Nuclear charge radius of 11Li

Hyperfine Interact (2006) 171:181–188DOI 10.1007/s10751-007-9507-6

Nuclear charge radius of 11Li

Rodolfo Sánchez · Wilfried Nörtershäuser ·Andreas Dax · Guido Ewald · Stefan Götte ·Reinhard Kirchner · H.-Jürgen Kluge ·Thomas Kühl · Agnieszka Wojtaszek ·Bruce A. Bushaw · Gordon W. F. Drake ·Zong-Chao Yan · Claus Zimmermann ·Daniel Albers · John Behr · Pierre Bricault ·Jens Dilling · Marik Dombsky · Jens Lassen ·C. D. Phil Levy · Matthew R. Pearson ·Erika J. Prime · Vladimir Ryjkov

Published online: 1 February 2007© Springer Science + Business Media B.V. 2007

R. Sánchez (B) · W. Nörtershäuser · A. Dax · G. Ewald · S. Götte · R. Kirchner ·H.-J. Kluge · T. Kühl · A. WojtaszekGesellschaft für Schwerionenforschung, D-64291 Darmstadt, Germany,e-mail: [email protected]

B. A. BushawPacific Northwest National Laboratory, P.O. Box 999, Richland, WA 99352, USA

G. W. F. DrakeDepartment of Physics, University of Windsor, Windsor, Ontario, Canada N9B 3P4

Z.-C. YanDepartment of Physics, University of New Brunswick, Fredericton, New Brunswick,Canada E3B 5A3

C. ZimmermannEberhard Karls Universität Tübingen, Physikalisches Institut, D-72076 Tübingen, Germany

D. Albers · J. Behr · P. Bricault · J. Dilling · M. Dombsky · J. Lassen · C. D. Phil Levy ·M. R. Pearson · E. J. Prime · V. RyjkovTri-University Meson Facility, Vancouver, British Columbia, Canada V6T 2A3

Present Address:A. DaxCERN, CH-1211 Geneva 23, Switzerland

Present Address:A. WojtaszekInstitute of Physics, Swietokrzyska Academy, PL-25-406 Kielce, Poland

182 R. Sánchez, et al.

Abstract We have determined the nuclear charge radius of 11Li by high-precisionlaser spectroscopy. The experiment was performed at the TRIUMF-ISAC facilitywhere the 7Li-11Li isotope shift (IS) was measured in the 2s → 3s electronic transi-tion using Doppler-free two-photon spectroscopy with a relative accuracy better than10−5. The accuracy for the IS of the other lithium isotopes was also improved. IS’sare mainly caused by differences in nuclear mass, but changes in proton distributionalso give small contributions. Comparing experimentally measured IS with advancedatomic calculation of purely mass-based shifts, including QED and relativistic effects,allows derivation of the nuclear charge radii. The radii are found to decreasemonotonically from 6Li to 9Li, and then increase with 11Li about 11% larger than9Li. These results are a benchmark for the open question as to whether nuclear coreexcitation by halo neutrons is necessary to explain the large nuclear matter radius of11Li; thus, the results are compared with a number of nuclear structure models.

Key words laser spectroscopy · nuclear charge radius · isotope shift ·halo nucleus · lithium

1 Introduction

Some of the lightest neutron-rich nuclei have been found to have much larger rootmean squared (rms) radius than their neighboring isotopes [1]. Very well known isthe neutron-drip-line nucleus 11Li. Its large nuclear matter radius has been explainedin terms of the small two-neutron separation energy (375 keV [2]), which allowsthe neutron wavefunction to extend far outside the nuclear core as a result ofquantum mechanical tunneling. Thus, 11Li is pictured as a 9Li-like core plus thetwo weakly-bound neutrons forming a “halo” around it. The size of this halo canbe as large as the matter radii found in heaviest naturally occurring elements. Inorder to understand this fascinating structure, extensive experimental and theoreticalstudies have been performed on this halo nucleus. However, the distribution of thethree protons inside the core is not yet completely understood. Experimentally, thiscan be probed by observing a change in the nuclear charge radius between 9Li and11Li. Combining isotope shifts determined by high-resolution laser spectroscopy withhigh-accuracy atomic theory calculations has lead to a new measurement principlefor the determination of nuclear charge radii for short-lived isotopes of very lightnuclei [6]. This method has already successfully been applied in two- (6He [3])and three-electron (8Li, 9Li [4]) systems, where recent advances in atomic theorycalculations have provided sufficiently accurate values for the mass effect. In thiscontribution, we discuss how this technique was used for the determination of the11Li nuclear charge radius [5], and the results are compared with a variety of nuclearstructure models.

2 Isotope shift and nuclear charge radius

The extraction of nuclear charge radii from optical isotope shifts (IS) is based on thefact that the frequency of an atomic transition is shifted between two isotopes A andA’ according to the change in their mean-square nuclear charge radii (〈rc

2〉) as,

δνA,A’FS = −2π

3Z e2�|�(0)|2 (〈rc

2〉A − 〈rc2〉A’

), (1)

Nuclear charge radius of lithium-11 183

Table 1 Values for the experimental isotope shift (δν A,7), calculated mass shift (δν A,7MS )∗, changes in

the mean-square nuclear charge radii (δ⟨r2

c⟩) [5] and root-mean-square nuclear charge radii (rc) for

the lithium isotopes

Quantity 6Li 8Li 9Li 11Li

δν A,7, MHz -11,453.983(20) 8,635.782(44) 15,333.226(39) 25, 101.226(124)δν

A,7MS , MHz -11,453.010(56) 8,635.113(42) 15,332.025(75) 25,101.812(123)

δ⟨r2

c⟩A,7, fm2 0.622(38) -0.427(39) -0.796(54) 0.374(54)

rc, fm 2.517(30) 2.299(32) 2.217(35) 2.467(37)

Uncertainties for rc are dominated by the uncertainty of the reference radius rc(7Li) = 2.39(3) fm.

∗After completing this report, more accurate mass shift calculations were published by M. Puchalskiet al. [7], which differ from those given in [6], and slightly shift the nuclear charge radii. The largestdifference is for 11Li, where it is reduced by 0.045 fm.

where �|�(0)|2 is the change of the expectation value of the electron density atthe nucleus of charge Z e. δν

A,A’FS is known as the field shift (FS). For the low-Z

elements, like lithium, the purely mass-based portion of the IS, the mass shift (MS),can be calculated with sufficient accuracy such that the residual difference betweenexperimental IS and calculated MS can be attributed to the FS, and thus be relatedto the changes in nuclear charge radii (1). These calculations solve the three-electronnon-relativistic Schrödinger equation with high accuracy and include relativistic andquantum electrodynamic effects by perturbation theory. Results of such calculations[6] are given for all lithium isotopes in Table 1.

3 Experimental

To determine the nuclear charge radius of 11Li, the experimental setup was installedat the TRIUMF-ISAC facility in Vancouver, Canada where one of the world’shighest yield (∼ 30,000/s) of low-energy (≈ 40 keV) 11Li ions is produced bybombarding a tantalum target with a 40 μA, 500 MeV continuous proton beam froma cyclotron. In our experiment (see Fig. 1) the 11Li ions are stopped inside a thincarbon foil (thickness ≈ 300 nm), and then thermally released as neutral atoms byheating the foil with a CO2 laser. The atoms cross the focus of two overlapping laserbeams: a titanium–sapphire (Ti:Sa) laser at 735 nm induces Doppler-free two-photontransitions from the 2s ground levels to the 3s excited levels. The atoms then decay tothe 2p levels, where they are resonantly excited to 3d levels by a dye laser at 610 nm,and then photoionized by absorption of another photon from either of the lasers.The resulting ions are mass separated with a quadrupole mass filter and detectedwith a continuous-dynode electron multiplier (CDEM). To maximize excitation andionization efficiency, both laser powers of several 100 mW are increased by a factorof 100 in a resonant optical cavity (30 cm length) built around the excitation region.To couple both lasers into the same cavity, the cavity is locked to the Ti:Sa laserwhile the dye laser is locked to the cavity. The Ti:Sa laser is stabilized by frequency-offset locking to a reference diode laser, which is locked to an I2 hyperfine lineby Doppler-free saturated absorption. The overall detection efficiency is greaterthan 10−4.


Fig. 1 Experimental setup and excitation scheme for the resonance ionization of lithium

4 Results

A typical spectrum recorded for 11Li is shown in Fig. 2. The nuclear spin of 11Li(I = 3/2) splits the 2s and 3s atomic energy levels into F = 1 and F = 2 hyperfinecomponents. Selection rules for s → s two-photon transitions allow only �F = 0 andthus only two transitions are observed. The isotope shift is calculated as the centerof gravity (cg) of the two hyperfine components, relative to the cg of a referenceisotope, 7Li. In total, twenty-four spectra like that in Fig. 2 were obtained for 11Li.Spectra were also obtained for the other lithium isotopes and resulting isotope shiftsare given in Table 1.

Root-mean-square nuclear charge radii extracted from these measurements incombination with mass shift calculations are also given in Table 1, where the nuclearcharge radius of 7Li (rc = 2.39(3) fm) measured by electron scattering [8] has beenused as the reference. These radii are represented by the black points in Fig. 3.Experimental uncertainties in nuclear charge radii for 6,8,9Li are dominated bythe uncertainty in the 7Li reference radius, while the spectroscopic measurementscontribute an uncertainty of ∼0.022 fm for 11Li. The values obtained for 6−9Li are inexcellent agreement with our previous measurements at GSI [4] but have improvedprecision. In Fig. 3 one can clearly see that the nuclear charge radius decreasescontinuously from 6Li to 9Li. This decrease can be understood in terms of clusteringof the different nuclei. For example, 6Li is known to be strongly clustered into anα-particle and a deuteron. Adding more neutrons, this cluster structure tends to“melt” and a more compact object is formed. Beyond the decrease of nuclear chargeradii from 6Li to 9Li, a significant increase is observed for 11Li, indicating a largechange in the behavior of the 9Li-like core.


Fig. 2 Typical resonance-ionization spectrum in the2s → 3s transition recordedfor 11Li. The x-axis representsthe beat frequencybetween the referencediode laser and thetitanium-sapphire laser

Nuclear charge radii predicted by different nuclear models are also shown inFig. 3. These theories construct the equations of motion that describe light nucleiusing realistic two- and three- nucleon potentials. These potentials are based onmeson-exchange interactions, with parameters that are usually determined by fittingto experimental nucleon-nucleon scattering data. Once these potentials are definedthey can be used to numerically calculate wave functions, ground state energies, anddensity and momentum distributions numerically. Such calculations require rapidlyexpanding computer resources for every nucleon added and are currently limited tosystems of mass number ≤ 12.

The nuclear charge radii predicted by models that treat protons and neutrons aspoint-like particles were converted from point-proton mean-square radii 〈rp

2〉 intomean-square nuclear charge radii 〈rc

2〉 by folding in the proton mean-square chargeradius [17], 〈Rp

2〉 = 0.801(32) fm2, and the neutron mean-square charge radius [18] ,〈Rn

2〉 = −0.117(4) fm2, according to [19]

〈rc2〉 = 〈rp

2〉 + 〈Rp2〉 + N

Z〈Rn

2〉 + 3�2

4Mp2c2

. (2)

The last term, 3�2/4Mp2c2 ∼ 0.033 fm2 , where Mp is the proton mass is the

Darwin–Foldy correction which accounts for “Zitterbewegung” of the protons.As shown in Fig. 3 most of the models give good predictions of nuclear charge

radius from 6Li to 9Li; however, the predictions for 11Li spread over a large range.The No-Core Shell Model [10] (�) and the Large-Basis Shell Model [9] (�) are

essentially the same models. In both cases calculations have been performed usingrealistic nucleon-nucleon potentials. While early calculations for 7−11Li treated thethree-body interactions as an effective phenomenological potential [9], the model hasbeen revised for 6,7Li to include microscopic three-body potentials [10]. As shown inFig. 3, neither the absolute charge radii nor the trend along the isotope chain agreewith the experimental results; in particular, the 9Li and 11Li charge radii are predictedto be nearly the same.

Greens-Function Monte Carlo Calculations [11, 12] () are in good agreementwith the experimental results for 6Li to 9Li, but the halo nucleus 11Li has not yet beentreated successfully by this technique. Due to the weak two-neutron binding energy


Fig. 3 Root-mean-square nuclear charge radii of the lithium isotopes: · · · � · · · this work, · · · � · · ·ab-initio no-Core Shell Model [10], · · · ♦ · · · Large-Basis Shell Model [9], · · · · · · Greens-FunctionMonte-Carlo Model [11, 12], · · · � · · · Stochastic Variational Multi-Cluster Model [13, 14], · · · ⊕ · · ·Fermionic Molecular Dynamics Model [15], · · · © · · · Dynamic Correlation Model [16]

of 375 eV [2], the calculated trial function in this approach usually does not convergeinto a bound state during the subsequent propagation procedure in the variationalMonte Carlo calculations.

Fermionic Molecular Dynamic Model [15] (⊕) values almost agree with theexperimental results for 6−9Li, but, so far, the nuclear charge radius for 11Li couldnot be obtained since the model does not deliver the right binding energy for thisnucleus.

The values for 6,7Li derived from the Dynamic Correlation Model [16] (©) agreewith the measurements but the values for 9,11Li are clearly overestimated. However,it does reproduce the increase from 9Li to 11Li.

The best overall agreement is observed for the Stochastic Variational Multi-Cluster Model [13, 14] (�), and the predicted 11Li charge radius is close to theexperimental value. It is interesting to note that a simple three-body model of 11Li,which does not include 9Li core excitation, gives a value of rc = 2.40(6) fm [20],(Zhukov, 2005, private communication) in good agreement with our result.

5 Summary and outlook

Laser spectroscopy measurements of isotope shifts combined with accurate theoret-ical mass shift calculations have allowed the determination of the nuclear chargeradius of the lightest short-lived isotopes.

Figure 4 schematically shows measured nuclear charge radii in the lower part ofthe chart of nuclides. Helium and Lithium are now the only elements with Z < 10


Fig. 4 Root-mean-square nuclear charge radii of the lightest stable and unstable isotopes. For thenaturally occurring isotopes nuclear charge radii were determined by electron scattering while forthe radioisotopes, like, 8−11Li [this work] and 6He [3] the isotope shift method has been applied

for which nuclear charge radii have been determined for radioactive isotopes. With11Li and 6He, two systems of two-neutron halos have been investigated. Furtherexperiments are planned to measure the nuclear charge radius of other short-livedlight isotopes. The BeTINa collaboration [21] will investigate the nuclear chargeradius of the one-neutron halo 11Be by laser spectroscopy of Be ions in a Paul trap,while the group from Argonne National Laboratory will extend their measurementsto the four-neutron halo nucleus 8He.

Acknowledgements This work is supported from BMBF Contract No. 06TU203 and EURONS(European Commission Contract No. 506065). Support from the U.S. DOE Office of Science(B.A.B.), NRC through TRIUMF, and NSERC and SHARCnet.(G.W.F.D. and Z.-C.Y.) is acknowl-edged. A.W. was supported by a Marie-Curie Fellowship of the European Community ProgrammeIHP under contract number HPMT-CT-2000-00197.

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Nuclear charge radius of 11Li