Hydrogen atom, muon substitution, helium muon fusion

[rayman123]

Homework Statement

Substitute an electron in a neutral hydrogen atom with a muon.
a) calculate the Bohr radius of the ground state for this myonic atom of atom. The answer must be right to at least 2 significant digits.
b) Calculate the fraction of the myon that is located inside the proton, which can be assumed to have radius of 1.3fm.
c) Helium can not form negative ions with electrons, although hydrogen can. Is it possible to form negatively charged helium atom if you try attach a myon to the atom?

Homework Equations

a) $$r_{n} = \frac{\epsilon_{0} \cdot h^2\cdot n^2}{\pi\cdot \mu \cdot e^2\cdot Z}= 2.56034\cdot 10^{-13}m$$

radius of the proton
$$r_{p}= 1.3\cdot 10^{-15}$$


b) not sure if I that is correct how to calculate that fraction...
$$\frac{2.56043\cdot 10^{-13}}{1.3\cdot 10^{-15}}= 196.9492$$
now the fraction of the muon which is located in the proton will be


$$196.9492 \cdot 2.56043\cdot 10^{-13}= 5.04275\cdot10^{-11}$$ ? is that correct?

c) I have just found some publications where actually such atoms have been formed, where muons where stuck to helium atoms forming negative charged atoms.

Thanks

 

[Mindscrape]

b) I think that's okay. Part b) is kind of strange because by the Bohr model there is no actual overlap because he assumed a planetary style of orbit. Quantum mechanics proved that wrong, and so you have probability densities that you have to consider, which even apply to multiple states. But the nature of the question deals with the Bohr model, so I don't know. I would do what you did. :)

c) Seems like it could be done. It's an interesting problem from the perspective of particle physics. Looks like you found some people who have done this, so looks like you're good.

 

[diazona]

Your answer to part (b) can't be right. If you keep track of the units in the calculation, you'll see why.

However, I agree with Mindscrape that the question (part b) is kind of confusing. My issue is that it doesn't say whether it wants the fraction of volume that is enclosed within the proton, or the fraction of muon probability density, or something else. In the latter case you would have to know (or assume) the state of the atom.

 

[rayman123]

Hi! Thank you both for your replies.
Well I have no idea how part b should be calculated, I suspected my answer was wrong...but had no other solution to that part...
I guess the question to part b does not involve complicated calculations since this is not an advance course but I just do not know how to calculate that fraction. Does anyone has any idea how this fraction can be calculated?

I guess the atom is in its ground state...

When I compare Bohrs radius and the calculated one for the muonic atom I can see that muonic radius is much smaller, that means that the muon is much closer to the nuclei than electron.
I found an interview where a scientisct from CERN claimed that the muon actually can pass through the nuclei (proton in my model)

 

[diazona]

Here's my question: have you learned (or could you be reasonably expected to look up) the wavefunction for hydrogen in its ground state? If so, the problem may be asking you to find the probability that the muon has a position within the proton. I would suggest asking your instructor to clarify what that part of the problem means.

By the way, even in normal hydrogen, the electron can be found within the nucleus (proton), but the probability of that happening is a lot smaller than with muonic hydrogen.