Quiz — Chapter 13: Alpha Decay

Instructions: Select the best answer for each question. Each question has exactly one correct answer.

Q1. Why does classical mechanics fail to explain alpha decay?

(A) The nuclear force is not strong enough to hold the alpha particle inside the nucleus.

(B) The alpha particle's kinetic energy is less than the height of the Coulomb barrier, so classical mechanics predicts it can never escape.

(C) The alpha particle is not a well-defined particle inside the nucleus.

(D) Classical mechanics does not account for the weak interaction, which causes alpha decay.

Q2. The Gamow factor $G$ for alpha decay through a Coulomb barrier depends on which of the following?

(A) The charge of the daughter nucleus and the alpha energy only.

(B) The nuclear temperature and the Fermi energy.

(C) The charge of the daughter nucleus, the alpha energy, the nuclear radius, and the reduced mass.

(D) The spin and parity of the parent nucleus only.

Q3. In the WKB approximation, the tunneling probability through a potential barrier is:

(A) $P = \exp\left(+\frac{2}{\hbar}\int_R^b\sqrt{2\mu(V-E)}\,dr\right)$

(B) $P = \exp\left(-\frac{2}{\hbar}\int_R^b\sqrt{2\mu(V-E)}\,dr\right)$

(C) $P = \exp\left(-\frac{1}{\hbar}\int_R^b\sqrt{2\mu(E-V)}\,dr\right)$

(D) $P = 1 - \exp\left(-\frac{2}{\hbar}\int_R^b\sqrt{2\mu(V-E)}\,dr\right)$

Q4. Alpha decay half-lives for known emitters span a range of approximately:

(A) A factor of 10 (one order of magnitude)

(B) A factor of $10^{5}$ (five orders of magnitude)

(C) A factor of $10^{20}$ or more (twenty-plus orders of magnitude)

(D) A factor of $10^{50}$ (fifty orders of magnitude)

Q5. The Geiger-Nuttall law states that, for alpha emitters of a given element:

(A) The half-life is proportional to the alpha energy.

(B) $\log_{10}t_{1/2}$ is approximately linearly related to $1/\sqrt{E_\alpha}$.

(C) The Q-value equals the Coulomb barrier height.

(D) The preformation factor is the same for all isotopes of that element.

Q6. Which of the following best describes the "preformation factor" $S$ in alpha decay?

(A) The probability that the alpha particle tunnels through the barrier on a single attempt.

(B) The probability that a preformed alpha cluster exists at the nuclear surface.

(C) The number of times per second the alpha particle strikes the barrier.

(D) The ratio of the Coulomb barrier height to the alpha kinetic energy.

Q7. For an even-even parent nucleus decaying to an excited $2^+$ state of the daughter (ground state is $0^+$), the alpha particle must carry angular momentum:

(A) $\ell = 0$

(B) $\ell = 1$

(C) $\ell = 2$

(D) Any even value of $\ell$

Q8. Fine structure in alpha decay refers to:

(A) The splitting of alpha energies due to the spin-orbit interaction.

(B) The observation of multiple discrete alpha energy groups from a single parent, corresponding to transitions to different states of the daughter.

(C) The relativistic correction to the alpha particle's kinetic energy.

(D) The difference between alpha energies of neighboring isotopes.

Q9. Proton radioactivity was first observed in:

(A) ${}^{4}\text{He}$ in 1911 by Rutherford.

(B) ${}^{151}\text{Lu}$ in 1981 by Hofmann et al.

(C) ${}^{223}\text{Ra}$ in 1984 by Rose and Jones.

(D) ${}^{238}\text{U}$ in 1896 by Becquerel.

Q10. In cluster radioactivity (e.g., ${}^{14}\text{C}$ emission from ${}^{223}\text{Ra}$), the daughter nucleus is nearly always close to:

(A) ${}^{56}\text{Fe}$, the most tightly bound nucleus per nucleon.

(B) ${}^{4}\text{He}$, the alpha particle.

(C) ${}^{208}\text{Pb}$, the doubly magic nucleus.

(D) ${}^{12}\text{C}$, the basis of the atomic mass unit.

Q11. If the alpha energy $E_\alpha$ increases by 10% while all other parameters remain fixed, the tunneling probability $P$ will:

(A) Increase by approximately 10%.

(B) Decrease by approximately 10%.

(C) Increase by several orders of magnitude.

(D) Remain approximately the same.

Q12. The "assault frequency" $f$ in the one-body alpha decay model is approximately:

(A) $10^{5}\,\text{s}^{-1}$

(B) $10^{12}\,\text{s}^{-1}$

(C) $10^{21}\,\text{s}^{-1}$

(D) $10^{30}\,\text{s}^{-1}$

Q13. Alpha decay is energetically possible ($Q_\alpha > 0$) for essentially all nuclides with:

(A) $A > 4$

(B) $A > 56$

(C) $A \gtrsim 150$

(D) $A > 238$

Q14. Which physicist(s) first explained alpha decay as quantum tunneling through the Coulomb barrier in 1928?

(A) Rutherford and Bohr

(B) Geiger and Nuttall

(C) Gamow, and independently Gurney and Condon

(D) Fermi and Dirac

Q15. The branching ratio for cluster radioactivity (${}^{14}\text{C}$ emission) relative to alpha emission in ${}^{223}\text{Ra}$ is approximately:

(A) $10^{-2}$ (1 in 100)

(B) $10^{-5}$ (1 in 100,000)

(C) $10^{-10}$ (1 in 10 billion)

(D) $10^{-20}$ (1 in $10^{20}$)

Answer Key

Q Answer Explanation
1 B The alpha energy (4–9 MeV) is far below the Coulomb barrier (25–30 MeV). Classical mechanics forbids escape; quantum tunneling permits it.
2 C $G$ depends on $Z_d$ (barrier height/width), $E_\alpha$ (outer turning point), $R$ (inner turning point), and $\mu$ (effective mass in the WKB integral).
3 B The WKB tunneling probability has a negative exponent: $P = \exp(-2\int\sqrt{2\mu(V-E)}/\hbar\,dr)$.
4 C From $\sim10^{-7}\,\text{s}$ (${}^{212}\text{Po}$) to $\sim10^{15}\,\text{yr}$ (${}^{148}\text{Sm}$), spanning over 20 orders of magnitude.
5 B The Geiger-Nuttall law: $\log_{10}t_{1/2} \propto Z_d/\sqrt{E_\alpha}$, which is linear in $1/\sqrt{E_\alpha}$ for fixed $Z_d$.
6 B $S$ measures the probability that four nucleons inside the nucleus form a coherent ${}^4\text{He}$ cluster at the nuclear surface.
7 C Since $\alpha$ has spin 0, $\ell = |I_p - I_d| = |0 - 2| = 2$, and parity is conserved: $\pi = (+1)(-1)^2 = +1$.
8 B Fine structure = multiple alpha groups from transitions to ground and excited states of the daughter.
9 B ${}^{151}\text{Lu}$ ground-state proton emission, Hofmann et al. (1981) at GSI Darmstadt.
10 C The doubly magic ${}^{208}\text{Pb}$ ($Z=82$, $N=126$) provides a large shell-closure Q-value enhancement.
11 C The tunneling probability $P = e^{-G}$ is exponentially sensitive to $E_\alpha$. A 10% increase reduces $G$ significantly, increasing $P$ by orders of magnitude.
12 C $f = v/(2R) \approx 0.1c/(2 \times 8\,\text{fm}) \approx 10^{21}\,\text{s}^{-1}$.
13 C The Coulomb term in the SEMF makes $Q_\alpha > 0$ for $A \gtrsim 150$. Below this, the Q-value is negative or negligibly small.
14 C Gamow (Göttingen) and Gurney & Condon (Princeton) independently published the tunneling explanation in 1928.
15 C The branching ratio is $\sim 8.5 \times 10^{-10}$, approximately 1 in 10 billion alpha decays.