Evaluate the energies of muonic K X rays in Fe

[Demon117]

Homework Statement

Using a 1 electron model, evaluate the energies of the muonic K X rays in Fe assuming a point nucleus.

Homework Equations

The $\Delta E$ is the difference between energy of the 1s state in an atom with a "point" nucleus and the 1s energy in an atom with uniformly charged nucleus. The equation is found to be

$\Delta E= \frac{2Z^{4}e^{2}R^{2}}{5(4 \pi \varepsilon_{o})a_{o}^{3}}$

$R=R_{o}A^{1/3}$

The Attempt at a Solution

The muon is a particle identical to the electron in all characteristics except its mass, which is 207 times the electron mass. Since the Bohr radius depends only on the mass, the muonic orbits have 1/207 the radius of the corresponding electron orbits.

So, this means that the Bohr radius of a muon will be $a_{\mu}=\frac{1}{207}a_{o}$. By that logic, we should have:


$\Delta E= \frac{2Z^{4}e^{2}(R_{o}A^{1/3})^{2}}{5(4 \pi \varepsilon_{o})(\frac{1}{207}a_{o})^{3}}$

By substitution. . . . when I do this, I end up with energies far below what is expected in the muonic k x rays in some Fe isotopes, maybe my thinking is incorrect. Could someone lend a hand? I have been doing a lot of HW this weekend and I think I am at my breaking point.

 

[Demon117]

Homework Statement

Using a 1 electron model, evaluate the energies of the muonic K X rays in Fe assuming a point nucleus.

Homework Equations

The $\Delta E$ is the difference between energy of the 1s state in an atom with a "point" nucleus and the 1s energy in an atom with uniformly charged nucleus. The equation is found to be

$\Delta E= \frac{2Z^{4}e^{2}R^{2}}{5(4 \pi \varepsilon_{o})a_{o}^{3}}$

$R=R_{o}A^{1/3}$

The Attempt at a Solution

The muon is a particle identical to the electron in all characteristics except its mass, which is 207 times the electron mass. Since the Bohr radius depends only on the mass, the muonic orbits have 1/207 the radius of the corresponding electron orbits.

So, this means that the Bohr radius of a muon will be $a_{\mu}=\frac{1}{207}a_{o}$. By that logic, we should have:


$\Delta E= \frac{2Z^{4}e^{2}(R_{o}A^{1/3})^{2}}{5(4 \pi \varepsilon_{o})(\frac{1}{207}a_{o})^{3}}$

By substitution. . . . when I do this, I end up with energies far below what is expected in the muonic k x rays in some Fe isotopes, maybe my thinking is incorrect. Could someone lend a hand? I have been doing a lot of HW this weekend and I think I am at my breaking point.

Realized I need to use Mosley's law for this and note that the transition is from the P(3/2) to the 1s(1/2) state, and adjusting for the muon mass in the Rydberg constant the energy falls right within where we expect, about 1.3 MeV +/- .1 MeV.