Differential analysis of incompressibility in neutron-rich nuclei
aa r X i v : . [ nu c l - t h ] F e b Differential analysis of incompressibility in neutron-rich nuclei
Bao-An Li ∗ and Wen-Jie Xie † Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, TX 75429, USA Department of Physics, Yuncheng University, Yuncheng 044000, China (Dated: February 23, 2021)Both the incompressibility K A of a finite nucleus of mass A and that ( K ∞ ) of infinite nuclear mat-ter are fundamentally important for many critical issues in nuclear physics and astrophysics. Whilesome consensus has been reached about the K ∞ , accurate theoretical predictions and experimentalextractions of K τ characterizing the isospin dependence of K A have been very difficult. We proposea differential approach to extract the K τ and K ∞ independently from the K A data of any two nucleiin a given isotope chain. Applying this novel method to the K A data from giant monopole reso-nances in even-even Sn, Cd, Ca, Mo and Zr isotopes taken by U. Garg et al. at the Research Centerfor Nuclear Physics (RCNP), Osaka University, Japan, we find that the Cd-
Cd and
Sn-
Sn pairs having the largest differences in isospin asymmetries in their respective isotope chainsmeasured so far provide consistently the most accurate up-to-date K τ value of K τ = − ± K τ = − ±
86 MeV, respectively, largely independent of the remaining uncertainties ofthe surface and Coulomb terms in expanding the K A . Introduction.
Because of its fundamental importance innuclear physics and broad impacts on astrophysics, theincompressibility K ∞ of infinite nuclear matter has beena long standing and major scientific goal of many exper-imental and theoretical researches. Since the pioneeringwork of Blaizot who determined K ∞ = (210 ±
30) MeVfrom analyzing the experimental data on giant monopoleresonance (GMR) energies in Ca, Zr and
Pb [1],extensive theoretical studies and systematic experimentson the incompressibility K A of finite nuclei extractedfrom GMR energies over the last four decades [1–6] haveled to the community consensus that the K ∞ is in therange of 220 MeV to 260 MeV [3, 6, 7] or around 235 ± K A .The incompressibility K A is usually parameterized inthe form of a leptodermous expansion in powers of A − / in typical macroscopic models as [1] K A ≈ K ∞ (1 + cA − / ) + K τ δ + K Cou Z A − / (1)for a nucleus of mass number A, charge number Z andisospin asymmetry δ = N − ZA , with c ≈ − . ± .
12 [10]and K Cou ≈ − . ± . K τ characterizingthe isospin dependence of K A has been the main focusof many recent experimental and theoretical investiga-tions. By moving the Coulomb term to the left side ofthe above equation, for all practical purposes [3] in ex-tracting the K τ from the experimental K A data [12–15], ∗ [email protected] † [email protected] the K A − K Cou Z A − / was fitted with a quadratic func-tion of the form a + K τ δ assuming a = K ∞ (1 + cA − / )is a constant. This approach resulted in an “experimen-tal” value of K τ = -550 ±
100 MeV from the K A dataof Sn isotopes and K τ = -555 ±
75 MeV from the Caisotopes, respectively. The mass dependence of a andthe known correlation between K ∞ and K τ neglectedin the above approach were found to affect significantlythe extracted K τ values [5, 16]. For example, using thesame c and K Cou parameters but preserving the massdependence of a and considering the correlation between K ∞ and K τ in the error minimization of a multivari-ate χ fit, Stone et al. found that K τ = -595 ±
177 MeV, K ∞ =209 ± K τ = -463 ± K ∞ =211 ±
11 MeV from the Cd isotopes [5],respectively. As it was stressed already [3, 5, 16, 17],the state of affairs in understanding and extracting the K τ has been very unsatisfactory for a long time. Givenits known importance in both nuclear physics and astro-physics, it is imperative to find more robust methods toextract accurately the K τ from K A data. Such methodsare also expected to play important roles in analyzing thecoming new data from measuring the K A of exotic, moreneutron-rich nuclei in long isotopic chains at advancedradioactive beam facilities.In this Letter, we propose a differential approach toextract exactly the values of K ∞ and K τ independentlyfrom the K A data of two nuclei in any isotopic chain.The nucleus-nucleus pair having the largest differencein their isospin asymmetries is found to give the mostaccurate K τ and K ∞ values simultaneously. First appli-cations of this novel method to the K A data of Cd andSn isotopes from RCNP reveal that the Cd-
Cd and
Sn-
Sn pairs give consistently the most accuratevalues of K τ = − ±
59 MeV and K ∞ =213 ± K τ = − ±
86 MeV and K ∞ =220 ± c = − . K Cou = − . K τ and K ∞ values withlarger errors but consistent with results from traditionalintegral analyses by minimizing the χ in fitting the K A data of all nuclei in a given isotope chain. Effectsof varying the c and K Cou parameters by ±
20% aroundtheir known most probable values on extracting both K τ and K ∞ are also examined. While the variations of c and K Cou lead the extracted K ∞ values to vary withinits current consensus range, they have almost no effecton extracting the K τ , indicating the robustness of thedifferential approach. The differential approach.
Applying the Eq.(1) to anytwo isospin asymmetric ( δ = 0) nuclei of mass and charge( A , Z ) and ( A , Z ) separately, the K τ and K ∞ can beexpressed exactly as K τ = " K A S − K A S − K Cou Z A − / S − Z A − / S ! δ S − δ S (cid:19) , (2) K ∞ = " K A δ − K A δ − K Cou Z A − / δ − Z A − / δ ! S δ − S δ (cid:19) (3)where S i = 1 + cA − / i is the reduced effective num-ber of nucleons in the nucleus-i with i=1 or 2 due tothe surface effect with respect to infinite nuclear mat-ter (S=1). The K A i /S i is thus the incompressibilityper effective-nucleon while the δ i /S i is the square ofits isospin asymmetry in the nucleus-i. One can under-stand intuitively the physical meanings of the above ex-pressions using the mathematical definitions of K τ and K ∞ based on Eq. (1). Namely, neglecting the Coulombcorrection, K τ ≡ (cid:0) ∂K A /∂δ (cid:1) S ≈ ∆( K A /S ) / ∆( δ /S ) =( K A1 S − K A2 S ) / ( δ S − δ S ) gives the leading term of K τ inEq. (2). It is simply the changing rate of K A with respectto δ per effective-nucleon evaluated by using the ratioof their finite changes. Similarly, K ∞ ≡ ( ∂K A /∂S ) δ ≈ ∆( K A /δ ) / ∆( S/δ ) = ( K A1 δ − K A2 δ ) / ( S δ − S δ ) gives theleading term of K ∞ in Eq. (3).We notice that while the K τ and K ∞ are determinedindependently by the K A data themselves of any twonuclei used, they both depend on the two relatively welldetermined surface parameter c and Coulomb parameter K Cou . These two parameters will naturally introduce anintrinsic correlation between the K τ and K ∞ when theyare evaluated using the K A data of many nucleus-nucleuspairs in a given isotopic chain as we shall demonstrate.The corresponding uncertainties of K τ and K ∞ can becalculated exactly according to the rules of error prop-agation using the experimental errors of K A data, i.e., σ K A1 and σ K A2 in the nucleus-1 and nucleus-2 considered.Nevertheless, to see analytically what nucleus-nucleus pairs may give the most accurate K τ and K ∞ values, wenotice that for heavy nuclei in the same isotope chain, S ≈ S ≈
1, the error bars are reduced to σ K τ ≈ q σ K A1 + σ K A2 (cid:30)(cid:12)(cid:12)(cid:12)(cid:12) δ − δ (cid:12)(cid:12)(cid:12)(cid:12) , (4) σ K ∞ ≈ q ( δ · σ K A1 ) + ( δ · σ K A2 ) (cid:30)(cid:12)(cid:12)(cid:12)(cid:12) δ − δ (cid:12)(cid:12)(cid:12)(cid:12) . (5)They both are inversely proportional to | δ − δ | , thus nu-clear pairs having the largest difference in their isospinasymmetries will give the most accurate K τ and K ∞ val-ues simultaneously. Moreover, because of the weightingof σ K A by δ ≪ σ K ∞ , the K ∞ canbe more precisely evaluated than the K τ , explaining therelatively larger errors of the extracted K τ values.While in principle the above formalisms can be appliedto any two nuclei, we shall restrict their applications tonuclei in the same isotopic chain. This will reduce notonly effects of systematic experimental errors as whatis being used is the difference in K A scaled by eitherthe surface factor S or isospin asymmetry δ of the twonuclei in the same isotopic chain, but also effects of thehigher-order terms neglected in expanding the K A in Eq.(1). This is also one of the reasons why the differentialapproach can more precisely extract both the K τ and K ∞ compared to typical integral approaches.In cases where one of the nuclei is isospin-symmetric,say δ = 0, its K A alone can be used to evaluate the K ∞ according to K ∞ = K A /S − K Cou Z A − / /S while the K τ can be evaluated from the Eq. (2) bymaking the nucleus-2 as neutron-rich as possible to getthe most accurate result, indicating the importance ofusing exotic heavy isotopes. As noticed already in theliterature, see, e.g., Ref. [17], the leptodermous expan-sion in Eq. (1) itself may not be a good approximationfor light nuclei, the differential approach should thuswork better for more heavy nuclei. Differential analyses of the K A data from GMRexperiments at RCNP. Shown in Fig. 1 are theresults of our differential analyses of the K A datain , , , , , , Sn, , , , , Cd, , , Ca, , , Mo and , Zr from the GMR ex-periments at RCNP done by U. Garg et al. [12–15]using c = − . K Cou = − . K τ and K ∞ values are shown as functions of thedifference ( δ − δ ) in isospin asymmetries of the twonuclei involved in each isotope chain. Several interestingobservations can be made: (1) The uncertainties of both K τ and K ∞ generally decreases while their mean valuesremain approximately constants with the increasing( δ − δ ) for each isotope chain. (2) The Cd-
Cd and
Sn-
Sn pairs give the most accurate and consistentvalues of K τ = − ±
59 MeV and K τ = − ± K τ values from analyzing therelatively light , , Ca, , , Mo and , Zr data aresomewhat higher but having larger error bars too. They
Differnce in isospin asymmetry ( δ −δ ) -1400-1200-1000-800-600-400-2000200 K τ ( M e V ) K ∞ ( M e V ) -625-626 Sn Cd Sn Cd Ca Mo Ca Mo Zr 236 218213241
FIG. 1: (Color online) The K τ (lower window) and K ∞ (up-per window) from differential analyses of the incompressibil-ities in finite nuclei as functions of the difference ( δ − δ )in isospin asymmetries of the isotope pairs used. The solidlines are the mean values of K τ and K ∞ for the respectiveisotope chains. The arrows indicate the Cd and Sn isotopepairs giving the most accurate K τ and K ∞ values. are generally consistent with the means from analyzingthe Sn and Cd isotopes within error bars, althoughthere are some doubts if the Ca, Zr and Mo isotopes areheavy enough for the K A expansion of Eq. (1) to workproperly. We notice that among all data available fromthe RCNP experiments, the − Ca has the highest( δ − δ ) = 0 . K τ = − ±
149 MeV(not shown here to clearly display other results with the δ scale used). (4) The Cd-
Cd and
Sn-
Sn pairsalso give the most accurate K ∞ values of K ∞ =213 ± K ∞ =220 ± K ∞ shows the well-known isotope dependencewhen the Eq. (1) is used in fitting the K A data [1, 3–6, 17]. Nevertheless, the variation of the K ∞ from Cd toZr isotopes is well within the uncertainty range of thecurrently accepted consensus on K ∞ .Shown in Fig. 2 is the correlation between K τ and K ∞ with each point representing one nucleus-nucleuspair in the Cd or Sn isotope chain corresponding to theresults shown in Fig. 1. Averaging over these results isequivalent to performing a typical integral analysis, e.g.,a multivariate χ fitting or Bayesian analysis. The solidlines are results of a χ fit to all points in each isotopechain. The means of the K τ are − ±
100 MeV for theCd isotopes and − ±
188 MeV for the Sn isotopes, re-spectively. The corresponding means of K ∞ are 213 ±
205 210 215 220 225 K ∞ (MeV) -1000-900-800-700-600-500-400-300 K τ ( M e V ) Sn CdMean: K ∞ =213.1±2.8 MeVK τ =-625.2±100.2 MeV Mean: K ∞ =218.2±6.3 MeVK τ =-626.0±188.1 MeV FIG. 2: (Color online) The correlation between K τ and K ∞ with each point representing one nucleus-nucleus pair inthe Cd or Sn isotope chain corresponding to the results shownin Fig. 1. The solid lines are results of a χ fit to all pointsin each isotope chain. MeV for the Cd isotopes and 218 ± χ analyses [3, 5] ofthe same K A data using essentially identical surface andCoulomb parameters within error bars.Interestingly, within the errors of the mean values thereis a clear anti-correlation between K τ and K ∞ as one canunderstand easily. With the surface and Coulomb param-eters fixed, for a given K A value, the K τ and K ∞ is ex-pected to be anti-correlated according to Eq. (1). Noticethat the K τ vs K ∞ correlations for the Cd and Sn iso-topes are almost in parallel in the direction of K ∞ as theygive approximately the same K τ values but slightly differ-ent (about 5 MeV) K ∞ values (notice the fine K ∞ scaleused). It is worth noting here that our four-parameterBayesian analyses of the same K A data have confirmedquantitatively both the means of K τ and K ∞ as well astheir correlations [18].We emphasize that the error bars of the mean valuesof both K τ and K ∞ in the integral approaches, i.e., byaveraging over all isotope pairs, χ fits and Bayesiananalyses of the K A data directly are all much largerthan those we found in the differential analyses of Cd-
Cd and
Sn-
Sn pairs. Besides the advantageof cancelling the systematic errors in the differentialanalyses, another reason is that the K τ δ contributionto K A is very small even for the most neutron-rich nucleiavailable. For instance, with δ = 0 . , K τ = −
600 MeV, K τ δ = −
24 MeV that is still only about 10% of theacceptable K ∞ values around 240 MeV. It is actuallysignificantly less than the current uncertainty of about40 MeV of the consensus value for K ∞ . A global χ fit to the K A data or Bayesian analysis of all K A dataavailable [18] thus can not reliably extract the value of K τ ( M e V ) c=-1.2c=-1.2*0.8c=-1.2*1.2 k Cou =-5.2k
Cou =-5.2*0.8k
Cou =-5.2*1.2 K ∞ ( M e V ) Coulomb effect, c=-1.2Surface effect, k
Cou =-5.2 -626218
Difference in isospin asymmetry ( δ - δ )Sn isotopes FIG. 3: (Color online) Variations of the K τ (lower windows) and K ∞ (upper windows) due to the variations of the surfaceparameter c (left windows) and Coulomb parameter K Cou (right windows), respectively, for the Sn isotopes. K τ from its small contribution relative to K ∞ to the K A . In turn, the uncertainty of extracting the K ∞ cannot be better than K τ δ /K ∞ in the integral analyses ofthe K A data. On the contrary, the differential approachdecouples completely the extractions of K τ and K ∞ foreach isotope pair used. Only the K τ and K ∞ extractedindependently for different isotope pairs show an ex-pected intrinsic correlation through the relatively welldetermined surface and Coulomb parameters withintheir respective error bars. Effects of the surface and Coulomb parameters.
We haveused above the known most probable values of c = − . K Cou = − . et al. [5], the Coulomb parameter is rather model in-dependent [11, 19] while the calculations [1, 20–24] ofthe surface parameter c show somewhat larger variationsaround c ∼ −
1. It is generally accepted that both the c and K Cou parameters have less than about (10 − K τ and K ∞ in the differen-tial analyses? To answer this question, we have carriedout systematic calculations by varying the two parame-ters independently by ±
20% around their most probablevalues. Results of these studies are also corroborated byour four-parameter Bayesian analyses [18].As an example, shown in Fig. 3 are the variations ofthe K τ (lower windows) and K ∞ (upper windows) dueto the variation of the surface parameter c (left windows)and Coulomb parameter K Cou (right windows) for the Sn isotopes. Qualitatively, effects of varying the K Cou andespecially the parameter c are much smaller on K τ thanon K ∞ . Quantitatively, for the Sn-
Sn pair, chang-ing the c parameter by 40% from − . × . − . × . K τ change by about 6% from − ±
84 MeVto − ±
87 MeV, while the K ∞ changes by about 13%from 205 ±
29 MeV to 236 ±
34 MeV, respectively. On theother hand, by changing the K Cou by 40% from − . × . − . × .
2, the K τ change by about 8% from − ± − ±
87 MeV, while the K ∞ changes by about6% from 213 ±
31 MeV to 227 ±
31 MeV. Thus, the (6-8)% uncertainty of K τ due to the ±
20% uncertainty ofthe surface parameter is much smaller than the approx-imately 14% uncertainty due to the experimental errorsof K A . While the (8-13)% uncertainty of K ∞ due tothe ±
20% uncertainty in the Coulomb parameter is com-patible with that due to the experimental errors of the K A data.The observed dependences of K τ and K ∞ on the vari-ations of the surface and Coulomb parameters can beunderstood analytically by further examining the expres-sions of K τ and K ∞ in Eq. 2 and Eq. 3, respectively.Firstly, we examine effects of the parameter c . The c-dependent part of K τ is K τ ∝ (1 + cA − / ) K A − (1 + cA − / ) K A (1 + cA − / ) δ − (1 + cA − / ) δ . (6)Because the parameter c appears in all terms, its effectlargely cancels out. Moreover, for heavy nuclei c/A / ≈
0, thus the K τ becomes independent of c , i.e., K τ → K A − K A δ − δ . (7)While the c-dependent part of K ∞ is K ∞ ∝ δ K A − δ K A c · ( δ A − / − δ A − / ) + δ − δ . (8)Both terms in the denominator are very small. A verysmall change in the parameter c can thus lead to a largechange in K ∞ . This also implies that the surface prop-erties of different nuclei may affect significantly the ex-traction of K ∞ from the K A data as already noticed inthe χ analyses in Ref. [5].Similar analyses can be done to understand effects ofthe Coulomb parameter K Cou . More specifically, K τ ∝ − K Cou Z (1 + cA − / ) A − / − (1 + cA − / ) A − / (1 + cA − / ) δ − (1 + cA − / ) δ . (9)Again, the parameter c has little effect as it appears inall terms. Considering c ˙ A − / ≈ K τ → − K Cou Z × A − / − A − / δ − δ . (10)As heavy nuclei are more neutron rich in a given chain ofisotopes, i.e., for A > A , δ > δ , the fraction is alwaysnegative. Thus, one obtains K τ ∝ K Cou . While the K Cou itself is negative, thus a larger negative K Cou decreasesthe value of K τ as seen in our numerical calculations.As the overall contribution of the Coulomb term to the K τ is small, its variation causes little change in the final K τ value. While for the Coulomb effect on K ∞ , a similaranalysis leads to K ∞ ∝ − K Cou Z A / A / × δ A / − δ A / δ − δ . (11)Since the last fraction is always positive, thus K τ ∝ − K Cou . Therefore, a larger negative K Cou increases the value of K ∞ . Moreover, the aboveanalysis clearly explains why the Coulomb parameterhas opposite effects on extracting the K τ and K ∞ values. Conclusions.
In conclusion, we proposed a differentialapproach to analyze the incompressibilities of neutron-rich nuclei. The nucleus-nucleus pair having the largestdifference in their isospin asymmetries in a given isotopechain is found to give the most accurate values ofboth K τ and K ∞ simultaneously. Applying this novelapproach to the K A data from RCNP, we found that the Cd-
Cd and
Sn-
Sn pairs give consistently themost accurate up-to-date K τ value of − ±
59 MeVand − ±
86 MeV, respectively, largely independent ofthe remaining uncertainties of the surface and Coulombparameters. These results can exclude many predictionsbased on various microscopic and/or phenomenologicalnuclear many-body theories in the literature.
Acknowledgments.
This work was supported in part bythe U.S. Department of Energy, Office of Science, underAward Number DE-SC0013702, the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei)under the US Department of Energy Grant No. DE-SC0009971, and Yuncheng University Research Projectunder Grant No. YQ-2017005. [1] J. P. Blaizot, Phys. Rep. 64 (1980) 171.[2] D. H. Youngblood, H. L. Clark, and Y.-W. Lui, Phys.Rev. Lett. , 691 (1999).[3] U. Garg and G. Col`o, Prog. Part. Nucl. Phys. , 55(2018).[4] J. Piekarewicz, J. Phys. G , 064038 (2010).[5] J. R. Stone, N. J. Stone, S. A. Moszkowski, Phys. Rev.C , 044316 (2014).[6] G. Col`o, U. Garg and H. Sagawa, Eur. Phys. J. A , 26(2014).[7] S. Shlomo, V.M. Kolomietz and G. Col`o, Eur. Phys. J.A , 23 (2006).[8] E. Khan, J. Margueron and I. Vida˜na, Phys. Rev. Lett. , 092501 (2012).[9] J. Margueron, C.R. Hoffmann and F. Gulminelli, Phys.Rev. C , 025805 (2018).[10] S. K. Patra, M. Centelles, X. Vi˜nas, M. Del Estal, Phys.Rev. C , 044304 (2002).[11] H. Sagawa et al., Phys. Rev. C , 034327 (2007).[12] T. Li et al., Phys. Rev. Lett. 99 (2007) 162503. [13] T. Li et al., Phys. Rev. C , 034309 (2010)[14] D. Patel et al., Phys. Lett. B 718 (2012) 447.[15] K. B. Howard, U. Garg, Y. K. Gupta, M. N. Harakeh,Eur. Phys. J. A , 228 (2019).[16] J. M. Pearson, N. Chamel, and S. Goriely, Phys. Rev. C , 037301 (2010).[17] S. Shlomo and D. H. Youngblood, Phys. Rev. C , 529(1992).[18] B.A. Li and W.J. Xie, in preparation (2021).[19] P. Vesely et al., Phys. Rev. C , 024303 (2012).[20] J. P. Blaizot, D. Gogny, and B. Grammaticos, Nucl.Phys. A 265 , 315 (1976).[21] J. Treiner et al., Nucl. Phys.
A 371 , 253 (1981).[22] J. P. Blaizot and B. Grammaticos, Nucl. Phys.
A 355 ,115 (1981).[23] W. Myers and W. Swiatecki, Ann. Phys. , 401 (1990).[24] W. Myers and W. Swiatecki, Nucl. Phys.