Constraining Nuclear Symmetry Energy parameters from Neutron skin thickness of 48 Ca
CConstraining Nuclear Symmetry Energy parameters from Neutron skin thickness of Ca S. K. Tripathy ∗ , D. Behera † , T. R. Routray ‡ and B. Behera § In the present work, we use a finite range effective interaction to calculate the neutron skinthickness in Ca and correlate these quantities with the parameters of nuclear symmetry energy.Available experimental data on the neutron skin thickness in Ca are used to deduce informationon the density slope parameter and the curvature symmetry parameter of the nuclear symmetryenergy at saturation and at subsaturation densities. We obtained the constraints such as 54 . ≤ L ( ρ ) ≤ . . ≤ L ( ρ c ) ≤ . − . ≤ K sym ( ρ ) ≤ − . − . ≤ K sym ( ρ c ) ≤ . Caand in
Pb is obtained.
PACS number : 21.65.Ef,24.30.Cz
I. INTRODUCTION
The nuclear symmetry energy (NSE), E s ( ρ ) is a fundamental quantity in the understanding of the equation of state(EoS) of isospin asymmetric nuclear matter (ANM). The density dependence of NSE plays an important role in nuclearphysics and astrophysics[1, 2]. Since E s ( ρ ) is not a directly measurable quantity, there have been attempts from boththe theoretical and experimental perspectives to understand the density dependence aspect of NSE [3–8]. In fact,density dependence of E s ( ρ ) is the most uncertain part in the EoS [9–11] and mostly relies upon the determinationof E s ( ρ ), its slope parameter L ( ρ ) at saturation density ρ and the curvature parameter K sym ( ρ ). While we havea fair knowledge on the value of E s ( ρ ) and its slope parameter L ( ρ ), our present knowledge of K sym ( ρ ) is ratherpoor. From different nuclear experiments and astrophysical observations prior to 2013, we have E s ( ρ ) = 31 . ± . L ( ρ ) = 58 . ±
16 MeV [12]. The values of E s ( ρ ) = 31 . ± . L ( ρ ) = 58 . ± . E s ( ρ ) = 31 . ± . L ( ρ ) = 59 . ± . E s ( ρ ) and L ( ρ ) at subsaturation densities. Fuchs andWolter from an analysis of different microscopic and phenomenological models, have obtained the NSE at a subsat-uration density around ρ c (cid:39) . ρ to be E s ( ρ c ) (cid:39)
24 MeV [15]. From the properties of doubly magic nuclei, Browntried to constrain the EoS at a density of ρ c = 0 . − [16]. Zhang and Chen have obtained a tighter constraint onthe symmetry energy at subsaturation density ρ c = 0 .
11 fm − i.e. E s ( ρ c ) = 26 . ± .
20 MeV [17] from an analysisof the binding energy difference of heavy isotope pairs. It is note here that the central density of heavy nucleus isaround 0 .
11 fm − and a knowledge of the density slope parameter and the curvature symmetry parameter at thisdensity is important in determining the density dependence of the NSE at low density region.The nuclear symmetry energy plays an important role in the formation of neutron skins in neutron-rich nuclei.The neutron skin thickness (NST), ∆r np = (cid:10) r (cid:11) / n − (cid:10) r (cid:11) / p , is used as a sensitive probe of NSE to improve ourknowledge in the isovector channels of nuclear effective interaction at least in the subsaturation density region [18–21]. In recent times, a lot of efforts have been made to correlate the NST with the parameters of nuclear symmetryenergy. In fact, the NST is observed to have a linear relationship with the density slope parameter L ( ρ ) [22, 23].There have been a lot of efforts made to obtain the neutron skin thickness in P b and to constrain the densitydependence of E s ( ρ ) from the results [20–22, 24–27]. The first run of the Lead Radius Experiment (PREX) measuredthe neutron skin thickness in Pb to be ∆r np = 0 . +0 . − . f m [28]. The PREX results have large error bars butthe proposed PREX II is expected to reduce the error by a factor of 3 [29]. The measurements from coherent pionphotoproduction (the Mainz experiment) provide ∆r np ( Pb)=0 . ± . stat ) +0 . − . ( syst ) f m [30]. The CalciumRadius Experiment (CREX) has also been approved and is ongoing at the Jefferson Lab. It is expected that, theCREX may reduce the error to 0 .
02 fm [29]. Using the coupled-cluster calculation, Hagen et al. obtained the neutronskin thickness ∆r np in Ca as 0 . − .
15 fm [31]. Also, from this calculation they have constrained the density ∗ Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India, E-mail: tripa-thy sunil@rediffmail.com †
1. Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India2. School of Physics, Sambalpur University, Jyotivihar, Sambalpur, Odisha-768019, India,E-mail:dipadolly@rediffmail.com ‡ Retired Professor, School of Physics, Sambalpur University, Jyotivihar, Sambalpur, Odisha-768019, India, E-mail:trr1@rediffmail.com § Retired Professor, School of Physics, Sambalpur University, Jyotivihar, Sambalpur, Odisha-768019, India a r X i v : . [ nu c l - t h ] A ug slope parameter as 37 . ≤ L ( ρ ) ≤ . Ca from proton inelastic scattering experiments at RCNP, Osaka [32]. From this experiments, they inferred theneutron skin thickness in Ca to be ∆r np ( Ca)=0 . ± .
03 fm. Very recently, Tanaka et al. [33] obtained theneutron skin thickness in the isotopes of Ca from the interaction cross sections for − Ca. The NST for Ca fromthese observations yielded ∆r np ( Ca)=0 . ± .
048 fm [33]. Tagami et al. recently used Gogny-D1S Hatree-Fock-Bogoliubov model with angular momentum projection to calculate the neutron skin thickness in some Calcium isotopesand obtained ∆r np ( Ca)=0 . − .
190 fm [34]. Xu et al. have carried out a Bayesian analysis on the measuredand some speculated values of the neutron skin thickness in Sn isotopes,
Pb and Ca to constrain the densitydependence of nuclear symmetry energy [35]. Amidst all these efforts to constrain the density slope parameter and thecurvature symmetry parameter, our knowledge on the density dependence of NSE is still very poor at subsaturationdensity region.In the present paper, we calculate the neutron skin thickness of the doubly magic nuclei Ca using the EoSsconstructed from a finite range effective interaction. Recent experimental constraints on the NST in Ca are used toconstrain the parameters of nuclear symmetry energy at the saturation density and at a subsaturation density. Thepaper is organised as follows: in Section II, the basic formalism of the nuclear equation of state as obtained fromthe finite range effective interaction is presented. The method of constraining the interaction parameters is discussedin brief. In Section III, we calculate the neutron skin thickness of Ca using the finite range effective interactionswithin the frame work of droplet model. The correlation of the neutron skin thickness with the parameters of nuclearsymmetry energy have been carried out. Experimental constraints on the neutron skin thickness in Ca are used toconstrain the nuclear symmetry energy parameters. A linear relationship is obtained between the NST of Ca and
Pb. The conclusion and summary of the present work are presented in Section-V.
II. BASIC FORMALISMA. Finite range effective interaction and Nuclear Symmetry Energy
We consider a finite range simple effective interaction (SEI)[36] v eff ( r ) = t (1 + x P σ ) δ ( r ) + 16 t (1 + x P σ ) (cid:20) ρ ( R )1 + bρ ( R ) (cid:21) γ δ ( r ) + ( W + BP σ − HP τ − M P σ P τ ) f ( r ) , (1)where f ( r ) is the form factor which may have either a Gaussian or Yukawa or an exponetial form. Here we considera Yukawa form factor f ( r ) = e − r/α r/α , α being the range of the interaction. r and R are respectively the relative andcentre of mass coordinates of the two interacting nucleons. W, B, H and M are the strength parameters of the Wigner,Bartlett, Heisenberg and Majorana components. P σ and P τ are the spin and isospin exchange operators respectively.The parameter b takes care of the supara lumious behaviour at high density and γ determines the stiffness of thenuclear equation of state in symmetric nuclear matter (SNM). Other parameters of the interactions, t , x , t , x , areadjusted so as to reproduce the saturation properties of SNM. This SEI has already been used to study the momentumand density dependence of the isoscalar part of the nuclear mean field at zero and finite temperature [37–39], isovectorpart of the nuclear mean field at zero temperature [40, 41], temperature dependence of nuclear symmetry energy[42, 43] and to calculate the half-lives of spherical proton emitters [44]. The SEI with a Gaussian form factor for thefinite range part of the effective interaction has been used in recent times to address the problem of binding energyand charge radii of spherical nuclei [45], spin polarized neutron matter [46], deformation properties of nuclei [47] andneutron star properties [48, 49].The energy density H ( ρ, y p , T ) in ANM at a density ρ , proton fraction y p and temperature T can be obtained fromSEI as H ( ρ, y p , T ) = (cid:90) [ f nT ( k ) + f pT ( k )] (cid:0) c (cid:126) k + M c (cid:1) d k + 12 (cid:34) ε l ρ + ε lγ ρ γ +10 (cid:18) ρ bρ (cid:19) γ (cid:35) (cid:0) ρ n + ρ p (cid:1) + (cid:34) ε ul ρ + ε ulγ ρ γ +10 (cid:18) ρ bρ (cid:19) γ (cid:35) ρ n ρ p + ε lex ρ (cid:90) (cid:90) (cid:2) f nT ( k ) f nT ( k (cid:48) ) + f pT ( k ) f pT ( k (cid:48) ) g ex ( | k − k (cid:48) | ) (cid:3) d k d k (cid:48) + ε ulex ρ (cid:90) (cid:90) (cid:2) f nT ( k ) f pT ( k (cid:48) ) + f pT ( k ) f nT ( k (cid:48) ) g ex ( | k − k (cid:48) | ) (cid:3) d k d k (cid:48) , (2)where f τT ( k ) , τ = n, p are the respective Fermi-Dirac distribution functions, Λ = α and g ex ( | k − k (cid:48) | ) = | k − k (cid:48)| . Thenew parameters ε l , ε ul , ε lγ , ε ulγ , ε lex and ε ulex are related to the interaction parameters as ε l = ρ (cid:20) t − x ) + 4 πα (cid:18) W + B − H − M (cid:19)(cid:21) , (3) ε ul = ρ (cid:20) t x ) + 4 πα (cid:18) W + B (cid:19)(cid:21) , (4) ε lγ = ρ γ +10 (cid:20) t
12 (1 − x ) (cid:21) , (5) ε ulγ = ρ γ +10 (cid:20) t
12 (2 + x ) (cid:21) , (6) ε lex = 4 πα ρ (cid:18) M − W − B + H (cid:19) , (7) ε ulγ = 4 πα ρ (cid:18) M + H (cid:19) . (8)The energy per particle in SNM is obtained at zero temperature ( T = 0) as e ( ρ ) = 3 M c x f (cid:2) x f u f − x f u f − ln ( x f + u f ) (cid:3) + ε ρρ + ε γ ρρ γ +10 (cid:18) ρ bρ (cid:19) γ + ε ex ρ ρJ ( ρ ) , (9)where x f = (cid:126) k f Mc , u f = (1 + x f ) . The Fermi momentum in SNM is given by k f = (cid:0) . π ρ (cid:1) . The functional J ( ρ )is given by J ( ρ ) = (cid:82) (cid:16) j ( k f r ) k f r (cid:17) e − r/α r/α d r (cid:82) e − r/α r/α d r , (10)where j ( k f r ) is the first order spherical Bessel function and ε = (cid:0) ε l + ε ul (cid:1) , ε γ = (cid:0) ε lγ + ε ulγ (cid:1) , ε ex = (cid:0) ε lex + ε ulex (cid:1) .The zero temperature EoS in pure neutron matter (PNM) is obtained as e n ( ρ ) = 3 M c x n (cid:2) x n u n − x n u f − ln ( x n + u n ) (cid:3) + ε l ρρ + ε lγ ρρ γ +10 (cid:18) ρ bρ (cid:19) γ + ε lex ρ ρJ n ( ρ ) , (11)where x n = (cid:126) k n Mc , u n = (1 + x n ) . k n = (cid:0) π ρ (cid:1) denotes the Fermi momentum in PNM. The functional J n ( ρ ) isexpressed as J n ( ρ ) = (cid:82) (cid:16) j ( k n r ) k n r (cid:17) e − r/α r/α d r (cid:82) e − r/α r/α d r , (12)where j ( k n r ) is the first order spherical Bessel function.The nuclear symmetry energy, E s ( ρ ), is defined as E s ( ρ ) = 12! ∂ e ( ρ, δ ) ∂δ | δ =0 , (13)and can also be expressed as the difference in the energy per particle in pure neutron matter e n ( ρ ) = e ( ρ, δ = 1) andthat in SNM, E s ( ρ ) = e n ( ρ ) − e ( ρ ) , (14)where the contribution from higher order terms in δ is assumed to be small. With this definition of NSE, we can havefrom Eqs.(9) and (11) E s ( ρ ) = 3 M c (cid:34) x n u n − x n u f − ln ( x n + u n ) x n − x f u f − x f u f − ln ( x f + u f ) x f (cid:35) + ( ε l − ε )2 ρρ + ( ε lγ − ε γ )2 ρρ γ +10 (cid:18) ρ bρ (cid:19) γ + [ ε lex J n ( ρ ) − ε ex J ( ρ )]2 ρρ . (15)An expansion of NSE around saturation density ρ reads as E s ( ρ ) = E s ( ρ ) + L (cid:18) ρ − ρ ρ (cid:19) + K sym (cid:18) ρ − ρ ρ (cid:19) + · · · , (16)where L ( ρ ) = 3 ρ ∂E s ( ρ ) ∂ρ | ρ = ρ and K sym = 9 ρ ∂ E s ( ρ ) ∂ρ | ρ = ρ are respectively the slope and curvature parameters of E s ( ρ ) at ρ . It is obvious from the above expansion that the density dependence of the nuclear symmetry energyrelies upon the exact determination of the parameters L ( ρ ) and K sym ( ρ ).We may expand the NSE around a subsaturation density ρ c < ρ as E s ( ρ ) ≈ E s ( ρ c ) + L ( ρ c ) ε + K sym ( ρ c )2! ε + O ( ε ) , (17)where ε = ρ − ρ c ρ c . L ( ρ c ) = 3 ρ c dE s ( ρ ) dρ | ρ = ρ c is the density slope parameter and K sym ( ρ c ) = 9 ρ c d E s ( ρ ) dρ | ρ = ρ c is thecurvature parameter at the reference density ρ c . B. Fixation of interaction parameters
The interaction parameters of SEI are adjusted so as to obtain viable equations of state for the SNM and PNM andto have a good description of the momentum dependence of nuclear mean field. The complete description of SNMrequires only the knowledge of six parameters γ, b, α, ε , ε γ and ε ex . However, the equation of state in PNM requiresthe splitting of the strength parameters ε , ε γ and ε ex into like ( l ) and unlike ( ul ) channels. We do not have anyavailable experimental or empirical constraints for this splitting. Behera et al. have constrained the parameter ε lex as ε lex = ε ex [42] which allows the neutron effective mass in neutron-rich matter to pass over the proton effectivemass. Once the splitting of ε ex into ε lex and ε ulex is fixed, we require the nuclear symmetry energy E s ( ρ ) and its slope E (cid:48) s ( ρ ) = ρ dE s ( ρ ) dρ | ρ = ρ = L ( ρ ) at saturation density to obtain the splitting of the other two strength parameters ε and ε γ into like and unlike components. The details of constraining the parameters required for SNM are givenin Refs. [36, 42] where the standard values M c = 939 M eV , energy per nucleon in SNM e ( ρ ) = 923 M eV , (cid:16) c (cid:126) k f + M c (cid:17) = 976 M eV corresponding to the saturation density ρ = 0 . f m − are used. In order toconstrain the parameters required for PNM, we follow the procedure as described in Refs. [36, 50]. The SEI predictsan incompressibility in normal nuclear matter, K = 240 M eV corresponding to γ = 0 . m ∗ M = 0 .
67. The nuclear symmetry energy from the constructed EoSs provide a good description of its densitydependence for a wide range of nuclear matter density. The NSE at a sub-saturation density ρ c (cid:39) ρ ≈ .
11 fm − for all the sets of interaction parameters is obtained to be E s ( ρ c ) = 26 .
65 MeV. At a density around twice the normalnuclear matter density, E s (2 ρ ) lies close to the limit E s (2 ρ ) = 46 . ± . M eV , a constraint obtained from theanalysis of astrophysical observations for a constant maximum mass of M max = 2 . M (cid:12) and radius R , = 12 . III. NEUTRON SKIN THICKNESS IN Ca In this section, we calculate the neutron skin thickness in Ca using the EoSs constructed from the finite rangeeffective interaction (SEI) within the framework of droplet model Myers and Swiatecki [53]. It is worth to mentionhere that, Myers and Swiatecki in their work [53] have argued that, the droplet model results of the neutron skinthickness are almost equal to the results obtained by Hatree-Fock (HF) calculations. The reason behind the strikingsimilarity between the DM and HF results lie in the fact that, the shell effects appearing in HF calculations maynot be important for the discussion of neutron skin thickness [53]. In a recent work, we have also calculated theneutron skin thickness of some nuclei using the finite range effective interactions in the framework of droplet modeland obtained similar results to that of the HF calculations [50].The neutron skin thickness of nuclei has been identified as a strong isovector indicator [54]. In general, NST isdefined as the difference between the rms radii for the density distribution of the neutrons and protons in the nucleus.Basing upon different contributions to NST, we can write ∆r np = (cid:114) (cid:20) t − e Z E s ( ρ ) + 52 R (cid:0) b n − b p (cid:1)(cid:21) , (18)where t is the distance between the neutron and proton radii of uniform sharp distributions, b n and b p are the surfacewidths of the neutron and proton profiles. Neglecting the shell correction within the purview of the droplet model,we can have p r e s e n t w o r k L i n e a r f i t D rnp (48Ca) [fm] b ( a ) - 8 - 7 - 6 - 5 - 4 - 3 - 20 . 1 20 . 1 30 . 1 40 . 1 50 . 1 60 . 1 7 K s y m ( r ) / E s ( r ) D rnp (48Ca) [fm] ( b ) FIG. 1: (a) The neutron skin thickness of Ca is shown as a function of β . A linear fit to the values is also shown in the figure. (b) The neutronskin thickness of Ca is shown as a function of
Ksym ( ρ Es ( ρ . t = 32 r E s ( ρ ) Q (cid:18) I − I c x A (cid:19) . (19)where r = (cid:0) πρ (cid:1) − / , I = N − ZA is the neutron-proton asymmetry in the nucleus and I c = e Z E s ( ρ ) R is the Coulombcorrection to the symmetry energy coefficient. The factor x A = E s ( ρ ) Q A − / is associated with the ratio of the surfacesymmetry energy to the volume symmetry energy of semi infinite nuclear matter. Q = (cid:16) E s ( ρ ) a sym ( A ) − (cid:17) − E s ( ρ ) A − / is the surface stiffness parameter that measures the resistance of the nucleus against separation of neutrons fromprotons to form a skin. Assuming the validity of a sym ( A ) = E s ( ρ A ) and using the expansion E s ( ρ A ) (cid:39) E s ( ρ ) + L ( ρ ) (cid:15) A + K sym ( ρ )2 (cid:15) A , Eq.(19) can be reduced to [36] t (cid:39) − r (cid:15) A β (cid:18) K sym ( ρ )2 L ( ρ ) (cid:15) A (cid:19) A / ( I − I c ) , (20)where β = L ( ρ ) / E s ( ρ ) = E (cid:48) s ( ρ ) E s ( ρ ) and (cid:15) A = ρ A − ρ ρ . There appears to be a clear linear correlation between between the bulkpart of the NST in finite nuclei and some isovector indicators such as 1 − a sym ( A ) E s ( ρ ) , β and K sym ( ρ ) E s ( ρ ) . In a recent work,we have used 16 sets of interaction parameters by varying the E s ( ρ ) and L ( ρ ) so as to reproduce the symmetryenergy at the central density of Pb as 26 .
65 MeV. In that work, we have found that, EoSs with same value of NSEat saturation density ρ may have different slopes. In view of this, the ratio β = L ( ρ )3 E s ( ρ ) has a critical role in decidingthe quantity t and consequently ∆r np rather than L ( ρ ). The surface contribution to the neutron skin thickness ∆r surfnp can be evaluated from the neutron and proton density profiles. Many authors have considered that b n (cid:39) b p ,so that ∆r surfnp (cid:39)
0. However, Warda et al. [25] have obtained a linear relation ∆r surfnp = (cid:16) . E s ( ρ ) Q + 0 . (cid:17) I fm forthe surface contribution to the neutron skin thickness. With the inclusion of the surface contribution as prescribedby Warda et al.[25], the NST can now be expressed as ∆r np = (cid:114) (cid:34) − r (cid:15) A β (cid:18) K sym ( ρ )2 L ( ρ ) (cid:15) A (cid:19) A / ( I − I c ) − e Z E s ( ρ ) + (cid:32)(cid:114) E s ( ρ )2 Q + 0 . (cid:33) I (cid:35) . (21)It is obvious from the above expression (21) that, ∆r np has a linear relationship with β and K sym ( ρ ) E s ( ρ ) . In Figures1(a) and 1(b), we plot the neutron skin thickness of Ca calculated using the SEI in the framework of droplet modelas function of β and K sym ( ρ ) E s ( ρ ) . Linear plots are obtained for these correlations. A linear fit provides us the relations P r e s e n t w o r k L i n e a r f i t D rnp (48Ca) [fm] L ( r ) [ M e V ] ( a )R C N PR I K E N - 3 0 0 - 2 5 0 - 2 0 0 - 1 5 0 - 1 0 0 - 5 0 00 . 0 80 . 1 00 . 1 20 . 1 40 . 1 60 . 1 80 . 2 00 . 2 2 R C N PR I K E N D rnp (48Ca) [fm] K s y m ( r ) [ M e V ] P r e s e n t w o r k L i n e a r f i t ( b )
FIG. 2: The neutron skin thickness in Ca is plotted as a function of (a) L ( ρ ) and (b) K sym ( ρ ). The experimental regions for ∆r np ( Ca)from the Osaka-RCNP measurements [32] and the RIKEN measurements [33] are shown for comparison. ∆r np ( Ca ) = 0 .
057 + 0 . β fm , (22) ∆r np ( Ca ) = 0 .
212 + 0 . K sym ( ρ ) fm . (23)A high resolution measurement of the electric dipole polarisability α D in Ca at RCNP, Osaka predicted theneutron skin thickness in Ca as ∆r np ( Ca)=0 . ± .
03 fm [32]. Very recently, by measuring the interaction crosssection for Ca scattering on a target at RIKEN, Tanaka et al. have obtained ∆r np ( Ca)=0 . ± .
048 fm [33].In a recent work [50], we have used the experimental values of the neutron skin thickness in
Pb to constrain thenuclear symmetry energy parameters. In the present work, we use similar methods to constrain the nuclear symmetryparameters from the experimental values of the ∆r np ( Ca). In Figure 2(a), we plot the neutron skin thickness ascalculated using the SEI as function of the density slope parameter L ( ρ ) at saturation density. The experimentallyextracted regions from the Osaka-RCNP and the RIKEN measurements are also shown in the figure for comparison.A comparison of our results with the Osaka-RCNP results constrains the slope parameter L ( ρ ) in the range 54 . − . ≤ L ( ρ ) ≤ . E s ( ρ ) ≥
34 MeV.In Figure 2(b), the neutron skin thickness in Ca calculated from SEI is shown as a function of the curvatureparameter at saturation density K sym ( ρ ) and compared with the results from Osaka-RCNP and RIKEN measure-ments. While the Osaka-RCNP results constrain the curvature parameter in the range − . ≤ K sym ( ρ ) ≤ − . − . ≤ K sym ( ρ ) ≤ − . Ca with the density slope parameter at a reference density ρ c < ρ . Replacing ρ by ρ in Eq. (17) and keeping upto 2nd order in ε , we get [36] t = 2 r εβ (cid:48) (cid:20) K sym ( ρ c ) L ( ρ c ) ε (cid:21) A / ( I − I c ) , (24)where β (cid:48) = L ( ρ c )3 E s ( ρ ) and ε = ρ − ρ c ρ c . Consequently, the neutron skin thickness is expressed as [36] ∆r np = (cid:114) (cid:34) r εβ (cid:48) (cid:18) K sym ( ρ c )2 L ( ρ c ) ε (cid:19) A / ( I − I c ) − e Z E s ( ρ ) + (cid:32)(cid:114) E s ( ρ )2 Q + 0 . (cid:33) I (cid:35) . (25)It is obvious from the above expression in Eq.(24) that, the neutron skin thickness has a linear relationship with theparameters β (cid:48) and K sym ( ρ c ). From the calculations of ∆r np ( Ca) using the SEI, we may infer the linear relations as ∆r np ( Ca ) = − .
104 + 0 . β (cid:48) fm , (26) ∆r np ( Ca ) = 0 .
215 + 0 . K sym ( ρ c ) fm . (27) P r e s e n t w o r k L i n e a r f i t D rnp (48Ca) [fm] L ( r c ) [ M e V ] ( a )R C N PR I K E N - 1 5 0 - 1 2 5 - 1 0 0 - 7 5 - 5 0 - 2 5 00 . 0 80 . 1 00 . 1 20 . 1 40 . 1 60 . 1 80 . 2 00 . 2 2 R C N PR I K E N ( D rnp (48Ca) [fm] K s y m ( r c ) [ M e V ] P r e s e n t w o r k L i n e a r f i t ( b ) FIG. 3: he neutron skin thickness in Ca is plotted as a function of (a) L ( ρ c ) and (b) K sym ( ρ c ). The experimental regions for ∆r np ( )Cafrom the Osaka-RCNP measurements [32] and the RIKEN measurements [33] are shown for comparison.. D rnp(48Ca) [fm] D r n p ( P b ) [ f m ]
FIG. 4: The neutron skin thickness in Ca is correlated with the that in
P b . The Eq.(26) can be easily translated as ∆r np ( Ca)= − .
017 + 0 . L ( ρ c ) which provides a linear relation betweenthe NST and the density slope parameter at a subsaturation density. In Figures 3(a) and (b), we show ∆r np ( Ca) asfunction of L ( ρ c ) and K sym ( ρ c ) respectively. The results of Osaka-RCNP and RIKEN measurements are also shownin the figure for comparison. The Osaka-RCNP results constraints the density slope parameter at the subsaturationdensity in a tighter range as 47 . ≤ L ( ρ c ) ≤ . ≤ L ( ρ c ) ≤ . K sym ( ρ c ) as − . ≤ K sym ( ρ c ) ≤ − . − ≤ K sym ( ρ c ) ≤ − . Ca as function of theNST in
Pb. An obvious linear correlation is obtained for these quantities in the figure. A linear fit to the results D rnp [fm] A E x p t . ( T a n a k a e t a l . ) E s ( r ) = 3 3 M e V E s ( r ) = 3 4 M e V E s ( r ) = 3 5 M e V E s ( r ) = 3 6 M e V FIG. 5: The neutron skin thickness in calcium isotopes plotted as function of mass number. The experimental results of Tanaka et al. [33] forthe NST in calcium isotopes are also shown in the figure for comparison. reads as ∆r np ( Ca ) = 0 . . ∆r np ( P b ) . (28)The central value of the NST in Pb as obtained in the first run of PREX [28] is 0 .
33 fm. Using this value inEq.(28), we may have a crude idea about the CREX result with the estimated error of CREX as ∆r np ( Ca ) = 0 . ± .
02 fm . (29)This result is large as compared to the experimental estimates from Osaka-RCNP and RIKEN. Experiments withhadronic probes constrained the NST in Pb as ∆r np = 0 . ± (0 . (stat) ± (0 . (syst fm [20] and ∆r np = 0 . +0 . − . (Osaka-RCNP)[21] and measurements from coherent pion photoproduction yield a value ∆r np ( P b ) = 0 . ± . ∆r np ( Ca)=0 . ± .
02 fm and ∆r np ( Ca)= 0 . ± .
02 fm respectively which are compatible to the Osaka-RCNP and RIKENresults.The experimental results of the neutron skin thickness in different calcium isotopes are available. Very recently,Tanaka et al. have determined the NST in − Ca from the measurement of interaction cross section [33]. Wecalculate the neutron skin thickness for some of the calcium isotopes by using the finite range effective interaction(SEI) and plot them as function of mass number in Figure 5 for four different sets of interaction parameters. Theexperimental results of Tanaka et al. are also shown in the figure for comparison. One may observe that, in general, theneutron skin thickness in calcium isotopes increases with an increase in the mass number. The theoretical calculationsfrom SEI follow the experimental trend of Tanaka et al. and reproduce the results for the isotopes − Ca. However,for the isotopes with mass number greater than 48, our results from SEI are underestimated as compared to theexperimental values. It appears from the figure that, the sets of the finite range effective interaction with highervalues of E s ( ρ ) are more favoured for the calculations of neutron skin thickness in calcium isotopes. IV. SUMMARY AND CONCLUSION
In the present work, we have calculated the neutron skin thickness of Ca using some recently constructed EoSsfrom finite range effective interaction (SEI) in the framework of droplet model. The EoSs from SEI provide a gooddescription of the nuclear symmetry energy at a subsaturation density ( ρ c < ρ ), saturation density ρ and at asuprasaturation density (2 ρ ) and therefore they are suitable for applications to a wider range of density. The finiterange effective interactions predict the neutron skin thickness of Ca in the range 0 . − .
169 fm with a spreadof about 0 .
04 fm. Experimental constraints on ∆r np ( Ca) are available from the Osaka-RCNP and the RIKENmeasurements. We used these experimental constraints to constrain some of the nuclear symmetry energy parameterssuch as the density slope parameter and the curvature symmetry energy parameter at the saturation density and at thesubsaturation density. While the results of the Osaka-RCNP measurements constrain the slope parameter L ( ρ ) in therange 54 . −
102 MeV, the RIKEN results constrain the slope parameter in the range 21 . ≤ L ( ρ ) ≤ . − . ≤ K sym ( ρ ) ≤ − . − . ≤ K sym ( ρ ) ≤ − . ρ c (cid:39) .
11 fm − , the constraints as obtained from a comparison of the experimental results and the present calculationsusing the finite range effective interactions are 47 . ≤ L ( ρ c ) ≤ . ≤ L ( ρ c ) ≤ . − . ≤ K sym ( ρ c ) ≤ − . − ≤ K sym ( ρ c ) ≤ − . E s ( ρ ) ≥
34 MeV.From the calculations of the NST in
Pb with the finite range effective interactions (SEI) within droplet model, weobtained a linear relationship between ∆r np ( Ca) and ∆r np ( Pb). This relation can be used to predict the resultof CREX as ∆r np ( Ca)=0 . ± .
02 fm. This result is somewhat larger as compared to the recent experimentalestimates from Osaka-RCNP and RIKEN which may be due to the large central value of ∆r np ( Pb) obtained inPREX. However, accurate determination of the neutron skin thickness in
Pb can predict the CREX results withsome accuracy from the obtained relationship.
References [1] J. Lattimer and M. Prakash,
Phys. Rep. , , 121 (2000).[2] A. W. Steiner, M. Prakash, J. Lattimer, P. Ellis, Phys. Rep. , , 325 (2005).[3] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch and A. W. Steiner, Phys. Rev. Lett. , , 122701(2009).[4] M. B. Tsang et al., Phys. Rev. C , 015803 (2012).[5] J. Peikarewicz J et al., Phys. Rev. C , , 041302(R) (2012).[6] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita, T. Adachi, C. A. Bertulani, J. Carter, M. Dozono, H. Fujita,K. Fujita et al., Phys. Rev. Lett. , , 062502 (2011).[7] Z. Zhang and L. W. Chen, Phys. Rev. C , , 064317 (2014).[8] X. Roca-Maza, X. Vinas, M. Centelles, B. K. Agrawal, G. Colo, N. Paar, J. Piekarewicz and D. Vretenar, Phys. Rev. C , , 064304 (2015).[9] B. A. Li, P. G. Krastev, D. H. Wen and N. B. Zhang, Eur. Phys. J. A , , 217 (2019).[10] B. A. Brown, Phys. Rev. Lett. , , 5296 (2000).[11] B. A. Li, L. W. Chen and C. M. Ko, Phys. Rep. , 113 (2008).[12] B. A. Li and X. Han,
Phys. Lett. B , , 276 (2013).[13] M. Oertel, M. Hempel, T. Kl¨ahn, S. Typel, Rev. Mod. Phys. , , 015007 (2017).[14] C. Drischler, R. J. Furnstahl, J. A. Melendez and D. R. Phillips, arXiv:2004.07232.[15] C. Fuchs and H. H. Wolter, Eur. Phys. J. A , , 5 (2006).[16] B. A. Brown, Phys. Rev. Lett. , , 232502 (2013).[17] Z. Zhang and L. W. Chen, Phys. Lett. B , , 234 (2013).[18] A. Trzcinska et al., Phys. Rev. Lett. , , 082501 (2001).[19] B. A. Brown, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C , , 034305 (2007).[20] B. Klos et al., Phys. Rev. C , , 014311 (2007).[21] J. Zenihiro, et al., Phys. Rev. C , , 044611 (2010).[22] X. Roca-Maza, M. Centelles, X. Vinas, M. Warda, Phys. Rev. Lett. , , 252501 (2011).[23] X. Roca-Maza and N. Paar, Prog. Part. Nucl. Phys. , , 96 (2018).[24] M. Centelles, X. Roca-Maza, X. Vinas and M. Warda, Phys. Rev. C , , 054314 (2010).[25] M. Warda, X. Vinas, X. Roca-Maza and M. Centelles, Phys. Rev. C , , 024316 (2009).[26] X. Vinas, M. Centelles, X. Roca-Maza, and M. Warda, Eur. Phys. J. A , , 27 (2014).[27] C. Mondal, B. K. Agrawal, M. Centelles, G. Colo, X. Roca-Maza, N. Paar, X. Vinas, S. K. Singh and S. K. Patra, Phys.Rev. C , , 064303 (2016).[28] PREX collaboration (S. Abrahamyan, Z. Ahmed et al.), Phys. Rev. Lett. , , 112502 (2012).[29] K. Paschke et al. , Jefferson Lab Experiment E12-11-101 (PREX-II) proposal at http://hallaweb.jlab.org/parity/prex(2014).[30] C. M. Tarbert et al. ( Crystal Ball at MAMI and A2 Collaboration), Phys. Rev. Lett. , , 242502 (2014).[31] G. Hagen et al. , Nature Phys. , , 186 (2015), arXiv:1509.07169.[32] J. Birkhan, M. Miorelli, S. Bacca, et al. , Phys. Rev. Lett. , , 252501 (2017).[33] M. Tanaka, et al. , Phys. Rev. Lett. , , 102501 (2020).[34] S. Tagami, J. Matsui, M. Takechi and M. Yahiro, arXiv:2005.13197.[35] J. Xu, W. J. Xie and B. A. Li, arXiv:2007.07669.[36] D. Behera, S.K. Tripathy, T. R. Routray and D. Behera, to appear in Physica Scripta , arXiv:2004.14205.[37] T. R. Routray, B. Sahoo, R. K. Satpathy and B. Behera,
J. Phys. G: Nucl.Part. Phys. , 887 (2000). [38] B. Behera, T. R. Routray and R. K. Satpathy, J. Phys. G: Nucl.Part. Phys. , , 2073 (1998).[39] B. Behera, T. R. Routray, B. Sahoo and R. K. Satpathy, Nucl. Phys. A , , 770 (2002).[40] B. Behera, T. R. Routray and A. Pradhan, Mod. Phys. Lett. A , , 2639 (2005).[41] B. Behera, T. R. Routray, A. Pradhan, S. K. Patra and P. K. Sahu, Nucl. Phys. A , , 132 (2009).[42] B. Behera, T. R. Routray and S. K. Tripathy, J. Phys. G: Nucl.Part. Phys. , , 125105 (2009).[43] B. Behera, T. R. Routray and S. K. Tripathy, J. Phys. G: Nucl.Part. Phys. , , 115104 (2011).[44] T. R. Routray, S. K. Tripathy, B. B Dash, B. Behera and D. N. Basu, Eur. Phys. J. A , , 92 (2011).[45] B. Behera, X. Vi˜nas, M. Bhuyan, T. R. Routray, B. K. Sharma and S. K. Patra, J. Phys. G: Nucl.Part. Phys. , , 095105(2013).[46] B. Behera, X. Vi˜nas, T. R. Routray and M. Centelles, J. Phys. G: Nucl.Part. Phys. , , 045103 (2015).[47] B. Behera, X. Vi˜nas, T. R. Routray, L. M. Robledo, M. Centelles and S. P. Pattnaik, J. Phys. G: Nucl.Part. Phys. , ,045115 (2016).[48] T. R. Routray, X. Vi˜nas, D. N. Basu, S. P. Pattnaik, M. Centelles, L. B. Robledo and B. Behera, J. Phys. G: Nucl.Part.Phys. , , 105101 (2016).[49] S. P. Pattnaik, T. R. Routray, X. Vi˜nas, D. N. Basu, M. Centelles, K. Madhuri and B. Behera, J. Phys. G: Nucl.Part.Phys. , , 055202 (2018).[50] D. Behera, S.K. Tripathy, T. R. Routray and D. Behera, communicated (2020).[51] N. B. Zhang and B. A. Li, Eur. Phys. J. A , , 39 (2019).[52] B. A. Li, P. G. Krastev, D. H. Wen, W. J. Xie and N. B. Zhang, AIP conference proceedings , , 020018 (2019).[53] W. D. Myers and W. J. Swiatecki, Nucl. Phys. A , , 267 (1980).[54] P. -G. Reinhard and W. Nazarewicz, Phys. Rev. C ,81