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 β π βe and mass 207 π π 207m e β ) and a boron-11 nucleus (of charge + 5 π +5e and mass 11 Γ 1840 π π 11Γ1840m e β ), where π e and π π m e β are the charge magnitude and mass of an electron. You may take the reduced mass of hydrogen to be equal to the electron mass, π π m e β .A muonic atom of boron-11 can be treated as a bound state of a muon (of charge β π βe and mass 207 π π 207m e β ) and a boron-11 nucleus (of charge + 5 π +5e and mass 11 Γ 1840 π π 11Γ1840m e β ), where π e and π π m e β are the charge magnitude and mass of an electron. You may take the reduced mass of hydrogen to be equal to the electron mass, π π m e β .
Answer
Muonic Atom of Boron-11 β 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 Γ me
Boron-11 nucleus: Charge = +5e (Z = 5), Mass = 11 Γ 1840 Γ me = 20,240 me
Step 1: Reduced Mass (ΞΌ)
Step 2: Bohr Radius Calculation
Part (b): Photon Energy for Transition (n = 3 β n = 1)
Step 1: Energy Level Formula
For hydrogen-like atoms:
En = -13.6 Γ 205.1 Γ 25 / nΒ² = -69718 / nΒ² 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 Γ 10β»ΒΉβ΄ m |
| (b) | Photon Energy (n = 3 β n = 1) | 61.97 keV |