Softness of Sn isotopes in relativistic semiclassical approximation
Abstract
Within the framework of relativistic ThomasFermi and relativistic extended ThomasFermi approximations, we calculate the giant monopole resonance (GMR) excitation energies for Sn and related nuclei. A large number of nonlinear relativistic force parameters are used in this calculations. We find that a parameter set is capable to reproduce the experimental monopole energy of Sn isotopes, when its nuclear matter compressibility lies within MeV, however fails to reproduce the GMR energy of other related nuclei. That means, simultaneously a parameter set can not reproduce the GMR values of Sn and other nuclei.
pacs:
24.30.Cz, 21.10.Re, 24.10.Jv, 21.65.f, 21.60.Ev, 21.65.MnI Introduction
Incompressibility of nuclear matter, also knows as compressional modulus has a special interest in pure nuclear and astronuclear physics, because of its fundamental role in guiding the equation of state (EOS) for nuclear matter. Compressional modulus can not be measured by any experimental technique directly, rather it depends indirectly on the experimental measurement of isoscalar giant monopole resonance (ISGMR) for its conformation blaizot (). This fact enriches the demand of correct measurement of excitation energy of ISGMR. The relativistic parameter with random phase approaches (RPA) constraint the range of the compressibility modulus MeV lala97 (); Vrete02 () for nuclear matter. Similarly, the nonrelativistic formalism with HartreeFock (HF) plus RPA allows the acceptable range of compressibility modulus MeV, which is less than relativistic one. It is believed that the part of this discrepancy in the acceptable range of compressional modulus comes from the diverse behavior of the density dependence of symmetry energy in relativistic and nonrelativistic formalism horow02 (); horowr01 (). Now, both the relativistic and nonrelativistic formalisms come with a general agreement on the value of nuclear compressibility i.e., MeV colo04 (); todd05 (); Agrawal05 (). But the new experiment on isotopic series i.e., rises the question ”why Tin is so fluffy ?”ugarg (); tli07 (); ugarg07 (). This question again finger towards the correct theoretical investigation of compressibility modulus.
Thus, it is worthy to investigate the compressibility modulus in various theoretical formalisms. Most of the relativistic and nonrelativistic theoretical models reproduce the strength distribution very well for medium and heavy nuclei, like and , respectively. But at the same time it overestimate the excitation energy of around 1 MeV. This low value of excitation energy demands lower value nuclear matter compressibility. This gives a new challenge to both the theoretical and experimental nuclear physicsts. Till date, lots of effort have been devoted to solve this problem like inclusion of pairing effect citav91 (); gang12 (); jun08 (); khan09 (), mutually enhancement effect (MEM) etc. khana09 (). But the pairing effect reduces the theoretical excitation energy only by 150 KeV in isotopic series, which may not be sufficient to overcome the puzzle. Similarly, new experimental data are not in favor of MEM effectdpatel13 (). Measurement on excitation energy of shows that the MEM effect should rule out in manifestation of stiffness of the isotopic series.
Here, in the present work, we use the relativistic ThomasFermi (RTF) and relativistic extended ThomasFermi (RETF) acentel93 (); acentel92 (); speich93 (); centel98 (); centell93 () with scaling and constraint approaches in the framework of nonlinear model bo77 (). The RETF is the correction to the RTF, where variation of density taken care properly mostly in the surface of the nucleus cetelles90 (). The RETF formalism is more towards the quantal Hartee approximation. It is also verified that the semiclassical approximation like ThomasFermi method is very useful in calculation of collective property of nucleus, like giant monopole resonance (GMR). In particular, for heavier mass nuclei, it gives almost similar results with the complicated quantal calculation. This is because of quantal correction are averaged out in heavier mass nuclei and results are inclined toward semiclassical one. So it is very much instructive to calculate excitation energy of various nucleus by this method and compared with experimental results. Since, last one decade, the softness of Sn isotopes remain a headache for both theorists and experimentalists, it is worthy to discuss the softness of Tin isotopes in semiclassical approximations like RETF and RTF.
Here, we calculate the excitation energy of Sn isotopes from Sn to Sn using the semiclassical RTF and RETF model. The different momentum ratio like and are compared with the scaling and constraint calculations. The theoretical results are computed in various parameter sets such as NLSH, NL1, NL2, Nl3 and FSUG. We analyzed the predictive power of these parameter sets and discussed various aspects of the compressibility modulus.
This paper is organized as follow: In Section II we have summarized the theoretical formalisms, which are useful for the present analysis. In section III, we have given the discussions of our results. Here, we have elaborated the giant monopole resonance obtained by various parameter sets and their connectivity with compressibility modulus. The last section is devoted to a summary and concluding remarks.
Ii Theoretical framework
The principle of scale invariance is used to obtain the virial theorem for the relativistic mean field serot86 () theory by working in the relativistic Thomas–Fermi (RTF) and relativistic extended ThomasFermi (RETF) approximations patra01 (); acentel93 (); acentel92 (); spei98 (); mario98 (); mario93a (). Although, the scaling and constrained calculations are not new, the present technique is developed first time by Patra et al patra01 (); mario10 (), which is different from other scaling formalisms.
The detail formalisms of the scaling method are given in Refs. patra01 (); patra02 (). For completeness, we have outlined briefly some of the essential expressions, which are needed for the present purpose. We have worked with the nonlinear Lagrangian of Boguta and Bodmer bo77 () to include the manybody correlation arises from the nonlinear terms of the meson selfinteraction schiff51 (); fujita57 (); steven01 () for nuclear manybody system. The nuclear matter compressibility modulus also reduces dramatically by the introduction of these terms, which motivates to work with this nonliner Lagrangian. The relativistic mean field Hamiltonian for a nucleonmeson interacting system is written by serot86 (); patra01 ():
(1)  
(2) 
Here , , and are the masses for the nucleon (with being the effective mass of the nucleon), , and mesons, respectively and is the Dirac spinor. The field for the meson is denoted by , for meson by , for meson by ( as the component of the isospin) and for photon by . , , and =1/137 are the coupling constants for the , , mesons and photon respectively. and are the nonlinear coupling constants for mesons. By using the classical variational principle we obtain the field equations for the nucleon and mesons. In semiclassical approximation, we can write the above Hamiltonian in term of density as:
(3) 
where
(4) 
(5) 
(6) 
(7) 
and is the free part of the Hamiltonian. The total density is the sum of proton and neutron densities. The semiclassical groundstate meson fields are obtained by solving the Euler–Lagrange equations ().
(8) 
(9)  
(10) 
(11) 
(12) 
The above field equations are solved selfconsistently in an iterative method.
(13) 
with
(14) 
In order to study the monopole vibration of the nucleus we have scaled the baryon density patra01 (). The normalized form of the baryon density is given by
(15) 
is the collective coordinate associated with the monopole vibration. As Fermi momentum and density are interrelated, the scaled Fermi momentum is given by
(16) 
Similarly , , and Coulomb fields are scaled due to selfconsistence eqs. (710). But the field can not be scaled simply like the density and momentum, because the source term of field contains the field itself. In semiclassical formalism, the energy and density are scaled like
(17)  
(18) 
The symbol shows an implicit dependence of . With all these scaled variables, we can write the Hamiltonian as:
(19)  
(20) 
Here we are interested to calculate the monopole excitation energy which is defined as with is the restoring force and is the mass parameter. In our calculations, is obtained from the double derivative of the scaled energy with respect to the scaled coordinate at and is defined as patra01 ():
(21)  
and the mass parameter of the monopole vibration can be expressed as the double derivative of the scaled energy with the collective velocity as
(22) 
where is the displacement field, which can be determined from the relation between collective velocity and velocity of the moving frame,
(23) 
with is the transition density defined as
(24) 
taking . Then the mass parameter can be written as . In nonrelativistic limit, and the scaled energy is . The expressions for and can be found in bohigas79 (). Along with the scaling calculation, the monopole vibration can also be studied with constrained approach bohigas79 (); maru89 (); boer91 (); stoi94 (); stoi94a (). In this method, one has to solve the constrained functional equation:
(25) 
Here the constrained is . The constrained energy can be expanded in a harmonic approximation as
(26) 
The second order derivative in the expansion is related with the constrained compressibility modulus for finite nucleus as
(27) 
and the constrained energy as
(28) 
In the nonrelativistic approach, the constrained energy is related by the sum rule weighted . Now the scaling and constrained excitation energies of the monopole vibration in terms of the nonrelativistic sum rules will help us to calculate , i.e. the resonance width bohigas79 (); centelle05 (),
(29) 
Iii Results and Discussions
It is interesting to apply the model to calculate the excitation energy of Sn isotopic series and compared with experimental results. Thus, we calculate the GMR energy using both the scaling and constraint methods in the framework of relativistic extended ThomasFermi approximation using various parameter sets for Z= 48 and 50 and compared with the excitation energy with momentum ratio and obtained from multipule decomposition analysis (MDA). The basic reason to take a number of parameter sets is that the infinite nuclear matter compressibility of these forces cover a wide range of values. For example, NLSH has compressibility 399 MeV, while that of NL1 is 210 MeV. From MDA analysis we get different momentum ratio, such as , and . These ratios are connected to scaling, centroid and constraint energies, respectively. That is why we compared our theoretical scaling result with and with the constrained calculations.
In Figures 1 and 2 we have shown the and ratio for isotopic chains of Cd and Sn. The results are also compared with experimental data obtained from (RCNP) ugarg (); tli07 (); ugarg07 (). From the figures, it is cleared that the experimental value lies between the results obtained from FUSG (FSUGold) and NL1 force parameters. It is to be noted that, throughout the calculations, we have used only the nonlinner parameter sets for their excellent prediction of nuclear observables with the experimental data. If one compares the experimental and theoretical results for Pb, the FSUG set gives better results amongst all. For example, the experimental and theoretical data are 14.170.1 and 14.04 MeV, respectively. These values are well matched with each other. From this, one could conclude that the infinite nuclear matter compressibility lies nearer to that of FSUG (230.28 MeV) parameter. But experimental result on ISGM in RCNP shows that the predictive power of FUSG is not good enough for the excitation energy of Sn isotopes. This observation is not only confined to RCNP formalism, but also persists in the more sophisticated RPA approach. In Table I, we have given the results for QRPA(T6), RETF(FSUG) and RETF(NL1). The experimental data are also given to compare all these theoretical results.
Nucleus  (MeV)  (MeV)  

QRPA(T6)  RETF(FSUG)  RETF(NL1)  Expt.  QRPA(T6)  RETF(FSU)  RETF(NL1)  Expt.  
Sn  17.3  17.42  15.86  16.7  17.0  17.2  15.39  16.1 
Sn  17.2  17.32  15.75  16.5  16.9  16.9  15.28  15.9 
Sn  17.1  17.19  15.63  16.3  16.8  16.77  15.15  15.7 
Sn  17.0  17.07  15.51  16.3  16.6  16.63  15.03  15.6 
Sn  16.9  16.94  15.38  16.2  16.5  16.44  14.89  15.5 
Sn  16.8  16.81  15.24  15.9  16.4  16.34  14.75  15.2 
Sn  16.7  16.67  15.1  15.8  16.2  16.19  14.6  15.1 
The infinite nuclear matter compressibility with T6 parameter set is 236 MeV and that of FSUG is 230.28 MeV. The difference in between these two sets is only 6 MeV. The similarity in compressibility (small difference in ) may be a reason for their prediction in equal value of GMR. The table shows that, there is only 0.1 MeV difference in QRPA(T5) and RETF(FSU) results in the GMR values for SnSn isotopes, but the results are exactly matched for the SnSn. This implies that for relatively higher mass nuclei, both the QRPA(T6) and RETF(FSUG) results are almost similar. If some one consider the experimental value of Sn isotopic series, then QRPA(T5) gives better result. For example, experimental value of for Sn is 16.7 0.2 MeV and that for QRPA(T5) is 16.6 MeV. These two values matches well with each other. The infinite nuclear matter compressibility of T5 set is 202 MeV. It is shown by V. Tselyaev et al. tsel09 () that the T5 parameter set with such compressibility, better explains the excitation energy of Sn isotopes, but fails to predict the excitation energy of Pb. It over estimates the data for Pb. The experimental data of ISGMR energies for Zr and Sn lies in between the calculated values of T5 and T6 forces. In summary, we can say that the RPA analysis predicts the symmetric nuclear matter compressibility within MeV and our semiclassical calculation gives it in the range MeV. These two predictions almost agree with each other in the acceptable limit.
Nuclear Mass  (MeV)  

pairing+MEM  Our work  Expt.  our work  Expt.  
13.4  13.6  13.70.1  2.02  3.30.2  
13.4  13.51  13.60.1  2.03  2.80.2  
13.4  13.44  13.50.1  2.03  3.30.2 
In Table II, we have displayed the data obtained from a recent experiment dpatel13 () and compared our results. Column two of the table is also devoted to the result obtained from pairing plus MEM effect khan09 (). The data show clearly that our result (extended ThomasFermi) has a priority over the pairing + MEM prediction. For example, the difference between the pairing+MEM results and experimental observation is 0.3 MeV for Pb isotopes, which is away from the experimental error, while it is only 0.1 MeV (within the error bar) in the RETF and data. This trend also followed by Pb and Pb nuclei. In our model, we have not included any pairing externally. But still our results are good enough in comparision with MEM+pairing. This implies two things: (i) pairing may not be impotant in calculation of excitation energy or (ii) pairing effect is automatically included in ThomasFermi calculations.
To our understanding, the second option seems to be more appropiate, because lots of work show that pairing must be included for the calculation of excitation energy of open shell nuclei. Fig. 3 shows the variation of excitation energy with protonneutron asymetry in Sn isotopes. Here, we want to know, how the excitation energy varies with asymetry or more specifically, ”is the variation of experimental excitation energy with asymetry same that of theoritical one ?”. The graph shows that the variation with both NL1 and FSUG are following similar partten as experimental one with a different mangnitued as shown in the figure.
In Figure 4, we have compared the results obtained from RETF and RTF with the experimental data for Sn isotopes. The celebraty NL3 parameter set is used in the calculations. The graph shows that there is only a small difference ( MeV) in RETF and RTF results. Interestingly, the RETF correction is additive to the RTF result instead of softening the excitation energy of Sn isotopes. Then the natural question arises: is it the brhavior for all the parameter sets in RETF approximation ?. To attend the question, we plotted Fig. 5, where we have shown the difference of obtained with RETF and RTF results (RETFRTF) for various parameter sets. For all sets, except NL1, we find RETFRTF as positive. Thus, it is a challenging task to antangle the term which is the responsible factor to determine the sign of RETFRTF. Surprisingly, for most of the parameter sets, RTF is more towards experimental data. Inspite of this, one cannot says anything about the qualitative behavior of RETF. Because, the variation of the density at the surface taken care properly by RETF formalism, which is essential. One more interesting observation is that, when one investigate the variation of RETFRTF in the isotopic chain of Sn, it remains almost constant for all the parameter sets, except FSUG. In this context, FSUG behaves differently.
Variation of RETFRTF with neutronproton asymmetry for FSUG set shows that, there may be some correlation of RETF with the symmetry energy. This is clearly absent in all other parameter sets. Now it is essential to know, in which respect the FSUG parameter set is different from other. The onetoone interaction terms for NL3, NL2, NL1 and NLSH all have similar couplings. However, the FSUG is different from the above parameters in two aspect, i.e., two new coupling constants are added. One corresponds to the selfinteraction of and other one corresponds to the isoscalarisovector meson coupling. It is known that selfinteraction of is responsible for softening the EOS gmuca (); toki94 (); bodmer91 () and the isoscalarisovector coupling takes care of the softening for symmetry energy of symmetric nuclear matterhorow02 (). The unique behavior shown by the FSUG parametrization may be due to the following three reasons:

introduction of isoscalarisovector meson coupling .

introduction of selfcoupling of meson.

Or simultaneous introduction of both these two terms with refitting of parameter set with new constraint.
In order to discuss the first possibility, we plotted NL3+(0.03) in Figure 6. The graph shows that there is no difference between NL3 and NL3+, except the later set predicts a more possitive RETFRTF. It is well known that, the addition of coupling, i.e., NL3+(0.03) gives of a softer symmetry energy patra9 (). This implies that, models with softer energy have greater difference in RETF and RTF. At a particular protonneutron asymmetry, RETFRTF has a larger value for a model with softer symmetry energy. This observation is not conclusive, because all the parameter sets do not follow this type of behavior. Quantitatively, the change of RETFRTF in the Sn isotopic series is about , while this is only in NL3 and other parameter set.
In Table III, we have listed the meson contribution to the total energy. From the analysis of our results, we find that only contribution to the total binding energy change much more than other quantity, when one goes from RTF to RETF. But this change is more prominent in FSUG parameter set than other sets like NL1, NL2, NL3 and NLSH. Simple assumption says that, may be the absent of term in other parameter is the reason behind this. But we have checked for the parameter NL3+, which does not follow. This also shows similar behavior like other sets. In Table IV, we have given the results for FSUG, NL3+ and NL1. The data show clearly that, there is a huge difference of monopole excitation energy in RETF and RTF with FSUG parameter set. For example, the meson contribution to the GMR in RETF for Sn is 21.85 MeV, while in RTF it is only 0.00467 MeV.
Mass  FSUG  NL3(0.03)  NL1  

RETF  RTF  RETF  RTF  RETF  RTF  RETF  RTF  
112  21.85  6.66  0.00130  0.00467  20.60  20.12  17.91  16.73 
114  28.72  8.64  0.00202  0.00664  27.11  26.48  23.51  22.00 
116  36.42  10.83  0.00248  0.00829  34.40  33.62  29.73  27.90 
118  44.87  13.23  0.00298  0.01013  42.42  41.499  36.52  34.37 
120  54.04  15.82  0.00353  0.01213  51.13  50.05  43.84  41.37 
122  63.88  18.58  0.00411  0.01429  60.48  59.24  51.63  48.85 
124  74.33  21.49  0.00473  0.01660  70.43  69.03  59.84  56.76 
However, this difference is nominal in NL3+ parameter set, i.e., it is only 0.48 MeV. Similarly, this value is 1.18 MeV in NL1 set. The contribution of meson to total energy comes from two terms: (i) one from and other (ii) from . We have explicitly shown that contribution comes from makes a huge difference between the GMR obtained from RETF and RTF formalisms. This type of contribution does not appear from NL3+. For example, in Sn the contribution of with RETF formalism is 6.0878 MeV, while with RTF formalism is 5.055 MeV.
The above discussion gives us a significant signiture that the contribution of may be responsible for this anomalous behavior. But an immediate question arises , why NL3+ parameter set does not show such type of effects, inspite of having term. This may be due to the procedure in which term is added in two parameters. In NL3+(0.03), the term is not added independently. The and are interdependent to each other to fix the binding energy BE and difference in neutron and proton rms radii . But in FSUGold, coupling constant is added independenly to reproduce the nuclear observables.
Nuclear Mass  NL3  FSUGOLD  

164.11  149.96  145  147.37  134.57  138.42  
164.64  155.39  131.57  147.11  139.71  127.64  
P  136.70  110.43  105  123.40  100.36  102.53 
Ca  145.32  134.47  105  130.93  123.15  102.53 
In Table IV, we have listed the compressibility of some of the selected nuclei in scaling and constraint icalculations. This results are compared with the computed values obtained from EOS model. To evaluate the compressibility from EOS, we have followed the procedure discussed in centelles09 (); shailesh13 (). M. Centelles et al centelles09 (), parameterised the density for finite nucleus as and obtained the asymmetry coefficient of the nucleus with mass A from the EOS at this particular density. Here also, we have used the same parametric from of the density and obtained the compressibility of finite nucleus from the EOS. For example, for in FSUG parameter set. We have calculated the compressibility from the EOS at this particular density, which comes around 145 MeV. We have also calculated the compressibility independently in ThomasFermi and extended ThomasFermi using scaling and constraint calculations, which are 161 MeV and 146.1 MeV, respectively.
Iv Summary and Conclusion
In summary, we analysed the predictive power of various force parameters, like NL1, NL2, NL3, NlSH and FSUG in the framework of relativistic ThomasFermi and relativistic extended ThomasFermi approaches for giant monopole excitation energy of Snisotopes. The calculation is then extended to some other relevant nuclei. The analysis shows that ThomasFermi approximation gives better resluts than pairing+MEM data. It exactly reproduces the experimental data for Sn isotopes, when the compressibility of the force parameter is within MeV. We also concluded that a parameter set can reproduce the excitation energy of Sn isotopes, if its infinite nuclear matter compressibility lies within MeV, however, fails to reproduce the GMR data for other nuclei within the same accuracy.
We have qualitatively analized the difference in GMR energies RETFRTF using RETF and RTF formalisms in various force parameters. The FSUGold parameter set shows different behavior from all other forces. Also, extended our calculations of monopole excitation energy for Sn isotopes with a force parametrization having softer symmetry energy (NL3+ ). The excitation energy decreses with the increse of protonneutron asymetry agreeing with the experimental trend. In conclusion, after all these thorough analysis, it seems that the softening of Sn isotopes is an open problem for nuclear theory and more work in this direction are needed.
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