PHYSICAL REVIEW CVOLUME 45, NUMBER 3MARCH 1992
Geome ic in e p e a ion o he e ec o he quad upole o ce in he collisions o de o med nuclei
M. V. And es and J.Gomez-Camacho
Depa amen o de FAMN, Uni e sidad de Se illa, Facul ad de Fs'sicas, Apa ado 1065, 41080Se illa, Spain
M. A. Naga ajan
Da esbu y Labo a o y, Wa ing on, WA4 4AD, Uni ed Kingdom
(Recei ed 15 Oc obe 1991)
The e ec o aquad upole o ce on ase o degene a e s a es o a o a ional band wi h a bi a y spin
p ojec ion along he symme y axis Kis s udied. Analy ic exp essions o he eigen alues and eigen ec-
o s a e ob ained in e ms o ase o o hogonal polynomials. This is applied o he collision o asphe i-
cal nucleus wi h ade o med one in which he coupling o agi en se o o a ional s a es is allowed, ig-
no ing exci a ion ene gies. The elas ic S-ma ix, ansi ion ampli udes, and he usion c oss sec ions a e
ob ained as aweigh ed a e age o he magni udes co esponding o ase o de ini e o ien a ions o he
axis o he de o med nucleus wi h espec o he ela i e coo dina e. Tha weigh ed a e age co esponds
o app oxima e he ex eme sudden esul , consis ing o an in eg al o e all he o ien a ions, by agen-
e alized Gaussian quad a u e.
PACS numbe (s): 24.10.Eq, 24.70.+s, 24.50.+g, 03.65.Nk
I. INTRODUCTION
The collec i e exci a ions o nuclei play a e y impo -
an ole in he eac ion mechanisms. The desc ip ion o
hese e ec s equi es he explici inclusion o he exci a-
ion by means o coupled-channels calcula ions. Fo he
case o o a ional nuclei, ap ope coupled-channels cal-
cula ion would equi e he inclusion o many s a es o he
o a ional band, making he calcula ion complica ed and
ime consuming, and complica ing he in e p e a ion o
he esul s. On he o he hand, when he o a ional
mo ion can be conside ed slow e sus he ela i e mo ion
(sudden app oxima ion) he sca e ing ampli udes can be
calcula ed as an in eg al ex ended o all he o ien a ions
o he o o o he sca e ing ampli udes calcula ed as i
he o ien a ion o he de o med nucleus was ozen du -
ing he eac ion, weigh ed wi h he p obabili y densi y
ha he g ound s a e has ha o ien a ion [l]. Also, i is
known ha when he e ec o he coupling o a es ic ed
se o exci ed s a es is conside ed, he exci a ion ene gy is
igno ed, he cen i ugal ba ie is sui ably app oxima ed
and he coupling o m ac o s ha e he same shape, he
coupled-channels sys em can be decoupled, and he
sca e ing ampli udes can be exp essed as acombina ion
o he ones co esponding o ase o uncoupled eigen-
channels [2—
5].
Naga ajan, Balan ekin, and Takigawa [6) con ibu ed
o b idge he gap be ween he coupled-channels calcula-
ion and he sudden esul demons a ing ha , o a
@=0 o a ional band, he coupled-channels e ec co e-
sponding o include he o a ional s a es om I=O o
I=2%—
2igno ing hei exci a ion ene gies, was
equi alen o do aweigh ed a e age o he ampli udes
co esponding o No ien a ions. These o ien a ions a e
cha ac e ized by he angle 0be ween he symme y axis
o he de o med nucleus and he ela i e mo ion, ha
akes he alues such ha P2&(cos0) =0. Mo eo e , ha
weigh ed a e age is p ecisely he combina ion ob ained
when he in eg al co esponding o he sudden app oxi-
ma ion is app oxima ed by an N-poin Gauss-Legend e
quad a u e.
The aim o his wo k is o sea ch o asimila esul
ha could be applied o a o a ional band o a bi a y E.
This is impo an i one is in e es ed in s udying he
inhuence o pola iza ion on he eac ion mechanisms, be-
cause, i he nucleus is o be pola ized, i s g ound s a e
needs o ha e spin di e en om ze o.
This pape is o ganized as ollows. In Sec. II we
e alua e he ma ix elemen o aquad upole in e ac ion
be ween o a ional s a es using he idal spin basis. In
Sec. III we map he o a ional s a es in o anew se o
s a es cha ac e ized by ase o o hogonal polynomials.
In Sec. IV we pe o m an analy ical diagonaliza ion in
he new basis. In Sec. Vwe discuss he meaning o eigen-
alues and eigen ec o s. In Sec. VI we apply his ea -
men o he calcula ion o sca e ing ampli udes. In Sec.
VII he ela ion o he geome ical limi is ob ained. Sec-
ion VIII is o summa y and conclusions.
II. COUPLING POTENTIALS
Le us conside he in e ac ion be ween a sphe ical nu-
cleus and an axially de o med one. The in e ac ion can
be w i en as
V( , ')=Vo( )+ V2( )P2( . g),
whe e is he ela i e coo dina e and g' s ands o he
di ec ion o he symme y axis o he de o med nucleus.
In his exp ession we ha e neglec ed spin-o bi e ms and
hexadecapole and highe -o de de o ma ion. The ma ix
elemen s o he quad upole in e ac ion be ween o a ion-
al s a es in he usual coupled-channels basis is gi en by
45 1339 1992 The Ame ican Physical Socie y
1340 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN
(IKLJI V2( )P~( g)II'K'L'J )
=V2( )5x ~,W(II'LL';2J)LI
x&I.o2olI. 'o) &IK20II'K), (2)
whe e Iis he in e nal angula momen um o he de-
o med nucleus and Kis he p ojec ion along he symme-
y axis, ha cha ac e izes he o a ional band. Lis he
o bi al angula momen um, and Jis he o al angula
momen um. I is known ha his in e ac ion can be pa -
ially diagonalized in he idal spin basis [7]
(IKMJI V,( )P,( g)II'K'M'J )
=V~( )5M ~5+ KI/I'(IM20II'M )(IK20II'K'),
whe e Mis he p ojec ion o he spin Ialong he ela i e
coo dina e ( idal spin). No e ha h oughou his pa-
pe , Mdeno es he idal spin while Kis he p ojec ion o
he spin along he symme y axis o he o o . Bo h mag-
ni udes a e conse ed by he coupling po en ial. No e
also ha he alue o Jdoes no a Fec he alues o he
coupling po en ial in he idal spin basis. So, in wha ol-
lows, we will d op he index Jin he cha ac e iza ion o
he s a es.
The coupling po en ials ha e acommon o m ac o
V2( ) and di e en s eng h ac o s. i he exci a ion en-
e gy is igno ed o agi en numbe o s a es, he coupling
ma ix could be diagonalized nume ically, and ase o
eigenchannels be ob ained, om which all he ele an e-
ac ion magni udes can be calcula ed [2—
S]. Howe e , he
eigen alues and eigens a es o he nume ical diagonaliza-
ion do no ha e any clea geome ical meaning.
The main di icul y in ex ending he analy ic diagonali-
za ion [6] o KPO bands is ha each s a e is coupled o
ou o he s a es, while o aK=0 band i is only cou-
pled o wo o he . Tha makes i di icul o make asui -
able unca ion in he s a es o he band included in he
calcula ion, and ye ob ain analy ic exp essions o eigen-
alues and eigen ec o s. An excep ion o his a e he
K=—,
'bands, because hey can be conside ed as K=O
bands o which aj=—,
'pa icle is coupled. Fo hese ana-
ly ic exp essions o eigen alues and eigen ec o s can be
ound.
III. MAPPING
To simpli y he complica ed coupling s uc u e o a
EWO band we wi11 in oduce ase o s a es ob ained ac -
ing wi h he P2( .g) ope a o o e he g ound s a e o he
band.
I1&=N, (P2lo& —
lo &&0IP, lo&),
n—
1
In &=N„P,ln —
1& —gli&&ilP, ln —
»
i=0
(4)
These s a es do no ha e, in gene al, good angula
momen um I. Explici exp essions o o e lap (nlIKM )
in he case K=—
', a e gi en in Table I(M =—
,
') and Table
II (M=—,
'). The s a es In )a e he combina ions o s a es
IIKM) ha a e mos ele an o he quad upole cou-
pling. %e expec ha he coupled-channels e Fec s due o
he o a ional s a es IIKM) om I=K o I=K+2N
would be e y simila o conside he s a es In) om
n=0 o n=¹In bo h cases, e ms o o de %+1in J'2
a e igno ed: No e ha coupling s uc u e is simpli ied,
because each o he s a es In )is only coupled o In +1)
and ln —
1). This way o gene a ing he ln )s a es is
known as he Lanczos me hod [8], which allows one o
ob ain some eigen alues and eigens a es qui e accu a ely
om only pa o he ull ma ix.
Al hough he s a es In)ha e acomplica ed expansion
in e ms o s a es o good angula momen um, hey ha e
a e y appealing exp ession when exp essed as asupe po-
si ion o s a es co esponding o de ini e o ien a ions o
he o o . The s a e IO) can be w i en as [9]
= (2K+1)/16m' Jdn[2P~~(n)InK )
+(—
i)2~gF, (n)lnK)],
(S)
whe e x=cosP and N(x) is ano maliza ion ac o . The
s a es lxKM) ha e he same pa i y as he s a es o he
o a ional band, and sa is y
whe e InK )is as a e ec o o he de o med nucleus
co esponding o an o ien a ion gi en by he Eule angles
n=(a,p, y) o he o o wi h espec o an ex e nal coo -
dina e sys em wi h he zaxis along he ela i e coo di-
na e, and ap ojec ion EC o he angula momen um along
he symme y axis. InK )is he ime e e sed s a e. We
can de ine he s a e lxEM )as
lxKM) =N(x) Jdady[2)xM(n)InK)
+(—
1)' 2) (n)lnK )], (6)
(nII )
TABLE I. Mapping coe icien o K=—,
',M=—. TABLE II. Mapping coe icien s o K=—
', ,M=—,
'.
&o
((I
&2 4465 7755
(o '-',
GEOMETRIC INTERPRETATION OF THE EFFECT OF THE. ..134i
TAHI.EIII. %eigh unc ions and o hogonal polynomials.
I( =—M=—
27 2K=—
',M= —
'
m(x) (1+3x )/4
1
Q—,
,
'2 (15x'—
7)
(133x —
126x +17)
(3—
3x )/4
I
Q—'
,(Sx'—
1)
Q64 (21x —
14x'+ 1)
I.-.lxKM &=MI.KM &,
(Ig') lxKM &=KlxKM &,
P2( .g)lxKM &=P2(x)lxKM&, '
(x'K'M'lxKM &=5x x5M ~5(x' x) .—
(7)
(8)
(10)
polynomials a e he e en Legend e polynomials. The an-
aly ic diagonaliza ion and he geome ic in e p e a ion
ob ained in [6] o he K=0 band elied on he o ho-
gonali y p ope ies o he Legend e polynomials. Thus,
we a e in he si ua ion o applying simila echniques o
ou case.
They can be in e p e ed as s a es o he o o in which
he axis o he o o and he ela i e coo dina e o m a
ixed angle p, bu hey a e a e aged o e all he Eule an-
gles aand yso ha Mand Ea e good quan um num-
be s. In e ms o hese s a es, he g ound-s a e wa e
unc ion is jus
IV. ANALYTIC DIAGONALIZATION
We will demons a e ha he (unno malized) s a e
I(()&= gQ„( p)l
n=0 (15)
lO&=lKKM&= dx w(x)' lxKM&,
whe e
w(x) =[[dhM(p)] +[d —
xM(p)] }.
4(12)
is an eigens a e o he ope a o P2( g) in he subspace
gene a ed by [ln &,n=O,N}, co esponding o he eigen-
alue P~(x&), i x& is aze o o QN+, .Fo ha , i should
be no iced ha he Q„,as o hogonal polynomials, sa is y
a ecu sion ela ion o ha can be w i en as (c . [10])
I is s aigh o wa d o see ha , in gene al,
In &= dx w(x)'~'g„(x')lxKM&, (13)
xQ„(x )=a„„,
Q„+,(x )+a„„Q„(x)
+a„„,
Q„,(x ), (16)
whe e Q„ is apolynomial o o de n ha sa is ies he
o hogonali y condi ion
(nlm &= dx w(x)Q„(x )Q (x )=5„. (14)
Using his equa ion and he ac ha Qo =1, one can gen-
e a e he polynomials. The explici exp essions o
w(x), go, g„and Q2 a e gi en in Table III, o K=—,
'and
2' 2'
No e ha , o aK=0band, he s a e ln &has good an-
gula momen um I=2n. The co esponding o hogonal
I
+b„„,
g„,(x ).
Using ha ela ion, i is s aigh o wa d o see ha
(17)
whe e ao, =0. The coeScien s in he ecu sion ela ion
depend on he no maliza ion o he polynomials. In ou
case, he polynomials a e no malized o 1[see Eq. (14)],i
can be seen [10] ha a„„+,=a„+,„Simila ly, one can
w i e
P2 (x)g„(x')=b„„+1Q„+1(x
')+b„„Q„(x
')
NN
gQ.(3') 2(x)Q.(x')= XQ.(3') 2(3)g (x )+bNN+1[QN(3 )QN+1(x )QN(x )QN 1(+3')] '
n=0 (18)
Using hese esul s, we can w i e
P,( g)lg&=P, (x&)lg&
+bNN+1[QN(xp)IN+1&
—
QN+1(xp)IN &] .
Thus, i he ope a o is es ic ed o he subspace gene a -
ed by [ln &,n=O,N} he e m p opo ional o lN+1&
cancels. I x& a e aken as he ze os o QN+„ he e m
p opo ional o lN & anishes, and we a e le wi h he e-
sul we wan ed o demons a e.
No e ha QN+, ,as an o hogonal polynomial, has
N+1ze os in he in e al (0,1), which would co espond
o N+1eigen alues and eigens a es o P2. The eigen al-
ues o P2 a e shown in Tables IV and V. Using he
Ch is o el-Da boux o mula, one ge s
(ply&= yg„(,
')g„( ', )=0
n=0
i x& and x& a e di e en ze os o QN+, .Thus, we
con6 m ha he eigens a es a e o hogonal. Finally, he
eigens a es can be no malized so ha
1342 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN 45
TABLE IV. Eigen alues o he P2 ope a o o K=2,M=2.
/=4
N=O
N=1
N=2
N=3
N=4
0.2000
—
0.2556
—
0.3871
—
0.4367
—
0.4599
0.6766
0.2303
—
0.0312
—
0.1811
0.8294
0.4984
0.2385 0.8953
0.6512 0.9294
n=O
(n~y) =Q„(xp) gQ(xy)
i=0
V. INTERPRETATION OF THE EIGENVALUES
AND EIGENSTATES
(21)
The alues o ~ „(13)~ ep esen he p obabili y densi y o
ha ing an angle Pbe ween he axis o he o o and he
ela i e coo dina e. They a e plo ed in Figs. 1and 2 o
K=—,
',n=0,1,2. They do no p esen any p e e ed
o ien a ion, and esemble he quali a i e beha io o he
modulus squa e o sphe ical ha monics.
I we use he equi alen exp ession o he eigens a es
I should be no iced ha he s a es ~n ) o m acom-
ple e basis o all he combina ions o s a es ~xKM ) ha
a e e en in x. Thus, as a e Q—,
'(~xKM)+~ —
xKM)),
ha co espond o an axially symme ic de o med nu-
cleus whose symme y axis o ms a ixed angle wi h he
ela i e coo dina e, can be expanded in e ms o he
s a es ~n )as
Q—,
'(~xKM )+~xKM ))=—
&2w (x)gQ„(x )~n).
n=0
(22)
I his expansion is unca ed up o n=X, we will ha e
he s a e in he subspace gene a ed by he basis
[~n),n=O,N] ha esembles mos closely o as a e
wi h a ixed o ien a ion. This unca ion is mo e accu a e
o he alues o xso ha Q~+~(x )=0. Thus, we can
in e p e he eigens a es o P2( g) in asubspace gene a -
ed by I~0) .~N)] as he combina ion o s a es ha
esemble mos o s a es o ade ini e o ien a ion. This in-
e p e a ion is s eng hened by he ac ha he eigen al-
ues o P2( g) in he unca ed subspace coincide wi h
hose co esponding o he s a es o de ini e o ien a ion
~x&KM ), ha a e P2(x&).
To illus a e his, we can exp ess he s a es (n )as a
combina ion o s a es co esponding o an angle Pbe-
ween he axis o he o o and he ela i e coo dina e:
/P) = dP ~(P)/PKM &(24)
we ind ha he alues o ~ &(p)~(see Figs. 3and 4) ha e
signi ican alues a o ien a ions close o he angles p~
such ha hei cosines gi e he ze os o QN+ ~(x )&»
ge s bigge , he unc ion ~ &(p)~becomes a 5 unc ion.
VI. APPLICATION TO SCATTERING
IN THE SUDDEN LIMIT
The in e ac ion be ween he nuclei conse es bo h he
idal spin Mand he p ojec ion o he spin along he sym-
me y axis o he o o E. Howe e , ha is no ue o
he u11 Hamil onian. The cen i ugal e m o he ela i e
mo ion changes M, while he Co iolis e m o he in e nal
Hamil onian o he o o changes E. Howe e , he
Co iolis e m can be neglec ed o no e y high in e nal
angula momen um I, and he cen i ugal e m can be
subs i u ed by an a e age alue (isocen i ugal app oxi-
ma ion [7]) o hea y-ion collisions. No e ha , when he
g ound-s a e angula momen um I=0 and one is discuss-
ing elas ic sca e ing, he isocen i ugal app oxima ion is
equi alen o igno ing he Co iolis o ce o he ela i e
mo ion [2]. The isocen i ugal app oxima ion implies
ha Mis conse ed in he sca e ing p ocess, while
neglec ing he Co iolis o ce o he in e nal deg ees o
eedom implies ha Kis conse ed.
Any magni ude ela ed o he sca e ing o wo pa i-
cles can be ob ained in e ms o he Sma ix. This, in
TABLE V. Eigen alues o he P2 ope a o o K=~,M=~.
N=0
N=1
N=2
N=3
N=4
—
0.2000
—
0.3780
—
0.4343
—
0.4590
—
0.4720
0.3780
0.0252
—
0.1574
—
0.2606
0.6399
0.3187
0.1008 0.7683
0.5068 0.8393
45 GEOMETRIC INTERPRETATION OF THE EFFECT OF THE. ..1343
1.01.0
0.80.8
0.40.4
0.20.2
0.0~/6 P(»d)
0.00
P( ad)
FIG. l. ~ „(P)~' s he angle P o K=—
', ,M=2. The solid
line co esponds o n=0, he dashed line o n=1,and he do -
dashed line o n=2.
FIG. 2. ~ „(p)~2 s he angle p o K=z,M= z. Same n«a-
ion as Fig. 1.
gene al, will be a unc ion o he o al angula momen um
J, he incoming and ou going o bi al angula momen a
L,L he incoming and ou going in insic angula mo-
men a I,I', and hei p ojec ion along he symme y axis
K,K'. When he Co iolis e m can be igno ed, he Sma-
ix is diagonal in K. Besides, when idal o ces dom-
ina e, and he isocen i ugal app oxima ion can be done
[7], he Sma ix elemen s can be w i en in e ms o he
idal spin Sma ices as
~J
~L'I'K', LIK
=5~x Ph g(LOIM~JM)(L'OI'M~JM)SI I
M
(25)
n, n'=0 (I'KM ~n')S, (n~IKM ).(26)
The coe icien s (nIKM )can be ob ained in as aigh -
o wa d (bu labo ious) way om he de ini ion o he
s a es ~n ). Some o hem a e p esen ed in Tables Iand
II.Now i one uses he basis ~(() ) ha diagonalizes he in-
e ac ion, i will also diagonalize he Sma ix, and one
ge s
N+1
S„„=y(n' ( )S (xy)(y~n ).
/=1
The Sma ix S(x&) is he one ob ained om aone-
channel op ical model calcula ion wi h he po en ial
whe e Mis he idal spin, L=(L+L')/2, and Ph is a
phase ac o in ol ing Coulomb phase shi s. Now, i we
conside he coupling o he o a ional s a es om I=K
o K+2N, igno ing he exci a ion ene gies, we can use
he mapping discussed p e iously o ge
2.0
V~( )= Vo( )+ V2( )P2(x&) .
The elas ic Sma ix o he g ound s a e is jus
N+1
SLY y~MKSL(
(I5 =1
whe e
(28)
(29)
1.5
2.0
1.0
0.5
0.00
/
/X
pp /ipe
e= e-
/37T/6
P(»d)
FIG. 3. &(P)~ s he angle P o K= 3,M= —'. The solid
line co esponds o /=0, he dashed line o P= I, and he do -
dashed line o /=2.
1.0
0.5
0.00 /6
P( ad)
FIG. 4. ~ &(P)~ s he angle II o K= 3,M= —.Same no a-
ion as Fig. 3.

1344 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN
TABLE VI. Weigh ac o s o E=—
', ,I=—
', .
N=O
N=1
N=2
N=3
N=4
1.0000
0.5113
0.3145
0.2238
0.1733
0.4887
0.4120
0.2973
0.2218
0.2735
0.3068
0.2587 0.1721
0.2286 0.1176
N
MK 22
i=0 (30)
A~,~(8)=Xd,M
M
AM(8)dK
(32)
The usion c oss sec ions, de ined as he c oss sec ion ha
does no appea in he channels included explici ly in he
calcula ion, o agi en alue he angula momen um J
and o he idal spin Mcan be calcula ed in e ms o
hose o he uncoupled eigenchannels and one ge s
N+1
JM MK J~
/=1 (33)
No e ha he usion c oss sec ions o agi en p ojec ion
o he spin o he p ojec ile along he beam di ec ion m
can be ela ed o he usion c oss sec ions o agi en idal
spin Musing idal symme y [11]
6JM
OK
J
K, m(34)
m, M
No e ha g&w4, =1. The weigh ac o s w& a e
shown in Tables VI and VII. As he weigh ac o s do
no depend upon L, he sum o ob ain he sca e ing am-
pli udes as a unc ion o he sca e ing angle o agi en
idal spin can be pe o med, and one ge s
N+1
Az~z(8) =gw& A(x&,8) .(31)
/=1
No e ha idal symme y leads o he ac ha he ansi-
ion ampli udes co esponding o p ojec ions m, m'o he
spin along he beam di ec ion a e gi en by
ha he momen o ine ia o he o o is e y la ge, and
so he o ien a ion o he o o is ixed du ing he sca e -
ing p ocess, he sca e ing ampli udes will be gi en in
e ms o he a e age o e all he o ien a ions, weigh ed
wi h he p obabili y densi y ha hey occu in he g ound
s a e [1]:
Sz~= Jdx w(x)S (x), (35)
Az z(8)= dx w(x) A(x,8),
1
oP= dx w(x)o (x) .(37)
—
1
(36)
Le P(x) be ase o polynomials ha a e o hogonal
wi h espec o he weigh unc ion w(x). One can ap-
p oxima e he in eg als as ollows:
1N'+ 1
Jdx w(x) (x)= gA~ (x~), (38)
—
11
N'
gP(x&)P (x)
L~(x)= gP(x~)
(40)
Using he o hogonali y o P(x) one ge s
N'
whe e x& a e he ze os o he polinomial PM+, (x), and
A&a e weigh s gi en by he exp ession
A&= Jdx w(x)L&(x) .(39)
—
1
L&(x) is he Lag ange mul iplie unc ion [12] ha can be
w i en as
A~= gP(x~) (41)
whe e OJ is he classical sca e ing angle ha co esponds
o an angula momen um J.
VII. RELATION TO THE GEOMETRICAL LIMIT
I he exci a ion ene gy o all he s a es o he o a ion-
al band can be igno ed, which is equi alen o assuming
m=0
The unca ion e o is o he o de o he 2N'+ 2de i a-
i e o (x)
In he case ha w(x) and (x) a e e en unc ions o x,
only he polynomials Q„(x )a e ele an , and one can
w i e he exp ession
TABLE VII. Weigh ac o s o I( =2,M=—,
'.
N=O
N=1
N=2
N=3
N=4
1.0000
0.7646
0.5916
0.4779
0.3995
0.2354
0.3325
0.3380
0.3167
0.0759
0.1532
0.1911 0.0309
0.0779 0.0147
GEOMETRIC INTERPRETATION OF THE EFFECT OF THE. .. 1345
1%+1
dx w(x) (x)= gB& (x&),
—
1P—
1
whe e x& a e he posi i e ze os o Q~+,(x ), and
1N
B~= dx w(x)L~(x )= gQ„(x~)
—
1n=0
(42)
(43}
The unca ion e o is o he o de o he 4N+4 de i a-
i e o (x).
Compa ing hese exp essions wi h he ones ob ained
be o e, we conclude ha he e ec o including he cou-
pling o he s a es in ) om n=0 o N, o he o a ional
s a es iIKM ) om I=K o K+2N, igno ing he exci a-
ion ene gies, is equi alen o pe o ming an (N+1)
poin Gaussian in eg a ion on he geome ical exp es-
sions o hese magni udes. Mo eo e , he coupling o he
s a es I=K+2N+1 and I=E+2N+2 is going o be
impo an o amagni ude such as he elas ic sca e ing
o he eac ion c oss sec ion when he geome ical ex-
p ession o ha magni ude as a unc ion o he o ien a-
ion angle has asigni ican con ibu ion o he 4N+4
de i a i e.
s a es o he o a ional band in acoupled-channels calcu-
la ion will a ec signi ican ly he elas ic magni udes (S
ma ix and ansi ion ampli udes} and he usion c oss
sec ions when he pa ame ic exp ession o hese magni-
udes as a unc ion o he cosine o he o ien a ion angle
p esen s signi ican con ibu ions o he 4N+4 de i a-
i e.
As a inal commen , we hink ha he analy ic diago-
naliza ion ound in his wo k and in o he cases discussed
in [6] is no jus ama hema ical cu iosi y. One should
expec o ha e i o o he kinds o collec i e exci a ion.
An algeb aic ea men o he diagonaliza ion o he col-
lec i e exci a ion, based on g oup heo y, can be an al e -
na i e o he analy ic app oach p esen ed he e, based on
he p ope ies o he o hogonal polynomials, and i can
p o ide adeepe unde s anding o he e ec o in e nal
deg ees o eedom on he eac ion mechanisms.
This wo k was pa ially suppo ed by he Acciones In-
eg adas HB-196 and he Spanish DGICYT PB89-0636.
APPENDIX: COMPARISON OF THE ANALYTIC
RESULTS WITH NUMERICAL DIAGONALIZATION
VIII. SUMMARY AND CONCLUSIONS
The quad upole coupling in a o a ional band wi h
KAO can be simpli ied i he s a es wi h de ini e angula
momen um a e mapped in o ase o s a es in )ob ained
ac ing wi h he P2 ope a o on he o a ional g ound
s a e and o hogonalizing. These s a es co espond o a
supe posi ion o s a es wi h de ini e o ien a ions, weigh -
ed wi h ase o o hogonal polynomials. In gene al, he
s a es in )do no co espond o s a es o good angula
momen um and hence he unca ion o he basis s a es
~n )is, excep o he case o K=0 and IC =—,
'bands, dis-
inc om he unca ion in e ms o he membe s o he
g ound-s a e o a ional band. Ne e heless, he esul s
ob ained when unca ing in hese wo di e en ways a e
e y simila , as i can be seen in he Appendix.
The eigen alues and eigen ec o s o he P2 ope a o in
basis de ined by he s a es in ) om n=0 o Na e ob-
ained in e ms o he ze os o he o hogonal polynomial
o o de N+1. The eigen ec o s can be in e p e ed as
he combina ion o s a es in ) om n=0 o N ha mos
esemble as a e wi h de ini e o ien a ion o he symme y
axis o he o o wi h espec o he ela i e coo dina e.
The eigen alues co espond o P2(x), whe e xis he
cosine o he o ien a ion angle.
The Sma ix, elas ic ansi ion ampli udes, and usion
c oss sec ions in he idal spin basis co espond o a
weigh ed a e age o hese magni udes co esponding o
he o ien a ions ha de ine he eigen alues and eigen ec-
o s o P2.
The classical sudden esul o hese magni udes co e-
spond o an in eg al o e all he o ien a ion angles,
weigh ed wi h he p obabili y ha agi en o ien a ion
occu s in he g ound s a e. The weigh ed a e age de-
sc ibed be o e co esponds o agene alized Gaussian
quad a u e o he sudden in eg al exp ession.
The e ec o he inclusion o he 2N+1 and 2N+2
AcoR bE—
Vb
o. =ln 1+exp 2m.
2E %co (A1)
The ba ie heigh depends on he o ien a ion o he o-
o wi h espec o he ela i e coo dina e as
Vb =Vp+ V2P2(cosg). Hence, wi hin he sudden app ox-
ima ion, he usion c oss sec ion will be gi en by
AcoR b
O = gW; 2E
E—
Vo —
V2P2
Xln 1+exp 2w (A2)
The coupling ma ix o Eq. (3) can be diagonalized nu-
me ically, including he o a ional s a es iIKM) om
I=K o I=K+2N. Thus, one ob ains 2N+1 eigen al-
ues and eigen ec o s, om which he sca e ing magni-
udes can be ob ained. Howe e , he analy ical diagonal-
iza ion including he s a es ~n) om n=0 o Ngene a es
N+1 eigen alues and eigen ec o s. Despi e his ac , we
will show ha he p edic ions o sca e ing magni udes
happen o e y simila in bo h cases.
When Ko M ake he alue 0, only he alues o I
wi h IKe en a e—
coupled. The s a es in) coincide
wi h he s a es o agi en angula momen um I=E+2n.
When Ko M ake he alue —,
', o he 2N+1 eigens a es
o P2, Na e o hogonal o he g ound s a e, and hence do
no a ec he eac ion mechanism. The emaining N+1
coincide wi h he eigens a es in he basis in) and ha e
he same eigen alues. This is due o he ac ha o a-
ional s a es wi h K(o M) equal o —,
'can be conside ed
as o a ional s a es wi h Ko Mequal o 0 o which a
pa icle wi h j=—,
'is coupled. Fo he o he cases, he
eigens a es and eigen alues in he iIKM) basis di e
om hose in he ~n)basis.
Le us conside he Wong o mula o he usion c oss
sec ion
M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN
TABLE VIII. Enhancemen ac o s o he usion c oss sec ion calcula ed in he ~n)basis and in he
~Il M )basis.
Imax
3
2
7
2
11
2
15
2
19
2
F{n)
7.3891
424.29
1098.3
1375.9
1435.2
E«V, F(I)
7.3891
488.36
1150.6
1390.3
1437.4
F(n)
3.0685
4.8265
4.7079
4.6719
4.6989
E=Vo F(I)
3.0685
4.7463
4.7081
4.6784
4.6950
F= gw;exp 2m V2 P2 (A3)
When he ene gy coincides wi h he unde o med ba ie
Vo
whe e P2 is he i h eigen alue o Pz and w, is he squa e
o he o e lap o he g ound s a e wi h he i h eigens a e.
Le us de6ne an enhancemen ac o Fas he a io o he
coupled-channels usion c oss sec ion o he uncoupled
c oss sec ions. When he ene gy is well below he ba ie ,
ha ac o is gi en by
N
F= gw, ln 1+exp 2m V2 P2 ln2 (A4)
In Table VIII we p esen he e alua ion o he ac o F
o anucleus wi h K=M =—,
', calcula ed in he basis ~n)
and in he basis ~IKM ). We ha e aken 2 V2/ iw =10,
ha is easonable o he sys em Na +Pb. The
ag eemen is ema kable, and jus i6es he mapping pe -
o med in Sec. III.
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