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Double cha ge-exchange eac ions and he e ec o
ans e
J A Lay1,†, S Bu ello2, J I Bellone2,3, M Colonna2and H Lenske4
wi hin he NUMEN p ojec
1Dp o. de F´ısica A ´omica, Molecula y Nuclea , Uni e sidad de Se illa, Se illa, Spain
2Is i u o Nazionale di Fisica Nuclea e, Labo a o i Nazionali del Sud, Ca ania, I aly
3Dipa imen o di Fisica e As onomia, Uni e si di Ca ania, I aly
4Ins i u ¨u Theo e ische Physik, Uni e si y o Giessen, Giessen, Ge many
E-mail: †[email protected]
Abs ac . The e is a ecen ly enewed in e es on he s udy o single and double cha ge
exchange eac ions wi h hea y ions. We epo he e a p elimina y heo e ical s udy o double
cha ge exchange eac ions in e ms o wo successi e cha ge exchanges in 2nd o de Dis o ed
Wa e Bo n app oxima ion. This allows us o include he e ec o he di e en ans e channels
in an app oxima e way. E idences show ha he cha ge exchange p ocess is dominan . The
ollowing one in impo ance is he combina ion o one single cha ge exchange, one neu on
pick-up and one p o on s ipping. This pa icula p ocess calls o u he in es iga ion.
1. In oduc ion
Nuclea eac ions a e one o he mos impo an ools o un a el he na u e o he in e ac ion
be ween neu ons and p o ons inside he a omic nucleus. In p inciple, we can ex ac a lo o
in o ma ion by making collide a nucleus o in e es wi h a ligh ion o wi h a hea y ion. In
he la e case, he heo e ical desc ip ion o he eac ion is la gely mo e in ol ed, since we
ha e o deal wi h he nucleon-nucleon in e ac ion o wo di e en many-body sys ems. In spi e
o his ex a di icul y, we ind in he li e a u e di e en cases in which we ha e imp o e ou
unde s anding o ce ain p oblems hanks o pa icula cha ac e is ics no a ailable wi h ligh
nuclei. As an example, we can ci e he s udy o b eakup o halo nuclei unde ex eme elec ic
ields supplied by a hea y a ge [1] o he sea ch o he Gian Pai ing Resonance, whe e a la ge
Q- alue is needed and only eachable wi h a hea y p ojec ile [2].
Fo he case o cha ge exchange eac ions, we can s udy he single cha ge exchange (SCE)
in he nucleus o in e es wi h bo h ligh and hea y ions. Howe e , pe o ming double cha ge
exchange (DCE) wi h ligh ions p esen s ce ain di icul ies like dealing wi h he measu emen
and econs uc ion o he ajec o y o h ee di e en inal p ojec ile pa icles as we would
ha e wi h a ligh ion like 3He o . We can use hea y ions o a oid such p oblems since bo h
he a ge -like and p ojec ile-like p oduc s o he eac ions will be bound in mos o he cases.
In addi ion, he inal c oss sec ion o double cha ge exchange will depend on he Fe mi and
Gamow Telle esponse o bo h he nucleus o in e es ( he a ge ) and he hea y ion used as
expe imen al p obe in he eac ion. By choosing ca e ully a hea y ion wi h a la ge Fe mi o
Gamow Telle esponse, we can inc ease he inal c oss sec ion, helping o he ex ac ion o he
pa icula esponse o he nucleus o in e es .
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Nowadays, he e is a la ge expe imen al in e es on double cha ge-exchange eac ions wi h
hea y ions which has gi en o igin o di e en campaigns mainly a he adioac i e beam acili ies
o RCNP a Osaka and RIKEN in Japan [3, 4] and a he LNS-INFN o Ca ania in I aly
[5, 6]. The o me ones a e ocused on s udying he Gamow-Telle esponse o nuclei and in he
p oduc ion o exo ic nuclei h ough double cha ge exchange eac ions. The la e is de o ed o
he s udy o double cha ge-exchange eac ions in connec ion wi h he neu inoless double-be a
decay.
In he p esen con ibu ion we will ocus in one o he eac ions pe o med a he LNS-INFN:
15 MeV/A 18O p ojec iles impinging on a 40Ca a ge [5]. A e he double cha ge-exchange we
will measu e 18Ne and 40A in di e en s a es. He e, we do no conside any di ec p ocess leading
om he ini ial o he inal coun e pa s. This means ha he nucleus-nucleus in e ac ion canno
change wo cha ges ac ing only once. Such a possibili y is unde in es iga ion [7]. The e o e,
he double cha ge-exchange is p oduced by a pa o he e ec i e nucleus-nucleus in e ac ion
able o change one cha ge a a ime bu ac ing wice, hus being called a second o de p ocess.
The inal si ua ion can be eached also h ough di e en combina ions o neu on pick-
up, p o on s ipping, and single cha ge-exchange p ocesses. Any o hese so-called compe ing
p ocesses is o a highe o de han he double cha ge-exchange, so ha his la e p ocess should
domina e. None heless, a low bomba ding ene gies, he di e ences can be oo small o neglec
he con ibu ion o he compe ing p ocesses. In any case, i we manage o con ol hem we can
ha e access o a complemen a y s uc u e in o ma ion o he nucleus conside ed which can be
o g ea in e es when s udying double-be a decay candida es.
The s udy o he compe i ion be ween cha ge exchange wi h hea y ion and he co esponding
ans e a di e en bomba ding ene gies has only been p ope ly done o he case o single
cha ge-exchange [8]. In single cha ge-exchange, he ans e consis s o one neu on (p o on)
pick-up ollowed by a one p o on (neu on) s ipping (o ice e sa) wha means ha can
be sol ed in e ms o 2nd o de Dis o ed Wa e Bo n App oxima ion (DWBA). Fo double
cha ge-exchange we can ha e ans e channels in ol ing ou pa icles, which means he
need o a 4 h o de DWBA. Non-o hogonali y be ween p io and pos ep esen a ion in
asn e gene a es a non-o hogonali y con ibu ion. This e m, oge he wi h simul aneous
and succesi e con ibu ions, con ibu es cohe en ly o he inal o al c oss sec ion. Full 2nd
o de DWBA including simul aneous, succesi e and non-o hogonali y e ms has been ecen ly
ully implemen ed, helping us o calcula e exac c oss sec ions o hese cases, whe eas o 4 h
o de DWBA he e is no implemen a ion ye o he co esponding non-o hogonali ies.
The e o e, ou pu pose is o p elimina y asses he impo ance o he di e en con ibu ions
in o de o s udy he possibili y o ex ac ing he desi ed esponse om he expe imen al da a.
We will es ima e he ans e con ibu ion o he 18O + 40Ca eac ion a 15 MeV/A and we
will compa e wi h he double cha ge-exchange expe imen al da a om [5].
2. Double cha ge-exchange and mul inucleon ans e c oss sec ions
On one hand, we only conside double cha ge exchange as a second o de cha ge exchange, so
ha he o m ac o s needed o he eac ion calcula ion a e hose o he eac ions: 18O+40Ca
→18F+40K and 18F+40K→18Ne+40A . Fo he calcula ion o such o m ac o s we can ely
on a double olding p ocedu e o a esidual nucleon-nucleon in e ac ion and he co esponding
ansi ion densi ies o a ge and p ojec ile.
On he o he hand, co esponding ans e con ibu ions mus be conside ed. The o malism
o mul inucleon ans e in DWBA has been known o se e al decades bu only ecen ly we
ha e s a ed o calcula e second o de DWBA ans e due o i s compu a ional demands. Fo
he case o double cha ge exchange we deal wi h he ans e o wo p o ons and wo neu ons.
Tha would mean he need o a 4 h o de DWBA which has no implemen a ion ye . Since ou
pu pose is o es ima e he o de o magni ude and i s ela i e impo ance wi h espec o he
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cha ge-exchange, we can app oxima e he ull c oss sec ion by he one ob ained conside ing a
second o de DWBA wo-neu on ans e o he g ound s a es o 16O+42Ca ollowed by a second
o de DWBA wo-p o on ans e o he inal p oduc s 18Ne+40A .
We ha e o add he e he possibili y o ha ing one neu on and one p o on ans e ollowed by
a single cha ge exchange and ice e sa. This p ocess is o hi d o de . The e o e, e en hough
i can be agains ou in ui ion o mix wo di e en esponses o nuclei, i could be la ge han
he ull ou h o de ans e .
The s uc u e inpu needed o he ans e a e he di e en o e laps be ween he di e en
a ge -like p oduc s in he di e en ini ial, in e media e and inal si ua ions conside ed. The
same eads o he p ojec ile like ones. Fo he sake o comple eness, hese o e laps should be
p o ided by he same s uc u e calcula ion p o iding ansi ion densi ies o he cha ge exchange
calcula ion. Howe e , hey a e usually app oxima ed by he o e lap ob ained wi h single pa icle
wa e unc ions mul iplied by he co esponding spec oscopic ac o . An al e na i e op ion is o
look o di e en spec oscopic ac o s ex ac ed om expe imen al ans e eac ions on he
conside ed nuclei.
The las ing edien o calcula ing he eac ion c oss sec ion a e he op ical po en ial be ween
he di e en coun e pa s o he ini ial, in e media e, and inal si ua ions. All o hem can
be ob ained again by a double olding p ocedu e wi h he co esponding cen al densi ies o
p ojec ile-like and a ge -like nuclei.
Finally, he o al c oss sec ion is he cohe en sum o all hese con ibu ions. The e o e, we
canno neglec he p ocesses whose c oss sec ions a e small when compu ed sepa a ely. All o
hem in e e e and, h ough his in e e ence, hei inal impac on he o al c oss sec ion can
be la ge han expec ed.
3. P elimina y Resul s
In o de o ackle he 18O + 40Ca eac ion a 15 MeV/A we combine wo p og ams HIDEX [9]
and FRESCO [10]. Wi h HIDEX we calcula e he single cha ge exchange o m ac o by
olding QRPA ansi ion densi ies o he di e en nuclei wi h he Lo e and F aney nucleon-
nucleon in e ac ion [11]. We app oxima e he 18F+40K→18Ne+40A pa by using he same
o m ac o s as in he i s single cha ge exchange, i.e. 18O+40Ca →18F+40K. Howe e ,
including all he cha ge exchange s eng h o all possible mul ipola i ies wi hin FRESCO is e y
demanding compu a ionally, so ha in his p elimina y app oach we ha e cons ain ou sel es
o he mul ipola i y leading o he g ound s a es o he in e media e coun e pa s: 18F(1+) and
40K(4−). Then we inc ease he o al s eng h o his speci ic pa o he o al esponse o he
nuclei in o de o be o he o de o he expe imen al c oss sec ion a ze o deg ees. In his way
he esul should be app oxima e bu s ill ep esen a i e o he s eng h o he double cha ge
exchange p ocess and i s compa ison wi h he compe ing channel emains meaning ul.
We use he same in e ac ion o calcula e he co esponding op ical po en ials. We
include he o m ac o s and op ical po en ials in o FRESCO whe e we can add he di e en
ans e con ibu ions. Fo his las pa , we ely on expe imen ally ex ac ed spec oscopic
ac o s [12, 13, 14, 15] o he di e en cases. Only g ound s a es a e conside ed o e en-e en
o odd-odd nuclei. Fo he odd-e en o e en-odd ones, we conside all he single pa icle le els
in he icini y o he Fe mi ene gy. The e o e, all hese le els p o ide a la ge basis and we only
lack o comple eness in he e en-e en and odd-odd nuclei.
A inal ema k has o be done ega ding he choice o p io o pos in e ac ions o each
s eps. We only conside he successi e pa o he ans e choosing pos in e ac ions o all
he s eps. In o de o ha e he ull ans e c oss sec ion, we will need o conside he di e en
non-o hogonali ies.The ans e leading o he same coun e pa s as DCE is a 4 h o de DWBA
calcula ion. The co esponding non-o hoganli y e ms ha e no been implemen ed so a . A
calcula ion including only succesi e e ms wi h pos in e ac ions, as he calcula ion p esen ed
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he e, should be a good es ima ion and mus each he exac esul i ou basis is inc eased
owa ds comple eness [10].
In o de o assess he ela i e impo ance o pu e ou pa icle ans e , he combina ion
o ans e and a single cha ge exchange and he successi e double cha ge exchange, we ha e
pe o med h ee di e en calcula ion. In he i s one we include all con ibu ions. In he second
one, only cha ge exchange and he combina ion o cha ge exchange and ans e . Finally, in a
hi d calcula ion, only cha ge exchange is included.
We show he p elimina y esul s in able 1. When only double cha ge exchange is conside ed,
we ge a di e en ial c oss sec ion o 18.3 µb/s a ze o deg ees. This alue is an 8% la ge han
he calcula ion ha includes all possible compe ing channels. This las calcula ion is en i led o
be he mos co ec one and he e o e his 8% can be unde s ood as he le el o accu acy ound
when neglec ing he e ec o ans e . Howe e , i we conside he double cha ge exchange in
combina ion wi h one-neu on and one-p o on ans e channels, he di e ence wi h espec o
he ull calcula ion is educed o less han a 5%.
Table 1. P elimina y calcula ions o he di e en ial c oss sec ion a ze o deg ees and he o al
c oss sec ion o he di e en p ocesses conside ed.
P ocesses conside ed dσ
dΩ(0) (µb/s ) In eg a ed σ(µb)
DCE only 18.3 0.060
DCE and (np ans e + SCE) 16.2 0.051
DCE and all ans e s conside ed 17.0 0.054
Expe imen al [5] 11.1 0.072
This educed di e ence is only p esen in he case o he di e en ial c oss sec ion a ze o
deg ees. The angula dis ibu ion also changes depending on he p ocesses conside ed. In ac ,
he esul s o he o al in eg a ed c oss sec ions show a la ge di e ence as can also be ound
in able 1.
4. Conclusions
We ha e s udied he double cha ge exchange eac ion 18O + 40Ca→18Ne+40A a 15 MeV/A
measu ed a he LNS-INFN [5] ocusing in he ela i e impo ance o he di e en compe ing
channels, i.e. double cha ge exchange and he di e en possible ans e combina ions leading
o he same inal nuclei.
We ha e ound ha double cha ge-exchange is he leading p ocess since he e is only a
di e ence o an 8% be ween he di e en ial c oss sec ion a ze o deg ees calcula ed including
all p ocesses conside ed and he one calcula ed by conside ing only he double cha ge-exchange.
This esul was as expec ed acco ding o he ac ha his p ocess is he lowes o de one
conside ed.
Among he p ocesses in ol ing ans e o pa icles, he combina ions o one single cha ge
exchange, one p o on s ipping and one neu on pick up is he ollowing in impo ance.
Conside ing his p ocess and he double cha ge-exchange we ob ain a di e en ial c oss sec ion
a ze o deg ees jus a 5% di e en om he one ob ained including all p ocesses.
Finally, he 4 h o de mul inucleon ans e is he lowes in impo ance. This is impo an
since he e is no a comple e de i a ion and implemen a ion o he co esponding non-
o hogonali y e ms needed o p ope ly calcula e his con ibu ion. This is also consis en wi h
wha ound o he 20Ne + 116Cd eac ion [16] whe e di e en calcula ions o he ans e o
wo neu on and wo p o ons p oduce s ongly supp essed c oss sec ions.
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The combina ions o one single cha ge exchange, one p o on s ipping and one neu on pick
up is a hi d o de p ocess. In o de o do a ull DWBA calcula ion one has o include non-
o hogonali y e ms o he pa ha includes ans e . Howe e , hese non-o hogonali ies a e
pa allel o hose o he second o de p ocess and i s de i a ion should be much simple han
hose o he 4 h o de ans e .
In any case, hese conclusions a e based on p elimina y, no comple e, calcula ions which
should be jus ep esen a i e o he o de o magni ude. Fu he in es iga ions a e demanded
in o de o co obo a e hese p elimina y esul s. Mo eo e , a ull de i a ion o he hi d o de
p ocess a ising om he combina ion o ans e and cha ge exchange could be o g ea in e es
in o de o imp o e accu acy when compa ing ou s uc u e models o neu inoless double-
be a decay candida es wi h he double cha ge-exchange eac ion expe imen al da a. Ano he
issue o in es iga e is whe he he choice o he hea y ion could educe e en mo e his ans e
con ibu ion. As shown in [17], he c oss sec ion o ans e o a pa icle depends o he
alignmen o i s spin wi h i s angula momen um in he ini ial and inal occupied o bi s. This
ea u e is no seen in ans e wi h ligh ions since he pa icles ans e ed a e ini ially in
s−o bi s. Howe e , he ele ance o he ans e can be e y di e en i we use a hea y ion wi h
alence neu ons and p o ons occupying a spin-aligned o a spin-an ialigned o bi .
Acknowledgmen s
The esea ch leading o hese esul s ecei ed unding om Spanish Minis e io de Econom´ıa y
Compe i i idad and FEDER unds unde P ojec No. FIS2014-53448-C2-1-P, by he Ayudas a
Consolidaci´on de G upos om Jun a de Andaluc´ıa, and by he Eu opean Union’s Ho izon 2020
esea ch and inno a ion p og am unde G an Ag eemen No. 654002.
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