Reading: This homework will cover beta decays. You should review your notes from these lessons.
Q1: Compute the Q value for β− and electron capture for 204Pb.
answer: -4.464,-0.7637
Q2: A subdominant source of solar neutrinos comes from decays of 7Be, 8B, 13N, 15O, and 17F. What
are the maximum energies that the neutrinos from these decays could have? For each, indicate if
the neutrino’s energy has a spectrum or is a ’line’ (mono-energetic neutrino) and why.
answer: 861 keV (line) ; 16.96 MeV (spect.); 1.2 MeV (spect.) ; 1.73 (spect.) ; 1.74 MeV (spect.)
Q3: Plot the mass parabola(s) for A=76 isobars using both the Bethe-Weizsacker formula and using
data from a table of nuclides. Consider values of Z between 27 and 34. For each of the following,
use your mass parabola(s) from both your calculations and from measurements to justify your
answers. Which nuclides are beta-stable? List all nuclides that are expected to decay through β−
decays. List all nuclides that are expected to decay through β+ decays or electron capture. Of the
beta-stable nuclides, is there any other way for these to decay?
Q4: Using the formula for the kinetic energy spectrum of the electron or positron in beta decays, plot
relative rate of various electron/positron kinetic energies.
Q5: One of the stages of the p-p fusion chain in the sun is the following reaction:
3He +3 He →4 He + p + p
What is the kinetic energy released by this reaction?
answer: 12.859 MeV
Q6: Credit: OpenStax, University Physics Volume 3, Another set of reactions that fuses hydrogen into
helium in the Sun and especially in hotter stars is called the CNO cycle:
12C +1 H →13 N+γ
13N →13 C+e+ + νe
13C +1 H →13 N+γ
13N +1 H →15 O+γ
15O →15 N+e+ + νe
15N +1 H →12 C +4 He
This process is a “cycle” because 12C appears at the beginning and end of these reactions. Write
down the overall energy released in this cycle. Assume that the positrons annihilate electrons to
form more γ rays.
Q7: Neutron moderators are designed to reduce the kinetic energy of neutrons. Neutron moderators
tend to be made of hydrogen-rich materials like plastics or water. In reactors, they help facilitate
fission, which starts with neutron absorption. The neutron absorption rate depends strongly on
the velocity of the neutrons. In radiation safety, plastics are used to shield people and sensitive
devices from neutron radiation. Moderators can do this by taking advantage of a simple kinematic
relationship, which we’ll explore here.
Consider 1D elastic scattering of a neutron with initial velocity v0 and a target particle at rest with
mass mX. Show that the ratio of the initial and final kinetic energy of the neutron is v2
N /v2
0 =
1
Week 5
(mN − mX )2/(mN + mX )2. What is this ratio if the target particle is a proton; what if it was an
oxygen atom? Which is a better moderator and why?
In reality, the target particles have a thermal distribution of energies and collisions in 3D are
complicated by the fact that the deflection angle alters the relationship you have derived. A more
accurate prediction can be expressed as the average logarithm of the reduction of neutron energy
per colliision: ln(E0/E) ∼ 2/(A + 2/3), where A is the atomic mass of the target. Compute this
for hydrogen and oxygen. answer: 1.2 ; 0.12
Q8: Calculate the energy released in the following rare spontaneous fission reaction:
238U →95 Sr +140 Xe + 3n. Use a table of nuclides to look up the relevant masses. answer: 171 MeV
Q9: Calculate the amount of energy produced by the fission of 1.00 kg of 235U given that the average
fission reaction of 235U produces 200 MeV. A mole of 235U is 235.04g/mol. answer: 8.2×1013 J
