Quiz — Chapter 14: Beta Decay: The Weak Interaction in the Nucleus

20 questions. Answers at the end.

Q1. In $\beta^-$ decay, a neutron converts to a proton. At the quark level, which process occurs?

(a) $u \to d + e^- + \bar{\nu}_e$ (b) $d \to u + e^- + \bar{\nu}_e$ (c) $d \to u + e^+ + \nu_e$ (d) $u \to d + e^+ + \nu_e$

Q2. The Q-value for $\beta^+$ decay, expressed in terms of atomic masses, includes an extra $2m_ec^2$ term compared to $\beta^-$ decay. This means:

(a) $\beta^+$ decay is always more energetically favorable than $\beta^-$ decay (b) $\beta^+$ decay requires the parent to be heavier than the daughter by at least $1.022\,\text{MeV}$ (c) Electron capture is always forbidden when $\beta^+$ decay is allowed (d) The positron has twice the rest mass of an electron

Q3. Electron capture competes with $\beta^+$ decay. Which statement is correct?

(a) EC is allowed only when $\beta^+$ is also allowed (b) EC is allowed whenever $\beta^+$ is allowed, but EC can also occur when $\beta^+$ is energetically forbidden (c) EC requires a higher Q-value than $\beta^+$ decay (d) EC and $\beta^+$ decay produce identical final states

Q4. The continuous energy spectrum of beta particles was first definitively demonstrated by the calorimetric experiment of:

(a) Chadwick (1914) (b) Rutherford (1911) (c) Ellis and Wooster (1927) (d) Fermi (1934)

Q5. Pauli proposed the neutrino in 1930 primarily to:

(a) Explain the discrete alpha particle energies (b) Save conservation of energy and angular momentum in beta decay (c) Provide a mechanism for nuclear fission (d) Explain parity violation in the weak interaction

Q6. In Fermi's theory of beta decay, the interaction is modeled as:

(a) Exchange of a virtual photon between the nucleus and the electron (b) A four-fermion point interaction with coupling constant $G_F$ (c) Tunneling through a Coulomb barrier, analogous to alpha decay (d) A three-body electromagnetic decay process

Q7. The allowed beta spectrum shape is $N(T_e) \propto F(Z', T_e) \cdot p_e \cdot E_e \cdot (Q - T_e)^2$. The factor $(Q - T_e)^2$ represents:

(a) The Coulomb correction for the daughter nucleus (b) The nuclear matrix element squared (c) The neutrino phase space (density of states for the neutrino) (d) The electron wavefunction probability at the nuclear surface

Q8. The Fermi function $F(Z', T_e)$ for $\beta^-$ decay satisfies $F > 1$ for all electron energies. Physically, this is because:

(a) The nuclear Coulomb field accelerates the emitted electron (b) The nuclear Coulomb field attracts the electron, increasing its wavefunction amplitude at the nucleus (c) The weak interaction is enhanced at low energies (d) Relativistic corrections always increase the decay rate

Q9. A Fermi transition ($\hat{O}_F = \sum_k \hat{\tau}_\pm(k)$) has selection rules:

(a) $\Delta J = 0, \pm 1$; $\Delta\pi = +$ (b) $\Delta J = 0$; $\Delta\pi = +$ (c) $\Delta J = 0$; $\Delta\pi = -$ (d) $\Delta J = 0, \pm 1$ (not $0 \to 0$); $\Delta\pi = +$

Q10. A transition with $J_i^\pi = 3^+ \to J_f^\pi = 2^-$ is classified as:

(a) Allowed Gamow-Teller (b) First forbidden (c) Second forbidden (d) Allowed Fermi

Q11. The Kurie plot is defined as $K(T_e) = \sqrt{N/(F \cdot p_e \cdot E_e)}$ plotted against $T_e$. For an allowed transition with $m_\nu = 0$, the Kurie plot is:

(a) A parabola opening downward (b) A straight line that intersects the $T_e$ axis at $T_e = Q$ (c) An exponential decay curve (d) A Gaussian centered at $T_e = Q/2$

Q12. The $ft$-value of a beta transition:

(a) Increases with increasing Q-value (b) Depends only on the nuclear matrix element (and fundamental constants) (c) Is always larger for Gamow-Teller transitions than Fermi transitions (d) Has units of energy

Q13. Superallowed $0^+ \to 0^+$ transitions are important because:

(a) They are the fastest beta decays known (b) They are pure Fermi transitions, providing a clean measurement of $V_{ud}$ (c) They always involve Gamow-Teller operators (d) They are the only beta decays that violate parity

Q14. A nuclear transition with $\log ft = 7.5$ is most likely classified as:

(a) Superallowed (b) Allowed (c) First forbidden (d) Second forbidden

Q15. In the Wu experiment, $^{60}$Co nuclei were:

(a) Heated to high temperatures to increase the decay rate (b) Cooled to millikelvin temperatures and aligned in a magnetic field to polarize the nuclear spins (c) Bombarded with neutrons to induce beta decay (d) Placed in an electric field to separate the emitted electrons

Q16. The Wu experiment demonstrated that:

(a) Energy is not conserved in beta decay (b) The neutrino has nonzero mass (c) Parity is violated in the weak interaction (d) The nuclear force is spin-dependent

Q17. In the V$-$A theory of the weak interaction, which particles participate in charged-current interactions?

(a) Left-handed particles and left-handed antiparticles (b) Right-handed particles and right-handed antiparticles (c) Left-handed particles and right-handed antiparticles (d) All helicity states participate equally

Q18. Two-neutrino double beta decay ($2\nu\beta\beta$) has been observed with typical half-lives of:

(a) $10^3 - 10^6$ years (b) $10^{10} - 10^{15}$ years (c) $10^{18} - 10^{24}$ years (d) $> 10^{30}$ years

Q19. The observation of neutrinoless double beta decay ($0\nu\beta\beta$) would prove that:

(a) Neutrinos have mass (b) The neutrino is a Majorana particle (its own antiparticle) (c) Parity is violated in the weak interaction (d) The Standard Model is complete

Q20. The experimental signature that distinguishes $0\nu\beta\beta$ from $2\nu\beta\beta$ is:

(a) The number of electrons emitted (one vs. two) (b) A sharp peak in the summed electron energy at $T = Q$, versus a continuous spectrum (c) The emission of gamma rays only in the $0\nu$ mode (d) A difference in the angular distribution of electrons

Answer Key

# Answer Explanation
1 (b) In $\beta^-$ decay, a $d$ quark converts to a $u$ quark via $W^-$ emission.
2 (b) The $2m_ec^2 = 1.022\,\text{MeV}$ accounts for the positron mass and the electron mass bookkeeping in atomic masses.
3 (b) EC requires only $M_\text{parent} > M_\text{daughter}$ (plus electron binding energy), while $\beta^+$ requires the additional $2m_ec^2$.
4 (c) Ellis and Wooster (1927) measured the average energy of beta particles calorimetrically, proving the spectrum was genuinely continuous.
5 (b) The continuous beta spectrum violated energy conservation unless a third, undetected particle carried away the missing energy and angular momentum.
6 (b) Fermi modeled beta decay as four fermions meeting at a point, with coupling strength $G_F$ — the low-energy limit of $W$ boson exchange.
7 (c) $(Q - T_e)^2 = E_\nu^2$ counts the available phase space for the massless neutrino. When the electron takes all the energy, no neutrino states remain.
8 (b) The Coulomb attraction increases the electron probability density at the nuclear surface, enhancing the transition rate — especially for slow electrons.
9 (b) The Fermi operator $\sum\hat{\tau}_\pm$ changes isospin but not spin or spatial wavefunction, so $\Delta J = 0$ and parity is unchanged.
10 (b) $\Delta J = 1$, $\Delta\pi = -$: the parity change requires $l \geq 1$, making this first forbidden.
11 (b) $K(T_e) \propto (Q - T_e)$, which is linear with x-intercept at $Q$.
12 (b) The $ft$-value absorbs the $Q$ and $Z$ dependences into the Fermi integral $f$, leaving only the nuclear matrix element.
13 (b) $0^+ \to 0^+$ excludes Gamow-Teller contributions, giving a pure Fermi transition with $|M_F|^2 = 2$ for $T=1$ analogs, directly measuring $g_V$ and $V_{ud}$.
14 (c) First forbidden transitions have $\log ft \approx 6-9$. A value of 7.5 falls squarely in this range.
15 (b) Polarizing the nuclear spins required both a strong magnetic field and millikelvin temperatures to overcome thermal randomization.
16 (c) The asymmetric angular distribution of electrons relative to the nuclear spin proved that the weak interaction distinguishes left from right.
17 (c) V$-$A couples only to left-handed fermions and right-handed antifermions — the defining feature of the charged-current weak interaction.
18 (c) $2\nu\beta\beta$ is a second-order weak process ($\propto G_F^4$), giving half-lives of $10^{18}-10^{24}$ years.
19 (b) $0\nu\beta\beta$ requires the virtual neutrino to be absorbed as an antineutrino at the second vertex, which is possible only if $\nu = \bar{\nu}$ (Majorana).
20 (b) In $0\nu\beta\beta$, no neutrinos carry away energy, so the summed electron energy equals $Q$ exactly — a monoenergetic peak versus the continuous $2\nu\beta\beta$ spectrum.