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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

--------------------------------------------------------------------

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.  

vvvvvvvvvvvvvvvvvv  

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.

vvvvvvvvvvvvvvv

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

Forward to:

Nuclear Shell Structure         Growth Structures         Rhodium and Iridium

               

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