Limit to the max size of a Nucleus

[daniel christ]

Hi Everybody,
I've seen that one of the reasons that elements past 137 can't be created is because any element past 137 would require electrons in inner orbitals to go faster than the speed of light, and past 137 they would have to. I have also heard that electrons don't literally orbit the nucleus, but instead pop in and out of existence because of some quantum mechanical witchcraft. Would the fact that electrons are not always particles, but energy as well allow for the mass while in energy form to travel at the speed of light, circumventing the problem.
This is reasoning is probably faulty somehow, but I'm interested to see what really happens.
Daniel Chriest

 

[daniel christ]

http://physics.info/atomic-models/ at the bottom of the page it shows what Dirac said

 

[DrChinese]

Hi Everybody,
I've seen that one of the reasons that elements past 137 can't be created is because any element past 137 would require electrons in inner orbitals to go faster than the speed of light, and past 137 they would have to. I have also heard that electrons don't literally orbit the nucleus, but instead pop in and out of existence because of some quantum mechanical witchcraft. Would the fact that electrons are not always particles, but energy as well allow for the mass while in energy form to travel at the speed of light, circumventing the problem.
This is reasoning is probably faulty somehow, but I'm interested to see what really happens.
Daniel Chriest

Welcome to PhysicsForums, Daniel!

The instability of the nucleus of elements with greater than 137 protons has nothing to do with the speed of light. Electrons do not really pop in and out of existence either.

Generally, protons (the number of which designates different elements) repulse each other due to their same charge ("opposites attract, like repel" as I'm sure you know). So how do elements have a bunch of same charged protons living together in the nucleus? There is a force called the strong force that binds them together even though their electrical charge makes them repel. Protons and neutrons both feel the strong force. However, the strong force has limits. Loosely speaking, the combined repulsive force of about 137 protons is enough to overcome even the strong force. In that case, a larger nucleus cannot be created and the protons repel. Neutrons - though electrically neutral - play a role in this too.

The details are quite complicated, but you would probably benefit from learning more about this. Try googling "proton strong force nucleus" and follow from there.

 

[dextercioby]

Dirac's analysis assumes pointlike electron and nucleus. As soon as the nucleus is of finite (but, of course, very tiny) size, the maximal atomic number can go up.
Read section 13.4 (p. 129 onwards) of Akhiezer and Berestetskii "Quantum Electrodynamics" (1965 translation from Russian). If you won't like this book because of the x4=ict part, you can find the same argument (but less detailed) in the volume 4 of the Landau-Lifschitz series, p. 134 in section 36 (1982, Pergamon Press).

EDIT: Dr. Chinese mentioned the so-called "strong force effects" which could prevent the stability of ultra-heavy nuclei. I was just addressing this issue from the purely semirelativistic treatment of Dirac (which could be seen as a certain limit of QED).

 

[DrChinese]

EDIT: Dr. Chinese mentioned the so-called "strong force effects" which could prevent the stability of ultra-heavy nuclei. I was just addressing this issue from the purely semirelativistic treatment of Dirac (which could be seen as a certain limit of QED).

Ah, I didn't really look much at the OP's reference - my bad. Thanks for your comment.

At any rate, to the OP: Past a point, as the number of protons (Z) goes up in a nucleus, the likelihood of the nucleus spontaneously breaking apart or (ejecting otherwise a proton) mostly increases. (I would not say it is completely independent of the electron configuration, but that plays a somewhat different role regarding stability of nuclei and is often ignored.)

https://www.quora.com/Is-there-a-th...cal-elements-that-could-exist-in-the-universe

And quantum mechanically, you normally do not refer to speed/velocity of a bound electron.

Putting all of this together: one thing essentially limits the number of protons in a nucleus (relative strength of the strong force and electromagnetic repulsive charge). Another thing limits the number of electrons that could be attached to a nucleus. I honestly don't know much about that, so maybe someone has a good reference for me.

 

[TeethWhitener]

The number Z=137 comes directly from the fine structure constant (~1/137). In a non-relativistic world (more specifically: if the Bohr model were true), it would be the point at which an electron in a 1s orbital would exceed the speed of light. In a relativistic world, it becomes the point at which the Dirac equation predicts a 1s electron binding energy of greater than 2m_ec². In other words, it's the hypothetical point at which the energy density of the electric field is enough to generate electron-positron pairs where the electron occupies a bound state orbital. With finite nuclear size effects taken into account, this bumps Z_max up to roughly 173.

 

[Vanadium 50]

In a relativistic world, it becomes the point at which the Dirac equation predicts a 1s electron binding energy of greater than 2mec2.

Are you sure about this? Where did the alpha go to get 2m?

 

[TeethWhitener]

Are you sure about this? Where did the alpha go to get 2m?

I should be more precise: Z=137 is where the energy of the ground 1s hydrogenlike state becomes negative (in other words, the ground state is in the "Dirac sea"). Check out this page: http://quantummechanics.ucsd.edu/ph130a/130_notes/node501.html
All the way at the bottom is the equation for the energy. Plugging in Z=137 (with j=1/2, n_r=0) gives zero in the denominator of the fraction under the radical sign. Of course what this really means is that the Dirac equation alone probably isn't enough to figure out what's actually going on in such a system. And (as someone already pointed out) a Z=137 or higher nucleus would probably disintegrate long before the timescale where atomic physics is meaningful. I think maybe someone did a condensed matter analogue recently.