Atomic shells and magic numbers
Electron shells hold 2, 8, 18 and 32 electrons, twice the square numbers. The first magic numbers of atomic nuclei, 2, 8 and 20, are twice the tetrahedral numbers 1, 4 and 10.
That matter is ordered by numbers was an article of faith for the Pythagoreans. The quantum physics of the 20th century seems to prove it right in a surprising way: the structure of atoms follows simple series of numbers, and they are the figurate numbers of the Pythagoreans.
The shells of the electron cloud
The electrons of an atom arrange themselves in shells. The first holds 2 electrons, the second 8, the third 18 and the fourth 32. That is , twice the square numbers 1, 4, 9, 16. The square number comes from the number of possible orbital shapes in a shell, the two from the fact that each electron has one of two spin directions. These numbers shape the periodic table: its rows have the lengths 2, 8, 8, 18, 18, 32, 32, because the shells partly overtake one another as they fill.
The magic numbers of atomic nuclei
The protons and neutrons in the nucleus also arrange themselves in shells. Nuclei with 2, 8, 20, 28, 50 or 82 protons or neutrons are especially stable, for neutrons also with 126. For protons 126 has so far only been predicted. Nuclear physicists call them magic numbers, a word the Pythagoreans would have liked. Maria Goeppert Mayer and Hans Jensen explained them in 1949 with the shell model of the nucleus and received the Nobel Prize for it in 1963.
In the simplest model, that of a particle in a spatial well, the first shell holds 2 particles, the second 6, the third 12, the fourth 20. Each is twice a triangular number: 2 × 1, 2 × 3, 2 × 6 and 2 × 10. Adding up the shells gives 2, 8, 20, 40, twice the tetrahedral numbers 1, 4, 10, 20.
Electron shells: 2 · n²
Nuclear shells: 2 × triangular number, summed
Left: the electron shells hold twice a square number. Right: in the simplest nuclear model each shell holds twice a triangular number, and the running sums are twice the tetrahedral numbers. The third magic number, 20, is twice ten.
So the first three magic numbers are exactly 2 × 1, 2 × 4 and 2 × 10. The ten of the tetractys, thought of as a stack of spheres in three layers 1 + 3 + 6, describes the structure of the first three shells of the atomic nucleus, taken twice for the two spin directions.
The tetrahedron of 1 + 3 + 6 = 10 spheres. Doubled it gives the magic number 20 of the nucleus, its layers correspond to the first three shells. Drag to rotate the model.
From the fourth shell on, nature departs from this simple picture. An additional force, the coupling of spin and orbit, shifts individual states, and instead of 40, 70 and 112, the numbers 28, 50, 82 and 126 become magic. This correction was precisely the discovery of Goeppert Mayer and Jensen. But the basic pattern it builds on is triangular and tetrahedral numbers.
What it means
The figurate numbers with which the Pythagoreans laid out pebbles (see The triangle of points) return in the innermost part of matter, as the numbers of states a particle can occupy in space. The reason is geometric: the states of a shell can be counted like points in a triangle, successive shells like layers of a tetrahedron. That ten of all numbers produces the third magic number is a fine example of how the old figure is connected with the structure of the world.
Sources and further reading
Sources
- Dmitri Mendeleev, Über die Beziehungen der Eigenschaften zu den Atomgewichten der Elemente, 1869
- Maria Goeppert Mayer, On Closed Shells in Nuclei II, Physical Review 75, 1949
- Otto Haxel, J. Hans D. Jensen and Hans E. Suess, On the „Magic Numbers“ in Nuclear Structure, Physical Review 75, 1949