Stability of alpha-chain States against Break-up and Binary Disintegrations
aa r X i v : . [ nu c l - t h ] S e p Stability of α -chain States against Break-up and Binary Disintegrations Akihiro Tohsaki and Naoyuki Itagaki Research Center for Nuclear Physics (RCNP), Osaka University,10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502 Kyoto, Japan (Dated: September 5, 2018)We focus upon the raison d’ˆetre of the α -chain states on the basis of the fully microscopic frame-work, where the Pauli principle among all the nucleons is fully taken into account. Our purpose is tofind out the limiting number of α clusters, on which the linear α -cluster state can stably exist. Howmany α clusters can stably make an α -chain state? We examine the properties of equally separated α clusters on a straight line and compare its stability with that on a circle. We also confirm itsstability in terms of break-up and binary disintegrations including α -decay and fission modes. Forthe effective nucleon-nucleon interaction, we employ the F1 force, which has finite-range three-bodyterms and guarantees overall saturation properties of nuclei. This interaction also gives reasonablebinding energy and size of the α particle and the α - α scattering phase shift. The result astonishesus because we can point out a possible existence of α -chain states with vast numbers of α clusters. There has been a long history of the theoretical andexperimental studies on the α -chain state. More thanhalf century ago, Morinaga, who was an experimentalist,pointed out that the second 0 + state of C just abovethe three- α threshold energy was a possible candidate ofit, and now the state is regarded as the α -gas state [1, 2].Since there is no stable nucleus with the mass number offive or eight, formation of C from three He nuclei ( α clusters) is a key process of the nucleosynthesis. Here,the second 0 + state at E x = 7 . C and locatedjust above the threshold energy to decay into three Henuclei [3]. The Tohsaki-Horiuchi-Schuck-R¨opke (THSR)wave function has been widely used for the studies ofsuch state [4]. Nevertheless, another excited state wasidentified as the α -chain-like state [5–7]. Furthermore,the European group has suggested the possibility of four- α and six- α chain states [8, 9]. From the theoreticalside, it has been discussed that giving angular momentumto the system or adding neutrons prevents the bendingmotion, which is the main decay path, and stabilizes thelinear-chain configurations [10–18].We can easily imagine that, if much longer linear α -chain exists, it could be more stable than the other shapeof aggregates with the same number of α particles due tothe weaker Coulomb repulsion. We examine the stabil-ity of longer linear chain state in terms of two kinds ofresistances with a microscopic framework; the resistanceagainst the complete break-up to nα particles and thatagainst binary disintegrations to { mα, ( n − m ) α } , where n and m are integer numbers. We have been enchantedby the simple beauty of α -chain state. In other words,we strongly want to know one of the various facets of α -particle aggregation.The aim of this report is to clarify the importance ofthe Pauli principle as well as the Coulomb interaction.For this purpose, it could be a necessary condition to re-move the ambiguity of the inter-nucleon force by adopt-ing the most reliable one suitable for the many α -clusteraggregate. We need the effective inter-nucleon force with- out any adjustable parameter; having adjustable param-eter prevents us from comprehensively understanding thenumber dependence of the α clusters. The following con-ditions must be satisfied: (i) to guarantee overall satu-ration properties of nuclei, (ii) to naturally explain thebinding energy and the size of the α particle, and (iii)to reproduce the phase shift of α - α elastic scattering. Inshort, we employ the effective interaction proposed byone of the authors (A.T.) a quarter century ago, whichis called F1 force [19],ˆ V = X i 31 0.75 0.002 1.8 − . 51 7.73 0.462 0.003 0.7 60 . 38 219.0 0.522 1.909 Now we prepare the most intuitive model space to ana-lyze the behavior of α -chain state, which D. M. Brink hasproposed a half-century ago [20]; a parameter space of R introduced to describe the geometrical configurations of α clusters, which is called Brink-Bloch parameter space.This is a fully microscopic framework, which takes fullaccount of the Pauli principle. The wave function for nα -clusters is written byΨ = A{ φ ( R ) φ ( R ) · · · φ ( R n ) } , (4)where A is the antisymmetrizer for all the nucleons, andthe wave function of a single α -particle wave function isdefined by φ ( R k ) = Y i,j =1 , (cid:18) πb (cid:19) exp {− b (cid:16) r ijk − R k (cid:17) } χ ijk , (5)where b is the nucleon width parameter, and χ ijk is aspin isospin wave function, where superscripts i and j are labels for the spin and isospin. The vectors { r ijk } are the real coordinate for four nucleons in the k th α -cluster, which shares the common Gaussian center R k .In this report, we adopt b = 1 . 415 fm, which leads tothe optimal binding energy of α particle; the quantityof 27.5 MeV is a little shallow but reasonable comparedwith the experimental data (28.3 MeV).The energy quantities are obtained by defining thenorm and energy kernels, h Ψ |H| Ψ i and h Ψ | Ψ i , where theHamiltonian is written by H = − ~ M X i ∇ i − T g + X i 511 4010 3.25 10.362 6.85 15.463 5.101 0 . 151 18015 3.28 11.301 6.73 16.305 5.005 − . 280 42020 3.30 11.917 6.65 16.801 4.884 − . 464 760... ... ... ... ... ... ... ...50 3.35 13.708 6.42 18.054 4.346 − . 756 490055 3.36 13.884 6.40 18.167 4.282 − . 772 594060 3.36 14.044 6.38 18.268 4.224 − . 785 7080(b) n d min Rd min E min d max Rd max E max E max − E min (fm) (fm) (MeV) (fm) (fm) (MeV)5 2.84 2.416 5.944 7.07 6.014 13.639 7.69510 3.21 5.194 10.210 6.77 10.955 15.834 5.62415 3.28 7.888 11.580 6.63 15.945 16.718 5.13820 3.31 10.580 12.381 6.54 20.903 17.235 4.854... ... ... ... ... ... ... ...50 3.38 26.915 14.464 6.31 50.246 18.532 4.06855 3.39 29.691 14.656 6.29 55.090 18.648 3.99260 3.39 32.387 14.829 6.27 59.901 18.752 3.923 2. The difference between the barrier height and en-ergy pocket per n ( E max − E min ) are converged to4.2 MeV and 3.9 MeV for the linear (Table II (a))and annular (Table II (b)) configurations, respec-tively. This deviation comes from the geometricaldifference of the configurations, which affects thecontributions of the Coulomb energy and the Pauliprinciple.3. The positions of the energy pocket ( d min ) are al-most the same in large n cases. The fact thatthe nearest neighbor distance of α -clusters ( d in ) isabout 3.3 fm means again the conservation of theproperty of the individual α - α cluster structure inthe free space.4. The four curves in Fig. 2 behave like P nk =1 (1 /k ) ∼ log( n ) + γ , where γ is Euler constant, 0.57721 · · · .In other words, as far as the long α -cluster chain ex-ists, the energy with respect to the number of the α clusters diverges as logarithmic behavior. Thereason comes from the fact that the other energiesexcept for the Coulomb energy, that is, the kineticand the interaction energy originating in the nu-clear force, are saturated as the number of α clus-ters increases.5. The linear configuration becomes more stable thanannular one beyond n = 12. The reason also comes TABLE III. The physical quantities of binary disintegrationsof the linear α -cluster chain. (a): the linear α -cluster chain(b): the annular α -cluster chain.(a) n d min E min d max E max E max − E min E inf (fm) (MeV) (fm) (MeV) (MeV) (MeV)5 3.16 − . 005 6.99 6.171 6.176 − . − . 024 6.86 5.978 6.002 − . − . 045 6.82 5.922 5.967 − . − . 063 6.80 5.877 5.940 − . − . 121 6.77 5.799 5.920 − . − . 135 6.77 5.788 5.923 − . − . 135 6.77 5.786 5.921 − . n d min E min d max E max E max − E min E inf (fm) (MeV) (fm) (MeV) (MeV) (MeV)5 3.25 − . 007 6.80 5.823 5.830 − . − . 008 6.35 4.791 4.799 − . − . 005 6.15 4.267 4.272 − . − . 005 6.02 3.901 3.906 − . − . 008 5.68 2.852 2.860 − . − . 006 5.65 2.765 2.771 − . − . 009 5.62 2.665 2.674 − . from the fact that the reduction of the Coulombenergy works well for the linear configuration com-pared with the annular one.We can point out that if the α -cluster chain is on var-ious lines, such as linear, annular, and spiral lines, thelinear configuration gives the lowest energy, as the num-ber of the α clusters increases. The column E min ( l ) − E min ( a ) in Table II (a) shows the difference betweenthe minimum energies of linear and annular chain states.Nevertheless, many kinds of long α -cluster chains canexist stably. We cannot find the limiting numbers of α -clusters for a complete break-up.Secondly, let us scrutinize the stability of the linear α -cluster chain in terms of binary disintegration. In Fig. 3,a schematic figure is shown. The n α -clusters are dividedinto two blocks, that is, one block with m α -clusters andthe other with ( n − m ) α -clusters. The distance, d , be-tween nearest neighbor α -clusters in the different blocksis a variable parameter to see the stability of the dis-integration. The case with m = 1 is regarded as themode corresponding to α -decay process, and the casewith m ∼ n/ m = n/ n , m = ( n − / n ) describes the symmetric fission. The innerdistance of two α -clusters, d in , is taken as that leads tothe minimum energy of linear nα -clusters. We show theenergy curves for 15 α -clusters case in Fig. 4. The seven ene r g y / n ( M e V ) d in (fm) n=15 annularn=15 linearn=10 annularn=10 linear FIG. 1. The energy per number of α clusters ( n ) versusthe nearest neighbor distance between equally separated α -clusters ( d in ). The cases of n = 10 , 15 are shown. 20 40 6001020 ene r g y / n ( M e V ) n energy pocketenergy barrier linearannularlinearannular FIG. 2. The systematics of the energy pocket and energybarrier as functions of number of equally separated α clusters( n ) for the linear and annular configurations. curves corresponding to different m are depicted versusthe distance d . Unlike Fig. 1, Fig. 2, and Table II, wherewe discussed the total energy per n , from here we employthe total energy (not per n ). The energy is measuredfrom the pocket of the equidistant linear chain, and at d → ∞ , the energy reaches an individual one dependingon m . We can learn that all the energies for the binarydisintegration have the energy minimum with the d value,which is close to the optimal d in value of the equidistant15 α clusters on a straight line. Furthermore, we canverify that the minimum energies come to the energy ofthe pocket of the equidistant linear chain. This meansthat 15 α -cluster chain is stable against any binary dis-integration. The maximum energy around d ∼ m ,and the α -decay mode seems to be more stable than the n-1 1(a) d in dn-m m(b) d FIG. 3. Schematic figure for the linear configuration of nα clusters. (a): α decay of one α cluster. (b): fission into ( n − m ) α cluster and mα cluster systems. The relative distancebetween the two systems is d , whereas the α - α distance is d in . ene r g y ( M e V ) m=1m=2m=3m=7m=7m=1 FIG. 4. The energy curves versus d (fm), the nearest neigh-bor distance between α -cluster chains in the different blocks.The 15 α clusters are separated to two blocks with m and15 − m α clusters. fission one, as the height decreases with increasing m . Indetail, the optimal d value for the α -decay mode ( m = 1)slightly deviates inwardly from the optimal d in value ofthe equidistant model. Therefore, the equidistant modelshould be a little modified, because the edge effect takesplace, which stabilizes the chain-state.Here we discuss the quanta of the harmonic oscillator, N . The lowest allowed quanta are the ones obtained atthe zero distance limit between the α clusters ( d in → N min of Table II (a). It is seen thatthe N min value decreases, when the linear chain state isseparated to two blocks; N min is 420 for n = 15, and itdecreases to 40 + 180 = 220 after separated into 5 α ’sand 10 α ’s ( m = 5). In the n = 15 case, this decreaseof N min is smallest (56) at m = 1 and largest (224) at m = 7. Indeed, the decrease of N min reaches the largestvalue in the case of symmetric fission, which is relatedto the decrease of the peak energy around d ∼ m .Focusing the α -decay mode ( m = 1) and the symmetricfission mode ( m = n/ 20 40 600510 ene r g y ( M e V ) n barrier of decay mode α barrier of fission mode FIG. 5. The barrier height of α -decay mode and the fissionmode. height, which is the height of the local peak around d ∼ α clusters; the boldline stands for the α -decay mode and the dotted one isfor the fission mode. The α -decay mode reveals almostthe same height through all the number of α clusters,but the fission mode looks like an exponential decrease.Therefore, the fission mode is more fragile than α -decaymode, and adding external forces may strongly affect thestability of α -cluster chain.The physical quantities for the binary disintegrationsare summarized in Table III, (a): the linear α -clusterchain (b): the annular α -cluster chain. Here, d min isthe d value for the energy pocket, E min is the minimumenergy there, d max is the d value for the local peak, and E max is the perk energy there. The final column ( E inf )is the total energy when two blocks are separated intotwo independent systems as d → ∞ .Let us summarize the energy properties of α -chainstates. As the first stage, we examined the adiabaticenergy curves with respect to two kinds of distances onthe Brink-Bloch parameter space within the microscopicframework. The Pauli principle and the exact treatmentof the Coulomb force are certainly important for under-standing the energy properties, which have been com-pletely taken into account. Furthermore, the most appro-priate effective inter-nucleon force, which can reproduceoverall saturation property of nuclei and the experimen-tal data of the α - α elastic scattering, has to be adopted,and here we introduced the F1 force. As for the α -cluster chain with equidistant distribu-tion, the nearest neighbor distance is a suitable measurefor the instant break-up of the chain. In this respect,we pointed out that the α -cluster chain is always sta-ble against the complete break-up. This is because eachterm of the Hamiltonian, except for the Coulomb energy,is saturated with increasing number of the α -clusters ifthe expectation value is divided by the number. Nev-ertheless, the effect of the Coulomb energy is too weakto break the chain, and the optimal α - α distance staysthe value almost the same as the free α - α system. Asfor the disintegration into two blocks of the linear chain,we examined the stability with respect to the distancebetween the edges of the two blocks. We can learn thatonce α -cluster linear chain is formed, it is surrounded bya long-tail wall owing to the Coulomb force between twoblocks. The Pauli principle works attractively inward inthe barrier. These effects stabilize the linear chain statesand prevent the break-up.In this report, we did not mention the bending mo-tion of the chain, which is the deviation from the one-dimensional configuration, but we already verified thestability of the linear chain of α -clusters against it. Inview of the reduction of the Coulomb energy, the linearchain always remains unchanged, and the reduction ofthe Coulomb energy is quite important as the numberof α clusters increases. Unfortunately, we do not knowany appropriate effective inter-nucleon force to study theoverall trends of α -cluster aggregate except for F1 force.We have estimated the energy property of the α -chainstate using the Brink-Boeker force [22], which also donot contain adjustable parameters. Quantitative energyobtained is different from each other, but the tendencyof the chain state is not so changed comparing with thatof the F1 force. We will report detailed results on thedependence of the effective inter-nucleon force elsewhere.We imagine that various kinds of α -cluster chain floatin the space, above all the linear chain, possibly exists.The chain could be plausibly disintegrated by the fissionmode when an external force acts on it. ACKNOWLEDGMENTS We are indebted to Prof. N. Tajima who has suggestedthe instability for the fission mode. The numerical cal-culation has been performed using the computer facilityof Yukawa Institute for Theoretical Physics, Kyoto Uni-versity. 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