Download Problem Set 7

Problem Set 7
Prof. J. Gerton
Due Tuesday October 25, 2011 at the beginning of class
Problem 1
(10 pts.) Quantization of the Bohr model
According to the atomic model from Bohr’s theory, which of the following velocities
for an electron in an atom is allowed? Explain your answer and show all your work.
(a) v1 = 4.375 × 106 m/s;
(b) v2 = 2.188 × 106 m/s;
(c) v3 = 1.683 × 106 m/s;
(d) v4 = 1.094 × 106 m/s.
Problem 2
(10 pts.) Lyman spectrum
The Lyman series is a particular set of spectral lines of the Hydrogen atom. When
the Hydrogen atom de-excites from an excited state with principal number ni to a
final state with n = 1, it emits a photon in the Lyman series, of wavelength
1
1
= R (1 − 2 ),
λ
ni
(1)
where R = 1.097 × 107 m−1 is called the Rydberg constant. Derive the Lyman
equation from the Bohr theory.
1
2
Problem 3
(10 pts.) Positronium
An electron and an anti-electron (also called a positron) can form a bound state
(called positronium), due to their mutual electromagnetic attraction. Compute the
energy levels En for positronium using the Bohr quantization rules. Hint: In the
derivation of the Bohr model, there was no dependence on the mass of the proton
because mp me and the proton has negligible effect on the orbital mechanics.
Now, this condition no longer holds true, the electron and the positron have the
same mass. To solve this problem, consider that the two particles rotate around their
mutual center of mass, so that you can consider a particle of mass me /2 rotating at
distance r/2 from the center of mass.
Problem 4
(10 pts.) Rutherford scattering
An α particle (Qα = +2e) of kinetic energy K = 27 MeV passes through a thin gold
(Au) foil. The gold nucleus has a charge QAu = +79e.
(a) What is the distance of closest approach D for a head-on collision?
(b) What is the impact parameter if the α particle is scattered at θ = π/3?
Problem 5
(10 pts.) Geiger-Marsden experiment
Geiger and Marsden used a monochromatic beam of α particles with kinetic energy
K = 7.7 MeV, and found they were scattered by gold atoms in a way that is precisely
described by Rutherford’s scattering formula.
(a) From this fact compute an upper limit on the radius of the gold nucleus.
(b) From this result and the fact that the atomic mass of gold is 197u, where u is the
atomic mass unit, compute a lower limit on the mass density of nuclear material.
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