Nuclear Astrophysics in Rare Isotope Facilities
ABSTRACT I discuss a few of the recent developments in nuclear reactions at very low energies with emphasis on the role of radioactive beam facilities. Comment: Proceedings 19th Winter Workshop on Nuclear Dynamics, Breckenridge, Colorado, USA, February 8-15, 2003
Proc. 19th Winter Workshop on
Nuclear Dynamics (2003) 000–000
19th Winter Workshop
on Nuclear Dynamics
Breckenridge, Colorado, USA
February 8–15, 2003
Nuclear Astrophysics in Rare Isotope Facilities
NSCL and Department of Physics and Astronomy, Michigan State University,
East Lansing, MI 48824
Abstract. I discuss a few of the recent developments in nuclear reactions at
very low energies with emphasis on the role of radioactive beam facilities.
Keywords: Nuclear Astrophysics, Radioactive Beams.
Present studies in nuclear astrophysics are focused on the opposite ends of the energy
scale for nuclear reactions: (a) the very high and (b) the very low relative energies
between the reacting nuclei. Projectiles with high bombarding energies produce
nuclear matter at high densities and temperatures. This is the main goal at the
RHIC accelerator at the Brookhaven National Laboratory and also the main subject
discussed in this Workshop. One expects that matter produced in central nuclear
collisions at RHIC for ∼ 104GeV/nucleon of relative energy, and at the planned
Large Hadron Collider at CERN, will undergo a phase transition and produce a
quark-gluon plasma. One can thus reproduce conditions existent in the first seconds
of the universe and also in the core of neutron stars. At the other end of the energy
scale are the low energy reactions of importance for stellar evolution. A chain of
nuclear reactions starting at ∼ 10 − 100 keV leads to complicated phenomena like
supernovae explosions or the energy production in the stars.
Nuclear astrophysics requires the knowledge of the reaction rate Rij between
the nuclei i and j. It is given by Rij= ninj< σv > /(1+δij), where σ is the cross
section, v is the relative velocity between the reaction partners, niis the number
density of the nuclide i, and <> stands for energy average.
In our Sun the reaction7Be(p,γ)8B plays a major role for the production of
high energy neutrinos originated from the β-decay of8B. These neutrinos come
directly from center of the Sun and are an ideal probe of the Sun’s structure. Long
ago, Barker  has emphasized that an analysis of the existing experimental data
yields an S-factor for this reaction at low energies which is uncertain by as much as
30%. This situation has changed recently, mainly due to the use of radioactive beam
facilities. The reaction12C (α,γ)16O is extremely relevant for the fate of massive
stars. It determines if the remnant of a supernova explosion becomes a black-hole
or a neutron star. It is argued that the cross section for this reaction should be
known to better than 20%, for a good modelling of the stars . This goal has not
yet been achieved.
Both the7Be(p,γ)8B and the12C (α,γ)16O reactions cannot be measured at
the energies occurring inside the stars (approximately 20 keV and 300 keV, respec-
tively). Direct experimental measurements at low energies are often plagued with
low-statistics and large error bars. Extrapolation procedures are often needed to
obtain cross sections in the energy region of astrophysical relevance. While non-
resonant cross sections can be rather well extrapolated to the low-energy region,
the presence of continuum, or subthreshold resonances, complicates these extrapo-
lations. Numerous radiative capture reactions pose the same experimental problem.
Approximately half of all stable nuclei observed in nature in the heavy element
region about A > 60 is produced in the r–process. This r–process occurs in environ-
ments with large neutron densities which lead to τn? τβ. The most neutron–rich
isotopes along the r–process path have lifetimes of less than one second; typically
10−2to 10−1s. Cross sections for most of the nuclei involved are hard to measure
experimentally. Sometimes, theoretical calculations of the capture cross sections as
well as the beta–decay half–lives are the only source of the nuclear physics input for
r–process calculations . For nuclei with about Z > 80 beta–delayed fission and
neutron–induced fission might also become important.
2. The Electron Screening Problem
Besides the Coulomb barrier, nucleosynthesis in stars is complicated by the pres-
ence of electrons. They screen the nuclear charges, therefore increasing the fusion
probability by reducing the Coulomb repulsion. Evidently, the fusion cross sections
measured in the laboratory have to be corrected by the electron screening when
used as inputs of a stellar model. This is a purely theoretical problem as one can
not reproduce the interior of stars in the laboratory. Applying the Debye-H¨ uckel,
or Salpeter’s, approach , one finds that the plasma enhances reaction rates, e.g.,
3He(3He, 2p)4He and7Be(p, γ)8B, by as much as 20%. This does not account
for the dynamic effect due to the motion of the electrons (see, e.g., [5,6]).
A simpler screening mechanism occurs in laboratory experiments due to the
bound atomic electrons in the nuclear targets. This case has been studied in great
details experimentally, as one can control different charge states of the projec-
tile+target system in the laboratory [7–11]. The experimental findings disagree
systematically by a factor of two with theory. This is surprising as the theory for
atomic screening in the laboratory relies on our basic knowledge of atomic physics.
At very low energies one can use the simple adiabatic model in which the atomic
electrons rapidly adjust their orbits to the relative motion between the nuclei prior
Nuclear Astrophysics in Rare Isotope Facilities 3
to the fusion process. Energy conservation requires that the larger electronic bind-
ing (due to a larger charge of the combined system) leads to an increase of the
relative motion between the nuclei, thus increasing the fusion cross section. As a
matter of fact, this enhancement has been observed experimentally. The measured
values are however not compatible with the adiabatic estimate [7–11]. Dynamical
calculations have been performed, but they obviously cannot explain the discrep-
ancy as they include atomic excitations and ionizations which reduce the energy
available for fusion. Other small effects, like vacuum polarization, atomic and nu-
clear polarizabilities, relativistic effects, etc., have also been considered . But
the discrepancy between experiment and theory remains [12,11].
0 0.51 1.52
Relative Energy [MeV]
Fig. 1: The stopping cross section of protons on H-targets. The dotted line gives the
energy transfer by means of nuclear stopping, while the solid line is the result for the
charge-exchange stopping mechanism . The data points are from the tabulation of
Andersen and Ziegler .
Fig. 2: Energy dependence of the Coulomb breakup cross section for
p +7Be + Pb at 84 MeV/nucleon. First-order perturbation calculations (PT) are shown
by the solid curve. The dashed curve is the result of a CDCC calculation including the
coupling between the ground state and the low-lying states with the giant dipole and
quadrupole resonances . The data points are from ref. .
8B + Pb −→
A possible solution of the laboratory screening problem was proposed in refs.
[14,15]. Experimentalists often use the extrapolation of the Andersen-Ziegler tables
 to obtain the average value of the projectile energy due to stopping in the target
material. The stopping is due to ionization, electron-exchange, and other atomic
mechanisms. However, the extrapolation is challenged by theoretical calculations
which predict a lower stopping. Smaller stopping was indeed verified experimentally
. At very low energies, it is thought that the stopping mechanism is mainly due
to electron exchange between projectile and target. This has been studied in ref.
 in the simplest situation; proton+hydrogen collisions. Two-center electronic
orbitals were used as input of a coupled-channels calculation. The final occupation
amplitudes were projected onto bound-states in the target and in the projectile.
The calculated stopping power was added to the nuclear stopping power mechanism,
4 C.A. Bertulani
i.e. to the energy loss by the Coulomb repulsion between the nuclei. The obtained
stopping power is proportional to vα, where v is the projectile velocity and α = 1.35.
The extrapolations from the Andersen-Ziegler table predict a larger value of α.
Although this result seems to indicate the stopping mechanism as a possible reason
for the laboratory screening problem, the theoretical calculations tend to disagree
on the power of v at low energy collisions. For example, ref.  found S ∼ v3.34
for protons in the energy range of 4 keV incident on helium targets. This is an even
larger deviation from the extrapolations of the Andersen-Ziegler tables.
We are faced here with a notorious case of obscurity in nuclear astrophysics.
The disturbing conclusion is that as long as we cannot understand the magnitude of
electron screening in stars or in the atomic electrons in the laboratory, it will be even
more difficult to understand color screening in a quark-gluon plasma, an important
tool in relativistic heavy ion physics (e.g., the J/Ψ suppression mechanism).
3. Radioactive Beam Facilities and Indirect Methods
Transfer reactions are a well established tool to obtain spin, parities, energy, and
spectroscopic factors of states in a nuclear system. Experimentally, (d, p) reactions
are mostly used due to the simplicity of the deuteron. Variations of this method
have been proposed by several authors. For example, the Trojan Horse Method
was proposed in ref.  as a way to overcome the Coulomb barrier. If the Fermi
momentum of the particle x inside a = (b+x) compensates for the initial projectile
velocity va, the low energy reaction A + x = B + c is induced at very low (even
vanishing) relative energy between A and x. Successful applications of this method
has been reported recently . Also recently the knockout reactions have been
demonstrated to be a useful tool to deduce spectroscopic factors in many reactions
of relevance for nuclear astrophysics .
At low energies the amplitude for the radiative capture cross section is dom-
inated by contributions from large relative distances of the participating nuclei.
Thus, what matters for the calculation of the direct capture matrix elements are
the asymptotic normalization coefficients (ANC). This coefficient is the product of
the spectroscopic factor and a normalization constant which depends on the details
of the wave function in the interior part of the potential. The normalization coeffi-
cients can be found from peripheral transfer reactions whose amplitudes contain the
same overlap function as the amplitude of the corresponding astrophysical radia-
tive capture cross section. This idea was proposed in ref.  and many successful
applications of the method have been obtained .
Charge exchange induced in (p,n) reactions are often used to obtain values
of Gamow-Teller matrix elements which cannot be extracted from beta-decay ex-
periments. This approach relies on the similarity in spin-isospin space of charge-
exchange reactions and β-decay operators. As a result of this similarity, the cross
section σ(p, n) at small momentum transfer q is closely proportional to B(GT)
for strong transitions . As shown in ref. , for important GT transitions
Nuclear Astrophysics in Rare Isotope Facilities 5
whose strength are a small fraction of the sum rule the direct relationship between
σ(p, n) and B(GT) values fails to exist. Similar discrepancies have been observed
 for reactions on some odd-A nuclei including13C,15N,35Cl, and39K and for
charge-exchange induced by heavy ions [27,28].
The (differential, or angle integrated) Coulomb breakup cross section for a +
A −→ b + x + A can be written as σπλ
energy transferred from the relative motion to the breakup, and σπλ
photo nuclear cross section for the multipolarity πλ and photon energy ω. The
function Fπλdepends on ω, the relative motion energy, and nuclear charges and
radii. They can be easily calculated  for each multipolarity πλ. Time reversal
allows one to deduce the radiative capture cross section b + x −→ a + γ from
number of reactions of interest for astrophysics ( and references therein). The
most celebrated case is the reaction7Be(p,γ)8B. It has been studied in numerous
experiments in the last decade. For a recent compilation of the results obtained with
the method, see the contribution of Moshe Gai to this workshop, and also ref. .
They have obtained an S17(0) value of 19.0 eV.b which is compatible with the value
commonly used in solar model calculations . To achieve the goal of applying
this method to many other radiative capture reactions (for a list, see, e.g. ),
detailed studies of dynamic contributions to the breakup have to be performed, as
shown in refs. [34,35]. The role of higher multipolarities (e.g., E2 contributions
[36–38] in the reaction7Be(p,γ)8B) and the coupling to high-lying states  has
also to be investigated carefully. In the later case, a recent work has shown that
the influence of giant resonance states is small (see figure 2). Studies of the role of
the nuclear interaction in the breakup process is also essential to determine if the
Coulomb dissociation method is useful for a given system .
In summary, radioactive beam facilities have opened a new paved way to dis-
closure many unknown features of reactions in stars and elsewhere in the universe.
C(ω) = Fπλ(ω) . σπλ
γ(ω), where ω is the
γ(ω) is the
γ(ω). This method was proposed in ref. . It has been tested successfully in a
I acknowledge discussions with B. Davids, M. Gai, H. Schatz, K. Suemmerer
and S. Typel. Work Supported by U.S. National Science Foundation under Grants
No. PHY-007091 and PHY-00-70818.
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