E2 strengths and transition radii difference of onephonon states of Zr from electron scattering at low momentum transfer
Abstract
 Background

Mixedsymmetry states in vibrational nuclei are characterized by a sign change between dominant proton and neutron valenceshell components with respect to the fully symmetric state. The sign can be measured by a decomposition of proton and neutron transition radii with a combination of inelastic electron and hadron scattering [C. Walz et al., Phys. Rev. Lett. 106, 062501 (2011)]. For the case of Zr, a difference could be experimentally established for the neutron components, while about equal proton transition radii were indicated by the data.
 Purpose

Determination of the groundstate (g.s.) transition strength of the mixedsymmetry state and verification of the expected vanishing of the proton transition radii difference between the onephonon states in Zr.
 Method

Differential cross sections for the excitation of onephonon and states in Zr have been measured with the () reaction at the SDALINAC in a momentum transfer range fm.
 Results

Transition strengths , and Weisskopf units are determined from a comparison of the experimental cross sections to quasiparticlephonon model (QPM) calculations. It is shown that a modelindependent plane wave Born approximation (PWBA) analysis can fix the ratio of transition strengths to the states with a precision of about 1%. The method furthermore allows to extract their proton transition radii difference. With the present data fm is obtained.
 Conclusions

Electron scattering at low momentum transfers can provide information on transition radii differences of onephonon states even in heavy nuclei. Proton transition radii for the states in Zr are found to be identical within uncertainties. The g.s. transition probability for the mixedsymmetry state can be determined with high precision limited only by the available experimental information on the ) value.
pacs:
21.10.Re, 23.20.Js, 25.30.Dh, 27.60.+jI Introduction
Collectivity, isospin symmetry and shell structure are generic features of the nuclear manybody quantum system. Collective nuclear valenceshell excitations are a key to understand how these features coexist, interplay and compete. In vibrational nuclei, the development of predominantly protonneutron symmetric collective nuclear structures at low excitation energies is governed by the strong residual protonneutron interaction. Their existence implies  due to quantummechanical orthogonality  the formation of collective states with, at least partial antisymmetry with respect to the contribution of proton and neutron valencespace components to their wave functions. Such excited states are said to have mixed symmetry (1). The investigation of mixedsymmetry states (MSS) is an important source of information on the effective protonneutron interaction in the valence shell of heavy atomic nuclei (2).
MSS have been defined in the framework of the protonneutron Interacting Boson Model (IBM2) (3). In analogy to the isospin symmetry of nucleons, the symmetry of a multiboson wave function formed by proton bosons and neutron bosons is quantified by the socalled spin. States with have wave functions that contain at least one pair of proton and neutron bosons antisymmetric under the exchange of proton and neutron labels. The signatures of MSS are () strong transitions to fully symmetric states (FSS) with matrix elements of about 1 and () weakly collective transitions to FSS.
The prediction of the IBM2 with respect to a multiphonon structure of MSS in vibrational nuclei was confirmed about ten years ago by the observation of large transition strengths between lowenergy states of Mo (4); (5); (6). The states were also investigated with electron scattering experiments at the superconducting electron accelerator SDALINAC and with proton scattering at iThemba LABS (7). The combined analysis supported a onephonon structure of the states of Mo.
In the neighboring eveneven isotone Zr with two neutrons outside the closed shell and with the subshell closure, a stronger configurational isospin polarization of the onephonon states than in Mo is expected (8); (9). Recent work showed that the collectivity of the lowlying symmetric and mixedsymmetric quadrupole excitation in vibrational nuclei originates from the coupling of the giant quadrupole resonance to the dominant valencespace configurations (10). Experimental evidence in Zr and Mo stems from the observation of a difference of the respective mattertransition radii (deduced from proton scattering) while chargetransition radii (deduced from electron scattering) were found to be about equal. The difference results from a sign change of the dominant valence neutron amplitude in MSS with respect to the FSS.
The present work provides an indepth study of the electron scattering results on Zr. In particular, we discuss a new method for a modelindependent determination of the ratio of the transition strengths of fully symmetric and mixedsymmetric onephonon excitations in heavy vibrational nuclei, which at the same time provides an estimate of the sensitivity to the transitionradius difference between these two states. The results are furthermore interpreted in the framework of the QPM (for an introduction to the model see (11)).
Ii Experiment
The experiment has been carried out at the Darmstadt superconducting electron linear accelerator SDALINAC. The LINTOTT spectrometer was used with a focalplane detector system based on four singlesided silicon detectors, each providing 96 strips with thickness of 500 m and a pitch of 650 m (12). Electrons with an incident beam energy MeV and beam currents ranging from 0.5 to 1 A impinged on a Zr target with an isotopic enrichment of 94.57% and an areal thickness of 9.75 mg/cm. Data were taken at five different scattering angles , , , , and covering roughly the first maximum of an form factor.
Examples of electronscattering spectra are shown in Fig. 1. The energy resolution was about 55 keV (full width at half maximum, FWHM). The observed peaks correspond to the elastic line, the collective onephonon ( MeV) and ( MeV) states, and the onephonon MSS (, MeV).
Peak areas of the transitions were obtained from a spectrum decomposition using the line shape described in Ref. (13). Absolute differential cross sections were determined from normalization to the elastic scattering peak. Theoretical elastic scattering cross sections were obtained from calculations with the code PHASHI (14) using charge density distributions from Ref. (15). The resulting inelastic cross sections with statistical uncertainties are given in Tab. 1 normalized to the Mott cross section. The overall systematic uncertainties of the normalization due to the model description of the charge density and experimental kinematic parameters (electron energy, scattering angle) were estimated to 5%, which were added in quadrature.
(fm)  

0.36  3.98(11)  2.02(7)  1.95(30)  1.0148 
0.41  5.19(4)  2.67(3)  3.30(13)  1.0146 
0.46  5.39(10)  2.95(8)  4.32(10)  1.0146 
0.55  7.94(21)  4.23(17)  10.5(5)  1.0143 
0.64  5.2(5)  4.4(4)    1.0143 
Iii DWBA Analysis
Figure 2 presents the results of Tab. 1 in comparison with QPM calculations as a function of momentum transfer
(1) 
In the present application (details are given in Ref. (10)), excited states in Zr are described by wave functions including one, two, and threephonon configurations. Note that the results are very similar to an earlier QPM study of the MSS in Zr (16). Theoretical () cross sections have been calculated within the Distorted Wave Born Approximation (DWBA) to account for Coulomb distortion effects. They provide a satisfactory description of the dependence. In order to extract the reduced transition probabilities, the calculations have been scaled to the data and extrapolated to the photon point, . The results are given in Tab. 2 labelled ”DWBA”. The quoted errors are those of the leastsquare fit to the data. Possible systematic uncertainties due to the extrapolation to the photon point are expected to be negligible. The absolute and strengths agree well within error bars with a previous experiment (17). This is also true for the transition strength but the present value is significantly more precise. Previous measurements based on lowenergy proton scattering show a large spread of results (14.7, 16.2, 18.9, 21.3, 23.6 W.u.) (18), most likely due to the model dependence of the extraction of an electromagnetic transition matrix element from hadronic scattering data.
Present work  Literature  

DWBA  PWBA  
)  6.18(23)  6.4(5)  
)  3.31(10)  3.32(27)  3.4(4) 
)  18.4(11)  19(6) 
Iv PWBA Analysis
The transition strengths derived from the data depend of course on the applied nuclear structure model. In light nuclei it has been shown that transition strengths can be extracted in a nearly modelindependent planewave Born approximation (PWBA) analysis (19). It assumes that Coulomb distortions of the electron wave function can be approximated by an overall correction factor determined from the g.s. charge distribution of the nucleus. For kinematics where transverse contributions can be neglected, the differential cross sections for electric transitions are then given by
(2) 
where , , and . The symbol denotes the fine structure constant, is the transition multipolarity, and is a kinematic function given e.g. in Ref. (19). The quantities B and the reduced transition probabilities B from realphoton experiments can be related by Siegert’s theorem (20)
The Coulomb correction factor
is determined from the ratio of DWBA and PWBA calculations employing the QPM transition densities.
The reduced transition probabilities can thus be related to the experimental differential cross section by
(3)  
For low momentum transfers, can be expanded in a power series of the momentum transfer
(4) 
Here, is defined by
(5) 
with desribing the transition density of multipolarity .
In general, the PWBA approximation is not valid for a heavy nucleus like Zr. However, it may hold for the ratio of cross sections populating the FSS and MSS for the following reasons: (i) the kinematics for both transitions are almost identical, and (ii) transition densities of collective transitions of a given multipolarity are similar (see, e.g., Fig. 2 in Ref. (10) for the cases studied here). Figure 3 shows the Coulombcorrection factors calculated with the QPM for the transitions to (middle) and (top) states in Zr together with their ratio (bottom) as a function of .
The ratio is unity to better than 1% over the range of the momentum transfer included in our experiments. Consequently, the effects from Coulomb distortion can be neglected in a relative analysis, and the extraction of the ratio can be achieved with improved accuracy, since systematic errors in the determination of absolute cross sections cancel.
Employing Eqs. (3) and (4) and defining the transition radius , the ratio of reduced transition strengths can be approximated by
(6) 
where the indices indicate the transitions to the and state, respectively. denotes the ratio of kinematic functions , given in Tab. 1, and the difference of the corresponding chargetransition radii. The experimental ratio depends on the square root of the ratio of the peak areas only.
For the approximation on the r.h.s. of Eq. (6) use is made of Siegert’s theorem and of the Tassie model (21) which provides a good description of the surface behaviour of transition densities for collective states. We have checked that the approximation employed in Eq. (6) within this model, yields very accurate results, and thus it is used in our analysis below. This approximation may be questionable for the results at the highest in Tab. 1 which also have poor statistics. The data point is thus omitted in the further analysis.
Figure 4 shows a plot of as a function of the squared elastic momentum transfer. A fit of Eq. (6) to the data has 3 parameters, viz. the ratio of strengths, and . In a first step , fm is fixed using the QPM results. A minimization of Eq.(6) to the data then determines
With the value from Tab. 2, we obtain W.u., in agreement with Ref. (17) and with the value obtained above from the DWBA analysis. While the ratio can be determined precisely with an uncertainty of about 1%, the accuracy of the absolute value is presently limited by the error of the value in the literature.
The second parameter in Eq. (6) provides information about the change of the proton transition radii between both states. This is particular interesting in view of the recent results of Walz et al. (10) providing evidence for a significant difference of the neutron transition radii for these two states, while their proton transition radii are expected to be very close based on QPM calculations. This may serve as a new experimental signature of MSS in vibrational nuclei with a specific shell structure. An experimental confirmation of this conjecture is of considerable interest. The fit of Eq. (6) leads to fm, where the uncertainty is dominated by the limited number of data points with small enough error bars at sufficiently low momentum transfers. One way to improve the fit is the inclusion of the results of Ref. (17) providing an additional data point at . The resulting fit (red curves) with error bars (green curves) shown in Fig. 4 corresponds to fm consistent with equal proton transition radii to about half a fm.
Finally, we briefly comment on a possible dependence of the result on a variation of the absolute size of the charge transition radius . We have repeated the analysis for 9 different values of between 4 and 7 fm, thereby overexhausting the range of possible values expected from model calculations and from the phenomenological finding that the transition radii of collective excitations differ not too much from the radius of the nuclear ground state (19). As demonstrated in Fig. 5, the deduced difference of the charge transition radii is independent of the choice of .
V Summary
To summarize, an investigation of the nature of onephonon symmetric and mixedsymmetric states in Zr has been performed using inelastic electron scattering at low momentum transfers. A comparison of the measured form factors with QPM calculations confirms the dominant onephonon structure of the transitions to the and states. It is shown that a PWBA analysis of the form factors, which usually fails for heavy nuclei, can nevertheless be applied to extract the ratio of the g.s. ) transition strengths in a relative analysis. This is a new promising approach to determine the g.s. transition strength of the MSS in vibrational nuclei with a precision limited only by the experimental information about the strength. The PWBA approach furthermore provides information about differences of the proton transition radii of the respective states, containing independent information about the mixedsymmetry character of states and the sign change of leading valence shell components between FSS and MSS (10). For Zr, the proton transition radii agree within about 0.5 fm, consistent with predictions that the sign change arises in this case from the neutron component.
Further analysis of the data indicates that an improved precision for the proton transition radii difference can be achieved by additional data, in particular in momentum transfer ranges presently not covered well (e.g., fm). Elementary to the present approach is an approximate cancellation of Coulomb corrections of the FSS and MSS. This may be questioned when moving away from shell closures, where the collectivity of the MSS groundstate decay decreases. Systematic investigations are necessary to establish the range of applicability of this new promising method. Work along these lines (e.g. studies of Zr and Mo isotopes) is underway.
We thank R. Eichhorn and the SDALINAC crew for their commitment in delivering electron beams. This work has been supported by the DFG under grant SFB 634.
References
 A. Arima, T. Otsuka, F. Iachello, and I. Talmi, Phys. Lett. B 66, 205 (1977).
 N. Pietralla, P. von Brentano, and A. F. Lisetskiy, Prog. Part. Nucl. Phys. 60, 225 (2008).
 F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987).
 N. Pietralla et al., Phys. Rev. Lett. 83, 1303 (1999).
 N. Pietralla, C. Fransen, P. von Brentano, A. Dewald, A. Fitzler, C. Friessner, and J. Gableske, Phys. Rev. Lett. 84, 3775 (2000).
 C. Fransen et al., Phys. Rev. C 67 024307 (2003).
 O. Burda et al., Phys. Rev. Lett. 99, 092503 (2007).
 V. Werner et al., Phys. Lett. B 550, 140 (2002).
 J. D. Holt, N. Pietralla, J. W. Holt, T. T. S. Kuo, and G. Rainovski, Phys. Rev. C 76, 034325 (2007).
 C. Walz, H. Fujita, A. Krugmann, P. von NeumannCosel, N. Pietralla, V. Yu. Ponomarev, A. Scheikh Obeid, and J. Wambach, Phys. Rev. Lett. 106, 062501 (2011).
 V. G. Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons (Institute of Physics Publishing, Bristol, 1992).
 A. W. Lenhardt, U. Bonnes, O. Burda, P. von NeumannCosel, M. Platz, A. Richter, and S. Watzlawik, Nucl. Instrum. Methods in Phys. Research A 562, 320 (2006).
 F. Hofmann, P. von NeumannCosel, F. Neumeyer, C. Rangacharyulu, B. Reitz, A. Richter, G. Schrieder, D. I. Sober, L. W. Fagg, and B. A. Brown, Phys. Rev. C 65, 024311 (2002).
 C. Bähr, code PHASHI, Technische Universität Darmstadt, unpublished.
 H. De Vries, C. W. De Jager, and C. De Vries, At. Data Nucl. Data Tables 36, 495 (1987).
 N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 69, 044312 (2004); ibid C 73, 037305 (2006).
 C. Fransen, V. Werner, D. Bandyopadhyay, N. Boukharouba, S. R. Lesher, M. T. McEllistrem, J. Jolie, N. Pietralla, P. von Brentano, and S. W.Yates, Phys. Rev. C 71 054304 (2005).
 T. Kibedi and R. H. Spear, At. Data Nucl. Data Tables 80, 35 (2002).
 H. Theissen, Springer Tracts in Mod. Phys. 65, 1 (1972).
 A. J. F. Siegert, Phys. Rev. 52, 787 (1937).
 L. J. Tassie, Aust. J. Phys. 9, 407 (1956).