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ELSWIER Nuclear Physics A731 (2004) 224-234 www.elsevier.com/locate/npe

Neutron-skin thickness in neutron-rich isotopes

A. Krasznahorkay,“” H. Akimune,c A.M. van den Berg,b N. Blasi,d S. Brandenburg,b M. Csatloqa M. Fujiwara,ef J. Gulyas,” M.N. Harakeh,b M. Hunyadi,b* M. de Huu,~ Z. Mate,” D. Sohler,” S.Y. van der Werf,b H.J. WGrtcheb and L. Zolnai”

aInstitute of Nuclear Research (ATOMKI), Debrecen, P.O. Box 51, H-4001, Hungary

bKernfysisch Ve rsneller Instituut, Zernikelaan 25, 9747 AA Groningen, The Netherlands

“Department of Physics, Konan University, Okamoto 8-9-1, Higashinada, Kobe 6588501, Japan

‘Istituto Nazionale di Fisica Nucleare, Via G. Celoria 16, 20133 Milano, Italy

eResearch Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan

f Japan Atomic Energy Research Institute, Advanced Science Research Center, Tokai 319-1195, Japan

After a short overview of the methods applied for measuring the neutron-skin thickness, we present the recent experimental results for the neutron-skin thicknesses of the 112-124Sn even-even isotopes and of “‘Pb We have used inelastic alpha scattering to excite the giant dipole resonance (GDR). The cross section of this process depends strongly on L1R,/R, the relative neutron-skin thickness. We have also measured the excitation of the spin-dipole resonance (SDR) t o e d d uce the neutron-skin thickness since the summed L=l strength of the SDR is sensitive to it. The results obtained are in good agreement with the previous experimental and theoretical ones.

1. INTRODUCTION

The difference between the neutron and proton radii of a heavy stable nucleus is of the order of a few percent. This general feature of nuclei, the neutron skin, may provide fundamental information on the properties of neutron-rich matter.

Nowadays, great efforts are devoted to produce highly neutron-rich isotopes and to study their properties. There exists a large part in the nuclear chart between the known neutron-rich isotopes and the neutron drip line, which has been denoted TERRA INCOG- NITA. Because of limited knowledge about the properties of such neutron-rich nuclear matter we can not determine even the border of the Terra Incognita, i.e. neutron drip

“Present address:Institute of Nuclear Research (ATOMKI), D b. e lecen, P.O. Box 51, H-4001, Hungary

0375-9474/$ - see front matter 0 2004 Elsevier B.V. All rights reserved, doi:10.1016/j.nuclphysa.2OO3.11.034

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line, with a precision better than 10 mass units around Z=50. A number of questions arises:

l Could we determine the neutron drip line more precisely?

l What do we know about the equation of state of neutron-rich nuclear matter?

l How does the nuclear force depend on isospin?

Neutron skin is already a kind of neutron-rich matter, which can be studied even in stable isotopes. Therefore, an intriguing question is: can we learn something about the equation of state (EOS) f o neutron-rich matter by measuring the thickness of the neutron skin? Furnstahl answered this question in his recent work [l]. He calculated the neutron- skin thickness in various models with different parameterizations, and investigated their sensitivity. One of the surprising results he found was that a well-defined correlation exists between the symmetry-energy term of the nucleus-energy function and the neutron-skin thickness. This correlation remained about the same in all the relativistic and non- relativistic models. According to this correlation, one can constrain the symmetry-energy term of the EOS by measuring the thickness of the neutron skin.

2. PREVIOUS RESULTS FOR THE NEUTRON-SKIN THICKNESSES

A straightforward way to determine the neutron-skin thickness of atomic nuclei is to measure the charge and neutron root mean square (rms) radii and subtract them from each other. The charge radius and the charge distribution can be measured very precisely by using electromagnetic probes like elastic electron scattering [a]. Since the electromagnetic interaction is precisely known and the wave length of high energy electrons can be much shorter than the size of the nucleus, one can study the charge distribution accurately.

The situation in determining the neutron or matter rms radius is much worse. We have to use probes whose interaction is mediated by the strong force, which is not known precisely. Furthermore, different nuclear models have to be employed, which makes the results less precise than in the case of the charge radius. In spite of these difficulties, there are different methods for determining the mass radius of nuclei [3].

2.1. Hadron scattering Hadron scattering has been used for studying the mass distribution of nuclei for over

three decades. Important progress was made when high-energy polarized protons became available. The Relativistic Impulse Approximation (RIA) with free nucleon-nucleon in- teractions could be applied for analyzing the data. After an elaborate analysis of the experimental data, Ray and co-authors [4,5] determined the neutron rms radii of a few stable spherical nuclei with better than 1 % error although it is difficult to judge about the model dependence of these values. Kelly et al. [6] p ro p osed a rather model-independent analysis, but the situation appeared to be too complicated to obtain reliable data from their analysis. Stradubski and Hintz [7] d e d uced also the neutron densities in 206j207,208Pb in a model-dependent state-of-art work for elastic proton scattering.

226 A. Krasznahorkay et al. /Nuclear Physics A731 (2004) 224-234

2.2. x- elastic scattering The cross section of n- elastic scattering on the nucleon is relatively large in the A(1332)

resonance region and is about three times larger for neutrons than for protons. This makes w elastic scattering a promising tool for studying the neutron distribution of nuclei. Unfortunately, a strong absorption occurs at the nuclear surface, making this method very sensitive to the tail of the distributions. The method was successfully used only for studying the neutron distributions of light stable nuclei. Takahashi [8] tried to increase the energy of the pions to decrease their cross sections. The result for “‘Pb agrees with those obtained by other probes.

2.3. Antiprotonic atoms A slow antiproton can be captured into an atom like an electron. Since its mass is

about 1800 times larger than that of the electron the radius of atomic orbits becomes extremely small. This means that antiproton reaches the surface of the nucleus already at n=9,10. Lubinski et al. and Trzcinska et al. [g-11] proposed a simple radiochemical method, and in-beam antiprotonic X-ray measurements for studying the nuclear stratosphere. The radiochemical method consists of the study of the annihilation residues with a mass number, one unit smaller than the target mass. The relative yields of the N-l and Z-l isotopes are related to the proton and neutron densities at the annihilation site. The position of the annihilation is calculated to be at about 2.5 fm larger than the half-density charge radius. Assuming the validity of two-parameter Fermi neutron and proton distributions they could determine the neutron-skin thicknesses. However, they stated that in heavy neutron-rich nuclei it is mostly the neutron diffuseness which increases and not the half-density radius. That statement disagrees with the recently available theoretical results [12].

2.4. Parity-violating electron scattering There is a proposal that has been accepted recently at Jefferson Laboratory, which

aims at determining the neutron distribution in a model-independent way using a high- intensity polarized electron beam from CEBAF [13,14]. The proposal is based on the idea that the Z-boson couples primarily to the neutron at the nuclear surface. By measuring the parity-violation in electron scattering, the weak-charge density can be mapped out and the neutron density can be determined. This proposal aims to measure the mass radius of ‘O*Pb with a precision of 1%. Although, this precision is not better than obtained by other methods, the great advantage of this method is its model independence.

2.5. The GDR method Although it is difficult to obtain the neutron density distribution, the difference in radii

of the neutron and proton density distributions is more easily accessible. In our previous work on inelastic alpha scattering, excitation of the giant dipole resonance (GDR) was used to extract the neutron-skin thickness of nuclei [15,16]. The cross section of this process depends strongly on ARPN/Ro, the relative neutron-skin thickness [17]. Unfor- tunately, the cross section of the GDR excitation is very small relative to those of other overlapping resonances. Thus, Q: - y coincidence measurements are needed to extract the small GDR cross sections.

A. Krasznahorkay et al. /Nuclear Physics A731 (2004) 224-234 227

2.6. The SDR method Another possible tool for studying the neutron-skin thickness is the excitation of the

spin-dipole resonance (SDR) [18]. The L=l strength of the SDR is sensitive to the neutron-skin thickness [19,20]. I n a previous work, we have demonstrated that the correlation between the SDR cross section and the neutron-skin thickness of nuclei is predictable. Normalizing the results for the case of 12’Sn the neutron-skin thicknesses obtained from the SDR strength measurements from the Sni3He,t) reactions are in good agreement with theoretical predictions [21,22] and previous measurements [3,16] along the stable Sn isotopes. These investigations can also be extended to unstable nuclei using (p,n) reactions with radioactive nuclear beams in inverse kinematics.

2.7. Summary of the obtained results The obtained results for the spherical Sn and Pb isotopes are summarized in Table 1.

Table 1 Summary of the neutron-skin thicknesses ((ri)“” - (r,) 2 V2 in fm) obtained in different methods.

I;;;ye (P,P) [4,51 (p,p) [71 GDR [16] SDR [18] antiproton [II] 0.09*0.02

‘14Sn SO.09 ?Sn 0.15f0.05 0.02f0.12 0.12&0.06 0.12&0.02 “‘Sn 0.1310.06 lzoSn 0.18”) 0.12f0.02 lz2Sn 0.22f0.07 124Sn 0.25ztO.05 0.21f0.11 0.19fO.07 0.19f0.02 208Pb 0.14f0.04 0.20f0.04 0.19f0.09 0.15~tO.02

“) Normalized to the theoretical value of Angeli et al. [21].

The most precise data for the neutron-skin thicknesses are provided by the antiproton method [ll], in which the composition of the nuclear stratosphere, where the nuclear density falls rapidly, is measured. However, by extrapolating the distributions from the tail part by assuming two-parameter Fermi (2pF) distributions, the systematic error can be very large.

Moreover, Trzcinska et al. [ll] assumed that the neutrons in the stratosphere are distributed more like a halo than a skin, and therefore the difference of the rms radii of the neutron and proton distributions is caused by the different shapes of the distributions rather than by the different radii. Using 2pF distributions they could explain their results by assuming different diffusenesses and similar radii.

In our previous work using the GDR method [16], we analyzed the experimental results by assuming similar neutron and proton distributions with slightly different radii (c) and could get consistent results for the neutron-skin thicknesses. This strongly suggests that we are dealing with a neutron skin and not a neutron halo.

In order to clarify the situation we repeated the experiment for “‘Pb with higher beam energy to get better statistics. We repeated also the SDR measurement for the Sn isotopes


with better statistics, which provides data directly for the relative neutron-skin thickness. The results were compared to the recent theoretical predictions.

3. THE PRESENT RESULTS OBTAINED WITH THE GDR METHOD

The new experiments were carried out at the Kernfysisch Versneller Instituut (KVI). A momentum-analyzed 196 MeV a-particle beam provided by the super-conducting cyclotron AGOR was used to bombard the enriched (99.0 %), self-supporting 208Pb target with a thickness of 20 mg/cm2.

The energy and the scattering angle of the a: particles were measured with the Big-Bite Spectrometer (BBS) [23]. BBS consists of two quadrupoles and a large dipole magnet. In conjunction with the BBS, a focal-plane detection system of the EuroSuperNova Collab- oration [24] was used, which was able to determine the position and the incidence angle of the a-particles. By applying ray-tracing techniques, it was possible to determine the scattering angles of the cy particles with a reliable precision.

In order to separate effectively the GDR f rom the GMR, GQR and the nuclear continuum excited in inelastic o-scattering, coincidence measurements were performed between the scattered a-particles and the emitted y rays decaying to the ground state.

The emitted y rays were detected by a large lO’x14” NaI(T1) crystal placed at 125” with respect to the beam direction equipped with a plastic anti-coincidence shield in the same way as described in Ref. [16]. Th e solid angle of the detector was 70 msr.

To improve further the efficiency of the y ray detection a four-element Ge Clover type detector surrounded by a BGO shield [25], pl aced at 55” to the beam axis was also applied in adding-back mode described in Ref. [26].

In order to normalize the cross sections, the K X-ray yield from the target was measured with a 20 cm2 x 10 mm planar Ge detector placed at 55” with respect to the beam direction. The accuracy of the K X-ray production cross sections was 5 10% [27-291.

The final-state spectra were obtained by summing up the energy of the inelastically scattered cy particles and the coincident y rays for the energy range 8 MeV<E,<15 MeV and for the spectrograph solid angle AR = 7.8 msr averaging over the scattering angle 0,~=1.5”-4.7”. Random coincidences were subtracted. The areas of the peaks in the final state spectra (shown in Fig. 1.) were determined by numerical integration. The obtained experimental o - 70 coincidence cross section for the GDR determined by the NaI(T1) and the Ge Clover detector is: 342518 pb/sr and 330618 pb/sr, respectively. Taking into account 10 % systematic error, their weighted average is 336rf34 ,ub/sr. The dominant component of the uncertainty arises mainly from the systematic errors of the X- ray normalization, the focal-plane detector system and the uncertainties of the efficiencies of the y-ray detectors.

To calculate the cross section for excitation of the GDR we used the usual approach, which connects the oscillations of the proton and neutron density distributions with the oscillations of the associated optical potential.

DWBA cross sections were calculated using the code ECIS [30] with the optical-model parameters determined by Goldberg et al. for “‘Pb [31]. In the derivation of the coupling potentials, which are the most crucial quantities in the calculations, the prescription of Satchler [17] described in Ref. [16] was used. For the density oscillations we adopt both

A. Krasznahovkay et al. /Nuclear Physics A731 (2004) 224-234 229

60

40

190 192 194 196 198 2 E, + E,(d) (MeV

0

190 192 194 196 198 : E, + EJreal) (Me\

r 1 200 ‘1

3 2 450

6 8 \

3 400 2

350

300

250

200 -1

+i 0.12+ 0.07 fm

0123456 AR,,+, (s

Figure 1. Final-state spectra of the 208Pb(Cu, cu’y) experiment constructed for the excitation energy region 8 MeV < E, 5 15 MeV. Decay y-rays are measured by the NaI(T1) detector (a) and by the clover detectors (b). The random coincidences have been subtracted from both spectra. The positions of the ground state and a few excited states in “‘Pb are denoted in the figures. The calculated a - 70 coincidence cross section as a function of the relative neutron-skin thickness compared to our experimental data (c).

the Goldhaber-Teller (GT) and the Jensen-Steinwedel (JS) macroscopic models. Coulomb excitation is included in both calculations by adding the usual [17] Coulomb transition potential. The cross sections a,,, ““(E) were calculated as a function of excitation energy by assuming 100% exhaustion of the TRK EWSR. Th e results were then folded with the photo-nuclear strength distribution (uY(E)) [32] as follows:

uaa’(E) = CT;?%(E) &4E) ) (1)

with gT(E) in barns. The ua,a’7o (E) coincidence cross sections were deduced following the procedure de-

scribed in Ref [16]. Fig. lc) shows the theoretical curve of the cross section as a function of the relative

neutron-skin thickness @.RPN/Ro. Comparing the obtained experimental cross section value with this curve, ARpN=0.12&0.07 fm was found for ‘08Pb. Our present result is within the uncertainties in agreement with the previous value provided by a similar experimental method [15], as well as with the (p,p) result [5].


4. THE PRESENT RESULTS OBTAINED IN THE SDR METHOD

A 177 MeV 3He beam from the AGOR cyclotron was used to bombard the self- supporting metallic ‘12Sn, ‘14Sn, 116Sn, 118Sn, 120Sn, 122Sn, and 124Sn targets with thicknesses of 5 - 13 mg/cm2 and isotopic enrichments of 75.0, 68.3, 98.0, 98.3, 99.6, 96.6, and 96.6 %, respectively. The typical beam intensity was 4 nA. The energy resolution of the beam was ~300 keV. Tritons were detected by the BBS, which was set at 0” with respect to the beam direction. The scattering angle of tritons was reliably determined by using ray-tracing techniques. A sample of the triton spectra is shown in Fig. 2.

@ 40000 2 3 35000 3 3 -9 30000

20000

t 1 ‘12Sn(3He,t)

0.29'

0 5 10 15 20 25 30 35 Ex WeV) Ex (MeV

0.29'

0.86'

- 5 10 15 20 25 30 35 L

Figure 2. Excitation-energy spectra measured in the 112Sn(3He,t) and in the 124Sn(3He,t) reactions.

The decomposition of the spectra into resonances and non-resonant components, as shown in Fig. 3a), makes use of a fitting procedure invoking analytical equations. The resonances are described by assuming Lorentzian shapes for the energy distributions involving three free parameters each: the centroid energy, the width, and the height. For the Quasi-Free Continuum (QFC) backg round, a parameterized analytic formula is used [33].

A. Kvasznahorkay et al. /Nuclear Physics A731 (2004) 224-234 231

(Js/quI) 15 P / Q P

VI

P

_:_;.

2 = .

. .

t 3

f-09 E 0”

t-4

3

0

Figure 3. a) (3He,t) energy spectra for “‘Sn taken at Bt = 3.2”. The solid lines in the spectrum represent the fits of the peaks for the SDR, GTR, IAS and the background due to the QFC process. b), c) Comparison of the calculated (solid lines) and measured (dots) differential cross sections of the IAS and SDR.


The spectra measured at different angles were fitted simultaneously using the same shape parameters for the resonances and for the background and only their amplitudes were considered to be independent in the spectra. The energy and the width of the spin- flip dipole resonance obtained from the present analysis are in good agreement with the previous experimental data [19,34]. F or each isotope, good agreement has been achieved also for the experimental and calculated differential cross sections of the IAS and SDR. In the case of “‘Sn, this is illustrated in Figs. 3b) and 3~).

The precise relative cross sections of the dipole resonances (dominated by the SDR) compared to that of the isobaric analogue state have been determined by analyzing both the excitation energy and the angular distribution of tritons. As the cross section of the IAS is proportional to N-Z [18] th e relative cross sections of the dipole resonances along the whole Sn isotopic chain could be determined very precisely.

The SDR method is based on the connection between the L = 1 strength and the neutron-skin thickness of the target nucleus [18]. The following sum rule is valid for the spin-dipole operator involving the difference between the p- and /3+ strengths [19,20],

where (T~)~ and (r”h p re resent the rms radii of the neutron and proton distributions, respectively. One can obtain a similar expression also for the charge-exchange non-spin- flip modes [35].

The experimental cross section of the L = 1 transitions measured in (p,n)-type reactions allows one to deduce the S- only, therefore a theoretical estimate of the Sf is needed. In this work, instead of using a simple model for the energy-weighted sum rule as we did previously [18] we took more precise S/S- ratios from continuum RPA calculations [36]. The model parameters were taken from Ref. [37] and effects of neutron pairing correlations have been neglected.

Using the calculated B = S/S- ratios the neutron-skin thicknesses can be deduce’d from eq. 2.:

(T2)1/2+2)l/2 = afle,(l- B) - (N - ZP2)P 72 P 2N(?q2

7

where ceaip is the experimental cross section of the SDR strengths and cy is a normalization constant.

The difference of the neutron-proton rms radii was calculated by using eq. 3. The values of rp = (r2)ki2 are taken from Ref. [38] and the CY normalization constant is determined by accepting the experimental result of Ref. [4] for the difference (r”)k/” - (r2)ii2 in ?Sn. The results obtained are compared in Fig. 4 as a function of mass number. The results of the previous measurements and theoretical values are also presented in Fig. 4.

5. CONCLUSION

Our recent results obtained for the neutron-skin thicknesses using the GDR and SDR methods agree well with both the previous experimental and theoretical ones. However,

A. Kvasznahor-kay et al. /Nuclear Physics A731 (2004) 224-234 233

Figure 4. The full dots with error bars show the neutron-skin thicknesses of the Sn isotopes determined in the present work as a function of the mass number. The experimental values determined by the (p,p) [3], and the antiprotonic methods are shown as full triangles and full squares with error bars, respectively. The numbered full lines represent the following theoretical results: (1) RHB/NLS, (3) RHB/NLSH, (4) HFB/SLy4, and (5) HFBjSkP are calculations performed by Mizutori et al. [la]. The line (2) is the calculation performed by Lalazissis et al. [39].

the value obtained for 208Pb (0.12&0.07) . 1s smaller than the best value obtained by the (p,p) method (0.2O~tO.04) [7]. W e are planning to recalculate the GDR cross sections assuming also the suggested [II] neutron halo. The SDR method gave the smallest (<O.Ol fm) error bars in the literature for the relative neutron-skin thicknesses, although the normalization is necessary.

6. ACKNOWLEDGEMENTS

The authors acknowledge the KVI cyclotron staff for their support during the course of the experiments. This work has been supported by the Hungarian OTKA Foundation No. TO38404 and the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).

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