Question

This is a challenging multi-step problem. Solve it on paper, writing out each step carefully. When doing calculations, do not round intermediate values. Note: If you have approached the problem in a principled way, do not abandon your approach if your numerical answer is not accepted; check your calculations! The Bohr model correctly predicts the main energy levels not only for atomic hydrogen but also for other "one-electron" atoms where all but one of the atomic electrons has been removed, such as in He⁺ (one electron removed) or Li⁺⁺ (two electrons removed).

(a) Predict the energy in eV of a photon emitted in a transition from the first excited state to the ground state in eV for a system consisting of a nucleus containing  Z = 51 protons and just one electron. You need not recapitulate the entire derivation for the Bohr model, but think carefully about the changes you have to make to take into account the factor Z.

(b) The negative muon (μ⁻) behaves like a heavy electron, with the same charge as the electron but with a mass 207 times as large as the electron mass. As a moving μ⁻ comes to rest in matter, it tends to knock electrons out of atoms and settle down onto a nucleus to form a "one-muon" atom. For a system consisting of a nucleus of osmium (Os¹⁹⁰ with 76 protons and 114 neutrons) and just one negative muon, predict the energy in eV of a photon emitted in a transition from the first excited state to the ground state. The high-energy photons emitted by transitions between energy levels in such "muonic atoms" are easily observed in experiments with muons.

(c) Calculate the radius of the smallest Bohr orbit for a μ⁻ bound to a nucleus of osmium (Os¹⁹⁰ with 76 protons and 114 neutrons). Compare with the approximate radius of the nucleus of osmium (remember that the radius of a proton or neutron is about 1 × 10⁻¹⁵ m, and the nucleons are packed closely together in the nucleus).

smallest Bohr orbit	r Bohr	=		m
approximate radius of nucleus	r nucleus	≈		m

Comments: This analysis in terms of the simple Bohr model hints at the result of a full quantum-mechanical analysis, which shows that in the ground state of the osmium-muon system there is a rather high probability for finding the muon inside the osmium nucleus. Nothing in quantum mechanics forbids this penetration, especially since the muon does not participate in the strong interaction. Electrons in an atom can also be found inside the nucleus, but the probability is very low, because on average the electrons are very far from the nucleus, unlike the muon. The eventual fate of the μ⁻ in a muonic atom is that it either decays into an electron, neutrino, and antineutrino, or it reacts through the weak interaction with a proton in the nucleus to produce a neutron and a neutrino. This "muon capture" reaction is more likely if the probability is high for the muon to be found inside the nucleus, as is the case with heavy nuclei such as osmium.

Answer 1

ton Energy - 13.6 2e En 13. 6 xS - 3 S3 3. 6 ev 2 - 3 S 33.6 e v E -35373. 6 ev 8843.4ev 4 26 S30.2ev ДЕ - = 20 me 13.6 2 En = х me n2 Reduced Mass o Miagne atam Мe 3 Mu 207me

J: 1836 X76 Mos 1838 1 14 me 34 9o 66 me Mos Mu 206.6 me Mos Mu - 13.6 X 76x 206.6 En = ev .6 25 X10ev En= M2 4.06 Xt0ev 2 4 e v E E 12.16 MeV AE

Rn= /m.) Z 0.511 X1o 20.8 x 76 15 3.2 S1XIo 3.251 fm 3. 25 Io m=
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