PHYSICAL REVIEW C 73, 014907 (2006)
Rela i is ic Coulomb exci a ion o he gian dipole esonance in nuclei: How o calcula e ansi ion
p obabili ies wi hou in oking he Li´
ena d-Wieche ela i is ic scala and ec o po en ials
C. H. Dasso and M. Galla do
Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea Facul ad de F´
ısica, Apa ado 1065, E-41080 Se illa, Spain
(Recei ed 4 Augus 2005; published 27 Janua y 2006)
The conclusions ex ac ed om a ecen s udy o he exci a ion o gian dipole esonances in nuclei a ela i is ic
bomba ding ene gies open he way o a u he simpli ica ion o he p oblem. I consis s in he elimina ion o
he ela i is ic scala and ec o elec omagne ic po en ials and he amilia nume ical di icul ies associa ed wi h
hei p esence in he calcula ion scheme. The inhe en ad an age o a e o mula ion o he p oblem o ela i is ic
Coulomb exci a ion o gian dipole esonances along hese lines is discussed.
DOI: 10.1103/PhysRe C.73.014907 PACS numbe (s): 21.60.Jz
I. INTRODUCTION
In a ecen a icle [1] i was shown how i is possible
o calcula e accu a e p obabili ies o exci a ion o he gian
dipole esonance a he one- and wo-phonon le els (GDR and
DGDR, espec i ely) in e y simple e ms. A key conclusion
o his e e ence was ha he esul s ob ained ollowing he
s anda d o malism o Win he and Alde [2,3] a e— o hose
pa ial wa es whe e hey can be conside ed eliable—much
less dependen on he speci ic s uc u al de ails p o ided by a
mic oscopic desc ip ion o he collec i e mode han p e iously
assumed.
In ac , o bomba ding ene gies up o a leas 5–10 GeV
pe nucleon and o si ua ions ha a e compa ible wi h a
unca ion o he model space a he le el o he DGDR i is
no compelling o ha e an elabo a e pic u e o he dis ibu ion
o cha ges and cu en s wi hin he nucleus. The ele an
in o ma ion is con ained in hei lowes momen s, namely he
posi ion o he cen e o cha ge and i s eloci y. A a ie y o
s uc u al models may sha e hese ea u es and his ealiza ion
makes he in es men o a signi ican e o in his di ec ion
a he imp ac ical. Fo he conc e e ask o unde s anding he
main ea u es exhibi ed by he da a ob ained wi hin p esen
expe imen al condi ions he simples , amilia model o an
oscilla ion o he p o on densi y agains he neu on densi y
does su ice. Le us he e ecall ha he basic pa ame e s ha
a e necessa y o implemen such pic u e a e in e ed om he
known exci a ion ene gy o he mode and he mass and cha ge
o he nucleus in ques ion.
Wi h his back ound in mind we p opose in his con ibu ion
o go a s ep u he and elimina e om he calcula ion scheme
o Re . [1] he p esence o he scala Li´
ena d-Wieche
po en ial o a poin like p ojec ile [4]
(x,y,z, )=γZ
Pe
(x−b)2+y2+γ2(z− P )2,(1)
and, e en mo e deside ably, i s ec o coun e pa

A= P
c. (2)
To his end we sugges o cas he p oblem in he e e ence
sys em o he p ojec ile, whe e he elec ic and magne ic ields
in he whole space and a any ins an o ime a e simply

E( )=ZPe
2
,
B( )=0,(3)
ha is, he same exp essions ha a e no mally used a
non ela i is ic ene gies.
The p ice o be paid o such simplici y and he absence
o ec o po en ials in he p oblem is ha we mus , o cou se,
ans o m he es o ing o ces associa ed wi h he gian dipole
esonance om a sys em in which he “sp ing” is a es
o ano he in which i mo es wi h a speed ≈c. This, as
we shall see, is no ha di icul o do. We would like o
s ess, howe e , ha he eason o posing his p oblem is no
en i ely “academic.” No ice ha i one manages o inco po a e
success ully such ans o ma ion laws in he o malism i
should be possible o compu e he exci a ion p obabili y o
he mode ia any o he ield whose exp ession in a sys em a
es is known.
Le us expand on he p e ious s a emen . The elec omag-
ne ic po en ials a e speci ically easy o ans o m be ween
ela i is ically mo ing sys ems o e e ence because he
combina ion ( 
A, ) o ms a ou - ec o whose change is
go e ned by he ou -by- ou Lo en z ans o ma ion ma ix.
In ac , he Li´
ena d-Wieche exp essions quo ed abo e a e
no mo e han he Lo en z- ans o med e sion o he elec ic
and magne ic po en ials gene a ed by a poin cha ge a es .
I is well known, howe e , ha he ans o ma ion law o
p ac ically any o he in e ac ion is no ha s aigh o wa d
o ob ain. I is because o his ecu en di icul y ha i
appea s p omising o inco po a e in he o malism, once and
o all, he ans o ma ion p ope ies o he in insic o ces ha
accoun o he esponse o he gian dipole esonance. Upon
a success ul comple ion o his p og am one would be able o
es he e ec s o a a ie y o exci a ion couplings ( o ins ance,
nuclea ) wi hou ha ing o wo y abou hei p ope ies unde
Lo en z ans o ma ions.
In his con ibu ion we se ou o explo e he p ac ical
implemen a ion o hese ideas. In Sec. II we wo k ou he
o mal aspec s o sol ing he p oblem o exci a ion o he
GDR as seen om he ame o e e ence o he p ojec ile.
The p oblem o he a ge ecoil, al eady ea ed in de ail
in Re . [1], is b ie ly e iewed in Sec. III. A compa ison
0556-2813/2006/73(1)/014907(5)/$23.00 014907-1 ©2006 The Ame ican Physical Socie y
C. H. DASSO AND M. GALLARDO PHYSICAL REVIEW C 73, 014907 (2006)
S S’
yy’
xx’
zz’
eloci y V = –V
a ge
b
p ojec ile z
R
FIG. 1. (Colo online) Re e ence ames used in he ex . The
p ojec ile ame is Sand he a ge ame ( he nucleus whose gian
dipole mode is exci ed) is S. The la e mo es owa d he le in he
zdi ec ion wi h a eloci y compa able o he speed o ligh . The
o igin o he wo sys ems coincide o =0 and he a ge -p ojec ile
ela i e coo dina e 
Rin he sys em Shas componen s (Rx,R
y,R
z)=
(−b, 0,− ), whe e bis he impac pa ame e .
o he esul s ob ained wi h he p ocedu es p esen ed in he
p e ious Sec. wi h esul s ex ac ed ollowing he con en ional
o malism o Alde and Win he is he subjec o Sec. IV. A
summa y o he a icle and some closing ema ks a e le o
Sec. V.
II. FORMALISM
Because in ela i is ic hea y-ion collisions he exci a ion
ene gies ha en e in o play (≈10–20 MeV) a e much smalle
han he bomba ding ene gies, we can conside he ela i e
mo ion be ween p ojec ile and a ge o be uni o m and assume
ha he classical ajec o ies a e well app oxima ed by s aigh
lines. Wi hou loss o gene ali y we ake ha di ec ion o
be along he zaxis. The p oblem we in end o app oach is
illus a ed in Fig. 1. He e we see wo sys ems o e e ence,
Sand S, which mo e wi h espec o each o he a a e y
la ge speed  . The sys em Sis chosen so ha he p ojec ile
is always a es in he posi ion (b, 0,0), whe e bplays he
ole o he classical impac pa ame e . I is in his ame o
e e ence ha we would like o w i e down he equa ions
o mo ion ha desc ibe he ex e nally d i en oscilla ion o
p o ons agains neu ons in he a ge . We exploi in his
sec ion he same assump ion made a he beginning o Re . [1],
namely ha he cen e o mass o he p o on and neu on
dis ibu ions—i.e., he o iginal “ a ge ”— emains a es a he
coo dina es (x,y,z
)≡(0,0,0), he o igin o sys em S.We
e i y in Sec. III ha , al hough ap io iunjus i ied, he use o
his no- ecoil app oxima ion does no a ec he conclusions o
he analysis in any signi ican way. The ime scales a e chosen
so ha he xaxes in Sand So e lap a = =0.
Unde he condi ions speci ied abo e, he coo dina es in
Sand Sa e connec ed by he ollowing:
x=x,
y=y, (4)
z=γ(z+ ),
=γ + z
c2,
whe e is posi i e and he ac o γassocia ed wi h he Lo en z
ans o ma ion be ween he wo e e ence sys ems is
γ=1/1−( /c)2.(5)
The se o equa ions (4) can be supplemen ed wi h he one
ela ing he spacelike eloci ies in he wo sys ems, namely
˙
x=˙
x
γ(1 +˙
z /c2),
˙
y=˙
y
γ(1 +˙
z /c2),(6)
˙
z=˙
z+
(1 +˙
z /c2).
I is use ul o ecall a couple o o mal ela ionships
in ol ing he γ ac o s o he mo ion o pa icles in S
and S(impo an a ela i is ic speeds in ei he sys em).
We use hem, below, o a i e a ou inal esul s. Calling
u2=(˙
x2+˙
y2+˙
z2) and u2=(˙
x2+˙
y2+˙
z2) we in oduce
˜γ=1/1−(u/c)2,(7)
˜γ=1/1−(u/c)2.
Wi h hese de ini ions i is possible o show ha he h ee
quan i ies γ, ˜γ, and ˜γsa is y
γ˜γ=˜γ/(1 +˙
z /c2),
γ˜γ=˜γ/(1 −˙
z /c2),(8)
γ2=1
(1 +˙
z /c2)(1 −˙
z /c2).
We can igno e om now on equa ions in ol ing one o
he spa ial o ien a ions because he mo ion is cons ained o
ake place on he [x,z](o [x,z
]) plane and y=y=0
h oughou . Keeping his in mind, we w i e he spacelike
componen s o he o ce ac ing a he end o he elonga ed
sp ing in he sys em S. They a e as ollows:
FHO
x=−Cx,
FHO
y=0,(9)
FHO
z=−Cz.
He e we use he same no a ion as in Re . [1]. The alue o
he es o ing o ce pa ame e C=D(¯hω)2is de i ed om he
exci a ion ene gy ¯hω o he mode and he educed mass D=
(ZTNT/AT)m, whe e mis a nucleon mass. S a ing om hese
exp essions we cons uc hecomponen s o Minkowski’s ou -
ec o o ce in S[8],
K
1=−˜γCx,
K
2=0,(10)
K
3=−˜γCz,
K
4=−i
c˜γC(x˙
x+z˙
z),
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which, Lo en z- ans o med in o S, yield
K1=−˜γCx,
K2=0,
K3=−Cγ ˜γγ(z+ )−
c2x˙
x
γ(1 +˙
z /c2)
(11)
+γ(z+ )˙
z+
(1 +˙
z /c2),
K4=−iCγ ˜γ
cγ(z+ )+x˙
x
γ(1 +˙
z /c2)
+γ(z+ )˙
z+
(1 +˙
z /c2).
F om he p e ious exp essions i is easy o iden i y he
in insic es o ing o ces ha should be used o cons uc
he ela i is ic equa ions o mo ion o he collec i e a iable
(x,y,z) in he sys em o coo dina es Sand ollow i s e olu ion
wi h espec o he a iable ,
FHO
x=−Cγ(1 +˙
z /c2)x,
FHO
y=0,(12)
FHO
z=−Cγ(z+ )+Cγ
c2x˙
x.
No ice he wo d ela i is ic in he p e ious pa ag aph. I
is qui e clea ha ypical eloci ies in he sys em Swill be
close o he speed o ligh and he e o e he classical (meaning
in his con ex “no quan al”) equa ions we need o sol e a e
Eins ein’s se
d
d D˙
x
1−(˙
x2+˙
z2)/c2=Fx=FHO
x+ZPZTe2δ
R2b
R,
d
d D˙
z
1−(˙
x2+˙
z2)/c2=Fz=FHO
z+ZPZTe2δ
R2
R.
(13)
To de ine he componen s o he o al o ce Fx,F
zwe
ha e added o he es o ing o ce [Eq. (12)] he (now i ial)
con ibu ion om he Coulomb in e ac ion. Wi hin he dipole
app oxima ion one can use he alue o he ield a any poin
in he neighbo hood o he o igin o Sand we speci ically
ake R=√b2+ 2 2. No ice he ac o δ=NT/(NT+ZT)
in he exp ession o he elec ic o ce. I appea s (c . Re . [1])
because we se up he e equa ions o mo ion o he collec i e
a iable  =(x,y,z) and igno e he o e all accele a ion o
he cen e o mass o he a ge . I should no en e in he
o mula ion (and in ac i does no ) i one in eg a es sepa a ely
o he p o on and neu on componen s, as i is la e done in
Sec. III.
The exp essions (13) a e no ye cas in a con enien o m o
be sol ed by s anda d in eg a ion me hods. Fo his pu pose we
need o isola e he second ime de i a i es o x,z. In oducing
he auxilia y quan i ies
A(˙
z)=1−˙
z2
c2,
B(˙
x, ˙
z)=˙
x˙
z
c2,(14)
C(˙
x)=1−˙
x2
c2,
we a i e o a e y compac se o ime-dependen , i s -o de
coupled di e en ial equa ions in he a iables x,z, ˙
x, ˙
z o be
nume ically p opaga ed om hei ini ial alues, namely
dx
d =˙
x,
dz
d =˙
z,
(15)
d˙
x
d =1
D˜γ3
FxC−FzB
AC−B2,
d˙
z
d =1
D˜γ3
FzA−FxB
AC−B2.
The inal exci a ion ene gy o he dipole mode is calcula ed
a e in eg a ion o [Eq. (15)] and—di iding by ¯hω—con e ed
in o an a e age numbe o phonons N∞. In u n, his numbe is
con e ed in o exci a ion p obabili ies o he g ound s a e and
he one- and wo-phonon le els, jus as i was done in Re . [1]
o si ua ions whe e N∞1.
A compu e code called REVRCE has been w i en o imple-
men his p esc ip ion. To illus a e he ou mos simplici y o
his app oach we show in Fig. 2 an o e iew o he comple e
lis ing o his Fo an p og am. The inse iden i ies he only
piece o he p og am whe e he Coulomb in e ac ion appea s
[c . Eq. (13)].
We compa e in Sec. IV he esul s ob ained using he
REVRCE code wi h s a e-o - he-a calcula ions ollowing he
o mula ion o Alde and Win he .
III. TARGET RECOIL
In Re . [1] i was discussed in de ail he impac o he
common p ac ice o calcula ing he exci a ion o he dipole
mode in a coo dina e sys em a ached o he a ge . Being his
a nucleus wi h a ne posi i e cha ge i is ac ually accele a ed
du ing he collision p ocess and he p esence o ine ial o ces
should no be igno ed wi hou a p ope in es iga ion. I is
no necessa y he e o adap he en i e line o a gumen s o
he p esen si ua ion. Ra he , we limi ou sel es o quo e he
equa ions o mo ion ha should be sol ed in sys em S o ollow
he sepa a e mo ion o he cha ged and neu al componen s o
he nuclea densi y.
In e ms o he wo independen collec i e a iables
 p=(xp,yp,z
p)(p o p o ons) and  n=(xn,yn,z
n)
(n o neu ons) he se o equa ions, now eigh in o al, is
014907-3
C. H. DASSO AND M. GALLARDO PHYSICAL REVIEW C 73, 014907 (2006)
FIG. 2. An o e iew o he o an p og am REVRCE ha cal-
cula es he exci a ion p obabili ies o he GDR and he DGDR
acco ding o he o malism p esen ed in Sec. II. The p og am is
comple e and sel -con ained wi h he excep ion o a single call o a
s anda d in eg a ion ou ine D02BAF [9]. The boxes a e included o
d aw a en ion o he only pa o he p og am whe e he elec o-
magne ic in e ac ion en e s in he equa ions o mo ion, assuming he
simple o m gene a ed by a cha ge a es in a neighbo hood o he
a ge (dipole app oxima ion).
as ollows:
dxp
d =˙
xp,
dzp
d =˙
zp,
(16)
d˙
xp
d =1
ZTm˜γ3
p
Fp
xCp−Fp
zBp
ApCp−Bp2,
d˙
zp
d =1
ZTm˜γ3
p
Fp
zAp−Fp
xBp
ApCp−Bp2.
and
dxn
d =˙
xn,
dzn
d =˙
zn,
(17)
d˙
xn
d =1
NTm˜γ3
n
Fn
xCn−Fn
zB
AnCn−Bn2,
d˙
zn
d =1
NTm˜γ3
n
Fn
zAn−Fn
xBn
AnCn−Bn2.
In hese exp essions
Ap(˙
zp)=1−˙
zp2
c2,
Bp(˙
xp,˙
zp)=˙
xp˙
zp
c2,
Cp(˙
xp)=1−˙
xp2
c2,
(18)
An(˙
zn)=1−˙
zn2
c2,
Bn(˙
xn,˙
zn)=˙
xn˙
zn
c2,
Cn(˙
xn)=1−˙
xn2
c2.
The o ces ac ing on he cha ged and neu al componen s a e,
espec i ely
Fp
x=−Cγ(1 +˙
zp /c2)(xp−xn)+ZPZTe2
R2b
R,
Fp
z=−Cγ(zp−zn)+Cγ
c2(xp−xn)˙
xp+ZPZTe2
R2
R,
(19)
and
Fn
x=−Cγ(1 +˙
zn /c2)(xn−xp),(20)
Fn
z=−Cγ(zn−zp)+Cγ
c2(xn−xp)˙
xn.
No ice ha he ac o δdoes no —as an icipa ed—scale he
Coulomb in e ac ion e m ac ing now only on he cha ged
densi y.
A compu e p og am called REVRCE2c (also e y simple)
has been w i en o implemen his p esc ip ion. We compa e
in he ollowing sec ion he esul s ob ained using his code
wi h hose o calcula ions pe o med aco ding o he s anda d
app oach.
IV. COMPARISON WITH THE STANDARD FORMALISM
In his b ie sec ion we compa e esul s o he p obabili y o
exci a ion o he gian dipole esonance o 40Ca in he eac ion
208Pb+40Ca a he ela i is ic bomba ding ene gies o 500,
1000, and 4000 MeV pe nucleon. On he one hand, we ha e
he p edic ions o he codes REVRCE and REVRCE2c acco ding
o he p esc ip ions discussed in Sec. II and III, espec i ely.
On he o he hand, he esul s o Bayman and Za di ob ained
wi h high compu a ional p ecision ollowing he app oach o
Alde and Win he [5,6].
Resul s o a la ge numbe o impac pa ame e s a e
collec ed in Fig. 3. The lowe limi in he ange o pa ial
wa es ep esen ed in he d awing is de e mined—as i was
al eady men ioned in Re . [1]—excluding om he se hose
alues ha a e incompa ible wi h a “sa e” unca ion o
he model Hamil onian a he wo-phonon le el. This is,
equi alen ly, he egime o alidi y o pe u ba ion heo y
whe e P0≈1 h oughou . On he la ge impac -pa ama e side
014907-4
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0.96
1
1.04
P0
Bayman-Za di
p og am REVRCE
p og am REVRCE2c
500 MeV/nucleon
-6
-5
-4
-3
-2
-1
log10 P1
40 80
-12
-10
-8
-6
-4
-2
log10 P2
40Ca + 208Pb
1000 MeV/nucleon
0 40 80 120
Impac Pa ame e ( m)
4000 MeV/nucleon
100 200 300
FIG. 3. P obabili ies o he exci a ion o he GDR (P1) and he
double GDR (P2)in40Ca in he eac ion 208Pb+40Ca a bomba ding
ene gies o 500, 1000, and 4000 MeV pe nucleon. The ene gy o he
GDR was assumed o be 11.6 MeV. In he i s ow he p obabili y
P0 o emaining in he elas ic channel is also shown. The h ee se s
o esul s gi e he p edic ions ob ained om he simples classical
model (REVRCE), wi h he inclusion o a ge ecoil (REVRCE2c) and
in he con en ional app oach (Bayman-Za di).
he p ac ical limi is se by he eliabili y o he adi ional cal-
cula ions, whose accu acy is, na u ally, much mo e di icul o
man ain.
We can see ha he di e en me hods yield, in all ci cum-
s ances, e y simila esul s. The ag eemen ob ained wi h he
codes REVRCE and REVRCE2c should no come as a su p ise,
hough, because he alidi y o he no- ecoil app oxima ion
had been es ablished in Re . [1] and is no a ec ed by a
e o mula ion o he p oblem h ough Lo en z ans o ma ions.
V. SUMMARY AND CONCLUSIONS
In his manusc ip we se ou o e i y ha an al e na i e
app oach o he p oblem o ela i is ic Coulomb exci a ion o
gian dipole esonances is possible. Ou aim was he e alua ion
o ansi ion p obabili ies a ela i is ic bomba ding ene gies
a oiding he in oduc ion in he calcula ion scheme o he
Li´
ena d-Wieche po en ials.
Such p ojec could ha e no been con empla ed wi hou
ha ing a ailable he esul s p e iously ob ained in Re . [1].
The e we lea ned ha —wi hin he ange o impac pa ame e s
whe e he ac i e eac ion channels in ol e only he g ound
s a e and he i s wo exci ed s a es— he in insic mo ion
can be sa is ac o ily modelled by a collec i e oscilla ion o
he cha ged and neu al componen s o he o al nuclea
densi y agains each o he . The p oblem is hen echnically
o mula ed in e ms o a mac oscopic a iable  ha ep esen s
he displacemen o cen e s o each dis ibu ion densi y wi h
espec o hei equilib ium posi ion. This is, o cou se, one
o he oldes isualiza ions o he in insic mo ion associa ed
wi h a gian dipole esonance in nuclei. Wha was no ob ious,
pe haps, is ha such simple scheme could yield accu a e
exci a ion p obabili ies and handle so well he cu en end
o push bombading ene gies up in o he ela i is ic egime.
Desc ibing he in insic mode in e ms o an ha monic
ib a ion i has been ela i ely s aigh o wa d o ecas he
solu ion o he p oblem h ough a new se o equa ions o
mo ion whe e he elec ic in e ac ion en e s in he simples
possible o m indica ed in Eqs. (13) and (19). The p ac ical
ad an age o he o mula ion can be app ecia ed in he
ema kable simplici y o he calcula ion ool shown in Fig. 2
and he excellen ag eemen s displayed in Fig. 3. No ice, also,
ha in he igh -hand sides o Eqs. (13) and (19) one could
easily add addi ional e ms o conside he e ec o o he ypes
o couplings.
In he case o he GDR he analogy wi h a classical
h ee-dimensional sp ing can be exploi ed o i s ulles ex en .
Le us men ion, howe e , ha he e a e well-es ablished
echniques ha employ a simila semiclassical language o
handling he exci a ion o collec i e ha monic ib a ions o
o he mul ipola i ies in eac ions wi h hea y ions [7] ha can
be a sou ce o inspi a ion o u u e de elopmen s.
ACKNOWLEDGMENTS
Suppo is acknowledged om he Minis y o Educac ion
and Science unde p ojec numbe s FIS2005-01105 and
FPA2005-04460.
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[3] K. Alde and A. Win he , Elec omagne ic Exci a ion (No h
Holland, Ams e dam, 1975).
[4] The esp essions co espond o he elec omagne ic po en ials
gene a ed by a classical mo ion o he p ojec ile along he zaxis
wi h impac pa ame e band a eloci y  P.
[5]B.F.BaymanandF.Za di,Phys.Re .C68, 014905 (2003).
[6]B.F.BaymanandF.Za di,Phys.Re .C59, 2189 (1999).
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ceedings o he In e na ional School o Physics “En ico
Fe mi” (No h Holland, Ams e dam, 1981), Cou se LXXVII,
p. 327.
[8] H. Golds ein, Classical Mechanics (Addison-Wesley, Reading,
MA, 1950).
[9] NAG Fo an Lib a y, The Nume ical Algo i hms G oup L d,
Ox o d.
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