Physics Le e s B 767 (2017) 480–484
Con en s lis s a ailable a ScienceDi ec
Physics Le e s B
www.else ie .com/loca e/physle b
E olu ion o iaxial shapes a la ge isospin: Rh iso opes
A. Na in a,∗, M. Rejmunda, S. Bha acha yyab, R. Pali c, G.H. Bha d, J.A. Sheikh d,
A. Lemassona, S. Bha acha yab, M. Caamaño e, E. Clémen a, O. Delaune a, F. Fa ge a,
G. de F ance a, B. Jacquo a
aGANIL, CEA/DRF – CNRS/IN2P3, Bd Hen i Becque el, BP 55027, F-14076 Caen Cedex 5, F ance
bVa iable Ene gy Cyclo on Cen e, 1/AF Bidhan Naga , Kolka a 700064, India
cDepa men o Nuclea and A omic Physics, Ta a Ins i u e o Fundamen al Resea ch, Colaba, Mumbai, 400 005, India
dDepa men o Physics, Uni e si y o Kashmi , S inaga , 190 006, India
eUSC, Uni e sidad de San iago de Compos ela, E-15706 San iago de Compos ela, Spain
a i c l e i n o a b s a c
A icle his o y:
Recei ed 26 June 2016
Recei ed in e ised o m 4 No embe 2016
Accep ed 13 No embe 2016
A ailable online 16 No embe 2016
Edi o : V. Me ag
Keywo ds:
Iso opic iden ifica ion o fission agmen s
Neu on- ich Rh iso opes
T iaxial p ojec ed shell model
Nuclea iaxiali y
The o a ional esponse as a unc ion o neu on–p o on asymme y o he e y neu on- ich iso opes o
Rh (116–119Rh) has been ob ained om he measu emen o p omp γ ays om iso opically iden ified
agmen s, p oduced in fission eac ions a ene gies a ound he Coulomb ba ie . The measu ed ene gy
“signa u e” spli ing o he y as bands, when compa ed wi h he T iaxial P ojec ed Shell Model (TPSM)
calcula ions, shows he need o la ge, nea ly cons an , iaxial de o ma ions. The p esen esul s a e
compa ed wi h global p edic ions o he exis ence o non axial shapes in he pe iodic able in he case
o e y neu on- ich nuclei Rh iso opes. The p edic ed end o a second local maximum o a iaxial
shape a ound N∼74 is no ound.
©2016 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3.
A basic p ope y o a omic nuclei is i s shape, which go e ns i s
a ious s a ic as well as dynamic p ope ies, and depends on he
in e ac ion among i s cons i uen s (Zp o ons and Nneu ons).
Shapes anging om sphe ical o e ahed al a e p edic ed ac oss
he nuclea landscape. The e olu ion o nuclea shapes as a unc-
ion o angula momen um and isospin is o p ime impo ance in
nuclea s uc u e s udies [1,2]. The e is a p edominance o p ola e
o e obla e shapes o he g ound s a e o e en-Ze en-Naxially
de o med nuclei [3]. De ia ions om axial symme y and he ex-
is ence o iaxial “ igidly de o med” nuclei, fi s p edic ed in he
la e 50’s, we e assumed o be commonly possible [4,5]. The e ha e
been a sus ained expe imen al and heo e ical e o s o es ablish
he signa u e o iaxial shapes o nuclei o a ious mass egions.
Mos o hese nuclei we e ound o exhibi ib a ional modes o
“so ness” wi h espec o he iaxiali y pa ame e γ[6–8]. In he
odd-Anuclei nea A ∼190 e idence o asymme ic shapes ha e
been sugges ed om he compa ison o he measu ed high spin
le els wi h calcula ions based on a model o asymme ic o o cou-
pled o nucleon in a single- jo bi al [9]. Fo a model independen
measu e o iaxiali y, he cubic shape pa ame e analysis was used
*Co esponding au ho .
E-mail add ess: na[email p o ec ed] (A. Na in).
o es ablish maximum e ec i e la ge iaxiali y o ce ain iso opes
o Os and P [10]. Mul iple Coulomb exci a ion s udies ha e been
used o s udy he b eaking o axial symme y [11]. Da a on he
spli ing o he iso- ec o GDR in ce ain nuclei was also shown o
be sensi i e o he de o ma ion and iaxiali y [12]. The o a ion
o iaxial nuclei has been sugges ed o be mani es ed as chi al
pa ne bands [13], which ini ia ed se e al expe imen al in es iga-
ions [14–16] in ecen yea s.
Mölle e al. [17] made a sys ema ic s udy o in es iga e he
sensi i i y o he g ound-s a e nuclea masses wi h he inclusion
o axial asymme y, o nuclei bo h nea and a om s abili y.
They showed he co ela ion be ween he need o axial asym-
me y and he p esence o expe imen ally known cha ac e is ic
γ-bands in nuclei a ound s abili y. Fo hese nuclei, asys ema ic
de ia ion be ween he calcula ed and he measu ed masses was
emo ed by he inclusion o axial asymme y. This s udy ga e a
su p isingly small numbe o nuclei (∼70 ou o 3000) whose cal-
cula ed masses we e a ec ed by axial-symme y b eaking. These
calcula ions showed ha he e ec o axial asymme y is maxi-
mum a ound Z=44, N=64 and also p edic ed ano he addi ional
egion o inc eased iaxiali y a ound N∼74 o hese elemen s.
Howe e in his egion, sys ema ic beyond mean field calcula-
ions, o e en–e en nuclei, using he Gogny D1S in e ac ion, ex-
ended by a 5-Dimensional Collec i e Hamil onian (5DCH), p edic
h p://dx.doi.o g/10.1016/j.physle b.2016.11.020
0370-2693/©2016 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by
SCOAP3.
A. Na in e al. / Physics Le e s B 767 (2017) 480–484 481
a smoo h e olu ion o γ o he ele an iso opes bu wi h a de-
c easing mean alue o he quad upole de o ma ion [18]. Thus he
measu emen s o e y neu on- ich Ru and Rh iso opes a e im-
po an o p obe he e olu ion o iaxiali y e y a om s abili y.
These iso opes a e in e es ing candida es o p obe whe he he el-
e an exci ed s a es and hei decay pa e ns p o ide a signa u e o
iaxiali y. This could also help in es ablishing he link be ween he
lowe ing o g ound s a e masses when axial symme y is b oken o
pa e n o exci ed s a es ha a e uniquely ela ed o iaxial shape.
Calcula ions ela ing nuclea masses o collec i e s uc u es ha e
also been epo ed [19].
The ene gies o exci ed s a es as a unc ion o angula momen a
p o ide an expe imen al handle o iden i y iaxial shapes. The sig-
na u e quan um numbe is associa ed wi h he in a iance o he
in insic Hamil onian o an axially-de o med nucleus wi h espec
o 180◦ o a ion a ound a p incipal axis. A signa u e-dependen
e m implies ha o a ional bands wi h K= 0(Kis he angula
momen um p ojec ion o he quasi pa icles along he symme-
y axis), in a sys em wi h o a ional in a iance, end o sepa a e
in o wo g oups, cha ac e ized by “ a o ed” and “un a o ed” signa-
u e quan um numbe [20]. The shi be ween hese ene ge ically
a o ed and un a o ed sequences o le els (ene gy signa u e spli -
ing), is used o unde s and non-axial shapes in nuclei [21,22].
We epo he e on he spec oscopy o he high spin s a es
in odd-ZRh neu on- ich iso opes
116–119Rh, which could no be
accessed ea lie expe imen ally, o unde s and he e olu ion o nu-
clea iaxiali y a om he alley o s abili y. These esul s a e
examined in he ligh o global p edic ions o iaxiali y in he
pe iodic able. I is impo an o no e ha he ea lie in es iga-
ions [10–12] p o ide expe imen al e idence o iaxiali y o nu-
clei, some o which a e in a iance wi h he p edic ion o he fini e
ange liquid d op model (FRLDM) [17].
The measu emen s we e pe o med a GANIL using a
238U beam
a 6.2 MeV/u (∼0.2pnA) on a 10-mic on hick
9Be a ge . The fis-
sion agmen s we e iden ified in mass numbe (A) and a omic
numbe (Z) in he VAMOS++ spec ome e [23] placed a 20◦wi h
espec o he beam axis. The ime o fligh was ob ained om
wo mul i wi e pa allel pla e a alanche coun e s MWPPACs, one
loca ed a e he a ge and he o he a he ocal plane (fligh pa h
∼7.5m). The a ious measu ed posi ions, ene gies, and imes,
along wi h he known magne ic field, we e used o de e mine,
on an e en -by-e en basis, A, cha ge s a e (q), Z, and he e-
loci y ec o (
) o he de ec ed agmen [23]. The p omp γ
ays we e measu ed in coincidence wi h hese iso opically iden-
ified agmen s using he EXOGAM a ay [24], which consis ed o
11 Comp on-supp essed segmen ed clo e HPGe de ec o s placed
15 cm om he a ge . The γ- ay ene gies o he agmen s in
hei es ame we e ob ained om he measu ed
along wi h he
known angle o he segmen o he ele an clo e de ec o [23,
25]. Mo e de ails o he me hod and measu emen can be ound in
Re . [26,27]
Fig. 1 shows he γ- ay spec a o he iso opically iden i-
fied
114–119Rh iso opes measu ed in his wo k. The de i ed le el
schemes o
114–119Rh ob ained using γ−γcoincidences, ene gy
sums, and ela i e in ensi ies, a e shown in Fig. 2 and Fig. 3 o
he odd-Aand e en-ARh iso opes espec i ely. The esul ing le el
schemes ex ac ed o
114, 115Rh confi m hose p e iously ob ained
in li e a u e [28,29], including hose o he side bands (no shown
in Fig. 2). Addi ionally, he 255.2 keV and 774.9 keV ansi ions
co esponding o he side bands o
114Rh and
115Rh espec i ely
and he 215.5 keV ansi ion o he main band in
115Rh we e
placed om he p esen measu emen . A ew mo e new ansi-
ions we e also seen in he mass ga ed spec a bu could no
be placed in he le el scheme due o lack o sufficien s a is ics
o γ−γco ela ions o hese ansi ions. A compa ison wi h
Fig. 1. A-and Z-iden ified Dopple -co ec ed γ- ay spec a o he
114–119Rh iso-
opes. The inse s in he espec i e panels shows he γ−γcoincidence spec a.
o he known ligh e e en-ARh iso opes sugges s ha he 143.4
and 310.7 keV ansi ions in
116Rh (Fig. 1c) could belong o a side
band. The spin-pa i ies a e p oposed based on he sys ema ics o
he known schemes o he lowe -A Rh iso opes [30]. In
114Rh,
ahigh spin 7−isome [31] and in
116Rh [32] a 6−isome we e
en a i ely sugges ed. A simila isome wi h J >4was p oposed
in
118Rh [33]. No coincidences we e possible o
119Rh due o he
low s a is ics in his e y exo ic nucleus. The γ ays o
116–119Rh
a e epo ed o he fi s ime (a single ansi ion om he de-
cay o a low spin isome ic decay s a e has been ea lie epo ed
in
117Rh [34]). De ec ion o he p omp γ ays in he Rh iso opic
chain up o N/Z=1.64, N=74, (as compa ed o N=70 ob ained
om he adi ional high- old γcoincidence me hod), was only
possible because o he inc eased sensi i i y and selec i i y o he
p esen se up.
The quan um mechanical iaxial p ojec ed shell model (TPSM)
is used o unde s and he band s uc u e and signa u e spli ing
o he neu on- ich Rh iso opes. Va ious phenomena ela ed o i-
axial shapes o nuclei, like γ- ib a ions [35], chi al symme y [36,
37], and wobbling modes [38], ha e been success ully desc ibed
using he TPSM. Mo e ecen ly, he TPSM along wi h Densi y Func-
ional Theo y (DFT) p o ided a consis en desc ip ion o da a o
e en–e en nuclei nea
110Ru [39]. In he TPSM, he quasi-pa icle
s a es a e gene a ed by sol ing he iaxial Nilsson and pai ing
(monopole and quad upole e ms) Hamil onian in he BCS app ox-
ima ion [35,37,40,41] wi h he ele an de o ma ion pa ame e s,
482 A. Na in e al. / Physics Le e s B 767 (2017) 480–484
Fig. 2. Y as bands o odd-A iso opes o Rh buil on he low-ene gy high-spin s a es
s udied in his wo k. Exci a ion ene gies (in keV) a e ela i e o he 7/2+le els.
The co esponding TPSM calcula ions a e also shown (see ex ).
and (γ= an−1(/)). To ob ain he s a es in he labo a o y
ame, he b oken o a ional symme ies in hese de o med mul i-
quasi-pa icle s a es a e p ojec ed on o good angula -momen um
s a es h ough a h ee-dimensional angula momen um p ojec ion
o malism. Each iaxial configu a ion is composed o se e al K
alues and he co esponding bands a e ob ained by assigning a
gi en K alue in he angula momen um p ojec ion ope a o [40].
These p ojec ed basis s a es a e hen employed o diagonalize he
shell model Hamil onian. The final s a es so ob ained, because
o he configu a ion mixing (di e en K alues), do no ha e a
well defined Kquan um numbe . The iaxial configu a ion, in he
case o an e en-Z–e en-Nsys em, is an admix u e o di e en K
s a es and he acuum configu a ion is composed o K=0, 2, 4, ...
s a es [40–42]. The de ails o he o malism o odd-Z–e en-Nand
odd-Z–odd-Nnuclei can be ound in Re s. [35,37].
Fo he Rh iso opes discussed he e, h ee majo shells, N=3, 4
and 5 (2, 3 and 4) o neu ons (p o ons), we e used in he calcu-
la ion. In he case o he odd-ARh iso opes, he one-quasi-pa icle
configu a ion (1−qpc) (p o on) and h ee-quasi-pa icle configu a-
ions (3−qpc) (one-p o on and wo-neu ons) we e used o com-
pu e he s a es measu ed in he p esen wo k. Simila ly, o he
e en-ARh iso opes, he wo-quasi-pa icle configu a ions (2−qpc)
(one p o on and one neu on) we e gene a ed using he same
me hod. The calcula ions used a s anda d se o pa ame e s o
his mass egion [36]. In he p esen wo k, he alues o and
we e adop ed so ha expe imen al da a is ep oduced eason-
ably well. The alue chosen in he TPSM analysis is close o hose
ob ained in mean-field calcula ions [43], howe e , de ia ions we e
no ed o . In should be no ed ha in he TPSM app oach, he
basis s a es a e gene a ed by sol ing he Nilsson po en ial wi h
he inpu de o ma ion alues o and , and hese de o ma ion
alues also fix he s eng h o he quad upole-quad upole pa o
he Hamil onian h ough sel -consis ency condi ions. Howe e , he
final de o ma ion (quad upole momen ) calcula ed using he p o-
jec ed wa e unc ions may be di e en om he inpu de o ma ion
used o he bases. The eason is ha he Hamil onian is diag-
onalized using he p ojec ed mul i-quasipa icle basis s a es and
many-body co ela ions en e ing hough his p ocess can al e ini-
ial de o ma ion alues.
Fig. 3. Y as bands o e en-A iso opes o Rh buil on he low-ene gy high-spin s a es
s udied in his wo k. Exci a ion ene gies (in keV) a e ela i e o he 7−le els. The
co esponding TPSM calcula ions a e also shown including a low lying exci ed band
(see ex ).
The p ojec ed ene gies o he di e en configu a ions be o e
configu a ion mixing p o ide insigh s in o he s uc u e o he
bands. The locus o he p ojec ed ene gies a a ious spins o a
pa icula configu a ion is e e ed o as a band. We fi s desc ibe
he calcula ions (pe o med up o Iπ=29/2+) o he odd-ARh
iso opes. The lowes band (g ound) is ound o be buil on a
1−qpc wi h K=7/2. A low spin, he nex a o able band, which
lies close o g ound band, is he γ-band wi h K=Kγ+2, Kγ
being he K- alue o he g ound band. The o he γ-band wi h
K=Kγ−2is ound o be ene ge ically less a o able. Fo highe
spin (Jπ≥15/2+), a 3−qpc wi h K=1/2is ound o ha e an en-
e gy simila o he 1−qpc K =7/2 and K=11/2 band. A e con-
figu a ion mixing, he ampli udes (which a y as a unc ion o he
angula momen um) o he a ious componen s o he low lying
y as s a es a e ound o be domina ed by he K=7/2 componen
along wi h addi ional small componen s a ising om he K=11/2
and he K=1/2s a es. The ene gies ob ained a e diagonaliza ion
o he y as band in he odd-A Rh iso opes a e shown (using he
de o ma ion alues gi en in Table 1) and a e compa ed wi h he
co esponding expe imen al da a in Fig. 2. The figu e shows ha
he TPSM calcula ions a e able o desc ibe he ene gies o he ex-
ci ed s a es well. The signa u e spli ing a lowe spin is ound o
a ise om K=7/2s a es. The inc ease in he signa u e spli ing
a highe spin can be a ibu ed o mixing o he K=1/2s a es.
Fig. 4 illus a es he sensi i i y o he calcula ions o he ene gy
di e ence be ween consecu i e spin le els (signa u e spli ing) o
a ious alues o γ o
115, 117Rh. In he case o he calcula ions
o he e en-ARh iso opes (pe o med up o Iπ=18−), di e -
en 2−qpc wi h K=1, 3, 5, 7 and 9 we e conside ed (only odd-K
configu a ions we e used since hey we e ene ge ically mo e a o -
able). The wo bands ha ing he lowes ene gy we e ound o be
based on configu a ions wi h K=7 and 9 espec i ely. Thei ex-
ci a ion ene gies we e ound o be wi hin 100 keV o he en i e
spin ange. The calcula ed y as s a es, ob ained a e band mix-
ing we e ound o gi e an o e all good desc ip ion o he da a and
a e shown in Fig. 3. Addi ionally, hese calcula ions p edic an ex-
ci ed band e y close o he y as band (see Fig. 3) and he ene gy
di e ence be ween he g ound and pa ne band is ound o be
minimum o
118Rh. Challenging measu emen s o hese exci ed
bands which could be signa u e o chi al symme y can be ad-
d essed wi h nex gene a ion γ- ay acking de ec o s [44] coupled
o a la ge spec ome e [23,45]. Fig. 5 illus a es he sensi i i y o
he calcula ions o a ious alues o γ o
114, 116, 118Rh. Fig. 4 and
A. Na in e al. / Physics Le e s B 767 (2017) 480–484 483
Fig. 4. Expe imen al ene gy signa u e spli ing (E(I) −E(I −1)) o (a)
115Rh and
(b)
117Rh compa ed wi h he TPSM esul s o a ious iaxial de o ma ion pa ame-
e γ.
Table 1
The axial de o ma ion pa ame e () and iaxial de o ma ion pa ame e () em-
ployed in he calcula ion o
114–119Rh iso opes (γ= an−1(/)).
114Rh 115Rh 116Rh 117Rh 118Rh 119Rh
0.230 0.190 0.221 0.210 0.210 0.180
0.130 0.108 0.130 0.120 0.150 0.105
γ29.5 29.6 30.5 29.7 35.5 30.3
Fig. 5 also illus a es he end o signa u e spli ing as a unc ion
o spin o he Rh iso opic chain, along wi h he co esponding cal-
cula ions using he alues which bes desc ibe he measu emen s
(Table 1). I can be seen ha he calcula ions ep oduce well he
measu ed signa u e spli ing. The spli ing is small a lowe spin
and inc eases wi h inc easing spin. As shown in he Figs. 4 and 5,
he calcula ed signa u e spli ing wi h small iaxial de o ma ion,
say, γ=10, has la ge de ia ion om he expe imen al alues. I
was also no ed ha he esul s o γ=0(no shown in he figu e)
is close o γ=10.
The s uc u e and he magni ude o signa u e spli ings a e now
used o unde s and he deg ee o iaxiali y in neu on- ich Rh
iso opes. Fig. 6 demons a es he need o la ge alues o iax-
ial de o ma ion (γ), shown in Table 1, o ep oduce he signa u e
spli ing in neu on- ich Rh iso opes. The p esen wo k indica es
ha hese neu on- ich nuclei ha e a simila end o he signa-
u e spli ing and he deg ee o iaxiali y emains almos cons an .
The signa u e spli ing o
115Rh (114Rh) was ea lie epo ed in
Re . [28,29]; i was ound o be simila o he ligh e odd-Aiso-
opes o Rh. As men ioned in he in oduc ion, he analysis o
Mölle e al. [17] o he exis ence o non-axial shapes in he pe i-
odic able p edic ed a egion o inc eased iaxiali y a ound
119Rh,
Fig. 5. Same as in Fig. 4 o he e en-A iso opes (a) 114Rh (b) 116Rh and (c) 118Rh.
in addi ion o he one known a ound
110Rh. The cu en wo k
p esen s he fi s compa ison o hese p edic ions wi h da a o
nuclei e y a om s abili y. As can be seen om Table 1, he
ex ac ed iaxial de o ma ions a e nea ly cons an be ween
114Rh
and
119Rh, and hus showing he absence o he exis ence o an-
o he egion o inc eased iaxiali y as p edic ed in Re . [17]. The
need o highe -o de shape pa ame e iza ion, which ha e been
shown o be equi ed in he ac inide egion [46] could be a possi-
ble eason o he disc epancy o Re . [17] wi h he expe imen al
esul s. Mean field calcula ions o e en–e en nuclei [47] p edic
changes in he S2nnea N ∼74 a ound Ru. Beyond mean field cal-
cula ions o Re . [18] p edic a smoo h e olu ion o γ o Ru and
Pd bu wi h a dec easing mean alue o β. Howe e Re . [48] men-
ions ha he mean alues o βand γde o ma ion pa ame e s
ob ained wi h 5-DCH calcula ions ha e o be aken wi h cau ion,
and ha small alues o βimply almos no de o ma ion and hus
li le ele ance o he pa ame e γ. Thus hese 5DCH calcula ions
using he Gogny HFB mean field p edic a anishing o de o ma ion
and iaxiali y o hese nuclei [48]. Hence he p esen wo k shows
he impo ance o such measu emen s o nuclei a om he al-
ley o s abili y and poin s owa ds he need o imp o emen s in
he a ious p edic ions o iaxiali y in his egion. The e olu ion
o hese de o ma ions could ha e an impac on he inpu s equi ed
o calcula ing he abundance o elemen s in his mass egion [49].
484 A. Na in e al. / Physics Le e s B 767 (2017) 480–484
Fig. 6. Expe imen al and calcula ed signa u e spli ing (wi h γ-de o ma ion gi en in
Table 1) o
115, 117, 119Rh (a) and
114, 116, 118Rh (b). Fo display pu poses, he da a
and calcula ion a e shi ed by −100 keV, 100 keV, and 300 keV o
115Rh (114Rh),
117Rh (116Rh), and
119Rh (118Rh) espec i ely.
In summa y, we ha e epo ed o he fi s ime he o a ional
esponse, ob ained om he measu emen o p omp γ ays, o
he e y neu on ich iso opes o
116–119Rh. The calcula ions o he
quan um mechanical model TPSM shows ha he obse ed le el
ene gy di e ences a e sensi i e o he γde o ma ion o
116–119Rh
iso opes, e en a low spin. Using he TPSM esul s, he need o
subs an ial and nea ly cons an iaxial de o ma ion o explain he
measu ed ene gy spec a o y as bands is shown o he e en-
mass Rh as well as odd-mass Rh iso opes beyond
114Rh and
115Rh
espec i ely. This wo k shows ha e olu ion o iaxiali y as a
unc ion o neu on numbe o he e y neu on ich Rh iso opes
di e s om he global p edic ions o he e olu ion o non-axial
shapes in he pe iodic able. In pa icula , o he p edic ed end
o a second local maxima o a iaxial shape a ound N∼74 is
no obse ed. The p esen wo k is a fi s s ep in unde s anding
he e olu ion o iaxial shapes a om s abili y. Fu he imp o e-
men s in he expe imen al sensi i i ies, o be achie ed in bo h
agmen and γ- ay de ec ion capabili ies a ene gies a ound he
Coulomb ba ie , could allow he cha ac e iza ion o he non-y as
s a es in hese nuclei o sea ch o he p esence o no el collec i e
modes associa ed wi h he iaxial o a ion, like chi al bands.
We would like o hank J. Goupil, G. F emon , L. Ménage ,
J. Rope , C. Spi aels, and he GANIL accele a o s a o hei ech-
nical con ibu ions and C. Schmi o help in a ious aspec s o
da a collec ion and he analyses. We acknowledge P. Mölle and
S. Pé u o use ul discussions and P. Van Isacke and S. Biswas o
a c i ical eading o he manusc ip . One o us (S.B.) acknowledges
pa ial financial suppo h ough he LIA F ance–India ag eemen .
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