arXiv:0709.0567v1 [nucl-ex] 5 Sep 2007
Isotopic dependence of the giant monopole resonance in the
even-A112−124Sn isotopes and the asymmetry term in nuclear
T. Li1, U. Garg1, Y. Liu1, R. Marks1, B.K. Nayak1, P.V. Madhusudhana Rao1,
M. Fujiwara2, H. Hashimoto2, K. Kawase2, K. Nakanishi2, S. Okumura2,
M. Yosoi2, M. Itoh3, M. Ichikawa3, R. Matsuo3, T. Terazono3, M.
Uchida4, T. Kawabata5, H. Akimune6, Y. Iwao7, T. Murakami7,
H. Sakaguchi7, S. Terashima7, Y. Yasuda7, J. Zenihiro7, M.N. Harakeh8
1Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
2Research Center for Nuclear Physics, Osaka 567-0047, Japan
3Cyclotron and Radioisotope Center,
Tohuku University, Sendai 980-8578, Japan
4Department of Physics, Tokyo Institute of Technology, Tokyo 152-8850, Japan
5Center for Nuclear Study, University of Tokyo, Tokyo 113-0033, Japan
6Department of Physics, Konan University, Kobe 658-8501, Japan
7Department of Physics, Kyoto University, Kyoto 606-8502, Japan
8Kernfysisch Versneller Instituut, University of Groningen,
9747 AA Groningen, The Netherlands
(Dated: February 1, 2008)
The strength distributions of the giant monopole resonance (GMR) have been measured in the
even-A Sn isotopes (A=112–124) with inelastic scattering of 400-MeV α particles in the angular
range 0◦–8.5◦. We find that the experimentally-observed GMR energies of the Sn isotopes are
lower than the values predicted by theoretical calculations that reproduce the GMR energies in
208Pb and90Zr very well. From the GMR data, a value of Kτ= −550 ± 100 MeV is obtained for
the asymmetry-term in the nuclear incompressibility.
PACS numbers: 24.30.Cz; 21.65.+f; 25.55.Ci; 27.40.+z
Incompressibility of nuclear matter remains a focus of experimental and theoretical in-
vestigations because of its fundamental importance in defining the equation of state (EOS)
for nuclear matter. The latter describes a number of interesting phenomena from collec-
tive excitations of nuclei to supernova explosions and radii of neutron stars . The Giant
Monopole Resonance (GMR) provides a direct means to experimentally determine the nu-
Experimental identification of the GMR requires inelastic scattering of an isoscalar
particle–the α particle, for example–at extremely forward angles, including 0◦, where the
cross section for exciting the GMR is maximal. Such measurements have improved con-
siderably over the years and it is now possible to obtain inelastic spectra virtually free of
all instrumental background directly  and in coincidence with proton- and neutron-decay
. In recent work, the GMR strength distributions have been extracted in a number of
nuclei from a multipole-decomposition analysis (MDA) of such “background-free” spectra
[2, 4, 5, 6, 7, 8, 9].
The excitation energy of the GMR is expressed in the scaling model  as:
EGMR= ¯ h
m < r2>
where m is the nucleon mass, < r2> is the ground-state mean-square radius, and KA,
the incompressibility of the nucleus. In order to determine the incompressibility of infinite
nuclear matter, K∞, from the experimental GMR energies, one builds a class of energy
functionals, E(ρ), with different parameters which allow calculations for nuclear matter and
finite nuclei in the same theoretical framework. The parameter-set for a given class of energy
functionals is characterized by a specific value of K∞. The GMR strength distributions are
obtained for different energy functionals in a self-consistent RPA calculation. The K∞
associated with the interaction that best reproduces the GMR energies is, then, considered
the “correct” value. This procedure, first proposed by Blaizot , is now accepted as
the best way to extract K∞ from the GMR data and, following this procedure, it has
been established that both relativistic and non-relativistic calculations are now in general
agreement with K∞= 240 ± 10 MeV [12, 13, 14].
The determination of the asymmetry term, Kτ, associated with the neutron excess (N-
Z), remains very important because this term is crucial in obtaining the radii of neutron
stars in EOS calculations [15, 16, 17, 18]. Indeed, the radius of a neutron star whose
mass is between about 1 and 1.5 solar masses (M⊙) is mostly determined by the density
dependence of the symmetry-energy term [19, 20]. Previous attempts to extract this term
from experimental GMR data have resulted in widely different values, from -320±180 MeV
in Ref.  to a range of -566±1350 MeV to 139±1617 MeV in Ref. . Measurements
of the nuclear incompressibility over a series of isotopes provide a way to “experimentally”
determine this asymmetry term in a direct manner. The Sn isotopes (A=112–124) afford
such an opportunity since the asymmetry ratio, ((N-Z)/A), changes by more than 80% over
this mass range.
In this Letter, we report on new measurements on GMR in the even-A Sn isotopes.
The GMR has been identified previously in some of the Sn isotopes as a compact peak in
measurements with inelastic α-scattering [7, 21, 23, 24] and although resonance parameters
for GMR in the Sn isotopes close to the values reported here have been extracted in the past
using less accurate techniques , the potentially large systematic errors in those values
necessitated the present measurements where such problems have been eliminated. We find
that the GMR energies in the Sn isotopes are lower than the values predicted in recent
theoretical calculations even though the interactions used in these calculations reproduce
the GMR energies in the “standard” nuclei,208Pb and90Zr, very well. Also, we obtain a
value Kτ= −550 ± 100 MeV from this data.
The experiment was performed at the ring cyclotron facility of the Research Center for
Nuclear Physics (RCNP), Osaka University, using inelastic scattering of 400-MeV α par-
ticles over the angular range 0◦–8.5◦. Details of the experimental technique and the data
analysis procedure have been provided previously [5, 6, 8] and are only briefly described
here. Inelastically-scattered α particles were momentum-analyzed with the high-resolution
magnetic spectrometer “Grand Raiden”  and detected in the focal-plane detector system
comprised of two multi-wire drift chambers and two scintillators, providing particle identi-
fication as well as the trajectories of the scattered particles. The vertical position spectrum
obtained in the double-focused mode of the spectrometer was exploited to eliminate all
instrumental background [5, 6, 8]. The background-free “0◦” inelastic spectra for the Sn
isotopes are presented in Fig. 1. In all cases, the spectrum is dominated by the GMR peak
near Ex∼ 15 MeV.
In order to extract the GMR strengths, we have employed the now standard MDA proce-
dure . The cross-section data were binned into 1-MeV energy intervals between 8.5–31.5
FIG. 1: Excitation-energy spectra for all even-A Sn isotopes, obtained from inelastic α scattering
at θlab= 0.69◦.
MeV and for each excitation energy bin, the experimental 17-point angular distribution
dΩ(θcm,Ex) was fitted by means of the least-square method with the linear combination
of calculated distributions
dΩ(θcm,Ex), so that:
dΩ(θcm,Ex) is the calculated distorted-wave Born approximation (DWBA) cross
section corresponding to 100% energy-weighted sum-sure (EWSR) for the L-th multipole.
This procedure provides strength distributions simultaneously for various multipoles.
The DWBA calculations were performed following the method of Satchler and Khoa
 using density-dependent single folding model, with a Gaussian α-nucleon potential for
the real part, and a Woods-Saxon imaginary term. We used the transition densities and
sum rules for various multipolarities as described in Ref. . The optical model (OM)
parameters were obtained from analysis of elastic scattering cross sections measured in a
Although all strength distributions up to L=3 have been reliably extracted from the
multipole decomposition, only the GMR strengths, the focus of this paper, are shown in
FIG. 2: GMR strength distributions obtained for the Sn isotopes in the present experiment. Error
bars represent the uncertainty due to the fitting of the angular distributions in MDA. The solid
lines show Lorentzian fits to the data.
Fig. 2. The solid lines in the figure represent Lorentzian fits to the observed strength
distributions. The choice of the Lorentzian shape is arbitrary; the final results are not
affected in any significant way by using, instead, a Gaussian shape, for example. The finite
strength at the higher excitation energies is attributable to the mimicking of L=0 angular
distribution by components of the continuum [4, 8]. The extracted GMR-peak parameters
and the various moment ratios typically used in theoretical calculations are presented in
The moment ratios, m1/m0, for the GMR strengths in the Sn isotopes are shown in
Fig. 3, and compared with recent theoretical results from Col` o (non-relativistic) [12, 29]
and Piekarewicz (relativistic) [13, 30]. As can be seen, the calculations overestimate the ex-
perimental GMR energies significantly (by almost 1 MeV in case of the higher-A isotopes).
This is very surprising since the interactions used in these calculations are those that very
closely reproduce the GMR centroid energies in208Pb and90Zr. Admittedly, there are uncer-
tainties associated with the range over which the experimental and theoretical distributions
are compared, and also with the assumptions inherent in the calculations regarding widths.
However, the calculations reported here are identical in all respects to those performed
TABLE I: Lorentzian-fit parameters and various moment-ratios for the GMR strength distributions
in the Sn isotopes, as extracted from MDA in the present work.mk is the k-th moment of
the strength distribution: mk =
xS(Ex)dEx. All moment ratios have been calculated over
Ex= 10.5–20.5 MeV. The errors quoted for EWSR are statistical only.
Target EGMR(MeV) Γ (MeV)EWSRm1/m0(MeV)
16.7 ± 0.2
16.1 ± 0.1
112Sn 16.1 ± 0.14.0 ± 0.4 0.92 ± 0.0416.2 ± 0.1
114Sn15.9 ± 0.14.1 ± 0.4 1.04 ± 0.06 16.1 ± 0.116.5 ± 0.215.9 ± 0.1
116Sn15.8 ± 0.14.1 ± 0.3 0.99 ± 0.0515.8 ± 0.1 16.3 ± 0.215.7 ± 0.1
118Sn15.6 ± 0.14.3 ± 0.4 0.95 ± 0.0515.8 ± 0.116.3 ± 0.1 15.6 ± 0.1
120Sn 15.4 ± 0.24.9 ± 0.5 1.08 ± 0.0715.7 ± 0.116.2 ± 0.2 15.5 ± 0.1
122Sn 15.0 ± 0.24.4 ± 0.4 1.06 ± 0.0515.4 ± 0.115.9 ± 0.2 15.2 ± 0.1
124Sn 14.8 ± 0.24.5 ± 0.5 1.03 ± 0.0615.3 ± 0.115.8 ± 0.1 15.1 ± 0.1
for208Pb and90Zr, and the experimental and theoretical centroids reported here have been
calculated over exactly the same excitation-energy range. This disagreement remains a chal-
lenge for the theory: Why are the tin isotopes so “soft”? Are there any nuclear structure
effects that need to be taken into account to describe the GMR energies in the Sn isotopes?
The incompressibility of a nucleus, KA, may be expressed as:
KA∼ Kvol(1 + cA−1/3) + Kτ((N − Z)/A)2+ KCoulZ2A−4/3
Here, c ≈ −1 , and KCoulis essentially model-independent (in the sense that the de-
viations from one theoretical model to another are quite small), so that the associated
term can be calculated for a given isotope. Thus, for a series of isotopes, the difference
KA − KCoulZ2A−4/3may be approximated to have a quadratic relationship with the asym-
metry parameter, of the type y = A+Bx2, with Kτbeing the coefficient, B, of the quadratic
term. It should be noted that it has been established previously [22, 32] that fits to the
above equation do not provide good constraints on the value of K∞. However, this expres-
sion is being used here not to obtain a value for K∞, but, rather, only to demonstrate the
approximately quadratic relationship between KAand the asymmetry parameter.
Fig. 4 shows the difference KA− KCoulZ2A−4/3for the Sn isotopes investigated in this
work vs. the asymmetry parameter, ((N −Z)/A). The values of KAhave been derived using
FIG. 3: Systematics of the moment ratios m1/m0for the GMR strength distributions in the Sn
isotopes. The experimental results (filled squares) are compared with results of non-relativistic RPA
calculations by Col` o  (filled circles) and relativistic calculations of Piekarewicz  (triangles).
Results for112Sn,116Sn and124Sn reported by the Texas A & M group [23, 24] are also shown
(inverse triangles). The differences between the present results and the Texas A & M results for
112,124Sn might be attributable to the background subtraction required in their analysis.
the customary moment ratio
m1/m−1for energy of the GMR in Eq. (1). A quadratic fit to
the data is also shown. The fit gives Kτ= −550 ± 40 MeV, with the uncertainty attributed
only to the fitting procedure. Including the uncertainties in KAin the fit adds another ∼25
MeV to this “error” (to ± 67 MeV) and the uncertainty in the value of KCoul(± 0.7 MeV;
see Ref. ) would contribute ∼ 15 MeV. Considering, further, the approximation made
in arriving at the quadratic expression, the actual total uncertainty would be somewhat
larger still; hence the rounded value Kτ = −550 ± 100 MeV quoted earlier in the text.
This result is consistent with the value Kτ = −500 ± 50 MeV obtained recently from an
analysis of the isotopic transport ratios in medium-energy heavy-ion reactions [34, 35]. As
shown in Ref. , this value provides constraints on the radius of a 1.4 M⊙neutron star
that are in rather good agreement with recent observational data. Thus, from the data
on the compressional-mode giant resonances, we now have “experimental” values of both
K∞and Kτ which, together, can provide a means of selecting the most appropriate of the
interactions used in EOS calculations. For example, this combination of values for K∞and
Kτessentially rules out a vast majority of the Skyrme-type interactions currently in use in
FIG. 4: Systematics of the difference KA− KCoulZ2A−4/3in the Sn isotopes as a function of the
“asymmetry-parameter” ((N-Z)/A); KCoul= -5.2 MeV . The solid line represents a least-square
quadratic fit to the data.
nuclear structure calculations . A similar conclusion was reached for EOS equations in
Ref. . Furthermore, a more precise determination of Kτ provides additional motivation
for measurement of isoscalar monopole strength in unstable nuclei, a focus of investigations
at RIKEN and GANIL, for example [37, 38].
In summary, we have measured the energies of the isoscalar giant monopole resonance
(GMR) in the even-A112−124Sn isotopes via inelastic scattering of 400-MeV α particles at
extremely forward angles, including 0◦. The GMR energies are significantly lower than those
predicted for these isotopes by recent calculations. Further, the asymmetry-term, Kτ, in the
expression for the nuclear incompressibility has been determined to be −550 ± 100 MeV.
We wish to express our gratitude to G. Col` o and J. Piekarewicz for providing results of
their calculations prior to publication. This work has been supported in part by the National
Science Foundation (Grants No. INT03-42942 and PHY04-57120), and by the Japan Society
for the Promotion of Science (JSPS).
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