PHYSICAL REVIEW CVOLUME 37, NUMBER 4APRIL 1988
Di ec e sus sequen ial ou -pa icle ans e in hea y ion collisions
wi h supe luid nuclei: Sn+Sn eac ion
C. H. Dasso
The Niels Boh Ins i u e, DK-2100 Copenhagen g, Denma k
M. Lozano
Depa amen o de Fssica A omica yNuclea , Uni e sidad de Seui lla, 4180Seuilla, Spain
A. Vi u i
Dipa imen o di Fisica "Galileo Galilei "and Is i u o Nazionale di Fisica Nuclea e, I-35131Padoua, I aly
(Recei ed 26 Augus 1987)
The di ec ans e o ou nucleons in eac ions wi h supe 6uid sys ems is compa ed wi h he
eeding o he same inal channel by asuccessi e ans e o co ela ed nucleon pai s. Aquan i a i e
analysis is ca ied ou o he eac ion 'Sn+" Sn which shows he o e whelming dominance o
he di ec mul ipa icle ans e a bomba ding ene gies below he Coulomb ba ie . Resul s o sim-
ple es ima es —
supplemen ed by ull coupled-channel calcula ions —
p o ide o ien a ion o selec -
ing he expe imen al condi ions which bes exhibi he speci ic cha ac e is ics o he supe conduc-
i e phase. The possibili y o de ec ing an in e e ence pa e n be ween he wo p ocesses is also dis-
cussed.
Amac oscopic pic u e o analyzing he ans e o
pai s o iden ical nucleons in he case o supe luid nuclei
has been ecen ly in es iga ed. Ea lie a emp s' ocused
in he leading p ocess which, o g azing collisions, is
known o in ol e he ans e o asingle nucleon pai . A
his le el, he weak cha ac e o he couplings allows o
exp essions which a e o mally iden ical o he ones p e-
iously exploi ed o pai ing modes in he icini y o
closed shells. Amo e gene al app oach, howe e , e-
qui es he iden i ica ion o amac oscopic o m o he
nuclea densi y as a unc ion o he gauge angle I). In
Re . 3 he exp ession
Po
p( , P)=(I+expI [ —
R(P)]/a J)
was p oposed which ollows na u ally om he pa a ne-
iza ion o he nuclea adius as a unc ion o Pacco d-
ing o
R(P)=Ra I+ cos2$
330
He e 8os ands o he unpe u bed nuclea adius, while
he quan i y P» indica es he magni ude o he s a ic de-
o ma ion o he supe luid sys em in gauge space. This
quan i y hus plays a ole en i ely analogous o he P» pa-
ame e in Re . 2, which ga e ameasu e o dynamical de-
ia ions in pa icle numbe o nuclei in hei no mal
phase.
An in e es ing consequence o he mac oscopic pic u e
o pai o a ional bands de eloped in Re . 3is he de e -
mina ion o ansi ion densi ies o p ocesses in which he
pa icle numbe changes by an a bi a y amoun hA =k.
These a e simply iden i ied wi h he Fou ie componen s
o he gene alized densi y in he in insic ame p( , P),
1.e.,
5( )= p( , P)e'"~dP .
Mul ipa icle ans e da a o collisions wi h
supe luid sys ems a e al eady a ailable in he li e a u e.
Indeed, in expe imen s o he eac ion 'Sn+ "Sn e-
po ed in Re . 4, he ans e o ou neu ons leading o
esidual "Sn nuclei has been eco ded. Acon en ional
iew would associa e hese e en s wi h a wo-s ep p ocess
in ol ing he successi e ans e o neu on pai s.
Wi hin he mac oscopic app oach, howe e , he con i-
bu ion o adi ec ou -neu on coupling eeding he inal
"Sn +"Sn channel should also be conside ed.
In his B ie Repo we make quan i a i e es ima es o
he ela i e impo ance o he wo compe ing p ocesses.
We ake as aspeci ic example he eac ion s udied in Re .
4. As i u ns ou , he esul s a e s ongly dependen on
he choice o expe imen al condi ions. We hus de elop
some guidelines o enhance he no el aspec s in mul ipa -
icle ans e which a e allowed by he supe luid cha ac-
e o he in iso opes.
As as a ing poin we conside he eac ion ampli udes
o bo h he di ec and sequen ial p ocesses. Se niclassi-
cal exp essions o he lowes o de s a e deemed ap-
p op ia e since he egime o in e es is es ic ed o g az-
ing collisions induced by e y weak couplings. We hen
ake
ad;, =— "F4( )e' 'd
and
~'I
a„= — F2( )e 'd F2( ')e 'd '
37 1774 1988 The Ame ican Physical Socie y
37 BRIEFREPORTS 1775
o he one- and wo-s ep p ocesses, espec i ely. In he
p e ious exp essions F4( ) and F2( ) s and o he o m
ac o s o ou - and wo-pa icle ans e . These can be
cons uc ed om ansi ion densi ies as ob ained om
he mac oscopic pic u e. Using he ion-ion in e ac ion as
a e e ence o la ge dis ances one can w i e
5pq( )
F~( )- U( ),
5po "
5p4( )
F~( )- U( ) .
5pp
We ecall ha he ela i e magni ude o he ansi ion
densi ies becomes independen o he dis ance in he ail
egion and is essen ially de e mined by he alue o P.3
The quan i ies co„co2 appea ing in he sequen ial ampli-
ude co espond o he exci a ion ene gy associa ed wi h
he wo-pa icle ans e channels 'Sn(" Sn, "Sn)" Sn
and "Sn(" Sn, "Sn)" Sn. These a e bo h posi i e Q-
alue ansi ions which, in ou case, yield ico& —
—
—
2.34
MeV and ico2 —
0.83 M—
e—
V. In he ampli ude ad;„
en e s, on he o he hand, he o al balance o ene gies in
he ou -pa icle channel 'Sn(" Sn, "Sn)" Sn, i.e.,
CO=N~+C02.
The e alua ion o hese in eg als can be easily pe -
o med exploi ing he sha p exponen ial cha ac e o he
couplings. Expanding he ajec o y o ela i e mo ion
a ounds i s u ning poin o one ge s
"o o' +d& (i /2a +—
i aqi
L
1—
2(&Q —
~Q)/ ~
uq= F2(Rp) e
+CQ —
(g /2~ )+jyP2 —
( ' /2c )+icoI
d
e'd 'e
whe e o=+a/ 'c is ela ed o an e ec i e collision ime
h ough =8(log2)o. The accele a ion ' 'c a he dis-
ance o closes app oach can be accu a ely es ima ed
om he Coulomb o bi a ene gies below he ba ie E~.
Fo hese bomba ding ene gies he la ges c oss sec ions
a e ound a back angles and co espond o head-on col-
lisions. As he ene gy is g adually inc eased o e Ez he
ele an pa ial wa e (and he sca e ing angle) mus be
adjus ed so as o main ain he g azing cha ac e o he
p ocess. In his egime o- z while o —
and conse-
quen ly he elec i e collision ime —
emains amode a e
unc ion o he bomba ding ene gy (e.g.,Re . 5).
In Fig. 1(a) he a io ~ad;, /a ~is displayed as a
unc ion o he bomba ding ene gy o acha ac e is ic
alue P=52. This quan i y can be used o measu e he
ela i e magni ude o he di ec and sequen ial p ocesses.
Nei he he a io no e en he absolu e c oss sec ions a e
expec ed o ha e as ong dependence o ene gies abo e
he ba ie E~. In ac , he coupling o m ac o s a e
essen ially p obed in his egime a acons an adius
- z. The e is, on he o he hand, asha p quali a i e
change in he a io as he ene gy d ops below he ba ie .
This pa o he cu e —
mo e eliable because i is no
a ec ed by unce ain ies in he op ical po en ial —
shows
ad ama ic gain o he di ec mul ipa icle ans e o e
i s sequen ial coun e pa .
The la ge alues o he a io ~ad;, /a~q ~a e o
cou se ob ained as bo h nume a o and denomina o ap-
idly dec ease wi h he bomba ding ene gy. I is impo -
an o no e, howe e , ha he subs an ial gain in o de s
o magni ude is achie ed a an accep able cos in he size
o he c oss sec ions o di ec ans e . To illus a e his
poin we show in Fig. 1(b) he quan i ies ~ad;, ~and
~a„q ~as a unc ion o he bomba ding ene gy, no mal-
izing hei alues o one o E, =Ez. The cu es in
Fig. 1(b) can hus be used o es ima e he loss in coun ing
a e o be expec ed as he ene gy is lowe ed below he
ba ie o a o he di ec mul ipa icle ans e . We no e
ha measu emen s o one- and wo-nucleon ans e ha e
been ecen ly ex ended o ene gies well below he ba ie
by he in oduc ion o new de ec ion echniques.
IO
II6 )IIS
C7 U
IOO
IO
Io0
200 l
220 1I
240
E(MOV)
l
'
I
II2S ~ICOS I16S ~II6S
l
260 280 l
300
R
Io
lo di ec
—
—
-sequen ial
-20 0
-Es (MOV)
20 40
FIG. l. (a) Ra io ~ad;, /a~ ~ o he ou -neu on ans e
p ocess in he eac ion "Sn('2 Sn,"oSn)" Sn as a unc ion o
he bomba ding ene gy. The alue P~ =52 has been assumed o
he pai ing de o ma ion pa ame e . (b) T ans e p obabili ies
~az;, ~and ~a~ (as a unc ion o he bomba ding ene gy.
Bo h quan i ies a e no malized o uni y o E, =Ez.
1776 BRIEFREPORTS 37
TABLE I. Di e en ial c oss sec ion a maximum angle (columns wo and h ee) and o al c oss sec-
ions (columns ou and i e) ob ained in he coupled channel calcula ion o he eac ion
"Sn(' Sn,"Sn)"~Sn a di e en bomba ding ene gies. The labels indica e whe he he esul s ha e
been ob ained o he di ec o he sequen ial p ocess.
(MeV)
200
220
240
260
d~„,/do
(mb/s )
1.5X10-'
1.9X10 '
7.4y 10
2.6y 10-~
de q/dQ
(mb/s )
7.8X 10-"
1.3g10-'
3.4x 10-4
1.6g10-'
(mb)
2.8x 10-'
3.9x10-4
1.9x 10-'
1.4x 10-'
(mb)
7.0x10-"
1.3X10-'
4.0X10-4
6.0)(10
The esul s in Fig. 1co espond o agi en alue o he
pa ame e p.In he e y weak egime o wo- o ou -
pa icle couplings (we ecall he equi alen de o ma ion
pa ame e s o inelas ic exci a ions would be in he o de
o 10 ) he choice o p~ is no , howe e , c i ical. In ac ,
he second-o de cha ac e o he sequen ial ans e
compensa es he a ia ion in he a io (Fz/F4) so ha he
esul s displayed in Fig. 1a e ac ually ep esen a i e o
o he ypical alues o p.We also poin ou ha a
di e en balance be ween he di ec and sequen ial p o-
cesses could be ob ained by allowing o acos4$ — e m
in he mac oscopic exp ession o he densi y (c .,e.g.,
Re . 3).
Absolu e alues o he ans e c oss sec ions can be
ob ained by mul iplying he p obabili y o he di Fe en
p ocesses by he elas ic c oss sec ions a he co espond-
ing angles. To p o ide an independen check, howe e ,
we ha e ca ied ou mo e comple e calcula ions wi hin a
coupled channel app oach, supplying he s eng h o he
couplings acco ding o he mac oscopic p esc ip ion as
discussed abo e. The ion-ion po en ial o Re . 8was
-l
10 '"Sn("oSn '"Sn& '"Sn
E& ~= 260 MeV
IO
C)
E
blO ~o~~~o 4IQN
~".-"~---""Sequen lel
lO l20 i@0 I60 l80
FIG. 2. Angula dis ibu ion (solid line) ob ained in he cou-
pled channel calcula ion o he eac ion "Sn(' Sn,"Sn)" Sn
a bomba ding ene gy E, =260 MeV, including bo h di ec
and sequen ial con ibu ions. Also shown a e he angula dis i-
bu ions ob ained o he di ec (do ed linc) o sequen ial
(dashed line) p ocesses.
used, adding an imagina y componen o he same
geome y and hal i s s eng h. E en hough his pa am-
e iza ion may be somewha unce ain, he choice o he
op ical po en ial should no be c i ical in he egion o in-
e es , i.e.,well below he ba ie . Resul s o calcula ions
pe o med o ou bomba ding ene gies a e collec ed in
Table I. These suppo he conclusions d awn om he
semiclassical es ima es. In pa icula , he di e en a es
a which he indi idual di ec and sequen ial c oss sec-
ions d op o he lowes ene gies.
The ene gy scale in Table Imay ha e o be adjus ed
somewha in o de o compa e he calcula ions o expe i-
men al da a like he ones epo ed in Re . 4 o highe en-
e gies. In ac , he absolu e alues o he c oss sec ions
a e ex emely sensi i e o he ac ual posi ion o he
Coulomb ba ie . We no e ha he alue Ez -260 MeV
esul ing om he po en ial o Re . 8may di e by as
much as 10-20MeV om sys ema ic pa ame iza ions o
he Coulomb ba ie like he one gi en in Re . 9. Thus,
an e en ual compa ison wi h da a would bes be o mu-
la ed in e ms o he a iable E, —
Ez.
Judging by he esul s epo ed abo e, one may con-
clude ha E, -Ez ep esen s he op imal condi ions a
which di ec and sequen ial p ocesses compe e. This
aises he possibili y o de ec ing an in e e ence be ween
he wo cohe en ways o popula ing he ou -pa icle
ans e channel. While s essing he specula i e cha ac-
e o his p oposi ion we show in Fig. 2angula dis ibu-
ions o he "Sn +"Sn channel ob ained o
E, =260 MeV-Ez. The e is an app eciable di e ence
in he angula dis ibu ions o he di ec and sequen ial
p ocesses, leading o adisce nible in e e ence pa e n.
In pa icula , one may no e he accele a ed a e in which
he sequen ial angula dis ibu ions d op owa ds he o -
'wa d angles. This is again amani es a ion o he
diScul ies encoun e ed by he wo-s ep p ocess o keep
up wi h he di ec one o he la ge adial dis ances. In
asense, he end which eme ges as one mo es in o mo e
o wa d angles (which equi e he con ibu ion o la ge
pa ial wa es) can be pu in co espondence wi h he pa -
e n ob ained o ixed backsca e ing as he bomba ding
ene gy is lowe ed.
This wo k was suppo ed in pa by g an s om he
Bundesminis e ium u Fo schung und Technologie,
Fede al Republic o Ge many (unde G an No. LM 06
177 II) and he Comision Aseso a de In es igacion
Cien i ica yTecnica, Spain (unde G an No. 2868-83).
37 BRIEFREPORTS )777
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