We study the effect of the spreading widths on the excitation probabilities
of the double giant dipole resonance. We solve the coupled-channels equations
for the excitation of the giant dipole resonance and the double giant dipole
resonance. Taking Pb+Pb collisions as example, we study the resulting effect on
the excitation amplitudes, and cross sections as a function of the width of the
states and of the bombarding energy.Comment: 8 pages, 3 figures, corrected typo

A. Müller

G. Kavermann

G. Lambrinidis

H. Bokemeyer

H. Folger

H. Schmidt-Böcking

J. Baumann

J. Bernhardt

J. Peter

J. Ullrich

K. Bethge

K.A. Müller

K.E. Stiebing

L. Schmidt

O. Fröhlich

O. Hohn

P. Senger

P. Thee

S. Runkel

English

arXiv

We study the effect of the spreading widths on the excitation 
probabilities of the double giant dipole resonance. We solve the 
coupled-channels equations for the excitation of the giant dipole 
resonance and the double giant dipole resonance. Taking $Pb+Pb$ collisions 
as example, we study the resulting effect on the excitation amplitudes, 
and cross sections as a function of the width of the states and of the 
bombarding energy

G. Baur

C.A. Bertulani

D. Dolci

EDP Sciences OAI-PMH repository (1.2.0)

Influence of damping on the excitation of 
the double giant resonance

Crossref

On the spectroscopy of atomic electrons (positrons) in the energy range of 10 to 50 MeV from heavy-ion collisions at intermediate energies

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Influence of damping on the excitation of the
double giant resonance
G. Baur(a), C.A. Bertulani(b) and D. Dolci(b)
(a) IKP, Forschunszentrum Juelich, Postfach 1913, D-52425 Juelich, Germany ∗
(b) Instituto de F́ısica, Universidade Federal do Rio de Janeiro
21945-970 Rio de Janeiro, RJ, Brazil †
Abstract
We study the effect of the spreading widths on the excitation prob-
abilities of the double giant dipole resonance. We solve the coupled-
channels equations for the excitation of the giant dipole resonance
and the double giant dipole resonance. Taking Pb + Pb collisions as
example, we study the resulting effect on the excitation amplitudes,
and cross sections as a function of the width of the states and of the
bombarding energy.
Double giant dipole resonances have been mainly studied in heavy ion
Coulomb excitation experiments at high energies (for a recent review, see
[1]). The feasibility of such experiments has been predicted in 1986 [2, 3]
where the magnitude of the cross sections for the excitation of the Double
Giant Dipole Resonance (DGDR) was calculated (see also [4]). In ref. [3] a
recipe was given for treating the effect of the width of the giant resonances
on the excitation probabilities and cross sections. In this letter we make a
quantitative prediction of this effect using a realistic coupled-channels calcu-
lation for the excitation amplitudes. The coupling interaction for the nuclear
excitation i −→ f in a semiclassical calculation for a electric (π = E), or
magnetic (π = M), multipolarity, is given by (eqs. (6-7) of ref. [5])
∗e-mail: G.Baur@fz-juelich.de
†e-mails: bertu@if.ufrj.br and dolci@if.ufrj.br
1
WC =
VC
ǫ0
=
∑
πλµ
Wπλµ (τ) , (1)
where
Wπλµ (τ) = (−1)λ+1
Z1e
~vbλ
1
λ
√
2π
(2λ+ 1)!!
Qπλµ (ξ, τ)M (πλ, µ) . (2)
Above, b is the impact parameter, γ = (1− β2)−1/2, β = v/c, τ = γvt/b
is a dimensionless time variable, ǫ0 = γ~v/b sets the energy scale and
Qπλµ (ξ, τ), with ξ = ξif = (Ef − Ei) /ǫ0 as an adiabatic parameter, de-
pends exclusively on the properties of the projectile-target relative motion.
The multipole operators, which act on the intrinsic degrees of freedom are,
as usual,
M(Eλ, µ) =
∫
d3r ρ(r) rλ Y1µ(r) , (3)
and
M(M1, µ) = − i
2c
∫
d3r J(r).L (rY1µ) , (4)
We treat the excitation problem by the method of Alder and Winther[6].
We solve a time-dependent Schrödinger equation for the intrinsic degrees
of freedom in which the time dependence arises from the projectile-target
motion, approximated by the classical trajectory. For relativistic energies,
a straight line trajectory is a good approximation. We expand the wave
function in the set {| k〉; k = 0, N} of eigenstates of the nuclear Hamiltonian,
where 0 denotes the ground state and N is the number of intrinsic excited
states included in the coupled-channels (CC) problem. We obtain a set of
coupled equations.
To simplify the expression we introduce the dimensionless parameter Θ
(λµ)
kj
by the relation
Θ
(λµ)
kj = (−1)
λ+1 Z1e
~vbλ
1
λ
√
2π
(2λ+ 1)!!
Mkj(Eλ) (5)
2
Then we write the coupled channels equations in the form [5]
dak(τ)
dτ
= −i
N∑
j=0
∑
πλµ
Qπλµ(ξkj, τ)Θ
(λµ)
kj exp (iξkjτ) aj(τ) . (6)
In what follows we concentrate on the E1 excitation mode. In this case,
we have
QE10(ξ, τ) = γ
√
2
[
τφ3(τ)− iξ
(v
c
)2
φ(τ)
]
; QE1±1(ξ, τ) = ∓φ3(τ) ,
(7)
where φ(τ) = (1 + τ 2)
−1/2
.
Following ref. [3] the inclusion of damping leads to the coupled-channels
equations with damping terms, i.e.,
dak(τ)
dτ
= −i
N∑
j=0
∑
µ
QE1µ(ξkj, τ)Θ
(1µ)
kj exp (iξkjτ) aj(τ) −
Γk
ǫ0
ak(τ). (8)
These equations lead to master equations for the occupation probabilities,
P̃j(τ) = |aj(τ)|2, in the form
dP̃k(τ)/dτ = Gk(τ)− Lk(τ) (9)
where
Gk(τ) = 2 ℑm
∑
j
∑
µ
QE1µ(ξkj, τ)Θ
(1µ)
kj exp(iξkjτ ) ak(τ) a
∗
j (τ) (10)
and
Lk(τ) = −
Γk
ǫ0
P̃k(τ) . (11)
These equations can be integrated, yielding the conservation law,
∑
k
(
P̃k(τ) + F̃k(τ)
)
= 1 , where F̃k(τ) =
Γk
ǫ0
∫ τ
−∞
P̃k(τ
′)dτ ′ (12)
3
Due to the exponential decay of the states with k > 1, we have for t → ∞
the limit P̃k(∞) = δj0P̃0(∞) and
P̃0(∞) +
∑
k
F̃k(∞) = 1 . (13)
This means that for t → ∞ there is a probability to find the system in
the ground state given by P̃0(∞) and a probability that it has been excited
and decayed through the channel j which is given by F̃j(∞). Thus, the set of
equations 8 are shown to correctly describe the contribution to the excitation
through channel j.
The excitation probability of an intrinsic state | j〉 in a collision with
impact parameter b is obtained from an average over the initial orientation
and a sum over the final orientation of the nucleus, as
Pj(b) =
1
2I0 + 1
∑
M0, Mj
|F̃j(∞)|2 , (14)
and the cross section is obtained by the classical expression
σj = 2π
∫
Pj(b) T (b) b db . (15)
Above, T (b) accounts for absorption according to the prescription of ref. [5],
using the nucleon-nucleon cross sections and the ground state density of Pb
from experimental data.
We consider the excitation of giant resonances in 208Pb projectiles, inci-
dent on 208Pb targets at 640 A·MeV. This reaction has been studied at the
GSI/SIS, Darmstadt [7]. For this system the excitation probabilities of the
isovector giant dipole (GDR) at 13.5 MeV are large and, consequently, high
order effects of channel coupling should be relevant. To assess the importance
of the damping effects, we calculate the matrix elements assuming that the
GDR is an isolated state depleting 100% of the energy-weighted sum-rule.
The matrix element for the GDR→ DGDR transition incorporates the boson
factor
√
2, as usual [3]. The energy location of the DGDR state is taken as
27 MeV, consistent with the experimental data. The spin and parities of the
states are given by 1− for the GDR, and 0+ and 2+ for the DGDR, respec-
tively. The distribution of the strength among the 0+ and 2+ DGDR states
are simply obtained from Clebsh-Gordan coefficients [8].
4
In figure 1 we plot the time-dependent occupation probabilities of the
ground state, in Pb, N = 0, of the GDR state, N = 1, and of the DGDR
state, N = 2, respectively. Figure 1(a) shows the occupation probabilities
with the widths equal to zero, ΓN = 0. In figure 1(b) we plot the occupation
probabilities of the GDR state, N = 1, and of the DGDR state, N = 2,
with ΓN = 0 (full lines), and with ΓGDR = 4 MeV (experimental), and
ΓDGDR = 5.7 MeV (dashed lines). The width of the DGDR is set to ΓDGDR =√
2ΓGDR, following the apparent trend of the experimental data [7]. Note,
that ΓDGDR = 2ΓGDR has a better (and simpler) theoretical explanation [9].
But, the time integrated population of the DGDR state will not be much
influenced by using the later parametrization.
We observe that the inclusion of damping leads to strong modifications in
the time-dependent occupation probabilities of the GDR and DGDR states.
One might wrongly deduce from figure 1 that the excitation probabilities
are reduced proportionally to the difference between the maximum value of
P̃N(τ), with and without damping. However, the quantities shown in figure
1(b) includes the loss of occupation probabilities of a given state, while it
is being populated by the time-dependent transitions. Thus, the reduction
of the excitation probabilities of the GDR and the DGDR due to damping
is smaller than deduced from figure 1. The relevant quantity to calculate
the excitation cross sections are the quantities F̃j(∞), which account for the
time-integrated transition probability, followed by decay, of the state j.
In figure 2 we plot the flux functions, or time-integrated transition prob-
abilities to the GDR and the DGDR states as a function of the width of
the collective state, ΓGDR, keeping constant the ratio ΓDGDR/ΓGDR =
√
2.
We keep the impact parameter fixed, b = 15 fm. We note that, varying the
width from 0 up to 4 MeV leads to a 10% decrease of the flux functions
into GDR and DGDR states. A similar tendency is observed for the total
cross section, integrated over impact parameters. The excitation probability
decreases with about 1/b4, therefore the grazing collisions are weighted most
strongly.
In figure 3(a) we plot the effect of damping in the total cross sections, as
a function of the bombarding energy. We take ΓGDR = 0 (dashed line) and
ΓGDR = 4 MeV (full line), keeping constant the ratio ΓDGDR/ΓGDR =
√
2.
We observe that the effect of damping disappears, as the bombarding energy
increases. At high energies the reaction is fast, and the system does not have
time to dissipate during its excitation. In this regime the sudden approxi-
mation is a valid approach to the calculation of the excitation amplitudes.
5
We note that our model is restricted to the excitation of isolated res-
onant states, including a time-dependent loss term on the far right of the
coupled-channels equations 8. This is different from a study of the influence
of the fragmentation of the resonances into many neighbouring states. In
this case, the Coulomb excitation of the giant resonances is obtained as a
superposition of excitations to states spread over an energy envelope, usually
taken as a Lorentzian shape. States at lower energy are more easily excited
than states at higher energies. Thus, the spreading of the resonances may
lead to another kind of effect of the widths, not obtainable in the above
treatment. To study this effect in a simple way, we use the harmonic model
of ref. [10]. The photo-nuclear cross sections which enter in these calcu-
lations are of Lorentzian shapes, with a cut at the low energy limit of 8
MeV, corresponding to the threshold for neutron emission, since most of
the DGDR manifestation in experiments come from neutron emission after
relativistic Coulomb excitation. The magnitude of the photo-nuclear cross
sections are obtained by using a 100% depletion of the Thomas-Reiche-Kuhn
energy-weighted sum-rule applied to the GDR.
The harmonic model provides a simple analytical formula to calculate the
excitation probabilities of the DGDR [10]. The resulting cross sections are
shown in figure 3(b) where in the solid line we take ΓGDR = ΓDGDR = 0, while
the dashed-lines are for ΓDGDR = 5.7 MeV and ΓGDR = 4 MeV. We observe
a similar effect as in figure 3(a). This is understood as a reduction due to
the spread of states at energies above the energy centroid of the Lorentzian
envelope. The excitation amplitude for these states are smaller, thus leading
to a net reduction of the energy integrated Coulomb excitation cross sections.
In conclusion, we have obtained the dependence of the excitation ampli-
tudes on the width of the giant resonance states. We show that the effect
reduces excitation probabilities, and cross sections. We have developed an
approach to solve this problem in realistic situations. It is demonstrated that
the dynamical effect of the widths of the GR’s in a time-dependent picture
leads to a decrease of the cross sections, more accentuated for low energy col-
lisions. The energy fragmentation of the giant resonances can be studied in a
simple fashion within the harmonic model. The net effect is also to decrease
the cross sections with increasing width, specially at low energy collisions.
Acknowledgments
This work was supported in part by the Brazilian funding agencies CNPq,
FAPERJ, FUJB/UFRJ, and PRONEX, under contract 41.96.0886.00.
6
References
[1] C.A. Bertulani, J. Phys. G24 (1998) 1
[2] G. Baur and C.A. Bertulani, Phys. Lett. B174 (1986) 23; G. Baur and
C.A. Bertulani, Phys. Rev. C34 (1986) 1654;
[3] C.A.Bertulani and G.Baur, Phys. Reports 163 (1988) 299
[4] P. Braun-Munzinger et al., Proposal 814 submitted to the AGS Program
Committee, SUNY at Stony Brook, accepted 1985 (unpublished)
[5] C.A. Bertulani, Computer Physics Communications 116 (1999) 345
[6] K. Alder and A. Winther, “Coulomb Excitation”, New York, Academic
Press, 1966.
[7] H. Emling, Prog. Part. Nucl. Phys. 3 (1994)
[8] C.A. Bertulani and V. Zelevinsky, Phys. Rev. Lett. 71 (1993) 967; Nucl.
Phys. A568 (1993) 931
[9] G.Baur and C.A.Bertulani, ”Proceedings of the Int. School of Heavy Ion
Physics”, Erice, Italy, October 1986, Plenum Press, ed. by R.A. Broglia
and G. F. Bertsch, p. 331.
[10] T. Aumann, C.A. Bertulani and K. Suemmerer, Phys. Rev. C51 (1995)
416
Figure Captions
Fig. 1 - Occupation probabilities in Coulomb excitation of Pb, in Pb+Pb
collisions at 640 MeV.A. N = 0 for the ground state, N = 1 for the GDR
state, N = 2, for the DGDR state, respectively. Figure 1(a) shows the
occupation probabilities with the widths equal to zero, ΓN = 0. In figure
1(b) we plot the occupation probabilities of the GDR state, N = 1, and of
the DGDR state, N = 2, with ΓN = 0 (full lines), and with ΓGDR = 4 MeV,
and ΓDGDR = 5.7 MeV (dashed lines). The width of the DGDR is set to
ΓDGDR =
√
2ΓGDR, according to the trend of the experimental data [7].
Fig. 2 - Flux functions, or time-integrated transition probabilities to the
GDR and the DGDR states in Pb as a function of the width of the collective
state, ΓGDR, keeping constant the ratio ΓDGDR/ΓGDR =
√
2. We keep the
7
impact parameter fixed, b = 15 fm. N = 1 for the GDR state, N = 2, for
the DGDR state.
Fig. 3 - (a) Total cross sections, as a function of the bombarding energy.
We take ΓGDR = 0 (dashed line) and ΓGDR = 4 MeV (full line), keeping
constant the ratio ΓDGDR/ΓGDR =
√
2. N = 1 for the GDR state, N = 2,
for the DGDR state. (b) Results of the harmonic model [10] for the Coulomb
excitation cross sections of the GDR (N = 1) and the DGDR (N = 2). The
solid lines use ΓGDR = ΓDGDR = 0, while the dashed-lines are for ΓDGDR =
5.7 MeV and ΓGDR = 4 MeV.
8
P
N
 (
t)
~
(a)
(b)
0.00
0.40
0.80
N = 0
N = 1
N = 2
0.00 4.00 8.00
γvt / b
0.00
0.20
0.40
N = 1
N = 2
1 2 3 4 5 6
ΓGDR (MeV)
0.0
0.1
1.0
F
N
N = 1
N = 2
10
100
1000
10000
N = 1
N = 2
σ  
 [m
b]
(a)
(b)
0 1 10
Elab [MeV/nucleon]
10
100
1000
10000
N = 1
N = 2
.1


We study the effect of the spreading widths on the excitation probabilities of the double giant dipole resonance. We solve the coupled-channels equations for the excitation of the giant dipole resonance and the double giant dipole resonance. Taking Pb+Pb collisions as example, we study the resulting effect on the excitation amplitudes, and cross sections as a function of the width of the states and of the bombarding energy

Influence of damping on the excitation of the double giant resonance

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