The nucleuses with so called "magic number" of nucleons have an extreme stability, i.e. the highest binding energy.
Shell model assumes this sequence of numbers:
(1)(2)(3)(4) (5) (6)
--------------------------
2 2 2 2 2 2
6 6 6 6 6
12 12 12 12
30 30 30
32 32
44
--------------------------
total 2 8 20 50 82 126
Let us rewrite the previous scheme another way:
(1)(2)(3)(4)(5)(6) (7)
--------------------------------------
2 2 x 2 2 2 2 2 Helium
6 6 x 6 6 6 6 Carbon
12 12 x 12 12 12 Magnesium
20 20 x 20 20 Calcium
30 30 x 30 Zinc
42 42 x Molybdenum
56 56 Barium
--------------------------------------
total 2 6 14 28 50 82 126
Missing numbers 8 and 20 can be written: 8=2+6 and 20=2+6+12 (similarly 40= 2+6+12+20, 38= 6+12+20, 64= 2+12+20+30). Here is an inverse of the previous diagram:
(1)(2)(3)(4)(5)(6)(7)
--------------------------------------
2 * 2 * * * * * Helium
6 * 6 * * * * Carbon
12 * 12 * * * Magnesium
20 * 20 * * Calcium
30 * 30 * Zinc
42 * 42 Molybdenum
56 * Barium
--------------------------------------
total 2 6 14 28 50 82 126
Let number s be a shell number. We define number M(s):
M(s) = m1(s)-m2(s) = 1/3 s(s+1)(s+2) - s(s-1)
s 0 1 2 3 4 5 6 7 8 9 10 --------------------------------------------------- m1(s) 0 2 8 20 40 70 112 168 240 330 440 m2(s) 0 0 2 6 12 20 30 42 56 72 90 --------------------------------------------------- M(s) 0 2 6 14 28 50 82 126 184 258 350
Numbers M(s) makes arithmetic progression of the 3-th degree.
0 2 6 14 28 50 82 126 184
2 4 8 14 22 32 44 58
2 4 6 8 10 12 14
2 2 2 2 2 2
Structure of periodic table is not definitive yet. (Even it is not known if it has finite or infinite number of elements.)
Sequence of electron shells:
0,2,8,18,32, .. i.e. 2*n^2
Particular orbits (s-p-d-f):
1s 00 0 0 1 -------------------------------------- 2s 00 1 0 2p 01 10 11 1 1 4 -------------------------------------- 3s 00 2 0 3p 01 10 11 2 1 3d 02 20 12 21 22 2 2 9 -------------------------------------- 4s 00 3 0 4p 01 10 11 3 1 4d 02 20 12 21 22 3 2 4f 03 30 13 31 23 32 33 3 3 16 -------------------------------------- .....Sequence of nuclear shells:
0,2,6,12,20,30,42, .. i.e. n*(n+1)= n^2+n
In the following diagram are the both sequences,
the first one: totals in vertical direction,
the second one: values in horizontal direction.
--+----------------------------------
0| 1-1
2| 1+1 4-2
6| 4+2 9-3
12| 9+3 16-4
20| 16+4 25-5
30| 25+5 36-6
42| 36+6 49-7
56| 49+7
--+----------------------------------
2 8 18 32 50 72 98
Also e.g. the beginning of table of stable isotopes is simple and well structured:
00 0 01 H 1 2 02 He 3 4 03 Li 6 7 -------------------- 04 Be 9 05 B 10 11 06 C 12 13 07 N 14 15 ------------------------- 08 O 16 17 18 09 F 19 10 Ne 20 21 22 11 Na 23 ------------------------- 12 Mg 24 25 26 13 Al 27 14 Si 28 29 30 15 P 31 --------------------------