Coulomb excitation of unstable nuclei at intermediate energies
C.A. Bertulani1∗, G. Cardella2, M. De Napoli2,3, G. Raciti2,3, and E. Rapisarda2,3
1 Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA
2 Istituto Nazionale di Fisica Nucleare, Sezione di Catania,
via Santa Sofia 64, I-95123, Catania, Italy
3 Dipartimento di Fisica e Astronomia, Universitá Catania,
via Santa Sofia 64, I-95123, Catania, Italy
Abstract
We investigate the Coulomb excitation of low-lying states of unstable nuclei in intermediate
energy collisions (Elab ∼ 10−500 MeV/nucleon). It is shown that the cross sections for the E1 and
E2 transitions are larger at lower energies, much less than 10 MeV/nucleon. Retardation effects
and Coulomb distortion are found to be both relevant for energies as low as 10 MeV/nucleon and
as high as 500 MeV/nucleon. Implications for studies at radioactive beam facilities are discussed.
PACS numbers: 25.60.-t, 25.70.-z, 25.70.De
Keywords: Coulomb excitation, cross sections, unstable nuclei.
∗ bertulanica@ornl.gov.
http://arxiv.org/abs/0704.0060v2
Unstable nuclei are often studied with reactions induced by secondary radioactive beams.
Examples of these reactions are elastic scattering, fragmentation and Coulomb excitation
by heavy targets. Coulomb excitation is specially useful since the interaction mechanism is
very well known [1]. It is the result of electromagnetic interactions of a projectile (ZP ,AP )
with a target (ZT ,AT ). One of the participating nuclei is excited as it passes through the
electromagnetic field of the other. Here we will only consider the excitation of the pro-
jectile as is of interest in studies carried out in heavy ion facilities around the world, e.g.
LNS/Catania, NSCL/MSU, GSI, GANIL, RIKEN, etc. In Coulomb excitation a virtual pho-
ton with energy E is absorbed by the projectile. Because in pure Coulomb excitation the
participating nuclei stay outside the range of the nuclear strong force, the excitation cross
section can be expressed in terms of the same multipole matrix elements that character-
ize excited-state gamma-ray decay, or the reduced transition probabilities, B(πλ; Ji → Jf).
Hence, Coulomb excitation amplitudes are strongly coupled with valuable nuclear struc-
ture information. Therefore, this mechanism has been used for many years to study the
electromagnetic properties of low-lying nuclear states [1].
Coulomb excitation cross sections are large if the adiabacity parameter satisfies the con-
dition
ξ = ωfi
< 1 , (1)
where a0 is half the distance of closest approach in a head-on collision for a projectile
velocity v, and Ex = ~ωfi is the excitation energy. This adiabatic cut-off limits the possible
excitation energies below 1-2 MeV in sub-barrier collisions. A possible way to overcome this
limitation, and to excite high-lying states, is to use higher projectile energies. In this case,
the closest approach distance, at which the nuclei still interact only electromagnetically, is
of order of the sum of the nuclear radii, R = RP + RT , where P refers to the projectile
and T to the target. For very high energies one has also to take into account the Lorentz
contraction of the interaction time by means of the Lorentz factor γ = (1− v2/c2)−1/2, with
c being the speed of light. For such collisions the adiabacity condition, Eq. (1), becomes
ξ(R) =
< 1 . (2)
From this relation one obtains that for bombarding energies around and above 100
MeV/nucleon, states with energy up to 10-20 MeV can be readily excited [3].
An appropriate description of Coulomb excitation at intermediate energies (Elab =
10 − 500 MeV/nucleon) has been described in ref. [2]. In this energy region neither the
non-relativistic Coulomb excitation formalism described in ref. [1], nor the relativistic one
formulated in refs. [3, 4] are appropriate. This is discussed in details in ref. [2] where it
is shown that the correct values of the Coulomb excitation cross sections differ by up to
30-40% when compared to the non-relativistic and relativistic treatments used to calculate
experimental observables (cross sections, gamma-ray angular distributions, etc.).
We follow the formalism of ref. [2] to calculate cross sections for Coulomb excitation from
energies varying from 10 to 500 MeV/nucleon. These are the energies where most radioactive
beam facilities are or will be operating around the world. The calculated cross sections will
be of useful guide for future experiments. We also compare the accurate calculations with
those obtained by using simple analytical formulas and test the regime of their validity.
The cross sections for the transition Ji → Jf in the projectile are calculated using the
equation [2]
dσi→f
4π2Z2T e
B(πλ, Ji → Jf)
(2λ+ 1)3
| S(πλ, µ) |2 , (3)
where π = E or M stands for the electric or magnetic multipolarity, and
B(πλ, Ji −→ Jf) =
2Ji + 1
|〈Jf ‖M(πλ)‖Ji〉|
are the reduced transition probabilities. In these equations, ǫ = 1/ sin(Θ/2), with Θ being
the deflection angle, a0 = ZPZT e
2/m0v
2 and a = a0/γ. The complex functions S(πλ, µ) are
integrals along Coulomb trajectories corrected for retardation. Their calculation and how
they relate to the non-relativistic and relativistic theories are described in details in ref. [2].
Here we will introduce another comparison tool for the total cross section, which is obtained
by integration of eq. 3 over scattering angles. The code COULINT [2] was used to calculate
the orbital integrals S(πλ, µ) and the cross sections of eq. 3 (for more details, see ref. [2]).
Using the theory described in ref. [4], it is easy to show that approximate values of the
cross sections for E1, E2, and M1 transitions can be obtained by means of the relations
(app)
B (E1)
ξK0K1 −
K21 −K
(app)
E3xB (E2)
K21 + ξ
K0K1 −
K21 −K
(app)
B (M1)
ξK0K1 −
K21 −K
, (5)
where Kn are the modified Bessel functions of the second order, as a function of ξ given by
eq. 2, with R corrected for recoil by the modification R → R + πa/2 [3].
Here we will only consider the excitation of the lowest lying states in light and medium
heavy nuclei. For nuclear masses A < 20, the TUNL nuclear data evaluation web site
was of great help [5]. The electromagnetic transition rates at the TUNL database are
given in Weisskopf units and are transformed to the appropriate B(πλ, Ii → If)-values
by means of the standard Weisskopf relations BW (E1; Ji → Jgs) = 0.06446A
2/3 e2fm2,
BW (E2; Ji → Jgs) = 0.05940A
4/3 e2fm4, and BW (M1; Ji → Jgs) = 1.79 (e~/2mnc)
. For
comparison, a few medium mass nuclei, as well as a few stable nuclei, were included in the
calculation. Other data were taken from refs. [6, 7, 8, 9].
Some cases of nuclei far from the stability line are very interesting and deserve further
study, possibly using the method of Coulomb excitation. For example, it is well known that
nuclei with open shells tend to have B(E2) values greater than 10 W.u., whereas nuclei with
shell closure of neutrons or protons tend to have distinctly smaller B(E2) values. Typical
examples of the latter category are the doubly magic nuclei, 16O and 48Ca, which B(E2)
values are 3.17 and 1.58 W.u., respectively. According to an empirical formula adjusted
to a global fit of the known transition rates, the values of first excited 2+ level, E2+ , and
B(E2; 0+ → 2+) are related by [10] (E2+ in keV)
B(E2; 0+ → 2+) = 26
A2/3E2+
e2fm4. (6)
The value of B(E2) for 16C based on this formula is at least one order of magnitude
larger than what is observed experimentally in a Coulomb dissociation experiment [9]. The
anomalously strong hindrance of the 16C transition is not well explained theoretically. This
is just an example of the power of Coulomb excitation as a tool to access the new physics
inherent of poorly known rare nuclear species.
Another example is the strong E1 transition in 11Be. 11Be is an archetype of a halo nu-
cleus and exhibits the fastest known dipole transition between bound states in nuclei. The
B(E1) transition strength between the ground and the only bound excited state (at 0.32
MeV) was determined from lifetime measurements by Millener et al. to be 0.116 e2fm2 [11].
However, Coulomb excitation experiments have obtained a much smaller value of the B(E1)
which is still a matter of investigation [12, 13, 14]. It is thus seems clear that predictions
based on traditional nuclear structure and reaction theory often yields results in disagree-
ment with experimental data. In spite of that, when proper corrections are accounted for
(e.g. channel-coupling, nuclear excitation, relativistic corrections), Coulomb excitation of ra-
dioactive beams is a powerful complementary tool to investigate electromagnetic properties
of nuclei far from the stability line.
In Table 1 we compare our calculations with several experimentally obtained cross sections
for Coulomb excitation of unstable nuclei. The units of energy are MeV, the laboratory
energy is in MeV/nucleon, the B-values are in units of e2 fm2λ, and the cross sections are
in millibarns. The last two columns give the calculated cross sections obtained by using
eqs. 3 and 5, respectively. Since the cross sections of eq. 5 are functions of the minimum
impact parameter, the values reported in the Table have been calculated according to the
experimental angular ranges reported in the seventh column. Except for the 11Be case, for
which the discrepancy between theory and experiment is known (see discussion above), the
calculated cross sections are close to the experimental values. Nonetheless, the calculated
cross sections tend to be smaller than the experimental ones for 17Ne, 32Mg, 38S, 40S, 42S,
44Ar, and 46Ar projectiles. This is worrisome because the B(πλ) values were extracted
from the experimentally obtained cross sections, using equations similar to eq. 5. These
experimental B-values would have to be larger by 10− 30% according to our calculations.
It is important to stress the fact that many experimental data on unstable nuclei collected
up to now have been analyzed by means of theoretical tools (DWBA and coupled-channels
codes) which do not include relativistic dynamics (the inclusion of relativistic kinematics
is straightforward). This problem was first addressed in ref. [15], where it was shown
that the analysis of experimental data at intermediate energies without a proper treatment
of relativistic dynamics leads to wrong values of electromagnetic transition probabilities.
We should stress that a full theoretical treatment of relativistic dynamics of strong and
electromagnetic interactions in many-body systems is very difficult and still does not exist
[15].
Data Projectile Target Elab πλ B(πλ) θrange Ex σexp σth σapp
1 [16] 11Be Pb 43. E1 0.115 < 5◦ 0.32 (1
) 191± 26 328. 323.
2 [16] 11Be Pb 59.4 E1 0.094 < 3.8◦ 0.32 (1
) 304± 43 213. 211.
3 [18] 11Be Au 57.6 E1 0.079 < 3.8◦ 0.32 (1
) 244± 31 170. 168.
4 [17] 11Be Pb 64. E1 0.099 < 3.8◦ 0.32 (1
) 302± 31 217. 215.
5 [19, 20] 17Ne Au 60. M1 0.163 < 4.5◦ 1.29 (1
) 12± 4 12.6 13.0
6 [6] 32Mg Pb 49.2 E2 454 < 4◦ 0.885 (0+ → 2+) 91.7± 14.4 137. 128.
7 [19] 38S Au 39.2 E2 235 < 4.1◦ 1.29 (0+ → 2+) 59± 7 48. 45.0
8 [19] 40S Au 39.5 E2 334 < 4.1◦ 0.91 (0+ → 2+) 94± 9 75.5 70.4
9 [19] 42S Au 40.6 E2 397 < 4.1◦ 0.89 (0+ → 2+) 128± 19 101. 94.3
10 [19] 44Ar Au 33.5 E2 345 < 4.1◦ 1.14 (0+ → 2+) 81± 9 62.3 58.3
11 [19] 46Ar Au 35.2 E2 196 < 4.1◦ 1.55 (0+ → 2+) 53± 10 40.9 38.2
12 [8] 46Ar Au 76.4 E2 212 < 2.9◦ 1.55 (0+ → 2+) 68± 8 50.0 47.4
Table 1. Cross sections for Coulomb excitation of unstable nuclei. The units of energy
are MeV, the laboratory energy is in MeV/nucleon, the B(πλ)-values are in units of e2fm2λ,
and the cross sections are in millibarns. The data for different experiments (numbered 1 to
12) were collected from the references listed in column 1. The last two columns give the
calculated cross sections obtained by using eqs. 3 and 5, respectively.
In figure 1 we show a comparison between the experimental data and our calculations.
We notice that the cross sections calculated with help of eq. 5 are not much different than
those calculated with eq. 3. They are systematically lower, up to 10%, than the exact
calculation following eq. 3. As we discuss below, this is not always the case, specially for
the excitation of high-lying states. In fact, this is a good check of eq. 3, which is done in a
very different way than the analytical calculations of eq. 5. But as we will see below, this
agreement is not always the case, specially when one includes small impact parameters for
which the sensitivity to the relativistic corrections is higher (see ref. [2]). The dashed curve
in figure 1 is a guide to the eye. It helps to see that the experimental cross sections are on
average larger than the calculated ones, either with eq. 3 (open circles), or with eq. 5 (open
triangles).
2 4 6 8 10 12
Data set
FIG. 1: Comparison between experimental Coulomb excitation cross sections (solid stars with
error bars) and theoretical ones, calculated either with eq. 3 (open circles), or with eq. 5 (open
triangles).
Ex [MeV] J
i → J
f πλ B(πλ) [e
2 fm2λ] 10 20 30 50 100 200 500
11Be 0.32 1
E1 0.115 1128 653 473 315 187 115 69.6
11B 2.21 3
M1 2.40×10−2 0.301 0.799 1.15 1.63 2.33 3.08 4.17
11C 2.00 3
M1 1.52×10−2 0.196 0.551 0.793 1.12 1.57 2.07 2.76
12B 0.953 1+ → 2+ M1 4.62×10−3 0.227 0.395 0.490 0.607 0.762 0.917 1.13
12C 4.44 0+ → 2+ E2 37.9 34.6 38.6 31.3 21.6 12.1 6.93 3.81
13C 3.09 1
E1 1.39×10−2 8.37 11.3 11.0 9.61 7.28 5.39 3.89
13N 2.37 1
E1 3.56×10−2 38.2 43.6 39.6 32.5 23.2 16.4 11.4
15C 0.74 1
E2 2.90 8.79 4.04 2.65 1.59 0.839 0.475 0.267
16C 1.77 0+ → 2+ E2 2.12 8.81 4.41 2.92 1.76 0.920 0.517 0.285
16N 0.12 0− → 2− E2 10.2 31.0 14.1 9.21 5.53 2.91 1.64 0.926
17N 1.37 1
M1 5.15×10−3 0.153 0.304 0.397 0.516 0.680 0.848 1.09
17O 0.87 5
E2 2.07 6.30 2.88 1.87 1.12 0.588 0.332 0.184
17F 0.5 5
E2 21.6 68.3 29.7 19.3 11.6 6.08 3.44 1.92
18O 1.98 0+ → 2+ E2 44.8 109 60.7 40.9 24.8 11.6 7.27 3.99
18F 0.94 1+ → 3+ E2 37.9 115 52.5 34.1 20.4 10.7 6.01 3.34
18Ne 1.89 0+ → 2+ E2 248 615 342 229 138 72.0 40.1 22.1
19O 0.1 5
M1 2.34×10−4 0.0495 0.0615 0.0673 0.0737 0.0799 0.779 0.799
19F 0.11 1
E1 5.51×10−4 8.07 4.36 3.06 1.97 1.10 0.592 0.337
19Ne 0.24 1
E2 119 361 157 102 61.6 32.5 18.5 10.5
20O 1.67 0+ → 2+ E2 28.0 72 37.4 24.9 15.1 7.86 4.41 2.43
20F 0.656 2+ → 3+ M1 3.56×10−3 0.237 0.385 0.465 0.560 0.683 0.803 0.959
20Ne 1.63 0+ → 2+ E2 319 834 433 287 173 89.8 50.3 27.6
30Ne 0.791 0+ → 2+ E2 460 1167 550 361 218 115 65.0 35.2
32Mg 0.885 0+ → 2+ E2 454 1151 541 355 214 112 63.0 36.7
42S 0.89 0+ → 2+ E2 397 945 445 292 175 91.9 52 29.7
46Ar 1.55 0+ → 2+ E2 190 399 209 140 84.4 44.1 24.7 13.6
54Ni 1.40 0+ → 2+ E2 626 1319 677 447 268 139 78.1 43.1
Table 2 - Cross sections (in mb) for Coulomb excitation of projectiles incident on Pb
targets at bombarding energies ranging from 10 to 500 MeV/nucleon. The energy units are
MeV, the laboratory energy is in MeV/nucleon, the B(πλ)-values are in units of e2fm2λ.
The cross sections for Coulomb excitation of numerous projectiles incident on Pb targets
at bombarding energies ranging from 10 to 500 MeV/nucleon are shown in Table 2. These
cross sections were calculated assuming that the detectors collect events from all possible
Coulomb scattering events. In a real experimental situation, the angular distribution is
restricted to angular windows, reducing the available cross sections. Only the lowest lying
transitions have been considered, i.e. from the ground to the first excited states. One
observes that some cross sections are very large, specially for 11Be, 18Ne, 30Ne and 54Ni.
For these and other similar cases, the measurements are easy to perform, with a large
number of events/second even with modest intensities. Cases such as 16C are well within
the experimental possibilities in most radioactive beam facilities.
Table 2 also shows that, except for M1 excitations, the Coulomb excitation cross sections
decrease steadily as the energy increases from 10 to 500 MeV/nucleon. Based on these
numbers alone, one could conclude that Coulomb excitation of low-lying states (in contrast
to the case of high-lying states, e.g. giant resonances [4]) are better suited for studies at low
energies. However, reactions at lower energies while are less influenced by contamination due
to nuclear breakup [12, 14] can give rise to large high-order effects [21]. The interpretation
of data could be distorted as in the case of Coulomb dissociation of 8B at low energy [24],
which was completely misinterpreted in terms of first-order calculations. In some situations,
when higher-order effects are relevant, the effect of the nuclear breakup cannot be neglected
either [22, 23]. Thus, the choice of the incident energy would depend on the experimental
conditions. Identification of gamma-rays from de-excitation using Doppler shift techniques
are often more advantageous at higher energies. Moreover, except for few cases (e.g. 11C), the
magnetic dipole transitions are much smaller than those for E1 and E2 transitions. Even
for M1 transitions the measurements are under the possibility of most new experimental
facilities.
The comparison of the exact calculations, using eq. 3 (solid lines), and the approximations
5 (dashed lines) are shown in figs. 2(a-d), for 11Be, 11B, 54Ni and 16O, respectively. The 16O
case (as well as for 12C in Table 2) was included for comparison, with a high-lying excited
state. We see from figs 2a and 2b that the approximations in eq. 5 work quite well for
the M1 multipolarity and reasonably well (within 20% at 10 MeV/nucleon and 5% at 50
MeV/nucleon) for the E1 cases. But they fail badly at low and intermediate energies for
(c) (d)
FIG. 2: Coulomb excitation cross section of the first excited state in 11Be, 11B and 54Ni and of the
13.05 MeV sate in 16O projectiles incident on Pb targets as a function of the laboratory energy.
the E2 ( fig. 2c). The reason is that the E2 Coulomb field (“tidal field”) is very sensitive
to the details of the collision dynamics at low energies. These conclusions can be deceiving
since even for the E1 and M1 cases the approximations in eq. 5 may strongly differ from
the exact calculations if the excitation energy is large (see discussion in ref. [2]). This is
shown in figure 2d, where we plot the Coulomb excitation cross section of the Ex = 13.09
MeV state in 16O. In this case, the cross sections based on eq. 5 is a factor of 10 smaller
than the exact calculation at 10 MeV/nucleon. At 100 MeV/nucleon this difference drops
to 10%, which still needs to be considered with care.
In summary, in this article we have used the formalism of ref. [2] to predict the cross
sections for Coulomb excitation of several light projectiles with electromagnetic transitions
found in the literature, listed in the TUNL database [5], and for a few other selected cases.
These estimates will be useful for planing Coulomb excitation experiments at present and
future heavy ion facilities. It is evident that the inclusion of relativistic effects combined
with Coulomb distortion are of the utmost relevance. The cross section inferred by using
non-relativistic or pure relativistic treatments can be wrong by up to 30% even at 100
MeV/nucleon, as shown here and in ref. [2]. Finally, the use of Coulomb excitation to
produce nuclei in high-lying states is an important tool to study particle emission processes.
For example, the excitation of 18Ne and its subsequent decay by two-proton emission is a
process of large theoretical and experimental interest. Experimental work in this direction
is in progress [25].
Acknowledgments
This research was supported by the U.S. Department of Energy under contract No. DE-
AC05-00OR22725 (Oak Ridge National Laboratory) with UT-Battelle, LLC., and by DE-
FC02-07ER41457 with the University of Washington (UNEDF, SciDAC-2).
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[4] C.A. Bertulani and G. Baur, Nucl. Phys. A 442 (1985) 739.
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	References

  We investigate the Coulomb excitation of low-lying states of unstable nuclei
in intermediate energy collisions ($E_{lab}\sim10-500$ MeV/nucleon). It is
shown that the cross sections for the $E1$ and $E2$ transitions are larger at
lower energies, much less than 10 MeV/nucleon. Retardation effects and Coulomb
distortion are found to be both relevant for energies as low as 10 MeV/nucleon
and as high as 500 MeV/nucleon. Implications for studies at radioactive beam
facilities are discussed.

Coulomb excitation of unstable nuclei at intermediate energies
C.A. Bertulani1∗, G. Cardella2, M. De Napoli2,3, G. Raciti2,3, and E. Rapisarda2,3
1 Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA
2 Istituto Nazionale di Fisica Nucleare, Sezione di Catania,
via Santa Sofia 64, I-95123, Catania, Italy
3 Dipartimento di Fisica e Astronomia, Universitá Catania,
via Santa Sofia 64, I-95123, Catania, Italy
Abstract
We investigate the Coulomb excitation of low-lying states of unstable nuclei in intermediate
energy collisions (Elab ∼ 10−500 MeV/nucleon). It is shown that the cross sections for the E1 and
E2 transitions are larger at lower energies, much less than 10 MeV/nucleon. Retardation effects
and Coulomb distortion are found to be both relevant for energies as low as 10 MeV/nucleon and
as high as 500 MeV/nucleon. Implications for studies at radioactive beam facilities are discussed.
PACS numbers: 25.60.-t, 25.70.-z, 25.70.De
Keywords: Coulomb excitation, cross sections, unstable nuclei.
∗ bertulanica@ornl.gov.
http://arxiv.org/abs/0704.0060v2
Unstable nuclei are often studied with reactions induced by secondary radioactive beams.
Examples of these reactions are elastic scattering, fragmentation and Coulomb excitation
by heavy targets. Coulomb excitation is specially useful since the interaction mechanism is
very well known [1]. It is the result of electromagnetic interactions of a projectile (ZP ,AP )
with a target (ZT ,AT ). One of the participating nuclei is excited as it passes through the
electromagnetic field of the other. Here we will only consider the excitation of the pro-
jectile as is of interest in studies carried out in heavy ion facilities around the world, e.g.
LNS/Catania, NSCL/MSU, GSI, GANIL, RIKEN, etc. In Coulomb excitation a virtual pho-
ton with energy E is absorbed by the projectile. Because in pure Coulomb excitation the
participating nuclei stay outside the range of the nuclear strong force, the excitation cross
section can be expressed in terms of the same multipole matrix elements that character-
ize excited-state gamma-ray decay, or the reduced transition probabilities, B(πλ; Ji → Jf).
Hence, Coulomb excitation amplitudes are strongly coupled with valuable nuclear struc-
ture information. Therefore, this mechanism has been used for many years to study the
electromagnetic properties of low-lying nuclear states [1].
Coulomb excitation cross sections are large if the adiabacity parameter satisfies the con-
dition
ξ = ωfi
< 1 , (1)
where a0 is half the distance of closest approach in a head-on collision for a projectile
velocity v, and Ex = ~ωfi is the excitation energy. This adiabatic cut-off limits the possible
excitation energies below 1-2 MeV in sub-barrier collisions. A possible way to overcome this
limitation, and to excite high-lying states, is to use higher projectile energies. In this case,
the closest approach distance, at which the nuclei still interact only electromagnetically, is
of order of the sum of the nuclear radii, R = RP + RT , where P refers to the projectile
and T to the target. For very high energies one has also to take into account the Lorentz
contraction of the interaction time by means of the Lorentz factor γ = (1− v2/c2)−1/2, with
c being the speed of light. For such collisions the adiabacity condition, Eq. (1), becomes
ξ(R) =
< 1 . (2)
From this relation one obtains that for bombarding energies around and above 100
MeV/nucleon, states with energy up to 10-20 MeV can be readily excited [3].
An appropriate description of Coulomb excitation at intermediate energies (Elab =
10 − 500 MeV/nucleon) has been described in ref. [2]. In this energy region neither the
non-relativistic Coulomb excitation formalism described in ref. [1], nor the relativistic one
formulated in refs. [3, 4] are appropriate. This is discussed in details in ref. [2] where it
is shown that the correct values of the Coulomb excitation cross sections differ by up to
30-40% when compared to the non-relativistic and relativistic treatments used to calculate
experimental observables (cross sections, gamma-ray angular distributions, etc.).
We follow the formalism of ref. [2] to calculate cross sections for Coulomb excitation from
energies varying from 10 to 500 MeV/nucleon. These are the energies where most radioactive
beam facilities are or will be operating around the world. The calculated cross sections will
be of useful guide for future experiments. We also compare the accurate calculations with
those obtained by using simple analytical formulas and test the regime of their validity.
The cross sections for the transition Ji → Jf in the projectile are calculated using the
equation [2]
dσi→f
4π2Z2T e
B(πλ, Ji → Jf)
(2λ+ 1)3
| S(πλ, µ) |2 , (3)
where π = E or M stands for the electric or magnetic multipolarity, and
B(πλ, Ji −→ Jf) =
2Ji + 1
|〈Jf ‖M(πλ)‖Ji〉|
are the reduced transition probabilities. In these equations, ǫ = 1/ sin(Θ/2), with Θ being
the deflection angle, a0 = ZPZT e
2/m0v
2 and a = a0/γ. The complex functions S(πλ, µ) are
integrals along Coulomb trajectories corrected for retardation. Their calculation and how
they relate to the non-relativistic and relativistic theories are described in details in ref. [2].
Here we will introduce another comparison tool for the total cross section, which is obtained
by integration of eq. 3 over scattering angles. The code COULINT [2] was used to calculate
the orbital integrals S(πλ, µ) and the cross sections of eq. 3 (for more details, see ref. [2]).
Using the theory described in ref. [4], it is easy to show that approximate values of the
cross sections for E1, E2, and M1 transitions can be obtained by means of the relations
(app)
B (E1)
ξK0K1 −
K21 −K
(app)
E3xB (E2)
K21 + ξ
K0K1 −
K21 −K
(app)
B (M1)
ξK0K1 −
K21 −K
, (5)
where Kn are the modified Bessel functions of the second order, as a function of ξ given by
eq. 2, with R corrected for recoil by the modification R → R + πa/2 [3].
Here we will only consider the excitation of the lowest lying states in light and medium
heavy nuclei. For nuclear masses A < 20, the TUNL nuclear data evaluation web site
was of great help [5]. The electromagnetic transition rates at the TUNL database are
given in Weisskopf units and are transformed to the appropriate B(πλ, Ii → If)-values
by means of the standard Weisskopf relations BW (E1; Ji → Jgs) = 0.06446A
2/3 e2fm2,
BW (E2; Ji → Jgs) = 0.05940A
4/3 e2fm4, and BW (M1; Ji → Jgs) = 1.79 (e~/2mnc)
. For
comparison, a few medium mass nuclei, as well as a few stable nuclei, were included in the
calculation. Other data were taken from refs. [6, 7, 8, 9].
Some cases of nuclei far from the stability line are very interesting and deserve further
study, possibly using the method of Coulomb excitation. For example, it is well known that
nuclei with open shells tend to have B(E2) values greater than 10 W.u., whereas nuclei with
shell closure of neutrons or protons tend to have distinctly smaller B(E2) values. Typical
examples of the latter category are the doubly magic nuclei, 16O and 48Ca, which B(E2)
values are 3.17 and 1.58 W.u., respectively. According to an empirical formula adjusted
to a global fit of the known transition rates, the values of first excited 2+ level, E2+ , and
B(E2; 0+ → 2+) are related by [10] (E2+ in keV)
B(E2; 0+ → 2+) = 26
A2/3E2+
e2fm4. (6)
The value of B(E2) for 16C based on this formula is at least one order of magnitude
larger than what is observed experimentally in a Coulomb dissociation experiment [9]. The
anomalously strong hindrance of the 16C transition is not well explained theoretically. This
is just an example of the power of Coulomb excitation as a tool to access the new physics
inherent of poorly known rare nuclear species.
Another example is the strong E1 transition in 11Be. 11Be is an archetype of a halo nu-
cleus and exhibits the fastest known dipole transition between bound states in nuclei. The
B(E1) transition strength between the ground and the only bound excited state (at 0.32
MeV) was determined from lifetime measurements by Millener et al. to be 0.116 e2fm2 [11].
However, Coulomb excitation experiments have obtained a much smaller value of the B(E1)
which is still a matter of investigation [12, 13, 14]. It is thus seems clear that predictions
based on traditional nuclear structure and reaction theory often yields results in disagree-
ment with experimental data. In spite of that, when proper corrections are accounted for
(e.g. channel-coupling, nuclear excitation, relativistic corrections), Coulomb excitation of ra-
dioactive beams is a powerful complementary tool to investigate electromagnetic properties
of nuclei far from the stability line.
In Table 1 we compare our calculations with several experimentally obtained cross sections
for Coulomb excitation of unstable nuclei. The units of energy are MeV, the laboratory
energy is in MeV/nucleon, the B-values are in units of e2 fm2λ, and the cross sections are
in millibarns. The last two columns give the calculated cross sections obtained by using
eqs. 3 and 5, respectively. Since the cross sections of eq. 5 are functions of the minimum
impact parameter, the values reported in the Table have been calculated according to the
experimental angular ranges reported in the seventh column. Except for the 11Be case, for
which the discrepancy between theory and experiment is known (see discussion above), the
calculated cross sections are close to the experimental values. Nonetheless, the calculated
cross sections tend to be smaller than the experimental ones for 17Ne, 32Mg, 38S, 40S, 42S,
44Ar, and 46Ar projectiles. This is worrisome because the B(πλ) values were extracted
from the experimentally obtained cross sections, using equations similar to eq. 5. These
experimental B-values would have to be larger by 10− 30% according to our calculations.
It is important to stress the fact that many experimental data on unstable nuclei collected
up to now have been analyzed by means of theoretical tools (DWBA and coupled-channels
codes) which do not include relativistic dynamics (the inclusion of relativistic kinematics
is straightforward). This problem was first addressed in ref. [15], where it was shown
that the analysis of experimental data at intermediate energies without a proper treatment
of relativistic dynamics leads to wrong values of electromagnetic transition probabilities.
We should stress that a full theoretical treatment of relativistic dynamics of strong and
electromagnetic interactions in many-body systems is very difficult and still does not exist
[15].
Data Projectile Target Elab πλ B(πλ) θrange Ex σexp σth σapp
1 [16] 11Be Pb 43. E1 0.115 < 5◦ 0.32 (1
) 191± 26 328. 323.
2 [16] 11Be Pb 59.4 E1 0.094 < 3.8◦ 0.32 (1
) 304± 43 213. 211.
3 [18] 11Be Au 57.6 E1 0.079 < 3.8◦ 0.32 (1
) 244± 31 170. 168.
4 [17] 11Be Pb 64. E1 0.099 < 3.8◦ 0.32 (1
) 302± 31 217. 215.
5 [19, 20] 17Ne Au 60. M1 0.163 < 4.5◦ 1.29 (1
) 12± 4 12.6 13.0
6 [6] 32Mg Pb 49.2 E2 454 < 4◦ 0.885 (0+ → 2+) 91.7± 14.4 137. 128.
7 [19] 38S Au 39.2 E2 235 < 4.1◦ 1.29 (0+ → 2+) 59± 7 48. 45.0
8 [19] 40S Au 39.5 E2 334 < 4.1◦ 0.91 (0+ → 2+) 94± 9 75.5 70.4
9 [19] 42S Au 40.6 E2 397 < 4.1◦ 0.89 (0+ → 2+) 128± 19 101. 94.3
10 [19] 44Ar Au 33.5 E2 345 < 4.1◦ 1.14 (0+ → 2+) 81± 9 62.3 58.3
11 [19] 46Ar Au 35.2 E2 196 < 4.1◦ 1.55 (0+ → 2+) 53± 10 40.9 38.2
12 [8] 46Ar Au 76.4 E2 212 < 2.9◦ 1.55 (0+ → 2+) 68± 8 50.0 47.4
Table 1. Cross sections for Coulomb excitation of unstable nuclei. The units of energy
are MeV, the laboratory energy is in MeV/nucleon, the B(πλ)-values are in units of e2fm2λ,
and the cross sections are in millibarns. The data for different experiments (numbered 1 to
12) were collected from the references listed in column 1. The last two columns give the
calculated cross sections obtained by using eqs. 3 and 5, respectively.
In figure 1 we show a comparison between the experimental data and our calculations.
We notice that the cross sections calculated with help of eq. 5 are not much different than
those calculated with eq. 3. They are systematically lower, up to 10%, than the exact
calculation following eq. 3. As we discuss below, this is not always the case, specially for
the excitation of high-lying states. In fact, this is a good check of eq. 3, which is done in a
very different way than the analytical calculations of eq. 5. But as we will see below, this
agreement is not always the case, specially when one includes small impact parameters for
which the sensitivity to the relativistic corrections is higher (see ref. [2]). The dashed curve
in figure 1 is a guide to the eye. It helps to see that the experimental cross sections are on
average larger than the calculated ones, either with eq. 3 (open circles), or with eq. 5 (open
triangles).
2 4 6 8 10 12
Data set
FIG. 1: Comparison between experimental Coulomb excitation cross sections (solid stars with
error bars) and theoretical ones, calculated either with eq. 3 (open circles), or with eq. 5 (open
triangles).
Ex [MeV] J
i → J
f πλ B(πλ) [e
2 fm2λ] 10 20 30 50 100 200 500
11Be 0.32 1
E1 0.115 1128 653 473 315 187 115 69.6
11B 2.21 3
M1 2.40×10−2 0.301 0.799 1.15 1.63 2.33 3.08 4.17
11C 2.00 3
M1 1.52×10−2 0.196 0.551 0.793 1.12 1.57 2.07 2.76
12B 0.953 1+ → 2+ M1 4.62×10−3 0.227 0.395 0.490 0.607 0.762 0.917 1.13
12C 4.44 0+ → 2+ E2 37.9 34.6 38.6 31.3 21.6 12.1 6.93 3.81
13C 3.09 1
E1 1.39×10−2 8.37 11.3 11.0 9.61 7.28 5.39 3.89
13N 2.37 1
E1 3.56×10−2 38.2 43.6 39.6 32.5 23.2 16.4 11.4
15C 0.74 1
E2 2.90 8.79 4.04 2.65 1.59 0.839 0.475 0.267
16C 1.77 0+ → 2+ E2 2.12 8.81 4.41 2.92 1.76 0.920 0.517 0.285
16N 0.12 0− → 2− E2 10.2 31.0 14.1 9.21 5.53 2.91 1.64 0.926
17N 1.37 1
M1 5.15×10−3 0.153 0.304 0.397 0.516 0.680 0.848 1.09
17O 0.87 5
E2 2.07 6.30 2.88 1.87 1.12 0.588 0.332 0.184
17F 0.5 5
E2 21.6 68.3 29.7 19.3 11.6 6.08 3.44 1.92
18O 1.98 0+ → 2+ E2 44.8 109 60.7 40.9 24.8 11.6 7.27 3.99
18F 0.94 1+ → 3+ E2 37.9 115 52.5 34.1 20.4 10.7 6.01 3.34
18Ne 1.89 0+ → 2+ E2 248 615 342 229 138 72.0 40.1 22.1
19O 0.1 5
M1 2.34×10−4 0.0495 0.0615 0.0673 0.0737 0.0799 0.779 0.799
19F 0.11 1
E1 5.51×10−4 8.07 4.36 3.06 1.97 1.10 0.592 0.337
19Ne 0.24 1
E2 119 361 157 102 61.6 32.5 18.5 10.5
20O 1.67 0+ → 2+ E2 28.0 72 37.4 24.9 15.1 7.86 4.41 2.43
20F 0.656 2+ → 3+ M1 3.56×10−3 0.237 0.385 0.465 0.560 0.683 0.803 0.959
20Ne 1.63 0+ → 2+ E2 319 834 433 287 173 89.8 50.3 27.6
30Ne 0.791 0+ → 2+ E2 460 1167 550 361 218 115 65.0 35.2
32Mg 0.885 0+ → 2+ E2 454 1151 541 355 214 112 63.0 36.7
42S 0.89 0+ → 2+ E2 397 945 445 292 175 91.9 52 29.7
46Ar 1.55 0+ → 2+ E2 190 399 209 140 84.4 44.1 24.7 13.6
54Ni 1.40 0+ → 2+ E2 626 1319 677 447 268 139 78.1 43.1
Table 2 - Cross sections (in mb) for Coulomb excitation of projectiles incident on Pb
targets at bombarding energies ranging from 10 to 500 MeV/nucleon. The energy units are
MeV, the laboratory energy is in MeV/nucleon, the B(πλ)-values are in units of e2fm2λ.
The cross sections for Coulomb excitation of numerous projectiles incident on Pb targets
at bombarding energies ranging from 10 to 500 MeV/nucleon are shown in Table 2. These
cross sections were calculated assuming that the detectors collect events from all possible
Coulomb scattering events. In a real experimental situation, the angular distribution is
restricted to angular windows, reducing the available cross sections. Only the lowest lying
transitions have been considered, i.e. from the ground to the first excited states. One
observes that some cross sections are very large, specially for 11Be, 18Ne, 30Ne and 54Ni.
For these and other similar cases, the measurements are easy to perform, with a large
number of events/second even with modest intensities. Cases such as 16C are well within
the experimental possibilities in most radioactive beam facilities.
Table 2 also shows that, except for M1 excitations, the Coulomb excitation cross sections
decrease steadily as the energy increases from 10 to 500 MeV/nucleon. Based on these
numbers alone, one could conclude that Coulomb excitation of low-lying states (in contrast
to the case of high-lying states, e.g. giant resonances [4]) are better suited for studies at low
energies. However, reactions at lower energies while are less influenced by contamination due
to nuclear breakup [12, 14] can give rise to large high-order effects [21]. The interpretation
of data could be distorted as in the case of Coulomb dissociation of 8B at low energy [24],
which was completely misinterpreted in terms of first-order calculations. In some situations,
when higher-order effects are relevant, the effect of the nuclear breakup cannot be neglected
either [22, 23]. Thus, the choice of the incident energy would depend on the experimental
conditions. Identification of gamma-rays from de-excitation using Doppler shift techniques
are often more advantageous at higher energies. Moreover, except for few cases (e.g. 11C), the
magnetic dipole transitions are much smaller than those for E1 and E2 transitions. Even
for M1 transitions the measurements are under the possibility of most new experimental
facilities.
The comparison of the exact calculations, using eq. 3 (solid lines), and the approximations
5 (dashed lines) are shown in figs. 2(a-d), for 11Be, 11B, 54Ni and 16O, respectively. The 16O
case (as well as for 12C in Table 2) was included for comparison, with a high-lying excited
state. We see from figs 2a and 2b that the approximations in eq. 5 work quite well for
the M1 multipolarity and reasonably well (within 20% at 10 MeV/nucleon and 5% at 50
MeV/nucleon) for the E1 cases. But they fail badly at low and intermediate energies for
(c) (d)
FIG. 2: Coulomb excitation cross section of the first excited state in 11Be, 11B and 54Ni and of the
13.05 MeV sate in 16O projectiles incident on Pb targets as a function of the laboratory energy.
the E2 ( fig. 2c). The reason is that the E2 Coulomb field (“tidal field”) is very sensitive
to the details of the collision dynamics at low energies. These conclusions can be deceiving
since even for the E1 and M1 cases the approximations in eq. 5 may strongly differ from
the exact calculations if the excitation energy is large (see discussion in ref. [2]). This is
shown in figure 2d, where we plot the Coulomb excitation cross section of the Ex = 13.09
MeV state in 16O. In this case, the cross sections based on eq. 5 is a factor of 10 smaller
than the exact calculation at 10 MeV/nucleon. At 100 MeV/nucleon this difference drops
to 10%, which still needs to be considered with care.
In summary, in this article we have used the formalism of ref. [2] to predict the cross
sections for Coulomb excitation of several light projectiles with electromagnetic transitions
found in the literature, listed in the TUNL database [5], and for a few other selected cases.
These estimates will be useful for planing Coulomb excitation experiments at present and
future heavy ion facilities. It is evident that the inclusion of relativistic effects combined
with Coulomb distortion are of the utmost relevance. The cross section inferred by using
non-relativistic or pure relativistic treatments can be wrong by up to 30% even at 100
MeV/nucleon, as shown here and in ref. [2]. Finally, the use of Coulomb excitation to
produce nuclei in high-lying states is an important tool to study particle emission processes.
For example, the excitation of 18Ne and its subsequent decay by two-proton emission is a
process of large theoretical and experimental interest. Experimental work in this direction
is in progress [25].
Acknowledgments
This research was supported by the U.S. Department of Energy under contract No. DE-
AC05-00OR22725 (Oak Ridge National Laboratory) with UT-Battelle, LLC., and by DE-
FC02-07ER41457 with the University of Washington (UNEDF, SciDAC-2).
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