$2\alpha+t$ cluster structure in $^{11}$B

The cluster structures of the excited states in $^{11}$B are studied by analyzing the isoscalar monopole and quadrupole strengths in the $^{11}$B($d$,$d'$) reaction at $E_d=200$ MeV. The excitation strengths are compared with the predictions by the shell-model and antisymmetrized molecular-dynamics (AMD) calculations. It is found that the large monopole strength for the $3/2^-_3$ state at $E_x=8.56$ MeV is well described by the AMD calculation and is an evidence for a developed $2\alpha+t$ cluster structure.

m onopol e strength for the 3=2 3 state at E x = 8: 56 M eV i s wel ldescri bed by the A M D cal cul ati on and i s suggested to be an evi dence for a wel ldevel oped 2 + t cl uster structure.

. + n
A l pha parti cl e cl usteri ng i s an i m portant concept i n nucl ear physi cs for l i ght nucl ei .O n the basi softhe Ikeda di agram [ 1] ,the cl usterstructure i sexpected to em erge near the -decay threshol d energy.It has been suggested that the 7. 65-M eV 0 + 2 state i n 12 C ,w hi ch l ocates at an exci tati on energy hi gher than the3 -decay threshol d by 0. 39 M eV ,hasa 3 -cl ustercon gurati on [ 2,3,4,5,6] .
T hi s 0 + 2 state i s theoreti cal l y descri bed by i ntroduci ng a novelconcept ofthe nucl ear structure, i.e.,thi s state has a di l ute-gas-l i ke structure w here three parti cl es are weakl y i nteracti ng and are condensed i nto the l owest s-orbi t [ 7,8,9,10,11] .It has been found that the wave functi on ofthe 0 + 2 state cal cul ated by theprevi ous3 cl usterm odel s [ 4,5,6]i sal m ostequi val entto thewave functi on ofthe 3 condensed state [ 8] .Si m i l ar di l ute-gas states of cl usters have been predi cted i n sel f-conjugate N = 4n nucl ei [ 12] .T he next natural questi on addressed i s w hether such a di l ute cl uster state exi sts i n the other R ecentl y, an exoti c character of the 3=2  For cl ari cati on ofthe cl uster structure,further i nform ati on on the naturalpari ty exci tati on strengths i s necessary.Especi al l y,the i soscal ar parts ofthe natural -pari ty transi ti on strengths are a key i ngredi ent because m ost ofthe cl uster states are exci ted by the i soscal ar natural -pari ty transi ti ons.
T he natural -pari ty transi ti on strengths have been extensi vel y exam i ned by m eans of -decay and (e;e 0 ) m easurem ents.H owever, such el ectrom agneti c probesare sensi ti ve onl y to the transi ti onsrel evantto protons.T herefore,the el ectri c transi ti on strength carri ed by protons i s general l y di erent from the i soscal ar strength carri ed by both protons and neutrons.
T he el ectri c [ B (E )] and i soscal ar [ B (E ;IS)] transi ti on strengths w i th a m ul ti pol ari ty of are descri bed by w here J i i s the spi n ofthe i ni ti alstate.M p (E ) and M n (E ) are the proton and neutron transi ti on m atri x el em ents.M p (E ) and M n (E ) are gi ven by w here z i s the i sospi n Paul i m atri x and Ô (E ) i s the transi ti on operator.
Ô (E )= r 2 and r 2 Y 2 are used for the m onopol e and quadrupol e transi ti ons, respecti vel y.
For l i ght sel f-conjugate nucl ei ,the i soscal ar strengths are deduced from the el ectri c transi ti on strengthsbecause the neutron transi ti on strengths are si mi l ar to the proton transi ti on strengths due to the approxi m atel y conserved charge sym m etry.Forthe othernucl ei ,however,the neutron and proton transi ti on strengthsare di erenteach other,and the i soscal arstrengthsshoul d be determ i ned by a vari ety ofdi erent m easurem ents.
O ne possi bl e m ethod to obtai n the i soscal artransi ti on strengthsi sto m easure theel ectri ctransi ti on strengthsform i rrornucl ei .In caseoftheA = 11 system , the el ectri c quadrupol e strengths have been m easured for l ow -l yi ng states i n the stabl e 11 B nucl eus, but no el ectri c quadrupol e strength i s reported for the exci ted states i n unstabl e 11 C .For the el ectri c m onopol e strengths, no experi m entalval ue i s know n for both the states i n 11 B and 11 C .
A nother possi bl e m ethod to obtai n the i soscal ar transi ti on strengths i s to m easure the hadron scatteri ng [ 17] .H adron scatteri ng at forward angl es and at i nterm edi ate energi es i s a good probe to obtai n such exci tati on strengths thanks to a good proporti onal i ty between the cross secti ons and the rel evant exci tati on strengths.Si nce both the i soscal ar and i sovector transi ti ons co-herentl y contri bute to the exci tati on strengths i n N 6 = Z nucl eiw i th non-zero ground-statei sospi n,i soscal arprobesl i kedeuteron or 4 H eareusefulto extract the i soscal ar exci tati on strengths.
In thepresentwork,thei soscal arm onopol eand quadrupol eexci tati on strengths In the tti ng procedure,the m ul ti pol e contri buti ons w i th J 3 were negl ected.
From the m ul ti pol e decom posi ti on anal ysi s (M D A ), the i soscal ar m onopol e and quadrupol estrengthswereextracted.Tabl e1 l i ststheobtai ned B (E 0;IS) and B (E 2;IS) val ues together w i th the B (E 2) val ues taken from R ef. [ 20] .
System ati cuncertai nti eson B (E 0;IS)and B (E 2;IS)arem ai nl y dueto errors i n the m odelcal cul ati on for the (d;d 0 ) reacti on.T he detai l ed expl anati on for M D A has been gi ven i n R ef. [ 13] .
A l though the 3=2 states are al l owed to be exci ted by the J = 0 + , 1 + , 2 + ,and 3 + transi ti ons,the J = 0 + contri buti on i n exci ti ng the 8. 56-M eV 3=2 3 state i s found to be extraordi nary l arge.Si nce the observed J = 0 + strength i sm uch strongerthan the expected J = 2 + strength,i ti sdi cul t to rel i abl y extractthe J = 2 + transi ti on strength forthe3=2 3 state.Forthe 5. 02-M eV 3=2 2 state,the m onopol e strength i snotrel i abl y extracted because the J = 0 + strength i sm uch weaker than the otherm ul ti pol e com ponents.
T he SM cal cul ati on was perform ed w i th the SFO (Suzuki -Fuji m oto-O tsuka) i nteracti ons [ 21]w i thi n the 0{2~!con gurati on space.T he l evelschem es for the negati ve-pari ty states are com pared w i th the experi m ent i n Fi g.A l though theSM cal cul ati on reasonabl y wel lexpl ai nstheexperi m entalB (E 2;IS) and B (E 2) val ues for the l ow -l yi ng states,the descri pti on for the transi ti on properti es for states at E x 9 M eV i s not reasonabl e.For the 5=2 2 state at E x = 8: 92 M eV , the SM cal cul ati on gi ves extrem el y sm al l quadrupol e strengths and underesti m ates the experi m ental data al though the spi n-i p M 1 strengths are wel ldescri bed i n R ef. [ 13] .
T he SM cal cul ati on al so fai l ed to descri be the observed 3=2 state i sexpected to have a di l ute cl usterstructure w i th a 2 + tcon gurati on i n the sam e m anner as that the 0 + 2 state i n 12 C has a di l ute 3 structure.
To exam i necl usternatureofexci ted statesi n 11 B,theexperi m entalresul tsare com pared w i th the A M D cal cul ati on.T he m ethod ofthe vari ati onalcal cul ati on afterthespi n-pari ty projecti on (VA P)wasused asdescri bed i n R efs.[ 22,23,24] w here the properti es of the exci ted states i n 12 C and 10 Be were reasonabl y wel lexpl ai ned.T he M V 1 [ 25]and G 3R S [ 26,27]i nteracti onswere used forthe centraland spi n-orbi t forces i n the cal cul ati on,respecti vel y.T he adopted i nteracti on param eters ofm = 0: 62,b = h = 0: 25,and u I = u II = 2800 M eV were the sam e as those i n R ef. [ 24] .
T he cal cul ated exci tati on energi es,m onopol e and quadrupol e strengthsi n 11 B are show n i n Fi g.2 and are l i sted i n Tabl e 1.In the A M D (VA P)cal cul ati ons, the 7=2 1 and 5=2 2 statesl ocate bel ow the 3=2 2 and 3=2 3 states,respecti vel y.
T he exci tati on energi es for the 3=2 3 and 5=2 2 states are hi gher than the experi m entalval uesby about2 M eV .T he predi cted l evelstructure i ssl i ghtl y di erent from the experi m entall evelschem e of 11 B.H owever,the exci tati on strengths are reasonabl y wel l reproduced w i thout i ntroduci ng any e ecti ve charges.In addi ti on,the l arge m onopol e strength forthe 3=2 3 state,w hi ch i s not predi cted by SM cal cul ati ons,i s successful l y predi cted.T he B (E 2) val ue for the 3=2 3 state i s predi cted to be 0. 84 e 2 fm 4 ,w hi l e the reported val ue from the (e;e 0 ) m easurem ent i s as l arge as 9: 4 0: 2 e 2 fm 4 [ 28] .
H owever, thi s reported val ue i s not rel i abl e because onl y the M 1 and E 2 transi ti ons were taken i nto account and the E 0 transi ti on was negl ected i n the previ ous anal ysi s [ 28,29] .W e have anal yzed the exi sti ng (e;e 0 )data agai n by taki ng the E 0 and M 1 transi ti ons i nto account and by negl ecti ng the E 2 transi ti on accordi ng to the suggesti on from the A M D (VA P) cal cul ati on.
A s the resul t, the l arge B (E 0) val ue of 18: 7 0: 7 e 2 fm 4 has been obtai ned.
A ssum i ng a si m pl e rel ati on ofM p (E 0)= (Z=N )M n (E 0),the B (E 0;IS)val ue of90 3 fm 4 i sobtai ned from thi sB (E 0)val ue.T hel argeB (E 0)and negl i gi bl y sm al l B (E 2) val ues are qui te consi stent w i th the present experi m ental and theoreti calresul ts i n Tabl e 1.
R egardi ng the el ectrom agneti c transi ti on between the 3=2 3 state and the ground state,i ti snoteworthy to di scussthe resul ton the correl ati on m easurem ent ofel ectron-posi tron pai rs em i tted i n the i nternalpai r form ati on decay [ 30] .A l though i t i s concl uded i n R ef. [ 30]that the E 2 transi ti on dom i nates the decay ofthe 3=2 3 state,the observed correl ati on ofthe el ectron-posi tron pai rsi sal so expl ai ned by assum i ng the m i xed E 0-M 1 transi ti on.T he E 0 and M 1 strengthsobtai ned from the present(e;e 0 )anal ysi sreasonabl y accountfor the correl ati on observed i n R ef. [ 30] .
In contrast to the l arge m onopol e strength for the 3=2 3 state,the m onopol e strength forthe 3=2 2 state i ssm al l .T hi sdi erence i n the m onopol e strengths i swel lexpl ai ned by theA M D (VA P)cal cul ati on.A ccordi ng to thecal cul ati on, the 3=2 3 state has a spati al l y wel l -devel oped cl uster structure w i th a l oosel y bound 2 + tcon gurati on,w hi l e the spati aldevel opm entofthe 2 + tcl uster structure i n the 3=2 2 state i s weak.T herefore,the l arge m onopol e strength forthe 3=2 3 state obtai ned i n the present study i s regarded to be evi dence of the devel oped 2 + tcl uster structure.
To eval uatethedi l uti on ofthedensi ty di stri buti on quanti tati vel y,wei ntroduce T he3 1 statei n 12 C al so possessesthel argerm sradi us.A ccordi ng to theA M D (VA P)cal cul ati on,however,the 3 1 state hasa devel oped 3 cl usterstructure w i th a tri angl e con gurati on,and i ts structure i s di erent from the di l utegas-l i ke structures ofthe 0 + 2 state i n 12 C and the 3=2 3 state i n 11 B w here the consti tuent cl usters are freel y m ovi ng [ 31] .
A m ani festati on ofthe di l ute structure ofthe 3=2 3 state m ay appear i n the i sotopi c shi fti n the exci tati on energi es si nce the spati ali ncrease i n si ze ofthe proton di stri buti on causes the reducti on ofthe C oul om b energy,al though the i sotopi c shi ft i s caused not onl y by the spati alexpansi on ofthe proton di stri buti on but al so by the nucl ear deform ati on and si ngl e-parti cl e exci tati on.T he i sotopi c shi fts for the 3=2 3 and 5=2 2 states are m uch l arger than those for the 3=2 2 and 7=2 1 states.T he 5=2 2 state,w hi ch i s predi cted to have a com pact structure,i s consi dered to exhi bi t a l arge i sotopi c shi ft due to the spi n-i p si ngl e-parti cl e exci tati on.O n the other hand,the l arge i sotopi c shi ft forthe 3=2 3 statei si nferred to bea re ecti on ofthe di l ute structure di scussed i n the present paper.
Si nce the A M D cal cul ati on tends to underesti m ate the l ong tai lofthe wave functi on ata l argeradi us,i ti sexpected thatthecal cul ati on i snotgood enough to quanti tati vel y reproduce the i sotopi c shi fts.A ctual l y,the theoreti calval ues ofthe i sotopi c shi ft are system ati cal l y sm al l er than the experi m entalval ues by a factorofabout2.H owever,the A M D (VA P) cal cul ati on,w hi ch suggests

j
D epartm ent ofPhysics,K yoto U niversity,K yoto 606-8502,Japan k Facul ty ofEngineering,M iyazakiU niversity,M iyazaki889-2192,Japan ' N ationalSuperconducting C ycl otron Laboratory, M ichigan State U niversity,East Lansing,M ichigan 48824,U SA m R IK EN (T he Institute for Physicaland C hem icalResearch),W ako,Saitam a 351-0198,Japan n D epartm ent ofPhysics,Tokyo Institute ofTechnol ogy,M eguro,Tokyo 152-8551, Japan o D epartm ent ofPhysics,K yushu U niversity,Fukuoka 812-8581,Japan p Institute ofPhysics,U niversity ofT sukuba,Ibaraki305-8571,Japan q Research and D evel opm ent C enter for H igher Education,K yushu U niversity, Fukuoka 810-8560,Japan A bstract T he cl uster structures of the exci ted states i n 11 B are studi ed by anal yzi ng the i soscal ar m onopol e and quadrupol e strengths i n the 11 B (d, d 0 ) reacti on at E d = 200 M eV .T he exci tati on strengths are com pared w i th the predi cti ons by the shel lm odel and anti sym m etri zed m ol ecul ar-dynam i cs (A M D ) cal cul ati ons.T he l arge

i n 11 B
have been obtai ned by scruti ni zi ng the data from the 11 B(d;d 0 ) reacti on at E d = 200 M eV previ ousl y m easured at the R esearch C enter for N ucl earPhysi cs,O saka U ni versi ty[ 13] .T he exci tati on strengthshave been compared w i th the theoreti calval uesby the anti sym m etri zed m ol ecul ar-dynam i cs (A M D ) and the SM cal cul ati ons.T he form er m ethod has been dem onstrated to be useful for descri bi ng the cl uster structure of the l i ght nucl ei[ 18,19] .By anal yzi ng the i soscal ar exci tati on strengths, the cl uster structure of the exci ted states i n 11 B are probed.T he 11 B(d;d 0 ) cross secti ons were anal yzed by sum m i ng up the cross secti onscal cul ated w i th vari ousm ul ti pol e transi ti onssi nce the spi n-pari ty ofthe ground state of 11 B i s 3=2 .T he cross secti on for each m ul ti pol e transi ti on was cal cul ated i n the fram ework ofthe m acroscopi c m odeli n w hi ch the transi ti on potenti al was obtai ned i n the prescri pti on of the deform ed potenti al m odel[ 17] .Si nce the angul ar di stri buti on ofthe cross secti on for each m ulti pol e transi ti on depends on i ts transferred angul ar m om entum ,i t i s possi bl e to decom pose the cross secti on i nto each m ul ti pol e com ponent by tti ng the m easured angul ardi stri buti on asshow n i n Fi g.1.
2.T he harm oni c osci l l ator potenti al was used to cal cul ate the si ngl e-parti cl e wave functi ons.T he osci l l ator l engths were obtai ned from the D W IA anal ysi s of the ( 3 H e, t) and (p;p 0 ) reacti ons[ 13] .Si nce the quadrupol e strengths w i th the bare charges ofe p = 1 and e n = 0 were m uch sm al l er than the experi m ental val ues by a factor of 2{3, the e ecti ve charges were i ntroduced to i m prove the theoreti cal predi cti on.T he best-t resul ts of the quadrupol e strengths were obtai ned w i th the e ecti ve charges of e eff p = 1: 24 and e eff n = 0: 22 as tabul ated i n Tabl e 1.T he obtai ned e ecti ve charges are sl i ghtl y sm al l er than the standard val ues ofe eff p 1: 3 and e eff n 0: 5 for l i ght stabl e nucl ei .

Forthe 5=2 2
state,the A M D (VA P)cal cul ati on reasonabl y expl ai nsboth the quadrupol e and spi n-i p M 1 strengths al though the experi m entaluncertai nti es for the quadrupol e strengths are l arge.T he cal cul ated wave functi on for the 5=2 2 state i s dom i nated by a SM -l i ke com ponent w i th a m i xi ng of the cl uster com ponent,w hi ch i s characteri zed by cl uster correl ati on at a SU (3) l i m i t.T he cl uster com ponent provi des al m ost no spi n-i p M 1 strength between the 3=2 ground state and the 5=2 2 state butenhances the quadrupol e strength,w hi l e the SM -l i ke com ponent provi des the si gni cant M 1 strength but al m ost no quadrupol e strength.T hi s suggests that both the SM -l i ke and cl uster structures shoul d coexi sti n the 5=2 2 state to expl ai n the si zabl e spi ni p M 1 and quadrupol e strengths si m ul taneousl y,and the cl uster correl ati on pl aysa rol e to enhance the E 2 strengthsforthe 5=2 2 state.To cl ari fy the detai l ed structure ofthe 5=2 2 state,the preci se m easurem entofthe E 2 strength i s desi red.

a
new quanti ty D w hi ch i sde ned by D = R d 3 r=A w hereA i sa m ass num ber, (r)i sa m atterdensi ty,and 0 i sa norm aldensi ty w hi ch i schosen to be 0 = 0: 16 nucl eons/fm3 .T he D val ue provi desa fracti on ofnucl eonsi n the l ow densi ty regi on w here the m atter densi ty i s l ower than 1/5 ofthe norm al densi ty.Tabl e 2 l i sts the root m ean square (rm s) radi iand the D val ues for severalstatesi n 11 B and 12 C esti m ated by the A M D (VA P)and condensatem odel(A C M )[ 9]wave functi ons.T he A C M wave functi on gi vesthe l arge rm s radi us and D val ue for the 0 + 2 state i n 12 C ,w hi ch attracts broad i nterest i n vi ew ofdi l utecl usterstates.Si nce theA M D fram ework i sa ki nd oftheboundstate approxi m ati on,the A M D cal cul ati on tends to underesti m ate the tai lof the densi ty di stri buti on ata l arge radi us.A ctual l y,the rm s radi us and the D val ue forthe 0 + 2 state esti m ated by the A M D (VA P)wave functi on are sm al l er than those by the A C M wave functi on.H owever,they are sti l lextraordi nary l argecom pared w i th thosefortheground stateof 12 C .A l though therm sradi us and D val ue forthe 3=2 3 state i n 11 B i n the A M D (VA P)cal cul ati on are al so sm al l er than those for the 0 + 2 state i n the A C M cal cul ati on,they are as l arge asthoseforthe0 + 2 statecal cul ated by A M D (VA P)and aresi gni cantl y l arger than those for the ground and 5=2 2 states.T he 5=2 2 and 3=2 3 states l ocate atthe sam e exci tati on energy (see Fi g.2),but the rm s radi us and D val ue of the 5=2 2 state are m uch sm al l er than those ofthe 3=2 3 state.O n the basi s ofthese argum ents,i t i s naturalto suggest that the 3=2 3 state i n 11 B has a di l ute structure.

Fi g. 3
show s the experi m entaland theoreti calval ues ofthe i sotopi c shi fts for the negati ve-pari ty statesi n the 11 B-11 C m i rrorsystem asa functi on oftheexperi m entalexci tati on energi esi n 11 B.T he theoreti calpredi cti onsare obtai ned by the A M D (VA P) cal cul ati on w here the energi es for the exci ted states i n 11 C are cal cul ated by usi ng the m i rror-sym m etri c A M D wave functi ons.
of the 3=2 3 state i s possi bl y due to the cl usteri ng.Si nce the theoreti cal descri pti on of the cl usteri ng phenom ena under the SM fram ework requi res 3state at E x = 8: 56 M eV .A l though the 3=2 3 state appears at E x = 11: 4 M eV i n the SM cal cul ati on,i tstransi ti on properti esare com pl etel y di erentfrom those ofthe observed 3=2 3 state.T he predi cted 3=2 3 state carri es al m ost no m onopol e exci tati on strength,w hi l e the observed 3=2 3 state i s dom i nantl y exci ted by the J = 0 + transi ti on [ see Fi g.1(e)] .H ence, the predi cted 3=2 3 state i s di erent from the observed 3=2 3 state.Iti snoteworthy to poi ntoutthe anal ogi esbetween the 3=2 3 state i n11B and the 0 + 2 state atE x = 7: 65 M eV i n 12 C w hi ch i s exci ted from the ground state w i th an i soscal arm onopol e strength ofB (E 0;IS)= 121 9 fm 4 [ 20] .Both the two states l ocate at the exci tati on energi es near the -decay threshol d,w hi l e they are not sati sfactori l y predi cted i n the SM cal cul ati on.T hey carry the l arge i soscal ar m onopol e strengths w i th a si m i l ar m agni tude.T herefore,the observed 3=2 3 state i n 11 B i si nferred to have a structure anal ogousw i th that ofthe 0 + 2 state i n 12 C from the vi ew poi ntofcl usterphysi cs.N am el y,the 3=2 3