# Isotopic mass curves

The mass number of a nuclide (isotope) is simply its nucleon count: the number of protons plus the number of neutrons. In the appropriate units, it’s a good approximation of the actual mass.

But it’s not exact. If we graph the measured “isotopic mass” of the nuclides with a given mass number, say 37, we get this:

Another example, with mass number 99:

For odd mass numbers, we get nice smooth curves. But that’s not the case for even mass numbers:

Atoms “like” to have an even number of protons. And they “like” to have an even number of neutrons. Even numbers allows the nucleus to be bound more tightly, putting more of the atom’s energy in the form of “binding energy” instead of mass, making it lighter.

For an odd mass number, each nuclide has an odd number of protons and an even number of neutrons, or vice versa. Whether it’s odd-even, or even-odd, makes little difference to the expected isotopic mass.

But when the mass number is even, each nuclide is either the best case (even-even), or the worst case (odd-odd). So the curve is jagged, and jumps up and down across the hypothetical smooth curve we would have gotten if the mass number were odd.

Generally speaking, the nuclides near the bottom of the curve are likely to be the stabler ones, with longer half-lives. The one with the very smallest mass is the only one that does not decay by any form of beta decay. (Note: I’m ignoring nuclear isomers in this post.) That doesn’t rule out other forms of decay (alpha decay or fission), but for mass numbers 9 through about 143, such decays are either impossible, or nearly so, for beta-stable nuclides.

An atom that decays by some form of beta decay jumps from a node on one of these graphs, to a lower node on the same graph — virtually always to an immediately adjacent node. A jump to the right is a beta-minus decay (electron emission), and a jump to the left is a beta-plus decay (which can be either electron capture, or positron emission).

If an atom can decay to an adjacent node (i.e. it is not at a local or global minimum), it usually does so relatively quickly (with a half-life less than, say, a hundred thousand years). But there are some exceptions, such as potassium-40 and iodine-129, that live longer.

It is possible for an atom to jump two steps to the left or right, potentially escaping from a local minimum. This is called double beta decay (or double electron capture). For example, argon-36 can decay to sulfur 36. But double beta decay is extremely rare. Half-lives that involve double beta decay are usually well over 1020 years. If the nuclide can also decay in a way that is somewhat less extremely rare, double beta decay effectively doesn’t happen.

I guess it must be theoretically possible for larger jumps to occur. I don’t see why that would be forbidden by the laws of physics, and in principle, anything not forbidden has a nonzero probability of happening. So, it’s “possible” for aluminum-36 to decay to calcium-36 by septuple beta decay. But it’s also “possible” for the Earth to spontaneously turn into a ball of live penguins. So, yeah.

At high mass numbers, a graph can make it clear that we might not yet have discovered the nuclides that are the most stable. Here’s 265. Its lightest member might be to the left of the known data points.

Despite the handicap faced by odd-odd nuclides, it is possible, for small mass numbers, for them to be at the bottom of the curve, and thus be beta-stable. The last time this happens is at 14:

If you squint, you might be able to tell that the odd-odd nitrogen-14 is just barely lighter than the even-even carbon-14. I guess this is possible because carbon-14’s 8 neutrons out of 14 nucleons is far from the natural optimum percentage of neutrons for atoms of about that mass (and on the other side, oxygen-14 has it even worse).

You should probably be glad that nitrogen wins the 14 contest, at least if you enjoy things like… being alive. If it didn’t, all the nitrogen-14 would turn into carbon, and we’d only be left with the much rarer nitrogen-15. Considering nitrogen’s critical role in biochemistry, and the fact that it makes up 78% of the Earth’s atmosphere, the universe’s prospects for life would be very different.