M Žáková

Z Andjelkovic

M L Bissell

K Blaum

G W F Drake

Ch Geppert

M Kowalska

J Krämer

A Krieger

M Lochmann

T Neff

R Neugart

W Nörtershäuser
email: w.noertershaeuser@uni-mainz.de

R Sánchez

F Schmidt-Kaler

D Tiedemann

Z.-C Yan

D T Yordanov

C Zimmermann

Isotope shift measurements in the

Introduction

Charge radii are among the basic properties of nuclei being used to test the predictions of nuclear structure models. Electron scattering experiments can to some degree provide the radial charge distribution and the nuclear charge radius, but they are not applicable for short-lived isotopes at present. For those, mainly two experimental approaches exist: nuclear reaction processes and optical precision spectroscopy. The matter radius can be extracted from nuclear interaction cross sections [1] and it was shown that the charge-changing cross sections show a Z dependency that is very similar to nuclear radii measured by electron scattering [2]. While charge-changing cross sections were investigated for nuclei close to the valley of stability [3] primarily for astrophysical applications, only a few attempts were made to extract charge distribution information for very exotic isotopes [4,5,6]. Isotope shift measurements using optical spectroscopy can provide nuclear-model independent information on the nuclear charge radius and, thus, on the proton distribution in the nucleus. The comparison of charge and matter radii is particularly interesting in systems which show a strongly different distribution of protons and neutrons inside the nucleus. These are namely nuclei with a neutron or proton skin or a so-called halo. Halo nuclei were discovered by Tanihata et al. in 1985 [1]. They performed nuclear reaction experiments with nuclei of light elements and observed significant changes in the interaction cross section between neighboring isotopes close to the neutron dripline. A notable increase for dripline nuclei was explained by a significantly extended matter distribution. It was proposed that this extension arises from weakly bound neutrons which form a region of diluted nuclear matter around a compact core with 'normal' nuclear density [7]. Since then, many experiments were carried out to understand the structure of these extraordinary systems. The most intensely studied halo nuclei are 6,8 He, 11 Li, 11 Be and 14 Be [8].

High-resolution optical spectroscopy [9] has considerably contributed to our understanding of halo nuclei. Collinear laser spectroscopy in combination with the β-NMR technique allowed the determination of the spin and the magnetic dipole moment of 11 Li [10] and 11 Be [11] as well as the determination of the electric quadrupole moment of 11 Li [12]. The results could be used to exclude a possible deformation of 11 Li because the spin 3/2 belongs to a spherical state, the magnetic moment lies close to the Schmidt value, and the quadrupole moment is similar to that of 9 Li. Both these values, which were improved recently [13,14], again indicate that the nucleus is nearly spherical.

The question of charge radii in these exotic nuclei has been raised and highlighted as important for a better understanding of the interaction between halo and core. At present only two groups have succeeded in measuring nuclear charge radii of the light halo systems in helium [15] and lithium [16] by high-precision laser spectroscopy. The extraction of nuclear charge radii from these isotope-shift measurements had become only possible because the mass-dependent part of the isotope shift could be calculated with sufficiently high accuracy, which is currently possible for atomic systems with up to three electrons. Yan and Drake published their first results with spectroscopic accuracy for transitions in lithium in 2000 [17] and the calculations have been continuously improved since then [18,19,20,21].

Standard collinear laser spectroscopy is not directly suitable for isotope shift measurements on the lightest elements. Until recently, 17 Ne was the lightest shortlived isotope for which so far charge radius information has been obtained using this technique [22]. The reason for that is an insufficient accuracy in the acceleration voltage measurements. Consequently, the uncertainty of the Doppler shift of the fast ion beam by far exceeds the finite-size contribution to the isotope shift. This extremely small contribution of about 10 -8 of the total transition frequency contains the essential information on the nuclear charge radius.

Recently, we described an advanced technique for spectroscopy on fast atoms and ions [23]. This approach combines collinear and anti-collinear laser spectroscopy and the absolute laser frequencies are determined with an accuracy better than 1 MHz employing a frequency comb. The technique was successfully applied for the measurements of the absolute transition frequencies in the 2s 1/2 → 2p 1/2 transition of 7,9,10,11 Be + ions [24]. In combination with accurate calculations of the mass-dependent isotope shift [20,21] these transition frequencies were used to determine the changes in the mean-square nuclear charge radii along this chain of isotopes [23].

In the measurement presented here, we determine the isotope shifts in the 2s 1/2 → 2p 3/2 transition with an accuracy of approximately 2 MHz, which is more than one order of magnitude higher than that in [25]. This allows the extraction of the nuclear charge radii with an accuracy that is comparable to that in the D 1 line [23] and our results confirm the charge radii extracted from those. Additionally, the splitting isotope shift, i.e., the difference in the isotope shift between the two fine structure levels, was obtained and is compared to theoretical predictions which are nearly independent of both QED and nuclear volume effects. This provides a consistency check of the calculations as well as of the measurements.

Unlike binding energies, charge radii provide a direct test of the wave functions obtained from nuclear structure calculations. As charge radii are sensitive to the asymptotic behavior of the wave function, they can be related to the evolution of clustering or the development of halos. In this paper the measured charge radii are compared to recent No-Core Shell Model (NCSM) results [26] and new calculations in the Fermionic Molecular Dynamics (FMD) approach are presented.

Experimental Setup

The experimental setup used for the isotope-shift measurements on beryllium isotopes has been described in previous publications [23,27] and will be summarized here only briefly. The measurements were performed at the on-line isotope separator (ISOLDE) at CERN. Radioactive beryllium atoms, produced by 1.4 GeV proton-induced fragmentation reactions in a uranium carbide target, were ionized by resonance laser ionization (RILIS) and mass separated after acceleration to an energy of about 50 keV. A Be + ion beam was then delivered to the beamline for collinear laser spectroscopy (COLLAPS) where it was superimposed with two laser beams -one in collinear and one in anti-collinear direction. Both lasers were locked at fixed frequencies, while the ion beam was accelerated or decelerated into the detection region by applying an additional high voltage between +10 kV and -10 kV. Resonance fluorescence was observed with photomultiplier tubes mounted at the beamline. The high voltage of ISOLDE and the voltage applied at the optical detection region were measured by utilising high-voltage dividers which were calibrated with a 1 ppm high-voltage divider developed and built at the University of Münster [28].

The final velocity of the ions shifts the rest-frame resonance frequency ν 0 in collinear and anti-collinear direction, respectively, to

ν c = ν 0 • γ • (1 + β) (1)
ν a = ν 0 • γ • (1 -β) (2)
with the time dilatation factor γ = 1/ 1 -β 2 . Changing the post-acceleration voltage at the detection region thus allows Doppler-tuning of the effective frequencies ν c and ν a across the fixed laser frequency. The laser frequencies for collinear and anti-collinear excitation are chosen such that the hyperfine structure patterns are covered for both directions within a small voltage range and that the centers of gravity (cg) almost coincide. With the cg coinciding for both laser beam directions the rest-frame transition frequency ν 0 can be determined solely from the laser frequencies according to

ν 2 0 = ν c • ν a . (3)
A simplified scheme of the laser setup is shown in figure 1. Two cw-single-mode ring dye lasers (Coherent 699-21) with a linewidth of about 1 MHz were used to produce the fundamental wavelengths of 628 nm and 624 nm for anti-collinear and collinear excitation, respectively. The laser for collinear excitation was stabilized to a suitable iodine hyperfine transition while the other was directly locked to a fiber-laser-based frequency comb (Menlo Systems, FC1500). Therewith it was possible to determine the absolute laser frequencies with a relative accuracy of 10 -9 . In total 12 iodine lines were chosen for the measurements. All lines are R branch lines with an even angular momentum J in the ground state. This is favourable, because they have a hyperfine structure exhibiting 15 components with the a 1 component well separated at the lower frequency side of the spectrum. This facilitates easy, reliable and reproducible locking during the beam time. The absolute locking frequencies for all selected iodine transitions were measured several times before, during and after the beam time with the frequency comb, to obtain accurate frequency information. For all lines very good agreement was found with the predicted frequencies from the iodine program IodineSpec (Toptica Photonics). In all cases deviations of less than 1 MHz were observed which is well below the quoted uncertainties of the calculated transition frequencies of a few MHz. The laser light was transported to the COLLAPS beamline by micro-structured fibers (LMA-20, . Simplified scheme of the experimental setup. The frequencies of the dye lasers for collinear and anti-collinear excitation were stabilized to a hyperfine transition of molecular iodine and to a frequency comb, respectively. The light from both lasers was frequency doubled and overlapped with the ion beam.

Crystal Fibers). Two frequency doublers (SpectraPhysics Wavetrain and Tekhnoscan FD-SF-07) were used for the generation of ultraviolet light which is required for the 2s 1/2 → 2p 3/2 transition in Be + (312 nm and 314 nm in collinear and anti-collinear direction, respectively). A power of up to 15 mW was achieved in the ultraviolet, but for spectroscopy this was attenuated to values below 5 mW to avoid saturation effects. The laser -laser and laser -ion beam overlap was optimized using two irises and a combination of small apertures with diameters down to 5 mm along the beamline. The irises could be closed during the alignment procedure to a diameter of 1 mm but were opened during measurements to avoid scattered light causing additional background. Systematic uncertainties due to, e.g., the ion -laser and laser -laser beam misalignment, the ion and laser beam divergence, or the momentum transfer to the ions by repetitive absorption of photons, were investigated in an off-line test run on the stable 9 Be. The main source of systematic uncertainty of about 500 kHz was found to be a possible misalignment between the two laser beams. Concerning absolute transition frequencies, an additional uncertainty of about 800 kHz arises from the uncertainty of the Rb-clock that is used as a reference for the frequency comb. However, the latter can be neglected in isotope-shift measurements. It contributes to all isotopes with the same sign and similar size and largely cancels when the difference of the transition frequencies between two isotopes is calculated. All measurements were consecutively repeated several times. Collinear and anti-collinear spectra were recorded quasi-simultaneously by alternately blocking one of the laser beams in a fast sequence and scanning the Doppler-tuning voltage across the resonances in typically 400 steps for the odd and 100 steps for the even isotopes. For one spectrum, 50-400 such scans, each taking about 4-8 s, were integrated depending on the production rate of the respective isotope. In total, 10-20 spectra with sufficient statistics were recorded for each isotope.

F = 2 F = 1 F = 2

Experimental Results

Figure 2 shows the collinear and anti-collinear spectra for 9 Be in the 2s 1/2 → 2p 3/2 transition as a function of the applied tuning voltage. The observed peaks are labeled according to the respective 2s 1/2 ground state hyperfine level F . The hyperfine structure in the upper 2p 3/2 level is not resolved, because the splitting is less than 10 MHz while the natural linewidth of the transition is 20 MHz. Furthermore, the observed linewidth in collinear spectroscopy is broadened to 50-70 MHz and, hence, only the ground state splitting is resolved, which ranges from 0.5-3.6 GHz for the different isotopes. The solid line represents a fit with a Voigt lineshape function and is discussed below. The dashed vertical lines mark the center of gravity voltages (U cg ) of both scans, which do not exactly coincide but are separated by ∆U cg which amounts to a few volts. The cg frequencies from the collinear and anti-collinear direction have to be combined to extract the rest-frame transition frequency. Since they do not appear at exactly the same acceleration voltage, the small correction ∆U cg must be introduced for either one of the two spectra. First, the scaling factor k between voltage and frequency is determined for the acceleration voltage around the center of gravity. Depending on the isotope, this factor is between 30 and 40 MHz/V. Using this scaling factor k, the frequency detuning ∆ν cg = k ×∆U cg can be calculated ‡. It is required to artificially shift one of the laser frequencies to observe both centers of gravity at the same acceleration voltage. With the shift applied to the anti-collinear laser frequency, the rest-frame transition frequency can be calculated by

ν 2 0 = ν c × (ν a + ∆ν cg ), (4)
where ν a and ν c are the doubled frequencies of the two locked dye lasers. The spectra of 9 Be + were fitted with a multiple peak function containing all hyperfine transitions calculated from the nuclear spin and the atomic angular momentum. Each transition was fitted with a Voigt profile and common linewidth for all transitions. The small magnetic dipole and electric quadrupole interval factors A 2p 3/2 and B 2p 3/2 , were taken from theoretical calculations [29]. They were introduced in the fitting routine as fixed parameters. The transition amplitudes were calculated relative to the strongest line according to the corresponding Racah intensity distributions. This procedure was necessary to ensure fitting results within physically meaningful constraints. In figure 2, a small peak on the left side of each hyperfine transition is visible. This peak arises from inelastic collisions of ions in the beam with residual gas atoms along the beamline. In such collisions Be + ions can be excited into the 2p states. The energy for this process is taken from the ions kinetic energy and is emitted in form of a photon. Hence, those ions have resonance positions 3.97 eV below the main peak. This effect was included in the fitting function and was treated as an additional Voigt peak at a fixed distance from the main peak with identical relative intensities compared to the main peak of the respective hyperfine component.

In order to test the sensitivity of the resulting transition frequency to the unresolved hyperfine splitting and the particular lineshape model, 9 Be spectra obtained in collinear and anti-collinear excitation were also fitted independently with different lineshape functions. Besides the fit with fixed 2p 3/2 hyperfine parameters described above, fits with single Voigt, Gaussian and Lorentzian profiles for each of the two hyperfine peaks, constrained to have common line widths parameters, were performed. For each lineshape function, the cg frequencies were then combined to calculate the rest-frame frequency. The standard deviation of all measurements for one isotope and lineshape model was used as the statistical uncertainty of this value. The results for ν 0 obtained for the different lineshapes, scattered only by a fraction of this statistical uncertainty, indicating a negligible dependence on the chosen lineshape. Hence, the mean value of ν 0 obtained for the different lineshape functions was taken as the experimental transition frequency whereas the largest statistical uncertainty observed in the different fittings was kept as the final uncertainty. ‡ The small error introduced by this linearization is insubstantial compared to the statistical and other sources of uncertainty and can be neglected.

-1,0 -0,5 0,0 For the radioactive isotopes a similar procedure was used. Spectra for 7,10,11 Be are shown in figure 3. Here the voltage axis is converted to a frequency scale. Solid lines show the fitting results with a Voigt line profile where the hyperfine splitting in the upper state was disregarded. A multi-line fit with individual hyperfine transitions as it was performed for 9 Be was only possible for 11 Be, since there is no information on the spectroscopic quadrupole moment of 7 Be and 10 Be does not exhibit any hyperfine structure (I = 0).

A special case is 10 Be, where the best counting statistics was obtained. For this isotope an additional asymmetry was observed in the line shapes. Such asymmetries shift the cg frequency in collinear and anti-collinear direction by approximately the same amount but with opposite sign in the frequency domain. Consequently, this contribution largely cancels when the rest-frame frequency ν 0 is calculated. Nevertheless, for comparison a third peak component beside the small inelastic collision contribution, residing on the higher frequency side of the resonance, was added to the fitting function, which resulted in a much better description of the profile as it is shown in the spectrum of 10 Be in figure 3. The possible origin of this structure is not fully understood yet. While the individual peak position in collinear and anti-collinear spectra obtained from Table 1. Extracted rms nuclear charge radii R c from the 2s 1/2 → 2p 3/2 (D 2 ) line compared to those obtained from the 2s 1/2 → 2p 1/2 (D 1 ) transition [23]. Differences in the mean-square nuclear charge radii δ r 2 c are calculated from isotope shifts δν 9,A IS . The field shifts δν 9,A FS (reference isotope is 9 Be) are calculated using the theoretical mass shifts δν 9,A MS taken from literature. Uncertainties of the absolute charge radii R c from the D 2 and D 1 line include the uncertainty of the reference radius R c ( 9 Be) = 2.519 (12)

* Ref [20]

⋄ Ref [30] the fits of 10 Be + with one, two and three Voigt peaks differ by as much as 15 MHz, the rest-frame frequencies calculated by combining the collinear and anti-collinear results agree within 500 kHz. This clearly demonstrates the robustness of the collinear -anticollinear approach with respect to the details of the lineshape function.

The isotope shifts relative to 9 Be were calculated from the fitting results and are listed in table 1. Using the precise mass-shift calculations from [20,30] the field shift, which is proportional to the change in the mean-square nuclear charge radius, can be calculated by

δ r 2 c 9,A ′ = δν 9,A ′ IS -δν 9,A ′ MS C (5)
with the field shift constant C ≈ 17 MHz/fm 2 . For the data analysis more precise values for C taken from [21] were used, which show small isotopic variations. The rms nuclear charge radii were calculated based on the charge radius R c ( 9 Be) = 2.519 (12) fm that was determined by elastic electron scattering [31]. A change of this reference charge radius would lead to a constant shift for all isotopes, not influencing the variations along the isotopic chain. The obtained rms charge radii as listed in table 1 are plotted in figure 4 and compared to the results from the 2s 1/2 → 2p 1/2 transition. Both data sets agree very well within their uncertainties, but due to the resolved hyperfine structure in the D 1 line these previous values are slightly more precise. Before we discuss the evaluated charge radii, a comparison between the isotope shifts in the D 1 and the D 2 transition should be made. These values are listed in table 2 and the difference between them is the so-called splitting isotope shift (SIS). The measured SIS are in a good agreement with the theoretical predictions but the uncertainties are still relatively large. An improved measurement of the isotope shift in [25,32] with stored and laser-cooled ions in an RF ion trap, could clearly improve this comparison. The best studied example for the SIS in a three-electron system is the lithium atom. Here, the value predicted from theory for the SIS between 6 Li and 7 Li is 0.396 MHz [20]. The most recent experimental values of -0.863(70) MHz [33] and 0.155(77) MHz [34] neither agree with each other nor with the predicted theoretical value. Therefore, the measured SIS for the 2p states in Be + is an important consistency check for theory and experiment.

Discussion

The charge radii presented here agree well with those reported earlier from our measurements in the D 1 transition [23]. There, we explained the general trend of the charge radii from 7 Be to 10 Be by the underlying cluster nature of these nuclei: The large nuclear charge radius of 7 Be (= α + t) arises from the low cluster-threshold of only 1.586 MeV, which implies an extended wave function for the relative motion of the clusters. While 8 Be (= α + α) is unbound, the additional neutron in 9 Be glues together the two α particles with a relatively small binding energy of only 1.573 MeV, which is reflected again in a correspondingly large charge radius. 10 Be, on the other hand, is well bound and the nucleus is stiff against a variation in the distance of the α clusters. The additional neutron in 11 Be can either occupy a 1p 1/2 orbit (in the excited 1/2 -state at 320 keV) or an orbit in the 2s 1/2 shell for the 1/2 + ground state. The experimental binding energy is only 504 keV and an extended halo is expected. Even if the core is not polarized the charge radius will change, as the 10 Be core will move with respect to the center of mass of the nucleus, according to

R 2 c ( 11 Be) = R 2 c (core) + 1 11 2 R 2 core-n + 1 4 R 2 n , (6)
with the core-neutron distance R core-n and the mean-square charge radius of the neutron R 2 n = -0.117(4) fm 2 [35]. In the following, we will discuss conclusions to be drawn from the combination of our results with other experiments and present details of fermionic molecular dynamics calculations on the beryllium isotopes.

Using the charge radii extracted from the D 1 transition, listed in table 1, and assuming that there is no polarization of the 10 Be core we obtain R core-n = 8.09(1.22) fm.

To get an idea how far the assumption of the absence of core polarization is valid, this number can be compared with data from electromagnetic and nuclear inelastic scattering on 11 Be: by measuring the one-neutron removal channel a radius for the valence neutron of 5.7(4) fm was deduced [36]. With this value, the center-of-mass motion of 10 Be would result in a 11 Be charge radius slightly smaller than observed in our measurements and an additional increase of the core charge radius to 2.41(2) fm is required to explain the experimental 11 Be charge radius. Hence, the charge radius of the core is expected to be about 2.2(1.8) % larger than that of the free 10 Be nucleus.

In our previous publication [23], we compared the experimental data to theoretical predictions from ab initio calculations in the Green's Function Monte Carlo (GFMC) approach [37,38](no result for 11 Be is so far available), No-Core Shell Model (NCSM) calculations and the Fermionic Molecular Dynamics (FMD) approach. Here, we compare them with updated NCSM results [26] using the CD-Bonn interaction and present new FMD calculations. Nuclear charge radii obtained from these theoretical models are shown in figure 4. Error bars for the NCSM radii were estimated based on model space size and oscillator frequency dependence [26]. Charge radii obtained with the INOY interaction, which provides binding energies closer to experiment than the CD-Bonn interaction, are significantly smaller and outside the CD-Bonn error bars. Although GFMC -Green's Function Monte-Carlo calculations [37,38] the charge radii are in general too small, the NCSM calculations are able to describe the decrease from 7 Be to 10 Be but they show only a small increase for the 11 Be charge radius. This is a deficiency of the NCSM model space which has difficulties to properly desribe the neutron halo in 11 Be. This problem is now being attacked by coupling the NCSM with the cluster model [39]. Root mean square nuclear charge radii obtained from theoretical calculations and isotope shift measurements are listed in table 3.

Fermionic Molecular Dynamics

Fermionic Molecular Dynamics (FMD) [40] is a many-body approach especially suited for the description of nuclei with cluster or halo structure. It has been used successfully to describe form factors and charge radii of light nuclei in the p shell and sd shell [22,41]. In the present calculation the intrinsic many-body basis states are given by Slater determinants |Q using a superposition of two Gaussian wave packets for each single-particle state. The parameters of the wave packets are mean position, mean momentum, width and orientation of the spin. The basis is very flexible, it contains the harmonic oscillator shell model states as well as Brink-type cluster model states as limiting cases. The variable width parameter allows an efficient description of spatially extended nucleon wave functions as appearing in halo nuclei. To restore the symmetries of the Hamiltonian, the Slater determinant is projected on parity, angular momentum and total linear momentum |Q; J π MK, P = 0 . Diagonalizing the Hamiltonian in the basis spanned by different K quantum numbers provides the states |Q; J π Mκ, P = 0 which are used as many-body basis states. The effective interaction is based on V UCOM derived from the realistic Argonne V18 interaction by explicit treatment of short-range central and tensor correlations in the unitary correlation operator method [42]. It also includes a phenomenological two-body correction term that has been fitted to the binding energies and radii of doubly-magic nuclei up to 48 Ca. No parameters have been adjusted for the beryllium isotopes.

The parameters of the wave function were obtained by variation of the energy after projection (VAP) on parity and angular momentum. In figure 5 cuts through the proton and neutron densities of the intrinsic states are shown. In contrast to mean field calculations the intrinsic states obtained in VAP calculations show a pronounced α-clustering (α particle-3 He in the case of 7 Be). As the cluster thresholds, especially in 7 Be and 9 Be, are low, we expect an extended wave function for the relative motion of the clusters. Already in a single Slater determinant the relative distance of the clusters has a Gaussian distribution, but for a realistic description a superposition of several manybody states is needed. We generated such many-body states by imposing constraints on the distance between the clusters. A good measure for the distance of the clusters (assuming no significant polarization of the clusters) is the rms radius of the proton distribution with respect to the center of charge (R pp ). This is different from the charge radius which measures the proton radius with respect to the center of mass and which is therefore sensitive not only to the distance of the clusters but also to the motion of the clusters relative to the center of mass.

For each nucleus we performed VAP calculations with a constraint on the radius R pp of the intrinsic states. In figure 6 the energies are shown as a function of R pp . The energy surfaces are very flat for 7 Be and 9 Be. Therefore configurations with large R pp will contribute to the ground state and will increase also the charge radius significantly compared to the VAP calculation using a single configuration. In 10 Be, on the other hand, the energy surface is rather steep and extended configurations will not significantly contribute to the ground state. Assuming no polarization of the clusters the R pp values can be translated into rms cluster distances of 4.23 fm ( 7 Be), 4.08 fm ( 9 Be) and 3.51 fm ( 10 Be).

For 11 Be the situation is different. The neutron separation energy is very small and as might be expected the FMD configuration obtained in the unconstrained VAP calculation essentially corresponds to the 10 Be VAP state plus an additional neutron in an extended s-orbit. Whereas R pp is almost identical in the 10 Be and 11 Be VAP states, the charge radius increases significantly. This is due to the motion of the 10 Be core with respect to the total center of mass. For an improved description of 11 Be we did not only use the R pp constraint which affects mainly the 10 Be core but also varied a second many-body state in a multiconfiguration calculation for each R pp . By this procedure configurations with a 10 Be core in the 2 + state and the valence neutron occupying a d-orbit are included. This admixture of core polarized configurations improves the 11 Be binding energy by about 1 MeV and is necessary to get 11 Be bound with respect to 10 Be. It leads to a slight increase of R pp from 2.10 fm in 10 Be to 2.19 fm in 11 Be. However the effect on the charge radius of the 10 Be core is very small, it changes from 2.52 fm to 2.51 fm.

In summary, all FMD charge radii are slightly too large compared to the experimental results (see figure 4) but capably reproduce the systematic trend. The FMD calculations presented in [23] did not include configurations with large cluster distances and therefore smaller radii, especially for 7 Be and 9 Be, were obtained. Absolute charge radii values deduced from δ r 2 c furthermore depend on the reliability of R c ( 9 Be) from electron scattering data, which should be kept in mind in comparison with theory.

To conclude, we have measured the isotope shifts in the 2s 1/2 → 2p 3/2 transition of Be + and extracted the nuclear charge radii for 7,10,11 Be. The radii were compared to results of our previous measurement in the 2s 1/2 → 2p 1/2 transition and good agreement was found. When combined with results from electromagnetic and nuclear inelastic scattering of 11 Be a small 2.2(1.8)% charge-radius increase of the core in 11 Be compared to the free 10 Be nucleus can be deduced. Results are also compared to calculations in the Fermionic Molecular Dynamics approach which were able to explain their systematic behavior. Thus, it can be concluded from theory and experiment that the dominant contribution to the change in the charge radius between 10 Be and 11 Be is the centerof-mass motion of the 10 Be core caused by the halo neutron, whereas the contribution of core polarization is rather small. Additionally, the splitting isotope shift in the 2p states was evaluated, showing good consistency with theoretical predictions.

Figure 1

 1 Figure1. Simplified scheme of the experimental setup. The frequencies of the dye lasers for collinear and anti-collinear excitation were stabilized to a hyperfine transition of molecular iodine and to a frequency comb, respectively. The light from both lasers was frequency doubled and overlapped with the ion beam.

Figure 2 .

 2 Figure 2. Collinear (top) and anti-collinear (bottom) spectrum recorded for the stable isotope 9 Be in the 2s 1/2 → 2s 3/2 transition as a function of the Doppler tuning voltage applied to the detection region. The solid line is the overall fitting function with a Voigt profile for each peak.

Figure 3 .

 3 Figure 3. Spectra recorded for radioactive7,10,11 Be isotopes in 2s 1/2 → 2p 3/2 transition. Solid lines are Voigt fit functions. Frequencies are relative to the center of gravity of the respective isotope.

Figure 4 .

 4 Figure 4. Nuclear charge radii of7,9,10,11 Be obtained from isotope shift measurements, FMD multiconfiguration mixing calculations (FMD) and calculations in the No-Core Shell Model (NCSM) with the CD-Bonn potential[26]. Radii extracted from the isotope shift in the D 1 line are represented by the black solid line and the grey area indicates the corresponding uncertainty. Radii extracted from the D 2 transition are shown as individual symbols with error bars.

Figure 5 .

 5 Figure 5. Cuts through the proton and neutron densities of the intrinsic states obtained in FMD VAP calculations. Contour lines are in units of half nuclear matter density.

  fm[31].

	Isotope	δν 9,A IS , MHz	δν 9,A MS , MHz	δν 9,A FS , MHz δ r 2 c , fm 2 R c (D 2 ), fm R c (D 1 ), fm
	7 Be	-49 242.1(2.0) -49 231.827(39)* -10.3(2.0)	0.61(12)	2.637(55)	2.647(17)
			-49 231.779(35) ⋄
	10 Be	17 326.8(2.0)	17 312.569(13)* 14.2(2.0)	-0.87(12)	2.339(53)	2.357(18)
			17 312.553(13) ⋄
	11 Be	31 568.8(2.3)	31 564.168(24)*	4.9(2.3)	-0.28(14)	2.463(62)	2.463(16)
			31 564.207(31) ⋄

Table 2 .

 2 Experimental isotope shifts δν 9,A IS (exp) in the 2s 1/2 → 2p 1/2 (D 1 ) and 2s 1/2 → 2p 3/2 (D 2 ) transitions and extracted splitting isotope shifts δν SIS (exp). In the last two columns the values for δν SIS from two independent theoretical calculations are listed. All values are given in MHz.Isotope δν IS D 1 (exp) δν IS D 2 (exp) δν SIS (exp) δν SIS[29] δν SIS[20]

	7 Be -49 236.94(80) -49 242.1(2.0) -5.1(2.2) -6.037(1) -6.049
	10 Be 17 323.82(82) 17 327.3(2.0)	3.5(2.4)	2.097(1) 2.13
	11 Be 31 565.04(86) 31 569.6(2.3)	3.6(2.5)	3.965(1) 3.878
	the 2s 1/2 → 2p 3/2 line, as it is planned at RIKEN

Table 3 .

 3 Root-mean-square nuclear charge radii R c for beryllium isotopes obtained from different theoretical calculations (th) and isotope shift measurements IS (exp). Experimental charge radii are the weighted averages of the values attained from the D 1[23] and D 2 transitions (this work). All values are given in fm.

	Isotope FMD (th) NCSM (th) GFMC (th) IS (exp)
	7 Be	2.68	2.56(10)	2.588	2.646(16)
	9 Be	2.57	2.41(11)	2.520	2.519(12)
	10 Be	2.38	2.34(9)	2.354	2.355(17)
	11 Be	2.51	2.37(11)	-	2.463(15)
	FMD -Fermionic Molecular Dynamics (this work)
	NCSM -No Core Shell Model with CD-Bonn potential [26]

  Energies of FMD configurations for7,9,10 Be obtained in variation after projection (VAP) calculations. The proton radius with respect to the center of charge R pp is used as a constraint for the intrinsic state. Arrows indicate the R pp expectation values obtained in the VAP minimum and from multiconfiguration mixing calculations.

		-20
		-30			VAP Multiconfig		7 Be
	VAP ) [MeV] pp (R	-50 -40						9 Be
	E
		-60				10 Be
		-70	1.75	2	2.25	2.5	2.75	3	3.25
						R pp [fm]
	Figure 6.

This work was supported by the Helmholtz Association (VH-NG-148), BMBF (06TU263I, 06MZ215, 06UL264I), and the EU (FP-6 EU RII3-CT-2004-506065), M.K. was supported by the EU (MEIF-CT-2006-042114). We gratefully acknowledge support by M. Fischer and R. Holzwarth from Menlo Systems and the ISOLDE technical group. We thank K. Pachucki and M. Puchalski for stimulating discussions and for providing their calculated D 2 mass shifts prior to publication.


G Orlandini
email: orlandin@science.unitn.it

W Leidemann

V D Efros

N Barnea

How to increase the applicability of integral transform approaches in physics?

Integral transform approaches are numerous in many fields of physics, but in most cases limited to the use of the Laplace kernel. However, it is well known that the inversion of the Laplace transform is very problematic, so that the function related to the physical observable is in most cases unaccessible. The great advantage of kernels of bell-shaped form has been demonstrated in few-body nuclear systems. In fact the use of the Lorentz kernel has allowed to overcome the stumbling block of the ab initio description of reactions to the full continuum of systems of more than three particles. The problem of finding kernels of similar form, applicable to many-body problems deserves particular attention. If this search were successful the integral transform approach might represent the only viable ab initio access to many observables that are not calculable directly.

Introduction

Integral transform approaches to physical problems are common to many fields in physics, e.g. nuclear physics, condensed matter physics and quantum chromo-dynamics. The motivation to use such approaches originates from the fact that, while one is often unable to evaluate a physical quantity of interest X , directly, one can calculate a few of its integral properties. Particularly interesting are integral transforms. They can give information on X . However, a sufficiently detailed reconstruction of X (inversion of the transform) is sometimes a formidable task or even impossible. One of the reasons is that in many cases the integral property known to be accessible by a numerical technique is the Laplace transform (LT ). This occurs in the numerous physical processes that are described by diffusion equations. Problems involving LT s also arise in computing statistical functions, which is very common in many fields.

From the Mellin's inverse formula (Bromwich integral) for the inverse Laplace transform [START_REF] Davies | Integral transforms and their application[END_REF], one sees that it is necessary that the LT is computable in the complex half-plane of convergence. In this case X can be computed by evaluation of a complex path integral, which can be a formidable task. However, in the common situation when the Laplace transform is computable (or measurable) on the real and positive axis only, the problem is extremely ill-posed. This case is much more complicated because of the absence of an exact inversion formula. In addition, as it will be explained below, one can say that among ill-posed problems, the inversion of the LT is one of the worst. There are some more or less efficient numerical methods available for computing the inverse transform [START_REF] Jaines | The Maximum Entropy Formalism ed R D Levune and M Tribus[END_REF][START_REF] Kryzhniy | [END_REF], however, in general when the input is numerically noisy and incomplete it may be impossible to obtain a stable result [START_REF] Acton | Numerical Methods that Work[END_REF].

In the last fifteen years it has been shown that an integral transform approach with a bell-shaped kernel, namely the Lorentzian function [START_REF] Efros | [END_REF], helps in solving a longstanding problem in few-body physics, i.e. the calculation of reactions involving states in the far continuum of four-([6]- [10]) and even six- [11] and seven- [12] body systems (for a review see [13]). The scope of this contribution is to (i) clarify why the Lorentzian function is a good kernel (section 1), (ii) discuss the limits of this approach (section 2) (iii) define the open problems (section 3) and (iv) suggest possible directions to explore for finding solutions (section 4).

Why is the Lorentzian function is a good kernel ?

To illustrate what is the intrinsic origin of the difficulties in inverting transforms with the Laplace kernel or, similarly, with the Stieltjes kernel [14,15], consider figure 1. The upper panel shows two functions r 1 (E) and r 2 (E), while in the middle panel their LT s are plotted, i.e.

Φ(σ) = ∫ K(σ, E) r(E) dE , ( 2.1)
with K(σ, E) = e -σE . As one can see the curves in figure 1(a) do not resemble the curves in figure 1(b). The functios r i (E) exhibit a peak, while the LT s are monotonically decreasing. Moreover, one sees that though r 1 (E) and r 2 (E) are rather different they lead to very similar LT s. Considering that both transforms could lie within the error of a numerical calculation one understands that it is impossible by inversion to distinguish between the curves in 1(a). The reason is that the Laplace kernel has spread the information over a very large domain. As shown in 1(c) the situation is different for the Lorentz integral transform (LIT), where the kernel of the transform is the Lorentzian function

K(σ R , σ I , E) = 1 (E -σ R ) 2 + σ 2 I .
(2.

2)

The form of the functions in 1(c) is preserved and the relative difference between the LITs, is of the same order as it is for the functions in 1(a).

We do not intend to discuss here the problem of the inversion of integral transforms, which are known in mathematics under the name of ill-posed problems. An extended literature exists about this topic (see e.g. [START_REF] Tikhonov | Solutions of Ill-Posed Problems[END_REF]). We only want to mention that, contrary to what the unfortunate name might suggest, they can be solved using regularization methods. Such methods are very efficient when the kernel is bell shaped with a controllable width. A clarifying discussion about this point can be found in [START_REF] Barnea | The Lorentz Integral Transform and its Inversion[END_REF].

In relation to what has been said above, one could think that the perfect kernel is a δ-function, i.e. K(σ R , E) = δ(σ R -E). In fact in this case the function and its transform would coincide. While this clearly does not help, the Lorentzian function, which up to normalization turns to the δ-function at σ I → 0, has the necessary characteristics, as have the Gaussian functions or other bell-shaped kernels.

However, it is clear that the other conditio sine qua non for applying an integral transform approach to physical problems is that the transform is calculable in practice.

In [START_REF] Efros | [END_REF] it was suggested that the Lorentzian kernel might satisfy both requisites when applied to a specially difficult and unsolved problem, i.e. the exact (ab initio) calculation of perturbation induced reactions to states in the continuum (scattering states) of a many-body system. In the following we will shortly summarize why this is the case. More extensive information can be found in [13], where the successful applications of what is now known as the Lorentz integral transform method have been reviewed and the procedure to apply it also to non-perturbative reactions has been outlined.

Short summary of the LIT method

Very often physical quantities of interest have (or may be reformulated to have) the following structure

r(E) = ∑ ∫ dγ Q|Ψ γ Ψ γ |Q δ(E γ -E), ( 2.3)
where the integration and summation go over all the complete and orthonormal set of eigenstates |Ψ γ of an hamiltonian Ĥ and the norms Q|Q and Q |Q are finite.

When the energy E and the number of particles in a system increases the direct calculation of the quantity r(E) becomes prohibitive. The difficulty is related to the fact that in these cases a great number of continuum spectrum states |Ψ γ contribute to r(E) and the structure of these states is very complicated.

In order to overcome the problem one can consider instead an integral transform with a smooth kernel K(σ, E). This yields

Φ(σ) = ∑ ∫ dγ Q|Ψ γ K(σ, E γ ) Ψ γ |Q = ∑ ∫ dγ Q| K(σ, Ĥ)|Ψ γ Ψ γ |Q . (2.4)
Using the closure property one obtains

Φ(σ) = Q| K(σ, Ĥ)|Q . ( 2.5)
The right-hand side of (2.5) requires only the knowledge of the finite norm functions |Q and |Q and avoids completely the full knowledge of the spectrum of Ĥ. (It is clear that what is presented here can be considered as a generalization of the sum rule, namely the moment method).

Choosing for K(σ, E) the Lorentzian function

K(σ, E) = 1 (E -E 0 -σ R ) 2 + σ 2 I .
(

where

σ = E 0 + σ R + iσ I σ I = 0, (2.7)
and E 0 is the ground-state energy, one has

L(σ R , σ I ) = Q| 1 ( Ĥ -E 0 -σ R + iσ I ) 1 ( Ĥ -E 0 -σ R -iσ I ) |Q ≡ Ψ| Ψ , ( 2.8)
where the 'LIT functions' Ψ and Ψ are solutions to the inhomogeneous equations This means that in contrast to continuum spectrum problems, in order to construct a solution, it is not necessary here to reproduce a complicated large distance asymptotic behaviour in the coordinate representation or singularity structure in the momentum representation. This is a very substantial simplification.

( Ĥ -E 0 -σ R -iσ I ) | Ψ = |Q , (2.9) ( Ĥ -E 0 -σ R -iσ I ) | Ψ = |Q . ( 2

What are the present limits of LIT applications ?

The LIT method has been successfully applied to quite a few physical problems. They regard electroweak reactions with few-body systems. An extension to many body systems, however, encounters at present some limits. They become clear in the following where we describe how one calculates the LIT L(σ R , σ I ) of equation (2.8).. Most of the LIT applications have been solved by using complete many-body basis and expanding | Ψ and | Ψ over N localized states. A convenient choice is to use N linear combinations of states that diagonalize the hamiltonian matrix. We denote these combinations |ϕ N ν and the eigenvalues N ν . The index N is to remind that they both depend on N . If the continuum starts at E = E th then at sufficiently high N the states |ϕ N having N ν < E th will represent approximately the bound states. The other states will gradually fill in the continuum as N increases. The expansions of our localized LIT functions read as

| Ψ = N ∑ ν ϕ N ν |Q N ν -E 0 -σ R -iσ I |ϕ N ν , ( 3.1)
| Ψ = ∑ ν ϕ N ν |Q N ν -E 0 -σ R -iσ I |ϕ N ν . (3.2)
Substituting (3.1) and (3.2) into (2.8) yields the following expression for the LIT,

L(σ R , σ I ) = ∑ ν Q|ϕ N ν ϕ N ν |Q ( N ν -E 0 -σ R ) 2 + σ 2 I , ( 3.3)
and for the common case

|Q = |Q L(σ R , σ I ) = ∑ ν | ϕ N ν |Q | 2 ( N ν -E 0 -σ R ) 2 + σ 2 I . ( 3.4)
From (3.3) and (3.5) it is clear that in this way L(σ R , σ I ) becomes a sum of Lorentzians. The spacing between the corresponding eigenvalues with N ν > E th depends on N and in a given energy interval the density of these eigenvalues increases with N . (Since the extension of the basis states grows with N , this resembles the increase of the density of states in a box, when its size increases). For the reliability of the inversion one needs to reach the regime where one has a sufficient number of levels

N ν in a [σ R -σ I , σ R + σ I ] interval.
In a similar way one can obtain other integral transforms with bell shaped kernels e.g. the Gauss integral transform (GIT)

K(σ R , σ I , E) = e -(E-σ R ) 2 /σ 2 I : G(σ R , σ I ) = ∑ ν | ϕ N ν |Q | 2 e -( N ν -σ R ) 2 /σ 2 I . (3.5)
It is clear that for a many-body system this is a formidable task, because of the larger and larger degeneracy of states with increasing N and the little chance that out of the many states a significant number is found within the σ I extension. In this resides the limit of the LIT method if one wants to extend it to many-body systems.

A more efficient procedure can be applied using the Lanczos approach. This consists in rewriting L(σ R , σ I ) as

L(σ) = - 1 σ I Im { Q| 1 σ R + iσ I + E 0 - Ĥ |Q } . (3.6)
and then as a continuous fraction in function of the Lanczos coefficients a i and b i

L(σ) = - 1 σ I Im        Q|Q z -a 0 - b 2 1 z-a 1 - b 2 2 z-a 2 -b 2 3 ...        . ( 3.7)
This has the big advantage that one has not to diagonalize or invert the hamiltonian matrix, however the problem of extending the method to many-body systems remains similar to the previous case.

Important remarks:

• We would like to emphasize that, contrary to discretization approaches to the continuum, here the use of a discrete basis is perfectly legitimate, because of the bound state like boundary conditions of | Ψ (| Ψ ). The continuum spectrum is recovered with a controlled resolution by the regularization when inverting the transform.

• One can notice that equation (3.5) corresponds to the following common procedure. This consists in diagonalizing the hamiltonian on a discrete basis and then folding the | ϕ N ν |Q | 2 with a Lorentzian function (or a Gaussian function), in order to simulate the "experimental window". However, as equation (3.5) clearly states, this does not represent r(E) itself, but its LIT (GIT), that has to be inverted (see [13], figure 5 and relative discussion).

• Considering our discussion one realizes that, within the eigenvalue approach, the difficulties in generalizing the integral transform method to many body systems do not lie in the kernels, that may be chosen conveniently, but in handling very large basis.

Open problems

The problem of calculating exactly (ab initio) reactions to continuum states (scattering states) of many-body systems is one of the oldest open problems in Quantum Mechanics.

Many important steps have been made in the last years towards a solution, thanks to the integral transform method. However, much has still to be done and we summarize the main problems in the following.

• Application of the LIT method to few-body non-perturbative processes (hadron scattering). The LIT method has been applied successfully only to inclusive and exclusive perturbation induced processes. The formalism for applications to non perturbative processes already exists. One can find it briefly discussed in section 2.3 of [13]. It is essentially a reformulation with the Lorentz kernel of the original idea in [14] with the Stieltjes kernel.

• Spectra presenting narrow resonances. From (2.9) and (2.10) it is clear that the smaller the width σ I the more slowly the LIT functions Ψ approach zero at large distances, making it very hard to find accurate transforms [START_REF] Leidemann | [END_REF]. In particular using expansions of | Ψ over basis states one has to make sure that a sufficient number of |φ N ν states have energies N ν in the resonance region Therefore very narrow resonances can be hardly resolved.

• Integral transforms and many-body methods. In section 3 it has been explained that for an extension of the LIT method to a larger number of particles one has to be able to handle very large basis. Investigating the use of the Coupled Cluster method to calculate the LIT is one of the interesting directions [19]. In alternative one could search for other kernels, that allow the calculation of the transform by some other methods. Due to the large use of MC approaches, not only in nuclear physics, but also in many other fields, from condensed matter to QCD, it would be very interesting to explore the applicability of Monte Carlo (MC) techniques to integral transforms with kernels other than the Laplace one. In the following section we suggest possible investigations along this line.

New directions to explore

As discussed in section 2 a good kernel has to be bell-shaped and has to generate a transform that can be calculated. The Laplace kernel leads to transforms of the kind

Φ(τ ) = Q|e [-( Ĥ-E 0 )τ ] |Q . (5.1)
that are usually calculated by MC techniques (evolution in imaginary time). However, the kernel does not have the desired form. One difference between the Laplace and the Lorentz kernel is that the former is a function of one parameter (besides the integration variable), while the latter is a function of two parameters that have a specific function: one allows to regulate the width of the bell, the other, is connected to its maximum and allows to scan the function of interest over its domain. Therefore one should think of other two or more parameter kernels that are such combinations of exponentials to generate bell shaped kernels. One of them could be e.g. This is a bell shaped function that, with suitable choices of the parameters, allows to vary at wish both the width and the centroid. The Heaviside step function Θ ensures both that the exponents remain negative and that K(σ, τ 1 , τ 2 , E) is positive definite. What is to investigate is whether a MC algorithm could be deviced to calculate Φ(σ, τ 1 , τ 2 , ) = Q|Θ( Ĥ -σ) where again τ and σ allow to change width and centroid of the kernel, respectively. The presence of the absolute value in the exponent again requires to deal with operators that contain the Θ operator. This might represent a problem.

Given the large number of physical problems in many different fields, that are connected to Laplace transforms, an effort to investigate the use of kernels like those proposed here, if successful, could open the way to many very interesting developments.

Figure 1 .

 1 Figure 1. (a): The functions r 1 and r 2 , (b): their Laplace transforms, (c) their Lorentz transforms with σ I = 10.

. 10 )

 10 Let us suppose that |Q = |Q ). In this case L(σ) equals to Ψ| Ψ ( Ψ | Ψ ). Since for σ I = 0 the integral in (2.1) does exist, the norm of | Ψ (| Ψ ) is finite. This implies that | Ψ and | Ψ are localized functions.

  [20] K(σ, τ 1 , τ 2 , E) = Θ(E -σ) [ e [-(E-σ) τ 1 ] -e [-(E-σ) τ 2 ] ] τ 2 > τ 1 > 0 . (5.2)

[

  e [-( Ĥ-σ) τ 1 ] -e [-( Ĥ-σ) τ 2 ] evolution as (5.1). Alternatively one can think of another representation of the δ-function, i.e.K(σ, τ, E) = τ e -|(E-σ)| τ(5.4)


P F Bortignon
email: bortignon@mi.infn.it

G Colò
email: colo@mi.infn.it

H Sagawa
email: sagawa@u-aizu.ac.jp

Nuclear single-particle states: dynamical shell model and energy density functional methods

We discuss different approaches to the problem of reproducing the observed features of nuclear single-particle spectra. In particular, we analyze the dominant energy peaks, and the single-particle strength fragmentation, using the example of neutron states in 208 Pb. Our main emphasis is the interpretation of that fragmentation as due to particle-vibration coupling (PVC). We compare with recent Energy Density Functional (EDF) approaches, and try to present a critical perspective.

Introduction

One of the most striking features of nuclei is the validity of the shell model where the nucleons move independently in a smooth single-particle (s.p.) potential. However, selfconsistent mean field (SCMF) approaches [1] are much more sucessful in reproducing the bulk properties that emerge from the mean field picture (like masses, that can nowadays be reproduced by Skyrme-HFB calculations with a remarkable accuracy of about 0.6 MeV [2]) than in accounting for the s.p. states.

If SCMF is viewed as an approximate realization of an exact, yet unknown and perhaps unfeasible, Density Functional Theory (DFT), it is not surprising that quantities like the total energy are well accounted for since they belong to the domain of this theory based on the well-known Kohn-Sham theorem. (We do not enter here the problem if this theorem can be expected to hold in a self-bound system since this discussion can be found elsewhere in the present volume).

If single-particle states are not, strictly speaking, within the DFT framework, two possible routes can be undertaken. Some authors [3,4,5,6] are currently trying to improve the accuracy of present DFT implementations aiming at functionals with socalled "spectroscopic" accuracy (referring to, of course, single-particle spectroscopy). Another point of view, which is the preferred one by the authors of the present contribution, is to generalize the shell model to the so-called "dynamical" shell model Confidential: not for distribution. Submitted to IOP Publishing for peer review 17 February 2010 [7,8,9,[START_REF] Colò | Proceedings of the Conference "Nuclear Structure and Dynamics (NSD09)[END_REF]. It has to be noted that the former attitude does not solve the problem of evaluating the so-called spectroscopic factors or, more generally, the well-known fragmentation of the s.p. strength.

The potential emerging from SCMF calculations is static, but in general nonlocal in space (or, in equivalent words, velocity-or momentum-dependent). The fluctuations of this average potential lead to collective modes, in particular surface vibrations. Taking into account the coupling of these modes to the s.p. motion, the shell model acquires a dynamical content. Thus, the average potential becomes also nonlocal in time, being characterized by an energy (or frequency) dependence.

The velocity dependence of the s.p. potential can be characterized by the so-called k-mass m k , and its frequency dependence by the ω-mass m ω [7]. Most experiments only probe the product of these two quantities, that is, the "total" effective mass m * (see the definitions in Sec. 2). The dynamics of the shell model affects different nuclear properties as: the fragmentation and related spectroscopic factors of the s.p. states, their density (which is proportional to m * near the Fermi energy), the s.p. spreading widths, and the imaginary component of the optical potential. A unified description of the s.p. motion at positive and negative energy eventually emerges.

The importance of the dynamical couplings was first demonstrated in the case of electrons in metals and of normal liquid 3 He. These results inspired the work of Bertsch and Kuo [START_REF] Bertsch | [END_REF] on the enhancement of the effective mass near the Fermi energy in finite nuclei. The review article [7] contains the formalism reported in the next Section as well as the many results obtained in the 80s. The expression "dynamics of the shell model" is after G.E. Brown [12].

The old calculations of the 80s are mostly not consistent. It is hard to extract quantitative conclusions because of the many approximations involved. More consistent calculations have been recently done. They can be confronted with modern EDF-based approaches. We shall do it in the simple case of 208 Pb, showing that the problem of the s.p. states is still an open one.

Formalism of the dynamical shell model

One can introduce using standard many-body techniques a "mass operator" M, defined by (see, e.g., Ref. [7])

ω + (2m) -1 ∇ 2 r G( r, r ; ω) = δ( r -r ) + d 3 r M( r, r ; ω)G( r, r ; ω) (1)
in terms of the one-body Green function (Lehmann representation)

G( r, r ; ω) = h φ h ( r)φ * h ( r ) ω -ω h -iη + p φ p ( r)φ * p ( r ) ω -ω p + iη . (2)
Here

φ h ( r) = ψ (A-1) h |a( r)|ψ (A) 0 , φ p ( r) = ψ (A) 0 |a( r)|ψ (A+1) p . (3)
ψ (A-1) h
is a suitably normalized eigenstate of the Hamiltonian of the (A-1)-particle system with energy

E (A-1) h , ψ (A+1) p
is a suitably normalized eigenstate of the Hamiltonian of the (A+1)-particle system with energy E (A+1) p , and

ω p = E (A+1) p -E (A) 0 ω h = E (A) 0 -E (A-1) h . (4)
Eqs. ( 1) and (2) give

ω + (2m) -1 ∇ 2 φ p ( r) -d 3 r M( r, r ; ω p )φ p ( r ) = 0. ( 5
)
This s.p. wave equation has two important properties: i) It has discrete eigenvalues corresponding to the energies of the bound states of the (A+1) and (A-1) systems, ii)

In the continuum gives the exact diagonal (elastic scattering) element of the S matrix [13].

The mass operator, in what follows, will contain the Hartree-Fock (HF) mean field and the contribution of the coupling of the s.p. HF states to the vibrational modes (particle-vibration coupling or PVC), in particular to collective modes obtained from self-consistent linear response theory (Random Phase Approximation or RPA).

Although one could solve (5) directly, the usual way has been so far to solve it at the HF level and add the contribution of the particle-vibration part of the mass operator within second-order perturbation theory. The expressions for this second-order correction can be found, e.g., in Refs. [7,8].

To work out the expressions for m k , m ω and m * , it is easier to consider a uniform system [7]. Thus, having the relation

E = k 2 2m + V(k; E) ( 6 )
between the energy E and the momentum k of the quasiparticle (dressed s.p.), and V being the real part of the complex mass operator, the effective mass is defined

dE dk = k m * . (7)
Then, it is easy to write

m k m = 1 + m k ∂V(k; ω) ∂k -1 ω=E(k) , m ω m = 1 - ∂ ∂ω V(k; ω) ω=E(k) , (8)
to obtain

m * m = m k m • m ω m . (9)

The contradicting results

Particle-vibration coupling: old and new results

In Tables 4.3a and 4.3b of Ref. [7], an extensive review of the results obtained for m ω in 208 Pb by nine groups in the period 1968-1983 can be found. Rather different frameworks had been adopted, no one being fully self-consistent: s.p. potentials range from harmonic oscillator (HO) to Woods-Saxon (WS) or HF with Skyrme forces; residual interactions at the particle-vibration vertex are either multipole-multipole forces, or forces of Landau-Migdal type, or Skyrme forces but with velocity-dependent terms dropped, or even G-matrix interactions.

The average value for m ω is in the range 1.2-1.4 for the neutron valence shells, and in the range 1.3-1.6 for the proton case. The inverse of m ω gives the spectroscopic factor reported in table 4.4 of Ref. [7]. In general, it has been impossible to find an accurate agreement between theory and experiment, either at the level of main energy centroids of the s.p. strength distribution, or at the level of spectroscopic factors. The problem of spectroscopic factors had been not considered too seriously, in view of the large experimental errors and ambiguities associated with them. Nowadays, new analysis are being carried out [14], but the theoretical ambiguities associated with the extraction of spectroscopic factors are still under discussion (see e.g. the contribution by R. J. Furnstahl and A. Schwenk in the present volume).

If we concentrate on s.p. energies, the most consistent among the calculations reported in [7], namely that by V. Bernard and N. Van Giai [8], still underestimates the density of levels. The most recent calculations for 208 Pb in the framework of dynamical shell model, or PVC model, have been reported in Refs. [9] and [START_REF] Colò | Proceedings of the Conference "Nuclear Structure and Dynamics (NSD09)[END_REF]. The results are displayed in Tables 1 and2 and compared with the results of Ref. [3] (cf. below) and with the experimental energies, in the case of the neutron valence particle and hole states. In Table 1 we display, for Refs. [9] and [START_REF] Colò | Proceedings of the Conference "Nuclear Structure and Dynamics (NSD09)[END_REF], the differences between the s.p. energies ε i obtained with PVC included, and the mean field energies ε (0) i . These differences ∆ε i will be called "shifts" in what follows. (See below for a of the shifts from Ref. [3]).

In the covariant theory of Ref. [9], the mean field is associated with the NL3 [15] effective Langrangian, which is also employed to obtain the phonons within the relativistic RPA (RRPA). The Dyson equation (2) for the s.p. Green's functions is solved in the diagonal approximation for the mass operator M. A rather simple particlevibration model is used, in which the coupling vertex is obtained with the matrix element of the residual interaction times the transition density of the vibrations.

The calculations of Ref. [START_REF] Colò | Proceedings of the Conference "Nuclear Structure and Dynamics (NSD09)[END_REF] are performed in the framework of Skyrme HF, with the PVC contributions to the mass operator M calculated within second-order perturbation theory. The phonons are obtained through fully self-consistent RPA, and all the terms of the particle-hole (p-h) interaction are kept also in the PVC vertex. There are strong cancellations due to the velocity dependent terms that lead to small shifts, especially for the hole states. The calculations have been done using the SLy5 Skyrme set [16]. Calculations with other Skyrme functionals are in progress. In Ref. [3], the HF problem is solved for the even-even and even-odd (A±1) systems using the SLy4 L interaction. The "dressed" energies ε i are obtained from an equation like the present Eq. ( 4). In this case E 0 is the total energy of the even-even system and E p (E h ) is the total energy of the odd system when a given occupancy (vacancy) is imposed. The "bare" energies are obtained by excluding the so-called mass, shape and spin polarization effects: in practice this corresponds, respectively, to neglecting the proper change of the center-of-mass correction, to imposing spherical symmetry, and to discarding time-odd terms.

One can see from Table 1 that shifts obtained by RMF plus PVC and Skyrme plus PVC are different, and it is hard to connect either of them to the calculations of [3]. If we look now at the final "dressed" energies obtained in the different approaches (Table 2), and compare with experimental values, we conclude that the open problem is that even for the "benchmark" nucleus 208 Pb we do not dispose of a consistent calculation which reproduces well the experimental data. This is also true for spectroscopic factors.

In the calculations of the spectroscopic factors, the effects of the short-range correlations (SRC) should also be included. On the one side, in a recent work [17], it is confirmed that the PVC involving long-range correlations (LRC) can be identified as the main mechanism, the SRC explaning only a small fraction (up to about 10%) of the deviations of the spectroscopic factors from the independent s.p. values. Consistently, from a qualitative point of view, in Ref. [7] it was concluded that the main contribution of highly excited core polarized states to m ω , and therefore to the spectroscopic factors, can be considered as a constant in the vicinity of the Fermi energy. The subtracted dispersion relation technique, with different choices for the behaviour of the imaginary part of the mass operator at high energy, was used.

EDF extensions

The lack of an accurate description of s.p. states is also clear for EDF practitioners not only in keeping with the findings of Ref. [3]. The authors of [4] have concluded that standard forms of the Skyrme energy density functional do not allow that accurate description. The effect of tensor forces has been studied very carefully by the authors of [18,19], but their conclusion is that the tensor terms, albeit important, cannot remedy the deficiencies of the central terms: in practice, they cannot produce a functional which is more accurate as far as s.p. states are concerned.

In Ref. [5], gradient corrections up to next-to-next-to-next-to-leading order (N 3 LO, in practice derivatives up to sixth power) have been added to a local EDF, in order to better reproduce surface effects, for example on the effective mass m ω (cf. also [20]). The precision on s.p. states is improving [6].

Conclusions

We still lack, despite the tremendous progress in many aspects of nuclear structure physics, an accurate description of s.p. strength distributions. While new radioactive beam facilities have already started to show how this kind of observables evolve with the decrease of the neutron or proton separation energies [21,22], even for standard nuclei we have troubles to get accuracy for dominant s.p. energies and spectroscopic factors (using models that have had big success in reproducing other kind of observables).

The addition of higher power gradient corrections to a local EDF may improve the s.p. energies. We deem, however, that a dynamical theory is necessary to describe quantities like the spectroscopic factors. The merging of the two approaches defines anyway, to some extent, the future of this fundamental field of nuclear structure research. In a many-body approach like the dynamical shell model one believes that the basic object should be a time-or energy-dependent quantity like the s.p. Green's function.

EDF practitioners aiming at functionals with spectroscopic accuracy believe that a static approximation, in which the Green's function reduces to a static density, may be a sound approximation. This issue asks for a clear answer based on comparision of results without uncontrolled approximations.

Table 1 .

 1 Shifts (that, is differences between dressed and bare single-particle energies) associated with the dominant s.p. neutron components in 208 Pb. See the text for a discussion of how they have been obtained.

		∆ε = ε i -ε	(0) i
		[9]	[10]	[3]
	3d 3/2 -0.61 -0.32 0.09
	4s 1/2 -0.56 -0.21 0.24
	3d 5/2 -0.76 -0.43 0.07
	1j 15/2 -1.36 -0.55
	2g 9/2 -0.79 -0.40 0.03
	1i 11/2 -0.31 -0.37 -0.14
	3p 1/2 -0.02 0.01 0.05
	2f 5/2	0.33 0.01 0.12
	3p 3/2	0.14 0.03 0.06
	1i 13/2	0.49 0.05 0.22
	1h 9/2	1.40 0.06
	2f 7/2	1.40 0.63 0.12

Table 2 .

 2 Dressed single-particle energies and spectroscopic factors S associated with the dominant s.p. neutron components in 208 Pb, compared with the experimental findings.

			ε th		ε exp	S th	S exp
		[9]	[10]	[3]		[9] [10]
	3d 3/2	-0.63	0.01	0.78	-1.40 0.89 0.94 0.88±0.1
	4s 1/2	-0.92	-0.31	0.80	-1.90 0.92 0.95 0.88±0.1
	3d 5/2	-1.39	-1.08	-0.43	-2.37 0.88 0.91 0.88±0.1
	1j 15/2	-1.84	-1.75		-2.51 0.66 0.86 0.53±0.1
	2g 9/2	-3.29	-3.62	-3.16	-3.94 0.84 0.93 0.78±0.1
	1i 11/2	-3.28	-1.38	-1.67	-3.16 0.88 0.94 0.96±0.2
	3p 1/2	-7.68	-8.04	-8.06	-7.37 0.90 0.93 1.07
	2f 5/2	-8.77	-8.94	-8.91	-7.94 0.86 0.94 1.13
	3p 3/2	-8.27	-9.16	-9.17	-8.27 0.86 0.91 1.00
	1i 13/2	-9.11 -10.13	-9.30	-9.00 0.81 0.94 1.04
	1h 9/2 -11.96 -12.01		-10.78 0.36 0.94 1.10
	2f 7/2	-9.71 -11.44 -11.90	-9.71 0.64 0.75 0.88

Acknowledgments

This work is partially supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology by Grant-in-Aid for Scientific Research under the program number (C(2)) 20540277.