Muon-Enhanced Atoms: Fusion Efficiencies Explored

[Garlic]

Hello everyone,
I know that electrons in heavy atoms move at relativistic speeds, resulting in heavier electrons and smaller radius.
If we replace an electron in a heavy atom with a muon, the muon would move to the center (occupying the lowest energy state possible), and making the radius of the atom smaller (similar to the behavior of lambda baryons in a hypernucleus). Since the muon would move at relativistic speeds, the relativistic half life would be longer.
I have read somewhere that fusion with atoms that have smaller radii is more efficient, so wouldn't this method be efficient?

 

[mfb]

Bound electrons and muons do not "move" in the usual sense. Their wave function is constant in time.

The binding reduces the available energy for the decay, which changes the phase space. In addition, those atoms can do muon capture (similar to electron capture, but with much more energy available). That means the lifetime can go down instead of up, see aluminium for example.

An additional muon in the center reduces the effective charge and increases the size of the other electron orbitals.

I have read somewhere that fusion with atoms that have smaller radii is more efficient

"I have read somewhere" is not a useful source. And what does "more efficient" mean? Also, atoms or nuclei?

There is muon-catalyzed fusion of hydrogen (with just one charge you don't have other electrons hanging around). It works, and a muon can catalyze on average up to ~100 H2 molecules until it sticks to a produced helium nucleus and gets lost. That is not enough to be used in a power plant, producing the muon costs too much energy.

 

[Garlic]

I understand.

Bound electrons and muons do not "move" in the usual sense. Their wave function is constant in time.

Doesn't the bohr radius get smaller in relativistic atoms because the electrons have higher relativistic masses?
Quote from wikipedia -relativistic quantum chemistry- "For gold with (Z = 79) the 1s electron will be going (α = 0.58c) 58% of the speed of light. Plugging this in for v/c for the relativistic mass one finds that mrel = 1.22me and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%."

Don't muons experience time even when they are delocalised? If they aren't, they wouldn't be decaying.

Where am I wrong?

 

[mfb]

Doesn't the bohr radius get smaller in relativistic atoms because the electrons have higher relativistic masses?

The inner orbitals shrink due to relativistic effects (relative to a world without special relativity), but that's not the situation you asked about here. You add a muon to an existing atom.

 

[Garlic]

I am sorry, but I don't understand what you mean by "relative to a world without special relativity".

 

[mfb]

Well, what does "the radius decreases" mean? Decreases relative to what? It was never larger in our world. The radius would be larger if there would be no relativistic effects. But there are.