Where the periodic table ends

There is a wonderful story in physics, with a rich history, that begins with this question:  What is the biggest possible atomic number?

In other words, where does the periodic table end?  We (as a species) have managed to observe or create nuclei with atomic number ranging from 1 to as large as 118.  But how far, in theory, could we keep going?

As it turns out, there is a scientific law that says that nuclei whose charge is greater than some particular critical value cannot exist.  What’s more, this critical value is related to the mysterious fine structure constant , one of the most fundamental and mysterious constants of nature.  (Here, is the electron charge, is Planck’s constant divided by , and is the speed of light.)

In particular, the periodic table should end at .

In this post I’ll explain where this law comes from, and why it is that no point object can have a charge greater than .

To begin with, imagine a point in space at which is localized a very large positive charge .  Like this:

I’ll call this point the nucleus.  You can now ask the question of what happens if you release an electron in the neighborhood of this nucleus.  Obviously, the electron gets strongly bound to the nucleus, and it settles into a compact state with some size around the nucleus.  To figure out how big should be, you can remember that its value is determined by a balance between the typical energy of attraction between the electron and the nucleus and the kinetic energy associated with confining the electron to within the distance (as was explained here for the hydrogen atom).

The tricky part here is that for a very large nuclear charge the electron kinetic energy gets big, and the electron ends up moving with speeds close to the speed of light.  To see this, consider that when the nuclear charge is large, the electron becomes tightly bound, which means is small.  From the uncertainty principle, confining the electron to within the distance gives it a momentum .  If is big enough (or is small enough) that (where is the electron mass), then the kinetic energy can be described using the relativistic formula .

Now if you put together the potential and kinetic energy, you’ll find that the total energy as a function of is

.

This is a disconcerting formula.  Unlike for the hydrogen atom (where the electron moves much slower than the speed of light), this energy has no minimum value as a function of .  In particular, if is large enough that , then the energy just keeps getting lower as is made smaller.

What this means is that for , the electron state is completely unstable, and the electron collapses onto the nucleus.

This may not seem like a particularly big problem to you.  You may think that perhaps one can just keep electrons away from the nucleus (at least for a little while), and the nuclear charge will sit happily in space.

However, when Nature wants electrons badly enough, it finds a way to get them.

In this case,  the nuclear charge creates an electron binding energy that is so large, it becomes even larger than the rest mass energy of the electron, .  With such a large energy at stake, the nucleus can literally rip apart the vacuum and pull an electron from it.

Or, more correctly, the nucleus can wait until random fluctuations of the electromagnetic field produce an electron and positron pair (which under normal circumstances would immediately disappear again), then greedily suck in the electron and spit out the positron.  Like this:

[This process is similar to the perhaps more famous phenomenon of Hawking radiation at the edge of a black hole.  In black holes the enormous gravitational field rips antiparticles from the vacuum and sucks them in, spitting out their (normal) particle partners.  The difference is that Hawking radiation is an extremely slow process, whereas the process described above would be nearly instantaneous.]

The ripping and devouring of vacuum electrons by the large nucleus continues until the charge has been reduced to the point where it becomes smaller than , and everything settles down again.

It’s a fascinating instability of the vacuum itself, and its result is to prohibit too much charge from existing at any one location.

This means an end to the periodic table.

Footnotes

1.  The derivation in this post was pretty schematic, and all I showed was that the critical value of the nuclear charge is proportional to .  Up until the 1970s it was believed that was exactly .  More recent works, however, have put this number closer to .

2.  Sadly, there’s no easy way to observe this vacuum instability at ; making nuclei with charge is no simple matter.  So one can think of this as yet another interesting fable of physics relegated to trivia by the fact that in our universe just happens to be a small number.  However, there are synthetic systems (like graphene) where the effective fine structure constant happens to not be a small number.  In such cases even charges as small as cannot exist stably.

3.  The picture, and the title, at the top of this post is of course taken from Shel Silverstein’s Where the Sidewalk Ends.