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Relativistic Bohr Model

November 10, 2007 – 4:20 pm

I recently needed to find the energy levels of hydrogen-like atoms (i.e. atoms with a single electron), taking into account relativistic effects. These effects become important for atoms with a large number of protons.

I only had a couple of days to do this, and I figured the easiest thing would be to create a relativistic version of the Bohr model. This is a toy model that predicts the energy levels of a non-relativistic hydrogen atom.

The first thing I guessed was that the basic assumption from the Bohr model regarding the angular momentum still holds. That is:

where is some integer and is hbar.

Also, the electron should still travel in a circular orbit, so that:

where is the orbit radius and is the linear momentum. It is easy to show that this is still correct relativistically (*).

Next we have the energy, which for a free particle now obeys the dispersion relation:

Our electron is in the electric field created by the nucleus, so this should be (**) :

Where is the electron charge and is its rest mass.

Let’s find the possible orbit radii. In the classical Bohr model this is usually done by calculating the force acting on the particle. But here we already wrote down this nice expression for the energy so we might as well use it… To do that, we will venture one last guess.

We assume that, given angular momentum , the system chooses the radius where it has minimal energy.

You can verify for example that when you make this assumption in the non-relativistic version (instead of using forces and such), you still get the same result.

To get the orbit radii we then need to solve:

which comes to:

Plugging this into the expression for the energy we get the atom’s energy levels:

And there you have it. I have reason to believe this model works, at least partially, because I used it on measurements of large-Z atoms and it fit the data splendidly (while the non-relativistic model failed miserably). In addition, taking recovers the non-relativistic energies, as expected.

Having said that, you may have noticed a problem in the last expression. For large enough Z, it is imaginary, and energy isn’t imaginary. How large a Z? Well if you’re a physicist may have also noticed that the coefficient is the fine structure constant:

So for the ground state , the largest possible Z is about 137. Above that, according to the model, the system is unstable.

You might be thinking that a more plausible explanation is that I made some mistake in my series of, uhm, guesses. That could be, but a quick look in the periodic table shows that it goes all the way up to… 118.

Well! Could this little toy model tell us something deep about the stability of atoms? Apparently it can. I googled for ’stability of large z atoms’ and came up with this review, where the instability is discussed. In page 10 it says:

The moral to be drawn from this is that relativistic kinematics plus quan-
tum mechanics is a ‘critical’ theory (in the mathematical sense). This fact
will plague any relativistic theory of electrons and the electromagnetic field
– primitive or sophisticated.

… we are on the ‘primitive’ side.

(*) This can be verified by taking and making a change of coordinates:

Place the electron on the intersection of the X axis and the circular orbit to obtain the answer.

(**) This guess can be justified by writing down the Lagrangian for a relativistic charged particle (c=1):

And converting to a Hamiltonian. Taking the Hamiltonian to be the energy, one arrives at this formula.