Interaction cross section study of the two-neutron halo nucleus $^{22}$C

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Interaction cross section study of the two-neutron halo

nucleus

22

_C

Y. Togano, T. Nakamura, Y. Kondo, J.A. Tostevin, A.T. Saito, J. Gibelin,

N.A. Orr, N.L. Achouri, T. Aumann, H. Baba, et al.

To cite this version:

Y. Togano, T. Nakamura, Y. Kondo, J.A. Tostevin, A.T. Saito, et al.. Interaction cross section

study of the two-neutron halo nucleus

22

_{C. Physics Letters B, Elsevier, 2016, 761, pp.412-418.}

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Interaction

cross

section

study

of

the

two-neutron

halo

nucleus

22

C

Y. Togano

a

,

∗

,

T. Nakamura

a

,

Y. Kondo

a

,

J.A. Tostevin

b

,

A.T. Saito

a

,

J. Gibelin

c

,

N.A. Orr

c

,

N.L. Achouri

c

,

T. Aumann

d

,

H. Baba

e

,

F. Delaunay

c

,

P. Doornenbal

e

,

N. Fukuda

e

,

J.W. Hwang

f

,

N. Inabe

e

,

T. Isobe

e

,

D. Kameda

e

,

D. Kanno

a

,

S. Kim

f

,

N. Kobayashi

a

,

T. Kobayashi

g

,

T. Kubo

e

,

S. Leblond

c

,

J. Lee

e

,

F.M. Marqués

c

,

R. Minakata

a

,

T. Motobayashi

e

,

D. Murai

h

,

T. Murakami

i

,

K. Muto

g

,

T. Nakashima

a

,

N. Nakatsuka

i

,

A. Navin

j

,

S. Nishi

a

,

S. Ogoshi

a

,

H. Otsu

e

,

H. Sato

e

,

Y. Satou

f

,

Y. Shimizu

e

,

H. Suzuki

e

,

K. Takahashi

g

,

H. Takeda

e

,

S. Takeuchi

e

,

R. Tanaka

a

,

A.G. Tuff

k

,

M. Vandebrouck

l

,

K. Yoneda

e

a_{DepartmentofPhysics,TokyoInstituteofTechnology,Tokyo152-8551,Japan}

b_{DepartmentofPhysics,FacultyofEngineeringandPhysicalSciences,UniversityofSurrey,GU27XH,UnitedKingdom} c_{LPCCaen,ENSICAEN,UniversitédeCaen,CNRS/IN2P3,14050CaenCedex,France}

d_{InstitutfürKernphysik,TechnischeUniversitätDarmstadt,D-64289Darmstadt,Germany} e_{RIKENNishinaCenter,Saitama351-0198,Japan}

f_{DepartmentofPhysicsandAstronomy,SeoulNationalUniversity,599Gwanak,Seoul151-742,RepublicofKorea} g_{DepartmentofPhysics,TohokuUniversity,Miyagi980-8578,Japan}

h_{DepartmentofPhysics,RikkyoUniversity,Tokyo171-8501,Japan} i_{DepartmentofPhysics,KyotoUniversity,Kyoto606-8502,Japan} j_{GANIL,CEA/DSM-CNRS/IN2P3,F-14076CaenCedex5,France}

k_{DepartmentofPhysics,UniversityofYork,Heslington,YorkYO105DD,UnitedKingdom} l_{IPNOrsay,UniversitéParisSud,IN2P3-CNRS,F-91406OrsayCedex,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received29December2015

Receivedinrevisedform4August2016 Accepted30August2016

Availableonline1September2016 Editor:V.Metag

Keywords:

Interactioncrosssection Matterradius

Two-neutronhalonucleus22_C

Theinteractioncrosssections(

σ

I)oftheveryneutron-richcarbonisotopes19C,20Cand22Chavebeen measuredonacarbontargetat307, 280,and235 MeV/nucleon, respectively.A

σ

I of1.280±0.023 b was obtained for22_{C,significantlylarger thanfor} 19,20_{C, supporting the halo characterof} 22_{C. A}22_C root-mean-squared matter radiusof 3.44±0.08 fm was deduced using afour-body Glauberreaction model.Thisvalueissmallerthananearlierestimate(of5.4±0.9 fm)derivedfroma

σ

I measurementon ahydrogentargetat40 MeV/nucleon.Thesenew,higher-precision

σ

I data providestrongerconstraints forassessingtheconsistencyoftheoriesdescribingweaklyboundnuclei.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

The nuclear halo, a valence neutron distribution that extends far beyond a well-bound core, is a characteristic feature of very weaklyboundnucleifarfromstability[1].Borromeantwo-neutron halosystems,suchas11Li, areofparticular interestforthestudy of many-body effects, such as possible strong neutron–neutron (nn) correlations [2–11]. In the case of 11_{Li, the analysis of}

in-teraction cross section data using simplified reaction and struc-turemodels was sufficient to identifya large difference between

*

Correspondingauthor.

E-mailaddress:togano@phys.titech.ac.jp(Y. Togano).

the root-mean-squared(rms) radii,

˜

rm, of11Li (3

.

34+₋00..0408 fm)and 9_Li(2

_.

₃₂

_±

₀

_.

_{02 fm)[12–14].Itisknownhoweverthat dynamical}

modelsthatincludebreakupdegreesoffreedomarenecessaryfor accuratequantitativeanalyses,andthattheseincreasethededuced

˜

rmvaluesforhalosystems[15,16].

Recently,themostneutron-richboundcarbonisotope22C (two-neutron separation energy S2n

= −

0

.

14

±

0

.

46 MeV [17]) has

drawn considerable attention, owing to a possibly very-extended two-neutronhalostructure–assuggestedbyalargereactioncross section (

σ

R

=

1

.

338

±

0

.

274 b) measured on a proton target at

40 MeV/nucleon [18].An associatedr

˜

m of5

.

4

±

0

.

9 fm, with

rel-atively large uncertainty, was deduced. Thisvalue issignificantly

http://dx.doi.org/10.1016/j.physletb.2016.08.062

largerthanthat forwell-knownhalonucleisuch as11_Li.The

nu-cleus22CisalsosignificantintermsofmagicityatN

=

16,aswas recently established for 24O [19–21]. Clarifying whether N

=

16 magicitypersistsin22_{Cwillshedlightontheshellevolutionalong}

theneutron drip-line. Importantly, ifthe N

=

16 shell closurein

22_{Cisconfirmed,thetwo-neutronvalenceconfigurationwouldbe}

[

2s1/2

]

2,optimalforhaloformation.Thestructureof22Cmayalso

providesome cluesastowhythedriplineendsatN

=

16 inthe carbon,nitrogen,andoxygenisotopes.Experimentally,evidencefor

N

=

16 magicityhasbeenfoundinthetwo-neutronremovalcross sectionfrom 22_{C andtheresulting}20_{C fragmentmomentum}

dis-tribution[22].

Thelarge uncertaintieson theearliermeasurement of

σ

R and

thededucedr

˜

m [18] donot significantlyconstrain thetheoretical

models.Amean-fieldapproach,usinganadjustedSkyrme interac-tion,predictedr

˜

m of3.89 fm[23].Three-bodymodelcalculations

for 22_{C, with dominant}

_[

_2s

1/2

]

2 valence neutron configurations

[24,25],deriver

˜

m inthe range3.50–3.70 fm.Giventhe increased

mass of 22C, and the resulting smaller fractional contribution of thetwo valencenucleonstor

˜

m,allthree-bodymodelcalculations

[24–27] require extremely weak two-valence-neutron binding to produceanenhancedr

˜

m.Giventhesignificanterroronthe

exper-imental estimate ofRef. [18] and that the theoretical values are within

∼

2

σ

ofthisvalue,moredefinitiveconclusionsrequiredata ofsignificantlyhigherprecision.

In this spirit, the present Letter reports a high precision 22_C

interactioncross section (

σ

I) measurement ona carbon target at

235 MeV/nucleon. At thisenergy, unlike the earlier 40 MeV/nu-cleonprotontargetmeasurement,theassumedforwardscattering dominance of the coreand valence particles, that underpins the Glauber (eikonal)modeldescription,is well satisfied.Inaddition, it has been shown that the optical limit (OL) approximation to Glauber theory provides an excellent description of the (reason-ablywell-bound)core-targetsystemswithouttheneedtoconsider additional(Fermi-motion[28] andPauliblocking[29])corrections –a resultofthehighlyabsorptivenatureofthecore-target inter-actionsinthecaseofalightnuclear(carbon)target.

Thereremains,however,a smalluncertaintydueto the differ-encebetween

σ

I, derived from the measurement, and the

reac-tioncrosssection

σ

R thatistypicallycalculatedusingtheGlauber

modelanalysis.As measured,

σ

I isthecrosssection forachange

ofnuclideinthecollision[30]while

σ

R

=

σ

I

+

σ

inel alsoincludes

anyinelasticcrosssectiontoboundexcitedstatesoftheprojectile andtarget.Since

σ

inel issmallatthepresentbeamenergy,aswill

be shownlater,

σ

I isa very good approximation to

σ

R withthe

presentexperimental(energyandtarget)conditions.

Here,the 19,20,22C projectile systems willbe treatedin a con-sistentframework where:(i) therelativelywell-bound(non-halo) corenuclei,18,20_{C,aredescribedusingHartree–Fock(HF)densities}

inwhichtheleast-boundorbitalsareconstrainedtotheempirical neutronseparation energies, and (ii) the structure anddynamics ofthe weakly-bound valenceneutron degrees of freedom in the halo-systems 19,22_{C are treated explicitly using few-body}

meth-ods.So,three- andfour-bodyGlaubermodeldescriptionsareused forthe19_Cand22_{C-targetsystems,respectively. The}

_˜

_r

m of22C is

thusderived from

σ

R bycomparisonswithfour-body Glauber

re-actionmodelcalculations(seee.g.[15,16]) thatincorporate three-body model 22C ground-state wave functions. Comparisons with themeasured

σ

R for20C(the 22Ccore)andtheone-neutronhalo

nucleus19Careusedtoassesstheconsistencyofthisapproach. TheexperimentwasperformedattheRadioactiveIsotopeBeam Factory(RIBF)acceleratorcomplexoperatedbytheRIKENNishina CenterandCenter forNuclear Study,University ofTokyo.A cock-tail beam of 19,20,22C was produced via projectile fragmentation ofa345 MeV/nucleon48_{Ca beamfromtheSuperconductingRing}

Fig. 1. (Coloronline.) Particle-identificationplotsofa)thesecondarybeamandb) outgoingreactionproductsfrom22_{C.Thecolorscaleislogarithmic.Theredsquares}

ina)show theconditionusedfortheanalyses.

Cyclotronincidentona20 mm thickBe target.The19,20,22_C

frag-ments were separatedusing the BigRIPS [31] withan aluminum degrader witha medium thicknessof15 mm placed at thefirst dispersive focal plane. The momentum acceptance was set to be

±

3%.Thecocktailbeamimpingedonacarbontargetwitha thick-nessof1

.

789 g

/

cm2_.Thetypical19,20,22_{Cintensitieswere6.0,290,}

and 6.6 Hz, respectively, which corresponded to approximately 0.36%,18%, and0.41% ofthe totalsecondary beamintensity.The meanenergiesofthe19,20,22Cbeamsatthetargetmid-pointwere 307,280,235 MeV/nucleon,respectively.

The secondary beamparticles were monitored event-by-event using two multi-wire drift chambers (BDC [32]) placed just up-stream of the carbon target, in order to ascertain the incident angle and position upon the target. The incident particles were identifiedbymeasuring thetime-of-flight(TOF),magneticrigidity (B

ρ

),andenergyloss(



E)withplasticscintillators, amulti-wire proportional counter,andan ionization chamber upstreamof the carbontarget.Fig. 1a)showstheatomicnumber( Z )andthemass to charge ratio ( A

/

Z ) reconstructed from the measured B

ρ

,TOF and



E.Asmallfractionofeventswere observedthatareshifted slightlytoasmaller A

/

Z for 20_{C.Theseeventswerefound to}

fol-low a different trajectory from the main one, and were mostly removed. Nevertheless, we still had such unwanted beam parti-cles ofabout0.03%of therespective isotope.Hence, by selecting the isotopes as shown in Fig. 1a), such events are not used in theanalysis. Contamination from Z

=

7 particlesin thegated re-gion of 19,20C is found to be less than 6

×

10−6 ofthe selected events, which contributes negligibly (

<

1 mb) to the current

σ

I

results.Thetargetwas surroundedbytheDALI2array [33]to de-tectde-excitation

γ

rays fromexcitedoutgoing fragments, which wereusedtoestimate

σ

inelasshownlater.

Table 1

Beamenergies(atmid-target)andmeasuredinteractioncrosssections(σI)forthe carbonisotopesstudiedhere.ThetwoerrorsshownonσIarethestatistical(first) andsystematic(second)uncertainties.

A E/A [MeV/nucleon] σI[barn]

19 307 1.125±0.025±0.013

20 280 1.111±0.008±0.009

22 235 1.280±0.022±0.007

measured using FDC1 was also employed for the identification. Fig. 1b)showstheparticleidentificationfortheproductsfromthe

22_C

₊

_Creaction.

The interaction crosssections

σ

I were derived usingthe

con-ventionaltransmissionmethod[30],where

σ

I canbewrittenas,

σ

I

= −

1 Nt log







0



.

(1)

Here, Nt isthenumber oftargetnucleiper unit area,while



is

theratioofthenumberofnon-interactingoutgoingparticlestothe numberofincomingparticles.



0 correspondsto



foran empty

target, to take into account reactions in the detectors [34]. The valueof



0 wasdeterminedtobe

∼

0

.

98,while



was 0.87–0.89.

The angular acceptancefor the ejectiles was

±

2

.

9 degree inthe vertical direction and

±

9

.

8 degree in the horizontal direction, which was sufficient to accept the non-reactedbeams, giventhe beamspread of0.5 degrees(FWHM) andthe multiple scattering of0.15degrees(1

σ

)inthecarbontarget.Thepositiononthe tar-getandincidentanglesofthe19,20,22_{Cbeamswereusedtorestrict}

theemittancesoastoguaranteethefullacceptanceofthe SAMU-RAIsetup.

In the following discussion

σ

I and

σ

R are compared directly

anditis assumedthatthe totalinelastic scatteringcrosssection,

σ

inel, to the bound excited states in the projectile and target is

negligiblecomparedto

σ

I.Thisassumptionwillbequantifiedand

validatedforthe280 MeV/nucleon20C

+

Cdata.

Table 1presents themeasured

σ

I for19,20,22C withthe

statis-ticalandsystematicuncertainties. The systematicerror ismainly governedby the variationofthe detectionefficiencyofFDC2 be-tweentheC targetandtheempty targetruns, sincethehit posi-tionoftheparticlesonFDC2wasdifferentowingtothedifferent

B

ρ

oftheparticlesaftertheenergyloss inthetarget.As isclear fromTable 1,the

σ

I valuefor22Cissignificantlyenhancedrelative

to those of 19,20C, consistent with a two-neutron halo character of22_C.

The assumption that

σ

R



σ

I is examined using the

γ

rays

emitted in the 20C

+

C

→

20C

+

X reaction. The

γ

-ray en-ergyspectra,measuredbyDALI2incoincidencewithoutgoing20C ions,are shown inFig. 2. The upperandlower panelsshow the spectra in the laboratory and projectile frames, respectively. The red solid and black dashed curves show fits to the spectra and the components of the fit, respectively. In the laboratory frame, a peakat4.4 MeV(withaccompanying single- anddouble-escape peaks) isobservedfrom thede-excitationofthe 12_C(2+

1) state of

the target [35]. The peaks are fitted using Gaussian lineshapes and an exponential background and the cross section is, subse-quently,deduced.Thefullenergypeakefficiencyisestimatedusing a Monte-Carlosimulation basedon theGEANT4. The 12_{C inelastic}

cross section is estimatedto be

∼

4 mb. Inthe projectile frame, Fig. 2b), apeak at1.6 MeV isclearly seenfromthe de-excitation oftheknown20C(2+₁) state[36,37].Fittingthepeak witha Gaus-sian lineshape and an exponential background, the 20_{C inelastic}

crosssection isestimated tobe

∼

3 mb. Importantly,the sumof thetargetandprojectileinelasticcrosssectionsissmallerthanthe uncertainty on the present 20_{C interaction cross section. Such a}

Fig. 2. (Coloronline.) Theγ-rayenergyspectrumin:a)thetargetframe,andb)the projectileframe,coincidentwithincoming20_Candoutgoing20_{Cions.Inthetarget}

frame,the4.4 MeVpeakoriginatesfromthe12_C2+

1 state(seetext).Theprojectile

framepeakat1.6MeVoriginatesfromthe20_C2+ 1 state.

small

σ

inel isconsistent withthose deduced forthe neutron-rich

Mgisotopes ataround250 MeV/nucleon, obtainedusinga differ-ent analysis method[38].The

σ

inel for19,22C are expected to be

smaller than for 20_{C since} 19,22_{C are much more weakly bound.}

Having demonstrated the validity of comparing

σ

R with

σ

I, we

nowturntotheinterpretationofthepresentresults.

Todescribethereactionsoftheone- andtwo-neutronhalo nu-clei 19C and22Cthree- and four-body Glauber dynamical models [15,16,25] are employed. For the relatively well-bound, non-halo projectileand22_Ccorenucleus20_C,the19_Ccorenucleus18_C,and 12_{C(usedasatest)wedescribethecollisionsusingthetwo-body}

optical limit (OL) Glauber model, based on the projectile/core-target S-matrices.TheOLapproachtakesonlytheleading termin theion–ionmultiplescatteringexpansionandneglectscorrelations amongthenucleons,otherthanthespatial correlationscontained inthe ground-statedensities.Thus,inall ofthesubsequent reac-tioncalculations,then- andcore-targetinteractionsattherelevant beam energy are calculated consistently using the single-folding (tN N

ρ

t,n-target) anddouble-folding (tN N

ρ

c

ρ

t, core-target)

mod-els,usingan effectivenucleon–nucleon(N N)interactiontN N.The

core (c) densities are taken from spherical Skyrme (SkX interac-tion [41]) Hartree–Fock(HF)calculations. The density ofthe car-bontargetnucleiwasassumedtobeGaussianwithanrmsradius of 2.32fm,consistent withtheempirical rms charge radius[14]. Regarding the effective N N interaction, tN N was assumed to be

zero-range. Its (complex)strength, ateach beamenergy, was de-termined from: (i) the free N N cross sections,and (ii) the ratio ofthereal-to-imaginarypartsoftheN N forwardscattering ampli-tude interpolatedfromthetabulated valuesofRay[42].Wenote that these folded n- and core-target interactions provide an ex-cellent description of the recent n-removal data from the same neutron-richcarbonisotopesat250 MeV/nucleon[22].

Apartialassessmentofthequalityofthiscalculationalscheme can be made using the previously measured energy dependence of the reaction cross sections for 12_C

₊

12_{C [28,39,40]. The}

cur-rent OLGlauber calculations(blue curve) are compared withthe available data in Fig. 3a) and show reasonable agreement above 200 MeV/nucleon.

Fig. 3. (Coloronline.) Energydependenceofthereactioncrosssectionsfor(a)12_C,

(b)19_C,and(c)20_Cona12_{Ctarget.Theopendiamonds,squaresandtrianglesin(a)}

showthe12_C+12_{CdatafrompreviousstudiesofRefs.[28,39,40],respectively.The}

filledredcirclesshowthepresentresults.Theerrorsshownarethequadraticsum ofthestatisticalandsystematicuncertainties(Table 1).Thebluecurvesinpanels(b) and(c)arecalculatedusingthree-bodyandtwo-bodyGlaubermodels,respectively. Detailsaregiveninthetext.

(

∼

900 MeV/nucleon) data [39]. The curve in Fig. 3b) shows the calculatedenergydependence of

σ

R forthe 19C

+

C system,

us-ing the three-body Glauber model of Refs. [15,16].The model is consistent withthe data points. We note that this calculation is essentiallyparameter free. The19Csystem istreatedasa bound, 2s1/2,18C

+

n halowiththeempirical neutronseparationenergy Sn

(

19C

)

=

0

.

58

(

9

)

MeV[43].InconstructingtheHFdensityofthe

18_{Ccore, the1d}

5/2 orbitalisconstrainedtotheknown Sn

(

18C

)

=

4

.

180 MeV,givingr

˜

m

(

18C

)

=

2

.

75 fm.So,combiningthermsradii

ofthes-waveneutronwavefunctionandthe18_Ccore[44],we

ob-tainr

˜

m

(

19C

)

=

3

.

10₋+₀0._.05₀₃ fm,wheretheuncertaintyoriginatesfrom

theerroron Sn

(

19C

)

.

For 20C, the present OL calculation results are presented in Fig. 3c). Both shell-model calculations of the



19C

|

20C



ground-statetoground-stateoverlapandthesignificant19C(1

/

2+) ground-stateyieldmeasuredintheneutronremovalreactionfrom20C[22] indicate a significant occupancy of the 2s1/2 orbital in the 20C

ground-state. The calculated neutron 2s1/2 orbital occupancy of

the20Cground-state(WBP)shell-modelwave function,asusedin Ref. [22], is1.18. In computingthe 20C density forthe OL calcu-lationswethusconstraintheHFcalculationsassumingarangeof neutron 2s1/2 orbital occupancies to assessthe sensitivityof the

model to admixtures of this configuration. The 1d5/2 and 2s1/2

orbitalsare assumedbound by Sn

(

20C

)

=

2

.

93 MeV [43]. Fig. 3c)

shows results with a 2s1/2 orbital occupancy of 0 (dotted), 1.2

(dashed)and1.7(solid). The resultsfor2s1/2 orbital occupancies

between 1.2 and 1.7 are consistent with the present, high pre-cision data point, whereas the earlier, higher-energy data point favours valuesat the lower endof thisrange. The

˜

rm

(

20C

)

value

of the HF density distribution with neutron 2s1/2 orbital

occu-pancyof1.2, thebestfittothebothpresentandthehigh-energy data, is 2

.

95

±

0

.

02 fm. The best fit to the present data using theHF,OLmodel,thesolid curve,isr

˜

m

(

20C

)

=

2

.

97+₋00..0305 fm,

hav-ing a 2s1/2 orbital occupancy of 1

.

7₋+0₀._.3₈. The associated errors

for

˜

rm and the 2s1/2 orbital occupancy arise from the

uncer-taintyinthepresentreactioncrosssectionandtheknownSn

(

20C

)

of 2

.

93

±

0

.

26 MeV. This result, 2

.

97₋+0₀._.03₀₅fm, is adopted as the present result ofr

˜

m

(

20C

)

. The reason why the calculated energy

dependenceofthesolid curve,whichbestreproducesthecurrent data point, overestimates the earlier higher energydata point is currentlyunclear.However,wenoteasimilartrendinthe12_C

₊

12_C

systemanddatafortheboronisotopes[45]andthesourceofthis differenceshouldbeinvestigatedfurther.Wealsonotethatthe

˜

rm

value extracted from Ref. [39], within an alternative framework, was(2

.

98

±

0

.

05 fm),inexcellent agreementwiththecurrent re-sult.Thisisindicativethatthereactioncrosssectionmeasurement, inisolation,hasinsufficientsensitivitytoaccuratelydeterminethis occupancy, howeverthededucedr

˜

m value appearsto berobustly

determined.

Forthe22_C

₊

_{Csystem,weuseafour-body(three-body}

projec-tileplustarget)Glauberreactionmodelanalysis[15,16],previously applied to 22C. Full details of the physical inputs and the con-structionofthethree-bodymodel22_{Cground-statewavefunctions}

can be found in Ref. [25]. In these calculations [25], the addi-tional pair of valence neutrons in 22_{C are found to occupy the}

2s1/2 orbital,withminimal1d3/2 orbitaloccupancy.So,unlikethe

discussion above of the2s1/2 occupancy ofthe free 20C nucleus,

in 22_{C the valencenucleons essentially block accessto the 2s} 1/2

orbital of nucleons from the 20_{C core; consistent with the core}

having a filled 1d5/2 sub-shell. With the 1d5/2 orbital bound by Sn

(

20C

)

=

2

.

93 MeV the HF 20C corerms radius (in 22C) isthen

2.89fm.Thisvalueisusedwhencomputingr

˜

m

(

22C

)

.

Solutionsofthe22Cthree-body wavefunctionsusetheGogny, PiresandDeTourreil (GPT)nn interaction.Then–20_{Cinteractions}

are describedby Woods–Saxonpotentialswithaspin–orbitterm. The parameters ofthese potentialsinthe 2s1/2 and 1d5/2,3/2

or-bitalscanbe foundinRef. [25].Toprovideafamilyof22_C

three-body wave functions with different ground-state eigenstates (i.e. binding energies E3B

(

= −

S2n

)

) and hence different sizes (rms

hyperradii) we use a family of three-body Hamiltonians. These Hamiltoniansdiffer inthechoice of(i) the(unbound) 2s1/2 state

potential depth (and their s-wave n–20_{C scattering length) and}

(ii) the strength V3B of the added attractive central hyperradial

three-bodyforce,V3B

(



)

= −

V3B

/(

1

+ [



/

5

]

3

)

,where



isthe

hy-perradius. Having fixed the Hamiltonian by choosing the above interaction strengths, the three-body wave function, its

˜

rm, and

S2n

(

22C

)

aredetermined fromthelowesteigenstate ofthe

eigen-value problem. The presentresults differ fromthose of Ref. [25] due to: (a) the incident beam energy, and (b) the use of the

Sn

(

20C

)

-constrainedHFcoredensitywithr

˜

m

(

20C

(

core

))

=

2

.

89 fm,

aswasdiscussedabove.

Fig. 4shows,asfilledsquares,thedependenceofthefour-body Glaubercalculationsof

σ

R onthe

˜

rm

(

22C

)

ofthecomputed

three-body ground-state wave functions. The points k1–k5 correspond

to the wave functionsof the same name tabulated in Table I of Ref. [25]. The remaining symbols extend this wave function set for morebound 22_{C systems,with smaller}

_˜

_r

m

(

22C

)

, from3.50 to

3.34 fm, by use of a more attractive three-body term, with V3B

values of1.5, 1.8, 2.0, 2.5, 3.0 and3.5 MeV. The symbolslie ap-proximately on a straight line, showing that the calculated

σ

R’s

arestronglysensitiveto

˜

rm(and/orS2n)butnottothefinerdetails

ofthemodelinteractions.The S2n valuesofthesewavefunctions

range from0.893–0.046 MeV as one movesfrom left to right in Fig. 4.Allofthewavefunctionshavedominant(92–94%)

[

2s1/2

]

2

andsmaller(2–4%)

[

1d3/2

]

2 two-neutronconfigurations.Thesolid

Fig. 4. DependenceofthecalculatedσRonthermsmatterradius˜rm(22C)withinthe four-bodyGlaubermodel.Thefilledsquaresarecalculatedresultsfromthefamily of22_{Cthree-bodywavefunctions.Thesolidcurveisasecondorderpolynomialfit}

tothecalculationsandthedashedlineandtheshadedregioncorrespondtothe measuredvalueand1σ erroronσRdeducedfromthepresentexperiment.

Fig. 5. Massnumberdependenceoftheroot-mean-squaredradiiofcarbonisotopes obtainedinthepresentstudy(filledcircles).Thedashedlineconnectsthe theoret-icalpredictionsofRef.[46].

experiment,respectively.Theerrorshownisthequadraticsumof the statistical and systematicuncertainties. From the model cal-culationsofthefigurewecanestimate

˜

rm

(

22C

)

and S2n

(

22C

)

from

thepresentdataof3

.

44

±

0

.

08 fm and0

.

56+₋0₀._.27₂₀MeV,respectively. The S2n

(

22C

)

valueisconsistentwiththeupperlimitof0.32 MeV

fromthedirectmassmeasurement[17].

Consistent with our earlier discussion, this deduced r

˜

m

(

22C

)

from the present few-body-model analysis is smaller, by about 2

σ

, than the previously reported value (5

.

4

±

0

.

9 fm [18]) with itslargeuncertainty.Aswasnotedearlier,thepresenthigher pre-cision and more absorptive and surface dominated higher beam energymeasurementon anuclear (carbon)target,places the un-derpinningGlauber (eikonal)dynamical model description ofthe reactionusedhereonamuchstrongerfootingregardingthe mag-nitudesofmultiple-scatteringandothercorrections[28,29].

Fig. 5showsthe A-dependenceof

˜

rm fortheneutron-rich

car-bonisotopesfromthepresentstudy,wheretheadoptedvaluesare shown.The one-neutronhalonature of19Cleads toan enhanced

˜

rm value relative to 20C. The

˜

rm

(

22C

)

isabout0.4 fm largerthan

that for20_{C,reflectingthetwo-neutronhalocharacterof}22_C.The

dashed lineshowsthe resultsofan alternativemean-field model for20_{Candfew-bodymodelsfor}19,22_{C[46].Ourdeduced}

_˜

_r

m

(

22C

)

isalsoinagreementwithvalues:(i)rangingfrom3.6–3.75 fmfor

S2n

(

22C

)

values between 400–600 keV, of the three-body model

calculation of Ref. [24], and (ii) r

˜

m

(

22C

)

=

3

.

4 and 3.6 fm, with

S2n

(

22C

)

of400and200 keV,ofRef. [26].Thesemodels, likethe

presentone,calculatean s-waveconfigurationofthetwovalence neutrons withprobability

≥

90%, inline witha picture inwhich the N

=

16 magicitypersistsat22_{C–assuggestedby othermore}

fundamental approaches [47,48]. However, as evidenced by our analysisfor20_{C,thereaction crosssection aloneissomewhat}

in-sensitivetodetails ofthemicroscopicstructure andmoreprecise, moreexclusive reactionexperimentsandmassmeasurements are required.

In summary,we have measured the interactioncross sections of 19,20,22_{C on a C target at 307, 280 and235 MeV/nucleon,}

re-spectively.Inelasticscatteringof20Ctoboundexcitedstatesofthe C target were quantified to verify the assumption that

σ

R



σ

I.

The 22C reactioncrosssection was found tobe muchlarger than that of 20_{C, supporting the two neutron halo nature of} 22_{C. In}

the case of 19C, the present and previously measured

σ

R (at

around 950 MeV/nucleon) are consistentwith thethree-body re-actionmodelcalculations,andthermsradiusof19Cobtainedwas 3

.

10+₋0_0..₀₃05fm. For20_{C,occupancyoftheneutron2s}

1/2 orbitalwas

neededtoreproducethe presentdata,andthermsmatter radius wasestimatedtobe2

.

97₋+0₀._.03₀₅ fm.Thededuced22Crmsmatter ra-dius, 3

.

44

±

0

.

08 fm, fromthe four-body Glauber reaction model analysis, isconsistent withothertheoretical predictionsbased on

22_{Cthree-bodymodelwavefunctions[24,26].Thepresentrms}

ra-dius issmaller andhasa muchreduceduncertainty thanthat of Ref.[18].Moreexclusivemeasurements, suchasCoulomb dissoci-ation and fastneutron-knockout reactions, will allow for a more detailedstudyofthestructureof22_{C,includingthepotentialrole}

ofcoreexcitationsontheN

=

16 magicity[49,50]. Acknowledgments

The authors thank the staff of RIKEN RIBF for their efforts in providing the 48_{Ca beam. We are grateful to Dr. M. Takechi}

for discussions and providing the experimental data ofRef. [28]. The presentwork was supported in partby JSPS KAKENHIGrant No. 24740154, MEXT KAKENHI Grant No. 24105005, the WCU (R32-2008-000-10155-0) and the GPF (NRF-2011-0006492) pro-grams of NRF Korea, and the HIC for FAIR. J.A.T. acknowledges support of the Science and Technology Facility Council (UK) grants ST/J000051 and ST/L005743. N.L.A., F.D., J.G., F.M.M. and N.A.O.acknowledgepartialsupportfromtheFranco–Japanese LIA-InternationalAssociatedLaboratoryforNuclearStructureProblems. A.N. and J.G. wouldalso like to acknowledge the JSPS Invitation Fellowship program forlong-termresearch in Japan attheTokyo InstituteofTechnologyandRIKENrespectively.

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