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due to its multifaceted roles in nuclear physics and astrophysics  @xcite as well as new physics beyond the standard model  @xcite , the symmetry energy has become a hot topic in current research frontiers of nuclear physics and astrophysics  @xcite . during the last decade , a lot of experimental , observational and theoretical efforts have been devoted to constraining the magnitude @xmath7 and density slope @xmath8 of the symmetry energy at nuclear saturation density @xmath9 ( @xmath10 @xmath5 ) , i.e. , @xmath11 and @xmath12 . although important progress has been made , large uncertainties on the values of @xmath11 and @xmath12 still exist ( see , e.g. , refs . for instance , while the @xmath11 is determined to be around @xmath13 mev , the extracted @xmath12 varies significantly from about @xmath14 to @xmath15 mev , depending on the observables and analysis methods . to better understand the model dependence and narrow the uncertainties of the constraints is thus of extreme importance .    while many studies on heavy ion collisions and neutron stars have significantly improved our knowledge on the symmetry energy , more and more constraints on the symmetry energy have been obtained in recent years from analyzing the properties of finite nuclei , such as the nuclear binding energy  @xcite , the neutron skin thickness  @xcite , and the resonances and excitations  @xcite . furthermore , it has been realized that the properties of finite nuclei usually provide more precise constraints on @xmath7 and @xmath8 at subsaturation densities rather than at saturation density @xmath9 . this feature is understandable since the characteristic ( average ) density of finite nuclei is less than @xmath9 . for example , the average density of heavy nuclei ( e.g. , @xmath1pb ) is about @xmath16 @xmath5 , and thus the properties of heavy nuclei most effectively probe the properties of nuclear matter around @xmath16 @xmath5  @xcite . indeed , a quite accurate constraint on the symmetry energy at the subsaturation cross density @xmath4 @xmath5 , i.e. , @xmath17mev , has been recently obtained from analyzing the binding energy difference of heavy isotope pairs  @xcite . in contrast to the fact that many and precise constraints on the magnitude of @xmath18 around @xmath19 have been obtained , to the best of our knowledge , so far there is only one experimental constraint on the density slope @xmath3 which was obtained from analyzing the neutron skin data of sn isotopes  @xcite . knowledge on @xmath3 is not only important for understanding the density dependence of the symmetry energy itself , but also plays a central role in determining the neutron skin thickness of heavy nuclei and the core - crust transition density in neutron stars . therefore , any new constraints on @xmath3 will be extremely useful .    in the present work , with the precise knowledge of @xmath2 , we demonstrate that the electric dipole polarizability @xmath20 in @xmath1pb measured at the research center for nuclear physics ( rcnp ) via polarized proton inelastic scattering at forward angles , can put a strong limit on the @xmath3 . we emphasize since at forward angles coulomb excitation dominates , the extracted @xmath20 at rcnp is expected to be a relatively clean isovector indicator with less uncertainties from strong interaction . the equation of state ( eos ) of asymmetric nuclear matter , given by its binding energy per nucleon , can be written as @xmath21where @xmath22 is the baryon density , @xmath23 is isospin asymmetry , @xmath24 is the eos of symmetric nuclear matter , and the symmetry energy is expressed as @xmath25around a reference density @xmath26 , the @xmath27 can be expanded in @xmath28 as @xmath29 where @xmath30 is the density slope parameter which characterizes the density dependence of the symmetry energy around @xmath26 . our calculations in the present work are based on the skyrme - hartree - fock ( shf ) approach with the so - called standard skyrme force ( see , e.g. , ref . @xcite ) which includes @xmath31 parameters , i.e. , the @xmath32 skyrme force parameters @xmath33 , @xmath34 , @xmath35 , and the spin - orbit coupling constant @xmath36 . instead of directly using the @xmath32 skyrme force parameters , we can express them explicitly in terms of @xmath32 macroscopic quantities , i.e. , @xmath37 , @xmath38 , the incompressibility @xmath39 , the isoscalar effective mass @xmath40 , the isovector effective mass @xmath41 , @xmath42 , @xmath43 , the gradient coefficient @xmath44 , and the symmetry - gradient coefficient @xmath45 . in this case , we can examine the correlation of properties of finite nuclei with each individual macroscopic quantity by varying individually these macroscopic quantities within their empirical ranges . recently , this correlation analysis method has been successfully applied to study nuclear matter properties from analyzing nuclear structure observables  @xcite , and will also be used in this work . the random - phase approximation ( rpa ) provides an important microscopic approach to calculate the electric dipole polarizability in finite nuclei . within the framework of rpa theory , for a given excitation operator @xmath46 , the reduced transition probability from rpa ground state @xmath47 to rpa excitation state @xmath48 is given by : @xmath49 where @xmath50 denotes the unoccupied ( occupied ) single nucleon state ; @xmath51 is the reduced matrix element of @xmath46 ; and @xmath52 and @xmath53 are the rpa amplitudes . the strength function then can be calculated as : @xmath54 where @xmath55 is the energy of rpa excitation state @xmath56 . thus the moments of strength function can be obtained as : @xmath57 in the case of electric dipole ( @xmath58 ) response , the excitation operator is defined as : @xmath59 where @xmath60 , @xmath61 and @xmath62 are proton , neutron and mass number , respectively ; @xmath63 is the nucleon s radial coordinate ; @xmath64 is the corresponding spherical harmonic function . for a given skyrme interaction , we can calculate the inverse energy - weighted moment @xmath65 using the hf - rpa method , and then obtain the electric dipole polarizability @xmath66 as @xmath67        for the theoretical calculations of electric dipole polarizability in @xmath1pb in the present work , we employ the skyrme - rpa program by col@xmath68 _ @xcite . in this program , the shf mean field and the rpa excitations are fully self - consistent . in particular , we calculate the isovector dipole strength in @xmath1pb with a spherical box extending up to @xmath69 fm , a radial mesh of @xmath70 fm and a cutoff energy of @xmath71 mev which denotes the maximum energy of the unoccupied single - particle states in the rpa model space . then the inverse energy - weighted moment @xmath65 is evaluated with an upper integration limit of @xmath72 mev according to the experimental energy range  @xcite , and thus the electric dipole polarizability @xmath20 can be calculated invoking eq .  . the electric dipole polarizability @xmath20 has been shown to be a sensitive probe of the symmetry energy  @xcite . in particular , based on the droplet model , roca - maza _ et al . _ @xcite obtained the following relation : @xmath73,\ ] ] where @xmath74 is the mean - square radius and @xmath75 is the symmetry energy coefficient of a finite nucleus of mass number @xmath62 . furthermore , using the empirical relation @xmath76  @xcite and expanding @xmath77 as @xmath78 roca maza _ et al . _ demonstrated that @xmath79 is correlated with both @xmath80 and @xmath12 . particularly , based on a large and representative set of relativistic and nonrelativistic nuclear mean - field models , they found a strong linear correlation between @xmath81 and @xmath12 and then extracted the constraint @xmath82 mev from the combination of the experimental determination of @xmath79 with the empirical estimate of @xmath83 mev . one can see that the uncertainty of the estimated @xmath80 leads to a large error of @xmath84 mev for @xmath12 .    instead of expressing @xmath77 in terms of @xmath80 and @xmath12 as in eq . ( [ esymra ] ) , one can also express @xmath80 in terms of @xmath85 and @xmath86 as @xmath87 noting @xmath88  @xcite , one can then see from eqs . ( [ alphaddm ] ) and ( [ esymr0rc ] ) that @xmath79 in @xmath1pb is also correlated with both @xmath86 and @xmath85 . as we will see in the following , the microscopic rpa calculations indeed show that @xmath89 is sensitive to @xmath80 and @xmath12 as well as to @xmath86 and @xmath85 . since @xmath85 has been stringently constrained recently ( see , e.g. , @xmath90 mev in ref.@xcite ) , the @xmath79 in @xmath1pb can thus be used to constrain the @xmath86 parameter . to examine the correlation of the @xmath20 in @xmath1pb with each macroscopic quantity , especially on @xmath91 and @xmath92 , we show in fig . [ alphad ] the @xmath20 in @xmath1pb from shf with the skyrme force msl0  @xcite by varying individually @xmath92 , @xmath45 , @xmath44 , @xmath93 , @xmath94 , @xmath39 , @xmath40 , @xmath41 , @xmath37 , and @xmath36 within their empirical uncertain ranges , namely , varying one quantity at a time while keeping all others at their default values in msl0 , for @xmath95 and @xmath96 @xmath5 , respectively . it is seen from fig . [ alphad ] that , as eq . ( [ alphaddm ] ) suggests , the @xmath66 in @xmath1pb exhibits strong correlations with both @xmath97 and @xmath98 , while much weaker correlation with other macroscopic quantities . particularly , the @xmath20 decreases sensitively with @xmath99 while increases rapidly with @xmath100 , implying a fixed value of @xmath20 will lead to a strong positive correlation between @xmath99 and @xmath100 . the results for @xmath101 @xmath5 just confirm the correlations of @xmath79 with @xmath80 and @xmath12 reported in ref.@xcite . for @xmath95 @xmath5 , given that the symmetry energy at @xmath102 @xmath5 has been well constrained as @xmath103 mev , one thus expects the @xmath20 in @xmath1pb can constrain stringently the parameter @xmath104 .    fixing the values of other @xmath105 macroscopic quantities , i.e. , @xmath45 , @xmath44 , @xmath93 , @xmath39 , @xmath40 , @xmath41 , @xmath37 and @xmath36 at their default values in msl0 , we illustrate in fig . [ alphadlc ] by open up - triangles ( down - triangles ) the @xmath20 in @xmath1pb as a function of @xmath3 for @xmath106 mev . as expected , it is seen from fig . [ alphadlc ] that the @xmath20 in @xmath1pb increases ( decreases ) with @xmath104 ( @xmath2 ) for a fixed @xmath107 ( @xmath104 ) . by comparing with the experimental data @xmath108 @xmath109 , one can extract a strong constraint of @xmath110 mev .     in @xmath1pb as a function of @xmath3 for fixed @xmath2 . the open ( solid ) up- and down - triangles represent the results with @xmath111 and @xmath112 mev , respectively , from shf - rpa calculations with the values of other parameters fixed in msl0 ( obtained in optimization ) . the band indicates the experimental value of @xmath113 @xmath109 from rcnp  @xcite . ] the above constraint of @xmath110 mev has been obtained by neglecting the weak correlations between the @xmath20 in @xmath1pb and other @xmath105 macroscopic quantities . to test the robustness of this constraint and to obtain a more precise constraint , for fixed @xmath2 and @xmath3 , we optimize all other @xmath105 parameters instead of simply fixing them at their default values in msl0 , by minimizing the weighted sum of @xmath114 evaluated from the difference between shf prediction and the experimental data for some selected observables using the simulated annealing technique  @xcite . in particular , in the optimization , we chose the following experimental data of spherical even - even nuclei , i.e. , ( i ) the binding energy @xmath115 of @xmath116o,@xmath117ca , @xmath118ni , @xmath119sr , @xmath120zr , @xmath121sn , @xmath122sm , @xmath1pb  @xcite ; ( ii ) the charge rms radii @xmath123 of @xmath116o , @xmath117ca , @xmath124ni , @xmath119sr , @xmath120zr , @xmath125sn , @xmath122sm , @xmath1pb  @xcite ; ( iii ) the breathing mode energy @xmath126 of @xmath120zr,@xmath127sn,@xmath122sm and @xmath1pb  @xcite . in the calculation of the breathing mode energy @xmath128 , we evaluate the inverse energy - weighted sum rule @xmath65 with the constrained hartree - fock ( chf ) method and obtain the energy - weighted sum rule @xmath129 using the double commutator sum rule  @xcite . in addition , in the optimization , we constrain the macroscopic parameters by requiring that ( i ) the neutron @xmath130 energy level splitting in @xmath1pb should lie in the range of @xmath131 mev ; ( ii)@xmath132 should be greater than @xmath133 and here we set @xmath134 ( @xmath135 is nucleon mass in vacuum ) to be consistent with the extraction from global nucleon optical potentials constrained by world data on nucleon - nucleus and ( p , n ) charge - exchange reactions  @xcite . as usual , in the optimization , we assign a theoretical error @xmath136 mev to e@xmath137 , @xmath138 fm to @xmath139 while use the experimental error for breathing mode energy @xmath140 with a weight factor @xmath141 , so that the respective @xmath142 evaluated from each sort of experimental data is roughly equal to the number of the corresponding data points  @xcite .    using the above optimization process , we evaluate the electric dipole polarizability @xmath20 in @xmath1pb as a function of @xmath3 for a fixed @xmath2 , and the results are shown in fig  [ alphadlc ] by solid up - triangles ( down - triangles ) for @xmath106 mev . it should be noted that for each pair of @xmath2 and @xmath3 with fixed values , the other @xmath105 macroscopic quantities have been optimized accordingly as described above . it is interesting to see that the values of @xmath20 with optimization are quite consistent with the results using the default values in msl0 without optimization and only show a small upward shift compared with the latter . comparing the results from optimization to the experimental data , one can obtain a constraint of @xmath6 mev , which is again in good agreement with the constraint @xmath110 mev extracted using the default values in msl0 . these features demonstrate the validity of neglecting the weak correlations between the @xmath20 in @xmath1pb and other @xmath105 macroscopic quantities . the present constraint on @xmath3 further agrees very well with the constraint @xmath143 mev extracted from analyzing the experimental data on the neutron skin thickness of sn isotopes  @xcite . this is a very interesting finding since these two constraints are obtained from two completely independent experimental observables . in addition , using the constrained @xmath2 and @xmath3 together with the corresponding @xmath105 other optimized quantities , one can easily extract the @xmath7 and @xmath144 at saturation density @xmath9 , and the results are @xmath145 mev and @xmath146 mev , which are essentially consistent with other constraints extracted from terrestrial experiments , astrophysical observations , and theoretical calculations with controlled uncertainties  @xcite . especially , our present results agree surprisingly well with the constraint of @xmath147-@xmath148 mev and @xmath149-@xmath150 mev ( at 95% confidence level ) obtained from analyzing the mass and radius of neutron stars  @xcite as well as that of @xmath151-@xmath152 mev and @xmath153-@xmath154 mev extracted from the experimental , theoretical and observational analyses  @xcite . our results are also in agreement with the constraint of @xmath155 mev and @xmath156 mev from analyzing pygmy dipole resonances ( pdr ) of @xmath157sn  @xcite and that of @xmath158 mev and @xmath159 mev from analyzing pdr of @xmath160ni and @xmath161sn  @xcite . in addition , our results are further consistent with the constraint of @xmath162 mev and @xmath163 mev extracted from analyzing the experimental data of the binding energy difference of heavy isotope pairs and the neutron skins of sn isotopes  @xcite as well as the constraint of @xmath164 mev and @xmath165 mev from a new finite - range droplet model analysis of the nuclear mass  @xcite .        given that the neutron skin thickness @xmath166 of @xmath1pb is uniquely fixed by the slope parameter @xmath3 at @xmath102 @xmath5  @xcite , we can also extract a constraint @xmath167 fm for @xmath1pb by using the optimized parameters together with @xmath103 mev and @xmath6 mev . our result is consistent with the estimated range @xmath168 fm in ref . @xcite obtained by analyzing the experimental @xmath20 in @xmath1pb with an empirical range of @xmath169 . one can see that our present constraint on @xmath166 of @xmath1pb has higher precision , indicating a more precise constraint on the symmetry energy at a subsaturation density is very helpful to extract @xmath166 of @xmath1pb from the electric dipole polarizability . our result further agrees with the constraint @xmath170 fm obtained from the @xmath1pb dipole polarizability by using an empirical correlation between @xmath20 and @xmath166 of @xmath1pb  @xcite , the constraint @xmath171 fm extracted very recently from coherent pion photoproduction cross sections  @xcite , and within the experimental error bar the constraint @xmath172 fm extracted from the prex at jlab  @xcite . furthermore , it has been well established that the core - crust transition density @xmath173 in neutron stars , which plays a crucial role in neutron star properties  @xcite , is strongly correlated with the density slope @xmath12 of the symmetry energy ( see , e.g. , ref . in particular , in ref . @xcite , the same correlation analysis method as in this work has been successfully applied to study the correlation between @xmath173 and the various macroscopic quantities , and indeed a strong correlation between @xmath173 and @xmath12 has been found . as mentioned in ref . @xcite , a similar strong correlation is also existed between @xmath173 and @xmath3 , and this is demonstrated in fig . [ rhot ] which shows the same correlations as fig . [ alphad ] but for the core - crust transition density @xmath173 in neutron stars . here , the transition density @xmath173 is calculated by using a dynamical approach ( see , e.g. , ref . one can see from fig . [ rhot ] that , for both @xmath174 and @xmath96 @xmath5 , @xmath173 exhibits a strong correlation with @xmath100 , a weak dependence on @xmath175 and @xmath176 , but almost no sensitivity to other macroscopic parameters . employing the optimized values for other macroscopic parameters as well as @xmath103 mev and @xmath6 mev , we then obtain a value of @xmath177 @xmath5 , which agrees well with the empirical values  @xcite . in summary , we have demonstrated that the electric dipole polarizability @xmath20 in @xmath1pb is sensitive to both the magnitude @xmath2 and density slope @xmath3 of the symmetry energy at a subsaturation cross density @xmath4 @xmath5 , and it decreases ( increases ) with @xmath2 ( @xmath3 ) , leading to a positive correlation between @xmath3 and @xmath2 for a fixed value of @xmath20 in @xmath1pb . using the experimental value of @xmath20 in @xmath1pb measured at rcnp and the very well - constrained range of @xmath2 , we have obtained a strong constraint on the slope parameter @xmath6 mev . this constraint is in surprisingly good agreement with the previous solely existing constraint @xmath143 mev from neutron skin data of sn isotopes , demonstrating the robustness of these constraints on the value of the @xmath3 parameter . the present constraint of @xmath3 further leads to @xmath145 mev and @xmath178 mev for the symmetry energy at saturation density , the neutron skin thickness @xmath179 fm for @xmath1pb , and @xmath180 @xmath5 for the core - crust transition density of neutron stars . these results are nicely consistent with many other constraints extracted from terrestrial experiments , astrophysical observations , and theoretical calculations with controlled uncertainties . our present results are based on the standard shf energy density functional . it will be interesting to see how the results change if different energy - density functionals , e.g. , the relativistic mean field model or the extended non - standard shf energy density functional , are applied . these works are in progress and will be reported elsewhere . we are grateful to li - gang cao for helpful discussions on the skyrme - rpa code . this work was supported in part by the major state basic research development program ( 973 program ) in china under contract nos . 2015cb856904 and 2013cb834405 , the nnsf of china under grant nos . 11135011 and 11275125 , the shu guang " project supported by shanghai municipal education commission and shanghai education development foundation , the program for professor of special appointment ( eastern scholar ) at shanghai institutions of higher learning , and the science and technology commission of shanghai municipality ( 11dz2260700 ) .
nuclear structure observables usually most effectively probe the properties of nuclear matter at subsaturation densities rather than at saturation density . we demonstrate that the electric dipole polarizibility @xmath0 in @xmath1pb is sensitive to both the magnitude @xmath2 and density slope @xmath3 of the symmetry energy at the subsaturation cross density @xmath4 @xmath5 . using the experimental data of @xmath0 in @xmath1pb from rcnp and the recent accurate constraint of @xmath2 from the binding energy difference of heavy isotope pairs , we extract a value of @xmath6 mev . 
 the implication of the present constraint of @xmath3 to the symmetry energy at saturation density , the neutron skin thickness of @xmath1pb and the core - crust transition density in neutron stars is discussed .