Double Magic Numbers: Not Always Guaranteed Stability

In the shell model of the nucleus, there are certain magic numbers of nucleons (protons or neutrons): 2, 8, 20, 28, 50, 82, 126 that are more tightly bound than the next higher integer. A magic number of nucleons is important because it increases the likelihood that isotopes will be stable. If the number of neutrons and protons are both  magic numbers (not necessarily the same one) we say the nucleus is doubly magic, and it is even more likely to be stable. Isotopes with doubly magic numbers include :

None of the above isotopes are radioactive except for nickel’s pair and ¹³²Sn, which, in spite of its double magic number of 50 protons and 82 neutrons, has a half life of less than a day. So a while ago, I contacted a scientist to ask him why ¹³²Sn is radioactive.
Here is the answer from Alex Brown of the National Superconducting Cyclotron Laboratory http://www.nscl.msu.edu

Sn-132 is doubly-magic, but being doubly-magic does not guarantee stability. There are other features that can make a nucleus unstable. In this case Sn-132 is too far from the “valley of stability” – it has too many neutrons. The neutrons in Sn-132 beta decay and turn into protons eventually leading to the most stable mass 132 nucleus Xe-132.
The “magic number” of 50 protons for Sn does show up by the fact that Sn has more stable isotopes than any other element – they are: Sn-112, Sn-114, Sn-115, Sn-116, Sn-117, Sn-118, Sn-119, Sn-120, Sn-122 and Sn-124. There are other nuclei that we predict to be doubly-magic that are unstable, such as Sn-100 and Ni-78.
At our National Superconducting Cyclotron Laboratory are are trying to produce these nuclei and study their properties.