A muonic atom of boron-11 can be treated as a bound state of a muon – Free B9
A muonic atom of boron-11 can be treated as a bound state of a muon (of charge โ ๐ โe and mass 207 ๐ ๐ 207m e โ ) and a boron-11 nucleus (of charge + 5 ๐ +5e and mass 11 ร 1840 ๐ ๐ 11ร1840m e โ ), where ๐ e and ๐ ๐ m e โ are the charge magnitude and mass of an electron. You may take the reduced mass of hydrogen to be equal to the electron mass, ๐ ๐ m e โ .A muonic atom of boron-11 can be treated as a bound state of a muon (of charge โ ๐ โe and mass 207 ๐ ๐ 207m e โ ) and a boron-11 nucleus (of charge + 5 ๐ +5e and mass 11 ร 1840 ๐ ๐ 11ร1840m e โ ), where ๐ e and ๐ ๐ m e โ are the charge magnitude and mass of an electron. You may take the reduced mass of hydrogen to be equal to the electron mass, ๐ ๐ m e โ .
Answer
Muonic Atom of Boron-11 โ Complete Solution
Part (a): Bohr Radius of Muonic Boron-11
A muonic atom is like hydrogen, but the electron is replaced by a muon, which is much heavier.
Muon mass: 207 ร me
Boron-11 nucleus: Charge = +5e (Z = 5), Mass = 11 ร 1840 ร me = 20,240 me
Step 1: Reduced Mass (ฮผ)
Step 2: Bohr Radius Calculation
Part (b): Photon Energy for Transition (n = 3 โ n = 1)
Step 1: Energy Level Formula
For hydrogen-like atoms:
En = -13.6 ร 205.1 ร 25 / nยฒ = -69718 / nยฒ eV
Step 2: Transition Energy (n = 3 to n = 1)
The negative sign indicates emission. Thus, photon energy is:
Summary
| Part | Quantity | Answer |
|---|---|---|
| (a) | Bohr Radius | 5.16 ร 10โปยนโด m |
| (b) | Photon Energy (n = 3 โ n = 1) | 61.97 keV |