﻿ Realm of atoms - Atom core

# Atom core

## Magic numbers

#### Shell model

The nucleuses with so called "magic number" of nucleons have an extreme stability, i.e. the highest binding energy.

• Magic numbers 2, 8, 20, (28), 50, 82, 126
• Submagic numbers 14, (28), 38(40), 64, ...

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 inversion 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*n2

```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)= n2+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
```
Periodic table reminds mathematical structures (groups of equivalence classes,...)

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
-------------------------
```