Proc. NatL Acad. Sci. USA Vol. 79, pp. 7073-7075, November 1982 Physics

Rules governing the composition of revolving clusters in quasiband and prolate-deformation states of atomic nuclei (nuclear sizes/36Kr/38Sr/4oZr/42Mo/'58Er/actinon nuclei) LINUS PAULING Linus Pauling Institute of Science and Medicine, 440 Page Mill Road, Palo Alto, California 94306 Contributed by Linus Pauling, August 23, 1982 ABSTRACT A set ofrules, involving the magic and semimagic Magic proton or neutron number values of neutron and proton numbers and the proton/neutron ratio, is formulated for the composition of the revolving clusters The magic numbers are 8, 20, 28, 50, 82, and 126. Nuclei with producing the values of the moment of inertia given by the dif- both Z and N magic have no ground-state band. ferences in energy ofthe adjacent levels in quasibands and bands RULE 1. The revolving cluster is n2 or p2 (mass 2)for the ofnuclei. The cluster compositions assigned with use ofthese rules 2+ level ofthe ground-state band ofa nucleus with, respectively, to isotopes of Kr, Sr, Zr, Mo, and the actinon nuclei and to suc- Z or N magic. cessive levels ofthe ground-state band of l"nEr lead to reasonable Values ofR, the distance from the center ofthe cluster to the values of the radius of revolution (the distance from the center of center of the rest of the molecule, have been given (3) for 44 the nonrevolving sphere to the center ofthe cluster). These values nuclei with Z or N equal to 28, 50, 82, or 126. The values all correspond to a spheron diameter of about 3.20 fm. agree with the assumed value ofthe spheron diameter and with the value of R for adjacent nuclei with revolving helion. An atomic nucleus containing N neutrons may be described by N/2 shell-model wave functions. These functions may be com- Proton or neutron number 2 smaller or greater than magic bined to form a set of N/2 localized ls functions. Maximum stability is achieved when these localized functions are iso- The values of the 2+ level for the isotopes of 52Te from N = tropic. Each ofthem is occupied by two neutrons with opposed 62-80 (5) lead to acceptable values ofR with a helion as the re- spins, and also, if Z = N, by two protons with opposed spins. volving cluster; that is, the diproton outside the Z = 50 com- For neutron-excess nuclei, the protons are distributed among pleted subshell combines with a dineutron to produce the re- the N/2 localized ls functions in the way determined by Z/2 volving helion. For no tellurium isotope is there evidence ofa shell-model wave functions. The nucleus can then be described larger cluster, such as p2n4 (two tritons). The same observation as consisting of close-packed spherons, usually helions (a par- is made for the isotopes of 18Cd (diproton hole) from N = 52 to ticles) and tritons (1). 74. The internucleonic forces are such that nuclei have nearly RULE 2. The revolving cluster is a helion (mass 4)for the constant density except for a skin, about 1.5 fm thick, where the 2+ level of the ground-state band of a nucleus with Z or N 2 density decreases nearly to zero. Accordingly, it is possible to smaller or greater than magic. assign a size to spherons: the spheron-spheron contact distance Values ofR have been given (3) for 75 nuclei ofthis sort, with (spheron diameter) is about 3.2 fin. The spherons, of course, the magic number 20, 28, 50, 82, or 126. Only one apparent are soft, so that there is some variation in effective diameter exception to the rule is observed, 8OHg120, for which R is 8% from nucleus to nucleus. larger than the average for the other seven isotopes. The small A quasiband is a series oflevels with increasing values ofthe difference suggests that there is some perturbation ofthe 0+ or angular-momentum quantum number J but with energy dif- 2+ level rather than a different revolving cluster. ferences that deviate greatly from the values for a rigid rotator. For many quasibands and bands, a revolving cluster has been Semimagic proton or neutron number assigned to the nucleus with composition (mass) such as to cor- respond to a reasonable value of the radius ofrevolution (2-4). RULE 3. A semimagic proton or neutron number does not The short range of intemucleonic (and hence of interspheronic) prevent the corresponding dinucleon from being part of the forces ofattraction prevents a cluster from getting far from the revolving cluster; there is one exception. rest of the nucleus. Analysis of the energy values of bands has The exception is 4OZr,%. It is observed that the 2+ energy is led to the conclusion that a cluster revolves either in the mantle about twice as great as the 2+ energies for adjacent isotopes and (the outer layer of spherons), if the mantle is loosely packed, that, in consequence, the value of R is approximately the same or on the surface ofthe nucleus, if the mantle is tightly packed when the mass of the revolving cluster (p or n2 rather than a) (5). is one-half as great. For this nucleus both Z and N are semi- Using as the criterion for correct assignment of the mass of magic-a unique situation. The revolving cluster may be a res- the revolving cluster the agreement ofthe corresponding value onance hybrid of p2 and n2. ofthe radius ofrevolution with other values and with the value RULE 4. The revolving cluster often is a helion for the 2+ of the spheron diameter, I now have formulated a set of rules level of the ground-state band of a nucleus when Z or N is 2 about the composition of these clusters, mainly in relation to smaller or greater than semimagic; there are exceptions to this the magic and semimagic numbers of the shell model. rule. An example ofRule 4 is provided by the isotopes ofmSr, 4oZr, The publication costs ofthis article were defrayed in part by page charge and 42Mo with N = 54 and 58. For these isotopes the rule in- payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. Abbreviation: Dal, dalton(s). 7073 Downloaded by guest on September 28, 2021 7074 Physics: Pauling Proc. Nati. Acad. Sci. USA 79 (1982) dicates a revolving helion because 56 is semimagic. (A revolving The importance of magic values in determining the nature helion is required for N = 48 and 52 by Rule 2.) It is seen in ofthe revolving cluster for proton or neutron number magic or Table 1 that this expectation is fulfilled except for 42Mo58, which differing by ±2 from magic has been discussed in earlier papers has a larger cluster. It is not fulfilled, however, by the corre- (2-4) and is mentioned above. The isotopes of Zr and Mo with sponding isotones of 36Kr and 44Ru, which also have larger large caps, N = 60, 62, and 64 (Table 1), provide other exam- clusters. ples. The proton number for Zr is 8 or 12 from an adjacent semi- magic number (32 or 28), with 2, 8, or 14 for Mo, and the neu- Composition of sphere and revolving cluster tron numbers 60, 62, and 64 are 10, 12, and 14 from the magic Diffraction studies of nuclei have shown that the protons and number 50. The clusters p8n'0 and p12n12 for Zr and p2n10, p8n 2) neutrons are similarly distributed over the volume ofa nucleus. and p8n14 for Mo accordingly all conform to Rule 6. The dramatic The theoretical resonance energy of tritons and helions is a increase in the value of the effective moments of inertia from maximum when all spherons participate. Accordingly, it is likely N = 58 to N = 60, first reported by Cheifetz et al (6), is ac- that the p/n ratio should tend to be approximately equal for the cordingly explained by Rule 6 and the other rules. An appar- sphere and the cluster. ently different but possibly closely related quantum mechanical RULE 5. The proton/neutron ratiofor the revolving cluster explanation, based on detailed consideration of the p-p, p-n, tends to approximate equality with thatfor the nucleus. and n-n interactions, has been presented by Khosa and Sharma A striking example is provided by the actinon nuclei. The (7). values of the energy of the 2+ level (3) for all 16 nuclei with Z The Superprolate Structure of 76Kr and Adjacent Nuclei in 2 92 and N . 142 are nearly constant, indicating that the re- High Rotational States. Recent studies of rotational bands of volving cluster (the cap) has the same mass for these nuclei. For nuclei in the mass range near A = 78 extending to high values 92U142 with cap pion1 (six tritons and two helions, presumably of the rotational quantum number have indicated a transition a ring of seven spherons around a central spheron) and doubly from a nearly spherical state with a small revolving cluster to magic sphere (p82n'26), the value ofR is 11.17 fm. For U142 and a highly deformed structure (8). For example, for 76Kr4o the U144 it is 10.96 fm and 10.98 fm; for five Pu isotopes, 11.05 fm energy difference between the excited 0+ state at 0.770 MeV (average); for four Cm isotopes, 11.22 fin; for two Cf isotopes, and the associated 2+ state at 1.688 MeV leads to the value I 11.03 fm; and for 1ooFm,152, 11.05 fin. For all ofthese nuclei, the = 136.6 Dal fM2 for the moment of inertia (Dal = daltons), p/n ratio ofthe cap, 0.625, is close to the values for the nuclei, which can be interpreted (2, 3) as resulting from the revolution which range from 0.63 to 0.66. ofahelion about a nearly spherical part ofthe nucleus at distance For other nuclei the cap involves all ofthe protons and neu- 6.00 fm from the center of the sphere. This value of the radius trons outside the Z = 82, N = 126 doubly magic shell; examples ofrevolution agrees well with the values found for other nuclei are Ui4o(cap po0n14), R = 11.07 fm; U138(plon12), 11.04 fm; in this mass region (3). The ground-state band shows steadily Th142(p8n16), 10.86 fm; Th14O(p8n14), 10.88 fm; and Thl38(p8n12), increasing values of I, from 296 Dal fin2 for 0+-2+ to 752 Dal 10.93 fin. fin2 for 10+-12+. The largest values of I are shown by two ex- For Th136 the ratio p/n for the cap p6n'0, 0.60, is closer to cited bands, one with 6- at 3.174, 8- at 3.899, 10- at 4.805, and that for the nucleus, 0.66, than is the ratio 0.80 for p8n'0, the 12- at 5.886 MeV, and the other with 5- at 2.684, 7- at 3.288, value for a doubly magic sphere. With p6n'0 the value of R is 9- at 4.073, i1- at 5.052, and 13- at 6.223 MeV, with sepa- 10.81 fm, in good agreement with values quoted above. Simi- rations corresponding to I = 865, 877, 889, 898, 905, 897, and larly, with p4n8 cap R equals 10.90 fm for Th,34. Accordingly, 892 Dal fM2, respectively. Values for other bands approach this the p/n ratio is more important than the doubly magic sphere limit, which is so large as to indicate that the structure is highly in determining the composition of the revolving cluster. deformed. The near constancy ofthe values ofI shows that the RULE 6. Magic andsemimagic values ofthe proton number structure is well defined. and the neutron number ofthe sphere (the part ofthe nucleus With 40 neutrons a polyspheron structure would involve the other than the revolving cluster) often lead to stable nuclei; packing of 20 spherons (1). The nearly spherical 20-spheron these completed-subshell structures may incorporate additional structure has a tetrahedral core of 4 spherons and a mantle of diprotons or dineutrons without contributing to the moment of 16 (1). The only deformed structure ofapproximately 20 spher- inertia. ons that has been described is the 19-spheron superprolate structure with a linear core of 3, two rings of 5 each, and two Table 1. Composition of the revolving clusters* and end caps of3 spherons each (1). A photograph ofa model ofthis corresponding values of the radius of revolution for structure has been published (1). With 40 neutrons the addi- the 0+-2+ ground-state bands of isotopes of tional spheron might enter a 5-ring, converting it to a 6-ring and 38Sr, 4OZr, and 42Mo giving the nucleus a semimagic central sphere with 28 neutrons surrounded by two opposite 3-spheron caps. With the protons 38Sr 40Zr 42Mo uniformly distributed over the spherons, there would be 10.9 N Cluster R,fm N Cluster R, fm N Cluster R,fm protons in the caps, and electrostatic repulsion might increase 40 a4 5.93 46 a 6.01 48 a 5.88 the numberto 12, eachcap then beingaclusterofthree helions. 42 a3 5.61 48 a 5.57 50 p2 6.52 The corresponding values ofR1 are 6.00, 6.04, and 6.09 fm for 44 a2 5.49 50 p2 5.41 52 a 6.13 the 6--12- band and 6.12, 6.14, 6.11, and 6.10 fm for the 46 p2n4 5.33 52 a 5.92 54 a 6.48 5--13- band, average 6.09 fm. This value, which should be 48 a 5.53 54 a 5.97 56 a 6.44 twice the spheron diameter, is acceptable. Less extensive in- 50 p2 5.91 56 p2 or n2 6.07 58 p2n4 6.44 formation about other nuclei (9, 10), such as 74'785'Kr, indicates 52 a 6.28 58 a 5.17 60 p2n'0, p4n8 6.33 that they also approach similar structures at high values ofJ. 54 a 6.34 60 p8n'0 6.32 62 p8Bn2 6.35 56 a 6.25 62 p12n12 6.71 64 p8n14 6.47 The ground-state band of 58 a 6.33 "6Ergo * Different structures (polyhelionic) have been assigned (3, 4) to some In a recent letter, Burde et al. have reported energy values for of these states. Energy values of 2+ are from ref. 5. states of 159Ergo up to spin J = 38+ from the reaction Downloaded by guest on September 28, 2021 Physics: Pauling Proc. Natl. Acad. Sci. USA 79 (1982) 7075 Table 2. Composition of revolving clusters and corresponding values of the radius of revolution for the ground-state band of '88Ergo JX E, MeV I, Dal fM2 Cluster R, fin J" E, MeV I, Dal fM2 Cluster R, fm 0+ 0.0000 20+ 4.8941 2,472 Pl8n22 7.86 2+ 0.1927 651 p4n8 7.66 22+ 5.6341 2,429 pl8n22 7.79 4+ 0.5284 872 p6n10 7.79 24+ 6.4403 2,437 p'8n22 7.81 6+ 0.9718 1,037 p8nl2 7.70 26+ 7.2863 2,520 pl8n22 7.94 8+ 1.4955 1,197 P0on14 7.67 28+ 8.1455 2,676 p18n24 7.98 10+ 2.0753 1,370 p12n16 7.71 30+ 9.0220 2,814 p'8n26 8.00 12+ 2.6841 1,579 p'12n20 7.87 32+ 9.9287 2,904 p18n26 8.12 14+ 3.1945 2,211 p12n22 8.06 34+ 10.8890 2,916 p18n26 8.14 16+ 3.6678 2,738 p'8n22 8.27 36 11.9077 2,913 p'8n26 8.14 18+ 4.2347 2,581 p 8n22 8.03 38+ 12.9690 2,954 p1n26 8.20

122Sn(40Ar,4ny) (11). They verified the previously known yrast sphere (1) is such that the distance from the center ofthe sphere transitions as J = 14+ and J = 28+ and found indication of an- to the center of the cap skimming over its surface is 3.5 times other beginning at J = 38+. They attribute the transitions to the spheron diameter, which accordingly is 3.16 fin. Similarly, alignment with the rotation axes ofhigh-Jparticle pairs, two i13/ the average value 8.02 fm for the large clusters of 158Er, for 2 neutrons atJ = 14+, two hll/2 protons atJ = 28+, and two which the radius ofrevolution is 2.5 spheron diameters (1), leads h9/2 neutrons at J = 38+. I suggest that these yrast transitions to 3.21 fin for the spheron diameter, in good agreement with may be described as resulting from changes in the composition the actinon value and the earlier estimates of3.24 fm (1), about ofthe revolving clusters, in accordance with the foregoing rules. 3.2 fm (3), and about 3.0 fm (4). For the nuclei in Table 1, the This is not alternative but is supplementary to the shell-model expected values of R lie between 4.8 fin, for movement of the interpretation. cluster in the mantle of the N - 50 sphere, and 8.0 fin, for the Values of the energy are given in Table 2. After an initial in- cluster skimming over the surface of the mantle. The values of crease in the values of the moment of inertia, they remain Z indicate movement in the mantle for the protons, and values nearly constant from J = 16+ to 26+, and again nearly constant ofN much greater than 50 indicate skimming over the surface; from J = 30+ to 38+. together they give 6.4 fm as the expected value of R. The 12 For Z = 68 the number of protons in the cap is 4 for semi- values of R for the larger values of N in Table 1 approximate magic number 64, 12 for semimagic number 56, and 18 for magic this value; for the others, with smaller R, the clusters drop far- number 50, and the number ofneutrons is 8 for magic number ther down into the mantle. The value of R, 6.09 fin, for 76Kr 82, 20 for semimagic number 70, 22 for semimagic number 68, corresponds to spheron diameter 3.05 fin. and The 0+-2+ difference 26 for semimagic number 64. energy 1. Pauling, L. (1965) Science 150, 297-305. with the smallest indicated cluster, p4n8, gives R = 7.66 fm, 2. Pauling, L. (1965) Proc. Nati Acad. Sci. USA 64, 807-809. which is in the expected range (2). The next cluster compatible 3. Pauling, L. & Robinson, A. B. (1975) Can. J. Phys. 53, 1953- with Rules 5 and 6 is p12n20, which when assigned to the 10+-12+ 1964. moment ofinertia leads to R = 7.87 fin. The intermediate values 4. Pauling, L. (1981) Proc. Natl. Acad. Sci. USA 78, 5296-5298. of I correspond to clusters involving successive additions of a 5. Sakai, M. (1975) At. Data Nucl. Data Tables 15, 513-542. to for 8+-10+. 6. Cheifetz, E., Jared, R. C., Thompson, S G. & Welhelmy, J. B. helion, up p12n16 (1970) Phys. Rev. Lett. 25, 38-43. The larger clusters, from p 8n22 to p8n26,all correspond to 7. Khosa, S. K. & Sharma, S. K. (1981) Phys. Rev. C 24, 2715-2719. magic proton number 50 and a semimagic neutron number. The 8. Piercey, R. B., Ramayya, A. V., Hamilton, J. H., Sun, X. J., rotational energy differences indicate that the revolving spher- Zhao, Z. Z., Robinson, R. L., Kim, H. J. & Wells, J. C. (1982) ons are divided into two caps, presumably opposed, with the Phys. Rev. C 25, 1941. reduced mass thus equal to the total mass ofthe caps. The cor- 9. Piercey, R. B., Hamilton, J. H., Soundranayagam, R., Ramayya, responding values ofR range from 7.79 fm to 8.27 fm, with av- A. V., Maguire, C. F., Sun, S.-J., Zhao, Z. Z., Robinson, R. L., Kim, H. J., Frauendorf, F., Doring, J., Funke, L., Winter, G., erage 8.02 fin, a not unreasonable value. Roth, J., Cleeman, L., Eberth, J., Neumann, W., Wells, J. C., At larger values ofJ the caps might become larger; the next Lin, J., Rester, A. C. & Carter, H. K. (1981) Phys. Rev. Lett. 47, one compatible with Rule 6 is pl'8n4, with I predicted to have 1514. the value 3,345 Dal fMi2. 10. Sastry, D. L., Ahmed, A., Ramayya, A. V., Piercey, R. B., Ka- wakami, H., Soundranayagam, R., Hamilton, J. H., Maguire, C. F., deLima, A. P., Ramavataram, S., Robinson, R. L., Kim, H. The spheron diameter J. & Wells, J. C. (1981) Phys. Rev. C 23, 2086.- 11. Burde, J., Dines, E. L., Shih, S., Diamond, R. M., Draper, J. The average of the 20 values of R for actinons with large caps, E., Lindenberger, K. H., Schtick, C. & Stephens, F. S. (1982) reported above, is 11.06 fm. The structure assigned to the Phys. Rev. Lett. 48, 530-533. Downloaded by guest on September 28, 2021