most accurate and detailed information on the nuclear matter distributions is obtained by proton elastic scattering at an energy of about 1 GeV [47]. This method can also be applied to study unstable nuclei by measuring the elastic cross sections in inverse kinematics, using RI beams and a hydrogen target. Recently, such a measurement was made at GSI by Alkhazov et al. [48]. They measured the differential cross sections for p-he elastic scattering in inverse kinematics at 700 A MeV and at small momentum transfer up to t =0.05 (GeV/c 2 ). The α-like core and significant neutron skins in 6,8 He have been confirmed. The root-mean-square matter radii as well as the matter-density distributions are consistent with the results of Ref. [13]. In addition, there have been many works for elastic-scattering measurements with RI beams at lower energies (< 100 A MeV). From eikonal calculations, the proton scattering from 6,8 He and 6 Li, in this energy range, is not sensitive to a difference in the neutron and proton distributions [49]. Although successful at high incident energies (700 A MeV), the Glauber-model overestimated the experimental data by about 40 50% when these

11 42 A. Ozawa et al. / Nuclear Physics A 693 (2001) calculations were extended to lower energies (< 100 A MeV) [49]. Thus, elastic scattering at low energies (< 100 A MeV) seems not to be a very promising tool to study the details of the density distributions for unstable nuclei with the present theoretical model of the nuclear reactions Fragmentation Measurements of the momentum distributions of fragments from the break-up of RI beams are good tools to search for some signature of a halo nucleus [50]. Also, the momentum distribution of the fragments provides in the first order the internal momentum distribution of removed nucleons, and thus provides means to study the wave function or density distribution of the valence nucleons. The fragmentation of unstable nuclei has been extensively measured at RI beam facilities in MSU [51], GSI [52] and GANIL [53] for various nuclei ( 6 He to 26 P) over a wide range of energies (30 to 1000 A MeV) after the pioneering work at LBL [54]. In general, narrow momentum distributions of fragments corresponds to a relatively large spatial extent of the valence nucleon(s). However, the break-up process seems not to be understood by a simple description. For example, the momentum distributions of 18 C from 19 C break-up have been measured at 30 A MeV [55] and 1 A GeV [52]. The two data sets do not show good consistency, although the data for 16 C from 17 C break-up shows a good agreement between the two energies. Measurements of the nucleon(s) removal cross sections (σ x ) give additional information concerning the size of valence nucleon(s). Two- and four-neutron removal cross sections have been used to estimate the difference in the proton and neutron density distributions in 6 He and 8 He [13]. Also, a combined analysis of σ x and σ I is used for identifying the core plus valence nucleon structure of a nucleus (see Sections and 5.4 for further discussion). σ I and the momentum distributions of fragments can be analyzed by the Glauber-model analysis with a core plus neutron(s) structure. However, for example, in 19 C, a comparative analysis of both these experimental data fails to yield a consistent conclusion regarding a possible ground-state configuration. It is suggested that a considerable change in the core of 19 C from a bare 18 C nucleus may take place [56] (see Section 5.4 for more discussion). 4. Method of analysis for experimental results A determination of the matter-density distribution of an unstable nucleus is not straightforward. Actually, we have no model-independent method for determining the density distribution at this moment. Until now, the most widely and effectively used model for the data of unstable nuclei is the Glauber model to relate the density distributions and cross sections. Although the model is simple, it shows reasonable results for many cases. We first show some phenomenological models, and then briefly show the Glaubermodel formalism. Recently, a cascade-type model, such as a quantum-molecular dynamical (QMD) model, has been applied for relating the σ R and density distribution [57]. No description of this model is presented here.

12 A. Ozawa et al. / Nuclear Physics A 693 (2001) Phenomenological analysis Kox formulae σ R and σ I can be compared with phenomenological formulae developed by Kox et al. [30]: σ R (E) = πr 2 0 ( A 1/3 P + A 1/3 t + a A1/3 P A1/3 t A 1/3 t ) 2 ( C(E) 1 V ) cb, (4.1) E CM P + A 1/3 where A P and A t are the projectile and target mass number, a = 1.85 is a mass symmetry parameter related to the volume overlap of the projectile and target, C(E) is an energydependent transparency, and V cb is the Coulomb barrier [33]. Using the formulae, we can deduce a strong absorption radius (r 0 ), which is independent of the target system and projectile energies. Especially for σ R measured by the associated-γ method, an analysis by the Kox formula is convenient to analyze a variety of targets and projectile energies. For stable nuclei, the best fit of the experimental data provided r 0 = 1.1 fm [30]. The r 0 deduced from different experiments are not consistent in some cases, for example 11 Li [33]. This may be due to the fact that r 0 is a quantity that depends on both the target and the projectile density distributions. Therefore, this equation is not relevant for a detailed discussion of the data, but is convenient to see quickly whether the density distribution is normal or not. A trend of increasing reduced radius with the neutron excess was found [31,37] Difference factor It is noticed that the energy dependence of the Glauber-model analysis can not completely reproduce the observed data. Thus, the following phenomenological analysis is applied to the data at intermediate energies to search for nuclei with an anomalous structure. As discussed in the following section, general Glauber model calculations always underestimate the cross sections at intermediate energies, if one assumes an HO-type density distribution, and determines the width parameter by reproducing σ I at relativistic energies. For a quantitative discussion, a difference factor (d) was defined as d = σ R(exp) σ R (Gl), (4.2) σ R (Gl) where σ R (exp) is the experimental σ R at intermediate energies and σ R (Gl) is the σ R calculated by the Glauber model at the same energies with the HO-type density distributions obtained by fitting the experimental σ R at relativistic energies [24]. The difference factors are deduced to Be, B and C isotopes [24,40]. In 11 Be, 14 Band 15 C, the difference factor largely deviates from those of other isotopes. The reason is that these nuclei have dominant l = 0 neutron single-particle components in their ground states Glauber model analysis The most-used model for σ I and σ R is the Glauber model. Let σ αβ be the cross section for the reaction [58]

13 44 A. Ozawa et al. / Nuclear Physics A 693 (2001) P ( Ψ 0 ) + T ( K,Θ 0 ) P ( q,ψ α ) + T ( K q,θ β ). (4.3) The initial projectile (P) and the target (T) are in their ground states. The relative momentum is hk. The projectile is excited by the reaction and goes to the state specified by α with a momentum transfer of hq. The targetnucleus receivesa momentumtransfer of hq and goes to state β. It is defined that α = 0andβ = 0 stand for the respective ground states. The σ R is obtained by summing σ αβ over the possible final states (αβ), except for αβ = 00: σ R σ αβ = db { 1 exp[iχpt (b)] 2}, (4.4) αβ 00 where the phase-shift function (χ PT ) for elastic scattering is defined by exp[iχ PT ]= Ψ 0 Θ 0 (1 Γ ij ) Ψ 0 Θ 0. (4.5) i P j T Here, Γ ij is the profile function for NN scattering and i, j indicate a proton or a neutron. The second term in the curly brackets of Eq. (4.4) can also be written as exp[ 2Imχ PT (b)], and represents the survival probability (also called the transmission function) of both the projectile and the target after a collision with impact parameter b. σ I is such a probability that the projectile loses at least one nucleon after a collision with a target nucleus, and can thus be obtained by summing σ αβ over all possible states (αβ) except for α = 0: σ I α 0,β σ αβ [ = db 1 Ψ 0 Θ 0 (1 Γij ) Ψ 0 Ψ 0 ] (1 Γ ij ) Ψ 0 Θ 0. (4.6) i P j T i P j T Using Eqs. (4.2) and (4.4), Ogawa et al. [58] estimated the difference between σ I and σ R ; it was found to be less than a few percent for a beam energy higher than several hundred MeV per nucleon. Therefore, at high energy, the calculation of σ R is used even for σ I in most of the cases. However, difference is expected to be larger at an energy lower than 100 A MeV. Equation (4.4) can be rewritten in other form: σ R = 2π 0 [ 1 T(b) ] b db, (4.7) where T(b)is the transmission for impact parameter b. The Glauber model is then reduced to a calculation of the transmission Glauber model with an optical-limit approximation One of the simplest approximation to calculate T(b) is the optical limit. In this approximation, the profile function is replaced by the NN cross sections under the zero-

14 A. Ozawa et al. / Nuclear Physics A 693 (2001) range limit. The σ R can then be calculated from the nucleon-density distribution and the total NN cross sections as { T(b)= exp } σ ij ρti z ( ) (s)ρz Pj b s ds, (4.8) ij where ρki z (s) is a z-direction integrated nucleon-density distribution, ρki z (s) = ρ ki ( s 2 + z 2 ) dz. (4.9) The index k = P (projectile) or T (target) and σ ij are the nucleon nucleon total cross sections in which indices i, j are used to distinguish a proton and a neutron. The nucleondensity distribution in the nucleus is written as ρ ki (r). Equation (4.9) was used by Karol [59] for estimating σ R of heavy-ion collisions, assuming surface-fitted Gaussian density distributions. He could reproduce the cross sections of stable nuclei reasonably well. This model was tested more precisely using a known density distribution. There, the proton-density distributions determined by electron scattering were used and the neutron distributions were assumed to be the same as that of the protons, except for the overall normalization. It was shown that these equations (Eqs. (4.7) (4.9)) reproduce the observed cross sections at 400 A and 800 A MeV within 2% for all reactions involving 7 Li, 9 Be, 12 C, and 27 Al. Therefore, this simple optical-limit calculation of the Glauber model has been proved to work well at high energies. The energy dependence of the cross sections has been studied by Kox et al. [30]. In the first analysis, the energy dependence of the 12 C + 12 C reaction cross sections were reproduced reasonably well if the Coulomb deflection was take into account for energies less than a few tenths of MeV per nucleon. However, recent studies by Ozawa [24] over a wider range of nuclei showed that this model over predicts the cross sections at low energy. No clear reason for this disagreement is yet known. The energy dependence of the cross section, in principle, provides information about the density distribution, particularly the distribution at the tail part of the density. Such a trial is presented in Section 5.4. The root-mean-square (RMS) matter radii deduced by Galuber-model with an opticallimit approximation and those by a similar analysis, but based on experimental data at intermediate energies [60], are summarized in Table Glauber model for a few-body system This optical limit of the Glauber model has been widely used for deducing the nuclear matter radii from the interaction and reaction cross sections. However, Al-Khalili and Tostevin [61] have pointed out that it may not be a good approximation if one applies the model to a loosely bound system, such as a nucleus with a neutron halo. This was also pointed out by Ogawa et al. in their calculation of 11 Li [58]. In this refined model, a halo nucleus is decomposed into a core part and a halo part. In such a model the wave function of a halo nucleus is written as

15 46 A. Ozawa et al. / Nuclear Physics A 693 (2001) Table 3 Summary of RMS matter radii deduced by a Glauber-model analysis RMS matter radii (fm) Nuclei OL FB Intermediate [60] 4 He 1.57 ± 0.04 [21] 6 He 2.48 ± 0.03 [21] 8 He 2.52 ± 0.03 [21] 6 Li 2.32 ± 0.03 [21] 2.46 ± Li 2.33 ± 0.02 [21] ± Li 2.37 ± 0.02 [21] ± Li 2.32 ± 0.02 [21] ± Li 3.12 ± 0.16 [21] ± Be 2.31 ± 0.02 [21] 9 Be 2.38 ± 0.01 [21] 2.53 ± Be 2.30 ± 0.02 [21] ± Be 2.73 ± 0.05 [21] 2.91 ± 0.05 [20] ± Be 2.59 ± 0.06 [21] ± Be 3.16 ± 0.38 [21] 3.36 ± ± 0.09 [22] 3.10 ± 0.15 [22] 8 B 2.38± 0.04 [21] 10 B 2.20± 0.06 [20] 2.56 ± B 2.09± 0.12 [20] ± B 2.39± 0.02 [21] ± B 2.46± 0.12 [21] ± B 2.44± 0.06 [21] 3.00 ± B 2.45± 0.27 [21] 2.61 ± ± 0.03 [24] 17 B 3.0± 0.6 [27] 4.10 ± ± 0.06 [22] 2.99 ± 0.09 [22] 19 B 3.11± 0.13 [22] 9 C 2.42± 0.03 [24] 10 C 2.27± 0.03 [24] 11 C 2.12± 0.06 [26] 2.46 ± C 2.35± 0.02 [25] ± ± 0.02 [20] 13 C 2.28± 0.04 [20] 2.42 ± C 2.30± 0.07 [20] ± C 2.40± 0.05 [24] 2.50 ± 0.08 [20] ± C 2.70± 0.03 [20] ± C 2.72± 0.03 [20] 2.73 ± 0.04 [20] 3.04 ± C 2.82± 0.04 [20] 2.90 ± C 3.13± 0.07 [20] 3.23 ± 0.08 [20] 2.74 ± C 2.98± 0.05 [20] 12 N 2.47± 0.07 [26]

16 A. Ozawa et al. / Nuclear Physics A 693 (2001) Table 3 continued RMS matter radii (fm) Nuclei OL FB Intermediate [60] 13 N 2.31± 0.04 [24] 14 N 2.47± 0.03 [20] 2.61 ± N 2.42± 0.10 [20] 16 N 2.50± 0.10 [20] 2.53 ± 0.14 [20] 2.71 ± N 2.48± 0.05 [27] ± N 2.65± 0.02 [20] 2.69 ± 0.05 [20] ± N 2.71± 0.03 [20] ± N 2.81± 0.04 [28] 2.82 ± 0.05 [20] 21 N 2.75± 0.03 [20] 22 N 3.07± 0.13 [20] 3.08 ± 0.13 [20] 23 N 3.41± 0.23 [20] 13 O 2.53± 0.05 [24] 14 O 2.40± 0.03 [24] 15 O 2.44± 0.04 [24] 2.70 ± O 2.54± 0.02 [20] ± O 2.59± 0.05 [20] 2.60 ± 0.05 [20] 18 O 2.61± 0.08 [20] 19 O 2.68± 0.03 [20] 2.68 ± 0.09 [20] 2.58 ± O 2.69± 0.03 [28] 3.00 ± O 2.71± 0.03 [20] 2.72 ± 0.04 [20] 2.76 ± O 2.88± 0.06 [20] 23 O 3.20± 0.04 [20] 3.24 ± 0.07 [20] 24 O 3.19± 0.13 [20] 17 F 2.54± 0.08 [27] 18 F 2.81± 0.14 [20] 19 F 2.61± 0.07 [20] 20 F 2.79± 0.03 [28] 2.81 ± 0.08 [20] 21 F 2.71± 0.03 [20] 23 F 2.79± 0.04 [20] 24 F 3.03± 0.06 [20] 3.04 ± 0.07 [20] 25 F 3.12± 0.08 [20] 26 F 3.23± 0.13 [20] 3.29 ± 0.15 [20] 17 Ne 2.75 ± 0.07 [27] 18 Ne 2.81 ± 0.14 [29] 19 Ne 2.57 ± 0.04 [29] 20 Ne 2.87 ± 0.03 [28] 21 Ne 2.83 ± 0.07 [29] 23 Ne 2.76 ± 0.04 [29] 24 Ne 2.79 ± 0.13 [29] 25 Ne 2.82 ± 0.04 [29] 26 Ne 2.86 ± 0.05 [29]

17 48 A. Ozawa et al. / Nuclear Physics A 693 (2001) Table 3 continued RMS matter radii (fm) Nuclei OL FB Intermediate [60] 28 Ne 2.92 ± 0.10 [29] 29 Ne 2.99 ± 0.08 [29] 20 Na 2.74 ± 0.03 [19] 21 Na 2.75 ± 0.02 [19] 22 Na 2.72 ± 0.04 [19] 23 Na 2.83 ± 0.03 [19] 25 Na 2.88 ± 0.02 [19] 26 Na 2.92 ± 0.03 [19] 27 Na 2.95 ± 0.04 [19] 28 Na 3.01 ± 0.02 [19] 29 Na 3.03 ± 0.04 [19] 30 Na 3.10 ± 0.03 [19] 31 Na 3.16 ± 0.08 [19] 32 Na 3.22 ± 0.11 [19] 20 Mg 2.88 ± 0.04 [28] 22 Mg 2.89 ± 0.06 [19] 23 Mg 2.96 ± 0.14 [19] 24 Mg 2.79 ± 0.15 [19] 27 Mg 2.90 ± 0.03 [19] 29 Mg 3.00 ± 0.03 [19] 30 Mg 3.06 ± 0.02 [19] 31 Mg 3.12 ± 0.06 [19] 32 Mg 3.12 ± 0.05 [19] OL and FB mean the Glauber-model analysis with an optical-limit (OL) approximation and that for a few-body (FB) system, respectively. Intermediate means the radii deduced from the data at intermediate energies. Ψ 0 = ϕ 0 Φ 0, (4.10) where Φ 0 is the core nucleus and ϕ 0 is the halo neutron wave function in the ground state of a halo nucleus (Ψ 0 ). In this case the reaction cross section is written as σ reac = { db 1 exp [ 2Imχ ct (b) ] ϕ 0 i 2} exp[iχ ni T(b + s i )] ϕ 0, (4.11) where i denotes the neutron in the halo and χ ct is a phase-shift function between the core nucleus and the target nucleus; χ nt is that between a neutron and the target nucleus. Because the core nucleus is usually well bound, this part is calculated by the optical limit. It is worth noting the method for testing the validity of the de-coupling of the core and the halo neutron. The same Glauber model predicts the relation between the neutron removal cross sections and the interaction cross sections, as follows:

18 A. Ozawa et al. / Nuclear Physics A 693 (2001) σ I (Ψ 0 ) σ I (Φ 0 ) = x σ xn, (4.12) where σ I (Ψ 0 ) is the interaction cross section of the halo nucleus and σ I (Φ 0 ) is that of the core nucleus. The right-hand side of the equation is the sum of the neutron and neutronremoval cross sections of the halo neutrons up to the all halo neutrons. The index x denotes the number of removed neutrons. Al-Khalili and Tostevin applied this core plus halo method to 11 Li, 11 Be, and 8 Bhalo nuclei for the first time [61] and found that this new model gives a smaller cross section than that obtained from the optical limit if one uses the same wave function. As a result, the deduced RMS matter radii of halo nuclei are 0.1 to 0.2 fm larger than that deduced from the optical limit. Therefore, this model is recently being commonly used to deduce the radii of halo nuclei. The validity of this method for determining the matter distribution is considered to be better, theoretically, than the optical-limit calculation. However, if one speaks strictly, it has not been tested by a model-independent method how well the density distribution can be determined by this method. The RMS matter radii deduced by the Galuber-model for a few-body system are also summarized in Table Effective density distribution In addition to the RMS radius, the effective density distribution of a nucleus can also be estimated from σ I measurements. As shown in Eq. (4.6), σ I is determined by three things: the nucleon nucleon cross sections, the target density distribution, and the density distribution of the projectile. By measuring either the energy dependence (change in the nucleon nucleon cross sections) or the target dependence of σ I, the density distribution of the projectile nucleus can be determined. This method has been applied for determining the density distribution of halo nuclei. Let us look at Eqs. (4.7) (4.9) in more detail concerning a halo nucleus. To understand the method easily, we just show the optical-limit description of the cross section. We first separate the density of 11 Li into the core and the halo parts, ρ(r)= ρ c (r) + ρ h (r). (4.13) This is the same assumption as that used in the few-body model shown in the last section. After substituting Eq. (4.13) into Eq. (4.7), σ I is decomposed into a contribution from the core (σ c ) and that from the halo (σ h ): σ I = 2π b db [ 1 T c (b) ] + 2π b dbt c (b) [ 1 T h (b) ] = σ c + σ h, (4.14) where the relation T(b)= T c (b) T h (b) (4.15) is used. Although σ c is exactly the σ I of the core, σ h is modified by T c (r), the transparency of the core. Therefore, σ h corresponds to the reaction of the halo neutrons by the target,

19 50 A. Ozawa et al. / Nuclear Physics A 693 (2001) but without the core reaction. Thus, only a reaction at a large impact parameter contributes to σ h. By changing the target nucleus or the NN cross sections (by changing the projectile energy), a different part of the halo distribution can be studied. With a smaller target, or one with a smaller NN cross section, the halo density at a smaller r contributes to σ I [62]. For this model, the necessary inputs are the density distribution of the core nucleus and the wave function of the halo. Usually, for nuclei with Z 8 the core-density distribution is assumed to have the HO-type density distribution shown in Eqs. (2.2) and (2.3). The size parameter is determined by the experimental value of σ I of the core nucleus. One can use a Fermi distribution (2.1) or folded Yukawa-type distribution. However, the diffuseness parameter and the size parameter must be determined by another method. Mainly for this reason, the HO-type distribution has been used so far. If we can give ρ h (or the wave-function of a halo for a few-body model), we can estimate the effective densities using a Glauber-model for few-body system. The single-nucleon density (wave-function) can be calculated by a simple potential model. When obtaining the single-particle density distribution the separation energy of the halo nucleon may be used. These methods were first applied for determining the density distribution of 11 Li and 11 Be nuclei [38,64]. It has also recently been applied to a wide range of nuclei by Ozawa et al. [20]. The recent application of such a model to 19 Cand 23 O shows a new type of anomaly, discussed in Section Discussion 5.1. Neutron skin Qualitatively, the term neutron skin describes an excess of neutrons at the nuclear surface, whereas neutron halo stands for such excess plus a tail of the neutron-density distribution. In other words, an excess of several neutrons builds up so that the neutron density actually extends out significantly further than that of the protons, resulting in a mantle of dominantly neutron matter. Recent experimental studies concerning neutron skins are reviewed in the following He and Na isotopes The presence of neutron skins in stable nuclei has been discussed since the mid 1950s [63]. No evidence of a thick neutron skin in stable nuclei has been observed, even if many of them have a large neutron excess (N Z) [65]. Some 8 years ago, a thick neutron skin with a thickness of 0.9 fm was reported by Tanihata et al. in the 8 He nucleus [13]. Recently, an α-like core and significant neutron skins in 6,8 He were confirmed by Alkhazov et al. [48]. The value of the RMS matter radii as well as the matter-density distributions are consistent with the results of Tanihata et al. The first direct comparison between nuclear matter and, charge radii over a wide range of neutron numbers was made in Na isotopes [14], where the RMS charge radii ( r 2 c 1/2 ) have been determined by isotope-shift measurements [66]. Suzuki et al. measured σ I at

20 A. Ozawa et al. / Nuclear Physics A 693 (2001) Fig. 3. The RMS neutron radii (RMS_n) (open symbols) [14] and the RMS proton radii (RMS_p) (solid circles) [66] as a function of the neutron number of Na isotopes. The dashed (solid) line is RMS_n (RMS_p) calculated by the RMF model, respectively. The difference between the open circles and the open triangles stems from the different assumed density distribution of Na isotopes. GSI, derived the effective RMS matter radii ( r 2 m 1/2 ), and deduced the RMS neutron radii ( r 2 n 1/2 ) for a chain of Na isotopes in the mass range from A = 20 to 32. Fig. 3 shows the empirical values of r 2 n 1/2 against the neutron number together with r 2 p 1/2 (the RMS proton radii). It indicates a gradual growth of the thickness of the neutron skin ( R = r 2 n 1/2 r 2 p 1/2 ) for neutron-rich Na isotopes up to 0.4 fm. The value of 0.4 fm is much larger than that observed in neutron-rich β-stable 48 Ca (0.12 fm) [67] Other isotopes Evidence has been found for the existence of a proton skin in 20 Mg and a neutron skin in 20 N from σ I measurements on A = 20 isobars [28]. However this conclusion was based on rp 2 1/2 values obtained by an extrapolation based on experimentally known rp 2 1/2 values of stable isotopes. A combined analysis of σ I and σ CC has been made by Bochkarev et al. [44], which shows evidence for the existence of a neutron skin in 20 N. They derived an upper limit for rp 2 1/2 ( rp 2 1/2 max) ofa = 20 isobars from σ I and σ CC using the Glauber model with an optical-limit approximation. Except for 20 N, all rp 2 1/2 max are greater than the corresponding rm 2 1/2 data. This feature reflects the fact that σ I are connected not only with the Glauber mechanism of projectile protons, but also with multiple scatterings and ablation [68]. For 20 N, however, rp 2 1/2 max is smaller than rm 2 1/2 by 0.4fm. Correspondingly, the neutron skin is thicker than 0.56 ± 0.29 fm. This agrees with the result of Ref. [28]. Therefore, they indicate the formation of a thick proton skin in the 20 N nucleus. At intermediate beam energies, ranging from 30 A to 100 A MeV, σ R for f p shell exotic nuclei were measured by Aissaoui et al. [36,37]. Based on the mass-number dependence of the reduced strong absorption radii in Eq. (4.1), the existence of thin neutron skins for the K and Sc isotopes is indicated. Elastic and inelastic proton scatterings have been measured in inverse kinematics on the unstable nucleus 40 S by Maréchal et al. [69]. A microscopic analysis suggests the presence of a neutron skin in the heavy sulfur isotopes of 40 S.

21 52 A. Ozawa et al. / Nuclear Physics A 693 (2001) As we have mentioned, the presence of a neutron-skin in β-unstable neutron-rich nuclei has been established. It implies the existence of new collective isovector modes at both low and high energy. In particular, the study of giant resonance (GDR) in exotic nuclei is an important area of research. Recently, a new method for measuring the neutronskin thickness of unstable nuclei has been suggested by Krasznahorkay et al. [70], using the (p,n) reaction in inverse kinematics. They demonstrated that there is a predictable correlation between the isovector spin-dipole resonance (SDR) cross section and the neutron-skin thickness General considerations As of now, much evidence has been accumulated showing that the neutron skin as well as the proton skin are quite common phenomena for unstable nuclei off the stability line. This conclusion agrees with recent predictions obtained by a relativistic mean-field (RMF) theory [71], a deformed Hartree Fock Bogoliubov model [72] and a spherical Hartree Fock (SHF) model [73]. In Fig. 4, R of the available data are plotted against the difference between the proton and neutron separation energy (S p S n ) or the Fermi-energy difference ( E F ). For nuclei having an even neutron number, we took S n as half the two-neutron separation energy. Here, S n and S p were taken from [74]. It can be seen that R has a strong correlation with S p S n. Such a correlation has been predicted by a RMF model [13]. Fig. 4. Two-dimensional plot of the difference between the proton separation energy and that of aneutron(s p S n ) versus the thickness of the neutron skin R. The experimental data are from [14] for Na isotopes, [13] for 6 He, and [44] for 20 N, respectively. The corresponding mass number is indicated at each data point for Na. The shaded area shows the calculated correlation for various isotopes ranging from helium up to lead. The inset shows the occupation of nucleons in a self-consistent potential. Protons see a shallower potential due to Coulomb interactions. The dark-gray area indicates the proton and neutron occupation in a stable nucleus. The lightly shadowed area shows the extra neutrons, which contribute to the formation of a neutron skin in β-unstable nuclei. The deepening of the proton potential in unstable nuclei due to the attractive p n interaction is not shown.

22 A. Ozawa et al. / Nuclear Physics A 693 (2001) In stable nuclei, the ratio of N/Z are restricted between 1 to 1.5. Also, the separation energy (S n ) of a nucleon is always 6 to 8 MeV. Due to the stability and these boundaries, (i) the observed central density is the same for all stable nuclei (ρ fm 3 ), and thus the radius of stable nuclei is proportional to A 1/3, and (ii) protons and neutrons have similar density distributions. In unstable nuclei, on the contrary, N/Z and S n can be varied from 0.6 to 4 MeV and 40 to 0 MeV, respectively. These variations manifest the decoupling of proton and neutron distributions as skins and halos. This situation is illustrated in the inset of Fig. 4. We thus understand the appearance of skins and halos, as mentioned at the beginning of this section Isospin dependence The isospin dependence of the nuclear sizes provides a new systematics that could not be studied without RI beams. Recent improvements of measurements allow seeing the isospin dependence of the radii for nuclei with A 20. The first isospin dependence analysis was made for isobars up to A = 12 [21]. It was seen that an isobar with larger isospin shows a larger radii than others. It was also observed that the mirror nuclei have the same radii, even if they have very different separation energies. It was shown from a comparison with Skyrme Hartree Fock calculations with various interactions and the RMF model, that the density-dependent interaction is important to explain the observed isospin dependence. For nuclei with larger A, similar behaviors of radii have been observed [24]. However, several anomalies were also seen. For the A = 17 system,a strongisospin dependencehas been observed [27]. Also, the mirror symmetry of radii is strongly broken in the 17 Ne 17 N pair. The radius of 17 Ne is much larger than that of 17 N. It shows a proton halo formation in 17 Ne and a different configuration of the ground states of these two nuclei. It is considered that the ground state of 17 Ne has a strong 2s 1/2 component in contrast to 1d 5/2 in 17 N. The increases of the radius in 17 B also suggested neutron halo formation. For the A = 20 system [28], the isobar radii show an irregular dependence on the isospin. It was considered that a change in the nuclear deformation is the main cause of the change of the radii. The largest difference in the radii has been obtained for mirror nuclei, 20 Oand 20 Mg. By evaluating the difference in the radii of the neutron and proton distributions, evidence has been found for the existence of a proton skin for 20 Mg and of a neutron skin for 20 N, as described in the previous section. With the naive nuclear model, nuclear radii for mirror nuclei are the same, since mirror symmetry is fulfilled in the nuclear force. Although the separation energy may be different due to a Coulomb energy shift, the height of the effective potential barrier for a proton is the same as that for a neutron if Coulomb barrier is included. Therefore, the radii of mirror pairs are the same unless their configuration is extremely different, as can be seen in the 17 Ne 17 N pair. This is also consistent with the data. However, according to the RMF calculations, the radius of proton-rich nucleus of a pair is always slightly larger than that of neutron rich nuclei, because of the Coulomb effects [75]. For a detailed comparison of the radii in the mirror pairs, we can define the ratio (δ)as