Nuclear Shell Model

In order to describe the internal workings of the nucleus, three nuclear structure models have been proposed, including the Liquid-Drop Model, first proposed in the 1940s by the Danish physicist, Niels Bohr, the Nuclear Shell Model, proposed independently by Maria Goeppert Mayer and Johannes H. D. Jenson in 1949, and the Collective Model, a variation on the liquid-drop idea. Many of the theories of nuclear physics use all three models in order to describe various and diverse phenomenon. As such, the three theories have a commonality not unlike the wave-particle duality of electromagnetism (i.e. light).

In order to understand the unique qualities of Rhodium and Iridium and the other six Precious Metals (aka the “platinum” metals) with respect to the ORME, it is useful to picture the nucleus as a series of roughly concentric shells. For smaller nuclei the nuclear shells are similar to the electronic shells, but then quickly diverge as specific levels (such as 1f) split. For higher levels of angular momentum, the energy levels and the number of nucleons increase (in Table 1 below, increasing from the bottom of the page to the top).

Note also that the total number of nucleons in the shells, given in [brackets] -- indicate level closures. More importantly, however, the numbers to the far right of the page, given in {bracketed parenthesis}, represent shell closures, and are often referred to in nuclear physics as the “magic numbers”. They were originally named “Magic” simply because it was not clear why such extraordinarily stable nuclei should appear in the table of elements where they did. There was an assumption of something being akin to the electron levels in the Periodic Table, but Nuclear Theory has always trailed Electronic Theory.

The details of the nuclear shells are presented in two formats.

Table 1 -- A structural ladder arrangement showing the diverse levels and shells.

Table 2 -- A listing of all the elements with their detailed shell structures

In addition, there are additional, technical notes following these tables, which include:

1 -- Specifics on the Precious Elements Group

2 -- General notes on Nuclear Shell Theory.

While this material can become highly technical, it is also highly visual, and thus -- much like our preference in newspapers for the comics over the front page -- the material doesn’t require quite as much mental effort to see and understand.

Acknowledgment should also be made to the people who first came up with these ideas, and who, through much experimentation, figured out what went where and why. Wow.

Table 2 -- It’s Element(ary), my dear Watson!

Shell Closures*

1 Hydrogen H 1s-1

2 Helium He 1s 2 {[2[}

3 Lithium Li 1s-2, 1p-1

4 Beryllium Be 1s-2, 1p-2

5 Boron B 1s-2, 1p-3

6 Carbon C 1s-2, 1p-4 6 [6]

7 Nitrogen N 1s-2, 1p-5

8 Oxygen O 1s-2, 1p6 {[8]}

9 Fluorine F 1s-2, 1p-6, 1d-1

10 Neon Ne 1s-2, 1p-6, 1d-2

11 Sodium Na 1s-2, 1p-6, 1d-3

12 Magnesium Mg 1s-2, 1p-6, 1d-4

13 Aluminum Al 1s-2, 1p-6, 1d-5

14 Silicon Si 1s-2, 1p-6, 1d-6 [14]

15 Phosphorus P 1s-2, 1p-6, 1d-6, 2s-1

16 Sulfur S 1s-2, 1p-6, 1d-6, 2s-2 [16]

17 Chlorine Cl 1s-2, 1p-6, 1d-7, 2s-2

18 Argon Ar 1s-2, 1p-6, 1d-8, 2s-2

19 Potassium K 1s-2, 1p-6, 1d-9, 2s-2

20 Calcium Ca 1s-2, 1p-6, 1d-10, 2s-2 {[20]}

21 Scandium Sc 1s-2, 1p-6, 1d-10, 2s-2,1f-1

22 Titanium Ti 1s-2, 1p-6, 1d-10, 2s-2,1f-2

23 Vanadium V 1s-2, 1p-6, 1d-10, 2s-2,1f-3

24 Chromium Cr 1s-2, 1p-6, 1d-10, 2s-2,1f-4

25 Manganese Mn 1s-2, 1p-6, 1d-10, 2s-2,1f-5

26 Iron Fe 1s-2, 1p-6, 1d-10, 2s-2,1f-6

27 Cobalt Co 1s-2, 1p-6, 1d-10, 2s-2,1f-7

28 Nickel Ni 1s-2, 1p-6, 1d-10, 2s-2,1f-8 {[28]}

29 Copper Cu 1s-2, 1p-6, 1d-10, 2s-2,1f-8, 2p-1

30 Zinc Zn 1s-2, 1p-6, 1d-10, 2s-2,1f-8, 2p-2

31 Gallium Ga 1s-2, 1p-6, 1d-10, 2s-2,1f-8, 2p-3

32 Germanium Ge 1s-2, 1p-6, 1d-10, 2s-2,1f-8, 2p-4 [32]

33 Arsenic As 1s-2, 1p-6, 1d-10, 2s-2,1f-9, 2p-4

34 Selenium Se 1s-2, 1p-6, 1d-10, 2s-2,1f-10, 2p-4

35 Bromine Br 1s-2, 1p-6, 1d-10, 2s-2,1f-11, 2p-4

36 Krypton Kr 1s-2, 1p-6, 1d-10, 2s-2,1f-12, 2p-4

37 Rubidium Rb 1s-2, 1p-6, 1d-10, 2s-2,1f-13, 2p-4

38 Strontium Sr 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-4 [38]

39 Yttrium Y 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-5

40 Zirconium Zr 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6 {[40]}

41 Niobium Nb 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-1

42 Molybdenum Mo 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-2

43 Technetium Tc 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-3

44 Ruthenium Ru 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-4

45 Rhodium Rh 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-5

46 Palladium Pd 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-6

47 Silver Ag 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-7

48 Cadmium Cd 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-8

49 Indium In 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-9

50 Tin Sn 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-10 {[50]}

51 Antimony Sb 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-11

52 Tellurium Te 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-12

53 Iodine I 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-13

54 Xenon Xe 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-14

55 Cesium Cs 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-15

56 Barium Ba 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-16

57 Lanthanum La 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-17

58 Cerium Ce 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18 (Rare Earths)

59 Praseodymium Pr 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-1

60 Neodymium Nd 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-2

61 Promethium Pm 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-3

62 Samarium Sm 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-4

63 Europium Eu 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-5

64 Gadolinium Gd 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-6 [64]

65 Terbium Tb 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-7

66 Dysprosium Dy 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-8

67 Holmium Ho 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-9

68 Erbium Er 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10

69 Thulium Tm 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-1

70 Ytterbium Yb 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2

71 Lutetium Lu 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-1

72 Hafnium Hf 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-2

73 Tantalum Ta 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-3

74 Tungsten W 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-4

75 Rhenium Re 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-5

76 Osmium Os 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-6

77 Iridium Ir 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-7

78 Platinum Pt 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-8

79 Gold Au 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-9

80 Mercury Hg 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-10

81 Thallium Tl 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-11

82 Lead Pb s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-12 {[82]}

83 Bismuth Bi 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-13

84 Polonium Po 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-14

85 Astatine At 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-15

86 Radon Rn 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-16

87 Francium Fr 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-17

88 Radium Ra 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-18

89 Actinium Ac 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-19

90 Thorium Th 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-20 (RE)

91 Protactinium Pa 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-21

92 Uranium U 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22

93 Neptunium Np 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-1

94 Plutonium Pu 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-2

95 Americium Am 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-3

96 Curium Cm 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-4

97 Berkelium Bk 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-5

98 Californium Cf 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-6

99 Einsteninium Es 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-7

100 Fermium Fm 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-8

101 Mendelevium Md 1s-2,1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-9

102 Nobelium No 1s-2,1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-10

103 Lawrencium Lr 1s-2,1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-11

104 Unnilquadium Unq 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-12

105 Unnilpentium Unp 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-13

106 Unnilhexium Unh 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14

107 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-1

108 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-2

109 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-3

110 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-4

111 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-5

112 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-6

113 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-6, 1I-1

and so forth up to

126 -- 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-22, 2f-14, 3p-6, 1i-1

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Footnotes to Table 2:

*Shell closures are denoted by brackets [ ], or in the case of a double shell closure by {[x]} -- the latter indicating extremely stable nuclei.

& Many of the elements past 106 do have names albeit very strange ones). If you’re really curious, link to Growth Structures.

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Appendix 1 -- Specifics on the Precious Elements Group

The so-called, “light platinum” group of four elements includes:

44 Ruthenium Ru 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-4

45 Rhodium Rh 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-5

46 Palladium Pd 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-6

47 Silver Ag 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-7

Any 1g 9/2 shell has 10 nucleons, while the 1g 7/2 shell has 8 nucleons. Nearby Magic Numbers are 40, 50. If we consider only the protons, the four elements of Ru through Ag are in the middle of a 1g 9/2 shell, near magic number 50.

Stable (non-radioactive isotopes) and the percentages of naturally occurring isotopes

[The fact Rhodium has one stable isotope is notable! The number of neutrons for Rh-103 is 58. Silver is almost as good, with number of neutrons for Ag-107, 60, and Ag-109, 62.]

44 Ruthenium Ru 96(5.5), 9(1.9), 99(12.7), 100(12.7), 101(17), 102(31.5), 104(18.7%)

45 Rhodium Rh 103 (100%)

46 Palladium Pd 102(.96),104(10.97),105(22.23),106(27.33) 108(26.71),110(11.81%)

47 Silver Ag 107 (51.35%) and 109 (48.65%)

Electronic Shell Structures -- Light Platinum Group

44 Ruthenium Ru [Kr] 4d-7, 5s-1

45 Rhodium Rh [Kr] 4d-8, 5s-1

46 Palladium Pd [Kr] 4d-10

47 Silver Ag [Kr] 4d-10, 5s-1

The so-called “heavy platinum” metal group include:

76 Osmium Os 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-6

77 Iridium Ir 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-7

78 Platinum Pt 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-8

79 Gold Au 1s-2, 1p-6, 1d-10, 2s-2,1f-14, 2p-6, 1g-18, 2d-10, 3s-2, 1h-9

The 1h 11/2 shell has 12 nucleons. The next shell is 1h 9/2 (10). Nearby Magic Numbers are 50 and 82. Considering only protons, Os-Au is in the middle of the second quarter of the 1h 11/2 and 9/2 shells (after magic number 50).

Stable (non-radioactive isotopes) and their naturally occurring percentages of isotopes

[The fact Gold has one stable isotope is notable! The number of neutrons for Au-197 is 118. Iridium is almost as good with number of neutrons for Ir-191, 114, and Ir-193, 116.]

76 Osmium Os 184(.02),186(1.6),187(1.6),188(13.3),189(16.1),190(26.4), 192(41%)

77 Iridium Ir 191 (38.5%) and 193 (61.5%)

78 Platinum Pt 190(.012), 192(.78), 194(32.8), 195(33.7), 196(25.4), and 198(7.2%)

79 Gold Au 197 (100%)

Electronic Shell Structures -- Heavy Platinum Group

76 Osmium Os [Xe] 4f-14, 5d-6, 6s-2

77 Iridium Ir [Xe] 4f-14, 5d-7, 6s-2

78 Platinum Pt [Xe] 4f-14, 5d-9, 6s-1

79 Gold Au [Xe] 4f-14, 5d-10, 6s-1

80 Mercury Hg [Xe] 4f-14, 5d-10, 6s-2

Electronic Shell Structure Theory

The magnitude of the orbital angular momentum vector is restricted to discrete values of:

(k)² = h² [k (k + 1)] / (2p)²

where h is Planck's constant, and k is a non-negative integer. The angular momentum component along a given axis (e.g. the z-axis) can only have values of: kz = m h / 2p, where m can take any integral value between -k and +k inclusive. Hence, a level of given k corresponds to 2k+1 different states, differing only in the orientation of the angular momentum. Since the field is isotropic, all these states have the same energy. The level is said to be (2k+1)-fold degenerate. The relation between the angular-momentum quantum number and the conventional notation of spectroscopy is:

k = 0 1 2 3 4 5 6

s p d f g h i

The above statements hold for any type of spherically symmetrical potential.

Electrons (as well as nucleons) have an intrinsic angular momentum, or spin, s, which can only have components of (+1/2) h/2p or (-1/2) h/2p along a given axis in space, such that

s² = 1/2 (1 + 1/2) h² / (2p)²

The total angular momentum of the system is j = k + s. The total angular momentum is also restricted in quantum mechanics to discrete values, i.e.:

j² = j (j + 1) h² / (2p)²

where j can be either one of the positive half integers, j = k + 1/2 or j = k - 1/2. In the first state, orbital and spin angular-momentum vectors are said to be “parallel”; in the other, “antiparallel”. The total number of states arising from a level of given k by the addition of spin is 2 (2k + 1).

The interaction between the spin and orbital angular momentum splits the energy of the two levels, j = k + 1/2 and j = k - 1/2. This spin-orbit interaction does not depend on the different directions of space (s-l, a dot product), and thus still requires the conservation of angular momentum, or quantization of j.

Proceeding from lighter to heavier elements with increasing nuclear charge Ze and the corresponding number Z of electrons, we have to fill the individual electron levels successively with as many electrons as the Pauli Exclusion Principle allows. Whenever two successive levels are wide apart, we speak of the closing of an atomic shell, because the next electron can be brought into the atom only at a much higher level, i.e. with much less binding energy. Both experimental and theoretical calculations show that the sequence of electronic levels in the atom is given by:

Levels: 1s | 2s 2p | 3s 3p | 4s 3d 4p | 5s 4d 5p | 6s 4f 5d 6p | 7s ...

Electrons: 2 | 2 6 | 2 6 | 2 10 6 | 2 10 6 | 2 14 10 6 | 2 ...

Cumulative: 2 | 4 10 | 12 18 | 20 30 36 | 38 48 54 | 56 70 80 86 | 88 ...

The vertical lines indicate shell closure, and explain the pronounced position of the Noble gases (such as Helium, Argon, Neon, Krypton...).

Nucleus Shell Structure Theory

The nucleus shell structure does not have the relatively simple electromagnetic structure of the atomic electron structure, but is a combination of the nuclear and electromagnetic forces. The energy levels, based on the isotropic harmonic oscillator potential, and disregarding the Zero-Point Energy, is:

e = h W / (2p) [ 2 (n - 1 ) + k ] = no h w (2p)

where W is the frequency and h is again Planck's Constant.

However, this does not account for spin-orbit coupling. Furthermore the energy of a level depends strongly on the alignment of spin and orbit with the anti-parallel spin-orbital connection having a higher energy than the parallel case. Furthermore, it is necessary to now characterize a level by both its k value and its j value. In effect, the filling of the levels with neutrons and protons will be such that, for a level with a given k, first the 2j+1 = 2k+1 states of j=k+1/2 are filled by one particle each, and later the 2k states of j=k-1/2.

The nuclear shells are then separated from other levels by reasonably wide energy gaps. Because of the spin-orbit coupling, the nuclear shells are not the same as the harmonic oscillator shells. In fact, at higher atomic number levels, the spin-orbit splitting becomes the dominant feature in the level arrangement.

Nuclei may have a charge distribution which is not spherically symmetric. A measure of the distortion from a spherical shape is the quadrupole moment. Quadrupole moments are conventionally defined as the integral of the expression 3z² - r² weighted with the nuclear charge for the state M = J. It is customary to give quadrupole moments in units of the charge of the proton. If ¶(r) is the charge density, and q the angle between the z axis and the radius vector to any point in the nucleus, the definition of the quadrupole moment is:

Q = 1/e I r² (3 cos² q - 1) ¶(r)M=J dV

The dimensions of a quadrupole are given as 10^-24 cm² or “barns” -- the latter derived from the idea of an elementary particle being able to hit the broad side of a barn! For a spherical distribution ¶(r) expression, Q vanishes. Positive quadrupole moments thus correspond to prolate spheroids (cigars); negative quadrupole moments, to oblate spheroids. Sample quadrupole moments are for Ir-191 and -193, +1.5, and for Au-197, +0.6. Measuring quadrupole moments consists in measuring differences of energy for different orientations of the quadrupole moments in an asymmetric potential field.

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All of the above is intended to identify the unique characteristics of Gold and Silver, and specifically, Rhodium and Iridium. The other Precious Metals may also be important, but Rhodium and Iridium seem to have a particularly important role in the ORME. This is shown in more detail in Nuclear Shell Structures, as well as Growth Structures.

ORME Physics David Radius Hudson ORME Tree of Life

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Nuclear Shell Structure Growth Structures Rhodium and Iridium

Nuclear Shell Model

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