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Spec oscopy o 46A by he ( ,p) wo-neu on ans e eac ion
Nowak, K.; Wimme , K.; Hellga ne , S.; Müche , D.; Bilds ein, V.; Di iken, J.; Else ie s,
J.; Ga ney, L. P.; Ge nhäuse , R.; Iwanicki, J.; Johansen, J. G.; Huyse, M.; Konki, Joonas;
K öll, T.; K ücken, R.; Lu e , R.; O landi, R.; Paka inen, Janne; Raabe, R.; Rei e , P.;
Roge , T.; Sch iede , G.; Seidli z, M.; So lin, O.; Van Duppen, P.; Wa , N.; De Wi e, H.;
Zielińska, M.
Nowak, K., Wimme , K., Hellga ne , S., Müche , D., Bilds ein, V., Di iken, J., Else ie s,
J., Ga ney, L. P., Ge nhäuse , R., Iwanicki, J., Johansen, J. G., Huyse, M., Konki, J.,
K öll, T., K ücken, R., Lu e , R., O landi, R., Paka inen, J., Raabe, R., . . . Zielińska, M.
(2016). Spec oscopy o 46A by he ( ,p) wo-neu on ans e eac ion. Physical
Re iew C, 93(4), A icle 044335. h ps://doi.o g/10.1103/PhysRe C.93.044335
2016
PHYSICAL REVIEW C 93, 044335 (2016)
Spec oscopy o 46A by he ( ,p) wo-neu on ans e eac ion
K. Nowak,1K. Wimme ,1,2,*S. Hellga ne ,1D. M¨
uche ,1V. Bilds ein,1,3J. Di iken,4J. Else ie s,4L. P. Ga ney,5,†
R. Ge nh¨
ause ,1J. Iwanicki,6J. G. Johansen,7M. Huyse,4J. Konki,8,9T. K ¨
oll,10 R. K ¨
ucken,1,11 R. Lu e ,12 R. O landi,13
J. Paka inen,8,9R. Raabe,4P. Rei e ,14 T. Roge ,4G. Sch iede ,10 M. Seidli z,14 O. So lin,15 P. Van Duppen,4N. Wa ,14
H. De Wi e,4and M. Zieli´
nska6
1Physik-Depa men E12, Technische Uni e si ¨
a M¨
unchen, D-85748 Ga ching, Ge many
2Depa men o Physics, The Uni e si y o Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
3Depa men o Physics, Uni e si y o Guelph, Guelph, On a io, N1G 2W1, Canada
4KU Leu en, Ins i uu oo Ke n- en S alings ysica, B-3001 He e lee, Belgium
5Oli e Lodge Labo a o y, Uni e si y o Li e pool, Li e pool L69 9ZE, Uni ed Kingdom
6Hea y Ion Labo a o y, Uni e si y o Wa saw, PL-02-093 Wa saw, Poland
7Depa men o Physics and As onomy, Aa hus Uni e si y, DK-8000 Aa hus C, Denma k
8ISOLDE, CERN, CH-1211 Gene a 23, Swi ze land
9Uni e si y o Jy askyla, Depa men o Physics, P.O. Box 35, FI-40014, Uni e si y o Jy askyla, Finland
and Helsinki Ins i u e o Physics, P.O. Box 64, FI-00014 Uni e si y o Helsinki, Finland
10Ins i u ¨
u Ke nphysik, Technische Uni e si ¨
a Da ms ad , D-64289 Da ms ad , Ge many
11Depa men o Physics and As onomy, Uni e si y o B i ish Columbia, Vancou e , B i ish Columbia, Canada V6T 1Z1
and TRIUMF, 4004 Wesb ook Mall, Vancou e , B i ish Columbia, Canada V6T 2A3
12Ludwig-Maximilians-Uni e si ¨
a -M¨
unchen, Schellings aße 4, D-80799 M¨
unchen, Ge many
13Ins i u o de Es uc u a de la Ma e ia, IEM-CSIC, Mad id E-28006, Spain
14Ins i u ¨
u Ke nphysik, Uni e si ¨
a zu K¨
oln, D-50937 K¨
oln, Ge many
15G and Acc´
el´
e a eu Na ional d’Ions Lou ds (GANIL), CEA/DSM - CNRS/IN2P3, B. P. 55027, F-14076 Caen Cedex 5, F ance
(Recei ed 1 Ma ch 2016; published 27 Ap il 2016)
S a es in he N=28 nucleus 46A ha e been s udied by a wo-neu on ans e eac ion a REX-ISOLDE
(CERN). A beam o adioac i e 44A a an ene gy o 2.16 AMeV and a i ium-loaded i anium a ge we e used
o popula e 46A by he 3H(44A , p) wo-neu on ans e eac ion. P o ons emi ed om he a ge we e iden i ied
in he T-REX silicon de ec o a ay. The exci a ion ene gies o s a es in 46A ha e been econs uc ed om he
measu ed angles and ene gies o ecoil p o ons. Angula dis ibu ions o h ee inal s a es we e measu ed and
based on he shape o he di e en ial c oss sec ion an exci ed s a e a 3695 keV was iden i ied as Jπ=0+.
The angula di e en ial c oss sec ion o he popula ion o di e en s a es a e compa ed o calcula ions using a
eac ion model employing bo h sequen ial and di ec ans e o wo neu ons. Resul s a e compa ed o shell-model
calcula ions using s a e-o - he-a e ec i e in e ac ions.
DOI: 10.1103/PhysRe C.93.044335
I. INTRODUCTION
Among he magic numbe s which desc ibe he shell
s uc u e o a omic nuclei, 28 is he i s main shell gap
c ea ed by he spin-o bi in e ac ion. The 1 7/2o bi al ge s
lowe ed in ene gy compa ed o he 1 5/2o bi al c ea ing his
gap wi hin he N=3 majo oscilla o shell. The e olu ion o
he shell gap a 28 nucleons, bo h as a unc ion o neu on and
p o on numbe , is in luenced by he na u e o he spin-o bi
in e ac ion. On he neu on- ich side o he alley o s abili y,
i was shown ha o he e ms in he nucleon in e ac ion also
play a ole in de e mining he size o he N=28 shell gap [1].
*Co esponding au ho : [email p o ec ed]
†P esen add ess: School o Enginee ing and Compu ing, Uni e si y
o he Wes o Sco land, Paisley, PA1 2BE, Uni ed Kingdom.
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 3.0 License. Fu he dis ibu ion o
his wo k mus main ain a ibu ion o he au ho (s) and he published
a icle’s i le, jou nal ci a ion, and DOI.
Th ee-body o ces ha e been success ully employed along he
Ca iso opic chain (Z=20) o desc ibe he high exci a ion
ene gy o he i s 2+s a e in 48Ca and he inc ease o he
N=28 gap be ween N=20 and 28 mic oscopically [2].
E olu ion o he gap be ween he neu on sd shell and he 1 7/2
o bi al along he N=28 iso ones is in luenced by he cen al
and enso in e ac ion be ween p o ons and neu ons [3]. Below
48Ca a a ie y o ea u es can be seen in he low-lying
exci a ions o he N=28 iso ones. These a ise om he
sub le in e play o he o ces, he b eakdown o he N=28
shell closu e, and he p o on subshell closu es a Z=16
(sul u ) and 14 (silicon). The B(E2; 2+
1→0+
gs) alue o 46A
(Z=18) was measu ed using Coulomb exci a ion a in e me-
dia e ene gies [4–6] as well as ex ac ed om he measu ed
li e ime [7] gi ing con lic ing esul s. The alue de e mined
in he Coulomb exci a ion expe imen s (B(E2; 2+
1→0+
gs)=
39(8) e2 m4[4], 44(6) e2 m4[5], and 54(5) e2 m4[6]) poin s
o a mode a e de o ma ion and collec i i y in 46A consis en
wi h he expec a ion o a semimagic nucleus. This is suppo ed
by ime-dependen Ha ee-Fock-Bogoliubo calcula ions [8]
ha link he inc ease in collec i i y wi h espec o 48Ca o a
2469-9985/2016/93(4)/044335(10) 044335-1 Published by he Ame ican Physical Socie y
K. NOWAK e al. PHYSICAL REVIEW C 93, 044335 (2016)
quenching o he N=28 shell gap. Shell-model calcula ions
on he o he hand a o he esul o a la ge B(E2) alue
as de e mined by he li e ime measu emen (B(E2; 2+
1→
0+
gs)=114+67
−32 e2 m4[7]). The neu on single-pa icle ene gies
o he 2p3/2,2p1/2, and 1 5/2o bi als in 47A ha e been
ex ac ed om a (d,p) ans e eac ion and compa ed o 49Ca
he N=28 shell gap is educed by 330(90) keV [9]. Mass
measu emen s also show a s ong gap a N=28 [10] and he
sepa a ion ene gies a e well desc ibed by calcula ions using
he SDPF-U [11] and SDPF-MU [12] e ec i e in e ac ions.
Below 46A he nucleus 44Sexhibi s a low-lying exci ed 0+
s a e [13] which was in e p e ed as a sign o shape coexis ence.
Measu emen s o o he low-lying s a es [14]aswellas
con igu a ion mixing calcula ions sugges an e osion o he
N=28 shell closu e a he han shape coexis ence [15]. 42Si
has a e y low i s exci ed s a e [16] and he R4/2 a io indica es
well-de eloped de o ma ion [17]. Shell-model calcula ions
p edic ha his nucleus is obla e in i s g ound s a e [11,12].
The single-pa icle s uc u e o 46A and i s neighbo s
was s udied in se e al expe imen s. Spec oscopic ac o s
ex ac ed om neu on emo al eac ions om 46A o 45A
ga e consis en esul s bo h in ans e [18] and knockou
eac ions [19]. These expe imen s show ha he g ound s a e
o 46A is domina ed by a 7/2con igu a ion. Spec oscopic
ac o s ex ac ed om he s udy o he N=27 iso ope 45A
by a one-neu on (d,p) ans e eac ion also ag ee wi h
shell-model esul s [20]. These esul s sugges ha he N=28
shell gap is s ill p onounced in 46A . E en hough he i s
exci ed s a e in 45A Jπ=3/2−is loca ed only a 542 keV,
he spec oscopic s eng h is la ge o he second exci ed 3/2−
s a e a 1416 keV. The low 3/2−
1s a e has likely a complica ed
s uc u e, in ol ing also p o on exci a ions [20] and can
he e o e no be ega ded as a sign o a educed shell gap.
The B(E2; 2+
1→0+
gs) as de e mined by in e media e beam
ene gy Coulomb exci a ion is somewha small [4–6], a esul
in disag eemen wi h he shell-model calcula ions [11,12,21]
as well as calcula ions using he gene a o coo dina e me hod
wi h he Gogny D1S in e ac ion [22]. The la e calcula ions
p edic a coexis ence o sphe ical and de o med s a es a low
exci a ion ene gy. The collec i e wa e unc ion calcula ed o
bo h he 0+
gs and 0+
2s a es show a mix u e o obla e and p ola e
componen s, on a e age his leads o a sligh ly obla e 0+
gs and
p ola e 0+
2a ∼2.75 MeV [22]. A ela i ely low-lying exci ed
0+s a e is also p edic ed by he shell-model calcula ions a
a ound 3 MeV (see Fig. 6). Expe imen ally, exci ed s a es
beyond he 2+
1s a e we e obse ed in in-beam expe imen s. In
a p o on inelas ic sca e ing expe imen [23] a candida e o
a3
−s a e a 4982 keV and se e al unassigned s a es a ound
4 MeV we e ound. Candida es o 0+
2,2
+
2, and 4+
1s a es
we e ound in agmen a ion eac ions [24]. The 0+
2s a e was
loca ed a 2710 keV and en a i ely assigned only based on he
obse a ion o a 1140-keV ansi ion in coincidence wi h he
2+
1→0+
gs ansi ion and he compa ison o calcula ions. F om
shell-model calcula ions in Re . [24] using he in e ac ion o
e e ence [25] he0
+g ound s a e is domina ed by a 0p-0h
con igu a ion. The i s exci ed 0+s a e on he o he hand has
a2p-2hs uc u e wi h wo neu ons p edominan ly loca ed in
he 2p3/2o bi al abo e N=28.
In his wo k he s uc u e o low-lying s a es in 46A was
s udied by a ( ,p) wo-neu on ans e eac ion in in e se kine-
ma ics. Two-neu on ans e eac ions a e an excellen ool o
s udy he na u e o 0+s a es caused by neu on exci a ions. The
angula dis ibu ion o p o ons om he eac ion is indica i e o
he ans e ed angula momen um o he eac ion. The e o e,
0+s a es can be iden i ied unambiguously. Fu he mo e, he
c oss sec ion o he wo-neu on ans e eac ion depends
on he de ails o he wa e unc ions o he s a es in ol ed,
allowing o p ecise es ing o heo e ical models.
II. EXPERIMENTAL SETUP
The expe imen was pe o med a he REX-ISOLDE acil-
i y a CERN [26]. Radioac i e 44A nuclei we e p oduced by
impinging he 1.4-GeV p o on beam om he PS boos e on o a
hick u anium ca bide (UCx) a ge . To educe con amina ion
om ca bon dioxide CO2a he same mass numbe 44 he
p ima y a ge was hea ed be o e he expe imen . A gon
as a noble gas is ola ile, eme ging easily om he hick
a ge ma e ial h ough a cooled ans e line o emo e less
ola ile con aminan s. A o ced elec on beam induced a c
discha ge (FEBIAD) ion sou ce [27] was used o achie e a
high ioniza ion e iciency o he 1+cha ge s a e o 44A .
A e accele a ion o 30 keV he beam is sen h ough
he high esolu ion sepa a o (HRS). The HRS p o ides
su icien esolu ion o disc imina e be ween 44A +and he
emaining CO+
2. Doubly cha ged 88K 2+could no ully be
sepa a ed and emained in he low ene gy beam. A e mass
sepa a ion, a adio equency quad upole coole and bunche
we e employed o imp o e beam emi ance. Ions we e hen
accumula ed and bunched in he REX ap o 60 ms be o e
anspo a ion o he elec on beam ion sou ce REX EBIS
o cha ge b eeding. Fo he 44A ions a maximum in he
cha ge s a e dis ibu ion a q=+13 was achie ed in 59-ms
cha ge b eeding ime. Be o e accele a ion in he REX linea
accele a o he ions a e sepa a ed by hei mass o cha ge
a io A/q. The cha ge s a e dis ibu ion o 88K is su icien ly
di e en such ha an A/q selec ion o 3.3846 p o ided a
clean beam o he expe imen . Selec ing a cha ge s a e o
q=+13 also elimina ed con amina ion om he 22Ne bu e
gas used in he EBIS. The ions we e accele a ed by he REX
LINAC consis ing o an RFQ, an IH s uc u e, h ee se en-gap
esona o s ollowed by a nine-gap esona o . Fo he p esen
expe imen he beam ene gy was limi ed o 2.16 AMeV, o
a oid usion eac ions wi h he a ge ca ie ma e ial, he e o e
he nine-gap esona o was no used.
The 44A beam a an a e age in ensi y o 2 ×105/s was
hen sen o he expe imen al s a ion whe e i impinged on a
i ia ed i anium oil. The a ge i sel is a 4.5-mm wide s ip
o i anium oil wi h a hickness o 0.5 mg/cm2. The i anium
is loaded wi h i ium a an a omic a io o 1.3 i ium a oms pe
i anium a om, co esponding o an e ec i e i ium hickness
o 36 μg/cm2. The a ge was he same one used in Re . [28]
and he decay o he i ium had educed he e ec i e hickness
since i s p oduc ion in Oc obe 2010. Ligh eac ion pa ne s
eme ging om he a ge we e de ec ed and iden i ied using
he T-REX silicon de ec o a ay [29]. The a ay consis s o
wo boxes o 140-μm hick silicon s ip de ec o s o measu e
044335-2
SPECTROSCOPY OF 46A BY THE ( ,p) TWO- . . . PHYSICAL REVIEW C 93, 044335 (2016)
he ene gy loss o ligh pa icles backed by 1-mm hick
unsegmen ed silicon de ec o s o o al ene gy measu emen .
In he mos backwa d di ec ion a double-sided annula silicon
s ip de ec o was moun ed. The de ec o s co e 65% o he
solid angle a ound he a ge . Recoil p o ons, deu e ons, and
i ons om elas ic and inelas ic sca e ing as well as ans e
eac ion channels a e iden i ied by hei cha ac e is ic ene gy
loss in he hin i s laye o he de ec o s ack h ough he
E-Eme hod. In backwa d di ec ion he ene gy o p o ons
is no su icien o punch h ough he i s laye o silicon,
howe e , he second laye can be used o disc imina e p o ons
om elec on om βdecay o beam pa icles acciden ally
s opped in he chambe . The e iciency and accep ance o he
a ay was modeled using a GEANT4 [30] simula ion o he
se up [29]. The silicon a ay is su ounded by he MINIBALL
ge manium de ec o a ay [31]. MINIBALL consis s o 24
high pu i y ge manium c ys als, each six old segmen ed o
imp o ed g anula i y, allowing o be e Dopple co ec ion
o de ec ed γ ays. Ene gy and e iciency calib a ions we e
pe o med using s anda d calib a ion sou ces.
III. DATA ANALYSIS
Ligh , cha ged ecoil pa icles, p o ons, i ons, and
deu e ons we e iden i ied using he ene gy loss E and o al
kine ic ene gy Emeasu emen s in he wo laye s o he
T-REX silicon de ec o s. Fo pa icles s opped in he E laye
addi ional kinema ic cu s ha e been applied. In labo a o y
backwa d di ec ion bo h p o ons and deu e ons ha e kine ic
ene gies below he iden i ica ion h eshold; hey a e s opped
in he i s laye , and he e o e no pa icle iden i ica ion is
possible. Howe e , he kine ic ene gy o deu e ons ollowing
he ( ,d) eac ion is e y low. The e o e, a condi ion on
sca e ing angle and pa icle ene gy can be used o elimina e
deu e ons in he spec um.
The spec um in Fig. 1shows he exci a ion ene gy o 46A
econs uc ed om he p o on angle and kine ic ene gy.
In addi ion o a s ong popula ion o exci ed s a es a ound
5 MeV (see below o de ails) h ee peaks a e obse ed in he
exci a ion ene gy spec um. They co espond o he g ound
s a e o 46A , he known i s exci ed 2+s a e a 1554 keV,
and a p e iously unknown s a e a an exci a ion ene gy o
3660(60) keV.
Figu e 2shows he Dopple co ec ed γ- ay ene gy
spec um o 46A assuming a sca e ing angle o 0◦in he
labo a o y sys em o 46A .
The ansi ions a 1554, 2318, 2518, and 2707 keV ha e
been p e iously obse ed [23,24]. A ansi ion a 1153 keV,
co esponding o he decay o he p e iously assigned 0+
2
s a e [24] was no obse ed. Newly obse ed a e he ansi ions
a 2141 and 3590 keV. The s a is ics a e no su icien o a γ-γ
coincidence analysis, bu he analysis o he exci a ion ene gy
spec um shows ha all ansi ions eed he i s exci ed s a e
and no o he s a e below 4 MeV was obse ed in Fig. 1.Fig-
u e 3shows he exci a ion ene gy o 46A econs uc ed om
he p o on angle and kine ic ene gy measu ed in coincidence
wi h he s onges γ- ay lines obse ed in Fig. 2.
The spec a ha e been i ed wi h a Gaussian unc ion,
and he esul ing mean exci a ion ene gy ag ees wi h he one
(MeV)
exc
E
1−012
coun s / 40 keV
0
5
10
15
20
25 (a)
° > 150
lab
ϑ
(MeV)
exc
E
1−012
345
345
coun s / 40 keV
0
5
10
15
20
25
30
(b)
° > 120
lab
ϑ
FIG. 1. Exci a ion ene gy o 46A econs uc ed om he p o on
angle and kine ic ene gy. The da a a e i wi h an exponen ial
unc ion ep esen ing he con inuum o highly exci ed s a es and
Gaussian unc ions co esponding o s a es in 46A . (a) Shows he
mos backwa d angles in he labo a o y sys em, whe e he esolu ion
is bes . (b) Also includes mo e o wa d angles, whe e he known 2+
s a e a 1554 keV is mo e p onounced. Because he exci a ion ene gy
esolu ion depends s ongly on he sca e ing angle, he i is only
used o ex ac he mean posi ion o he peaks, no he c oss sec ion.
de e mined om he sum o γ- ay ene gies wi hin he e o .
Figu e 3(a) shows ha he main con ibu ion o he 2+s a e
comes om indi ec eeding h ough exci ed s a es be ween 3
and 6 MeV. A ga e on he 2141-keV ansi ion e eals a single
s a e a an exci a ion ene gy o 3670(100) keV [Fig. 3(b)]. This
s a e co esponds o he p e iously discussed s a e o Fig. 1a
3660(60) keV. F om he sum o γ- ay ansi ion ene gies he
exci a ion ene gy o his s a e is de e mined o 3695(4) keV.
(keV)
γ
E
500 1000 1500 2000 2500 3000 3500
coun s / 10 keV
1
10
2
10
1554(1)
2141(3)
2318(3)
2518(2)
2707(2)
3590(3)
FIG. 2. Dopple co ec ed γ- ay ene gy spec um measu ed in
coincidence wi h ecoil p o ons iden i ied in T-REX. All p o on angles
ha e been included. T ansi ions a e labeled by hei ene gy in keV.
The g een a ow indica es 3695 keV, whe e a di ec g ound-s a e
decay o he p oposed 0+s a e would be loca ed.
044335-3
K. NOWAK e al. PHYSICAL REVIEW C 93, 044335 (2016)
(MeV)
exc
E
0 123456
coun s / 200 keV
0
5
10
15
20
25
30 (a)ga e 1554 keV
(MeV)
exc
E
0 123456
coun s / 200 keV
1−
0
1
2
3
4
5
6
7(b)ga e 2141 keV
(MeV)
exc
E
0 123456
coun s / 200 keV
0
2
4
6
8
10
12 (c)ga e 2707 keV
(MeV)
exc
E
0123456
coun s / 360 keV
0
2
4
6
8
10 (d)ga e 3590 keV
FIG. 3. Exci a ion ene gy o 46A econs uc ed om he p o on
angle and kine ic ene gy ga ed on se e al γ- ay ansi ions. All p o on
angles a e aken in o accoun . Random backg ound was sub ac ed.
Exci a ion ene gies ex ac ed om Gaussian i s ag ee wi h he sums
o γ- ay ansi ion ene gies.
Simila ly, we place s a es a 4255(4) and 5144(4) keV which
decay by 2707- and 3590-keV ansi ions o he i s exci ed
s a e. Fo he ansi ions a 2318 and 2518 keV he s a is ics
is no su icien o de e mine he eeding le el om p o on-γ
FIG. 4. Le el scheme o 46A as de e mined in his wo k. A ows
indica e he obse ed γ- ay ansi ions; hei wid h co esponds o
he ela i e in ensi y.
(deg)ϑ
0 20 40 60 80 100 120 140 160 180
(
m
b/
s
)
Ω
/d
σ
d
3−
10
2−
10
1−
10
1(a)0 keV
(deg)ϑ
0 20 40 60 80 100 120 140 160 180
(
m
b/
s
)
Ω
/d
σ
d
3−
10
2−
10
1−
10
1(b)
SDPF-MU
SDPF-U
EPQQM
1554 keV
(deg)ϑ
0 20 40 60 80 100 120 140 160 180
(
m
b/
s
)
Ω
/d
σ
d
3−
10
2−
10
1−
10
1(c)3695 keV
FIG. 5. Angula dis ibu ion o p o ons om he wo-neu on
ans e eac ion o 46A . (a) G ound s a e o 46A ; (b) i s exci ed 2+
s a e; (c) exci ed 0+
2s a e. Lines ep esen he heo e ical calcula ions
using he DWBA eac ion model desc ibed in Sec. IV B using
ampli udes calcula ed wi h h ee di e en shell-model e ec i e
in e ac ions, SDPF-MU ( ed, dashed), SDPF-U (g een, solid), and
EPQQM (blue, do -dashed).
coincidences p ecisely, howe e , hey a ise om s a es a ound
4 MeV. These ansi ions a e placed on op o he 2+s a e. The
esul ing le el scheme o 46A is shown in Fig. 4.
This le el scheme is consis en wi h he one ob ained om
he p o on inelas ic sca e ing expe imen [23]. Because (p,p)
does no popula e he exci ed 0+s a e di ec ly, and a wo-
neu on ans e eac ion o a 3−s a e is no expec ed, he wo
expe imen s a e complemen a y and in good ag eemen . In
044335-4

SPECTROSCOPY OF 46A BY THE ( ,p) TWO- . . . PHYSICAL REVIEW C 93, 044335 (2016)
0
+
0
1506
+
2
2928
+
0
3464
+
43510
+
0
3587
+
2
3649
+
33946
+
04046
+
4
4051
+
2
SDPF-NR
0
+
0
1594
+
2
3328
+
0
3529
+
43643
+
33775
+
23900
+
04125
+
0
4222
+
14236
+
2
SDPF-U
0
+
0
1443
+
2
3495
+
03741
+
43790
+
23891
+
34027
+
44075
+
04366
+
14533
+
0
SDPF-MU
0
+
0
1904
+
2
3098
+
0
3876
+
03978
+
24172
+
14295
+
2
4830
+
4
EPQQM
FIG. 6. Calcula ed le el schemes o 46A using a ious e ec i e in e ac ions in he shell model [11,12,21,25].
addi ion o he s a es shown in Fig. 4 he exci a ion ene gy
spec um (Fig. 1) indica es ha se e al o he s a es abo e 4-
MeV exci a ion ene gy ha e been popula ed. The le el densi y
inc eases wi h exci a ion ene gy and many indi idual s a es a e
popula ed wi h small c oss sec ions, he e o e disc e e lines
we e no iden i ied.
Because he beam in ensi y luc ua ed du ing he ex-
pe imen , he luminosi y was de e mined using he elas ic
sca e ing o i ons. These da a we e also used o cons ain he
op ical model pa ame e s o he DWBA analysis. The angula
dis ibu ions we e ob ained by ga ing on he exci a ion ene gy
(Fig. 1) and co ec ing o he geome ical accep ance o he
T-REX a ay [29]. Figu e 5shows he angula dis ibu ion o
p o ons om he wo-neu on ans e eac ion o he g ound
s a e and exci ed s a es o 46A a 1554 and 3695 keV. To
a oid sys ema ic unce ain ies da a om he annula de ec o
a backwa d angles we e excluded because o an un esol ed
p oblem wi h he ime-dependen e iciency o i s mul iplexed
eadou [29].
The compa ison wi h he DWBA calcula ions (Sec. IV B)
show ha p o ons om he ans e eac ion o he g ound s a e
o 46A ollow he calcula ed di e en ial c oss sec ion wi h he
cha ac e is ic L=0 minimum a a sca e ing angle ϑcm ∼20◦.
The angula dis ibu ion co esponding o he popula ion o
he 2+s a e displays a shallow maximum a ound ϑcm ∼30◦,
indica i e o he o bi al angula momen um ans e o L=2.
The di e en ial c oss sec ion o he newly obse ed exci ed
s a e a 3695 keV shows he same end as he g ound s a e.
This cha ac e is ic L=0 shape as well as he γdecay only
o he 2+s a e and no di ec ly o he g ound s a e, and he
somewha la ge wo-neu on ans e eac ion (see Sec. IV B)
indica e a spin and pa i y Jπ=0+ o hiss a e.
IV. THEORETICAL CALCULATIONS
Fo he heo e ical calcula ion o he wo-neu on ans e
eac ion c oss sec ion bo h nuclea s uc u e and eac ion
inpu s a e equi ed. Shell-model calcula ions a e employed
o ob ain he spec oscopic ampli udes (A) o one-neu on
ans e s eps as well as wo-nucleon ampli udes (TNA) o
he di ec pai ans e . In Sec. IV B he dependence o he
di e en ial c oss sec ion on he op ical model pa ame e s and
he in luence o he wo eac ion p ocesses a e analyzed.
A. Shell-model calcula ions
To ge insigh s in he unde lying s uc u e causing he la ge
c oss sec ion o he i s exci ed 0+s a e shell-model calcu-
la ions ha e been pe o med using he code NUSHELLX [32].
The model space comp ises he sd shell o he p o ons and he
p shell o he neu ons. Th ee s a e-o - he-a e ec i e in e -
ac ions ha e been compa ed, SDPF-U [11], SDPF-MU [12],
and EPQQM [21]. The SDPF-MU and SDPF-U in e ac ions
a e cons uc ed om h ee ing edien s. Bo h use he USD [33]
e ec i e in e ac ion o he sd p o on-p o on ma ix elemen s.
The neu on-neu on in e ac ion in he p shell is based on he
KB3 [34] ma ix elemen s o he SDPF-U in e ac ions and
he GXPF1B [35] in e ac ion o SDPF-MU, espec i ely. The
sd- p c oss-shell p o on-neu on ma ix elemen s a e aken
om G-Ma ix [36] o SDPF-U in e ac ion and om VMU [3]
in he case o SDPF-MU. The SDPF-U in e ac ion di e s
om he olde e sion, SDPF-NR [25], in ha expe imen al
in o ma ion on N=21 and po assium nuclei was used o
cons ain he monopole pa s. Fo he p esen calcula ion he
e sion o Z>14 nuclei was chosen. The EPQQM e ec i e
in e ac ion is based on pai ing plus quad upole-quad upole
o ces wi h a monopole e m [37]. I was buil o consis en ly
desc ibe nuclei be ween Ca and Si [21]. The calcula ed le el
schemes o 46A a e shown in Fig. 6. Addi ionally we also
ep esen he calcula ions wi h he o iginal SDPF-NR [25]
in e ac ion, which was p e iously [24] used o assign spin and
pa i y 0+ o a p oposed s a e a 2710 keV. The le el scheme
calcula ed wi h he SDPF-U and SDPF-MU in e ac ions a e
e y simila , while he EPQQM calcula ion p edic s a highe
ene gy o he i s 2+and 4+s a es. The i s exci ed 0+s a e
is ound a lowe exci a ion ene gy.
The wo-neu on ans e eac ion can p oceed ei he by a
successi e ans e o wo single neu ons o by a one-s ep
di ec ans e o a neu on pai . To compa e he esul ing wo-
neu on ans e c oss sec ion, he spec oscopic ampli udes A
o he 44A +n|45A and 45A +n|46A s eps as well as
wo-nucleon ampli udes TNA o he di ec one-s ep ans e o
044335-5
K. NOWAK e al. PHYSICAL REVIEW C 93, 044335 (2016)
A
0.6−
0.4−
0.2−
0
0.2
0.4
0.6
0.8 A (a)
45
-
1
7/2
-
1
3/2
-
2
3/2
-
1
1/2
-
2
1/2
-
3
1/2
-
1
5/2
-
2
5/2
A
2−
1−
0
1
2 (b)
+
gs
0
7/2
⊗
-
1
7/2
3/2
p⊗
-
1
3/2
3/2
p⊗
-
2
3
/2
1/2
p⊗
-
1
1/2
1/2
p⊗
-
2
1/2
1/2
p⊗
-
3
1/2
5/2
⊗
-
1
5/2
5/2
⊗
-
2
5/2
SDPF-MU
SDPF-U
EPQQM
A
0.8−
0.6−
0.4−
0.2−
0
0.2
0.4
0.6
0.8 (c)
+
1
2
7/2
⊗
-
1
3/2
3/2
p⊗
-
1
7/2
7/2
⊗
-
1
7/2
A
0.8−
0.6−
0.4−
0.2−
0
0.2
0.4
0.6
0.8
1
1.2 (d)
+
ex
0
7/2
⊗
-
1
7/2
3/2
p⊗
-
1
3/2
3/2
p⊗
-
2
3
/2
1/2
p⊗
-
1
1/2
1/2
p⊗
-
2
1/2
1/2
p⊗
-
3
1/2
5/2
⊗
-
1
5/2
5/2
⊗
-
2
5/2
FIG. 7. Calcula ed spec oscopic ampli udes (A) o he ans e
o s a es in 45A (a), and om a ious s a es in 45A o he g ound
s a e (b), he 2+
1s a e (c), and he i s exci ed 0+s a e (d) o 46A .
a pai ha e been calcula ed. Figu e 7shows he spec oscopic
ampli udes calcula ed in he shell model using he h ee
di e en e ec i e in e ac ions.
Only s a es which ha e a calcula ed spec oscopic ac o
C2S=A2>0.05 a e included in he igu e. The c oss sec ion
o a single-neu on ans e eac ion such as he 3H(44A , d)
eac ion o s a es in 45A depends only on he squa e o he
ampli ude; he phase has no e ec . The calcula ion wi h he
SDPF-U e ec i e in e ac ion p edic s wo 5/2−s a es wi h
signi ican spec oscopic ac o s, which a e bo h included in
he calcula ion. Because o he high exci a ion ene gy, he
wo-s ep ans e eac ion c oss sec ion h ough hese s a es
is negligible. Fo he calcula ion o he wo-neu on ans e
eac ion, howe e , he ela i e signs ma e s. All he ampli udes
depic ed in Figs. 7(b)–7(c), o a gi en e ec i e in e ac ion,
in e e e o con ibu e o he sequen ial ans e c oss sec ion.
The wo-nucleon ampli udes a e shown in Fig. 8.
Simila o he spec oscopic ampli udes he ela i e phase
o he ampli udes con ibu ing o he c oss sec ion o one s a e
de e mines he in e e ence.
B. Reac ion model
The wo-neu on ans e eac ion c oss sec ions and an-
gula dis ibu ions we e calcula ed using he FRESCO DWBA
code [38]. Op ical model pa ame e s o he incoming, in e -
media e, and ou going channel we e aken om global i s o
i ons [39–41], deu e ons [39,42], and p o ons [39,43,44]. The
global pa ame e se o Re . [39] is he one which is ex ended
o he lowes p ojec ile ene gies, he e o e his pa ame e se
is conside ed as he base line o a compa ison. The nume ical
alues o he pa ame e s a e lis ed in Table I.
TNA
0.8−
0.6−
0.4−
0.2−
0
0.2
0.4
0.6
0.8 (a)
+
gs
0
2
)
7/2
(1
2
)
5/2
(1
2
)
3/2
(2p
2
)
1/2
(2p
TNA
0.1−
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 (b)
+
ex
0
2
)
7/2
(1
2
)
5/2
(1
2
)
3/2
(2p
2
)
1/2
(2p
TNA
0.1−
0
0.1
0.2
0.3
0.4 (c)
+
1
2
2
)
7/2
(1
)
5/2
)(1
7/2
(1
)
3/2
)(2p
7/2
(1
2
)
5/2
(1
)
3/2
)(2p
5/2
(1
)
1/2
)(2p
5/2
(1
2
)
3/2
(2p
)
1/2
)(2p
3/2
(2p
SDPF-MU
SDPF-U
EPQQM
FIG. 8. Calcula ed wo-nucleon ampli udes (TNA) o he ans-
e o he g ound s a e (a), he i s exci ed 0+s a e (b), and he 2+
1
s a e (c).
To es ima e he e ec o he po en ial pa ame e s, calcula-
ions ha e been pe o med wi h di e en combina ions. The
esul o he wo-neu on ans e eac ion o he g ound s a e
o 46A is shown in Fig. 9 o selec ed po en ials.
Wi h he excep ion o he pa ame e se in ended o highe
deu e on ene gies (Ed>12 MeV) [39] all pa ame iza ions
ag ee in hei shape. The bigges impac on he shape, as well
as he in eg a ed c oss sec ion ha e changes in he in e media e
45A +d channel. Because he da a a e no su icien o i he
elas ic sca e ing o i ons and p o ons o ob ain cons ain s
on he pa ame e s, and o he elas ic deu e on channel no
da a ha e been measu ed, in he ollowing he op ical po en ial
pa ame e s a e ixed o he alues lis ed in Table I. Wi hin
he angula ange co e ed by he silicon de ec o a ay, he
calcula ed angula dis ibu ion o elas ic sca e ed i ons
ag ees wi h he obse a ion.
TABLE I. Pa ame e s o he op ical model om Re . [39].
44A + 45A +d46A +p
V(MeV) 162.73 102.26 59.14
( m) 1.17 1.05 1.20
a( m) 0.75 0.86 0.72
WV(MeV) 23.85
WD(MeV) 17.23 12.78
i( m) 1.40 1.43 1.32
ai( m) 0.84 0.66 0.66
VSO (MeV) 2.5 7.0 6.2
SO ( m) 1.20 0.75 1.01
aSO ( m) 0.72 0.50 0.75
C( m) 1.30 1.30 1.25
044335-6
SPECTROSCOPY OF 46A BY THE ( ,p) TWO- . . . PHYSICAL REVIEW C 93, 044335 (2016)
(deg)ϑ
0 20 40 60 80 100 120 140 160 180
(mb/s )Ω/dσd
3−
10
2−
10
1−
10
1Pe ey
Va ne p
Li
FIG. 9. Angula dis ibu ion o p o ons om he wo-neu on
ans e eac ion o he g ound s a e o 46A . The solid ed line
ep esen s he calcula ion wi h he pa ame e s shown in Table I.Fo
compa ison we show calcula ions wi h he p o on pa ame iza ion
om [44] (blue, dashed) i on op ical model pa ame e s om [40]
(g een, do -dashed). Spec oscopic one- and wo-nucleon ampli udes
a e aken om he shell-model calcula ion using he SDPF-MU [12]
e ec i e in e ac ion.
As al eady indica ed abo e, he wo-neu on ans e eac-
ion can p oceed wo ways, as a sequen ial ans e h ough
he in e media e (45A +d) sys em o as a simul aneous di ec
ans e o a neu on pai . Bo h p ocesses con ibu e o he
c oss sec ion and hei in e e ence de e mines he o al c oss
sec ion. Fo he calcula ions p esen ed in his pape he
ollowing model was adop ed.
Fo he in e media e 45A nucleus he g ound s a e was
es ablished as Jπ=7/2− om ans e [20] and knockou
eac ions [19]. The i s exci ed s a e is 3/2−[24]. In he
d(44A , p) eac ion, h ee o he L=1 s a es ha e been
obse ed [20]. Based on he shell-model calcula ions in
Sec. IV A wo 3/2−and h ee s a es wi h Jπ=1/2−a e
expec ed o be popula ed s ongly [Fig. 7(a)]. The eac ion
model includes s a es wi h calcula ed spec oscopic ac o s
la ge han 0.05. The hi d 1/2−s a e has no expe imen ally
obse ed equi alen , he e o e he exci a ion ene gy o his
le el is se o 3619 keV, he esul o he shell-model calcula ion
using he SDPF-MU e ec i e in e ac ion [12]. Shell-model
calcula ions also p edic a 5/2−s a e wi h a signi ican
spec oscopic ac o ; o his he ene gy alue o he L=3
candida e om ans e eac ions [20], 4.8 MeV, is adop ed.
The le el lies e y closely o he neu on sepa a ion ene gy
o 45A (Sn=5.169 MeV), he e o e i is supp essed by he
( ,d) eac ion (Q alue −1088 keV) and ans e h ough i
is negligible. Fo he second s ep o he eac ion, he (d,p)
ans e o s a es in 46A ; he ansi ions om all le els in
45A which ha e a subs an ial spec oscopic ac o calcula ed
(C2S>0.05) a e included in he eac ion model. Figu e 10
shows he pa hs included in he calcula ion o he eac ion
o he exci ed 0+
2s a e in 46A . All spec oscopic ampli udes
a e implemen ed wi h hei espec i e phase. The sequen ial
ans e was calcula ed using “pos -pos ” couplings [38,45]; i
o he combina ions o “p io ” and “pos ” couplings a e used,
he magni ude and shape o he di e en ial c oss sec ion a ies
0 keV
+
0
0 keV
-
7/2
542 keV
-
3/2
1416 keV
-
3/2 1876 keV
-
1/2
2510 keV
-
1/2
3619 keV
-
1/2
4800 keV
-
5/2
0 keV
+
0
1554 keV
+
2
3695 keV
+
0
A +
44 A + d
45 A + p
46
FIG. 10. Reac ion model employed in he analysis. Fo he i s
s ep o he sequen ial ( ,d)(d,p) wo-neu on ans e eac ion (blue
a ows) se e al s a es in 45A ha e been conside ed as in e media e
s a es. Expe imen al candida es o he heo e ically calcula ed s a es
wi h spec oscopic ac o s la ge han 0.05 ha e been aken om he
d(44A , p) measu emen o Re . [20]. The igu e shows as an example
he channels o he second s ep o which a spec oscopic ac o la ge
han 0.05 was calcula ed wi h he SDPF-MU in e ac ion [12] o he
exci ed 0+
2s a e. See ex o de ails.
less han i di e en pa ame iza ions o he op ical model a e
used. Fo he di ec one-s ep ans e wo-nucleon ampli udes
(TNA) a e calcula ed. The esul s o he wo componen s and
hei in e e ence is shown in Fig. 11.
Fo all h ee s a es he wo-s ep p ocess domina es he
c oss sec ion, howe e , he in e e ence o one- and wo-s ep
eac ion ampli udes is c i ical o he magni ude and shape o
he di e en ial c oss sec ion. Fo he g ound s a e he di ec
ans e has a la ge in luence han o he exci ed 0+
2s a e,
because wo-s ep eac ions a e inhibi ed by he eac ion Q
alue. In he case o he 2+s a e he di ec ans e alone is
abou wo o de s o magni ude smalle han he sequen ial one,
and he e o e plays a mino ole.
V. DISCUSSION
While shell model p edic s a la ge B(E2; 2+
1→0+
gs) alue
(105 e2 m4calcula ed wi h he SDPF-U in e ac ion) o 46A
han obse ed in Coulomb exci a ion, he wo-neu on ans e
c oss sec ion seems o be well ep esen ed. The calcula ion
wi h he SDPF-U e ec i e in e ac ion [11] gi es a be e
ep esen a ion o he c oss sec ion o he 2+
1s a e when
s anda d op ical model pa ame e s a e used (see Fig. 5). E en
i di e en se s o pa ame e s a e used, he angula di e en ial
c oss sec ion using he ampli udes calcula ed wi h he SDPF-U
e ec i e in e ac ion ep oduces he da a bes .
Expe imen ally he c oss sec ion o he popula ion o
he g ound and i s exci ed 0+s a es a e simila in mag-
ni ude. Neu on emo al eac ions om he g ound s a e
o 46A [18,19] showed ha i is domina ed by 0p-0h
con igu a ions wi h all alence neu ons in he 1 7/2o bi al
( 7/2)8. The s uc u e o he wo 0+s a es is e y di e en .
This can be seen by looking a he con ibu ion o di e en
044335-7
K. NOWAK e al. PHYSICAL REVIEW C 93, 044335 (2016)
(deg)ϑ
0 20 40 60 80 100 120 140 160 180
(
m
b/
s
)
Ω
/d
σ
d
3−
10
2−
10
1−
10
1(a)
ull
di ec only
wo-s ep only
(a)
+
gs
0 keV 0
(deg)ϑ
0 20 40 60 80 100 120 140 160 180
(
m
b/
s
)
Ω
/d
σ
d
3−
10
2−
10
1−
10
1(b)
+
1
1554 keV 2
(deg)ϑ
0 20 40 60 80 100 120 140 160 180
(
m
b/
s
)
Ω
/d
σ
d
3−
10
2−
10
1−
10
1(c)
+
2
3695 keV 0
FIG. 11. Calcula ed di e en ial c oss sec ion o he wo-neu on
ans e o he g ound s a e (a), he 2+
1s a e (b), and he i s exci ed
0+s a e (c) in compa ison wi h expe imen al da a. The SDPF-U
e ec i e in e ac ion was used o calcula e he one- and wo-nucleon
ampli udes. G een do -dashed lines ep esen he esul including only
he di ec wo-nucleon ans e ; blue dashed lines he wo-s ep p ocess
h ough s a es in 45A . The solid ed lines include he in e e ence o
bo h con ibu ions.
neu on pa icle-hole con igu a ions o he o al wa e unc ion
shown in Fig. 12.
The g ound-s a e neu on con igu a ions calcula ed wi h
di e en e ec i e in e ac ions a e e y simila wi h a ound
50% 0p-0hand a ound 20% o 1p-1hand 2p-2hexci a ions
o he 2p3/2o bi al. The con igu a ion o he exci ed 0+
2s a e
is domina ed by pa icle-hole exci a ions. He e a s iking
neu on ampli ude (%)
0
10
20
30
40
50
60 (a)
+
gs
A 0
44
6
)
7/2
(
1
)
3/2
(p
5
)
7/2
(
2
)
3/2
(p
4
)
7/2
(
4
)
3/2
(p
2
)
7/2
(
2
)
5/2
(
4
)
7/2
(
2
)
1/2
(p
4
)
7/2
(
neu on ampli ude (%)
10
20
30
40
50
60 (b)
+
gs
A 0
46
SDPF-MU
SDPF-U
EPQQM
neu on ampli ude (%
)
0
10
20
30
40
50
60 (c)
+
ex
A 0
46
8
)
7/2
(
1
)
3/2
(p
7
)
7/2
(
2
)
3/2
(p
6
)
7/2
(
3
)
3/2
(p
5
)
7/2
(
4
)
3/2
(p
4
)
7/2
(
2
)
5/2
(
6
)
7/2
(
1
)
1/2
(p
1
)
5/2
(
6
)
7/2
(
1
)
1/2
(p
1
)
3/2
(p
6
)
7/2
(
1
)
1/2
(p
2
)
3/2
(p
5
)
7/2
(
2
)
1/2
(p
6
)
7/2
(
FIG. 12. Calcula ed wa e unc ions o he g ound s a e o 44A
(a) and wo 0+s a es in 46A , (b) o he g ound s a e, and (c) o he
i s exci ed 0+s a e. Only he la ges neu on componen s a e shown.
di e ence be ween he h ee e ec i e in e ac ions can be
seen. In calcula ions wi h he SDPF-MU in e ac ion [12]
( 7/2)8 emains he la ges componen ; 2p-2h,3p-3h, and
4p-4hexci a ions o he 2p3/2o bi al con ibu e abou 50%.
The EPQQM calcula ions p edic a s ongly mixed wa e
unc ion, wi h many componen s wi h signi ican ampli udes.
In e es ingly, he 0p-0hcomponen is absen . Fo he SDPF-U
in e ac ion a la ge componen o ( 7/2)6(p3/2)2is domina -
ing he wa e unc ion. The e olu ion in collec i i y below
48Ca was a ibu ed o he enso componen o he nuclea
in e ac ion [12]. The monopole componen o he enso
in e ac ion causes a educ ion o he spli ing be ween he
1 7/2and 2p3/2o bi als esul ing in a educ ion o he N=28
shell gap. In 46A jus wo p o ons below he doubly magic
48Ca he si ua ion is unclea . The s onges among he T=0
c oss-shell monopole e ms is he a ac i e ν1 7/2-π1d3/2,
he e o e a educed occupa ion o he p o on 1d3/2o bi al
will cause ising o he ν1 7/2wi h espec o he 2p3/2
o bi al compa ed o 48Ca and educe he N=28 shell gap.
We ha e pe o med calcula ions wi hou he c oss-shell enso
044335-8