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 
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 
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 
The tricky part here is that for a very large nuclear charge 
Now if you put together the potential and kinetic energy, you’ll find that the total energy as a function of
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 
What this means is that for 
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 
However, when Nature wants electrons badly enough, it finds a way to get them.
In this case, the nuclear charge 
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
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 
2. Sadly, there’s no easy way to observe this vacuum instability at 
3. The picture, and the title, at the top of this post is of course taken from Shel Silverstein’s Where the Sidewalk Ends.
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).
Yes, you’re right. That’s why I put the symbol “Fy” on the silly periodic table picture above. :)
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.
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
, 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
rather than 
.
BTW, does someone have access to the Greiner et. al. classical paper in which they conjecture about Z_c=172????
Not in the public domain, I’m afraid.