PHYSICAL REVIEW C 87, 054617 (2013)
S uc u e o 8B om elas ic and inelas ic 7Be +psca e ing
J. P. Mi chell,1,2,3,*G. V. Rogache ,1,4,†E. D. Johnson,1L. T. Baby,1K. W. Kempe ,1A. M. Mo o,5P. Peplowski,1,6
A. S. Volya,1and I. Wiedenh¨
o e 1
1Depa men o Physics, Flo ida S a e Uni e si y, Tallahassee, Flo ida 32306, USA
2Depa men o de As onom´
ıa y As o ´
ısica, Pon i icia Uni e sidad Ca ´
olica de Chile, Vicu˜
na Mackenna 4860, Macul, San iago, Chile
3A gelande Ins i u ¨
u As onomie, Uni e si ¨
a Bonn, Au dem H¨
ugel 71, 53121 Bonn, Ge many
4Na ional Supe conduc ing Cyclo on Labo a o y, Michigan S a e Uni e si y, Eas Lansing, Michigan 48824, USA
5Depa men o Physics, Uni e si y o Se ille, Spain
6Johns Hopkins Uni e si y Applied Physics Labo a o y, Lau el, Ma yland 20723, USA
(Recei ed 1 Ma ch 2013; published 23 May 2013)
Backg ound: De ailed expe imen al knowledge o he le el s uc u e o ligh weakly bound nuclei is necessa y
o guide he de elopmen o new heo e ical app oaches ha combine nuclea s uc u e wi h eac ion dynamics.
Pu pose: The esonan s uc u e o 8B is s udied in his wo k.
Me hod: Exci a ion unc ions o elas ic and inelas ic 7Be +psca e ing we e measu ed using a 7Be a e iso ope
beam. Exci a ion ene gies anging be ween 1.6 and 3.4 MeV we e in es iga ed. An R-ma ix analysis o he
exci a ion unc ions was pe o med.
Resul s: New low-lying esonances a 1.9, 2.54, and 3.3 MeV in 8B a e epo ed wi h spin-pa i y assignmen
0+,2
+,and1
+, espec i ely. Compa ison o he ime-dependen con inuum shell (TDCSM) model and ab ini io
no-co e shell model/ esona ing-g oup me hod (NCSM/RGM) calcula ions is pe o med. This wo k is a mo e
de ailed analysis o he da a i s published as a Rapid Communica ion.J. P. Mi chell, G. V. Rogache , E. D.
Johnson, L. T. Baby, K. W. Kempe e al.,[Phys.Re .C82, 011601(R) (2010)].
Conclusions: Iden i ica ion o he 0+,2
+,1
+s a es ha we e p edic ed by some models a ela i ely low
ene gy bu ne e obse ed expe imen ally is an impo an s ep owa d unde s anding he s uc u e o 8B. Thei
iden i ica ion was aided by ha ing bo h elas ic and inelas ic sca e ing da a. Di ec compa ison o he c oss
sec ions and phase shi s p edic ed by he TDCSM and ab ini io no-co e shell model coupled wi h he esona ing
g oup me hod is o pa icula in e es and p o ides a good es o hese heo e ical app oaches.
DOI: 10.1103/PhysRe C.87.054617 PACS numbe (s): 21.10.−k, 24.30.− , 25.60.−
I. INTRODUCTION
One o he main goals o mode n nuclea heo y is o
combine he nuclea eac ion models wi h nuclea s uc u e
calcula ions o p o ide he uni ied amewo k ha allows he
calcula ion o le el spec oscopy and eac ion c oss sec ions
s a ing om he same es ablished nuclea Hamil onian.
Se e al heo e ical app oaches ha e been sugges ed o ad ance
his goal. B oadly, wo majo di ec ions can be iden i ied,
phenomenological and ab ini io. The i s one uses he
well-es ablished shell-model Hamil onian and couples i wi h
he co esponding eac ion channels. The ecoil co ec ed
con inuum shell model (RCCSM) [1] and he ime-dependen
con inuum shell model (TDCSM) [2] a e examples o hese
app oaches. The second majo di ec ion is he a emp o
calcula e he c oss sec ion s a ing om bo h ba e nucleon-
nucleon o ces and h ee-nucleon o ces. One example o
his app oach is he no-co e shell model combined wi h
he esona ing-g oup me hod (NCSM/RGM) [3]. The e y
a ac i e ea u e o hese de elopmen s is ha he exci a ion
unc ions o he esonance eac ions, such as elas ic and
inelas ic nucleon sca e ing, (p,n) and (p,α) eac ions, e c.,
can, in p inciple, be calcula ed and di ec ly compa ed o he
*[email p o ec ed]
†[email p o ec ed]
expe imen al da a. This is in addi ion o all known s uc u e
da a. Howe e , his compa ison is no as s aigh o wa d as i
may appea . Because o model space unca ion, limi a ions
om nucleonic deg ees o eedom and nume ical complexi y,
i is na u al o expec ha he nuclea spec um a he low
exci a ion ene gy is ep oduced be e han he spec um o
he highe lying exci ed s a es by any model. The e o e, i is
desi able o e i y he heo e ical p edic ions in he egion o
low exci a ion ene gy i s and weakly bound nuclei p o ide
a good es o hese models. He e, he con inuum appea s
a low ene gy, hus pe mi ing examina ion o he s uc u e-
eac ion ansi ion. Mo eo e , because o unca ion o he
model space, pa ame e s o models a e adjus ed o he well
known spec um o s able nuclei, esul ing in unsu p isingly
easonable ag eemen wi h he expe imen al da a o hese
nuclei. The be e es is p o ided by exo ic, weakly bound
nuclei. The neu on de icien bo on iso ope, 8B, is o pa icula
in e es . I s p o on sepa a ion ene gy is only 137 keV and
all o i s exci ed s a es a e in he con inuum, as can be seen
in i s le el s uc u e in Fig. 1. In addi ion, his nucleus has
been a subjec o nume ous heo e ical s udies. In he ecen
ab ini io NCSM/RGM analysis [3]o 8B he p o on +7Be
elas ic sca e ing phase shi s as well as he c oss sec ion o
he 7Be(p,p) and he 7Be(p, γ ) eac ions we e calcula ed.
Di ec compa ison o he expe imen al esul s on he 7Be(p,p)
and 7Be(p,p) eac ions wi h hese calcula ions and also he
analysis o he expe imen al da a using he TDCSM app oach
is he main objec i e o his wo k.
054617-1
0556-2813/2013/87(5)/054617(13) ©2013 Ame ican Physical Socie y
J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013)
Li
8
0.98 +
1
2.26 +
3
3.21 +
1
+
2
B
8
+
2
0.77 +
1
2.32 +
3
3.2 -
2
(a) (b)
3.3 +
1
2.55 +
2
1.9 +
0Li+n
7
2.03
Li*+n
7
2.51
Be+p
7
0.14
Be*+p
7
0.57
FIG. 1. (Colo online) The le el schemes o 8B (a) and i s mi o
8Li (b). S a es om ou p e ious wo k [8] a e in ed. The dashed-
do ed line indica es ha he s a e is en a i e.
The exci a ion unc ion o 7Be +phas been p e iously
measu ed in se e al expe imen s [4–7]. Howe e , he hick
a ge in e se kinema ics expe imen al me hod used in all
o hese measu emen s did no allow o sepa a ion be ween
elas ic and inelas ic sca e ing excep o he da a om [6],
whe e measu emen s we e pe o med a ene gies below he
inelas ic sca e ing h eshold. In Re . [7], an a emp has been
made o use γ-p o on coincidence o iden i y he inelas ic
sca e ing e en s, howe e , he elas ic exci a ion unc ion s ill
appea s o be con amina ed wi h inelas ic e en s (see Sec. II
o addi ional commen s). The expe imen desc ibed he e does
no su e om such de iciency because he in e media e a ge
hickness app oach has been applied. This app oach allowed
o measu emen o a signi ican ac ion o he 7Be +p
exci a ion unc ion, while simul aneously de ec ing he 7Be
ecoil in coincidence wi h p o ons in o de o dis inguish
be ween elas ic and inelas ic sca e ing e en s kinema ically.
The e o e, we did no use expe imen al da a om he p e ious
highe ene gy measu emen s [4,5,7] in he analysis bu we
included he low ene gy 7Be(p,p) elas ic sca e ing da a
be ween 0.3 and 0.75 MeV measu ed in Re . [6]. The subse
o he da a epo ed he e was i s published in [8].
This pape con ains a mo e de ailed desc ip ion o he
expe imen al esul s and also ex ends he p e iously measu ed
exci a ion ene gy egion o highe ene gies. A desc ip ion
o he expe imen al me hod ha was used o measu e he
exci a ion unc ions o 7Be +pelas ic and inelas ic sca e ing
be ween 1.6 o 3.4 MeV is gi en in Sec. II. The analysis o
he expe imen al da a was pe o med using he mul ichannel
mul ile el R-ma ix app oach and is desc ibed in Sec. III.
Sec ion IV con ains a discussion o his inding and i s
consis ency wi h he p e ious expe imen al da a on he 8B and
8Li nuclei and discusses whe he i is possible o explain he
new expe imen al da a wi hou in oducing he new esonances
in 8B. Analysis o he new expe imen al da a in he amewo k
o he ime-dependen con inuum shell model is p esen ed in
Sec. V. De ailed compa ison o he phase shi s ex ac ed
om he analysis o he p+7Be expe imen al da a o he
p edic ions o he ab ini io NCSM/RGM model is gi en in
Sec. IV. Conclusions a e gi en in Sec. VII.
II. EXPERIMENT
The exci a ion unc ion o p+7Be elas ic and inelas ic
sca e ing be ween 1.6 and 3.4 MeV in he cen e o mass
sys em (c.m.s.) was measu ed a he John D. Fox Supe con-
duc ing Accele a o Labo a o y a Flo ida S a e Uni e si y. A
adioac i e beam o 7Be was p oduced using he 1H(7Li,7Be)n
eac ion. A p ima y 7Li beam was accele a ed by a 9 MV
Supe FN Tandem Van de G aa accele a o ollowed by
a LINAC boos e . The p ima y a ge was a 4 cm long
hyd ogen gas cell wi h 2.5 μm Ha a en ance and exi
windows. The gas cell was cooled by liquid ni ogen and
had a gas p essu e o 390 mBa . The in- ligh p oduc ion
a e iso ope beam acili y RESOLUT was used o sepa a e
7Be om o he eac ion p oduc s and he p ima y beam.
RESOLUT is a se o wo supe conduc ing solenoids, dipole
and quad upole magne s and a supe conduc ing esona o .
Th ee 7Be beam ene gies we e used in his expe imen : 27.2,
22.0, and 18.5 MeV. The ypical in ensi y o he 7Be beam
was 105pps. The composi ion o he beam was 70% 7Be
and 30% 7Li con aminan . Diagnos ics o he seconda y beam
we e pe o med using a posi ion sensi i e mic ochannel pla e
de ec o ins alled be ween he dipole magne and he second
solenoid (2.7 m be o e he C2H4 a ge ) and he E-E
elescope consis ing o an ioniza ion chambe (used as E
de ec o ) backed by a 50 ×50 mm216 ×16 silicon s ip
de ec o posi ioned 66 cm downs eam om he seconda y
a ge .
A ske ch o he expe imen al se up is shown in Fig. 2.A
solid polye hylene (C2H4) a ge o hickness op imized o he
gi en beam ene gy (see desc ip ion below) was used. A se o
h ee annula mic on semiconduc o silicon s ip de ec o s (S2
design) o he p o on ecoils we e posi ioned 5, 6, and 7 cm
downs eam om he a ge , espec i ely. Ano he S2 de ec o
o he 7Be ecoils was posi ioned 24.5 cm om he a ge . The
Li
7
Be,
7
RESOLUT
Ta ge
2
CH
5 cm
6 cm
7 cm
24.5 cm
Si Ion De ec o
BEAM
21.7 MeV
p
p
Be
7
Be
7
2
1.96 mg/cm
FIG. 2. (Colo online) The expe imen al se up. The 7Be beam was
deli e ed by he RESOLUT acili y (on he le ). The C2H4 a ge s
o a ious hicknesses we e used. The p o ons we e de ec ed in an
a ay o h ee mic on semiconduc o s S2 de ec o s and he 7Be’s we e
measu ed in an S2 downs eam. (The inse p o ides a mo e de ailed
iew o he de ec o a angemen .)
054617-2
STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013)
S2 de ec o has annula geome y and consis s o 16 segmen s
and a side o ings ha allow o he sca e ing angle o he
p oduc s o be de e mined. The i s in he se o h ee p o on
de ec o s was a E de ec o o 65 μm, while he o he wo
and he 7Be de ec o we e 500 μm each.
The a ge hickness was op imized o maximum ene gy
losseso he7Be ions in he a ge while ensu ing ha all
7Be ecoils make i ou o he a ge wi h enough kine ic
ene gy le o be de ec ed in he downs eam S2 de ec o .
Kinema ic coincidence be ween p o ons in he a ay o h ee
S2 de ec o s and he 7Be ecoils in he downs eam S2 de ec o
we e hen used o iden i y he sca e ing e en s. The 65 μm
E S2 de ec o was used only in he ini ial s age o he
expe imen o e i y ha kinema ic coincidence be ween ligh
and hea y ecoils a e enough o clean iden i ica ion o he
p+7Be elas ic and inelas ic sca e ing e en s. This de ec o
was hen emo ed. Measu emen s a he beam ene gies o 22
and 18.5 MeV we e pe o med wi hou he 65 μm de ec o ,
while ha o he 27.2 MeV ene gy included he 65 μm
E de ec o . Time be ween he e en s in he p o on and
7Be de ec o s was measu ed wi h esolu ion o abou 3
ns in o de o elimina e andom coincidence backg ound.
Elas ic and inelas ic sca e ing p ocesses can be dis inguished,
because comple e kinema ics o he e en s a e measu ed. Mo e
speci ically, e en s ha ha e wo equal ene gy p o ons would
ha e di e en ene gy o 7Be ecoils i hey o igina e om
di e en (elas ic/inelas ic) p ocesses. This is due o di e en
eac ion Q- alue and kinema ics, and also e ec i e a ge
hicknesses (and hence ene gy losses) expe ienced by he
hea y ecoils. The inelas ic e en s ha p oduce p o ons wi h
he same kine ic ene gy as elas ic e en s ake place ea lie
(ups eam) in he a ge , whe e a nega i e eac ion Q- alue
is compensa ed by he highe ene gy o he 7Be p ojec ile.
(See also Re . [9] o de ails on his expe imen al echnique.)
The 2D sca e plo o he kinema ic coincidence be ween
p o ons and 7Be is shown in Fig. 3. The kinema ic loci which
co espond o elas ic and inelas ic sca e ing p ocesses a e
labeled and ou lined wi h con ou s. Kinema ically comple e
measu emen s allow o unambiguous iden i ica ion o he
e en s and do no equi e pa icle iden i ica ion. We con-
side ed he possibili y o nonbina y p ocesses con ibu ion.
Fo example, exci a ion o 7Be on hyd ogen o ene gies
Be Ene gy (MeV)
7
67891011
P o on Ene gy (MeV)
4
4.5
5
5.5
6
6.5
7
7.5
Be(p,p)
7
Be(p,p’)
7
FIG. 3. (Colo online) Sca e plo o kinema ic coincidence
be ween p o ons and 7Be ions. Regions which co espond o elas ic
and inelas ic sca e ing a e labeled.
abo e α+3He decay h eshold would esul in h ee pa icle
con inuum p+α+3He wi h αand 3He possibly p oducing
signals in he wo S2 de ec o s simul aneously. De ailed Mon e
Ca lo simula ion ha akes in o accoun he geome y o he
expe imen and o he expe imen al pa ame e s was pe o med.
I was ound ha α+3He coincidence would esul in a
2D ene gy- s-ene gy co ela ion ha is e y di e en om
well de ined ene gy- s-ene gy co ela ion o p+7Be bina y
p ocess.
Polye hylene a ge hicknesses used in his expe imen
we e 2.6, 2.5, and 1.5 mg/cm2 o he 27.2, 22, and 18.5
MeV beam ene gies, espec i ely. In addi ion, a sepa a e un
a 18.5 MeV o 7Be beam ene gy was pe o med wi h a sligh ly
hicke (2 mg/cm2) a ge , o ex end he measu ed exci a ion
unc ion o lowe ene gies wi hou changing he ene gy o he
beam. Unde his condi ion coincidence be ween he highes
ene gy p o ons and he 7Be ecoils a e los ( he hea y ecoils
p oduced a he beginning o he a ge do no make i h ough).
Only he lowe ene gy pa o his spec um was used in he
analysis.
Figu e 4shows exci a ion unc ions o esonance elas ic
and inelas ic sca e ing o 7Be +pmeasu ed in ou di e en
uns. Ene gy bins a e 30 keV. The e ex up iangles
co espond o he 7Be una 18.5MeVwi h he2mg/cm2
a ge , he squa es a e 18.5 MeV 7Be wi h he 1.5 mg/cm2
a ge da a, he ci cles a e 22 MeV 7Be wi h 2.5 mg/cm2da a,
he e ex down iangles a e om he 27.2 MeV un wi h
he 2.6 mg/cm3 a ge , and in all cases, he illed ma ke s
a e o elas ic sca e ing and he hollow ma ke s o inelas ic
sca e ing. The angula esolu ion o he expe imen al se up,
as de e mined by he pi ch o he ings in he S2 de ec o ,
dis ance om he a ge and he size o he beam spo on
he seconda y a ge , was 1.25◦. We used binning o 4◦in
he labo a o y ame, combining e en s eco ded by 12 ings
o he S2 de ec o in o one spec um. Exci a ion unc ions
(MeV)
exc
E
1.5 2 2.5 3 3.5
(mb/s )Ω
/dσd
0
20
40
60
80
100
120
o
Be(p,p) 148
7
o
Be(p,p’) 146
7
FIG. 4. The exci a ion unc ion o 7Be +pelas ic and inelas ic
sca e ing a 148 ±4◦and 146 ±4◦deg ees, espec i ely. Resul s
om uns a h ee di e en ene gies o 7Be beam a e shown. The
squa es co espond o he un a 18.5 MeV o 7Be wi h a 1.5 mg/cm2
a ge , he e ex up iangles a e da a aken a 18.5 MeV wi h he 2
mg/cm2 a ge , he ci cles a e he 22 MeV da a wi h he 2.5 mg/cm2
a ge , and he e ex down iangles a e om he 27.2 MeV un
wi h a 2.6 mg/cm2wi h solid ma ke s ep esen ing 7Be +pelas ic
sca e ing and open ma ke s he inelas ic p(7Be,p)7Be(1
2
−) sca e ing
exci a ion unc ions.
054617-3
J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013)
Exci a ion Ene gy (MeV)
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1
(mb/s )Ω/dσd
20
30
40
50
60
70
80
90
FIG. 5. (Colo online) The exci a ion unc ion o 7Li +pelas ic
sca e ing a 148 ±4◦is shown wi h solid ci cles. This exci a ion
unc ion was measu ed simul aneously wi h 7Be +p( he a e iso ope
beam composi ion was 70% 7Be and 30% 7Li) and used o absolu e
no maliza ion. The same exci a ion unc ion om [10,11]isshown
o compa ison wi h open ci cles.
a h ee sca e ing angles we e ob ained his way. These
angles a e 148 ±4◦, 140 ±4◦132 ±4◦in he c.m.s. o
elas ic sca e ing and 146 ±4◦, 138 ±4◦130 ±4◦ o inelas ic
sca e ing. Absolu e no maliza ion o he c oss sec ion was
pe o med using he known exci a ion unc ions o 7Li +p
elas ic sca e ing. These exci a ion unc ions we e ex ac ed
om he expe imen al da a using he same p ocedu e as o
he 7Be +pelas ic sca e ing, he e o e, by no malizing he
7Li +pda a o he known 7Li +pc oss sec ion and aking
in o accoun he a io o he 7Be ions o he 7Li ions in
he seconda y beam (as measu ed by he 0 deg ee ioniza ion
chambe and silicon s ip de ec o ), accu a e no maliza ion is
achie ed. No e ha his no maliza ion p ocedu e au oma ically
akes in o accoun he e iciency o he expe imen al se up. A
sample o he 7Li +pexci a ion unc ion measu ed in his
expe imen is shown in Fig. 5(solid ci cles) and compa ed
o he expe imen al da a om [10,11]. Exci a ion unc ions
ex ac ed om ou da a ag ee well wi h he di e en ial c oss
sec ion o elas ic and inelas ic sca e ing o 7Be +pmeasu ed
a se e al ene gies o 7Be using a hin a ge app oach and
epo ed by G ei e e al.,[12]. The exci a ion unc ions o
Yamaguchi e al. [7] howe e , di e om ou s, especially in
he inelas ic channel whe e hey ound he exci a ion unc ion
o be ai ly la ac oss hei en i e ene gy ange measu ed, while
ou esul s ha e a la ge peak a an exci a ion ene gy o 2.5 MeV.
This disc epancy may be ela ed o he backg ound in he NaI
scin illa o de ec o s used in Re . [7] ha could ha e p e en ed
a clean γ-p o on coincidence spec um o be ex ac ed.
III. R-MATRIX ANALYSIS
The exci a ion unc ions o elas ic 1H(7Be,p)7Be(g.s.) and
inelas ic 1H(7Be,p)7Be(1/2−; 0.43 MeV) sca e ing we e
analyzed using a wo channel, mul ile el R-ma ix app oach.
The na u al s a ing poin o he analysis is o in oduce
only he exci ed s a es o 8B ha we e iden i ied in p e ious
expe imen s [13], he 1+a 0.77 MeV, he 3+a 2.32 MeV,
and he b oad 2−a ∼3 MeV. These h ee s a es ep oduce
he exci a ion unc ion o p+7Be elas ic sca e ing be ween
(mb/s )Ω/dσd
20
40
60
80
100
120
140
o
Be(p,p) 148
7
(a)
B Exci a ion Ene gy (MeV)
8
1.6 1.8 2 2.2 2.4 2.6 2.8
(mb/s )Ω/dσd
0
5
10
15
20
25
30
35
40 o
Be(p,p’) 146
7(b)
a 2.50 MeV
+
1
a 3.0 MeV
+
1
-
and 2
+
Only 3
added
-
and 1
+
1
coupled channel
FIG. 6. (Colo online) R-ma ix i o he elas ic and inelas ic
7Be +psca e ing wi h known 3+and 2−s a es, a second exci ed
1+seen in 8Li, and he “backg ound” 1−s a e in oduced a highe
ene gy. The solid cu e co esponds o only 3+and 2−s a es a
2.3 and 3.5 MeV, espec i ely. The ed sho -dashed cu e includes
he con ibu ion o he highe lying 1+s a es assumed a 3.0 MeV.
Dash-do ed pu ple cu e shows he 1+s a e shi ed o 2.5 MeV and
he long dashed g een cu e also includes he 1−s a e in oduced a
5MeV.
0.5 and 3.5 MeV easonably well, as shown in Fig. 6(a) by
he solid line. Howe e , i is no possible o explain 30 mb/s
inelas ic c oss sec ion a 2.5 MeV i only known s a es in
8B a e conside ed [Fig. 6(b)]. This ailu e can be unde s ood
om he ollowing simple conside a ions. The i s exci ed
1+s a e a 0.77 MeV is oo na ow o ha e any signi ican
impac on he exci a ion unc ions a ene gies abo e 1.6 MeV.
The second exci ed s a e, 3+a 2.32 MeV, can only decay o
he 3/2−g ound s a e o 7Be because decay o he 1/2− i s
exci ed s a e equi es angula momen um o =3. The e o e,
e en i he co esponding educed wid h is la ge he inelas ic
pa ial p o on wid h, p=2P(kR)γ2, would s ill be small
compa ed o he elas ic pa ial p o on wid h due o a small
pene abili y ac o o high angula momen um decay. Hence,
he c oss sec ion o popula ion o he i s exci ed s a e in 7Be
due o he 3+ esonance in 8B, de e mined by he pp/2
o
a io, is small. The same is ue o he b oad 2−s a e in 8B
a ≈3 MeV as i can only decay o he i s exci ed s a e
in 7Be wi h angula momen um =2 while decay o he
g.s. p oceeds wi h =0. Figu e 6shows he esul s o an
R-ma ix calcula ion wi h only p e iously known 1+,3
+, and
2−s a es a 0.77, 2.32, and 3.7 MeV wi h educed wid h
pa ame e s e alua ed using he TDCSM (mo e de ails on
TDCSM calcula ions a e gi en in Sec. V) and known o al
wid hs o hese s a es. (Exci a ion ene gy and wid h o he
2−we e adjus ed sligh ly o p oduce a be e i .) I is clea
ha while he elas ic sca e ing da a is well ep oduced, he
inelas ic sca e ing da a canno be explained by he known
s a es.
Based on he le el scheme o 8Li (Fig. 1) i is na u al
o in oduce he second 1+s a e in 8B a an exci a ion
ene gy a ound 3 MeV. Reduced wid hs o his s a e we e
chosen acco ding o TDCSM calcula ions ca ied ou wi h
054617-4
STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013)
he Cohen-Ku a h CKI in e ac ion [14]. I was e i ied ha
hese educed wid hs ep oduce he known wid h o his s a e
in 8Li (∼1 MeV). The sho dashed cu e ( ed) in Fig. 6
shows he e ec o he 1+s a e on he i . While he elas ic
exci a ion unc ion is i ed well, he inelas ic c oss sec ion is
s ill unde es ima ed. E en i his s a e is shi ed o 2.5 MeV,
whe e inelas ic sca e ing has i s maximum c oss sec ion,
i s ill unde es ima es he da a [dash-do ed (blue) cu e in
Fig. 6]. Finally, in an a emp o inc ease he inelas ic c oss
sec ion wi hou using new s a es below 3 MeV we in oduced
a1
−“backg ound” s a e. This s a e can decay o he i s
exci ed s a e o 7Be wi h =0, he e o e i may con ibu e
signi ican ly o he inelas ic c oss sec ion. The educed wid hs
o he 1−s a e we e e alua ed using he shell model, and he
s a e was in oduced a 5 MeV. As expec ed, he 1−backg ound
s a e inc eased he inelas ic c oss sec ion o e all (long-dashed
g een cu e in Fig. 6). Bu e en wi h his s a e included he
inelas ic c oss sec ion canno be ep oduced.
The ab ini io calcula ions o 8B[3,15–17] p edic h ee
mo e posi i e pa i y (p-shell) s a es a low exci a ion ene gy.
Thesea e he0
+1,1
+2, and 2+2. The exci a ion ene gies
o hese s a es a y be ween 2 and 6 MeV depending on
he h ee-body o ce pa ame iza ion and he speci ics o
he calcula ions. Simila esul s a e ob ained in shell model
calcula ions (exci a ion ene gies o hese “missing” s a es
a y be ween 2 and 6 MeV in he shell model as well,
depending on he esidual in e ac ion used). The e o e, i is
na u al o in oduce hese s a es in an a emp o ep oduce
he la ge inelas ic sca e ing c oss sec ion. The 1+2s a e has
al eady been in oduced. Tha lea es only he 0+1and 2+2.
In oduc ion o a new 2+s a e placed a 2.5 MeV, ep oduces
bo h he magni ude and angula dependence o he obse ed
peak in he inelas ic c oss sec ion while keeping he elas ic
exci a ion unc ion in ag eemen wi h he expe imen al da a
(blue dashed cu e in Fig. 7). Howe e , e en wi h his new
s a e he c oss sec ion o inelas ic sca e ing below 2.3 MeV
is s ill unde es ima ed. The 2+s a e should ha e a ela i ely
0
20
40
60
80
100
120
140
dσ/dΩ (mb/s )
7Be(p,p)
148°
7Be(p,p)
140°
7Be(p,p)
132°
7Be(p,p’)
146°
7Be(p,p’)
138°
7Be(p,p’)
130°
7Be(p,p)
148°
7Be(p,p)
140°
7Be(p,p)
132°
7Be(p,p’)
146°
7Be(p,p’)
138°
7Be(p,p’)
130°
7Be(p,p)
148°
7Be(p,p)
140°
7Be(p,p)
132°
7Be(p,p’)
146°
7Be(p,p’)
138°
7Be(p,p’)
130°
0
5
10
15
20
25
30
35
40
1.5 2 2.5
dσ/dΩ (mb/s )
Eexc (MeV)
7Be(p,p)
148°
7Be(p,p)
140°
7Be(p,p)
132°
7Be(p,p’)
146°
7Be(p,p’)
138°
7Be(p,p’)
130°
1.5 2 2.5
Eexc (MeV)
7Be(p,p)
148°
7Be(p,p)
140°
7Be(p,p)
132°
7Be(p,p’)
146°
7Be(p,p’)
138°
7Be(p,p’)
130°
1.5 2 2.5 3 3.5
Eexc (MeV)
7Be(p,p)
148°
(a) 7Be(p,p)
140°
(b) 7Be(p,p)
132°
(c)
7Be(p,p’)
146°
(d)
7Be(p,p’)
138°
(e)
7Be(p,p’)
130°
( )
FIG. 7. (Colo online) Elas ic and inelas ic exci a ion unc ions
o 7Be +psca e ing. The blue dashed cu e is a i wi h he
p e iously known 1+and 3+s a es as well as a 2+a 2.54 MeV
o ep oduce he peak in he inelas ic da a. The 2−and 1−phase shi s
we e a ied. The ed solid cu e is he bes i wi h he R-ma ix
pa ame e s om Table I.
100
150
200
250
300
dσ/dΩ (mb/s )
7Be(p,p) 123.6°
7Be(p,p) 163.8°
(a)
(b)
50
100
150
200
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
dσ/dΩ (mb/s )
Eexc (MeV)
7Be(p,p) 123.6°
7Be(p,p) 163.8°
(a)
(b)
FIG. 8. (Colo online) Exci a ion unc ion o 7Be +pelas ic
sca e ing a low ene gies om [6] a 123.6◦(a) and 163.8◦(b).
The bes i is a solid black cu e. The calcula ed c oss sec ion was
con olu ed o accoun o 30 keV expe imen al esolu ion epo ed
in Re . [6]. Sys ema ic e o s we e included in o he e o ba s. The
dashed ed cu e is he R-ma ix i wi h he ab ini io 1+phase shi s
om [17] o he1
+
1s a e (channel spins 1 and 2 con ibu e abou
equally).
small wid h (270 ±40 keV) o i he obse ed peak-like
s uc u e in he inelas ic exci a ion unc ion a 2.5 MeV and
i s in luence below 2.3 MeV is small. In oducing he 0+s a e
a an exci a ion ene gy o 1.9±0.1 MeV wi h a wid h o
530+600
−100 keV allows he inelas ic sca e ing da a o be i below
2.3 MeV wi hou des oying he i o he elas ic sca e ing da a
(solid ed line in Fig. 7). I was e i ied ha a 1+spin-pa i y
assignmen o his s a e would esul in sligh ly wo se χ2,bu
mos impo an ly he elas ic educed wid h ampli ude o he
1+a 2.0 MeV would ha e o be e y small (∼0.1) o i he
elas ic and inelas ic da a simul aneously. This would make i
impossible o ep oduce he o al 7Li(n,n) c oss sec ion, o
which he 1+
2s a e plays dominan ole (see Fig. 10).
The low-ene gy da a om [6] we e used (Fig. 8) o p o ide
addi ional cons ain on he beha io o he phase shi s a low
ene gy. I p o ed o be pa icula ly impo an o he nega i e
pa i y phase shi s. We used he p edic ions o he ab ini io
calcula ions [17] o he2
−and 1−phase shi s as he s a ing
poin , bu he bes i was achie ed wi h he nega i e pa i y
phase shi s di e en om [17]. (I is discussed in mo e de ail
in Sec. VI.) The bes i ha included he low ene gy da a om
[6] and da a om his expe imen was achie ed using R-ma ix
pa ame e s gi en in Table I. The no malized χ2 o he bes i
was 0.89. S a es shown in pa en hesis in Table Ia e he b oad
“backg ound” s a es ha a e used in R-ma ix o malism o
p oduce he desi ed beha io o he co esponding phase shi s.
IV. NEW STATES IN LIGHT OF PREVIOUS
EXPERIMENTAL DATA
He e we s udy/assess i he low lying 0+
1and 2+
2s a es
a e consis en wi h he a ailable expe imen al da a on 8B and
8Li nuclei. The s uc u e o 8B has been ex ensi ely s udied
in p+7Be esonance elas ic sca e ing expe imen s [4–7]. In
Re . [6] he7Be +pexci a ion unc ion o elas ic sca e ing
was measu ed in he c.m. ene gy ange om 0.3 o 0.75 MeV.
054617-5

J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013)
TABLE I. Pa ame e s o esonances in 8B om he R-ma ix bes i . S a es in pa en hesis a e ou side o he measu ed exci a ion ene gy
ange bu p o ide essen ial “backg ound” h ough low ene gy ails. The ene gy eigen alue and he educed wid hs ampli udes o 7Be(p,p)
and 7Be(p,p)7Be(1/2−) sca e ing wi h channel spins 1 and 2 o he o me and 0 and 1 o he la e used in he R-ma ix i a e shown in
columns 6–10. We used 4.20 m as he channel adius o bo h he elas ic S=1,2 and inelas ic S=0,1 channels.
JπEex (MeV)  o (MeV) p(MeV) p(MeV) Eeigen γel S=1γel S=2γ1/2−S=0γ1/2−S=1
2+0–––−0.657 −0.793 −0.531 0.000 0.430
1+0.768(4) 0.027(6) 0.026(6) 0.001 0.276 0.718 0.130 −0.875 −0.335
0+1.9(1) 0.53+0.6
−0.10.06+0.3
−0.02 0.47+0.4
−0.12.102 0.353 0.000 0.000 1.303
3+2.31(2) 0.33(3) 0.33(3) 0.0 2.305 0.000 0.607 0.000 0.000
2+2.50(4) 0.27(4) 0.05 0.22 2.471 0.224 0.000 0.000 0.534
1+3.3(2) 3.2(9) 2.8 0.4 4.740 0.937 −1.179 0.029 0.664
(1−) – – – – 5.548 1.664 0.000 0.000 2.827
(2−) – – – – 12.059 0.000 3.15 0.000 0.000
The new s a es a e a 1.9, 2.5, and 3.3 MeV exci a ion ene gies
and hei in luence on he low ene gy pa o he exci a ion
unc ion is e y small. In gene al, he i o he elas ic sca e ing
da a does no equi e he low lying 0+
1and 2+
2s a es. The
expe imen al da a in Re s. [4,5,7] we e i ed wi h only 1+and
3+s a es a 0.77 and 2.32 MeV and a 2−s a e a ∼3MeV.
(P esence o a 1+s a e a ∼3 MeV was sugges ed in Re . [4].)
Howe e , he new s a es ha e li le in luence on he exci a ion
unc ion o elas ic sca e ing. The e o e, he absence o hese
s a es in he R-ma ix analysis o he elas ic sca e ing da a
canno be used as an a gumen agains hese s a es. I is
in e es ing o no e ha in all h ee p e ious measu emen s
[4,5,7] he c oss sec ion a he esonance ene gy o he 3+
s a e was measu ed o be ≈190 mb/s a 180◦.TheR-ma ix
i o ou elas ic sca e ing da a p oduces a lowe c oss sec ion
a 180◦,≈160 mb/s . This is no su p ising because he
expe imen al echnique used in p e ious measu emen s did
no sepa a e elas ic om inelas ic sca e ing. P o ons om
inelas ic sca e ing we e con ibu ing o he “elas ic” exci a ion
unc ions which esul ed in highe c oss sec ion alues o he
measu ed “elas ic” exci a ion unc ions.
Mo e expe imen al in o ma ion is a ailable ega ding he
s uc u e o he mi o nucleus, 8Li. One and wo neu on
ans e eac ions, 7Li(d,p)[18] and 6Li( ,p)[19]we eused
o popula e s a es in 8Li. I is e y unlikely ha bound s a es in
8Li could ha e been missed in hese expe imen s. The e o e,
he 0+
1and 2+
2s a es a e p obably abo e he neu on decay
h eshold (2.03 MeV) in 8Li.
The exci a ion unc ion o he 7Li(n,γ )8Li eac ion was
measu ed a low c.m. ene gies (up o 1 MeV) [20–23]. Only
he 3+s a e a 2.25 MeV (0.22 MeV abo e he neu on decay
h eshold) has been obse ed. In p inciple, lack o e idence
o he 0+and he 2+s a es in he 7Li(n,γ )8Li exci a ion
unc ion canno be conside ed as a decisi e a gumen agains
hei p esence. I he pa ial γwid h (γ) o hese s a es is
small hen hey can be ha d o iden i y wi hin he backg ound
om di ec neu on cap u e and esonance cap u e due o he
3+s a e. Figu e 9shows TDCSM calcula ions o he (n,γ )
exci a ion unc ion wi h he known 3+and 1+s a es and he
new 0+and 2+s a es a 2.4 and 2.5 MeV ( op panel) and a
2.8 and 3.3 MeV (bo om panel). I is clea om his igu e
ha obse a ion o he new s a es in he 7Li(n,γ ) eac ion is
di icul .
Resonances in 8Li a exci a ion ene gies o up o 9.0 MeV
ha e been s udied in elas ic and inelas ic n+7Li sca e ing
and analyzed using he R-ma ix app oach in Re . [24], whe e
he new low-lying s a es we e sugges ed. Fo example, he 0+
s a e a 3.02 MeV was in oduced. Un o una ely, he n+7Li
exci a ion unc ion is ela i ely ea u eless, which makes
R-ma ix analysis ambiguous. The con empo a y ( o 1987)
shell model p edic ions we e used in Re . [24] as guidance
o he i . We pe o med ou own R-ma ix analysis o he
n+7Li exci a ion unc ions and a emp ed o inco po a e he
new low-lying 0+and 2+s a es in o he n+7Li i . I appea s
ha he low ene gy n+7Li exci a ion unc ion o elas ic
sca e ing can be ep oduced wi h he 0+,1
+, and 2+s a es i
hey a e placed a exci a ion ene gies abo e 2.8 MeV wi hou
any modi ica ions o hei educed wid hs. The o al c oss
sec ion o he 7Li(n,n)7Li(g.s.) eac ion is shown in Fig. 10.
The solid line is he R-ma ix i wi h he s a es men ioned
0
5
10
15
20
25
30
35
40
2.2 2.4 2.6 2.8 3 3.2 3.4
c oss sec ion [μb]
Exci a ion ene gy [MeV]
3+1+
0+2+
0
5
10
15
20
25
30
35
40
3+1+
0+2+
7Li(n,γ)
s-wa e
FIG. 9. (Colo online) The 7Li(n,γ) eac ion exci a ion unc ion
calcula ed using TDCSM app oach. The known 3+and 1+s a es and
he new 0+and 2+s a es a 2.4 and 2.5 MeV ( op panel) and a 2.8
and 3.3 MeV (bo om panel) a e shown.
054617-6
STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013)
1
10
0.2 0.4 0.6 0.8 1 1.2 1.4
σ (b)
Ec.m. (MeV)
FIG. 10. (Colo online) To al 7Li(n,n)7Li eac ion exci a ion
unc ion om [24]. The solid line is he R-ma ix i wi h he known
3+s a e a 2.25 MeV and he new 0+,1
+,and2
+s a es a 2.8, 3.1,
and 3.3 MeV. The dashed line shows he e ec o shi ing he 0+s a e
down by 200 keV.
abo e, he backg ound s a es om Table I, and he known 3+
a 0.22 MeV. No e ha i he 0+is shi ed down by as li le
as 200 keV i would appea as ela i ely na ow peak, which
is no obse ed expe imen ally (do ed line in Fig. 10). F om
he conside a ions abo e we can conclude ha exis ence o he
new low lying 0+and 2+s a es in 8Li does no , in p inciple,
con adic a ailable n+7Li elas ic sca e ing expe imen al
da a. Howe e , hese s a es ha e o be shi ed up in exci a ion
ene gy by ∼800 keV compa ed o hei sugges ed loca ion in
8B. The e is also s ong e idence agains degene acy o he
new s a e(s) wi h he 3+s a e. I such degene acy exis s hen
he expe imen al c oss sec ion a he maximum o he 3+peak
(0.22 MeV) would be highe han can be accoun ed o by he
3+s a e alone. Ou R-ma ix i shows ha his is no he case.
I is mo e di icul o econcile he new s a es in 8B and
he a ailable 7Li(n,n)7Li(1/2−) expe imen al da a. I educed
wid hs pa ame e s om Table I o hese s a es a e used hen
he 7Li(n,n)7Li(1/2−) c oss sec ion is o e es ima ed due o
oo s ong con ibu ion om he 2+s a e (do ed blue cu e in
Fig. 11). The elas ic educed wid h ampli ude o he 2+s a e
has o be educed om 0.276 o <0.1 in o de o p oduce a good
i o he 7Li(n,n)7Li(1/2−) da a ( ed solid cu e in Fig. 11).
All o he pa ame e s o he 2+and also all pa ame e s o he
0+and 1+do no equi e any modi ica ion. The 7Be(g.s.) +p
spec oscopic ac o o he 2+s a e is al eady small (4%) in
8B bu i appea s ha i needs u he educ ion o less han
1% in 8Li o ep oduce he 7Li(n,n)7Li(1/2−) da a. We do no
ha e a good explana ion o his si ua ion.
The exci a ion ene gy shi o 800 keV be ween s a es in
mi o nuclei (Thomas-Eh man shi [25,26]) is e y la ge.
While no unique ( o example, he shi be ween he 1/2+
second exci ed s a e in 19O and 19Na is 730 keV [27,28]) i is
gene ally associa ed wi h single pa icle s uc u e, whe e he
alence nucleon is in he s-wa e s a e. A la ge Thomas-Eh man
shi esul s om di e en asymp o ic beha io o he alence
nucleon wa e unc ion be ween bound and unbound s a es
in mi o nuclei (Nolen-Schi e e ec [29]). The 0+and 2+
a e p-shell s a es, he e o e, a la ge Thomas-Eh man shi
is no expec ed. Realizing ha his is an unusual si ua ion,
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.2 0.4 0.6 0.8 1 1.2 1.4
σ (b)
Ene gy in cms (MeV)
FIG. 11. (Colo online) To al 7Li(n,n)7Li(1/2-) eac ion exci a-
ion unc ion om [36]. The dashed blue cu e is he R-ma ix i
wi h he educed wid h pa ame e s o Table I.The2
+
2 om he able
o e es ima es he c oss sec ion a a ound 1.2 MeV in he c.m.s. The
solid ed cu e is he same R-ma ix i , wi h he elas ic componen
o he 2+
2 educed o be e i he da a.
we a emp ed o ep oduce he obse ed p+7Be inelas ic
sca e ing exci a ion unc ion wi hou in oducing he new
esonances in 8B bu assuming a di ec exci a ion mechanism
o he 7Be i s exci ed s a e in p+7Be sca e ing.
These calcula ions we e pe o med using he coupled-
channels app oach. The po en ial o Kim e al. [30]was
i s conside ed o he ba e pa o he 7Be in e ac ion. The
ansi ion po en ial o he coupling be ween he g ound and he
i s exci ed s a e was gene a ed de o ming he ba e po en ial
and assuming ha hese wo s a es o he 7Be nucleus a e
membe s o a K=1/2 o a ional band wi h a quad upole
de o ma ion leng h o δ2=2 m. Besides he ansi ion
po en ial, his p ocedu e gi es ise also o eo ien a ion e ms,
which we e also aken in o accoun in he calcula ions. The
coupled equa ions we e sol ed o all o de s using he compu e
code FRESCO [31]. In Fig. 12(a) we show he exci a ion unc ion
o a θc.m.=146◦as a unc ion o he p+7Be c.m. ene gy.
FIG. 12. (Colo online) Inelas ic sca e ing di e en ial c oss
sec ion om he 7Be(p,p) eac ion calcula ed wi hin he coupled-
channels app oach, assuming a di ec mechanism. The le panel uses
ap+7Be po en ial which does no con ain esonances wi hin his
ene gy in e al. The igh panel shows he esul o he calcula ion
using a po en ial ha con ains a esonance.
054617-7
J. P. MITCHELL e al. PHYSICAL REVIEW C 87, 054617 (2013)
Clea ly, he con ibu ion o he di ec mechanism is e y small
in his ene gy ange, sugges ing ha he magni ude o he
measu ed inelas ic c oss sec ion a hese ene gies canno be
explained by a pu e di ec eac ion mechanism. We pe o med
a second coupled-channels calcula ion using a po en ial ha
p oduces a esonance a hese ene gies. This po en ial was
pa ame ized using a Woods-Saxon shape, wi h adius R=
2.23 m (deduced om he ma e adius o he 7Li nucleus),
di useness a=0.65 m and he dep h adjus ed o p oduce
a esonance a ound Ec.m.=2 MeV. The calcula ed inelas ic
exci a ion unc ion ob ained wi h his po en ial is gi en by he
solid line in Fig. 12(b). The p esence o he esonance p oduces
a p onounced maximum abou 2 MeV and a signi ican
inc ease o he magni ude o he c oss sec ion. So, based on he
coupled-channels analysis we conclude ha he high inelas ic
sca e ing c oss sec ion canno be ep oduced unless eso-
nance(s) is(a e) in oduced in he co esponding ene gy ange.
Finally, we ha e o make an impo an dis inc ion be ween
he 0+and he 2+s a es. While exis ence o he 2+s a e is ha d
o dismiss, he case o he 0+s a e is somewha weake . In spi e
o he ac ha wi hou his esonance he inelas ic c oss sec ion
a 2.0 MeV due o di ec exci a ion o he i s exci ed s a e in
7Be is 3 imes smalle han he expe imen al alue, one should
be ca e ul making he inal call based on such e idence. Fu he
in es iga ion is wa an ed. Speci ically, accu a e measu emen
o he p+7Be exci a ion unc ion o inelas ic sca e ing in he
ene gy ange om 0.7 o 2.0 MeV and in a b oad angula ange
should p o ide a de ini i e answe on he exis ence o he 0+.
A his poin we can only ega d his s a e as en a i e.
V. THE CONTINUUM SHELL MODEL ANALYSIS
OF THE p+7BE DATA.
The ime-dependen con inuum shell model [2] was used
as an al e na i e and mo e mic oscopically cons ained way o
analyze he p+7Be da a. This model ex ends he adi ional
shell model in o he domain o eac ion physics. I inco po a es
he many-body dynamics wi h all essen ial s uc u e and
eac ion componen s, and allows one o p edic he eac ion
obse ables. Some ea u es, such as he angula dependence
o c oss sec ions and in e e ence be ween esonances a e
pa icula ly sensi i e o he many-body s uc u e. The TDCSM
is buil upon one o he well-es ablished Hamil onians o he
adi ional shell model coupled o eac ion channels, whe e
a Woods-Saxon shaped po en ial is aken om a global
Woods-Saxon pa ame iza ion [32]. This heo e ical ea men
o 8B using he WBP shell model Hamil onian [33] is epo ed
in Re . [2]. The WBP Hamil onian was selec ed because unlike
mos in e ac ions i esul s in low-lying 1+
2,0+
1,and 2+
2s a es
in 8B, a exci a ion ene gies below 3 MeV. To conside a
ull spec um o possible Hamil onians in his in es iga ion,
in addi ion o WBP, we use PWT [33] and CKI [14] shell
model in e ac ions. The compa ison o he expe imen al
spec oscopic ac o s o he posi i e pa i y s a es in 8B o
he p edic ions o he shell model wi h di e en in e ac ions
is gi en in Table II. The expe imen al spec oscopic ac o s
we e calcula ed as he a io be ween he pa ial wid h and he
single pa icle wid h calcula ed using a Woods-Saxon po en ial
wi h a global Woods-Saxon pa ame iza ion [32]. In Table II
TABLE II. Expe imen al spec oscopic ac o s o posi i e pa i y
s a es compa ed o he shell model p edic ions.
Jπ1+0+3+2+1+
Eex (MeV) 0.768 1.9 2.31 2.50 3.3
S7Be(g.s.)+pa0.38 0.05 0.20 0.04 ≈1
S7Be(1/2−)+pa– 0.94 – 0.19 0.14
1+
10+
13+
12+
21+
22+
3
ECKI 1.08 4.95 1.69 4.24 2.77 5.15
CKI el. 0.44 0.34 0.33 0.56 1.10 0.06
CKI in. 0.87 0.90 – 0.04 0.14 0.27
EPWT 1.54 4.01 2.14 4.39 3.80 6.06
PWT el. 0.45 0.27 0.30 0.55 0.95 0.11
PWT in. 0.84 0.96 – 0.03 0 0.22
EWBP 0.55 1.75 1.99 2.40 1.73 3.25
WBP el. 0.40 0.48 0.37 0.13 1.0 0.40
WBP in. 0.77 0.84 – 0.40 0.14 0.03
aExpe imen al alues.
i can be seen ha all h ee esidual in e ac ions a e in good
ag eemen wi h he expe imen al spec oscopic ac o s o he
1+
1and 3+
1 esonances. All h ee in e ac ions ep oduce he
inelas ic spec oscopic ac o o he 0+
1, and bo h he elas ic
and inelas ic componen s o he 1+
2. The WBP in e ac ion is
he only in e ac ion ha p edic s a 2+
2 ha is domina ed by
an inelas ic componen , as is seen expe imen ally, while he
PWT and CKI in e ac ions bo h p edic a 2+s a e wi h a simila
inelas ically domina ed componen as he 2+
3.
The bes alida ion o he heo e ical model p edic ions
can be pe o med i he measu ed c oss sec ion is calcula ed
di ec ly om he model. Un o una ely, he eac ion physics
is e y sensi i e o kinema ics and o he exac posi ion o
le els in he spec um because o he phase space and ba ie
pene abili y. While he adi ional shell model may, in gene al,
be good in desc ibing posi ions and o de ing o s a es, o en
i s p ecision is no close o wha is equi ed by he eac ion
physics. Thus, i is common p ac ice o se he exac eac ion
kinema ics based on obse a ion. In ou app oach all known
s a es and h esholds a e adjus ed om expe imen al da a and
we ea he ene gies o unknown 1+
2,0+
1,and 2+
2s a es as
pa ame e s. In ou s udy we a y hese h ee pa ame e s o
bes i he obse ed c oss sec ion. The TDCSM p o ides an
e ec i e mechanism o modi y he posi ion o any s a e in he
Hamil onian while keeping all s uc u al aspec s unchanged.
This is done by adding o a shell model Hamil onian a
ac o izable e m δE|αα|, whe e |αis he eigens a e o be
shi ed and δE is he ene gy shi . The co esponding change
in he many-body p opaga o is pe o med exac ly wi h he
help o Dyson’s equa ion, o de ails see Re . [2].
In Fig. 13 he inelas ic sca e ing c oss sec ion o 7Be(p,p)
ob ained wi h TDCSM is compa ed o expe imen . Panels
(a), (b), and (c) co espond o calcula ions wi h WBP, PWT,
and CKI in e ac ions, espec i ely. The spin and pa i ies o
esonances in he ene gy egion plo ed a e ma ked. The
inelas ic c oss sec ion is no sensi i e o he 3+s a e which
is seen in he elas ic sca e ing c oss sec ion. All models
p edic a simila s uc u e o he 3+s a e and he e o e p oduce
054617-8
STRUCTURE OF 8B FROM ELASTIC AND ... PHYSICAL REVIEW C 87, 054617 (2013)
0
5
10
15
20
25
30
35
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Exci a ion ene gy [MeV]
(c)
3+1+
0+2+
0
5
10
15
20
25
30
35
C oss Sec ion [mb/s ]
(b)
3+
0+1+
2+
0
5
10
15
20
25
30
35 (a)
1+0+3+2+
129o
138o
EXP-129o
EXP-138o
FIG. 13. (Colo online) Inelas ic sca e ing di e en ial c oss
sec ions o 7Be(p,p) eac ion ob ained wi h he TDCSM ha uses
h ee di e en Hamil onians is compa ed o he expe imen al da a.
(a), (b), and (c) co espond o WBP, PWT, and CKI in e ac ions.
a compa able elas ic c oss sec ion which ag ees well wi h
expe imen . The elas ic 7Be(p,p) c oss sec ion wi h WBP
in e ac ion is demons a ed in Re . [2]. Posi ions o 1+
2,0+
1,
and 2+
2 esonances, indica ed in Fig. 13, a e no known ap io i;
he e hey a e adjus ed by isual examina ion o bes ep oduce
he expe imen al da a. The main peak in he 7Be(p,p)c oss
sec ion is due o he 2+
2 esonance a a ound 2.5 MeV o
exci a ion. I was ound ha ag eemen wi h he expe imen al
da a is good i he 0+
1is placed a ound 2 MeV and he
1+
2is mo ed o highe exci a ion ene gy. (Sensi i i y o he
inelas ic c oss sec ion o he posi ion o he 1+
2s a e is weak.
Howe e , he 1+
2s a e a exci a ion ene gies below 2.3 MeV
would p oduce a peak in he elas ic c oss sec ion, which is no
obse ed expe imen ally. See discussion in Sec. III.) In he case
o he WBP in e ac ion, Fig. 13(a), no posi ion adjus men was
made o he 1+
2and 0+
1s a es and he 2+
2is only mo ed down
by abou 140 keV. The CKI Hamil onian gi es wo 2+exci ed
s a es a 4.2 and 5.1 MeV o exci a ion. Bo h o hese s a es
ha e been ied as candida es o he 2.5 MeV esonance and
i was de e mined ha he second 5.1 MeV s a e in he CKI
Hamil onian has he co ec s uc u e. Ou main conclusion
om he calcula ions shown in Fig. 13 is ha he CKI in e ac-
ion appea s o be bes in ep oducing he c oss sec ion. The
s a es ob ained wi h he CKI appea o ha e s uc u e which
ag ees wi h he obse ed in e e ence and angula dependence
ea u es. In pa icula , only he CKI in e ac ion is able o
ep oduce he obse ed inc ease in he c oss sec ion a 2.5
MeV o highe angle [Fig. 13(c)]. The heigh o he esonance
peak a 2.5 MeV is he p ima y di e ence be ween heo y and
obse a ion. We a ibu e his di e ence pa ially o he 1−
s a e, which was no included in he shell model analysis (only
p-shell s a es we e conside ed) and also o he somewha di -
e en a io be ween he elas ic and inelas ic pa ial wid hs o
he 2+
2.
The ampli udes om he inal R-ma ix i wi h he CKI
in e ac ion a e summa ized in Table III ( esonance educed
wid h pa ame e s om he R-ma ix i we e ecoupled om
heLS o hejj coupling scheme o di ec compa ison wi h
TDCSM ampli udes). I should be no ed ha he choice o
channel adius in he R-ma ix calcula ions will ha e he a ec
o scaling he educed wid h pa ame e s, hus one should
no di ec ly compa e absolu e alues, bu a he sign and
ela i e alues o he educed wid hs. We no e ha he i
only sligh ly modi ies he ampli udes o he 3+
1and 2+
2s a es
lea ing he gene al ea u es o CKI unchanged. Howe e , he e
is a signi ican di e ence be ween expe imen al exci a ion
ene gies and CKI p edic ions o he 2+
2s a e. The 0+
1also has
a signi ican shi be ween he expe imen al and CKI p edic ed
ene gy.
TABLE III. Decay ampli udes om he inal R-ma ix i and om he CKI based TDCSM. The i s column deno es he spin and pa i y
o he esonance, ollowing a e exci a ion ene gy and ou ampli udes om he R-ma ix i . Exci a ion ene gy and ampli udes o s a es in 8B
om he CKI Hamil onian a e lis ed in he emaining i e columns.
R-ma ix i TDCSM wi h CKI in e ac ion
JπE[MeV] 7Be 3/2−g.s. 7Be 1/2−E[MeV] 7Be 3/2−g.s. 7Be 1/2−
p1/2p3/2p1/2p3/2p1/2p3/2p1/2p3/2
2+
10 0.19 −0.94 0.43 0.00 0.23 −0.98 −0.43
1+
10.768 −0.17 0.71 0.24 −0.91 1.08 −0.35 0.57 0.24 −0.91
0+
11.9 0.35 1.30 4.95 −0.59 0.95
3+
12.31 0.61 1.69 0.58
2+
22.50 −0.17 0.17 0.53 5.15 −0.17 0.16 −0.52
1+
23.3 −1.46 0.37 0.53 0.41 2.77 0.84 0.62 0.33 0.18
054617-9