Odessa Astronomical Publications, vol. 17 (2004) 511 On discussion NUCLEAR ISOTOPES AND MAGIC NUMBERS V.P. Bezdenezhnyi Astronomical observatory, Odessa National University T.G.Shevchenko Park, Odessa 65014 Ukraine, [email protected] Dedicated to D.I.Mendeleev's 170-th anniversary, to the 135-th anniversary of Periodic Law discovery and to G.A.Gamov's 100-th anniversary ABSTRACT. In addition to the formulas (1) and nucleon envelopes. According to the Pauli principle, (2) from work of the author (Bezdenezhnyi, 1999, here- a certain number of nucleons of a given kind can be after Paper I) alternative formulas (3){(6) are derived placed on each envelope. Whenever an envelope is for description of two sequences of magic numbers. ¯lled, it corresponds to the magic nucleus formation Their properties are described. Dependences of some with the respective magic number. Nucleon envelopes physical parameters as a function of the number of of protons and neutrons are ¯lled independently. chemical element (Z), number of neutrons (N) and the The simultaneous ¯lling up of proton and neutron mass number (A) are analysed. It is shown that magic envelopes is followed by a corresponding formation of numbers of protons (Z) and neutrons (N) are displayed particularly stable twice magic nucleus. At the same in them. The concept of magic mass numbers (A) is time the saturation of forces of nuclear interaction proposed, and it is shown that they are also present in is attained - a high value of nuclear package (mass some dependences. The list of magic and twice magic defect). (on Z and N) elements is extended, as well as the list In previous author's work (Paper I), the Table of twice magic ones on A and Z or N. We propose to (117.8) from "Quantum Mechanics" (Landau and refer a nucleus that contains magic numbers of protons Lifshits, 1963) has been analysed and two recur- and neutrons to thrice magic nucleus if their sum is rentformulas have been found connecting all the set a magic number too. It is suggested that during the of magic numbers and corresponding to their two decay of "double -uranium" super-nucleus one of three sequences - large and small ones: twice magic elements ( X(112,168), Y(126,184) and ¡ Z(168,258) or Z'(168,240) ) predicted (in Paper I) will Mi = Mi¡1 + X J(i 2) + J(i); where i=1, 2,... (1) be formed fragment accompanying in areas of twice magic nuclei: Yb(70,112), Sn(50,70) and S(16,16) mi¡2 = Mi ¡ J(i); where i = 3, 4,... (2) respectively. The numeral values, counted up on these formulas, Key words: nuclear astrophysics, r-, s-processes, are the following: Mendeleev's periodic system, isotopes, magic numbers Mi: 2, 6, 14, 28, 50, 82, 126, 184, 258, 350, 462, ... mi: - - 8, 20, 40, 70, 112, 168, 240, 330, 440, ... 1. Introduction In that way, new magic numbers: 6, 14, 40, 70, 112, 168, 240, 258, 330, 350, 440, 462 were added to the before known ones. Numbers of neutrons As is generally known from nuclear physics (Mukhin, 16, 96, 114, 152, appearing in the literature, are 1974), atomic nuclei containing certain numbers (2, 8, not magic ones: three from them are numbers of 20, 28, 50, 82, 126) of protons (P) or neutrons (N) show alpha-particle con¯gurations, and 114 is a di®erence an enhanced stability that makes them distinguished of magic numbers 184 and 70. The quantity of the among other nuclei neighbouring them. They are nuclei referred to magic and twice magic ones was named as magic nuclei (isotopes of chemical elements), increased in connection with the increasing of the and the numbers of their protons and neutrons are quantity of magic numbers. The author added (6,6), called "magic numbers". Their explanation lies in the Si(14,14), Ni(28,28), Zr(40,50) Sn(50,70) to the envelope nucleus model, in which the nucleons form previously known twice magic nuclei of isotopes: He the ¯lled envelopes separately for P and N. According (2,2), O(8,8), Ca(20,20), Pb(82,126). All of them have to the model of nucleus envelopes the nucleon energy high peaks in abundance curve of isotopes (Figure 1 levels with close values of energy are grouped into from author's quoted work according to Lang's (1978) series far apart from each other which are called data). Small double peaks in the well known curve 12 Odessa Astronomical Publications, vol. 17 (2004) of elemental abundances (Zyuss and Yuri) at magic between the physical parameters of isotopes. So, in numbers of neutrons 28, 40, 70, 112 have been added Figure D2 taken from Luc Valentin's (1986), the e®ects to double peaks of r-, s-processes of neutron capture of shells and magic numbers are superimposed upon at magic numbers 50, 82, 126. They convincingly the smooth dependence of binding energy as a function con¯rm an existence of these two capture processes. of the mass number (A) at N=28 and 50, Z=50, N=82 (marked there) and at twice magic nucleus Pb(82,126), but also distinct peaks are seen (Figure 1) at the twice 2. New results magic O(8,8), thrice magic Si(14,14), twice magic (on A and N or Z): N=28 (A=50), Z=40 (N=50), Z=50 Recently the author added the nuclei of isotopes (N=70), Z=50 (N=82), Z=70 (N=82) and Z=112 Si (14,20), Ca (20,28), Ni (28,40), C (6,8), O (8,6), (N=126). Noticeable declinations of the theoretical He(2,6) to twice magic ones. It is logically to name curve for binding energy from the experimental values three last isotopes, and also Ca(20,20) and Si(14,14) as for the lightest nuclei (Figure 4.3, Luc Valentine, 1986) thrice magic nuclei, as they have simultaneously magic are also explained by the e®ect of magic numbers: at numbers of protons (Z) and neutrons (N) and their sum A=4 (Z=N=2) and A=12 (Z=N=6). (mass number A=Z+N) is a magic number too. Magic The physicists discuss daring ideas of producing mass numbers (for example =258) also are proved in new nuclei by ¯ssion of super-heavy unstable nucleus the plots of dependences of some physical parameters. of the "double- uranium" type R(184,292) obtained Combining both sequences in one series, we have the by bombardment of uranium target with ion U(92). ¯rst 20 magic numbers: 2, 6, 8, 14, 20, 28, 40, 50, The ¯ssion usually occurs in the fragments of unequal 70, 82, 112, 126, 168, 184, 240, 258, 330, 350, 440, 462. masses. Therefore, at this ¯ssion new transuranium Taking the di®erences between neighbouring values, we elements can arise as well as new isotopes lighter than get series of 19 members: 4, 2, 6, 6, 8, 12, 10, 20, 12, uranium. 30, 14, 42, 16, 56, 18, 72, 20, 90. It is easily to notice Owing to this we had to extend the boundaries that this series consists of two sequences of numbers - of Table 2 (Paper I). The following terms of both odd D n¡ and even D n, the members of which are 2 1 2 sequences of magic numbers: M10=350, M11=462 and described by the following formulas: m8=330, m9=440 were calculated by formulas (1) and (2). As a result of this, it is suggested that the D n¡ = 2(n + 1); where n = 1, 2, ..., 10 (3) 2 1 super-nucleus may decay into one of four mentioned above twice magic isotopes X(112,168), Y(126,184) D n = n(n + 1); where n = 1, 2, .., 9 (4) 2 and Z(168,258) or Z'(168,240) plus fragment in areas Thus, every subsequent member is expressed of twice magic nuclei Yb(70,112), Sn(50,70) and through previous one by formulas: S(16,16) respectively. It could serve as a test for presence of these unobserved twice magic isotopes by M1 = 2; (5) means of analysis of those fragments. Mi+1 = Mi + Di; where i = 1, 2, 3, ... (6) 3. Properties of magic numbers are determined as i = 2n or i = 2n ¡ 1 for n in formulas (3) and (4). For example, at n = 1 D2n¡1 = D1 = 2(1 + 1) = 4 and M2 = M1 + D1=2+4=6, 1) All magic numbers are even. D2n=D2=2 and M3=M2 + D2=6+2=8 and etc. 2) They ¯t to the recurrent formulas (1)-(2) or (3)-(6). In the model of nucleus with nucleon orbits, the 3) They are the sums of two or three other magic num- Quantum Mechanics calculations for rectangular bers: 8=2+6, 14=6+8, 20=6+14, 28=20+8, 50=40+ potential well give (Table 21, Enrico Fermi, 1951) the 8+2 (or 50=28+14+8); 70=50+20, 82=40+28+14, total numbers of ¯lled states for closed shells: 2, 8, 20, 112=70+28+14 (or 112=70+40+2), 126=112+14; 40, 70, 112, 168, ... Numbers 40, 70 and 112 coincide 168=126+40+2 (or 168=126+28+14); 184=168+14+2 with those obtained by the author (Paper I), but the (or 184=126+50+8); 240=184+50+6 (or 240=168+ numbers 6, 14 and 28 are not present in this table. 70+2); 258=112+126+20, 330=184+126+20 (or 330= The numbers 2, 8, 20, 50, 82, 126 were referred to 168+112+50), 350=330+20 (or 350=184+126+40), the magic ones, because nuclei with such number of 440=350+82+8 (or 440=350+50+40), 462=350+112.