Shell Model of Nucleus

           Fibonacci numbers and the Pascal Triangle

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Shell Model of Nucleus

Shell Model of Atom

Jovanovic`s Model: Periodicity of nuclei properties

The proton configuration of an atom describes the orbitals occupied by protons on the nucleus. The basis of this prediction is the aufbau principle, which assumes that protons are added to an nucleus, one at a time, starting with the lowest energy orbital, until all of the protons have been placed in an appropriate orbital.
An easy way to calculate the total number of protons that can be held by a given energy level is to use the triangular numbers formula (n+1)*n , where n equals the number of the protons shell. For example, for the 1st proton shell n=1 and 2*1 = 2, telling us that the capacity of the 1st shell is 2 protons. For the 2nd shell ( n=2 ) and 3*2 = 6. For an atom to fill its 2nd proton shell, 8 protons would be needed : 2 to fill the 1st shell and 6 to fill the 2nd.


Principle energy
level ( n )

Maximum number
of protons (n+1)*n

1

2

2

6

3

12

4

20

5

30

6

42

7

56

      The number of sublevels that an energy level can contain is equal to the principle quantum number of that level. The first sublevel is called s sublevel. s sublevels have one orbital, which can hold up to two protons. The second sublevel is called a p sublevel. p sublevels have two orbitals, each of which can hold 2 protons, for a total of 4. The third sublevel is called a d sublevel and d sublevels have 3 orbitals, for a possible total of 6 protons. The fourth sublevel is called an f sublevel. f sublevels, with 4 orbitals, can hold up to 8 protons. The 6-th sublevel is called an h sublevel. h sublevels, with 6 orbitals, can hold up to 12 protons. The 7-th sublevel with 7 orbitals, can hold up to 14 protons. Although energy levels that are lower than 7 would contain additional sublevels, these sublevels have not been named because no known atom in its ground state would have protons that occupy them.


Orbital and Proton Capacity for the Sublevels

Sublevel # of orbitals Maximum number of protons

s 1 2

p 2 4

d 3 6

f 4
8

g 5 10

h 6 12

... 7 14

... 8 16


An easy way to calculate the total number of protons that can be held by a given energy sublevel is to use the formula for natural numbers: 2*k. where k=1,2,3,4,5,6, ....

Tables of sublevels in atom

sublevel

# of protons

# of neutrons

suma

# of electrons

suma

s

1

1

2

1

1

2

p

2

2

1

1

6

3

3

6

d

3

3

2

2

10

5

5

10

f

4

4

3

3

14

7

7

14

g

5

5

4

4

18

9

9

18

h

6

6

5

5

22

11

11

22

In the atomic shell model, the shells are filled with protons in order of increasing natural numbers series, producing the magic numbers. These elements have highly stable properties:

#

Case

  Proton Configuration

Geometrical Number
of protons

Real Number
of protons

1

Geometrical

2

2



He

2



2

2

Geometrical

2,2-4

8



C

2,*-4



6

3

Geometrical

2,2-4,2-4-6

20



Si

2,2-4,*-*-6



14

4

Geometrical

2,2-4,2-4-6,2-4-6-8

40



Ni

2,2-4,2-4-6,*-*-*-8



28

5

Geometrical

2,2-4,2-4-6,2-4-6-8,2-4-6-8-10

70



Sn

2,2-4,2-4-6,2-4-6-8,*-*-*-*-10



50

6

Geometrical

2,2-4,2-4-6,2-4-6-8,2-4-6-8-10,2-4-6-8-10-12

112



Pb

2,2-4,2-4-6,2-4-6-8,2-4-6-8-10,*-*-*-*-*-12



82

The ground electronic and proton configurations of the elements H through U are given below.

Configuration

Z Symbol Electron Configuration Proton Configuration

1 H 1s¹ 1s¹

2 He 1s² 1s²

3 Li [He] 2s¹ [He] 2p¹

4 Be [He] 2s² [He] 2p²

5 B [He] 2s² 2p¹ [He] 2p³

6 C [He] 2s² 2p² [He] 2p⁴

7 N [He] 2s² 2p³ [He] 2p⁴ 2s¹

8 O [He] 2s² 2p⁴ [He] 2p⁴ 2s²

9 F [He] 2s² 2p⁵ [O] 3d¹

10 Ne [He] 2s² 2p⁶ [O] 3d²

11 Na [Ne] 3s¹ [O] 3d³

12 Mg [Ne] 3s² [O] 3d⁴

13 Al [Ne] 3s² 3p¹ [O] 3d⁵

14 Si [Ne] 3s² 3p² [O] 3d⁶

15 P [Ne] 3s² 3p³ [O] 3d⁶ 3p¹

16 S [Ne]3s² 3p⁴ [O] 3d⁶ 3p²

17 Cl [Ne] 3s² 3p⁵ [O] 3d⁶ 3p³

18 Ar [Ne] 3s² 3p⁶ [O] 3d⁶ 3p⁴

19 K [Ar] 4s¹ [O] 3d⁶ 3p⁴ 3s¹

20 Ca [Ar] 4s² [O] 3d⁶ 3p⁴ 3s²

21 Sc [Ar] 3d² 4s² [Ca] 4f¹

22 Ti [Ar] 3d² 4s² [Ca] 4f²

23 V [Ar] 3d³ 4s² [Ca] 4f³

24 Cr [Ar] 3d⁵ 4s¹ [Ca] 4f⁴

25 Mn [Ar] 3d⁵ 4s² [Ca] 4f⁵

26 Fe [Ar] 3d⁶ 4s² [Ca] 4f⁶

27 Co [Ar] 3d⁷ 4s [Ca] 4f⁷

28 Ni [Ar] 3d⁸ 4s² [Ca] 4f⁸

29 Cu [Ar] 3d¹⁰ 4s¹ [Ca] 4f⁸ 4d¹

30 Zn [Ar] 3d¹⁰ 4s² [Ca] 4f⁸ 4d²

31 Ga [Ar] 3d¹⁰ 4s² 4p¹ [Ca] 4f⁸ 4d³

32 Ge [Ar] 3d¹⁰ 4s² 4p² [Ca] 4f⁸ 4d⁴

33 As [Ar] 3d¹⁰ 4s² 4p³ [Ca] 4f⁸ 4d⁵

34 Se [Ar] 3d¹⁰ 4s² 4p⁴ [Ca] 4f⁸ 4d⁶

35 Br [Ar] 3d¹⁰ 4s² 4p⁵ [Ca] 4f⁸ 4d⁶ 4p¹

36 Kr [Ar] 3d¹⁰ 4s² 4p⁶ [Ca] 4f⁸ 4d⁶ 4p²

37 Rb [Kr] 5s¹ [Ca] 4f⁸ 4d⁶ 4p³

38 Sr [Kr] 5s² [Ca] 4f⁸ 4d⁶ 4p⁴

39 Y [Kr] 4d¹ 5s² [Ca] 4f⁸ 4d⁶ 4p⁴ 4s¹

40 Zr [Kr] 4d² 5s² [Ca] 4f⁸ 4d⁶ 4p⁴ 4s²

41 Nb [Kr] 4d⁴ 5s¹ [Zr] 5g¹

42 Mo [Kr] 4d⁵ 5s¹ [Zr] 5g²

43 Tc [Kr] 4d⁵ 5s² [Zr] 5g³

44 Ru [Kr] 4d⁷ 5s¹ [Zr] 5g⁴

45 Rh [Kr] 4d⁸ 5s¹ [Zr] 5g⁵

46 Pd [Kr] 4d¹⁰ [Zr] 5g⁶

47 Ag [Kr] 4d¹⁰ 5s¹ [Zr] 5g⁷

48 Cd [Kr] 4d¹⁰ 5s² [Zr] 5g⁸

49 In [Kr] 4d¹⁰ 5s² 5p¹ [Zr] 5g⁹

50 Sn [Kr] 4d¹⁰ 5s² 5p² [Zr] 5g¹⁰

51 Sb [Kr] 4d¹⁰ 5s² 5p³ [Zr] 5g ¹⁰ 5f¹

52 Te [Kr] 4d¹⁰ 5s² 5p⁴ [Zr] 5g ¹⁰ 5f²

53 I [Kr] 4d¹⁰ 5s² 5p⁵ [Zr] 5g ¹⁰ 5f³

54 Xe [Kr] 4d¹⁰ 5s² 5p⁶ [Zr] 5g¹⁰ 5f⁴

55 Cs [Xe] 6s¹ [Zr] 5g¹⁰ 5f⁵

56 Ba [Xe] 6s² [Zr] 5g¹⁰ 5f⁶

57 La [Xe] 5d¹ 6s² [Zr] 5g¹⁰ 5f⁷

58 Ce [Xe] 4f¹ 5d¹ 6s² [Zr] 5g ¹⁰ 5f⁸

59 Pr [Xe] 4f³ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d¹

60 Nd [Xe] 4f⁴ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d²

61 Pm [Xe] 4f⁵ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d³

62 Sm [Xe] 4f⁶ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁴

63 Eu [Xe] 4f⁷ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁵

64 Gd [Xe] 4f⁷ 5d¹ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁶

65 Tb [Xe] 4f⁹ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁶ 5p¹

66 Dy [Xe] 4f¹⁰ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁶ 5p²

67 Ho [Xe] 4f¹¹ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁶ 5p³

68 Er [Xe] 4f¹² 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁶ 5p⁴

69 Tm [Xe] 4f¹³ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁶ 5p⁴ 5s¹

70 Yb [Xe] 4f¹⁴ 6s² [Zr] 5g ¹⁰ 5f⁸ 5d⁶ 5p⁴ 5s²

71 Lu [Xe] 4f¹⁴ 5d¹ 6s² [Yb] 6h¹

72 Hf [Xe] 4f¹⁴ 5d² 6s² [Yb] 6h²

73 Ta [Xe] 4f¹⁴ 5d³ 6s² [Yb] 6h³

74 W [Xe] 4f¹⁴ 5d⁴ 6s² [Yb] 6h⁴

75 Re [Xe] 4f¹⁴ 5d⁵ 6s² [Yb] 6h⁵

76 Os [Xe] 4f¹⁴ 5d⁶ 6s² [Yb] 6h⁶

77 Ir [Xe] 4f¹⁴ 5d⁷ 6s² [Yb] 6h⁷

78 Pt [Xe] 4f¹⁴ 5d⁹ 6s¹ [Yb] 6h⁸

79 Au [Xe] 4f¹⁴ 5d¹⁰ 6s¹ [Yb] 6h⁹

80 Hg [Xe] 4f¹⁴ 5d¹⁰ 6s² [Yb] 6h¹⁰

81 Tl [Xe] 4f¹⁴ 5d¹⁰ 6s² 6p¹ [Yb] 6h¹¹

82 Pb [Xe] 4f¹⁴ 5d¹⁰ 6s² 6p² [Yb] 6h¹²

83 Bi [Xe] 4f¹⁴ 5d¹⁰ 6s² 6p³ [Yb] 6h¹² 6g¹

84 Po [Xe] 4f¹⁴ 5d¹⁰ 6s² 6p⁴ [Yb] 6h¹² 6g²

85 At [Xe] 4f¹⁴ 5d¹⁰ 6s² 6p⁵ [Yb] 6h¹² 6g³

86 Rn [Xe] 4f¹⁴ 5d¹⁰ 6s² 6p⁶ [Yb] 6h¹² 6g⁴

87 Fr [Rn] 7s¹ [Yb] 6h¹² 6g⁵

88 Ra [Rn] 7s² [Yb] 6h¹² 6g⁶

89 Ac [Rn] 6d¹ 7s² [Yb] 6h¹² 6g⁷

90 Th [Rn] 6d² 7s² [Yb] 6h¹² 6g⁸

91 Pa [Rn] 5f² 6d¹ 7s² [Yb] 6h¹² 6g⁹

92 U [Rn] 5f³ 6d¹ 7s² [Yb] 6h¹² 6g¹⁰



Today, our knowledge about nuclei, is restricted to about 2500 of the potentially existing 6000 combinations of protons and neutrons (see Figure ).

Black squares indicate the stable nuclei.





Atomic Structure and the Pascal Triangle,Electron Configuration and the Pascal Triangle, Magic Numbers and the Pascal Triangle.

Linus Pauling. Research Notebooks http://osulibrary.orst.edu/specialcollections/rnb/index.html Research Notebook 25. 26.
The MIT Laboratory for Nuclear Science, USA
Lawrence Berkeley National Laboratory, USA
R.A.Brightsen The Nucleon Cluster Model
The collected works of Bert Schreiber
The double tetrahedron structure of the nucleus
Pythagorean approach to problems of periodicity in fermionic system
Mayer M G, Jensen J H D 1955 Elementary Theory of Nuclear Shell Structure
Cook N D 1999 Proceedings of the St. Andrews Conf. on Fission (Singapore: World Scientific)



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2001-2005 Radoslav Jovanovic                 created: September 2005.

Shell Model of Atom

Principle energy level ( n )

Maximum number of protons (n+1)*n

1

2

2

6

3

12

4

20

5

30

6

42

7

56

Principle energy
level ( n )

Maximum number
of protons **(n+1)*n**