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Home , Magic number (physics), Nuclear shell model, Table of nuclides

OF BASED ON THE

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 ( and ) 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 configuration in and the 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 have a physical basis. Even the periodic table of Mendeleev apparently, as it turned out later, was built up on a property of the , 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 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 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 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 He is repeated and appears twice, because the helium He 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 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 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 ( 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 , 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 plus one electron (pe). (see Ref. [19], p. 315). Harkins [19] found that the most abundant isotopes in meteorites are 16 O, 24 Mg, 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 Be or B with α-particles from a source of 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. Heisenberg [29] indicated to the quantum mechanical consequences of the assumption that the atomic nuclei are composed only of interacting protons and neutrons as building stones, without participation of electrons. Bartlett [30, 31] discussed the possible structure of light elements with small atomic number and came to the conclusion that a regularity ends at the nucleus 16 O and this may be interpreted as a result of the formation of a closed shell. He indicated an analogy with the system of external electrons. Similar closed shells for protons and for neutrons can exist also for heavier elements. The principal is of smaller importance, because the central field is not characterized by a Coulomb potential [31]. However, Bartlett [32], following a suggestion put forward by Dirac, adopted the view that beside protons and neutrons, also electrons are present in nuclei, because the β-type decay exists. Gapon and Iwanenko [33] considered the successive building up of the nuclei in shells of protons and neutrons as being in analogy with the periodic system of elements. Elsasser [34, 35], while referring to the suggestion of Bartlett [31, 32] on the existence of consecutive shells in nuclei, pointed to the significance of this for the formulation of the role of separate shells for protons and neutrons, respectively, in the number of isotopes for atomic numbers of light elements. Elsasser [35, 36] was citing the results obtained by Guggenheimer [37], who pointed to periodic regularity of the

5 Periodic Table of Nuclides Article no. 303 structure of nuclei in groups of consecutive chemical elements. Guggenheimer [37] placed emphasis on the numbers of neutrons N = 50 and N = 82, for which there exist a higher number of and the shells became closed. The term ‘isotones’ was introduced by Guggenheimer [37]. Elsasser [36, 38], after analyzing the models of isotopes and isotones, supplemented these two distinguished numbers with the third number N = 126.

3.3. NUCLEAR SHELL MODEL ACCORDING TO GOEPPERT MAYER, HAXEL, JENSEN AND SUESS

Suess [39] and Goeppert Mayer [40] discussed the of naturally occurring chemical elements and drew the attention to isotopes and isotones with higher stability and abundance. Especially stable were elements having the special, named distinguished or excellent numbers ( ausgezeichnete Zahlen [39]) of 28, 50, 82 and 126 neutrons or protons. Goeppert Mayer [40] suggested that this phenomenon is an effect of filling up the closed shells in nuclei. Almost simultaneously and independently of each other, Goeppert Mayer [41–43] and Haxel, Jensen and Suess [44–48] have assumed for the calculation of the energy levels of protons and neutrons the single particle orbit model in which nuclear potential energy has a shape between a square well and a three-dimensional harmonic oscillator with strong -orbit ( l, s) coupling of the nucleons (protons or neutrons). The single particle energy states are filled up according to the Pauli exclusion principle. For each nucleon the authors calculated the total quantum numbers j = l + s ( s = ± ½), the multiplicities 2 j + 1, and the sum of multiplicities on each shell. The state with j = l + ½ has a lower energy than the state with j = l – ½ and the energy level with higher j is filled up first. When the number of identical nucleons is even, they couple giving a spin zero and do not contribute to the . When the number of identical nucleons is odd, they couple giving a spin j and a non-zero magnetic moment occurs. This scheme of occupation is the same for the shells of protons and for the shells of neutrons. The differences between the scheme for protons and neutrons are only of quantitative character. The shell model allowed the prediction of the nuclear spins, the and the magnetic moments (with a few exceptions). Goeppert Mayer [41] presented the spin terms of energy levels, the number of states on each energy level and the maximal occupation of each shell. The sum of the maximal occupations of shells gave the number of nucleons in closed shells; these numbers were named ‘magic numbers’: 2, 8, 20, 28, 50, 82, 126 (see [41], Table I). The distance between the energy level of closed shells filled with a of nucleons and the next energy level is greater than the distances between the energy levels of other shells. Haxel, Jensen and Suess [44] obtained the following numbers for the occupation of closed shells: 2, 6, 8, 14, 20 , 28 , 40, 50 , 70, 82 , 92, 112, 126

Article no. 303 Andrzej B. Wi ęckowski 6

(numbers written in bold are particularly representative; the numbers not put in bold are sometimes denoted as ‘semi-magic numbers’ or ‘sub-magic numbers’). The founders of the closed shell structure of nuclei presented their works later in the publications written by Haxel, Jensen and Suess [49] and by Goeppert Mayer and Jensen [50]. Brueckner and Levinson [51] have developed approximated solutions based on a self-consistent field method for a many-body problem which justified the application of the nuclear shell model in the case of strong nucleon- nucleon interactions. A further overview of the nuclear shell models has been presented by Jensen [52]. and J. Hans D. Jensen were awarded with the 1963 for their discoveries concerning nuclear shell structure (Nobel lectures: Goeppert Mayer [53], Jensen [54]). More detailed descriptions of earlier works in connection with the historical development of the nuclear shell model were presented by Zacharias [55], Kragh [56] and Johnson [57].

3.4. NUMBER THEORETICAL ASPECTS OF THE NUCLEAR SHELL MODEL

Bagge [58] has found that the magic numbers in the sequence N1(n) = 2, 6, 14, 28, 50, 82, 126, are given by the formula:

n  m 3  +  N1(n) = ( n + 5 n)/3 = 2 ∑1  (3) m=1  2  where n = 1, 2, 3, 4, 5, 6, 7. A similar formula was proposed by Valente [59, 60]. Bagge [61] found also that the magic numbers in the sequence N2(n) = 2, 8, 20, 40, 70, 112, are given by the formula:

n 3 m N2(n) = ( n – n)/3 = 2 ∑  (4) m=1 2  where n = 2, 3, 4, 5, 6, 7. Lepsius [62, 63] has shown, how the magic numbers can be derived from the binomial coefficients occurring in the Pascal’s triangle. Pauling [64, 65] explained the magic numbers by assuming a structure of atomic nuclei build up by layers of a mantle, an outer core and an inner core having completed shells and completed sub-shells filled up by nucleons. In this scheme the main magic numbers are composed of numbers having the form 2 k2 (similarly to the closed shells of electrons in atoms):

7 Periodic Table of Nuclides Article no. 303

2 = 2·1 2 8 = 2·2 2 20 = 2 + 18 = 2·1 2 + 2·3 2 (28 = 2 + 18 + 8 = 2·1 2 + 2·3 2 + 8) 50 = 8 + 32 + 10 = 2·2 2 + 2·4 2 + 10 82 = 2 + 18 + 50 + 12 = 2·1 2 + 2·3 2 + 2·5 2 + 12 126 = 8 + 32 + 72 + 14 = 2·2 2 + 2·4 2 + 2·6 2 + 14

(the magic number 28 has not been included in the scheme given by Pauling [64, 65]). This explanation of the magic numbers has not found a wider acceptance. For the calculation of all the nuclear magic numbers Weise [66] proposed to use the formula:

MN (m, k) = k·( m2 – m) + ( m3 + 5 m)/3 (5) where m = 1, 2, 3, 4, … with k = 1 if m = 1, 2, 3; k = 0 if m > 3. Herrmann [67, 68] has proved that the magic numbers in nuclear shells can be derived from group theoretical considerations of symmetry without defining the field potential shape. He found the following two sets of magic numbers:

nmagic 1 = ( N + 1)( N + 2)( N + 3)/3 = (2), 8, 20, 40, 70, 112, 168, 240 (6) for N = (0), 1, 2, 3, …

n = n – N(N + 1) = ( N + 1)[( N + 2)( N + 3) – 3 N] /3 = magic 2 magic 1 (7) = ( N + 1)[( N + 1) 2 + 5]/3 = 2, 6, 14, 28, 50, 82, 126, 184, 258 for N = 0, 1, 2, 3, … Further number theoretical discussion on general laws of the structure of stable nuclei can be found in the monograph by Boeyens and Levendis [69].

4. PERIODIC TABLE OF NUCLIDES

4.1. CONSTRUCTION PRINCIPLE OF THE PERIODIC TABLE OF NUCLIDES The construction principle ( Aufbauprinzip ) for building up the periodic table of nuclides is based on the scheme of energy levels of nucleons in the nuclear shell model given by Goeppert Mayer [41]. In Fig. 2, a comparison of the construction principles of the periodic table of chemical elements (a) and of the periodic table of nuclides (b) is shown. In the cells of the right part (b) of the comparison presented are the spin terms of nucleon shells and (in parentheses) the numbers of nucleons (protons or neutrons) filling up the closed shells. The ordering of the spin terms is in full agreement with the scheme of energy levels given by Goeppert Mayer [41]. The locations of the terms of closed shells, which are ordered in two sequences: N1(n) = 2, 6, 14, 28, 50, 82, 126, 184 and N2(n) = 2, 8, 20, 40, 70, 112, 168, 240

Article no. 303 Andrzej B. Wi ęckowski 8 can be easy seen in Fig. 2. The numbers 2, 8, 20, 28, 50, 82, 126, 184 are the nuclear magic numbers, whereas the numbers 2, 6, 14, 40, 70, 112, 168, 240 are nuclear semi-magic numbers.

(a) (b) n

2 2 1 1s 1s 1/2 (2)

2 2 4 2 2 1s 2s 1p 3/2 1p 1/2 (2) (6) (8)

6 2 6 4 2 3 2p 3s 1d 5/2 1d 3/2 2s 1/2 (10) (14) (20)

6 2 8 6 4 2 4 3p 4s 1f 7/2 1f 5/2 2p 3/2 2p 1/2 (18) (28) (40)

10 6 2 10 8 6 4 2 5 3d 4p 5s 1g 9/2 1g 7/2 2d 5/2 2d 3/2 3s 1/2 (36) (50) (70)

10 6 2 12 10 8 6 4 2 6 4d 5p 6s 1h 11/2 1h 9/2 2f 7/2 2f 5/2 3p 3/2 3p 1/2 (54) (82) (112)

14 10 6 2 14 12 10 8 6 4 2 7 4f 5d 6p 7s 1i 13/2 1i 11/2 2g 9/2 2g 7/2 3d 5/2 3d 3/2 4s 1/2 (86) (126) (168)

14 10 6 2 16 14 12 10 8 6 4 2 8 5f 6d 7p 8s 1j 15/2 1j 13/2 2h 11/2 2h 9/2 3f 7/2 3f 5/2 4p 3/2 4p 1/2 (118) (184) (240)

Fig. 2 – (Colour online). Comparison of the construction principle ( Aufbauprinzip ) of (a) the periodic table of chemical elements and (b) the periodic table of nuclides.

4.2. LENGTHS OF THE PERIODS AND CALCULATION OF MAGIC AND SEMI-MAGIC NUMBERS For each energy level with the angular momentum quantum number j the number of occupying nucleons is equal to (2 j + 1). In each period with the ordinal number of the period (row) n the highest value of j is equal to:

jmax = n – ½ (8)

Calculating the number of nuclides L(n) (length of the period) in each row n we obtain: j max n n + 1 L(n) = 2( j + )1 = 2 i = 2   = ∑ ∑   (9) j=½ i=1  2  = n(n + 1) = 2, 6, 12, 20, 30, 42, 56, 72 for n = 1, 2, 3, 4, 5, 6, 7, 8.

9 Periodic Table of Nuclides Article no. 303

The binomial coefficient

n + 1   = n(n + 1) /2 (10)  2  is known as the nth triangular number. The sequence of the magic numbers N1(n) can be calculated with the formula:

n n n  i  − + − N1(n) = ∑ ([ iL )1 ]2 = ∑ (iL )1 + 2 n = 2 ∑  + 2 n = i=1 i=1 i=1 2 (11) + n 1 2 2 = 2   + 2 n = n(n – 1) /3 + 2 n = n(n + 5) /3  3  or with the formula:

n n n i + 1 n − − N1(n) = ∑ iL )([ (2 i )]1 = ∑ iL )( – L (n – 1) = 2 ∑  – 2   = i=1 i=1 i=1  2   2 (12) + n 2 n 2 = 2   – 2   = n(n + 1)( n + 2) /3 – n(n – 1) = n(n + 5) /3  3   2

Finally, by applying any of the two equivalent ways of deriving the above formulae for N1(n) we obtain the sequence:

2 N1(n) = n(n + 5)/3 = 2, 6, 14, 28, 50, 82, 126, 184 (13) for n = 1, 2, 3, 4, 5, 6, 7, 8. The sequence of the magic numbers N2(n) can be calculated with the formula:

n n i + 1 n + 2 N (n) = iL )( = 2   = 2   = 2 ∑ ∑    (14) i=1 i=1  2   3  = n(n + 1)( n + 2) /3 = 2, 8, 20, 40, 70, 112, 168, 240 for n = 1, 2, 3, 4, 5, 6, 7, 8. The binomial coefficient

+ n 2 3   = n(n + 1)( n + 2) /6 = [( n + 1) – ( n + 1)] /6 (15)  3  is known as the nth tetrahedral number.

Article no. 303 Andrzej B. Wi ęckowski 10

We have also

n N2(n) – N1(n) = 2   = n(n – 1) (16)  2 and

N1(n) – N2(n – 1) = 2 n (17)

4.3. TWO VERSIONS OF THE PERIODIC TABLE OF NUCLIDES By putting each set of nucleons (protons or neutrons) in a given energy state into the respective cells of Fig. 2 we obtain the periodic table of nuclides, which is shown in Fig. 3. Because the order of the energy levels for protons and for neutrons is the same, the form of the periodic table is valid both for protons and for neutrons. The cells in rose colour (darker in print version) correspond to magic or semi-magic numbers of nucleons. The two lowest strips (below the periodic table) represent the number of nucleons occupying the respective energy level. The spin terms in the strips allow to give for each its nucleon configuration. Additionally, below to the primary version of the periodic Table of Nuclides, a modified version is presented. This version shows graphically the similarities and the connection between the two sequences of the magic and semi-magic numbers N1(n) and N2(n).

4.4. EXAMPLES OF APPLICATION OF THE PERIODIC TABLE OF NUCLIDES The form of the periodic Table of Nuclides can be used in practice by inserting the nuclides with a chosen property into the numbered cells of the periodic tables for protons and neutrons, respectively. Some examples are given below. In Fig. 4, part (a) for protons and part (b) for neutrons, the primordial (stable and nearly stable having natural abundances) nuclides are presented. It is seen that for odd numbers of nucleons (protons or neutrons) the number of primordial isotopes and isotones does not exceed two (only in the case of K with Z = 19, the number of primordial isotopes is equal to three). For even numbers of nucleons the number of isotopes and isotones is in general higher. As expected, for magic and semi-magic numbers of nucleons the number of isotopes and isotones is noticeable high. It is well known that particularly stable are the primordial doubly magic and semi-magic nuclides containing a number of nucleons from the sequences N1(n) or/and N2(n), which are listed in Table 1. All these nuclides have simultaneously the highest abundance from among the component nuclides of the considered . Normally, to present the properties of nuclides in the periodic table, the nuclides have to be shown in two tables – for protons and for neutrons, separately. In Fig. 5 an excerpt is shown from the periodic Table for Nuclides with N = Z. In this case, it is sufficient to present the nuclear properties in one table only. The values of nuclear spin quantum number J and the parity π are presented. The numerical data are taken from

11 Periodic Table of Nuclides Article no. 303 the Table of Nuclides presented by KAERI [70]. In Fig. 5 we find all four stable low mass odd-odd nuclides, which have a non-zero nuclear spin J: hydrogen () 2H ( J π = 1 +), 6Li ( J π = 1 +), boron 10 B ( J π = 3 +), 14 N ( J π = 1 +).

n (a) 0

1 1 2 H He 2 3 4 5 6 7 8 Li Be B C N O 3 9 10 11 12 13 14 15 16 17 18 19 20 F Ne Na Mg Al Si P S Cl Ar K Ca 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr 5 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb 6 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 … Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs Mt Ds Rg Cn 7 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 …

6 71 72 73 74 75 76 77 78 79 80 81 82 Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb … 7 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 Nh Fl Mc Lv Ts Og Uue Ubn …

n (b) 0

1 0 1 2 H He 2 2 3 4 5 6 7 8 He Li Be B C N O 3 8 9 10 11 12 13 14 15 16 17 18 19 20 O F Ne Na Mg Al Si P S Cl Ar K Ca 4 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr 5 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb 6 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 … Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs Mt Ds Rg Cn 7 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 …

40 Zr 6 70 71 72 73 74 75 76 77 78 79 80 81 82 Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb … 7 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 Nh Fl Mc Lv Ts Og Uue Ubn …

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 1 2 3 4 5 6 1 2 3 4 1 2 … h9/2 h9/2 h9/2 h9/2 h9/2 h9/2 h9/2 h9/2 h9/2 h9/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f5/2 f5/2 f5/2 f5/2 f5/2 f5/2 p3/2 p3/2 p3/2 p3/2 p1/2 p1/2 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 1 2 3 4 5 6 1 2 3 4 1 2 … g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g7/2 g7/2 g7/2 g7/2 g7/2 g7/2 g7/2 g7/2 d5/2 d5/2 d5/2 d5/2 d5/2 d5/2 d3/2 d3/2 d3/2 d3/2 s1/2 s1/2

1 2 3 4 5 6 7 8 9 10 11 12 h11/2 h11/2 h11/2 h11/2 h11/2 h11/2 h11/2 h11/2 h11/2 h11/2 h11/2 h11/2 … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i13/2 i11/2 i11/2 i11/2 i11/2 i11/2 i11/2 i11/2 i11/2 i11/2 i11/2 i11/2 i11/2 …

Fig. 3 – (Colour online). Periodic tables of nuclides based on the nuclear shell model of nucleons (protons or neutrons) ( Z, N ≤ 168): (a) primary version of the periodic table of nuclides, (b) modified version of the periodic table of nuclides. For protons in the cells of the periodic tables the numbers and the chemical symbols are valid; for neutrons only the numbers are valid. The cells with darker (rose in colour version) background correspond to magic and semi-magic numbers of nucleons from the sequences N1(n) and N2(n) [ n – period (row) number]. In part (b) the cells of nuclides with a magic or semi-magic number of nucleons from the sequence N2(n) are repeated and appear twice. The 0 cells are added only for setting the pairs of magic and semi-magic numbers N1(n) and N2(n) in order. Two lowest strips represent the number of nucleons occupying the respective energy level.

Article no. 303 Andrzej B. Wi ęckowski 12

In Fig. 6 and Fig. 7 presented are fragments of the periodic tables of selected sequences of nuclides with their values of the spin quantum number J and the parity π for protons and for neutrons. Following periodic tables were pairwise juxtaposed:

n 0 (a) n 1 2 1 1H 3He 2H 4He 3 4 5 6 7 8 6Li 9Be 10 B 12 C 14 N 16 O 2 7Li 11 B 13 C 15 N 17 O 18 O 9 10 11 12 13 14 15 16 17 18 19 20 19 F 20 Ne 23 Na 24 Mg 27 Al 28 Si 31 P 32 S 35 Cl 36 Ar 39 K 40 Ca 21 Ne 25 Mg 29 Si 33 S 37 Cl 38 Ar 40 K 42 Ca 3 22 Ne 26 Mg 30 Si 34 S 40 Ar 41 K 43 Ca 36 S 44 Ca 46 Ca 48 Ca 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 45 Sc 46 Ti 50 V 50 Cr 55 Mn 54 Fe 59 Co 58 Ni 63 Cu 64 Zn 69 Ga 70 Ge 75 As 74 Se 79 Br 78 Kr 85 Rb 84 Sr 89 Y 90 Zr 47 Ti 51 V 52 Cr 56 Fe 60 Ni 65 Cu 66 Zn 71 Ga 72 Ge 76 Se 81 Br 80 Kr 87 Rb 86 Sr 91 Zr 48 Ti 53 Cr 57 Fe 61 Ni 67 Zn 73 Ge 77 Se 82 Kr 87 Sr 92 Zr 4 49 Ti 54 Cr 58 Fe 62 Ni 68 Zn 74 Ge 78 Se 83 Kr 88 Sr 94 Zr 50 Ti 64 Ni 70 Zn 76 Ge 80 Se 84 Kr 96 Zr 82 Se 86 Kr 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 93 Nb 92 Mo (Tc) 96 Ru 103 Rh 102 Pd 107 Ag 106 Cd 113 In 112 Sn 121 Sb 120 Te 127 I 124 Xe 133 Cs 130 Ba 138 La 136 Ce 141 Pr 142 Nd (Pm) 144 Sm 151 Eu 152 Gd 159 Tb 156 Dy 165 Ho 162 Er 169 Tm 168 Yb 94 Mo 98 Ru 104 Pd 109 Ag 108 Cd 115 In 114 Sn 123 Sb 122 Te 126 Xe 132 Ba 139 La 138 Ce 143 Nd 147 Sm 153 Eu 154 Gd 158 Dy 164 Er 170 Yb 95 Mo 99 Ru 105 Pd 110 Cd 115 Sn 123 Te 128 Xe 134 Ba 140 Ce 144 Nd 148 Sm 155 Gd 160 Dy 166 Er 171 Yb 96 Mo 100 Ru 106 Pd 111 Cd 116 Sn 124 Te 129 Xe 135 Ba 142 Ce 145 Nd 149 Sm 156 Gd 161 Dy 167 Er 172 Yb 5 97 Mo 101 Ru 108 Pd 112 Cd 117 Sn 125 Te 130 Xe 136 Ba 146 Nd 150 Sm 157 Gd 162 Dy 168 Er 173 Yb 98 Mo 102 Ru 110Pd 113 Cd 118 Sn 126 Te 131 Xe 137 Ba 148 Nd 152 Sm 158 Gd 163 Dy 170 Er 174 Yb 100 Mo 104 Ru 114 Cd 119 Sn 128 Te 132 Xe 138 Ba 150 Nd 154 Sm 160 Gd 164 Dy 176 Yb 116 Cd 120 Sn 130 Te 134 Xe 122 Sn 136 Xe 124 Sn

n 0 (b) 1H 1 2 1 2H 4He 3He 3 4 5 6 7 8 6Li 7Li 9Be 11 B 13 C 15 N 2 10 B 12 C 14 N 16 O

9 10 11 12 13 14 15 16 17 18 19 20 17 O 18 O 21 Ne 22 Ne 25 Mg 26 Mg 29 Si 30 Si 33 S 34 S 36 S 19 F 23 Na 27 Al 31 P 35 Cl 37 Cl 3 20 Ne 24 Mg 28 Si 32 S 36 Ar 38 Ar 39 K 40 Ca

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 40 K 40 Ar 43 Ca 44 Ca 47 Ti 46 Ca 49 Ti 48 Ca 53 Cr 54 Cr 57 Fe 58 Fe 61 Ni 62 Ni 64 Ni 67 Zn 68 Zn 70 Zn 41 K 45 Sc 48 Ti 50 V 50 Ti 55 Mn 59 Co 63 Cu 65 Cu 69 Ga 71 Ga 42 Ca 46 Ti 50 Cr 51 V 56 Fe 60 Ni 64 Zn 66 Zn 70 Ge 72 Ge 4 52 Cr 58 Ni 74 Se 54 Fe

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 73 Ge 74 Ge 77 Se 76 Ge 80 Se 83 Kr 82 Se 87 Sr 86 Kr 91 Zr 92 Zr 95 Mo 94 Zr 97 Mo 96 Zr 101 Ru 100 Mo 105 Pd 104 Ru 108 Pd 111 Cd 110 Pd 113 Cd 114 Cd 117 Sn 116 Cd 119 Sn 120 Sn 75 As 78 Se 81 Br 84 Kr 87 Rb 93 Nb 96 Mo 99 Ru 98 Mo 102 Ru 106 Pd 109 Ag 112 Cd 115 Sn 115 In 118 Sn 121 Sb 76 Se 79 Br 82 Kr 85 Rb 88 Sr 94 Mo 98 Ru 100 Ru 103 Rh 107 Ag 110 Cd 113 In 116 Sn 120 Te 122 Te 78 Kr 80 Kr 84 Sr 86 Sr 89 Y 96 Ru 102 Pd 104 Pd 108 Cd 112 Sn 114 Sn 124 Xe 5 90 Zr 106 Cd 92 Mo

1 2 3 4 5 6 7 8 1 2 3 4 5 6 1 2 3 4 1 2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f5/2 f5/2 f5/2 f5/2 f5/2 f5/2 p3/2 p3/2 p3/2 p3/2 p1/2 p1/2 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 1 2 3 4 5 6 1 2 3 4 1 2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g9/2 g7/2 g7/2 g7/2 g7/2 g7/2 g7/2 g7/2 g7/2 d5/2 d5/2 d5/2 d5/2 d5/2 d5/2 d3/2 d3/2 d3/2 d3/2 s1/2 s1/2 Fig. 4 – (Colour online). Periodic tables of primordial (stable or nearly stable) nuclides; (a) table of isotopes for protons (Z ≤ 70) and (b) table of isotones for neutrons (N ≤ 70), respectively.

13 Periodic Table of Nuclides Article no. 303

Table 1 The primordial doubly magic and semi-magic nuclides, number of protons Z, number of neutrons N, and abundance [%]

Nuclide Z N Abundance [%]

4 He N1(1) = N2(1) = 2 N1(1) = N2(1) = 2 99.999866 12 C N1(2) = 6 N1(2) = 6 98.93 16 O N2(2) = 8 N2(2) = 8 99.757 28 Si N1(3) = 14 N1(3) = 14 92.223 40 Ca N2(3) = 20 N2(3) = 20 96.94 48 ( Ca) N2(3) = 20 N1(4) = 28 (0.187) 90 Zr N2(4) = 40 N1(5) = 50 51.45 120 Sn N1(5) = 50 N2(5) = 70 32.58 208 Pb N1(6) = 82 N1(7) = 126 52.4

– periodic Table of Nuclides with N = Z – 1 for protons and periodic Table of Nuclides with Z = N – 1 for neutrons, – periodic Table of Nuclides with N = Z + 1 for protons and periodic Table of Nuclides with Z = N + 1 for neutrons, – periodic Table of Nuclides with N = Z – 2 for protons and periodic Table of Nuclides with Z = N – 2 for neutrons, – periodic Table of Nuclides with N = Z + 2 for protons and periodic Table of Nuclides with Z = N + 2 for neutrons,

1 N = Z 1 2 2H 4He 1+ 0+ 2 3 4 5 6 7 8 6Li 8Be 10 B 12 C 14 N 16 O 1+ 0+ 3+ 0+ 1+ 0+ 3 9 10 11 12 13 14 15 16 17 18 19 20 18 F 20 Ne 22 Na 24 Mg 26 Al 28 Si 30 P 32 S 34 Cl 36 Ar 38 K 40 Ca 1+ 0+ 3+ 0+ 5+ 0+ 1+ 0+ 0+ 0+ 3+ 0+ 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 42 Sc 44 Ti 46 V 48 Cr 50 Mn 52 Fe 54 Co 56 Ni 58 Cu 60 Zn 62 Ga 64 Ge 66 As 68 Se 70 Br 72 Kr 74 Rb 76Sr 78 Y 80 Zr 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+ 1+ 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+

1 2 3 4 5 6 1 2 3 4 1 2 d5/2 d5/2 d5/2 d5/2 d5/2 d5/2 d3/2 d3/2 d3/2 d3/2 s1/2 s1/2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 1 2 3 4 1 2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f5/2 f5/2 f5/2 f5/2 f5/2 f5/2 p3/2 p3/2 p3/2 p3/2 p1/2 p1/2 Fig. 5 – (Colour online). Periodic Table of Nuclides (Z, N ≤ 40) fulfilling the condition N = Z. The values of the nuclear spin quantum number J and the parity π are shown.

Article no. 303 Andrzej B. Więckowski 14

1 N=Z–1 1 2 1H 3He 1/2+ 1/2+ 2 3 4 5 6 7 8 5Li 7Be 9B 11C 13N 15O 3/2– 3/2– 3/2– 3/2– 1/2– 1/2– 3 9 10 11 12 13 14 15 16 17 18 19 20 17F 19Ne 21Na 23Mg 25Al 27Si 29P 31S 33Cl 35Ar 37K 39Ca 5/2+ 1/2+ 3/2+ 3/2+ 5/2+ 5/2+ 1/2+ 1/2+ 3/2+ 3/2+ 3/2+ 3/2+ 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41Sc 43Ti 45V 47Cr 49Mn 51Fe 53Co 55Ni 57Cu 59Zn 61Ga 63Ge 65As 67Se 69Br 71Kr 73Rb 75Sr 77Y 79Zr 7/2– 7/2– 7/2– 3/2– 5/2– 5/2– 7/2– 7/2– 3/2– 3/2– 3/2– 3/2– 3/2– 5/2– 1/2– 5/2– 3/2– 3/2– 5/2+ 5/2+

1 Z=N–1 1 2 1n 3H 1/2+ 1/2+ 2 3 4 5 6 7 8 5He 7Li 9Be 11B 13C 15N 3/2– 3/2– 3/2– 3/2– 1/2– 1/2– 3 9 10 11 12 13 14 15 16 17 18 19 20 17O 19F 21Ne 23Na 25Mg 27Al 29Si 31P 33S 35Cl 37Ar 39K 5/2+ 1/2+ 3/2+ 3/2+ 5/2+ 5/2+ 1/2+ 1/2+ 3/2+ 3/2+ 3/2+ 3/2+ 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41Ca 43Sc 45Ti 47V 49Cr 51Mn 53Fe 55Co 57Ni 59Cu 61Zn 63Ga 65Ge 67As 69Se 71Br 73Kr 75Rb 77Sr 79Y 7/2– 7/2– 7/2– 3/2– 5/2– 5/2– 7/2– 7/2– 3/2– 3/2– 3/2– 3/2– 3/2– 5/2– 1/2– 5/2– 3/2– 3/2– 5/2+ 5/2+

1 N=Z+1 1 2 3H 5He 1/2+ 3/2– 2 3 4 5 6 7 8 7Li 9Be 11B 13C 15N 17O 3/2– 3/2– 3/2– 1/2– 1/2– 5/2+ 3 9 10 11 12 13 14 15 16 17 18 19 20 19F 21Ne 23Na 25Mg 27Al 29Si 31P 33S 35Cl 37Ar 39K 41Ca 1/2+ 3/2+ 3/2+ 5/2+ 5/2+ 1/2+ 1/2+ 3/2+ 3/2+ 3/2+ 3/2+ 7/2– 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 43Sc 45Ti 47V 49Cr 51Mn 53Fe 55Co 57Ni 59Cu 61Zn 63Ga 65Ge 67As 69Se 71Br 73Kr 75Rb 77Sr 79Y 81Zr 7/2– 7/2– 3/2– 5/2– 5/2– 7/2– 7/2– 3/2– 3/2– 3/2– 3/2– 3/2– 5/2– 1/2– 5/2– 3/2– 3/2– 5/2+ 5/2+ 3/2–

1 Z=N+1 1 2 3He 5Li 1/2+ 3/2– 2 3 4 5 6 7 8 7Be 9B 11C 13N 15O 17F 3/2– 3/2– 3/2– 1/2– 1/2– 5/2+ 3 9 10 11 12 13 14 15 16 17 18 19 20 19Ne 21Na 23Mg 25Al 27Si 29P 31S 33Cl 35Ar 37K 39Ca 41Sc 1/2+ 3/2+ 3/2+ 5/2+ 5/2+ 1/2+ 1/2+ 3/2+ 3/2+ 3/2+ 3/2+ 7/2– 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 43Ti 45V 47Cr 49Mn 51Fe 53Co 55Ni 57Cu 59Zn 61Ga 63Ge 65As 67Se 69Br 71Kr 73Rb 75Sr 77Y 79Zr 81Nb 7/2– 7/2– 3/2– 5/2– 5/2– 7/2– 7/2– 3/2– 3/2– 3/2– 3/2– 3/2– 5/2– 1/2– 5/2– 3/2– 3/2– 5/2+ 5/2+ 3/2–

1 2 3 4 5 6 1 2 3 4 1 2 d5/2 d5/2 d5/2 d5/2 d5/2 d5/2 d3/2 d3/2 d3/2 d3/2 s1/2 s1/2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 1 2 3 4 1 2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f5/2 f5/2 f5/2 f5/2 f5/2 f5/2 p3/2 p3/2 p3/2 p3/2 p1/2 p1/2 Fig. 6 – (Colour online). Periodic Tables of Nuclides (Z, N≤ 40) fulfilling the condition: |N – Z| = 1. 15 Periodic Table of Nuclides Article no. 303

1 N = Z–2 1 2 – – – – 2 3 4 5 6 7 8 4Li 6Be 8B 10 C 12 N 14 O 2– 0+ 2+ 0+ 1+ 0+ 3 9 10 11 12 13 14 15 16 17 18 19 20 16 F 18 Ne 20 Na 22 Mg 24 Al 26 Si 28 P 30 S 32 Cl 34 Ar 36 K 38 Ca 0– 0+ 2+ 0+ 4+ 0+ 3+ 0+ 1+ 0+ 2+ 0+ 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 40 Sc 42 Ti 44 V 46 Cr 48 Mn 50 Fe 52 Co 54 Ni 56 Cu 58 Zn 60 Ga 62 Ge 64 As 66 Se 68 Br 70 Kr 72 Rb 74 Sr 76 Y 78 Zr 4– 0+ 2+ 0+ 4+ 0+ 6+ 0+ 4+ 0+ 2+ 0+ 0+ 0+ 3+ 0+ 1+ 0+ 1– 0+

1 Z = N–2 1 2 – – – – 2 3 4 5 6 7 8 4H 6He 8Li 10 Be 12 B 14 C 2– 0+ 2+ 0+ 1+ 0+ 3 9 10 11 12 13 14 15 16 17 18 19 20 16 N 18 O 20 F 22 Ne 24 Na 26 Mg 28 Al 30 Si 32 P 34 S 36 Cl 38 Ar 2– 0+ 2+ 0+ 4+ 0+ 3+ 0+ 1+ 0+ 2+ 0+ 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 40 K 42 Ca 44 Sc 46 Ti 48 V 50 Cr 52 Mn 54 Fe 56 Co 58 Ni 60 Cu 62 Zn 64 Ga 66 Ge 68 As 70 Se 72 Br 74 Kr 76 Rb 78 Sr 4– 0+ 2+ 0+ 4+ 0+ 6+ 0+ 4+ 0+ 2+ 0+ 0+ 0+ 3+ 0+ 1+ 0+ 1– 0+

1 N = Z+2 1 2 4H 6He 2– 0+ 2 3 4 5 6 7 8 8Li 10 Be 12 B 14 C 16 N 18 O 2+ 0+ 1+ 0+ 2– 0+ 3 9 10 11 12 13 14 15 16 17 18 19 20 20 F 22 Ne 24 Na 26 Mg 28 Al 30 Si 32 P 34 S 36 Cl 38 Ar 40 K 42 Ca 2+ 0+ 4+ 0+ 3+ 0+ 1+ 0+ 2+ 0+ 4– 0+ 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 44 Sc 46 Ti 48 V 50 Cr 52 Mn 54 Fe 56 Co 58 Ni 60 Cu 62 Zn 64 Ga 66 Ge 68 As 70 Se 72 Br 74 Kr 76 Rb 78 Sr 80 Y 82 Zr 2+ 0+ 4+ 0+ 6+ 0+ 4+ 0+ 2+ 0+ 0+ 0+ 3+ 0+ 1+ 0+ 1– 0+ 4– 0+

1 1 2 4Li 6Be 2– 0+ 2 3 4 5 6 7 8 8B 10 C 12 N 14 O 16 F 18 Ne 2+ 0+ 1+ 0+ 0– 0+ 3 Z = N+2 9 10 11 12 13 14 15 16 17 18 19 20 20 Na 22 Mg 24 Al 26 Si 28 P 30 S 32 Cl 34 Ar 36 K 38 Ca 40 Sc 42 Ti 2+ 0+ 4+ 0+ 3+ 0+ 1+ 0+ 2+ 0+ 4– 0+ 4 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 44 V 46 Cr 48 Mn 50 Fe 52 Co 54 Ni 56 Cu 58 Zn 60 Ga 62 Ge 64 As 66 Se 68 Br 70 Kr 72 Rb 74 Sr 76 Y 78 Zr 80 Nb 82 Mo 2+ 0+ 4+ 0+ 6+ 0+ 4+ 0+ 2+ 0+ 0+ 0+ 3+ 0+ 1+ 0+ 1– 0+ – –

1 2 3 4 5 6 1 2 3 4 1 2 d5/2 d5/2 d5/2 d5/2 d5/2 d5/2 d3/2 d3/2 d3/2 d3/2 s1/2 s1/2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 1 2 3 4 1 2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f7/2 f5/2 f5/2 f5/2 f5/2 f5/2 f5/2 p3/2 p3/2 p3/2 p3/2 p1/2 p1/2 Fig. 7 – (Colour online). Periodic Tables of Nuclides (Z, N ≤ 40) fulfilling the condition: | N – Z | = 2.

Article no. 303 Andrzej B. Wi ęckowski 16

In each set of two consecutive tables for protons and for neutrons the respective values of the nuclear spin J and the parity π are identical and the set contains isobars being mirror nuclides (an incompatibility is for J π(16 N) = 2 – and J π(16 F) = 0 –). The numerical data are taken from the Table of Nuclides presented by KAERI [70].

5. CONCLUSIONS

It has been shown that the properties of nuclides can be adequately presented in the periodic table of nuclides by showing them in two tables identical in form: one for protons and another for neutrons. The application of the periodic Table of Nuclides was exemplified by presentation of primordial nuclides and isobars being mirror nuclides. The use of the periodic Table of Nuclides forms an alternative method for presentation of the properties of nuclei in scientific and educational applications. The periodic Table of Nuclides gives an informative graphical impression and a visual understanding of the regularities in nuclear shells. The periodic table has also an educational significance and can be helpful in giving a more understandable and adequate illustration of the properties of nuclides in teaching and . A disadvantage of the proposed periodic table of nuclides is that the properties of nuclides should be presented in two tables, one for protons and another for neutrons. The author is aware that the presented periodic Table of Nuclides is an idealized one and it should undergo further modifications in the future.

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