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?

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As it turns out, there is a scientific law that says that nuclei whose charge $Z$ is greater than some particular critical value $Z_c$ cannot exist.  What’s more, this critical value $Z_c$ is related to the mysterious fine structure constant $\alpha = e^2/\hbar c \approx 1/137$, one of the most fundamental and mysterious constants of nature.  (Here, $e$ is the electron charge, $\hbar$ is Planck’s constant divided by $2 \pi$, and $c$ is the speed of light.)

In particular, the periodic table should end at $Z \approx 1/\alpha \approx 137$.

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 $\sim 137$.

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To begin with, imagine a point in space at which is localized a very large positive charge $Z e$.  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 $r$ around the nucleus.  To figure out how big $r$ should be, you can remember that its value is determined by a balance between the typical energy of attraction $-Ze^2/r$ between the electron and the nucleus and the kinetic energy $K$ associated with confining the electron to within the distance $r$ (as was explained here for the hydrogen atom).

The tricky part here is that for a very large nuclear charge $Z$ 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 $Z$ is large, the electron becomes tightly bound, which means $r$ is small.  From the uncertainty principle, confining the electron to within the distance $r$ gives it a momentum $p \sim \hbar/r$.  If $p$ is big enough (or $r$ is small enough) that $p \gg m c$ (where $m$ is the electron mass), then the kinetic energy can be described using the relativistic formula $K = p c \sim \hbar c/r$.

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

$E \sim \hbar c/r - Z e^2/r$.

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 $r$.  In particular, if $Z$ is large enough that $Z \gtrsim \hbar c/e^2 \sim 1/\alpha \sim 137$, then the energy just keeps getting lower as $r$ is made smaller.

What this means is that for $Z \gtrsim 137$, the electron state is completely unstable, and the electron collapses onto the nucleus.

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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 $Z > 137$ 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 $Z > 137$ creates an electron binding energy that is so large, it becomes even larger than the rest mass energy of the electron, $mc^2$.  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 $Z$ has been reduced to the point where it becomes smaller than $\sim 137$, 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.

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### Footnotes

1.  The derivation in this post was pretty schematic, and all I showed was that the critical value $Z_c$ of the nuclear charge is proportional to $1/\alpha \approx 137$.  Up until the 1970s it was believed that $Z_c$ was exactly $137$.  More recent works, however, have put this number closer to $170$.

2.  Sadly, there’s no easy way to observe this vacuum instability at $Z > 1/\alpha$; making nuclei with charge $137$ 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 $\alpha$ 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 $Z = 2$ 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.

April 27, 2013 9:12 am

If I am not wrong, alpha was first calculated by Arthur Eddington and he determined that it was 1/136. Now it is known that alpha is 1/137 and sometimes element (hypothetical) number 137 is called Feynmanium (after Richard Feynman).

April 27, 2013 9:18 am

Yes, you’re right. That’s why I put the symbol “Fy” on the silly periodic table picture above. :)

April 27, 2013 9:22 am

Nucleus isn’t a point charge.
Your e-n force equation is a first order approximation, and is nowhere near correct for a high charge nucleus.

April 27, 2013 9:31 am

Hi Frank. Generally speaking, approximating the nucleus as a point charge is valid so long as the size of the nucleus is much smaller than the size of the electron cloud. For normal atoms, the ratio of the electron cloud size to the nucleus size is nearly 100,000, and this is a fine approximation. It only becomes bad very close to the critical nuclear charge $Z_c$, where the electron begins to collapse onto the nucleus.

But technically speaking, you’re right. And this is in fact the (only eventually-understood) reason why $Z_c \approx 170$ rather than $137$.

4. July 8, 2013 12:06 pm

BTW, does someone have access to the Greiner et. al. classical paper in which they conjecture about Z_c=172????