EMPIRICAL EVIDENCE FOR
 MAGIC NUMBERS IN NUCLEI
 
So what was the experimental evidence in the 1930's? There 
was, indeed some evidence for closed shells in nuclei, but it was
very fragmentary, and not convincing.  On the other hand, there 
was, what looked like good evidence, for something like a liquid 
drop model from slow neutron crosssections. So the conventional 
wisdom was strongly in favor of the latter, and it was strengthened 
in 1938-39 with the discovery of nuclear fission. The liquid drop
model gave a very simple explanation of the fact that about 200 MeV
is released per fission.  It is ironic that Maria Mayer got her start
in nuclear physics by working on the Manhattan project. In the
development of the nuclear program, many new radioactive isotopes
were made. I don't know the exact story, but after 1945, Maria  Mayer
began to look at systematics of the binding energies, and she 
discovered that they clearly indicated the presence of closed shells, 
as had been suspected before, but now in a much more convincing 
way. I don't think she was all that familiar with nuclear physics at the 
time, in particular with the arguments that there should be no 
independent particle model in nuclei. But the closed shells, for
nuclei with 2,8,20,28,50.82,126 neutrons (or protons)  were clearly in 
evidence. Note that all these numbers are even. Indeed, it had been 
known even before, (and about this there was no argument) that 
nuclei for which both N and Z are even, are slightly more stable 
than those where one or both of there numbers are odd.  
This is a manifestation of the pairing effect.
Pairs of neutrons and also pairs of protons like to couple to 
total spin 0, with a little extra binding. That feature, which accounts 
for the fact that all known even-even nuclei have ground state
angular momentum of 0, is due to the basically attractive nature of 
the nuclear force, and by the way, it does NOT occur in atoms. 
Now the existence of closed shells goes beyond that.  
Thus nuclei with 82 neutrons are, of course, more stable than those 
with 81 or 83, but they are also slightly more stable than those with
80 or 84. So that, the discovery of the magic numbers (actually 
rediscovery in a more convincing way) was the first major 
contribution of Maria Mayer to the nuclear shell model. Even the
skeptics had to admit that the evidence could not be swept under
the rug. Now let me mention Eugene Wigner, who had done
fundamental work on nuclear forces and symmetries  which 
earned him the Nobel Prize in 1963 (shared with Mayer and 
Jensen) . Wigner believed in the liquid drop model, but he 
recognized the very strong evidence for the closed shells. 
It seemed a little like magic to him, and that, apparently, is how 
the words "Magic Numbers" were coined.  
The next step for nuclear theorists was to explain why one got the 
particular values 2,8,20,28,50, 82 and 126 for the closed shells. 
Now the lowest values 2, 8, and 20 are not difficult to get, if you 
believe in the idea of independent particle motion. Put the 
nucleons into a reasonably looking single particle potential, 
like a harmonic oscillator, and they come right out. But the higher 
numbers were something of a challenge. Several groups worked 
on this problem in 1948, trying clever ways to modify the
potential so that the closed shells would appear for 28, 50, 82, 
and 126. Maria Mayer (following a suggestion by Fermi)  hit
on the answer. The idea is that, in addition to the central 
potential, there is also a strong spin-orbit coupling.  
 
 
  Go to next section(Spin-Orbit)
 
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