Magic Numbers

Nuclear physics researchers at the Max Planck Institute have methodically been studying the mass of nuclei with atomic numbers 102 and 103 and varying numbers of neutrons, and they predict a new “magic number” for nuclei based on the result.

Atoms have a nucleus with electrons around it. Patterns of atomic behavior emerged pretty early on. For example, atoms with 2, 10, 18, 36, and 54 electrons were remarkably stable and nonreactive. This turned out to reflect some fundamental symmetries of the atom. The factor two came from the number of spin states of an electron (+1/2 and -1/2). Then the pattern looks like

• 2*1
  
• 2*(1+2*2)
  
• 2*(1+2*2+2*2)
  
• 2*(1+2*2+2*2+3*3)

The magic became clearer when looking at the higher energy states of hydrogen: the simplest representations of the groups of rotations, together with the two electron states, had numbers of states that corresponded to the numbers of electron states possible. This pattern showed why oxygen would have two “bonds” and what the excited states would be. 2, 8, 18, 32, 50; easy new magic numbers, and the numbers of electrons in a "shell": all side effects of that rotational symmetry.

The nice simple bond predictions come apart at the seams when the atoms become very big: all the other electrons interacting with each other make the system much blurrier. The nice simple "outer shell" we learned about in chemistry isn’t always quite on the "outside" of the atom, and counting bonds doesn’t make nearly so much sense when the atom can have 1, 2, 3, 4 or 5 bonds without breaking a sweat. Still, the pattern works OK for small atoms, and the symmetry is a fundamental law for all of them.

Nuclei are more of a mess. The protons and neutrons interact with strong forces, exchange identities, the protons repel electromagnetically, and they all barge around with startlingly high energies. This is thanks to Heisenberg’s principle: Δp Δx ≥ h means that when the nucleon is constrained to lie somewhere in a space the size of a nucleus, the uncertainty in its energy is of the order of 30MeV, or more than a million times the energy required to ionize most atoms.

It doesn’t help that heavier nuclei are less stable overall than lighter ones (on the average), and by analogy with the electrons around an atom you expect any patterns to be less clear. Even so, there are some patterns. Most nuclei are unstable and decay to stable nuclei. If either the number of protons or the number of neutrons is 2, 8, 20, 28, 50, 82, or 126 the nucleus is much more likely to be stable, and is more stable still if the number of protons and number of neutrons are both one of these magic numbers. For example He_4 has 2 protons and 2 neutrons and is extremely stable: it not only doesn’t decay but it isn’t that easy to disrupt.

2, 8, 20, 28, 50, 82, 126. That pattern isn’t so easy to figure out an origin for. A factor of 2 we understand: protons have two spin states also. But beyond that the sequence looks rather peculiar. At this point, the student is apt to beg: can you please tell me the next number in the sequence?

The answer, according to Blaum et al, is 156.

Have fun! A Nobel may have your name on it.

corrected a number