# Size of radius - HELP

1. Jul 26, 2007

### touley

Hi

I'm stuck on a problem that I've been working on for a very long time and I was hoping someone could help. The problem reads as follows:

A muonic atom consists of a muon (m = 106 MeV/c2) in place of an electron. For the muon in a doubly ionized lithium-7 ion, calculate the following quantities.
a) the smallest radius in the ground state
(b) the binding energy of the muon in the ground state
(c) the series limit of the wavelength for the first three series
nm (first series)
nm (second series)
nm (third series)

I know I need to use the equation:

4*pi*epsilon*h-bar^{2} / me^{2}

Because the atom is lithium, I know I have to include Z (Z=3) next to the e^{2}. I derived the formula a billion times - I don't know what I'm doing wrong!
I also know that I need to find the reduced mass of a muon in lithium. I think I did this right.

I've been trying to work this out for an obscene amount of time. I feel like I'm overlooking something very basic. Can anyone help me please?

2. Jul 26, 2007

### Dick

Ok, that's the Bohr radius. And yes, you should change one of the e's to Ze and use the reduced mass. So what makes you think you are doing something wrong?

3. Jul 26, 2007

### touley

its online homework. every time i put my answer in, its wrong. i dont have very many chances left.

maybe i'm calculating the reduced mass wrong?

the equation is: m*M/(m+M)

for m I'm using the mass of the muon, 106MeV.
for M I'm using the mass of the nucleus, which is 3 times the mass of a proton, because I'm dealing with lithium. Mass of a proton is 938 Mev. 3 times that is 2814 Mev. I also tried it with kg-masses and I still get a wrong answer.
It's been PAST TIME for me to move on, but I need this one to figure out the rest.

4. Jul 26, 2007

### touley

lol, i also tried it using the reduced mass of one muon and one proton. i still dont get the right answer.

5. Jul 26, 2007

### Dick

Three protons is not a stable lithium nucleus. The most common isotope has mass 7.

6. Jul 26, 2007

### touley

still not getting it right.

reduced mass of muon = m * 7M / (m + 7M) ?

7. Jul 26, 2007

### Dick

Using mass of Li from a periodic table of 6.941 amu (I don't have a list of isotope masses). I get 1.85*10^(-28) kg for the reduced mass. Just a couple percent below the muon mass.

8. Jul 26, 2007

### touley

Thank You! You're The Best! :)

9. Jul 26, 2007

### Dick

So what were you doing wrong?

10. Jul 26, 2007

### touley

i kept getting 1.82 * 10^-28 for my reduced mass and it gave me a slightly larger radius. i guess it wasn't close enough for 'em. thanks again :-D