PHYSICAL REVIEW CVOLUME 52, NUMBER 6
Tenso analyzing powe s o Li b eakup
DECEMBER 1995
N. J. Da is, C. H. Shephe d-Themis ocleous, *A. C. Sho e , T. Da inson, D. G. I eland, ~K. Li ings on, ~
E. W. Macdonald, ~R. D. Page, ~P. J. Sellin, and P. J. Woods
Depa men o Physics and As onomy, Uni e si y o Edinbu gh, May ield Road, Edinbu gh EH9 3JZ, Sco land
N. M. Cla ke, G. Tunga e, J.A. R. G i i h, S. J. Hall, O. Ka ban, ~I. Ma el-B a o, **and J. M. Nelson
School o Physics and Space Resea ch, Uni e si y o Bi mingham, Edgbas on, Bi mingham B152TT, England
K. Rusek
Sol an Ins i u e o Nuclea S udies, Zaklad I, Hoza 69, 00 68I Wa saw, Poland
J. Gomez-Camacho
Depa men o de FAMN, Facul ad de Fisicas, Uni e sidad de Se illa, Ap do. 1065, 41080 Se illa, Spain
(Recei ed 5July 1995)
Di e en ial c oss sec ions and Tzo and T20 analyzing powe s ha e been measu ed o 70 MeV Li b eakup
in o he apa icle plus i on channel, on a'Sn a ge . Measu emen s we e made o bo h con inuum b eakup
and sequen ial b eakup ia he 4.63 MeV s a e in Li. The T20 da a o he con inuum b eakup do no ag ee
wi h asemiclassical Coulomb model, indica ing ha he b eakup a small angles does no p oceed solely ia a
Coulomb o ce. The da a gene ally show asomewha be e ag eemen wi h con inuum disc e ized coupled
channels calcula ions, indica ing he impo ance o he nuclea o ce and channel coupling in he eac ion
mechanism.
PACS numbe (s): 25.70.Mn, 24.10.Eq, 24.70.+s
I.INTRODUCTION
The nplus i on clus e s uc u e o Li esul s in ala ge
b eakup yield o hese agmen s. Ade ailed s udy o he
b eakup is o in e es because wo mechanisms ha e been
obse ed [1—
3], sequen ial b eakup ollowing exci a ion o
he Li, in pa icula o he 4.63MeV 7/2 s a e, and
b eakup in o he apa icle plus i on ene gy con inuum.
In addi ion o he nuclea physics in e es , b eakup eac-
ions may also be used o in e low ene gy pho ocap u e
c oss sec ions necessa y o an unde s anding o he c ea ion
o Li in he big bang s anda d model. I is di icul o ob ain
he pho ocap u e da a di ec ly a he low ela i e ene gies
which a e mos impo an because Coulomb epulsion be-
ween he upa icle and he i on esul s in e y small c oss
sec ions. ACoulomb b eakup eac ion may be used ins ead,
bu i is essen ial o de e mine he impo ance o he nuclea
o ce in he eac ion mechanism, as any con ibu ion om i
may in alida e he in e ence o pho ocap u e c oss sec ions
P esen add ess: CERN, CH-1211, Gene a, Swi ze land.
~P esen add ess: Depa men o Physics, Uni e si y o Glasgow,
Glasgow, UK.
~P esen add ess: Gene al Acciden Insu ance, Pe h, UK.
~P esen add ess: Depa men o Physics, Uni e si y o Li e pool,
Li e pool, UK.
P esen add ess: Depa men o Physics, Uni e si y o SheNeld,
She ield, UK.
~P esen add ess: Velice 26, 37351 D i en Czech Republic.
P esen add ess: Depa men o de FAMN, Facul ad de Fisicas,
Uni e sidad de Se illa, Ap do. 1065, 41080 Se illa, Spain.
The con inuum b eakup yield is s ong a o wa d angles
and alls o apidly a la ge angles [2]. The con inuum
b eakup can be explained by he di e en ial s ong nuclea
o ce be ween he a ge and agmen s [5], in an npa icle
plus i on clus e desc ip ion o Li. Howe e , he Coulomb
o ce becomes impo an a ex eme o wa d angles [6].
Di e en ial c oss sec ions o he Li con inuum b eakup
on a'Sn a ge ha e p e iously been measu ed and ound
o ag ee a small angles wi h acalcula ion which assumes a
pu e Coulomb o ce [2,3,6,7]. The c oss sec ions a e calcu-
la ed using asemiclassical app oxima ion [8] in which he
mo ion o he p ojec ile along ahype bolic ajec o y is pa-
ame ized in he amewo k o i s o de pe u ba ion heo y
by dimensionless o bi al in eg als. In o de o de e mine he
di e en ial c oss sec ion he educed ansi ion p obabili y as
a unc ion o ela i e ene gy is equi ed. This may be ob-
ained [9] om da a on he in e se usion eac ion [10].Thus
he semiclassical Coulomb calcula ions o di e en ial c oss
sec ion equi e inpu based on o he expe imen al da a. Semi-
classical calcula ions o analyzing powe s, howe e , do no
ely on addi ional expe imen al da a. This is because he e-
duced ansi ion p obabili y is independen o spin subs a e
and he e o e cancels. Semiclassical Coulomb calcula ions
ha e been success ully applied o Li elas ic and quasielas ic
sca e ing Tzo analyzing powe s [11].In he cu en wo k
semiclassical Coulomb calcual ions a e de eloped o
b eakup analyzing powe s. The analyzing powe s may be
mo e sensi i e o con ibu ions om di e en o ces han he
di e en ial c oss sec ion because hey depend on in e e -
ence e ms be ween sca e ing ampli udes. Acompa ison o
calcula ions wi h he da a he e o e p o ides an addi ional
es o he impo ance o he di e en o ces. In pa icula , i
he Coulomb o ce domina es, de ia ion o he con inuum
0556-2813/95/52(6)/3201(11)/$06. 00 3201 0& 1995 The Ame ican Physical Socie y
3202 N. J. DAVIS e al. 52
b eakup da a om he semiclassical Coulomb calcula ions
could occu o e en asmall con ibu ion om he nuclea
o ce in he b eakup eac ion mechanism.
Con inuum disc e ized coupled channels (CDCC) calcula-
ions, which include anuclea o ce, may also be pe o med
o compa e wi h he da a. Such calcula ions ha e been ound
o p o ide a e y good desc ip ion o c oss sec ions and ana-
lyzing powe s o elas ic and inelas ic sca e ing o Li [12—
14].CDCC calcula ions, in ol ing anuclea o ce only, ha e
p e iously been compa ed wi h di e en ial c oss sec ions o
Li con inuum and sequen ial b eakup on a'Sn a ge [12].
Reasonable ag eemen was ob ained excep a he o wa d
angles. An in es iga ion o he e ec o he Coulomb o ce
on he sequen ial b eakup was pe o med by including a
Coulomb in e ac ion de i ed by olding he Coulomb po en-
ial be ween anucleon and he a ge wi h he ansi ion den-
si ies be ween Li s a es. The Coulomb o ce was ound o
con ibu e signi ican ly a o wa d angles. Asimila in es i-
ga ion o he e ec o he Coulomb o ce on he con inuum
b eakup was no , howe e , pe o med. In he cu en wo k
CDCC calcula ions a e applied o b eakup analyzing powe s,
p o iding a e y impo an new es o he CDCC app oach.
Any b eakup p ocess esul ing in wo agmen s is a
h ee-body eac ion wi h wo eac ion planes de ined by he
inciden pa icle and he de ec ed agmen s. Consequen ly
he choice o coo dina e sys em is gene ally no ob ious,
al hough i is impo an o he heo e ical in e p e a ion o
he da a ha i be as simple as possible, pa icula ly when
pola iza ion e ec s a e in ol ed. Clea ly he simples si ua-
ion co esponds o an expe imen al a angemen when he
wo eac ion planes nea ly coincide (see Sec. IV) and as an-
da d wo-body eac ion coo dina e sys em can be de ined. I
should be explained ha in he semiclassical and CDCC cal-
cula ions he unde lying h ee-body eac ion is in e p e ed by
means o a wo-body eac ion. In o de o do his assump-
ions abou he basic eac ion mechanism ha e o be made.
These a e ha he inciden Li wi h spin s=3/2 and p ojec-
ion nis exci ed o acon inuum o esonan s a e cha ac e -
ised by spin s'and p ojec ion yand ha his decays spa ially
in o an npa icle and a i on wi h a ela i e angula mo-
men um L, whe e L+1/2=s', and asubs a e mz, whe e
mz+m, =yand m, is he i on spin subs a e. Only in he
case o J=0, when he b eakup is iso opic, can he analyz-
ing powe s calcula ed by ei he inelas ic sca e ing heo y be
compa ed di ec ly wi h he expe imen al da a. Gene ally, he
measu ed analyzing powe s ha e o be compa ed wi h p e-
dic ed quan i ies in ol ing pola iza ion ans e coe icien s
and co ela ion unc ions (see Sec. III).
The aim o he cu en wo k is ameasu emen o he
Tpp and Tzp analyzing powe s o Li b eakup, in o de o
de e mine he impo ance o he Coulomb and nuclea o ces
in he con inuum b eakup eac ion mechanism and in es i-
ga e he applicabili y o semiclassical and CDCC calcula-
ions.
II.SEMICLASSICAL COULOMB CALCULATIONS
Tkq
(s s
Xp~ssk( —
1)'
~.p,slF,sPI'
F~P*F~'P
y6 y6
whe e n' =u+ qand he usual no a ion k= /2k+ 1is used.
Fo aspin ze o a ge and esidual nucleus, his exp ession is
simpli ied because P= 8=0.
I he mo ion and Coulomb in e ac ion be ween he p o-
jec ile and a ge can be desc ibed semiclassically, i may be
pa ame ized in he amewo k o i s o de pe u ba ion
heo y by dimensionless o bi al in eg als Rz (H, j) [8].The
po en ial used in he ime-dependen Sch odinge equa ion
o he pa icle mo ion may be expanded in e ms o compo-
nen s o di e en mul ipoles kwi h p ojec ions p, .The o -
bi al in eg als a e ela ed o he coe icien s o his expansion
and depend only on he sca e ing angle 0and he dimen-
sionless adiaba ici y pa ame e j.The la e depends on he
ene gy Fo he ansi ion and he beam ene gy Ez i bo h a e
in uni s o MeV by
ZZA F( A~
1+—
I,
12.7E~' (A, ]'(2)
whe e Zis he a omic numbe , Ais he mass numbe , and he
su ixes pand indica e p ojec ile and a ge , espec i ely.
Using he semiclassical app oach, he sca e ing ampli ude
o Coulomb exci a ion o as a e wi h spin s', may be ex-
p essed in e ms o he o bi al in eg als by [16]
(sZs')
F=—
i(—
1)' kX, ,R„(~.C) (3)
(
whe e y, ,is he s eng h pa ame e o mul ipole exci a-
ion o o de k o a inal ejec ile s a e o spin s'. The a ge
emains in i s g ound s a e. Subs i u ing Eq. (3) in o Eq. (1)
gi es
A. Gene al o malism
Analyzing powe s may be exp essed in e ms o sca e ing
ampli udes Fps whe e pand 6a e he spin subs a es o he
a ge and esidual nucleus. The analyzing powe s a e gi en
by [15]
(s
P. skX „(—
1)' (a+q k~( sZs' ~( s'
sR~ R),
a—
qp, +q a—
p)
TENSOR ANALYZING POWERS FOR Li BREAKUP 3203
whe e p'=/L+q and he (O, j) dependence o he o bi al
in eg als has been omi ed o he sake o b e i y. The
s eng h pa ame e is independen o spin subs a e and he e-
o e cancels in he exp ession o he analyzing powe s. The
analyzing powe s o aCoulomb p ocess consequen ly de-
pend only on he ini ial and inal spins, he mul ipole o he
ansi ion, and he o bi al in eg als.
Coo dina e
Sys em A:
B.Applica ion o Li con inuum b eakup
Fo applica ion o he calcula ions o he con inuum
b eakup o Li in o he upa icle plus i on channel, he
ela i e mo ion o he agmen s mus be assumed o ha e a
single angula momen um. The mos signi ican Coulomb
mul ipole e m leading o b eakup is an E1 ansi ion, co e-
sponding o L=O and leading o an npa icle plus i on
inal s a e which has spin pa i y 1/2+. The e o e s=3/2,
s'=1/2, and X=1. Using Eq. (4) he second ank analyzing
powe s a e calcula ed in e ms o he o bi al in eg als as
Coo dina e
Sys em H:
in
in
ou
IR»l'+ IRi-il' —
2IRio '
"2(IR»l'+IRi- l'+IRiol') 'FIG. 1. Coo dina e sys ems Aand Hused in he semiclassical
Coulomb calcula ions o analyzing powe s.
+3(R ~R o —
R ]R~p)
2(IRzz +IR& — l +IRiol )'(6) powe in coo dina e sys em Hmay be exp essed in e ms o
he second ank analyzing powe s in coo dina e sys em Aas
3R )R)
P~(IR»l'+ IR1—1I'+IR101 )
In o de o p oceed u he wi h he analyzing powe cal-
cula ions, exp essions mus be ound o he o bi al in eg als;
hese a e coo dina e sys em dependen . The helici y coo di-
na e sys em [17],H, was used o he T2p measu emen s.
Howe e , o bi al in eg als a e mo e eadily calcula ed in a
coo dina e sys em [8],A, wi h he zaxis no mal o he eac-
ion plane and he xaxis bisec ing inciden , k;„, and ou go-
ing, k,„„wa e ec o s. The coo dina e sys ems Aand Ha e
shown in Fig. 1.Analyzing powe s in Hmay be de e mined
by i s calcula ing hem in Aand hen o a ing o H.
The o bi al in eg als in Acan be exp essed in e ms o
Coulomb exci a ion unc ions [8] Ix, which a e eal and
coo dina e sys em independen , such ha
Tk q=( —
1)~Tk*, (13)
esul s in
AA
2—
2220 (14)
and consequen ly Eq. (12) becomes
—
[T2p+ +6T22cos( 8)]
T20 2(15)
whe e he o a ion ma ix elemen s D o~ (( +8)/2,
/2, 37 /2) a e unc ions o he Eule angles [19] o he
o a ion. I should be no ed ha he gene al p ope y o ana-
lyzing powe s [15],
+Ii=i
R) 2+ '
Ri0—
0. (9)
Subs i u ing Eqs. (10) and (11)in o Eq. (15) gi es
6I),I) ) cos( 8)—
I, —
I
22
To4(I2 +I2 )(16)
Subs i u ing Eqs. (8) and (9) in o Eqs. (5) and (7) gi es sec-
ond ank analyzing powe s in coo dina e sys em A:
A
T20 2' (10)
—
+3I»Ii
T22= +2(I ( I+, )
Analyzing powe s a e sphe ical enso s which ans o m in a
simple manne [18]unde o a ions. Thus he T2p analyzing
T20 is simply T20 e e ed o an axis no mal o he eac-
ion plane [20]. Consequen ly T2p= T2p, and so T2p is
gi en by Eq. (10).
In o de o calcula e angula dis ibu ions o T20 he Cou-
lomb exci a ion unc ions mus be calcula ed. This canno be
done analy ically bu may be done nume ically [21].Values
o he adiaba ici y pa ame e , de e mined om Eq. (2), a e
used in he calcula ions. Fo 70MeV Li beam on a'Sn
a ge , Z„=3,Z, =50, A~=7, A, =120, and E~=70 in Eq.
(2). Emay ake a ange o alues om 2.47, he b eakup
h eshold in MeV, upwa ds, and so calcula ions may be pe -
o med using di e en adiaba ici y pa ame e s co esponding
3204 N. J. DAVIS e al. 52
TABLE I. Woods-Saxon op ical po en ial pa ame e s used in he CDCC calcula ions. and aindica e
adius and di useness pa ame e s espec i ely and suNces , i, and Cindica e eal, imagina y, and Cou-
lomb po en ials, espec i ely. Vis he eal po en ial dep h and Wand Wd a e imagina y and de i a i e
imagina y po en ials, espec i ely.
+120S a
+ 120S b
Li= + o.
(MeV)
58.0
145.0
90'
1.46
1.25
1.39
a„
0.708
0.690
0.700
W
(MeV)
28.0
0.0
Wd
(MeV)
0.0
27.5
( m)
1.46
1~11
a;
0.708
0.920
( m)
1.4
1.4
1.3
'C. M. Pe cy and F. G. Pe cy, A . Da a Nucl. Da a Tables 17, 1(1976).
"R. P. Wa d and P. R. Hayes, A . Da a Nucl. Da a Tables 49, 315 (1991).
'Fo bound s a es he po en ial dep h was adjus ed in o de o ob ain he co ec binding ene gy.
o di e en agmen ela i e ene gies. The analyzing powe s
a e hen de e mined by using he calcula ed Coulomb exci-
a ion unc ions in Eq. (16).
III. CONTINUUM DISCRETIXED COUPLED CHANNELS
CALCULATIONS
The CDCC calcula ions we e pe o med using he com-
pu e code FREsco [22]. Clus e olding po en ials we e in-
co po a ed. The Coulomb as well as he nuclea in e ac ion
was included in he CDCC calcula ions. Bo h diagonal po-
en ials and coupling in e ac ions we e aken in o accoun .
The nuclea and Coulomb po en ials we e ob ained using he
same me hod, by olding po en ials be ween upa icle clus-
e and a ge and be ween i on clus e and a ge . The po-
en ials o he channel coupling a e de i ed om empi ical
op ical model po en ials o a'Sn a ge . These a e Woods-
Saxon in shape and a e lis ed in Table I.Ideally he po en ials
should be o 40MeV npa icles [23], as is used, and
30MeV i ons, o co espond o beam eloci y agmen s
om 70 MeV Li b eakup. Ho~e e a20 MeV i on po en-
ial [24] was adop ed because none was a ailable a 30 MeV.
The Li 3/2 g ound s a e and 1/2 i s exci ed s a e
clus e wa e unc ions we e calcula ed in aWoods-Saxon
po en ial well ha ing geome y pa ame e s as lis ed in Table
I. These we e chosen so as o ep oduce he empi ical alue
o he educed ansi ion p obabili y B(E2;3/2 ~1/2 )
[25].The wa e unc ion o he 7/2 second exci ed s a e was
calcula ed using aweak binding ene gy app oxima ion [26],
in which a e y small binding ene gy is assumed o his
s a e. The 5/2 hi d exci ed s a e was ea ed as an ene gy
bin o 5MeV wid h.
Tes calcula ions we e pe o med in o de o es ablish he
bes disc e iza ion and unca ion o he model space. The
Li con inuum was ini ially disc e ized as in ap e ious
CDCC s udy by Saku agi e al. [12],wi h alues o he ela-
i e o bi al angula momen um L o he a+ clus e s lim-
i ed o L= 1,3. The es calcula ions we e pe o med o he
Li+ Pb sys em since expe imen al da a o he elas ic,
inelas ic, and b eakup channels exis o his sys em [1,6].
Fou channel calcula ions, including he g ound s a e and he
i s h ee exci ed s a es, p oduced an angula dis ibu ion o
he di e en ial c oss sec ion o he 7/2 s a e simila in
shape o he calcula ions pe o med by Saku agi e al. [12],
bu o abou 30% lowe magni ude. This may be because he
la e we e pe o med wi h double olding po en ials. The
inclusion o he non esonan con inuum educed his by less
han 10%.
The es calcula ions e ealed ha he in luence o he
wa e numbe k=(0.75—
1.00) m 'bin on he inal esul s
is e y small and ha he model space can in ac be limi ed
o he ange k=(0.25 —
0.75) m '. The esul s o he low-
es bin k=(0.00—
0.25) m 'unde es ima ed he expe imen-
al alues [1] especially a he mos o wa d angles. This
ene gy ange was domina ed by he bin wi h L= 1and spin
pa i y I=3/2 .Saku agi e al. [12]a gued ha he coupling
o he e en Lb eakup s a es mus be much weake han o
he odd Ls a es and he e o e he L alues can be es ic ed
o L= 1,3. In o de o in es iga e his u he , he L=0bins
we e included in he calcula ions. I was ound ha , al hough
he esul s o he elas ic and inelas ic sca e ing and o he
7/2 s a e we e no a ec ed by he inclusion o he L=O
s a es, he c oss sec ion calcuIa ed o he lowes L=O,
k=(0.00—
0.25) m ', con inuum bin was la ge a sca e ing
angles anging om 10' o 35'. I o e es ima ed he mea-
su ed [I] alues. The la ge c oss sec ion o he L=0bin is
in ag eemen wi h adia i e cap u e s udies [27] which e-
ealed ha F.1con ibu ions domina e. The same was shown
in an analysis o 63 MeV b eakup da a [28].I was he e o e
conside ed impo an o include e en Lb eakup s a es in he
CDCC calcula ions, especially because he o wa d angle
da a a e o pa icula in e es . The a gumen by Saku agi
e al. [12] o neglec ing e en L elies on a h ee-body model
o npa icle, i on, and a ge in which he npa icle and
i on po en ials wi h espec o he a ge a e simila . In he
p esen s udy he upa icle and i on op ical po en ials used
a e e y di e en , which may explain he la ge L=O c oss
sec ion.
CDCC calcula ions o 70MeV Li b eakup on 'Sn
we e pe o med using he model space shown in Fig. 2. The
wid h o he lowes bin was se o 0.38 m ' o co espond o
he con inuum b eakup da a measu ed in he expe imen . The
model space was unca ed o amaximum o 0.8 m 'and o
L=0,1,2,3. Fo L=2, only he lowes k=(0.0—
0.38) m
bin was aken in o accoun , in o de o educe he numbe o
channels. This limi a ion is no expec ed o signi ican ly a -
ec he analysis, since calcula ed con inuum c oss sec ions
we e gene ally ound o dec ease wi h inc easing ela i e en-
e gy. Fo acompa ison wi h he con inuum b eakup da a i
was necessa y o sum he con ibu ions o he calcula ions
52 TENSOR ANALYZING POWERS FOR Li BREAKUP 3205
k( m-& )
0.8
0.75-- 6.08 6.08
Ene gy ela i e o 7Li~e+ b eakup h eshold (Me ) whe e he
A~=(kcm. s'ylAlk s~) (18)
0.6
0.5-- 3.16 -—3.16 -—3.16
0.4
0.38
4.94
2.16
4.21 a e ampli udes calcula ed by he FRESCO code, s and m, a e
he spin and p ojec ion o he i on, k, k, ,and k; a e he
upa icle, i on, and inciden momen a, espec i ely, Ais an
app op ia e o m ac o , and
0.86 0.86 0.86 0.86 0.86 0.86 0.86 c.m. ka+ ~ - (19)
Equa ion (17) may be w i en in e ms o asum o e sphe i-
cal ha monics o he b eakup angula momen um L=3, p o-
jec ion mz.'
(k,k, ,s,m, iA ik, ;su)
L:
1+
2
-2.47
3
2
-1.99
I
2
5+
2
3+
2
5
2whe e
=g Ag(s,m,Lmiis'y)YI (k„),
ym~
4k, —
3k
k„=
(20)
(21)
FIG. 2. The disc e iza ion o he Li=a+ b eakup con inuum
used in he coupled channels calcula ions.
o di e en spin pa i y alues o ob ain o als o he di e -
en ial c oss sec ion and analyzing powe s. An incohe en
summa ion o e he con ibu ions was aken and analyzing
powe con ibu ions we e weigh ed by he co esponding di -
e en ial c oss sec ions. In o de o in es iga e he oles o
he nuclea and Coulomb o ces in he CDCC calcula ions,
addi ional calcula ions we e pe o med wi h anuclea in e -
ac ion only. I was ound ha , subjec o limi a ions on he
nume ical in eg a ions o he coupled equa ions o 40 m and
on he pa ial wa es o 150A,, he esul s we e e y simila ,
indica ing adominan e ec o he nuclea in e ac ion.
To achie e aCDCC esul o consis en compa ison wi h
he measu ed analyzing powe s, some conside a ion has o
be made o he phase space de ec ed in he expe imen . Fo
he con inuum b eakup, he la ge con ibu ion om L=O
esul s in he phase space ha ing li le e ec on he calcula-
ions; hence, esul s di ec om he FRESCO code a e com-
pa ed wi h he da a. Fo he L=3sequen ial b eakup, phase
space is a mo e impo an . In o de o ake accoun o his,
conside he sca e ing ampli ude
(k,k, ,s,m, iAik;;su) =g(k,k, ,s,m, ik, ,s' y)A
(17)
This ela i e momen um aken as i on ela i e o c pa icle
could equally well be aken as apa icle ela i e o i on.
In o de o calcula e app op ia e analyzing powe s om
he sca e ing ampli udes, he coincidence de ec ion p obabil-
i y (0„,$„) o agmen s wi h ela i e momen um di ec-
ion gi en by sphe ical pola coo dina es (0„,@„)in he he-
lici y coo dina e sys em Hneeds o be inco po a ed. This
p obabili y unc ion was calcula ed using aMon e Ca lo
simula ion code [29]in which he collima o posi ions o he
coincidence de ec ion we e de ined. Li nuclei we e exci ed
o he 4.63 MeV s a e wi h aLo en zian dis ibu ion and
we e sca e ed iso opically o e asolid angle la ge enough
o include he collima o s. The p obabili y unc ion
(8„,P„) was ex ac ed om he Mon e Ca lo calcula ion in
he o m o a wo-dimensional spec um o numbe o coun s
e sus O„and P„, o which he apa icle and i on we e
de ec ed in coincidence. Fo his spec um he O„ ange was
di ided in o 5' bins and he P„ ange was di ided in o 10'
bins. (0„,$„) is an e en unc ion o @„because o he
symme y o he coincidence de ec ion sys em abo e and
below he plane in which he sca e ing angle is de ined.
The double di e en ial c oss sec ion do.(k, ,k„)/
dA, md'„and analyZing POWe S Tkq CO eSPOnding O haV-
ing he cen e o mass momen um o he upa icle plus
i on in asolid angle 0, and he ela i e momen um in a
solid angle A„a e gi en by
dc (k, ,k„) 1Lk'
(2s+ 1)TI,q+XI,qqiqis L
dQ, dA„4, ,q' qI00
L)
0) W(LLs's';k's, )Yg q(k„), (22)
whe e he pola iza ion ans e coe icien s a e gi en by
Xlq ~q=g (s~&qis~')&(s'y&'q'is'y')&'AP (23)

3206 N. J. DAVIS e al.
Howe e , he expe imen does no esol e in de ail he double di e en ial c oss sec ion. Ins ead, hose ele an e en s whe e
an upa icle and a i on a e de ec ed in coincidence a e coun ed as sequen ial b eakup (SBU), which happens wi h he
p obabili y (8„,P„).Thus
(2s+ 1)T c.m.
2do.(k, ,k„)
(9„,@„)(2s+1)Tk 'sing„d g„d@„
0@~=0J8„=0 c.m.
1.,I.
gXlq ks'L W(LLs's';k's, )Ik (24)
whe e
(2m m
co ding o Eq. (23). The analyzing powe s a e hen calcu-
la ed om Eq. (24), no malizing such ha Tpp= 1.
I„= Yl,«(9„,@„)
(8„,P„)sing„d O„d@„.
qb„=0 39„=p (25)
1.0
0.5
0.0
-0 5"
/j'
/:/
/:/
'J'
20 1.0
05"
0.0-
—
0.5
—
3.0IIIII
030 60 90 120 150 180
8„(deg ees)
FIG. 3. CDCC calcula ed analyzing powe s as a unc ion o
cen e o mass ela i e momen um di ec ion, o 15 labo a o y
sca e ing angle. The solid, dashed, do dashed, and do ed cu es
a e o P„= 0, 30', 60', and 90', espec i ely.
This in eg al is necessa y because o he sensi i i y o he
calcula ed analyzing powe s o he angles O„and P„.This is
illus a ed in Fig. 3, which shows analyzing powe s calcu-
la ed assuming single H„and P„ alues o alabo a o y sca -
e ing angle o 15'. Iki ~ educes o a eal numbe since
(8„,$„) is e en in P„and he in eg a ion is aken o e
2q in P„. In o de o calcula e he analyzing powe s, he
in eg als Ik qa e calcula ed acco ding o Eq. (25) using he
p obabili y unc ion (0„,P„) ob ained om he Mon e
Ca lo calcula ion and he pola iza ion ans e coe icien s
Xkq p q&a e calcula ed om he FREsm ampli udes Ayac-
IV. EXPERIMENT
The expe imen was pe o med using 70 MeV pola ized
7Li beams om he pola ized hea y ion sou ce [30] and
accele a ed by he andem Van de G aa accele a o , a he
Nuclea S uc u e Facili y a Da esbu y Labo a o y in he
UK. FOI he T2o measu emen s pola iza ion o he beam
was achie ed using ansi ions be ween he 2-8 and 4-6
hype ine a omic le els [31]in amagne ic ield, esul ing in
s a es wi h equal magni ude bu opposi e signs o he enso
pola iza ion and equal odd ank pola iza ions. This equali y
was e i ied by compa ison wi h an unpola ized beam. The
T20 measu emen s we e made a ala e da e when op ical
pumping [32] was a ailable o pola iza ion o he beam,
doubling he heo e ical maximum pola iza ion ob ainable.
Ions wi h each o he ou spin subs a es we e selec ed in
u n, using ahigh equency ansi ion in amagne ic ield o
swi ch be ween subs a es. Fo bo h T20 and T20 measu e-
men s, he pola iza ion s a es we e swi ched e e y ew sec-
onds, a e aspeci ied in eg a ed beam cu en was measu ed,
o minimize sys ema ic e o s due o beam d i o pola iza-
ion luc ua ions. AWien il e was used o o ien he pola -
iza ion symme y axis along he beam di ec ion a he a ge
o he T2O measu emen s and no mal o aplane bisec ing
agmen coincidence de ec o cen e s o he T2o measu e-
men s. The beam pola iza ion was de e mined om he
'H( Li,n) He eac ion [16] using apu pose buil down-
s eam pola ime e [33].The measu ed magni udes o second
ank beam pola iza ions we e ypically 2o= 0.4 o he
T20 measu emen s and 20=0.6 o he T20 measu emen s.
Measu emen s o i s and hi d ank pola iza ions in he op-
ically pumped beam esul ed in magni udes no la ge han
0.05 each.
Fo he b eakup eac ions a2mg cm 'Sn a ge was
used. The yield o con inuum b eakup o Li in o an n
pa icle and a i on is la ge o small ela i e ene gies o he
agmen s [2].Fo his eason agmen s wi h asmall open-
ing angle mus be de ec ed. The de ec ion sys em consis ed
o wo pai s o AF. Ede ec o elescopes placed symme i-
cally, one pai ei he side o he beam, as shown in Fig. 4.
The symme ic a angemen was used so ha da a om bo h
sides o he beam could be summed, hus elimina ing sys-
ema ic e o s a ising om any shi in posi ion o he beam
on a ge and he e ec s o odd ank pola iza ion componen s
in he beam. The elescopes comp ised o 230 p,m hick p-n
TENSOR ANALYZING POWERS FOR Li BREAKUP 3207
Plan iew
120 S
a ge
7Li beam
Side iew
agmen s
e'
6
agmen s
AE
Ve o Pai o
de ec o
elescopes
Pai o
de ec o
elescopes
Ve o
Ti(H)
pola ime e
a ge Pola ime e
~de ec o
elescope
I I
AE E
Ta beam s op
Coun s 40
30"
20"
0
0.00.51.01.52.02.53.0
Rela i e ene gy (MeV)
l20 S
a ge
7Li beam
EVe o
FIG. 4. De ec o sys em.
FIG. 6. F agmen ela i e ene gy spec um o he 'Sn(7Li,
n ) 'oSng, eac ion a alabo a o y angle o 15'.
b eakup ia he 4.63 MeV s a e in Li and he da a be ween
he peaks a ise om con inuum b eakup [2]. The ela i e
ene gy ebe ween he i on and npa icle may be calcula ed
acco ding o
V. RESULTS
The as coincidence da a we e ga ed on he g ound s a e
o 'Sn by summing i on and npa icle ene gies. An en-
e gy esolu ion o 0.4MeV was achie ed. Acoincidence
spec um o i on ene gy is shown in Fig. 5. The sha p peaks
co espond o he wo kinema ic solu ions o sequen ial
Coun s 50
40"
30"
20"
10" ~ ga a I L. .
10 20 30 40
Ene gy (MeV}
FIG. 5. Ene gy spec um o i on s om he
Sn( Li, ) 'Sng, eac ion a alabo a o y angle o 15'.
junc ion silicon AE de ec o s and 4mm hick li hium d i ed
silicon Ede ec o s. Simila de ec o s placed behind he E
de ec o s ac ed as e os o elimina e high ene gy cha ge 1
pa icles which pass h ough he Ede ec o s. The de ec o
collima o s we e 8mm wide and 6mm high wi h he cen e s
o agi en pai 12 mm apa . The de ec o s in each pai o
elescopes we e moun ed symme ically abo e and below he
beam axis and 150mm om he a ge .
The de ec o s we e ene gy calib a ed using 5.486MeV
npa icles om 'Am sou ces moun ed close o he de ec-
o s. Pa icle iden i ica ion was achie ed using he AE and E
signals. Fas iming was achie ed by signals gene a ed om
he AE p eampli ie s, used o s a and s op a ime o ampli-
ude con e e o each pai o elescopes. Da a we e ans-
mi ed om analogue o digi al con e e s o aGEC 4190
compu e and eco ded e en by e en on ape.
T2p and "T2p da a we e ob ained o a ange o angles o
he beam di ec ion, 8, shown in Fig. 4, om 9 o 25 in he
labo a o y ame.
4E,+3E 4$3E—
E cos(@)
7(26)
whe e E, and Ea e he i on and npa icle ene gies e-
spec i ely and @is he angle be ween hem. Figu e 6shows
a ela i e ene gy spec um, whe e Phas been aken o be he
angle be ween he de ec o collima o cen es. The peak is
om he sequen ial b eakup and he da a a lowe ela i e
ene gies a e om con inuum b eakup. Yields we e ob ained
o each pola iza ion s a e o he beam. This enabled da a o
he con inuum b eakup o be ob ained om a h eshold o
100keV o 1.7MeV in ela i e ene gy.
The measu ed analyzing powe s o he
'oSn( Li,n ) 'Sng, con inuum b eakup a e compa ed wi h
semiclassical Coulomb and CDCC calcula ions in Fig. 7.The
small ela i e ene gy dependence o he semiclassical Cou-
lomb calcula ions o T2p is shown by he dashed cu e which
co esponds o ze o ela i e ene gy and he do ed cu e
which co esponds o a ela i e ene gy o 2.16 MeV, equi a-
len o he alue o sequen ial b eakup ia he 4.63 MeV
s a e in Li. The co esponding calcula ion o T2p is no
ela i e ene gy dependen and is shown by he dashed cu e.
The "T2p da a ag ee well wi h he semiclassical calcula ion
a small angles bu de ia e om i a he la ge angles. The
T2p da a clea ly do no ag ee wi h hese calcula ions. The
angula end o he da a opposes ha o he calcula ions and
la ge magni ude T2p alues a e measu ed, in disag eemen
wi h he small magni udes p edic ed by he calcula ions. This
esul is in con as wi h he esul s o di e en ial c oss sec-
ion [2,3,6] measu emen s which indica ed ha he Coulomb
o ce is esponsible o he con inuum b eakup a small
angles. Analyzing powe s a e mo e sensi i e han di e en ial
c oss sec ions o he con ibu ions o di e en o ces in he
eac ion mechanism and hus will be a ec ed by anuclea
o ce p esen in he eac ion mechanism e en i he Coulomb
o ce also plays ala ge ole. The ex en o he disag eemen
be ween he T2p da a and he calcula ions indica es, howe e ,
ha he nuclea con ibu ion o he con inuum b eakup is
signi ican . The T2p measu emen p o ides amo e igo ous
es o he semiclassical Coulomb model han he T2p mea-
su emen . This is because he calcula ion o T2p depends di-
3208 N. J. DAVIS e al. 52
20 1.0
0.5
T~ Q4
20
0.2
0.00.0
20 1.0
05"
—
0.5"
20
—
0.2"
0.2
01I~
L=O, I=1/2
L=2, I=3/2
L=2, I=5/2
—
0.5'—
0.1"
0510 15 20 25 30
Cen e o mass angle (deg ees) 20
—
0.2
0.4510 15 20 25 30
FIG. 7. Tzo and Tqo o 'Sn( Li,n ) 'Sns, con inuum
b eakup o e~ 1.7MeV. The dashed cu es show semiclassical cal-
cula ions o Tzp and o Tpp wi h a=0 MeV and he do ed cu e
shows he semiclassical calcula ion o Tzp wi h e=2.16MeV. The
solid cu es show he CDCC calcula ions wi h all L alues om 0
o 3included.
ec ly on he Coulomb exci a ion unc ions whe eas he cal-
cula ion o Tzp does no . The semiclassical calcula ion o
Tpp only depends on he assump ion o mul ipola i y 1,
which is good o aCoulomb o ce bu is no necessa ily so
o anuclea o ce. Ne e heless, Tpp can no dis inguish
he mul ipola i y 1componen o anuclea o ce om aCou-
lomb o ce. The Tzp da a hus expose he disag eemen wi h
he semiclassical Coulomb model be e han he "Tzp da a.
I aCoulomb o ce we e esponsible o he con inuum
b eakup asmall de ia ion o he da a om he calcula ions
migh be expec ed due o he app oxima ions used in he
semiclassical model. Howe e , he ex en o he de ia ion o
Tpp indica es ha he disag eemen is mo e undamen al and
asigni ican nuclea o ce con ibu ion o he con inuum
b eakup eac ion mechanism is he e o e expec ed o be
p esen . The poo ag eemen o he Tzp da a wi h he semi-
classical Coulomb calcula ions, obse ed oge he wi h he
good ag eemen o he Tpp da a wi h he calcula ions a
small angles sugges s ha he nuclea o ce con ibu ion o
he con inuum b eakup eac ion mechanism is no only sig-
ni ican , bu also has ala ge mul ipola i y 1componen .
CDCC calcula ions o ~oSn(7Li, n ) 'oSn, con inuum
b eakup analyzing powe s a e shown o he di e en Lcon-
ibu ions in Fig. 8. The L=O con ibu ions a e he mos
impo an , because o he la ge associa ed di e en ial c oss
sec ion, wi h he L=2and L=3con ibu ions insigni ican .
The L=0 componen o he CDCC calcula ion o Tpp akes
alues be ween 0.4and 0.6o e an angula ange o
0'—
25, and so is e y simila o he semiclassical calcula-
ion which gi es a alue o 0.5. The L=1con ibu ions ha e
—
0.2"
0.2
20
0.1'
L=1, I=1/2
L=1, I=3/2
L=3, I=5/2
L=3, I=7/2
0.0
—
0.1"
R
0 5 10 15 20 25 30
Cen e o mass angle (deg ees)
FIG. 8. CDCC calcula ions o Tzp and T~p o
Sn( Li,n ) 'Sns, con inuum b eakup, o di e en L alues.
mos e ec o he la ge angles, 20' —
30 .The esul o
incohe en ly combining all componen s o he CDCC calcu-
la ion gi es he solid cu es shown in Fig. 7.Agood desc ip-
ion o he Tzp da a is ob ained o e he angula ange mea-
su ed, wi h he signi ican con ibu ion om he L=1
componen s a la ge angles enhancing ag eemen wi h he
da a. The CDCC and semiclassical calcula ions o Tzp a e
qui e simila , wi h he CDCC gene ally ali le lowe han he
semiclassical esul o 0.5. Howe e , he oscilla ions in he
CDCC calcula ion, which a e no p esen in he semiclassical
calcula ion, imp o e he ag eemen wi h he da a. The CDCC
calcula ion is wi hin e o ba s o h ee o he i e da a poin s
while he semiclassical calcula ion only exhibi s his quali y
52 TENSOR ANALYZING POWERS FOR Li BREAKUP 3209
3
10 „
10
(mbs )10'.
10 '
20 1.0
05
0.0
Q,o.
(mbs )10';
10.
10 '"
'—
——
Sum o e L= 0 o 3
No malised by 0.187
L=O, l=1/2
—-—-L=2, l=3/2 20 1.0
05"
0.0--
—
0.5"
10' —
0.5
10',
dQ, 10'...
{mbs )10'.
10;
10';
l=1/2
l=3/2
l=5/2
l'=7/2
-1.00 5 10 15 20 25 30
Cen e o mass angle (deg ees)
FIG. 10. T20 and T2O o 'Sn( Li,a ) 'Sns, sequen ial
b eakup ia he Li 4.63 MeV s a e. The do ed cu es show he
CDCC calcula ions ob ained di ec ly om FREsCo and he solid
cu es ha e phase space e ec s included.
10 0510 15 20 25 30
Cen e o mass angle (deg ees)
FIG. 9. Di e en ial c oss sec ions o 'Sn( Li,a ) 'Sns,
con inuum b eakup o m~1.7MeV. The cu es show CDCC cal-
cula ions o di e en L alues.
o ag eemen o he smalles angle da a poin . The o al
CDCC calcula ion o T2p shown in Fig. 7 ep oduces he da a
a he smalle angles, la gely om he L=0con ibu ion. A
he la ge angles whe e he L= 1, I"=3I2 componen has a
signi ican e ec and is o opposi e sign o he da a, he
ag eemen is poo . De ec ion phase space e ec s would be
mo e impo an a he la ge angles bu a e imp ac ical o
inco po a e in o he con inuum b eakup calcula ions. The
CDCC calcula ion does, howe e , gi e be e ag eemen wi h
he small angle T2p da a han he semiclassical Coulomb cal-
cula ion does. The CDCC calcula ion ep oduces wi hin e -
o s he end be ween he wo smalles angle da a poin s,
while he semiclassical calcula ion me ely c osses his end
a abou 12'.
I was no he aim o he expe imen o measu e absolu e
di e en ial c oss sec ions o he b eakup and no maliza ion
di icul ies p e en a eliable esul . Howe e , angula dis i-
bu ions o di e en ial c oss sec ions, o which he end is
accu a e, we e ob ained and i is in e es ing o compa e hese
wi h CDCC p edic ions. Di e en ial c oss sec ion da a o
Sn( Li,a ) 'Sns, con inuum b eakup a e compa ed wi h
he esul s o CDCC calcula ions in Fig. 9.I is in e es ing o
no e ha L=0gi es he dominan con ibu ion o he calcu-
la ed di e en ial c oss sec ion o 7—
19 .Beyond 19' he
L=1, I=3)2 con ibu ion domina es. The o al CDCC cal-
cula ion exhibi s e y good ag eemen wi h he da a i no -
malized by a ac o o 0.187. The e a e se e al explana ions
o why his no maliza ion is necessa y. The da a may ha e
sys ema ic unce ain ies due o inaccu acies in he a ge
hickness and in he solid angle which had o be de e mined
by aMon e Ca lo calcula ion [29]in o which some assump-
ion o he con inuum b eakup yield a ia ion wi h ela i e
ene gy had o be inco po a ed, he esul o asemiclassical
Coulomb calcula ion being used. Acalcula ion using he
FRESCO code could no easonably be pe o med because o
he small ene gy bins which would be equi ed. Some da a
we e also los due o de ec o dead egions and elec onic
h esholds. The CDCC calcula ions may no gi e he co ec
absolu e magni ude o he di e en ial c oss sec ions because
o hei clus e model basis. The ag eemen o he angula
dis ibu ion o he CDCC calcula ions wi h he di e en ial
c oss sec ion da a is, howe e , excellen .
Analyzing powe da a o 'Sn( Li,n ) 'Sns, sequen-
ial b eakup ia he 4.63 MeV s a e in Li a e compa ed wi h
CDCC calcula ions in Fig. 10. The esul s ob ained di ec ly
om he FRESCO code, shown by he do ed cu es, a e no
su icien o ep oduce he da a. Howe e , when he phase
space e ec s a e included, as shown by he solid cu es, he
calcula ions exhibi excellen ag eemen wi h he da a, wi h
he excep ion o he p edic ed 13 maximum in T2p which is
no seen in he da a. This is mos p obably because he da a
a e smea ed ali le o e a ange o angles because o ini e
de ec o sizes. The p edic ed maximum in T2p co esponds o
aminimum in he di e en ial c oss sec ion, as shown in Fig.
11.This means such smea ing will ha e ala ge e ec in his
case, b inging he measu ed T2p owa ds alues o nea by
angles.