Global trends of nuclear d 2,3,4 5/2 configurations: Application of a simple effective-interaction model
aa r X i v : . [ nu c l - e x ] A ug Global trends of nuclear d , , / configurations: Application of a simpleeffective-interaction model M. Wiedeking
1, 2, ∗ and A. O. Macchiavelli † iThemba LABS, P.O. Box 722, 7129 Somerset West, South Africa School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: August 12, 2020)With new experimental information on nuclei far from stability being available, a systematicinvestigation of excitation energies and electromagnetic properties along the N = 10 , ,
12 isotonesand Z = 10 , ,
12 isotopes is presented. The experimental data are discussed in the context ofthe appearance and disappearance of shell closures at N = Z = 8 , , ,
20, and compared to aneffective-interaction approach applied to neutrons and protons in d , , / configurations. In spite ofits simplicity the model is able to explain the observed properties. I. INTRODUCTION
The interaction between valence neutrons and protonsplays a prominent role in understanding how nuclearstructure evolves with changing nucleon numbers [1–5].The monopole average of the neutron-proton interactioncan cause large changes in the effective single-particle en-ergies (ESPE) that may significantly alter the underlyingshell structure, allowing the pairing and quadrupole com-ponents to develop superfluidity and deformation awayfrom closed shells [6–9]. Indeed, changes in the ESPE’scan modify the available valence space and thus influ-ence the onset and strength of collective modes (e.g.quadrupole versus pairing). Measuring and understand-ing the evolution of ESPE’s from stability to the driplines is essential for a complete understanding of nuclearstructure effects and has been a central theme of studyfor the last two decades [10]. With rapid advances in de-tector systems, exotic beams production, and large-scalecomputing it is now possible to investigate the evolu-tion of shell-structure and collectivity for long sequencesof nuclei in isotopic or isotonic chains reaching far awayfrom stability. Trends obtained from such sequences of-ten contradict our understanding from stable nuclei, im-plying that the observed differences are due to the isospindependence of the nuclear force, notably the tensor com-ponent [7]. Studies on the emergence (erosion) of new(traditional) shell gaps have featured predominantly inexperimental and theoretical nuclear structure physicsresearch in nuclei away from the line of β stability.In light nuclei, examples of changes in the shell struc-ture include but are not limited to i) the emergence ofnew shell closures [11, 12] at N = Z = 14 [13, 14] and N = Z = 16 [15, 16], ii) the weakening of the proton p − sd shell closure as neutrons are added to Z = 8 iso-topes [17], and iii) the island of inversion at N = 20,where 2 p h , 4 p h , ... excitations from the sd -shell to ∗ [email protected] † [email protected] intruder orbits from the f p shell become energeticallyaccessible and yield deformed ground-state structures[8, 9, 18, 19].Fundamental quantities in even-even nuclei such as theexcitation energy of the first-excited 2 + state E (2 +1 ), theelectric quadrupole transition strength B ( E , +1 → + ),and the ratio R = E (4 +1 )/ E (2 +1 ) with E (4 +1 ) being theenergy of the first-excited 4 + state, have been shown toprovide important insight into the structural evolution ofnuclei [20, 21].For instance, the scale of E (2 +1 ) and R already pro-vide information on the collectivity and allow for a clas-sification of nuclear shapes. An increase in B ( E , +1 → + ), compared to that of a single-proton excitation, sug-gests that more active nucleons contribute coherentlyto the transition and signals deformations (dynamic orstatic) of the nuclear shape [22]. Similarly, the excitationenergies and ordering of low-lying states in odd-A nucleican also be effectively used to determine the migrationand inversion of orbits as a function of the number ofprotons and neutrons.With the wealth of available experimental data in the13 < A <
41 region it is instructive to investigate trendsof these fundamental quantities and correlate them to thenumber of available valence nucleons (holes) [5, 23] and tosimple effective interactions and coupling configurations.The latter approach has been proposed and discussed indetail by Talmi [24, 25] and is applied here to describegeneral properties of d n / configurations.In section II we compile the available experimentaldata on E (2 +1 ), E (4 +1 ), and B ( E , +1 → + ), as wellas the energies of the first-excited 3/2 + ( E (3 / +1 )), 5/2 + ( E (5 / +1 )), and 9/2 + ( E (9 / +1 )) states for Z = N =10 , ,
12 isotopes and isotones. The effective interactionmodel is briefly reviewed in section III. A comparisonwith the data from d , , / configurations, the discussionof the results and the connection to the N p N n scheme [5]are presented in section IV. !" !" ν ! !" !" !()*+, π ! !" FIG. 1. (Color online) Region of the Segr`e Chart showingthe isotones (blue) and isotopes (red) considered in the d , , / analysis. R , B ( E ) , E + Z(b) N=12 4 8 12 16 20 24 28N (d) Z=12 R , B ( E ) , E + R B(E2)E (a) N=10 (c) Z=10 FIG. 2. (Color online) Energies E (2 +1 ) in MeV (black dia-monds and dashed lines), ratios R (red squares and solidlines), and B ( E , +1 → + ) in 10 W.U. (blue triangles andlong-dashed lines) as a function of number of protons andneutrons in the d / and d / configurations for N = 10 (a), N = 12 (b), Z = 10 (c), and Z = 12 (d) nuclei. II. EXPERIMENTAL DATA
Experimental data are available for a large number ofnuclei in d , , / configurations. Fig. 1 shows the isotopesand isotones which have been considered in this work.Values of E (2 +1 ), E (4 +1 ), and B ( E , +1 → + ) are sum-marized in Table I for the N = 10 ,
12 and Z = 10 , E (3 / +1 ), E (5 / +1 ), and E(9 / +1 )in the Z = N = 11 odd-A isotopes and isotones. Thisinformation is also presented in graphical form in Figs. 2and 3. -1.5 -0.5 E n e r g y ( M e V ) Z (a) N=11 E9/2 + ̶ E3/2 + E5/2 + ̶ E3/2 + -101 N (b) Z=11 FIG. 3. Energy differences E (5 / +1 ) − E (3 / +1 ) (black dia-monds) and E (9 / +1 ) − E (3 / +1 ) (red squares) for nucleons inthe d / configuration for the N = 11 (a) and Z = 11 (b)nuclei. III. EFFECTIVE INTERACTION MODEL
Igal Talmi has discussed in depth the effective in-teraction model in j n configurations [24, 25]. Theenergy spacings and properties of low-lying states aredescribed by a two-body effective interaction obtainedfrom experiment. In spite of its simplicity the modelseems to capture the main physical ingredients and isable to correlate well the empirical data. Here, we arespecifically interested in the ( d / ) , , configurations forwhich there are analytical solutions. In these systemswe can generate I = 0 + , 2 + , and 4 + states for even-evenand J =3/2 + , 5/2 + , and 9/2 + states for odd-A nuclei.Following Talmi, given the 2-body matrix elements: V I = h d / IM | V | d / IM i (1)the energies of the states in the odd-A nuclei can be read-ily obtained: E / = 157 V + 67 V E / = 23 V + 56 V + 32 V E / = 914 V + 3314 V (2)in terms of V I in Eq. 1.We use a simple interaction consisting of short-rangeand long-range terms, written in the form: V I = − Gδ I, + χI ( I + 1) (3)which could be considered as a simplified version of aPairing-plus-Quadrupole force in a single- j shell [94]. Interms of the strengths, G and χ , we have for two particles(holes) in the d / orbit: V = − GV = 35 χV = 2 χ. (4)The results for the d / configurations follow from Eqs. 2and 4 and show that in a pairing dominated system TABLE I. Energies of the first 2 + ( E (2 +1 )) and 4 + ( E (4 +1 )) states, the ratio R = E (4 +1 ) /E (2 +1 ), the half-life T / (whenthe B ( E , +1 → + ) is calculated from it), and the electric quadrupole transition strength B ( E , +1 → + ) for neutrons andprotons in the d / and d / configurations. The data is compiled from evaluations as indicated by the references in the firstcolumn except for instances where data is taken from other sources as included next to the relevant entries. Blank entries areindicative of the absence of experimental data.Nucleus E (2 +1 ) (keV) E (4 +1 ) (keV) R T / (ps) B ( E , +1 → + ) (W.U.) Be [26] 1540(13) [27] C [28] 1760(2) a a +15 − ) a +33 − ) O [29] 1982.07(9) 3554.84(40) 1.79350(22) 1.94(5) 3.40(9)N=10 isotones Ne [30] 1633.674(15) 4247.7(11) 2.60009(67) 0.73(4) 20.7(1.1) Mg [31] 1247.02(3) 3308.22(6) 2.652901(80) 20.8(26) † [32] Si [33] 1874(3) [34] 3471(6) b [34] 1.8522(44) 4.7(15) † [35] C [36] 1585(15) c ⋆ c O [30] 1673.68(15) 3570.5(9) d Ne [31] 1274.537(7) 3357.2(5) 2.63406(39) 3.60(5) 12.76(18)N=12 isotones Mg [33] 1368.672(5) 4122.889(12) 3.012328(14) 1.33(6) 21.5(10) Si [37] 1797.3(1) 4446.37(18) be S [38] 1507(7) 7.2(12) † [39] Ar [40] 530(22) b [41] Ne [28] 1733(53) f Ne [29] 1887.3(2) 3376.2(4) 1.78890(28) 0.46(4) 18.3(16) Ne [30] 1633.674(15) 4247.7(11) 2.60009(67) 0.73(4) 20.7(1.1) Ne [31] 1274.537(7) 3357.2(5) 2.63405(39) 3.60(5) 12.76(18)Z=10 isotopes Ne [33] 1981.6(4) 3962.0(8) [42] 1.99939(57) 0.66(15) 6.8(16) Ne [37] 2018.28(10) 3516.8(5) [43] 1.74247(26) 0.60(8) 6.16(82) Ne [38] 1304(3) 3010(6) b † [44] Ne [45] 792(4) 2235(12) b † [46] Ne [47] 715.5(106) bg b [48] 2.962(51) Mg [49] 1598(10) 3700(20) b Mg [31] 1247.02(3) 3308.22(6) 2.652901(80) 20.8(26) † [32] Mg [33] 1368.672(5) 4122.889(12) 3.012328(14) 1.33(6) 21.5(10) Mg [37] 1808.74(4) 4318.89(5) 2.387789(60) 0.476(21) 13.42(59) Mg [38] 1473.54(10) 4021.0(5) 2.72880(39) 1.07(13) h Mg [45] 1482.8(3) 3379.0(8) [50] 2.27880(71) 1.5(2) 9.5(13) Mg [47] 885.3(1) 2322.3(3) 2.62318(45) 11.4(20) 14.7(16) i Mg [51] 660(7) 2047(16) b [52] 3.102(41) 40(8) 17.3(35) Mg [53] 660.6(78) j k † [46] Mg [54] 645.5(60) l l Mg [55] 500(14) b [56] † Converted to Weisskopf Units with B ( E sp = 5 .
94 10 − A / e b . a Average of values measured with high-purity Germanium detectors reported in [57] and [58]. b Tentative spin and parity assignment. c Average from values reported in [59] and [60]. ⋆ Spin and parity has not been assigned and it is only assumed that this is the 4 + state c.f. [61]. d Average from values reported in [17] and [62]. e A tentative assignment of 3842.2 (18) for the 4 +1 state is given in the evaluation [37] but its existence is doubtful since thelevel is not observed in fusion evaporation [63], ( He,n) [64, 65], (p,t) [66, 67], or fragmentation reaction data [68]. Instead thelevels at 4446.37(18) is considered to be the 4 +1 state [63–68] which has been corroborated [69] f Average from values reported in [70–72]. g Average from values reported in [47] and [48]. h Average from values reported in the evaluation [38] and measurement [39]. i Average of evaluated value from [47] and [73]. j Average from values reported in [46, 52, 53, 56, 74]. k Average from values reported in [75] and [56]. l Average from values reported in [52, 54, 56].
TABLE II. Energies of the first-excited 5/2 + ( E (5 / +1 )), 3/2 + ( E (3 / +1 )), and 9/2 + ( E (9 / +1 )) states, the ratios R = E (3 / +1 ) − E (5 / +1 ) E (9 / +1 ) − E (5 / +1 ) and R = E (3 / +1 ) − E (5 / +1 ) E (2 +1 ) − E (0 +1 ) for neutrons and protons in the d / configuration. The E (2 +1 ) values in R areaverages of adjacent even-even nuclei where possible. Energies are compiled from published evaluations as indicated by thereferences in the first column. For instances where data is obtained from other sources, those are shown next to the relevantentries. Blank entries indicate the absence of data.Nucleus E (5 / +1 ) (keV) E (3 / +1 ) (keV) E (9 / +1 ) (keV) R R Be [76] 0 a a,b [77] C [78] 332(2) 0 3085(25) -0.1206(13) -0.1985(22) O [29] 0 96.0(5) 2371.5(10) 0.04048(21) 0.05252(27)N=11 isotones Ne [79] 350.727(8) 0 2866.6(2) -0.139406(12) -0.2411978(61) Mg [80] 450.71(15) 0 2713.3(7) [81] -0.199201(91) -0.34462(12) Si [82] 0 42.5(55) c a [83] 0.0180(23) 0.02315(30) S [84] 0 a ≤ a [85] 0.066357(31) Na [76] 0 a a Na [79] 331.9(1) 0 2829.1(7) -0.132909(55) -0.230431(70) Na [80] 439.990(9) 0 2703.500(25) -0.1943839(46) -0.3329211(72) Na [82] 0 89.53(10) 2418.5(8) a [86] 0.037019(43) 0.047241(54)Z=11 isotopes Na [84] 0 62.9(6) a a Na [87] 72.0(5) ad Na [88] 375(4) a Na [89] 427(5) e a a [90] -0.2949(53) -0.620(14) Na [91] 373(5) a [90] 0 a -0.565(10) a Tentative spin and parity assignment. b The state and its energy has not been firmly accepted [76]. c Average from values reported in [92] and [83]. d Uncertainty not given in evaluated data base [87] but discussed to be 0.5 keV or less [93]. e Average from values reported in [89] and [90]. ( χ/G ≪
1) the paired 5/2 + state is favored and the3/2 + and 9/2 + unpaired states are degenerate. As thestrength χ is increased relative to G , the 3/2 + and9/2 + states move further apart, with the unpaired 3/2 + state approaching the 5/2 + state and becoming theground stated for χ/G & N and showed that O favors the5 / + paired neutron d / configuration for the groundstate in contrasts to C where the 3 / + unpaired stateis favored.Using the experimental data the χ/G values in the effec-tive model can be extracted. To do so, it is convenient topresent the data in terms of dimensionless energy ratiosversus χ/G : R = 1 /R = ( E (2 +1 ) − E (0 +1 ) / ( E (4 +1 ) − E (0 +1 ))for the even-even d , / configurations, R = ( E (3 / +1 ) − E (5 / +1 )) / ( E (9 / +1 ) − E (5 / +1 ))for the odd-A d / configurations, and R = ( E (3 / +1 ) − E (5 / +1 )) / ( E (2 +1 ) − E (0 +1 ))for a combination of odd-A d / and even-even d , / con-figurations where the average E (2 +1 ) value from two ad-jacent even-even nuclei for a given odd-A nucleus was used where possible[ ? ].The results are shown in Fig. 4for all isotones and isotopes under consideration. As areference, the limits of the perfect rotor ( χ/G → ∞ ) arealso indicated at for R , - for R , and - for R .When inspecting the data in Fig. 4 it is apparent thatthe majority of nuclei with d , , / configurations residewithin a transitional regime, stressing the fact that thereis a delicate balance between the pairing and quadrupoleforces in defining the structure of these nuclei. The limitsof a ”perfect” rotor are approached primarily for Z =11 ,
12 nuclei which lie in the N = 20 island of inversion.While the d / nucleus Na appears to exhibit the mostestablished rotor behaviour, based on R , of any nucleusconsidered in this work, we have to keep in mind thatthe (9 / + ) spin assignment to the 1875(19) keV level istentative and will be further discussed in the next section. IV. RESULTS AND DISCUSSION
Some interesting features are apparent when compar-ing chains of isotopes and isotones in the d , , / config-urations. The nuclear properties under considerationin Fig. 2 for the even-even nuclei clearly show the N = Z = 8 shell closure. Adding one or two pairs ofprotons or neutrons leads to an increase in collectivity -0.35 -0.25-0.15-0.050.05 0 1 2 3 4 5 6 R N=11Z=11
Rotor-1.0-0.8-0.6-0.4 -0.2 R χ /G N=11Z=11Rotor0.250.350.450.55 0 2 4 6 8 10 12 14 16 R Z=10Z=12N=10N=12Rotor(a)(b)(c) 30.7
FIG. 4. (Color online) Ratios R , R , and R versus χ/G for the d , / configurations (a) and for the d / configurations(b) and (c). For Na χ/G = 30 . χ/G = 5 . for R , - for R , and - for R . before the N = Z = 14 ,
16 shell closures are reached.The N = 16 shell closure at Z = 12 shows E (2 +1 ) , R and B ( E , +1 → + ) values which suggest a somewhatweakened N = 16 gap compared to the Z = 10 isotopesor N = 10 ,
12 isotones.For the lower-mass nuclei C and C the B ( E , +1 → + ) values appear suppressed which may imply magicityat Z = 6 which is contradicted by the increase in R anddecrease in E (2 +1 ) values. This apparent conundrum hasbeen discussed to be due to the states being dominatedby neutron excitation and the small B ( E , +1 → + )values are based on polarization effects [57, 96].Inspecting Ne and Mg a trend towards magicity isobserved just prior to the rapid onset of deformation for Ne and Mg [44, 50]. Indeed, the island of inversionat N =20 for the Z = 10 ,
12 isotopes is clearly visiblewith increasing B ( E , +1 → + ) and R and decreasing E (2 +1 ) values which remain relatively constant for N > d / orbit the effects of the”spectator” nucleons, due to the neutron-proton corre-lations, should be reflected in the behavior of the cou-pling constants. Furthermore, given the short- and long-range components used in our schematic force in Eq.(3), it is appealing to correlate our empirically derivedstrengths with the number of valence particles ( N p , N n )(holes) away from the shell closures, the so-called N p N n scheme [1, 4, 5].In fact, Fig. 5 shows that the χ/G values consistentlydecrease for nuclei at the 8, 14, and 16 shell closureswith the notable exception of Mg at N = 16. Awayfrom closed shells, where the long-range force gains im-portance, larger χ/G values are observed except for Be.For the N = Z = 11 nuclei the proton-neutron va-lence interaction plays a key role in determining the ex-cited and ground-state properties and the inversion froma paired to unpaired neutron (proton) coupling schemedue to proton-neutron interaction and can be quantita-tively explained by the model of effective interactions.With three valence neutrons (protons) in the j = 5 / d / . This leads to the unpaired con-figuration (3 / + state) becoming the ground state awayfrom shell closures compared to the paired configuration(5 / + state) being the ground state for nuclei with amagic number of neutrons (protons). The switch of theground state from 5 / + to 3 / + is due to the increasingimportance of the long-range relative to pairing strength.From these, the N = Z = 8, 14, and 16 shell closures areseen to be well established while the N = 20 magicity hasbeen eroded with the island of inversion clearly favouringthe unpaired 3 / + ground-state configuration.The limited study on C, N, O mentioned ear-lier [95] suggests C to be an open nucleus. The odd-odd N has a 1 − ground state with unpaired d / neutronsand is a transitional nucleus exhibiting increased impor-tance of the long-range component as compared to O.The location of the χ/G value for the Z = 4 nucleus Be in Fig. 5 (b) is indicative of significant uncertaintiesregarding the existence of the 3 / + state as also notedin table II. Should the 5 / + state indeed be the groundstate in Be the model (which otherwise reproduces theglobal trends quite satisfactorily) would be contradictedsince a spherical shell closure at Z = 4 is difficult if notimpossible to envisage. While it seems prudent to notethat the spin and parity assignments of the ground andexcited states in Be are tentative, we also realize thatthe resonance nature of these levels might limit the appli-cation of our model. Nevertheless, further experimentalinvestigations into the spin assignments of the states of Be are desirable.For the odd-even nuclei N = Z = 11 the energy differ-ences E (5 / +1 ) − E (3 / +1 ) and E (9 / +1 ) − E (3 / +1 ) clearlytrack the shell closures accurately, as shown in Fig. 3,with 5 / + ground states at closed-shells and 3 / + groundstates away from magicity as discussed above. As men-tioned earlier, the notable exception is Na, which notonly appears to exhibit a rotor behavior for R , butalso an unexpectedly reduced E (9 / +1 ) − E (3 / +1 value.This may be indicative that the tentative (9 / + ) spin as-signment needs to be revisited, in particular, since the R value, which is independent of E (9 / +1 ) (data pointat χ/G = 10 .
15 in Fig. 4 (c)), does not exhibit the samebehavior. For Na the E (5 / +1 ) − E (3 / +1 ) energy differ-ence suggests that this nucleus with a 3 / + ground statelies at the shores of the island of inversion, supportingthe findings from β -decay studies [93, 97].The overall systematic behavior is in agreement withthe expectations that follow from the factor P = N p N n / ( N p + N n ) [5] , which in heavier nuclei has beenshown to be ≈ . ǫ ~ ω / ∆. The dimensionless ratio ǫ ~ ω / ∆, the Migdal parameter, measures the competi-tion between the deformation, ǫ , and the pairing gap, ∆,and is key in our understanding of the moments of inertiaof nuclei [22, 98]. In our approach, the ratio χ/G plays asimilar role to ǫ ~ ω / ∆ and the two should be correlated.This is actually seen in Fig. 6 where we present the val-ues of χ/G vs. the Migdal parameter, providing furthersupport as to the applicability (at least qualitatively) ofthe effective interaction approach. Note that Ne with ǫ ~ ω / ∆ = 0 .
65 and Mg with ǫ ~ ω / ∆ = 0 .
63 (panel(a)), and Na with ǫ ~ ω / ∆ = 0 .
67 (panel (b)) are lo-cated at the interfaces of the different structural regimesand exhibit a transitional behaviour.
V. SUMMARY
With the major advances we have witnessed in thedevelopment of large shell model and ab-initio meth-ods for nuclear structure, it is perhaps opportune to askourselves about the value of the work presented above.Talmi’s vision of explaining complex nuclei with simplemodels is still very much relevant as it usually offers anintuitive approach to understand and correlate structuralinformation in terms of basic physical ingredients (whichsometimes get lost in the more sophisticated approaches).Our analysis fits well within this context.We have shown that a simple model for d , , / pro-tons and neutrons, with an effective-interaction that in-cludes both short- and long-range terms is able to ex-plain the experimental data with the appearance anddisappearance of shell closures at N = Z = 8 , , , χ/G , is in line with the N p N n -scheme expectations and appears to be responsi-ble for the inversion of the ground states in the odd-Asystems. (a)(b) χ / G PZ=10 R24Z=12 R24N=10 R24N=12 R2402468 χ / G N,ZPN=11 R39Z=11 R39N=11 R32Z=11 R32
FIG. 5. (Color online) Trend of the χ/G values from R of d , / protons (blue solid circles and red solid squares) andneutrons (black solid triangles and green solid diamonds) (a)and from R and R from a combination of d , / and d / protons (red solid and open squares) and neutrons (blue solidand open circles) (b). In each panel the scaled N p N n -scheme, P , (solid black line) with shell closures at N = Z = 2 , , , Z = 4 in panel (b) and the data point χ/G = 30.7 for Na has been omitted (see text for details). Thevery good agreement observed in the χ/G values for some ofthe N = Z = 11, N = Z = 10, and N = Z = 12 nuclei isnaturally due to mirror symmetry. VI. ACKNOWLEDGMENTS
This work is based on the research supported in part bythe National Research Foundation of South Africa (GrantNumber 118846) and by the Director, Office of Science,Office of Nuclear Physics, of the U.S. Department of En-ergy under Contract No. DE-AC02-05CH11231 (LBNL). χ / G Z=10 R24Z=12 R24N=10 R24N=12 R24024681012 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 χ / G εħω / Δ N=11 R39Z=11 R39
N=11 R32Z=11 R32 (a)(b)
FIG. 6. (Color online) Correlation between χ/G and ǫ ~ ω / ∆for (a) R values from d , / protons (blue solid circles and redsolid squares) and neutrons (black solid triangles and greensolid diamonds) and for (b) R and R values from a combi-nation of d , / and d / protons (red open and solid squares)and neutrons (blue open and solid circles). The data pointsfor Be and Na have been omitted from panel (b). (seetext for details)[1] I. Hamamoto, Nucl. Phys. , 225 (1965).[2] P. Federman and S. Pittel, Phys. Rev. C , 820 (1979).[3] P. Federman, S. Pittel, and A. Etchegoyen, Phys. Lett. B 140 , 269 (1984).[4] R. Casten, Phys. Lett.
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