Large deformed structures in Ne-S nuclei near neutron drip-line
aa r X i v : . [ nu c l - t h ] F e b Large deformed structures in
N e − S nuclei near neutron drip-line S.K. Patra and C.R. Praharaj
Institute of Physics, Sachivalaya Marg, Bhubaneswar-751 005, India (Dated: October 12, 2018)The structure of Ne, Na, Mg, Al, Si and S nuclei near the neutron drip-line region is investigated in the frame-work of relativistic meannfield (RMF) and non-relativistic Skyrme Hartree-Fock formalisms. The drip-line ofthese nuclei are pointed out. We analysed the large deformation structures and many of these neutron rich nucleiare quite deformed. New magic number are seen for these deformed nuclei.
PACS numbers: 21.10.-k, 21.10.dr, 21.10.Ft, 21.30.-x, 24.10.-1, 24.10.Jv
I. INTRODUCTION
The structure of light nuclei near the neutron drip-line isvery interesting for a good number of exotic phenomena. Nu-clei in this region are very different in collectivity and clus-tering features than the stable counterpart in the nuclear chart.For example, the neutron magicity is lost for the N=8 nucleusfor Be [1] and N=20 for Mg [2]. The discovery of largecollectivity of Mg by Iwasaki et al. [3] is another example ofsuch properties. The deformed structures and core excitationsof Mg and neighboring nuclei and location of drip-line in thismass region is an important matter [4]. On the other hand, theappearance of N=16 magic number for O is well established[5]. The discovery of the two isotopes Mg and Al, oncepredicted to be drip-line nuclei [6, 7] gives indication that theneutron drip-line is located towards the heavier mass region.The existence of neutron halo in Li is well established andthe possibility of proton halo in B and the neutron halo in Be and B are very interesting phenomena for the drip-linenuclei. In addition to these above exciting properties, the clus-ter structure of entire light mass nuclei and the skin formationin neutron-drip nuclei provide us features for the study of lightmass drip-line nuclei. Also, the exotic neutron drip-line nucleiplay a role in many astrophysical studies. In this paper, ouraim is to study the neutron drip-line for Ne-S isotopic chainin the frame-work of a relativistic mean field (RMF) and non-relativistic Skyrme Hartree-Fock formalism and analyse thelarge deformation of these isotopes.The paper is organised as follows: The relativistic and non-relativistic mean field formalisms are described very brieflyin Section II. The results obtained from the relativistic meanfield (RMF) and Skyrme-Hartree-Fock (SHF) formalisms, anda discussion of these results, are presented in Section III. Fi-nally summary and concluding remarks are given in SectionIV.
II. THEORETICAL FRAMEWORK
Mean field methods have been widely used in the study ofbinding energies and other properties of nuclei [8, 9]. Al-though the older version of the SHF and RMF models havesome limitation to reproduce some of the observables, the re-cent formalisms are quite efficient to predict the bulk prop-erties of nuclei not only near the stability valley, but also for the nuclei near the proton and neutron drip-lines. We use heretwo of the successful mean field models [8, 10–16] (SkyrmeHartree-Fock and the Relativistic Mean Field ) to learn aboutthe properties of drip-line nuclei
N e − S . A. The Skyrme Hartree-Fock (SHF) Method
There are many known parametrizations of Skyrme interac-tion which reproduce the experimental data for ground-stateproperties of finite nuclei and for the observables of infinitenuclear matter at saturation densities, giving more or lesscomparable agreements with the experimental or expectedempirical data. The general form of the Skyrme effective in-teraction, used in the mean-field models, can be expressed asa density functional H [10, 11], given as a function of someempirical parameters, as H = K + H + H + H eff + · · · (1)where K is the kinetic energy term, H the zero range, H the density dependent and H eff the effective-mass depen-dent terms, which are relevant for calculating the propertiesof nuclear matter. These are functions of 9 parameters t i , x i ( i = 0 , , , ) and η , and are given as H = 14 t (cid:2) (2 + x ) ρ − (2 x + 1)( ρ p + ρ n ) (cid:3) , (2) H = 124 t ρ η (cid:2) (2 + x ) ρ − (2 x + 1)( ρ p + ρ n ) (cid:3) , (3) H eff = 18 [ t (2 + x ) + t (2 + x )] τ ρ + 18 [ t (2 x + 1) − t (2 x + 1)] ( τ p ρ p + τ n ρ n ) . (4)The kinetic energy K = ~ m τ , a form used in the Fermi gasmodel for non-interacting fermions. The other terms, repre-senting the surface contributions of a finite nucleus with b and b ′ as additional parameters, are H Sρ = 116 (cid:20) t (1 + 12 x ) − t (1 + 12 x ) (cid:21) ( ~ ∇ ρ ) − (cid:20) t ( x + 12 ) + t ( x + 12 ) (cid:21) × h ( ~ ∇ ρ n ) + ( ~ ∇ ρ p ) i , and (5) H S ~J = − h b ρ~ ∇ · ~J + b ′ ( ρ n ~ ∇ · ~J n + ρ p ~ ∇ · ~J p ) i . (6)Here, the total nucleon number density ρ = ρ n + ρ p , the ki-netic energy density τ = τ n + τ p , and the spin-orbit density ~J = ~J n + ~J p . The subscripts n and p refer to neutron andproton, respectively, and m is the nucleon mass. The ~J q = 0 , q = n or p , for spin-saturated nuclei, i.e., for nuclei with ma-jor oscillator shells completely filled. The total binding energy(BE) of a nucleus is the integral of the density functional H .At least eighty-seven parametrizations of the Skyrme inter-action are published since 1972 [12] where b = b ′ = W , wehave used here the Skyrme SkI4 set with b = b ′ [13]. Thisparameter set is designed for considerations of proper spin-orbit interaction in finite nuclei, related to the isotope shiftsin Pb region and is better suited for the study of exotic nu-clei. Several more recent Skyrme parameters such as SLy1-10, SkX, SkI5 and SkI6 are obtained by fitting the Hartree-Fock (HF) results with experimental data for nuclei startingfrom the valley of stability to neutron and proton drip-lines[10, 13, 14, 17]. B. The Relativistic Mean Field (RMF) Method
The relativistic mean field approach is well-known and thetheory is well documented [15, 16]. Here we start with the rel-ativistic Lagrangian density for a nucleon-meson many-bodysystem, as L = ψ i { iγ µ ∂ µ − M } ψ i + 12 ∂ µ σ∂ µ σ − m σ σ − g σ − g σ − g s ψ i ψ i σ −
14 Ω µν Ω µν + 12 m w V µ V µ + 14 c ( V µ V µ ) − g w ψ i γ µ ψ i V µ − ~B µν . ~B µν + 12 m ρ ~R µ . ~R µ − g ρ ψ i γ µ ~τ ψ i . ~R µ − F µν F µν − eψ i γ µ (1 − τ i )2 ψ i A µ . (7)All the quantities have their usual well known meanings.From the relativistic Lagrangian we obtain the field equationsfor the nucleons and mesons. These equations are solvedby expanding the upper and lower components of the Diracspinors and the boson fields in an axially deformed harmonicoscillator basis with an initial deformation. The set of coupledequations is solved numerically by a self-consistent iterationmethod.The centre-of-mass motion energy correction is esti-mated by the usual harmonic oscillator formula E c.m. = (41 A − / ) . The constant gap BCS pairing is used to add thepairing effects for the open shell nuclei. It is to be noted thatin the present work only intrinsic state solutions are presented.Each of these deformed intrinsic states is a superposition ofvarious angular momenta states. To obtain the good angu-lar momentum states and spectroscopic predictions for thesenuclei near neutron drip-line we need to project out states ofgood angular momenta. Such calculation will be consideredas a future extension of this work. The quadrupole momentdeformation parameter β is evaluated from the resulting pro-ton and neutron quadrupole moments, as Q = Q n + Q p = q π ( π AR β ) . The root mean square (rms) matter ra-dius is defined as h r m i = A R ρ ( r ⊥ , z ) r dτ ; here A is themass number, and ρ ( r ⊥ , z ) is the deformed density. The to-tal binding energy and other observables are also obtained byusing the standard relations, given in [16]. We use the wellknown NL3 parameter set [18]. This set not only reproducesthe properties of stable nuclei but also well predicts for thosefar from the β -stability valley. Also, the isoscalar monopoleenergy agrees excellently with the experimental values for dif-ferent regions of the Periodic Table. The measured superde-formed minimum in Hg is 6.02 MeV above the groundstate, whereas in RMF calculation with NL3 set, this num-ber is 5.99 MeV [18]. All these facts give us confidence touse this older, though very much still in use, NL3 set for thepresent investigation.
III. RESULTS AND DISCUSSIONA. Ground state properties from the SHF and RMF models
There exists a number of parameter sets for the standardSHF and RMF Hamiltonians and Lagrangians. In some of ourprevious papers and of other authors [16, 18–21] the groundstate properties, like the binding energies (BE), quadrupolemoment deformation parameters β , charge radii ( r c ) andother bulk properties are evaluated by using the various non-relativistic and relativistic parameter sets. It is found that,more or less, most of the recent parameters reproduce wellthe ground state properties, not only of stable normal nucleibut also of exotic nuclei which are away from the valley ofbeta-stability. So, if one uses a reasonably acceptable param-eter set the prediction of the results will remain nearly forceindependent. This is valid both for SHF and RMF formalisms.However, with a careful inspection of these parametrizations,some of the SHF and RMF sets can not reproduce the empir-ical data. In this context we can cite the deviation of isotopicshifts than the experimental data [22] for Pb nuclei while us-ing SHF forces like, SkM* values [23]. However, the RMFsets reproduce the kink quite nicely [24]. On the other hand,most of the RMF sets over estimate the nuclear matter incom-pressibility. In general, the predictive power of both the for-malisms are reasonably well and can be comparable to eachother, which can be seen in the subsequent subsections. In ad-dition to this, the general results SHF (SkI4) and RMF (NL3)forces are similar for the considered region. Thus in the sub- TABLE I:
The calculated ground state binding energy obtained fromSHF and RMF theory are compared with the experimentally knownheaviest isotope for Ne, Na, Mg, Al, Si and S [25]. nucleus RMF SHF Expt. nucleus RMF SHF Expt. Ne 215.1 210.6 211.2 Na 237.9 234.5 232.8 Mg 257.7 255.1 256.2 Al 285.2 281.9 283.1 Si 310.1 305.1 306.6 S 353.5 350.2 354.2TABLE II:
The predicted neutron drip-line for Ne, Na, Mg, Al, Siand S in RMF (NL3) and SHF (SKI4) parameter sets are comparedwith prediction of infinite nuclear matter (INM) mass model [26],finite range droplet model (FRDM) [27] and experimental data [25]along with the number shown in parenthesis are the experimentallyextrapolated values. nucleus RMF SHF INM FRDM Expt.Ne 34 34 36 32 30 (34)Na 40 37 40 36 33 (37)Mg 40 40 46 40 34 (40)Al 48 48 48 45 39 (42)Si 54 48 50 48 41 (44)S 55 55 53 50 45 (49) sequent results during our discussion we will refer the resultsof RMF (NL3) calculations, except some specific cases. Thusthe result of SHF (SkI4) are not displayed in Tables.
B. Binding energy and neutron drip-line
The ground state binding energy (BE) are calculated for Ne,Na, Mg, Al, Si and S isotopes near the neutron drip-line. Thisis done by comparing the prolate, oblate and spherical solu-tion of binding energy for a particular nucleus. For a given nu-cleus, the maximum binding energy corresponds to the groundstate and other solutions are obtained as various excited intrin-sic states. In Table I, the ground state binding energy for theheaviest isotopes for the nuclei discussed are compared withthe experimental data [25]. From the Table it is observed thatthe calculated binding energies are comparable with SHF andRMF results. We have listed the neutron drip-lines in TableII, which are obtained from the ground state binding energyfor neutron rich Ne, Na, Mg, Al, Si and S nuclei. The nu-clei with the largest neutron numbers so far experimentallydetected in an isotopic chain till date, known as experimentalneutron drip-line are also displayed in this Table for compari-son. The numbers given in the parenthesis are the experimen-tally extrapolated values[25]. To get a qualitative understand-ing of the prediction of neutron drip-line, we have comparedour results with the infinite nuclear matter (INM) [26] and fi-nite range droplet model (FRDM) [27] mass estimation. Fromthe table, it is clear that all the predictions for neutron drip-lineare comparable to each other.The drip-lines are very important after discovery of the twoisotopes Mg and Al [6] that here once predicted to be be-yond the drip-line [7, 28]. This suggests that the drip-line issomewhere in the heavier side of the mass prediction whichare beyond the scope of the present mass models [7, 28]. In
TABLE III:
The calculated value of charge radius ( r ch ), quadrupolemoment deformation parameter β and binding energy (BE) for Ne,Na and Mg nuclei in RMF (NL3) formalism. The maximum bindingenergy is the ground state solution and all other values are the in-trinsic excited state solution. The radius r ch is in fm and the bindingenergy is in MeV. nucleus r ch β BE(MeV) nucleus r ch β BE(MeV) Ne 2.970 0.535 156.7 Ne 2.901 -0.244 152.0 Ne 2.953 0.516 165.9 Ne 2.889 -0.241 161.1 Ne 2.940 0.502 175.7 Ne 2.881 -0.242 170.5 Ne 2.913 0.386 181.8 Ne 2.880 -0.249 179.7 Ne 2.890 0.278 188.9 Ne 2.881 -0.259 188.9 Ne 2.907 0.272 194.2 Ne 2.886 -0.206 194.2 Ne 2.926 0.277 199.9 Ne 2.893 -0.159 199.9 Ne 2.945 0.247 203.9 Ne 2.925 -0.183 203.9 Ne 2.965 0.225 208.2 Ne 2.957 -0.203 208.2 Ne 2.981 0.161 211.2 Ne 2.974 -0.133 211.2 Ne 2.998 0.100 215.0 Ne 2.995 -0.081 215.0 Ne 3.031 0.244 216.0 Ne 3.013 -0.133 216.0 Ne 3.071 0.373 218.6 Ne 3.033 -0.180 218.6 Ne 3.095 0.424 219.5 Ne 3.044 -0.230 219.5 Ne 3.119 0.473 220.9 Ne 3.054 -0.275 220.9 Ne 3.132 0.505 220.4 Ne 3.064 -0.315 215.7 Ne 3.146 0.539 220.3 Ne 3.075 -0.352 220.4 Na 2.939 - 0.250 189.4 Na 2.964 0.379 192.3 Na 2.938 -0.258 200.3 Na 2.937 0.273 200.6 Na 2.940 -0.202 206.3 Na 2.965 0.295 207.1 Na 2.946 -0.157 212.5 Na 2.993 0.323 214.2 Na 2.980 -0.184 217.7 Na 2.993 0.272 219.0 Na 3.012 -0.205 223.4 Na 3.004 0.232 224.3 Na 3.025 -0.131 227.5 Na 3.031 0.169 228.1 Na 3.043 -0.074 232.5 Na 3.047 0.108 232.7 Na 3.061 -0.129 233.0 Na 3.077 0.237 234.5 Na 3.082 -0.179 234.3 Na 3.113 0.356 237.9 Na 3.095 -0.226 234.8 Na 3.137 0.404 239.8 Na 3.108 -0.270 235.9 Na 3.161 0.450 242.3 Na 3.121 -0.308 236.9 Na 3.175 0.481 242.5 Na 3.135 -0.345 238.4 Na 3.190 0.512 243.1 Na 3.156 -0.359 240.0 Na 3.199 0.491 243.4 Na 3.180 -0.375 241.8 Na 3.209 0.472 244.1 Na 3.184 -0.358 241.3 Na 3.228 0.477 243.4 Mg 3.043 0.487 194.3 Mg 3.001 -0.256 186.8 Mg 3.009 0.376 202.9 Mg 2.993 -0.261 199.1 Mg 2.978 0.273 212.5 Mg 2.990 -0.268 211.6 Mg 3.015 0.310 220.2 Mg 2.988 -0.204 218.2 Mg 3.048 0.345 228.7 Mg 2.992 -0.154 225.6 Mg 3.055 0.289 234.3 Mg 3.027 -0.186 232.0 Mg 3.062 0.241 240.5 Mg 3.059 -0.207 239.0 Mg 3.131 0.599 237.7 Mg 3.075 0.179 245.1 Mg 3.068 -0.128 244.0 Mg 3.090 0.119 250.5 Mg 3.085 -0.067 249.9 Mg 3.131 0.471 248.8 Mg 3.117 0.233 253.1 Mg 3.102 -0.126 251.5 Mg 3.150 0.343 257.3 Mg 3.124 -0.181 253.9 Mg 3.184 0.588 254.1 Mg 3.173 0.388 260.5 Mg 3.141 -0.196 255.3 Mg 3.198 0.432 263.9 Mg 3.160 -0.213 257.0 Mg 3.212 0.462 264.9 Mg 3.179 -0.258 258.8 Mg 3.227 0.492 266.3 Mg 3.198 -0.300 261.1 Mg 3.237 0.473 267.8 Mg 3.216 -0.338 263.4 Mg 3.247 0.456 269.7 Mg 3.234 -0.374 266.4
TABLE IV:
Same as Table III, but for Al, Si and S. nucleus r ch β BE(MeV) nucleus r ch β BE(MeV) Al 3.097 0.388 182.3 Al 3.077 -0.258 179.4 Al 3.072 0.381 197.7 Al 3.060 -0.266 193.9 Al 3.191 0.550 206.6 Al 3.052 -0.275 207.8 Al 3.215 0.572 217.0 Al 3.053 -0.292 221.9 Al 3.178 0.471 226.7 Al 3.037 -0.208 238.6 Al 3.061 0.251 239.3 Al 3.033 -0.141 245.6 Al 3.073 0.207 246.2 Al 3.070 -0.184 253.8 Al 3.085 0.170 253.6 Al 3.101 -0.205 259.8 Al 3.101 0.113 260.0 Al 3.103 -0.111 261.2 Al 3.118 0.057 267.2 Al 3.165 -0.333 269.4 Al 3.139 0.159 269.9 Al 3.134 -0.108 275.1 Al 3.167 0.268 274.1 Al 3.157 -0.172 272.8 Al 3.187 0.313 277.6 Al 3.173 -0.189 277.7 Al 3.208 0.355 281.5 Al 3.191 -0.208 280.3 Al 3.275 0.418 282.7 Al 3.214 -0.254 283.5 Al 3.285 0.406 285.1 Al 3.236 -0.299 286.7 Al 3.304 0.441 287.6 Al 3.257 -0.336 290.4 Al 3.325 0.474 290.5 Al 3.278 -0.370 290.6 Al 3.348 0.483 290.7 Al 3.281 -0.355 291.2 Al 3.371 0.491 291.3 Al 3.282 -0.338 292.2 Al 3.375 0.456 291.0 Al 3.274 -0.288 293.6 Al 3.378 0.420 291.0 Al 3.271 -0.263 293.5 Al 3.359 0.341 294.5 Al 3.346 -0.296 294.0 Al 3.246 0.125 293.6 Al 3.432 0.660 290.8 Al 3.246 0.090 294.8 Al 3.335 -0.319 293.6 Al 3.447 0.653 290.4 Al 3.276 0.117 293.8 Al 3.319 -0.252 294.0 Si 3.162 0.294 169.1 Si 3.170 -0.278 169.3 Si 3.127 0.286 185.1 Si 3.139 -0.274 184.8 Si 3.099 0.282 201.8 Si 3.118 -0.280 200.9 Si 3.054 0.168 215.8 Si 3.114 -0.299 216.4 Si 3.017 0.001 231.4 Si 3.122 -0.331 232.1 Si 3.035 0.001 240.7 Si 3.093 -0.237 240.7 Si 3.070 0.148 250.6 Si 3.054 -0.060 250.4 Si 3.089 0.120 258.7 Si 3.108 -0.180 259.1 Si 3.109 0.104 267.2 Si 3.137 -0.201 268.5 Si 3.126 0.050 275.4 Si 3.131 -0.084 275.6 Si 3.148 0.000 284.4 Si 3.204 -0.336 278.5 Si 3.160 0.085 287.3 Si 3.161 -0.083 287.4 Si 3.184 0.193 291.4 Si 3.186 -0.162 291.5 Si 3.200 0.238 295.4 Si 3.201 -0.181 294.8 Si 3.218 0.281 299.8 Si 3.219 -0.204 298.8 Si 3.224 0.263 302.4 Si 3.245 -0.254 301.9 Si 3.232 0.244 305.4 Si 3.272 -0.301 306.0 Si 3.230 0.167 307.1 Si 3.295 -0.336 310.1 Si 3.228 0.013 309.8 Si 3.318 -0.369 314.6 Si 3.240 0.123 311.8 Si 3.320 -0.356 315.2 Si 3.252 0.172 314.3 Si 3.322 -0.342 316.2 Si 3.252 0.117 315.8 Si 3.316 -0.308 317.5 Si 3.253 0.053 317.9 Si 3.303 -0.262 319.3 Si 3.258 0.005 319.7 Si 3.345 -0.298 319.8 Si 3.263 0.001 321.8 Si 3.381 -0.321 320.8 Si 3.290 0.045 321.1 Si 3.366 -0.251 320.7 Si 3.319 0.074 321.1 Si 3.341 -0.159 321.5 Si 3.345 0.078 321.1 Si 3.358 -0.135 321.2 Si 3.371 0.082 321.4 Si 3.377 -0.112 321.2 Si 3.391 0.042 321.6 Si 3.391 -0.052 321.3 Si 3.415 0.000 322.3 Si 3.415 -0.010 322.0 TABLE V:
Same as Table III, but for S. nucleus r ch β BE(MeV) nucleus r ch β BE(MeV) S 3.241 0.197 275.5 S 3.233 -0.116 275.1 S 3.248 0.140 285.8 S 3.257 -0.168 286.5 S 3.260 0.077 295.6 S 3.260 -0.078 295.7 S 3.273 0.002 306.2 S 3.309 -0.308 299.7 S 3.285 0.152 311.6 S 3.287 -0.116 310.1 S 3.300 0.228 318.6 S 3.300 -0.164 316.9 S 3.312 0.264 325.3 S 3.307 -0.173 322.6 S 3.325 0.299 332.4 S 3.316 -0.181 328.5 S 3.331 0.287 337.7 S 3.324 -0.189 333.6 S 3.338 0.277 343.2 S 3.335 -0.207 339.2 S 3.359 0.318 347.2 S 3.348 -0.229 344.1 S 3.381 0.367 351.0 S 3.366 -0.263 349.5 S 3.375 0.312 353.4 S 3.367 -0.240 351.8 S 3.371 0.258 356.6 S 3.375 -0.237 355.1 S 3.385 0.257 358.5 S 3.380 -0.230 358.0 S 3.400 0.259 360.8 S 3.389 -0.230 360.8 S 3.403 0.227 362.9 S 3.420 -0.257 362.9 S 3.403 0.189 365.5 S 3.451 -0.277 365.0 S 3.427 0.188 366.4 S 3.451 -0.231 365.9 S 3.451 0.183 367.6 S 3.447 -0.178 367.6 S 3.463 0.158 369.1 S 3.466 -0.172 368.4 S 3.477 0.139 371.0 S 3.486 -0.142 369.8 S 3.494 0.105 371.4 S 3.497 -0.090 370.5 this calculations the newly discovered nuclei Mg and Alare well within the prediction both in the SHF and RMF for-malisms. Again a further comparison of the drip-line withRMF and SHF prediction, we find the drip-line predictions inboth calculations are well comparable, except for a few ex-ceptions in Na and Si as shown in Table II.
C. Neutron configuration
Analysing the neutron configuration for these exotic nuclei,we notice that, for lighter isotopes of Ne, Na, Mg, Al, Siand S the oscillator shell N osc = 3 is empty. However, the N osc = 3 shell gets occupied gradually with increase of neu-tron number. In case of Na, N osc = 3 starts filling up at Nawith quadrupole moment deformation parameter β = 0 . and − . with occupied orbits [330]1 / − and [303]7 / − ,respectively. The filling of N osc = 3 goes on increasingfor Na with neutron number and it is [330]1 / − , [310]1 / − , [321]3 / − and [312]5 / − at β = 0 . for Na. Again forthe oblate solution the occupation is [301]1 / − , [301]3 / − , [303]5 / − and [303]7 / − for β = − . for Na. Inthe case of Mg isotopes, even for , Mg, the N osc = 3 shell have some occupation for the low-lying excited statesnear the Fermi surface for M g (at β = 0.599 with Be =237.721 MeV the N osc =3 orbit is [330]7 / − and for M g : β = [330]1 / − , BE=248.804 MeV at β = 0 . ). With theincrease of neutron number in Mg and Si isotopic chain, theoscillator shell with N osc = 3 gets occupied more and more.For most of the Si isotopes, the oblate solutions are the dom-inating ones than the low-lying prolate superdeformed states, TABLE VI:
The calculated value of charge radii ( r ch ), quadrupoledeformation parameter β and binding energy (BE) for Ne, Mg, Siand S even-even nuclei in SHF (SkI4) formalism. The maximum bind-ing energy is the ground state solution and all other values are theintrinsic excited state solution. The radius r ch is in fm and the bind-ing energy is in MeV. nucleus r ch β BE(MeV) r ch β BE(MeV) Ne 3.029 0.5481 156.817 2.950 -0.1356 154.474 Ne 3.005 0.5223 175.758 2.943 -0.1989 172.758 Ne 2.952 0.2546 188.354 2.951 -0.2541 188.538 Ne 2.953 0.1233 199.380 2.944 0.0060 199.389 Ne 3.013 0.1623 206.524 3.010 -0.1334 206.433 Ne 3.054 0.0030 213.721 Ne 3.103 0.3808 213.118 3.118 0.3716 213.215 Ne 3.179 0.4880 213.483 3.108 -0.1462 209.695 Ne 3.203 0.6015 212.230 3.147 -0.2789 208.770 Mg 3.128 0.5248 195.174 3.077 -0.252 189.946 Mg 3.090 0.3623 212.885 3.079 -0.2988 213.153 Mg 3.111 0.3419 228.997 3.056 -0.1076 227.899 Mg 3.119 0.2022 240.328 3.117 -0.1835 240.514 Mg 3.145 0.0000 252.033 3.145 0.0000 252.033 Mg 3.209 0.3263 255.067 3.175 -0.1196 253.455 Mg 3.295 0.4884 201.512 3.252 -0.289 257.634 Mg 3.265 0.4413 259.899 3.213 -0.2124 255.368 Mg 3.321 0.4741 262.796 3.299 -0.3538 260.200 Si 3.117 0.009 231.037 3.194 -0.3494 233.590 Si 3.145 0.1477 252.146 3.168 -0.2102 252.625 Si 3.179 0.007 269.479 3.199 -0.1990 270.483 Si 3.216 0.000 286.332 Si 3.146 0.1549 292.418 3.241 -0.009 292.425 Si 3.291 0.3051 298.173 3.279 -0.1978 298.173 Si 3.325 0.2990 230.450 3.309 -0.2817 303.969 Si 3.349 0.3592 307.399 3.334 -0.3508 310.023 Si 3.334 0.2119 309.712 3.377 -0.3031 311.601 Si 3.337 0.009 312.451 3.372 -0.2405 313.508 Si 3.348 0.002 315.425 3.438 -0.2893 313.995 S 3.262 0.1491 241.434 3.178 -0.1856 241.385 S 3.271 0.200 208.173 3.256 -0.1700 267.975 S 3.288 0.1212 288.804 3.295 -0.1566 289.304 S 3.314 -0.003 309.619 S 3.341 0.2144 320.168 3.331 -0.1289 318.951 S 3.374 0.3042 332.097 3.350 -0.1529 327.809 S 3.392 0.2898 341.033 3.379 -0.2195 337.031 S 3.436 0.3677 348.266 3.410 -0.2714 346.445 S 3.423 0.2517 352.486 3.413 -0.2090 351.578 S 3.450 0.2379 356.188 3.430 -0.2032 356.589 S 3.441 0.1229 360.815 3.498 -0.2673 359.011 S 3.482 0.1099 362.347 3.487 -0.1356 362.531 S 3.528 0.003 364.650 3.524 -0.1023 363.926 S 3.558 0.1105 366.031 3.556 -0.0100 366.033 i.e. mass of the oblate solutions are the ground state solutionsand the prolate and some superdefomed are the excited con-figurations. Again, in S-isotopes, the prolate are the groundstate and the oblate are the extreme excited states. Note thatin many cases, there exist low laying superdeformed statesand some of them are listed in the Tables.
24 28 32 36 40 44 48 52 56-0.4-0.200.20.4 NeNaMgAlSiSmass number (A) d e f o r m a ti on p a r a m e t e r ( β ) NL3
FIG. 1:
The ground state quadrupole deformation parameter β ver-sus mass number A for Ne, Na, Mg, Al, Si and S isotopes near thedrip-line with NL3 parameter set. D. Quadrupole deformation
The ground and low-lying excited state deformation sys-tematics for some of the representative nuclei for Ne, Na,Mg, Al, Si and S are analysed. In Fig. 1, the ground statequadrupole deformation parameter β is shown as a functionof mass number for Ne, Na, Mg, Al, Si and S. The β valuegoes on increasing with mass number for Ne, Na and Mgisotopes near the drip-line. The calculated quadrupole defor-mation parameter β for Mg is 0.59 which compares wellwith the recent experimental measurement of Iwasaki et al [3]( β = 0 . ± ). Note that this superdeformed states in 3.2MeV above than the ground band. Again, the magnitude of β for the drip-line nuclei reduces with neutron number N andagain increases. A region of maximum deformation is foundfor almost all the nuclei as shown in the figure. It so happensin cases like, Ne, Na, Mg and Al that the isotopes are maxi-mum deformed which may be comparabled to superdeformednear the drip-line. For Si isotopes, in general, we find oblatesolution in the ground configurations. In many of the cases,the low-lying superdeformed configuration are clearly visibleand some of them are available in the Tables. E. Shape coexistence
One of the most interesting phenomena in nuclear structurephysics is the shape coexistence [29–31]. In many of the casesfor the nuclei considered here near the drip-line isotopes, theground state configuration accompanies a low-lying excitedstate. In few cases, it so happens that these two solutionsare almost degenerate. That means we predict almost similarbinding energy for two different configurations. For example,in the RMF calculation, the ground state binding energy of Ne is 189.093 MeV with β = − . and the binding en-ergy of the excited low-lying configuration at β = 0 . is188.914 MeV. The difference in BE of these two solutions isonly 0.179 MeV. Similarly the solution of prolate-oblate bind-ing energy difference in SkI4 is 0.186 MeV for Mg with
24 30 36 42 48 54 60-6-4-202468 NeMgSiS24 30 36 42 48 54 60 B E p - B E o ( M e V ) ARMF (NL3) SHF (SkI4)
FIG. 2:
The difference in binding energy between the prolate-oblatesolutions is shown for even-even Ne, Mg, Si and S isotopes near theneutron drip-line with NL3 and SkI4 parameter sets.
20 24 28 32 36 4005101520 MgSiSNe20 24 28 32 36 40RMF (NL3) SHF (SkI4)N S ( M e V ) FIG. 3:
The two-neutron separation S n energy versus neutron num-ber N for neutron-rich Ne, Mg, Si and S isotopes. β = − . and 0.202. This phenomenon is clearly avail-able in most of the isotopes near the drip-line. To show it ina more quantitative way, we have plotted the prolate-oblatebinding energy difference in Figure 2. The left hand side ofthe figure is for relativistic and the right side is the nonrela-tivistic SkI4 results. From the figure, it is clear that an islandof shape coexistence isotopes are available for Mg and Si iso-topes. These shape coexistence solutions are predicted takinginto account the intrinsic binding energy. However the actualquantitative energy difference of ground and excited config-uration can be given by performing the angular momentumprojection, which is be an interesting problem for future. F. Two neutron separation energy and new magic number
The appearance of new and the disappearance of knownmagic number near the neutron drip-line is a well discussedtopic currently in nuclear structure physics [5, 32]. Some ofthe calculations in recent past predicted the disappearance ofthe known magic number N=28 for the drip-line isotopes ofMg and S [33, 34]. However, magic number 20 retains its -60-50-40-30-20-10010 Mg β=0.588 neutron proton neutron proton β=0.343 β=0.471 neutron proton neutron proton β=0.119 Mg ε i ( M e V ) (1/2 + & 1/2 - )* ** * FIG. 4:
The / + and / − intrinsic single-particle states for thenormal and superdeformed state for Mg and Mg. A few of thelowest energy parity-doublet states of the superdeformed (SD) so-lutions are shown by asterisk for the SD configuration. More suchdoublets are noticed for the SD intrinsic states. The ± / − statesare denoted by shorter (and green) lines and the ± / + states aredenoted by longer ( and black). magic properties even for the drip-line region. In one of ourearlier publications, [35] we analysed the spherical shell gapat N=28 in S and its neighboring Mg and Si using NL-SH [22] and TM2 parameter sets [36]. The spherical shellgap at N=28 in S was found to be intact for the TM2 andis broken for NL-SH parametrization. Here, we plot the two-neutron separation energy S n of Ne, Mg, Si and S for theeven-even nuclei near the drip-line (fig 3). The known magicnumber N=28 is noticed to be absent in S. On the other handthe appearance of steep 2n-separation energy at N=34 both inRMF and SHF calculation looks quite prominent, and this isjust two units ahead than the experimental shell closure N=32[37].
G. Superdeformation and Low Ω parity doublets The deformation-driving m = 1 / − orbits come down inenergy in superdeformed solutions from the shell above, incontrast to the normal deformed solutions. The occurrence ofapproximate / + / − parity dobulets (degeneracy of | m | π = / + / − states) for the superdeformed solutions are clearlyseen in Figs. 4 and 5 where excited superdeformed configura-tions for Mg and Mg and for Al and Al are given.For each nucleus we have compared the normal deformed ( β ∼ . − . and the superdeformed configurations andanalysed the deformed orbits.The / + and / − states for the single particle levels areshown in Fig. 4 (for Mg and Mg). From the analysisof the results of this calculation, we have found a systematicbehaviour of the low Ω (particularly / + and / − ) prolatedeformed orbits for the superdeformed solutions. As repre- -60-50-40-30-20-10010 Al β=0.653 neutron proton neutron proton β=0.090 β=0.660 neutron proton neutronproton β=0.125 Al ε i ( M e V ) (1/2 + & 1/2 - )* ** * FIG. 5:
Same as Fig. 4 for Al and Al. sentive cases, we present here results for ( Mg − Mg) and( Al − Al) and plot the / + and / − orbits for the su-perdeformed and normal deformed shapes of these nuclei. Wenotice from the plot of the orbits that there is occurrence of / + and / − orbits very closeby in energy for the superde-formed (SD) shape. Two such / + and / − doublet struc-tures, marked in asterisk are shown in Figs. 4 and 5 for theSD solutions. Such / + / − degenerate orbits occur notonly for the well-bound orbits but also for the unbound con-tinuum states. As example, the doublet neutrons [220]1 / + and [101]1 / − states is 4 MeV apart in energy in the normaldeformed prolate solutions tend to become degenerate in theSD solution [220]1 / + and [101]1 / − states (prolate) belong-ing to two different major shells, so close to each other in thesuperdeformed solution (shown in Figs. 4 for Mg). Moresuch doublets are easily identified (Figs. 4 and 5) for superde-formed solutions of , Mg and , Al. In fact it is noticedthat the
Ω = 1 / states of unique parity, seen clearly wellseparated in the normal deformed solutions, get quite close toeach other for the SD states, suggesting degenerate parity dou-blet structure. This can lead to parity mixing and octupole de-formed shapes for the SD structures [38]. Parity doublets andoctupole deformation for superdeformed solutions have been discussed for neutron-rich Ba and Zr nuclei [39 ? ]. Thereis much interest for the experimental study of the spectra ofneutron-rich nuclei in this mass region [40]. The highly de-formed structures for the neutron-rich N e − N a − M g − Al nuclei are interesting and signature of such superdeformedconfigurations should be looked for. IV. SUMMARY AND CONCLUSION
In summary, we calculate the ground and low-lying excitedstate properties, like binding energy and quadrupole deforma-tion β using NL3 parameter set for Ne, Na, Mg, Si and Sisotopes, near the neutron drip-line region. In general, we findlarge deformed solutions for the neutron-drip nuclei whichagree well with the experimental measurement. We have donethe calculation using the nonrelativistic Hartree-Fock formal-ism with Skyrme interaction SkI4. Both the relativistic andnon-relativistic results were found comparable to each otherfor the considered mass region. In the present calculationsa large number of low-lying intrinsic superdeformed excitedstates are observed for many of the isotopes and some of themare reported. The breaking of N=28 magic number and theappearance of a new magic number at N=34 appears in ourcalculations. A proper angular momentum projection may tellus the exact lowering of binding energy and it may happenthat the superdeformed would be the ground band of some ofthe neutron-rich nuclei. Work in this direction is worth doingbecause of the present interest in the topic of the drip-line nu-clei. In this study we find that, for the SD shape, the low Ω orbits (particularly Ω = 1 / ) become more bound and show aparity doublet structure. Closelying parity-doublet band struc-tures and enhanced electromagnetic transition rates are a clearpossibility for the superdeformed shapes. V. ACKNOWLEDGMENTS
This work has been supported in part by Council of Sci-entific & Industrial Research (No. 03(1060)06/EMR-II) aswell as projects No. SR/S2/HEP-16/2005 and SR/S2/HEP-037/2008, Department of Science and Technology, Govt. ofIndia. [1] A. Navin et al., Phys. Rev. Lett.
266 (2000); H. Iwasaki etal.,
B491 , 8 (2000); H. Iwasaki et al.,
B481
B346
B522
227 (2001).[4] S. K. Patra and C. R. Praharaj, Phys. Lett.
B273
13 (1991);Phys. Rev.
C47
A565
442 (1993).[5] A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida and I. Tanihata,Phys. Rev. Lett.
185 (1995).[8] D. Vautherin and D.M. Brink, Phys. Rev
C 5
626 (1972). [9] P.G. Reinhard, Rep. Prog. Phys.
439 (1989).[10] E. Chabanat, P. Bonche, P. Hansel, J. Meyer, and R. Schaeffer,Nucl. Phys. A
710 (1997).[11] J.R. Stone and P.-G. Reinhard, Prog. Part. Nucl. Phys. , 467 (1995).[14] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer,Nucl. Phys. A635
231 (1998).[15] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys.
132 (1990).[17] B.A. Brown, Phys. Rev.
C58
220 (1998).[18] G.A. Lalazissis, J. K¨onig, and P. Ring, Phys. Rev.
C55 A651
117 (1999).[21] S.K. Patra, R.K. Gupta, B.K. Sharma, P.D. Stevenson, and W.Greiner, J. Phys.
G34
B312
377 (1993).[23] N. Tajima, P. Bonchc, H. Flocard, P.-H. Heenen and M.S.Weiss, Nucl. Phys.
A551
434 (1993).[24] M.M. Sharma, G.A. Lalazissis and P. Ring, Phys. Lett.
B317
A729
213 (1999).[27] P. M¨oller, J. R. Nix and K. -L. Kratz, At. Data and Nucl. DataTables,
131 (1997).[28] M. Samyn, S. Goriely, M. Bender and J.M. Pearson, Phys. Rev.
C70
C46
R1163 (1992); S.K. Patra and C.R. Praharaj, Phys. Rev.
C47
517 (2003); L. Satpathy and S.K. Pa-tra, Nucl. Phys.
A722
24c (2003); R.K. Gupta, M. Balasub-ramaniam, Sushil Kumar, S.K. Patra, G. M¨unzenberg and W.Greiner, J. Phys.
G32
565 (2006); Raj K. Gupta, S. K. Patraand W. Greiner, Mod. Phys. Lett.
A12
B335
259 (1994).[34] Zhongzhou Ren, Z.Y. Zhub, Y.H. Cai and Gongou Xu, Phys.Lett.
B380
241 (1994).[35] R. K. Gupta, S. K. Patra and W. Greiner, Mod. Phys. Lett.
A12
A579
557 (1994).[37] R. Kanungo, I. Tanihata and A. Ozawa, Phys. Lett.
B528
INT Workshop on ”Nuclear Many-Body Theoriesfor 21st Century” , University of Washington, Seattle (2007).[39] C.R. Praharaj,
Structure of Atomic Nuclei , Ch. 4, page 108,Edited by L. Satpathy; C.R. Praharaj, J. Phys.
G12
L139(1986).[40] D. Miller et al, Phys. Rev.