PERIODIC TABLE OF NUCLIDES BASED ON THE NUCLEAR SHELL MODEL ANDRZEJ B. WI ĘCKOWSKI 1,2 1 Institute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, ul. Szafrana 4a, PL-65-516 Zielona Góra, Poland 2 Institute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17, PL-60-179 Pozna ń, Poland E-mail : [email protected] Received February 12, 2019 Abstract . A historical overview of works involved in the development of the shell model of nuclei has been made. A special attention was given to physical, geochemical and number theoretical aspects. Two sequences of nuclear magic and semi-magic numbers are known to be of major importance to the graphical construction of the periodic table of nuclides. On the basis of ordering the nucleons (protons and neutrons) according to their energy state a periodic Table of Nuclides was built up. For comparison a version of the periodic table of chemical elements and two slightly differing versions of the periodic table of nuclides are presented. Key words: Nuclear magic and semi-magic numbers, primordial nuclides, mirror nuclides. 1. INTRODUCTION While the periodic table of chemical elements is known since the nineteenth century, an analogous periodic Table of Nuclides based on the nuclear shell model remains unknown till now. The electron configuration in atoms and the nucleon configuration in nuclides are described similarly by the respective shell models. In comparison with the structure of electron shells, the structure of nucleon shells is more complex, because there are two separate kinds of nucleons (protons and neutrons) and different forces are involved. The aim of this paper is to work out a graphical presentation of the periodic table of nuclides. The centre of attention will be drawn to the shell model of nuclei. 2. PERIODIC TABLE OF CHEMICAL ELEMENTS The principles of chemistry have a physical basis. Even the periodic table of Mendeleev apparently, as it turned out later, was built up on a property of the atomic nucleus, namely its mass, because at the time of developing the table of chemical elements, the electron structure of atoms was unknown. The building up Romanian Journal of Physics 64 , 303 (2019) Article no. 303 Andrzej B. Wi ęckowski 2 of the periodic table of chemical elements by Meyer [1, 2] and Mendeleev [3–7] was a milestone in the development of chemistry. In 1882 both scientists, Dmitri Mendeleev and Lothar Meyer, were honoured jointly with the Davy Medal by the Royal Society of London for their discovery of the periodic relations of the atomic weights . Moseley [8, 9], while investigating the X-ray spectra of different elements, gave a physical basis of the periodic table by modern truly ordering chemical elements according to their atomic numbers Z. Later with the development of the quantum mechanics (and the quantum chemistry) the location of the elements in the periodic table was connected with their electron configuration in atomic shells. The periodic table of chemical elements is undergoing further development and study. A broad band of historical, physical, chemical and mathematical aspects of the periodic table was presented in two books edited by Kaji, Kragh and Palló [10] and by Scerri and Restrepo [11]. A version of the periodic table of chemical elements is presented in Fig. 1. n 1 1 2 H He 2 2 3 4 He Li Be 3 5 6 7 8 9 10 11 12 B C N O F Ne Na Mg 4 13 14 15 16 17 18 19 20 Al Si P S Cl Ar K Ca 5 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr 6 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba 7 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra 8 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs Mt Ds Rg Cn Nh Fl Mc Lv Ts Og Uue Ubn f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 ff4 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 p1 p2 p3 p4 p5 p6 s1 s2 Fig. 1 – (Colour online). A version of the periodic table of chemical elements. The blocks of electron shells s, p, d, f, in atoms are in the order from right to left. The cells with darker (rose in colour) background (column p 6) correspond to the noble gases [ n – period (row) number]. The lowest strip (under the periodic table) represents the number of electrons occupying the respective energy level of electron shell. For each period with the ordinal number of the row n the value of the length of the period L(n) is equal to: L(n) = [2 n + 1 – (–1) n]2 /8 = 2, 2, 8, 8, 18, 18, 32, 32 (1) for n = 1, 2, 3, 4, 5, 6, 7, 8. The atomic number of the noble gases Z(n) can be calculated with the formula: 3 Periodic Table of Nuclides Article no. 303 n Z(n) = ∑ iL )( – 2 = [2 n3 + 6n2 + 7 n – 21 – 3( n + 1)(–1) n] /12 = i=1 (2) = (0), 2, 10, 18, 36, 54, 86, 118 for n = (1), 2, 3, 4, 5, 6, 7, 8. In Fig.1 the cell of helium He is repeated and appears twice, because the helium He atom having the electron configuration 1s2 belongs to the sequence of the noble gases Z(n) [ Z(2) = 2]. 3. DEVELOPMENT OF THE SHELL MODEL OF NUCLEI 3.1. WORKS BEFORE DISCOVERY OF THE NEUTRON BY CHADWICK It was Harkins [12–16], who has studied the problem of the structure of atomic nuclei very early. Harkins [12–14] adopted the hypothesis that during building up the atomic nuclei the most stable nuclei were formed with the biggest abundance. By analyzing the abundance of chemical elements in iron meteorites, stone meteorites and on the surface of the Earth, he came to the conclusion that the elements with even atomic number are much more abundant than the odd numbered elements. At that time the view dominated that the nuclei of elements consist of helium (α++ ) and hydrogen ( π+) nuclei, as well as of binding and cementing electrons ( β–) (see Harkins [15, 16]). Harkins [17] and Rutherford [18], while discussing the constitution of nuclei in isotopes, postulated the possible existence of an atom having one binding electron in the hydrogen nucleus, which has a mass equal to unity and a zero electric charge. For this hypothetical particle Harkins [19, 20] used the name ‘neutron’; he wrote: Here the term neutron represents one proton plus one electron (pe). (see Ref. [19], p. 315). Harkins [19] found that the most abundant isotopes in meteorites are oxygen 16 O, magnesium 24 Mg, silicon 28 Si, sulphur 32 S and iron 56 Fe. Niggli [21] has investigated the chemical composition of eruption rocks and found that in the Earth’s crust the distribution curve of petrogenic elements demonstrates maxima: ( 1H), 8O, 14 Si, 20 Ca, 26 Fe, having a difference of the atomic numbers equal to 6. On the other hand, the distribution curve of metallogenic elements demonstrate maxima: 26 Fe – 28 Ni, 48 Cd – 50 Sn, 80 Hg – 82 Pb, where in each pair the difference in atomic numbers is equal to 2. Later, the atomic numbers 28, 50, 82 were denoted ‘magic numbers’ of nucleons (protons and neutrons). Sonder [22] proposed an expansion of the first series by the following elements: …, 26 Fe, 38 Sr, 50 Sn, 56 Ba, 74 W, ( 80 Hg), 92 U, where the differences in atomic numbers are 2·6, 2·6, 1·6, 3·6, 1·6, 2·6, respectively. By analysis of Article no. 303 Andrzej B. Wi ęckowski 4 the abundance values of elements and by some speculations on the structure and symmetry of nuclei, Sonder [22–24] postulated the existence of a nuclear periodicity. Beck [25] built up a scheme of known isotopes and showed regularities in the structure of atomic nuclei. He was the first who, from the distribution of some series of isotopes, postulated the possibility of the existence of nuclear shells, similar to the shells of electrons. 3.2. WORKS BEFORE THE FORMULATION OF THE NUCLEAR SHELL MODEL BY GOEPPERT MAYER, HAXEL, JENSEN AND SUESS The discovery of the neutron has been made by Chadwick [26, 27]. He produced neutrons of mass 1 and charge 0 by the bombardment of beryllium Be or boron B with α-particles from a source of polonium Po. He has carried out the following nuclear reactions: 9Be + 4He → 12 C + 1n and 11 B + 4He → 14 N + 1n. Chadwick [26] supposed that the neutron is a constituent of atomic nuclei. In a comment to this discovery Iwanenko [28] also considered the neutron as being a component of the nucleus and mused of whether the neutron can be an elementary particle like the electron and the proton.