Dynamic e ec i e po en ial o
a
-pa icle bound and quasibound s a es
C. H. Dasso,1R. J. Lio a,2and M. Lozano3
1The Niels Boh Ins i u e, Uni e si y o Copenhagen, Blegdams ej 17, DK-2100 Copenhagen O”, Denma k
2Royal Ins i u e o Technology, F esca i a
¨gen 24, S-10405 S ockholm, Sweden
and No di a, Blegdams ej 17, DK-2100 Copenhagen O”, Denma k
3Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea Uni e sidad de Se illa, AP. 1065, E-41080 Se illa, Spain
~Recei ed 21 No embe 1995!
We exploi analy ic p ope ies o he op ical po en ial o elas ic sca e ing o
a
pa icles on nuclei o ex ac
in o ma ion on he e ec i e in e ac ion ha should be used o desc ibe he mo ion o a clus e o wo neu ons
and wo p o ons bound o a nuclea sys em. This p esc ip ion sol es long-s anding ambigui ies in he o mal-
isms used o he s udy o
a
decay. @S0556-2813~96!01308-8#
PACS numbe ~s!: 24.10.H , 21.60.Gx, 23.60.1e, 27.80.1w
Alpha decay was one o he i s nuclea phenomena ob-
se ed in na u e. No su p isingly, a emp s o desc ibe his
p ocess go back o he ea lies days o quan um mechanics.
Pe haps he mos anspa en o mula ion was ha o Gamow
@1#, who ecognized he possibili ies o quan al unneling o
explain he many o de s o magni ude spanned by he expe i-
men ally measu ed li e imes. As he gene al knowledge o
nuclea s uc u e imp o ed, howe e , his simple pic u e an
in o oubles. These p ima ily had o do wi h he need o
unde s and mic oscopically he o ma ion inside o he
mo he nucleus o an alpha pa icle as a clus e o wo p o-
ons and wo neu ons. This goal in oduced all kinds o di -
icul ies in he p oblem, anging om complex coupling
schemes down o he ea men o he con inuum equi ed o
sa is y he app op ia e bounda y condi ions. Especially no e-
wo hy a e ques ions aised abou he iabili y o a scheme
ha equi es he wa e unc ion o he alpha pa icle no only
a he nuclea su ace, bu in he in e io as well @2#. The
ese a ions cen e he e mos ly on he inhibi ing ole o he
Pauli p inciple, which, ine i ably, en e ed in o he pic u e as
soon as he alpha pa icle was hough o in e ms o i s
cons i uen nucleons.
In his pape we e u n o his ques ion ollowing a pa h
ha aims o bypass hese di icul ies. To his end we ely on
he wo k by Bell and Squi es @3#~c . also Re . @4#! who
in es iga ed he analy ic p ope ies o he complex nucleon-
nucleus in e ac ions. In hei con ibu ion hey cla i ied he
in ima e connec ion be ween he op ical po en ial o sca e -
ing s a es and he e ec i e in e ac ion ha binds he nucle-
ons o he co e. In ac , hey es ablished he concep ual iden-
i y o hese quan i ies, o he ex en ha he la e is none
o he han he analy ic con inua ion o he o me in o he
nega i e ~o , mo e p ecisely, bound-s a e!ene gy domain.
When Bell and Squi es p esen ed hei esul s, much was
al eady known abou he binding po en ial o single nucle-
ons. This in o ma ion had been mainly ex ac ed om he
analysis o nuclea sys ems ei he di ec ly a ailable in na u e
o a i icially p oduced. Thus i could be a gued ha hei
wo k was mos ly o academic in e es . Thei conclusions a e,
howe e , po en ially much mo e ele an in he case o alpha
pa icles. Michel e al. @5#and Delba e al. @6#made signi i-
can s eps in he unde s anding o he op ical po en ial o he
sca e ing o alpha pa icles by ligh e sys ems, such as oxy-
gen and calcium. Mic oscopic es ima ion o he s a ic po en-
ial be ween an
a
pa icle and a nuclea co e can also be
con empla ed by olding p ocedu es; see, o ins ance, Re s.
@7, 8#. Unce ain ies emain, howe e , abou he couplings
ha should be used o accoun o he mo ion o he ou -
nucleon clus e in a quasibound s a e o a hea ie nucleus
ha e en ually
a
decays. In ac , o calcula e mic oscopically
he eno maliza ion o he ba e in e ac ion ha gi es ise o
he ac ual ene gy dependence o he
a
-nucleus po en ial un-
de hese ci cums ances is a a he o midable ask. One can
inspec , o ins ance, in Re . @9#, he nume ous diag ams ha
ha e o be aken in o accoun o cons uc in a pe u ba i e
app oach he co ela ion and pola iza ion e ms ha dynami-
cally ‘‘d ess’’ he mass ope a o o o dina y nucleons.
Needless o say, he si ua ion would ge e en mo e compli-
ca ed in he case o alpha pa icles.
The p ospec s a e as ly imp o ed, howe e , i one akes
a phenomenological app oach in which he ene gy depen-
dence o he op ical po en ial is used. Indeed, an ex apola-
ion o he empi ically de e mined couplings o nega i e en-
e gies, i easible, would yield a binding po en ial whe e
Pauli blocking, su ace e ec s, and o he ele an co ec ions
a e p ope ly included o all pe u ba i e o de s.
I is no necessa y o ‘‘p opose’’ such measu emen s since
expe imen al s udies o asymp o ically ee s a es ha e been
a ailable o a long ime. In ac , one can simply go back o
he li e a u e and sea ch o exis ing op ical-model i s o he
elas ic c oss sec ions o
a
pa icles on nuclei. We shall
hence o h ocus ou a en ion only on he eal pa Vo he
in e ac ion, as i is expec ed ha his componen will ha e a
smoo he ansi ion om posi i e o nega i e ene gies.1We
show in Table I a collec ion o op ical model pa ame e s o
1E ec i e in e ac ions ge con ibu ions ela ed o nonlocali y in
space and ime. I is well known ha he ene gy dependence o he
eal pa o he la e , DV, has a coun e pa in he imagina y pa
W. The connec ion be ween hese wo unc ional dependences o
hea y ions has been success ully app oached in e ms o dispe sion
ela ions ~see, o ins ance, Re . @10#!. The cha ac e is ic shape o
W(E) o E'0, howe e , is no sui able o ex apola ion.
PHYSICAL REVIEW C SEPTEMBER 1996VOLUME 54, NUMBER 3
54
0556-2813/96/54~3!/1217~4!/$10.00 1217 © 1996 The Ame ican Physical Socie y
he speci ic eac ion
a
1208Pb. I may be somewha puzzling
a i s o see he di e si y o dep hs, adii, and di useness
alues ha such a compila ion e eals. One mus , howe e ,
ecognize ha he numbe s ep esen comple ely independen
e o s whe e no a emp was made o pu in e idence an
o e all ene gy dependence o he in e ac ion.
I is clea ha ou i s ask is hen o ep ocess he in o -
ma ion con ained in Table I, so as o es ablish a poin o
con ac be ween he di e en measu emen s. The e a e se -
e al ways o inco po a e he ene gy dependence o he in e -
ac ion. We choose, o ou p esen pu pose, o de ine a com-
mon Woods-Saxon geome y
V~ !52 V
¯
0
11exp@~ 2R!/a#,~1!
whe e R5 0~2081/3141/3!, and o exhibi he dynamical con-
en o he e ec i e in e ac ion in e ms o an ene gy-
dependen s eng h V
¯
05V
¯
0(E). Fo he adius and di use-
ness pa ame e s, we ake 050.98 m, and a50.80 m. This
choice is inspi ed by he con enien pa ame iza ion o he
ene gy dependence o he in e ac ion gi en in @11#, namely,
V0~E!5~182.1641.3!1~20.24860.047!E,~2!
which is alid o E.80 MeV. Compa ing po en ials a a
s ong-in e ac ion adius ~in m! s51.07~2081/3
141/3)12.722aln@112~Ec.m.2Eb!/Eb#, whe e Ebis he
Coulomb ba ie @12#, we ha e ex ac ed om he op ical-
model pa ame e s lis ed in Table I he po en ial dep hs
V
¯
0(E). These a e plo ed in Fig. 1; eassu ingly, all poin s
now all in a a he o de ed pa e n. The s aigh lines co e-
spond o he pa ame iza ion ~2!, allowing o he unce ain y
in he coe icien s. Fo he many da a poin s a E'20 MeV,
we ha e op ed o show hei span in V
¯
0, which i s qui e well
wi hin he end exhibi ed a o he bomba ding ene gies.
The choice o 208Pb as a a ge nucleus was gea ed o he
e en ual in es iga ion o he adioac i e
a
decay o 212Po.
The sepa a ion ene gy o an alpha pa icle om his nucleus
is abou 8 MeV. I ollows om he sys ema ics displayed in
Fig. 1 ha he
a
-208Pb po en ial in his ene gy ange can be
app oxima ed by
TABLE I. Real po en ial dep h V0, adius pa ame e 0(AT
1/3),
and di useness a o a Woods-Saxon geome y as a unc ion o
bomba ding ene gy Ein he elas ic sca e ing o
a
pa icles by
208Pb. Each ow gi es esul s o op ical-po en ial i s om di e en
expe imen s, as quo ed in he co esponding e e ences.
E~MeV!V0~MeV! 0~ m!a~ m!Re .
16.0 35.0 1.550 0.570 @22#
18.0 35.0 1.550 0.570 @23#
19.0 96.4 1.376 0.625 @24#
19.5 35.0 1.550 0.570 @27#
20.0 96.4 1.376 0.625 @24#
20.0 174.0 1.470 0.470 @27#
20.0 35.0 1.550 0.570 @23#
21.0 32.0 1.550 0.570 @27#
22.0 96.4 1.376 0.625 @24#
22.0 35.0 1.550 0.570 @27#
22.0 94.6 1.364 0.650 @24#
22.0 117.5 1.298 0.700 @24#
22.0 86.9 1.449 0.550 @24#
22.0 159.7 1.387 0.560 @24#
22.0 110.9 1.464 0.500 @24#
22.0 35.0 1.518 0.571 @23#
22.0 92.5 1.384 0.625 @24#
22.0 100.4 1.444 0.542 @24#
22.0 90.3 1.411 0.600 @24#
22.0 200.2 1.390 0.529 @24#
22.5 33.0 1.550 0.570 @27#
23.5 117.5 1.298 0.700 @25#
23.5 86.9 1.449 0.550 @25#
23.5 92.5 1.384 0.625 @25#
23.5 110.9 1.464 0.500 @25#
24.5 37.0 1.550 0.570 @27#
25.5 35.0 1.550 0.570 @23#
25.5 41.0 1.550 0.570 @27#
27.0 98.4 1.371 0.621 @22#
96.0 89.3 1.350 0.710 @26#
104.0 60.0 1.392 0.656 @27#
129.0 89.3 1.350 0.710 @26#
139.0 110.0 1.315 0.705 @27#
139.0 155.0 1.282 0.677 @27#
139.0 200.0 1.261 0.657 @27#
166.0 119.0 1.260 0.740 @27#
172.0 155.0 1.282 0.677 @26#
218.0 119.9 1.260 0.740 @26#
340.0 67.2 1.325 0.800 @28#
480.0 63.0 1.300 0.950 @28#
FIG. 1. Po en ial dep h V
¯
0(E) ex ac ed om he expe imen s
quo ed in Table I and con e ed o a common Woods-Saxon geom-
e y wi h adius pa ame e 050.98 m and di useness a50.80 m.
The solid line gi es he ene gy dependence sugges ed in Re . @1#,
while he dashed lines gi e he expe imen al ma gins de ined by he
e o ba s.
1218 54
C. H. DASSO, R. J. LIOTTA, AND M. LOZANO
V
a
-Pb~ !5186
11exp@~ 27.36!/0.80#,~3!
whe e Vand a e exp essed in MeV and m, espec i ely.
The shape o his po en ial is qui e di e en om he one o
McFadden and Sa chle @13#, in spi e o he close sa u a ion
alue. A ai knowledge o he e ec i e in e ac ion ~3!is a
necessa y ing edien o es ima e he ela i e mo ion o he
alpha pa icle p io o i s unneling decay om 212Po. Spe-
ci ically, i ei he de e mines he wa e unc ion o he alpha
pa icle nea he nuclea su ace in he R-ma ix heo y o
Teichman and Wigne @14#~which we will apply he e; c .
also Re . @15#! o bo h bulk eloci y and ba ie p o ile in
Gamow’s. Fo cla i y o p esen a ion, we b ie ly e iew be-
low he R-ma ix app oach o decay p ocesses.
A dominan ea u e o eac ion p ocesses in which wo
nuclei collide a low ene gies o o m a compound nucleus
ha subsequen ly disin eg a es is he p esence o esonances.
In a ime-independen amewo k, and assuming ha he
esonances do no o e lap and a e no close o h esholds, he
Sma ix as a unc ion o ene gy can be pa ame ized wi h a
many-le el B ei -Wigne o mula
Scc8~E!5ei~
d
c1
d
c
8!
F
d
cc82i(
n
Gn,c
1/2Gn,c8
1/2
E2
n
G
,~4!
whe e
d
cis he phase shi in channel c, he pa ial decay
wid h Gn,cis p opo ional o he p obabili y ha he eso-
nance labeled by ndecays in o he channel c, and En5Re
n
and Gn522Im
na e he posi ions and wid hs o he eso-
nances. Fo isola ed esonances only one e m domina es Eq.
~4!, and he uni a i y o he Sma ix equi es ha i s esidues
Gn,c
1/2Gn,c8
1/2 be eal. In ha case each Gn,ccan also be aken o
be eal, and (cGn,c5Gn. This si ua ion does no occu e-
quen ly ~e.g., among neu on-uns able s a es!, and he physi-
cal meaning o he pa ial wid h can become ques ionable
because hey a e uly complex quan i ies. Howe e , o he
a
decay o low-lying s a es o hea y nuclei, his p oblem
does no a ise because he alues o Gn/En ange be ween
10210 and 10230.
In he R-ma ix heo y one di ides he con igu a ion space
o he
a
1daugh e -nucleus sys em in o wo egions: an
‘‘in e nal egion’’ V o which he compound s a e is e-
s ic ed and an ‘‘ex e nal egion’’ ha encompasses he es
o he space. The bounda y be ween he egions should be
chosen such ha all nuclea in e ac ions and e ec s induced
by he Pauli p inciple a e only non-negligible in he in e nal
egion. In he ex e nal egion he mo ion o he agmen s is
go e ned by he Coulomb in e ac ion and he daugh e
nucleus and he
a
pa icle beha e like a bina y sys em mo -
ing ou wa ds wi h asymp o ic ene gy En. The solu ion inside
he olume Vsa is ies a sel -adjoin bounda y condi ion on
he su ace So V. In he decay o sphe ical nuclei he su -
ace Scan be chosen o be a sphe e o adius ccen e ed
a ound he esidual nucleus. The esidues o he Sma ix can
hen be compu ed in e ms o quan i ies e alua ed on he
su ace S, and one inds @15# ha he li e ime o he decay is
gi en by
~ c!5
2P~ c!
g
2~ c!,~5!
whe e Pis he pene abili y h ough he ba ie and
g
is he
educed wid h ampli ude.
We conclude wi h a conc e e calcula ion o he li e ime
o
a
decay o 212Po ha exploi s he p ocedu es ad oca ed
he e ~ his quan i y has also been ecen ly s udied in de ail
using a di e en app oach o he s ong in e ac ion in Re .
@16#!. Fo ou p esen pu pose we ha e used he compu e
code GAMOW @17# o sol e he Sch o
¨dinge equa ion o he
a
-Pb binding po en ial ~3!, imposing ou going bounda y con-
di ions o he egula solu ions. The eigen alue co espond-
ing o en adial nodes u ns ou o be 8.96 MeV wi h negli-
gible imagina y pa , a alue ema kably close o he kine ic
ene gy o he emi ed alpha pa icle. We no e ha one could
also in p inciple es ima e he li e ime using ha alue o he
imagina y pa o he complex ene gy. Howe e , as poin ed
ou abo e, he a io be ween he imagina y and eal pa s o
he ene gy is so small ha i canno be calcula ed nume i-
cally wi h su icien accu acy.
Wi h he po en ial ~3!and o >9 m, only he unneling
h ough he Coulomb ield is ele an . Since he esul ing
unc ion P( c) is s ongly dependen on c, one may hink
ha i is possible o ob ain p ac ically any alue o he li e-
ime by adjus ing he dis ance a which he pene a ion o he
alpha clus e occu s. Ac ually, a consis ency check on he
heo y is ha
( c) should be a he s a iona y o dis ances
a ound he nuclea su ace. The educed wid h ampli ude can
be w i en as
g
~ c!5
A
2 c
2
m
C~ c!,~6!
FIG. 2. Li e ime a io be ween heo y and expe imen and pen-
e a ion p obabili y as a unc ion o he adius pa ame e c o he
a
decay o 212Po. No e ha he la e quan i y has been scaled up by
a ac o o 1010. The do ed line ~uni y!is gi en as a e e ence.
54 1219DYNAMIC EFFECTIVE POTENTIAL FOR
a
-PARTICLE...
whe e Cis he wa e unc ion o he
a
pa icle. The exp es-
sion o he li e ime quo ed abo e does no ake in o accoun
ha only a ac ion o he mo he nucleus wa e unc ion
co esponds o he mo ion o an
a
clus e a ound he daugh-
e nucleus. This ac o mus be included o compa e wi h
expe imen ally measu ed li e imes. A ecen de ailed mic o-
scopic calcula ion @2#shows ha he
a
clus e spec oscopic
ac o is 0.025. We adop his alue, which is also in good
ag eemen @18#wi h he
a
- o ma ion ampli ude ex ac ed
om a compa ison be ween he Thomas heo y @15#and ex-
pe imen . We no e ha an olde , la ge-scale shell model cal-
cula ion @19#p oduced a alue o ;1023 o ha quan i y.
In Fig. 2 we p esen he a io be ween he expe imen al
and calcula ed li e imes as a unc ion o he pa ame e c.
We also show in he igu e he pene abili y Pas a unc ion
o dis ance. One can see ha he heo e ical calcula ion
ag ees ex emely well wi h he expe imen in a egion close
o he nuclea su ace ~ he quali y o his p edic ion may be
bes app ecia ed by poin ing ou ha disc epancies be ween
heo y and expe imen ega ding he absolu e alues o
a
-decay wid hs can easily span o de s o magni ude!. No ice
ha he li e ime esul s a e indeed p ac ically independen o
co e a la ge in e al. While he pene a ion changes by
nine o de s o magni ude in he depic ed ange, he calcu-
la ed a io emains uni y wi hin, a wo s , a ac o o 2.
The success o his simple calcula ion e eals he po en-
ial o he ideas in oduced in his con ibu ion. I should be
no ed ha he p esc ip ion applies equally well o he in es-
iga ion o mo e exo ic emission p ocesses ha ha e also
been con empla ed ~c ., e.g., Re . @20#and e e ences quo ed
he ein!. These in ol e he decay o hea ie nucleon clus e s
such as ca bon, oxygen, e c., which, a he mic oscopic dy-
namical le el, would be e en ha de o app oach. The in o -
ma ion p esen ly a ailable o he elas ic sca e ing o some
o hese sys ems may no be su icien o ex ac a eliable
ene gy dependence o he co esponding op ical po en ials.
Wi h he use o mode n de ec ion equipmen , howe e , hese
measu emen s a e no ime consuming, as a ecen de e mi-
na ion o diagonal and o -diagonal ma ix elemen s o he
e ec i e in e ac ions has shown @21#.
Discussions wi h T. Ve se a e g a e ully acknowledged.
This wo k was suppo ed in pa by G an No. DFG Bo 1109,
EU Human Capi al and Mobili y p og am G an No.
ERBCHRX-CT92-0075, and Spanish CICYT P ojec No.
PB92-0663.
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Buene d, Nucl. Phys. A430, 349 ~1984!.
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C. H. DASSO, R. J. LIOTTA, AND M. LOZANO