Complete Electric Dipole Response and the Neutron Skin in Pb
A benchmark experiment on Pb shows that polarized proton inelastic scattering at very forward angles including is a powerful tool for high-resolution studies of electric dipole () and spin magnetic dipole () modes in nuclei over a broad excitation energy range to test up-to-date nuclear models. The extracted polarizability leads to a neutron skin thickness fm in Pb derived within a mean-field model [Phys. Rev. C 81, 051303 (2010)], thereby constraining the symmetry energy and its density dependence, relevant to the description of neutron stars.
pacs:25.40.Ep, 21.10.Re, 21.60.Jz, 27.80.+w
The electric dipole () response of nuclei is dominated by the giant dipole resonance (GDR), a highly excited collective mode above the particle emission threshold ber75 (). Its properties are well understood but recent interest focusses on evidence for a soft mode in neutron-rich nuclei below the GDR termed pygmy dipole resonance (PDR). Because of the saturation of nuclear density, excess neutrons might form a skin whose oscillations against an isospin-saturated core should give rise to a low-energy mode paa07 (). Therefore, the PDR may shed light onto the formation of neutron skins in nuclei pie06 (). Another quantity related to nuclear modes is the symmetry energy acting as restoring force. The strength distribution carries information on its poorly known magnitude and density dependence car10 (), indispensable ingredients for the modeling of the equilibrium properties of neutron stars hor01 ().
A case of special interest is the doubly magic nucleus Pb. In a measurement of parity-violating elastic electron scattering at JLAB, the PREX collaboration prex () aimed at the first model-independent determination of the neutron skin thickness in Pb. However, the recent result fm suffers still from limited statistics. Studies of energy density functionals (EDFs) ben03 () using Skyrme forces rei10 () or a relativistic framework pie10 () suggest the nuclear dipole polarizability as an alternative observable constraining both neutron skin and symmetry energy. The polarizability is related to the photoabsorption cross section boh81 ()
where denotes the photon energy. Because of the inverse energy weighting, depends on the strength at low energies. Theoretically, advanced methods exist in closed-shell nuclei for a realistic description of the strength distributions spe91 ().
The centroid of the PDR lies typically in the vicinity of the neutron emission threshold (). Data on the PDR in very neutron-rich nuclei are still scarce adr05 (); kli07 (); wie09 (). Stable nuclei at different shell closures have been explored with the reaction (Ref. sav08 () and refs. therein). While this technique provides important information on the fine structure of the PDR, it is essentially limited to excitation energies up to , and unobserved branching ratios of the decay to excited states may require corrections for the total strength rus08 (). Measurements of decay neutrons are constrained to energies and uncertainties in the vicinity of are large. We present here a new experimental tool, viz. polarized proton scattering at angles close to and including , to provide the complete E1 response in nuclei up to excitation energies well above the region of the GDR. At proton energies of MeV the cross sections at small momentum transfers are dominated by isovector spinflip- transitions (the analog of the Gamow-Teller mode) and by Coulomb excitation of non-spinflip transitions fre90 (); hey10 (). A separation of these two contributions, necessary for an extraction of the response, is achieved by two independent methods: a multipole decomposition analysis of the angular distributions (MDA) and the measurement of polarization transfer (PT) observables.
The Pb() experiment was performed at RCNP, Osaka, Japan. Details of the technique can be found in tam09 (). In the present work pol11 (), a proton beam of 295 MeV with intensities nA and an average polarization bombarded an isotopically enriched Pb foil with an areal density of 5.2 mg/cm. Data were taken with the Grand Raiden spectrometer fuj99 () in an angular range and for excitation energies MeV. Sideway () and longitudinally () polarized proton beams were used to measure the polarization transfer coefficients ohl79 () and , respectively. Additional data with unpolarized protons were taken at angles up to . Utilizing dispersion matching techniques, a high energy resolution keV (full width at half maximum) could be achieved.
Figure 1(a) displays a spectrum at . Strong transitions at low excitation energies, a resonance-like structure close to MeV and the prominent isovector giant dipole resonance (IVGDR), peaked at MeV with pronounced fine structure, are observed. The total spin transfer can be extracted from the measured PT observables which at takes a value of one for spinflip () and zero for non-spinflip () transitions suz00 (). Figure 1(b) shows for MeV. Values between 0 and 1 result from a summation over partially unresolved transitions with different spinflip character. The data reveal a concentration of spinflip strength in the energy region MeV, where the spin- resonance in Pb is located hey10 (), while the bump between 10 and 16 MeV has character consistent with the excitation of the GDR. The strength above the GDR may result from the components of the onsetting quasifree scattering cross section bak97 ().
A multipole decomposition was performed for angular distributions of the cross sections in the PDR and GDR regions. Theoretical angular distributions were calculated with the code DWBA07 raynal () using microscopic quasiparticle-phonon model (QPM) wave functions rye02 () and the Love-Franey effective proton-nucleus interaction fra85 (). The interference of Coulomb and nuclear contributions to the cross sections was taken into account for transitions. For a satisfactory description of the data it was sufficient to include, besides and , one higher multipole representative for all other contributions. The latter was chosen to be either or in the region of the PDR. In the GDR region the contribution was zero within error bars and was replaced by a phenomenological background describing the data at high excitation energies. The weight of each component was determined by a least-squares fit to the data.
Cross sections for and 1 from the MDA and PT analysis for MeV are compared in Fig. 2. Within uncertainties the correspondence between the two completely independent decomposition methods is excellent. This puts confidence in the MDA results discussed in the following, which provide much better resolution because of the superior statistics compared to a double scattering measurement of PT. In the GDR region no direct comparison is possible because of the unknown content of the phenomenological background. However, both methods agree that contributions are very small.
Next we show that reliable B() strengths can be extracted from the data. While the angular dependence of transitions is generally state-dependent because of the Coulomb-nuclear interference, cross sections at very small angles () arise purely from Coulomb excitation. Thus the conversion from cross section to strength is straightforward using semiclassical theory ber88 (). The B() distribution up to 8.2 MeV is compared in Fig. 3(a) with an average over all available Pb and Pb data (Refs. rye02 (); end03 (); shi08 (); sch10 () and refs. therein). Excellent agreement is obtained up to . The excess strength in the data above the neutron threshold can be attributed to previously unknown neutron decay widths of the excited states, which modify the branching ratios in the -decay experiments and thus the extracted B values. Figure 3(b) shows the photoabsorption cross sections in the GDR region together with results from a vey70 () and a total photoabsorption sch88 () experiment. Again, very satisfactory agreement of all three measurements is observed.
Figure 4(a) displays the experimental B distribution. From the numerous computations of the response in Pb we show in Fig. 4(b) recent results from the QPM rye02 (), and (c) the relativistic time-blocking approximation (RTBA) lit07 (). The QPM calculations contain up to 3-phonon configurations for MeV and 2-phonon configurations in the GDR region. Although the RTBA has recently been extended to include the full set of 2-phonon states lit10 (), the results shown are based on a particle-holephonon model space lit07 (). In the low-energy region, the QPM provides a realistic description of the fragmentation but the overall strength is somewhat too small, while the RTBA model space is not yet sufficient to reproduce the fine structure, and the strength is somewhat too large. The width of the GDR is roughly reproduced by both models. Within the QPM the effective isovector interaction strength is adjusted to the experimental GDR centroid at 13.4 MeV. The RTBA calculations are fully self-consistent and the GDR centroid determined by the covariant EDF parametrization amounts to 12.9 MeV for the NL3 parameter set used. Such a comparison between high-precision data and the 3-phonon version of the QPM guides the next generation of self-consistent extensions of the covariant EDF. Taking into account higher-order configurations, ground state correlations, and pairing vibrations should improve agreement with the data.
Finally, as discussed above, an important quantity is the electric dipole polarizability. We find fm/ for the strength up to 20 MeV. By taking an average of all available data including excitation energies up to 130 MeV vey70 (); sch88 (), a result with further reduced uncertainty fm/ is obtained. The covariance ellipsoid of the correlation between and the neutron skin thickness in the approach of Ref. rei10 () is shown in Fig. 5. Only with the present precision for (hatched band) one can constrain the neutron skin thickness to fm. The hitherto most precise determinations of this quantity for Pb fri07 (); zen10 () deduced from exotic atoms ( fm) and hadron scattering ( fm), respectively, are in excellent agreement with our result based on a totally independent method. Recent calculations of neutron matter and neutron star properties heb10 () in the framework of chiral effective field theory suggest fm. The predictions are sensitive to three-nucleon forces, which may be further constrained by the present results. Since the correlation between polarizability, neutron skin thickness and symmetry energy is model-dependent, viz. sat06 (), a systematic study with a variety of EDFs as well as experimental tests in other nuclei would be important.
To summarize, polarized proton scattering at very forward angles is a tool to study, with high resolution, the complete electric dipole response of nuclei from low excitation energies up to the GDR. The strength distribution deduced in a benchmark experiment on Pb is in excellent agreement with available data. It provides, however, new information in the region around the neutron emission threshold where all previous experiments had limited accuracy. A precise value for the polarizability can be extracted with important consequences for a determination of the neutron skin and the symmetry energy in neutron-rich nuclei. Although controversially discussed rei10 (), may independently be derived from a similar correlation with the PDR strength pie10 (); kli07 (), which is accurately determined by the present data as well. Beyond these results, the experiment also confirms the spin- resonance in Pb. Furthermore, the fine structure of the dipole modes contains information on level densities kal06 () and characteristic scales she04 (), giving insight into their dominant damping mechanisms.
We are indebted to the RCNP for providing excellent beams. Discussions with P.-G. Reinhard and A. Schwenk are appreciated. This work was supported by JSPS (Grant No. 14740154), DFG (contracts SFB 634 and 446 JAP 113/267/0-2). B. R. acknowledges support by the JSPS-CSIC collaboration program and E. L. by the LOEWE program of the State of Hesse (HIC for FAIR).
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