Phymcs Le e s B 300 (1993) 303-307
No h-Holland PHYSICS LETTERS B
Th eshold anomaly in non-cen al o ces
J.
G6mez-Camacho and M.A. Naga ajan
'
Depa amen o de Fiswa A 6mwa. Molecula y Nuclea . Un e sMad de Se lla. 4pdo 1065. E-41OSO Se lla. Spa e
Recewed 9 No embe 1992
The beha mu o he h eshold anomaly o non-cen al po en ials, wh,ch accoun o collec we excl a ,ons m hea y-~on colh-
s~ons. ~s in es iga ed. I ~s shown hai he non-cen al po en mls should exhibi an ene g5 dependence a ene gies m he icini y o
he Coulomb ba ie Th~s ene gy dependence is, howe e , di e en om ha o he elas kc op ical po enual, occu ing a lowe
ene gies I u he shown hai he e a e co ec mns o he admonal collec wc model such ha . he ans~ mn po en ial ~s
exp essed as he de i a i e o he op ical po en ial, he co esponding de o ma ion leng h will be complex and ene gy-dependen
Simple model calculauons a e p esen ed
Du mg ecen yea s, e idence o he ene gy-depen-
dence o he eal ( he h eshold anomaly) and ima-
gina y pa s o nucleus-nucleus op ical po en ials has
been ound by ca e ul analyses o he elas xc sca e -
ing a ene gies m he cml y o he Coulomb ba e
[ 1-5] This ene gy dependence has been a ibu ed
o he couphng o non-clas c channels o he elas ic
channel n his ene gy egion [6.7]. The h eshold
anomaly has also been ela ed o he apid mc ease
o he su ace imagina y po en ial as he ene gy s m-
c eased abo e he Coulomb ba ie and he conse-
quen co ec ion o he eal po en ial h ough a dis-
pe sion ela ion [8-10]. The op ical po en ial can
hus be w i en as
U(E)= V o+AI'(E) + H(E), ( I )
whe e go is an ene gy-independen po en ial (dou-
ble- olded po en ial, o example), I,I'(E) s he m-
agma y po en ial and he dispe si e eal po enual
AI'(E) s de ined as
AV(E)= P I4'(E')
~dE', (2)
whe e P deno es a p mclpal alue m eg al.
Gene ahzlng his o he case o a se o s ong-cou-
pled channels, i was sugges ed by Sa chle [ 1 1 ] ha
Pe manen add ess SERC Da esbu y Labo a o 3., Da esbu s,
Wa mg on WA4 4AD, UK
each elemen o he po en ial ma ix, which de ines
he coupled channels equa ion, should be expec ed o
exh b a h eshold anomaly. In such a case, he en-
e gy dependence o he di e en diagonal and o -di-
agonal elemen s o he po en al ma ix will be a ib-
u ed o couphngs o o he channels no mcluded m
he subse o coupled channels. Howe e . a p io i,
he e s no eason o all he e ms o he po en ial
ma ix o ha e he same ene gy dependence. Sa chle
[ 1 1 ] a gued ha , m he case o collec i e exci a ions,
one would expec a simila ene gy dependence m he
coupling m e ac lon ( ansi ion po en ial) and he
elas ic op ical po en al, hough one could expec a
shi m he h eshold o he coupling po en ial by ap-
p ox ma el,, hal o he exci a ion ene gy. He u he
concludes ha one uses an op ical po en ial ha s
consis en wi h he dispe sion ela on bu s ill needs
an ene gy-dependen de o ma ion leng h, hen ei he
he op ical model i sel o he nuclea s uc u e model.
o bo h, a e madequa e
The exci a ion unc ions o he i s exci ed 3 - s a e
o 2°8Pb by ~60 a ene gies a ound he Coulomb ba -
e ha e been measu ed and analysed [ 12,13] and
mo e ecen ly de ailed angula d s bu ions o he
same sys em we e measu ed a ene gies om abo e
o below he Coulomb ba ie [14]. These angula
dis ibu ions we e analyzed and he esul s hin ed a
a poss b h y ha he ansi ion po en ial may ha e an
0370-2693/93/$ 06 00 © 1993 Else ie Science Pubhshe s B V All igh s ese ed 303
Volume 300, numbe 4 PHYSICS LETTERS B 18 Feb ua y 1993
ene gy dependence which is di e en om ha o he
elas ic opucal po en ml [ 14 ].
We p esen he e a sxmple model which shows ha ,
o collec i e exci a mns, he cen al and couphng
po en mls, al hough consis en sepa a ely wi h dis-
pe sion ela ions, could exhibi di e en ene gy de-
pendences No e ha coupling po en mls include he
case o conen auon po en ials o he case o elas ic
sca e ing o nuclm wq h 1>/I Le us conside he
usual collec i e model dese lp mn o inelas ic exc~-
a mn. The op ical po en ml ~_" ~s conside ed as a
unc mn o he collec i e a iable 6, which is a mea-
su e o he change m dis ance o he su aces o he
colhdmg nuclm. The eal and mlagma y pa s o he
po en ial can be expanded m a Taylo se ies m pow-
e s o (~, and o he i s o de md.
dl(~ dll.
I(,J)=l-~7 " , I~'(6)=~- d o. (3)
I he po en mls a e exponen ial in he egmn o m-
e es , hen
l'(d)=V(l+~/a), '(,5)=H'(l+d/a), (4)
whe e a is he di useness pa ame e , whlch has been
assumed o be he same o he eal and imagina y
po en ials. No e ha , when exp essing he ~magma y
po enual as a unc ion o & we a e imphc~ l~ assum-
ing ha he collec we a iable changes slow b, com-
pa ed
wHh
he p ocesses ha con ibu e o he ~ma-
gma y po en ml Tha implies also ha he ene gy o
he ba ie B will be a unc ion old. App oxima ely.
we can w i e
B(d)..-B( I-~/R) .
(5)
whe e B is he heigh o he unde o med bame and
R ~s he co esponding adms. To explo e he ~mph-
ca ons o he e ec o ~ on bo h he geome .~ o he
po en ml as well as on he bame , we conside a ac-
onzable o m o
I '(E)
H'(I..') = 14"l-'(. ). (`6)
whe e I " is a unc mn o he adms while F(x) ex-
p esses he ene gy dependence m e ms o he pa ana-
e e x=
(B-E)~.1. F(x)
~s assumed o be a Fe mi
unc mn F(x)= [ 1 + exp(. ) ]-~. The pa ame e J is
a measu e o he ene gy ange o e wh~ch he ,magl-
na y po en ml inc eases. Acco ding o eqs (4) and
(5), bo h W and "(x) a e dependen on he collec-
i e a iable 6.
Wi h he ansa z o I '(E) o eq. (6), he dispe -
si e po enual AI '(E) o eq. ( I ) becomes
aXI'(E) = I 'G(. ), (7)
whe e G (. ) is ela ed o F(x) h ough he dispe sion
ela mn
G(.x)=- p i ---=-F(V)dy. (8)
7/' I'--.V
-x
The op ical po en ial can be w i en as
/,,'(E. d)= I'(E. d)+11 '(E, d). (9)
To i s o de in d, one ob ains
I'(E, d)= I'o+ l '(/(. )
d
+ - [ I'+ I 'G(x)+
I 'eG'(. )I,
(10)
a
W(E. 6) = J l.(x)
d
+ - [ liT(x)+ I iF'(. )] , (l 1 )
a
whe e we ha e in oduced he pa ame e c =
aB/RA
and used eq. (4). In hese equauons, he diagonal po-
en ml U0(E) s he e m independen on d and he
ansmon po enual _',(E) is he e m linea in ~. We
can wine h~s exphcl l~ as
Uo(E) = I ~, + l!'G(x) +ll!'F(x). ( 12 )
,( E)
d(dU°(E)a )
=- ---+a- ~l'(G'(. )+d:'(x)) .
(13)
The i s e m on he igh o eq. (13) ~s he usual
p esc ip ion o he collec i e model whe e he an-
smon po en ml ~s ela ed o he de i a i e o he cen-
al po emml. The new ea u e o his model is he
second e m on he igh which a ises om he shi
o he ba ie due o de o mauon. Eqs ( 12 ) and ( 13 )
sugges ha , while he cen al e ms will exhibi he
h eshold anomaly a ound an ene gy
B,
he non-cen-
al e ms will exhibi i a ound an ene gy B( 1
-a/
R ). Besides, all he non-cen al e ms will exhibi he
h eshold anomaly a he same ene gy, indepen-
den ly o he mul ~pola l y o he couphng s eng h
304
Volume 300, numbe 4 PHYSICS LETTERS B 18 Feb ua y 1993
(p o ided ha a i s o de ea men s alid)
I is o m e es o poin ou ha exac ly he same
exp essions ( 12 ) and ( 13 ) a e ob amcd o he ccn-
al and anss on po en als i one conside s he adi-
aba ic hml when he coupled equa ons can be de-
coupled m o elgenchannels. In he case o including
he g ound s a e and a one-phonon s a e, he wo c~-
gcnchannels desc ibe he sca e ing by po en ials o
adix R + i, wi h he co esponding ba ie s B( I g i/
R). The ansi ion po en ial is hal he di e ence o
he wo cxgenchannel po en ials and up o o de d
yields exac ly eq. ( 13 ).
I one s dl de ines he coupling po en ial by
_ dU(E)
U (E)=-6¢ ~- " (14)
he a io o he c ec l e o he s a ic de o ma ion
leng h o he po en ial becomes complex and ene gy
dependen :
G' (A) " - I/~" (X)
R(E)= =l-c
l)~/W+G(x)+ F(x)"
(15)
Eq. (15) shows sepa a ely he con lbu ons o he
e ec i e dc o ma on o he s a ic de o ma ion and
o he dispe si e e ec due o he change in he ba -
le ene gies. R (E) acqui es an i magma y pa in he
VlCln y o he ba ie , o he o de o e W/4I'(B). The
eal pa o R (E) has an ene gy dependence a ound
he bame ha is consis en w h he magma y pa
h ough dispe sion ela ions.
The size o he co ec ion o he con en ional model
depends c uc ally on Ihc pa ame e e, ha is de e -
mined p ima ily by he ange A o e which he ma-
glna y po en ial ises ( o
E=B-A,
he imagina y
po en ial is 27%, while o
E=B+A,
i is 73% o he
maximum alue). Depending on he alue o ~, one
ob ains di e en shapes o he ene gy depcndcnce o
he imagina y pa o he ansi ion po en ial. Fo
< 1, he ~magina y ansmon po en ial ises mono-
omcally o a cons an alue 6d W/d . Fo ~ > 1. one
obse es a cha ac e is ic maximum in he imagina y
ans on po en ial
jus
abo e he ba ie , and hcn a
g adual dec ease o he alue dd
W/d
a la ge cne -
gies. Wc llus a e his in igs. 1 and 2, whe e wc show
bo h he elas ic (cen al) and ansi ion po en ials o
he inelas ic sca e ing o ~60 on z°8pb o he i s 3-
s a e o he a ge . Calcula mns ha e been done ak-
ing B=80 MeV, a=0 6 m and R= 12 m. Two al-
.... I
Fig I Cen al ( ull hne) and non-cen al (dashed hne) po en-
als e sus he ene gy o A=2 5 MeV The eal pa s a e he up-
pe hnes, and he imagina y pa s a e he lowe hnes
~,.? ?() 1(I¢'. I .'0 I 1~
l (g!W, :
F~g 2 Sameas ig llo A=455MeV.
ues o d. which a c J=4.55 MeV, aken om e . [9],
and A= 2.50 MeV, ha e been used. In bo h cases, he
op ical po en ial is aken as 1.5 + 11.2 MeV o E= 150
MeV. Bo h calcula ions a e consis en wi h he em-
pi ical da a o he op ical po en ial o
160+2°8Pb.
These yield alues o e o 0 88 and 1.6 espec i ely.
The ansi ion po en ial is no malized so ha i co-
incides wi h he cen al po en ial a high ene gies. I
is clea ly seen ha he ene gy dependence o he an-
si ion and cen al po en ials a e di e en in an en-
e gy egion a ound he Coulomb ba ie . E en hough
om igs. 1 and 2 i may appea ha ~< I, he a-
di ional collec i e model, whe e he ansl on po en-
305
Volume 300, numbe 4 PHYSICS LETTERS B 18 Feb ua y 1993
lal is aken as he p oduc o a s a ic de o ma ion
leng h and he de a we o he complex po en ial, is
alid, his is no s ic ly ue In ig. 3 we show he
eal and imagina y pa s o R(E) as a unc ion o he
ene gy o he wo cases 3=4.55 McV and 3=2.5
MeV I is seen ha e en o , =4.55 MeV (~< I ),
he de o ma ion leng h becomes complex. I is e i-
den , o cou se, ha e y high quah 3 inelas ic da a
will be necessa y o ex ac his complex de o ma ion
leng h a ene gies a ound he Coulomb ba ie
The pa ame e e can be unde s ood m a ime-de-
penden pic u e, =,5//I has bccn desc ibed as a e-
a da ion ime in he ime-dependen p~c u e o he
1magmas.' po en ial [ 15 ]. I is he cha ac e is ic ime
o he p ocesses ha con ibu e o he maagma y po-
en ial, ha is, o he couphng o s a es no ela ed
o he collec we a iable d.
aB/R
is app oxlma eb
equal o I" Classically, he hamil oman o a lb a-
Uonal collec i e a iable d is
II=P]/2M+Kd2/2-
l'cxp(d/a) ,
(16)
whe e
M
and K a e, espec wel. ., he ine ia pa ame-
e and he ha monic cons an o he collec i e a i-
able We can de ine he cha ac e is ic ime o collec-
i e exci a ion as
dl~/d
I/T= -- (17)
',~
"7
.
:z'6(] --" ~(; 1(;(: : ~;" I 1('
? I *d, ', :
Fig 3 Real (uppe hnes) and imagina y, (lo , e hnes) pa s o
lhe auo o he e ec i e and slallc de o ma mn leng hs as a une-
I on o he ene gy The ull hnes co espond o 3=2 5 MeV and
he dashed hnes o A=4 55 MeV
dP,,/d
call be ob ained om eq. (16) using Hamil-
on's
equa ion
dP~ dH d I"
- dd C + - a exp(:i/a).
(18)
d
The second e m on he RHS domina es o e he i s
in he adlabauc limi . On he o he hand, as 6 will
ake alues o he o de
o a,
we can es ima e he alue
o
P,,
as
h/a.
Thus,
T=hR/aB
is he cha ac e is ic
ime o collec i e exci a ions. Hence he pa ame e
= /T is he a ]o o he ~me o p ocesses ha ake
away lux om he coupled channels sys em o he
mae o he p ocesses ha couple mhm he coupled
channels sys em I < 7" hc couphng o non-collec-
we channels is as compa ed o he couphng o col-
lec we channels, and hen i s e ec on he diagonal
and ansi ion po en ials is simila , p oducing a s~m-
la ene gy dependence o bo h. I > 7-, he couphng
o non-collec i e channels Is slow compa ed o ha
o collec i e channels, and so a di e en ene gy de-
pendence appea s. No e ha hese condmons a e no
incompa ible wi h he assump ion ha he collec i e
a iable changes slowly compa ed wi h p ocesses ha
con ibu e o he imagma po en ial. The cha ac e -
is ic ime o change old can be de ined as
dd/d
1/7-- (19)
d
dd/d
can be ob ained om eq. (16) using Hamil-
on's equa ion
dd _ dll _ l',s
. (20)
dl
dP~ .ll
Hence. using he p e ious es ima es o (5 and/~, one
ge s T'=
31a'/h.
which can be la ge han i he in-
e ia pa ame e ~s la ge enough.
To conclude, we ha e shown ha he ansi ion po-
cn mls o collecm.e exci a ion ha e a di e en en-
e gy dependence o he diagonal po en ials. The de-
o ma mn leng h o he po en ml becomes complex
and ene gy-dependen a ound he Coulomb bame .
The size o his e ec depends c ucially on how as
he imagina y op ical po en ial ises o i s maximum
alue as he ene gy inc eases. These e ec s could be
in es iga ed measu ing he sca e ing o pola ized
p ojec iles a ene gies a ound he Coulomb ba ie .
306
Volume 300, numbe 4 PHYSICS LETTERS B 18 Fcb ua 3' 1993
J.G.C. acknowledges pa ial suppo om he
Spanish DGICYT unde con ac PB89-0636. M.A.N
wishes o hank he Spamsh DGICYT o enabhng a
sabba ical s ay a he Um e sl y o Se illa.
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307