It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let’s analyze the decay of a free neutron that is at rest in the laboratory. We’ll ignore the antineutrino and consider the decay n → p⁺ + e^–. The analysis requires the use of relativistic energy and momentum, from Chapter 36.

a. What is the total kinetic energy, in MeV, of the proton and electron?

b. Write the equation that expresses the conservation of relativistic energy for this decay. Your equation will be in terms of the three masses m_n, m_p, and m_e and the relativistic factors γ_p and γ_e .

c. Write the equation that expresses the conservation of relativistic momentum for this decay. Let v represent speed, rather than velocity, then write any minus signs explicitly.

d. You have two simultaneous equations in the two unknowns v_p and v_e. To help in solving these, first prove that γv = (γ₂ – 1)^1/2c.

e. Solve for v_p and v_e. (It’s easiest to solve for γ_p and γ_e, then find v from γ.) First get an algebraic expression for each, in terms of the masses. Then evaluate each, giving v as a fraction of c.

f. Calculate the kinetic energy in MeV of the proton and the electron. Verify that their sum matches your answer to part a.

g. Now explain why the electron is ejected in beta decay while the proton remains in the nucleus.