Question Statement
Why do heavier alpha-emitting radioisotopes (such as $^{212}\text{Po}$ with $E_\alpha = 8.78\text{ MeV}$, $T_{1/2} = 0.3\text{ }\mu\text{s}$) have half-lives that are vastly shorter than lighter ones (such as $^{238}\text{U}$ with $E_\alpha = 4.19\text{ MeV}$, $T_{1/2} = 4.5 \times 10^9\text{ years}$)?
(A)
Quantum tunneling probability through the Coulomb barrier depends exponentially on the barrier height and width, which both decrease rapidly with increasing alpha energy
(B)
The assault frequency of alpha particles against the nuclear wall increases by twenty orders of magnitude
(C)
Alpha particles in heavier nuclei are relativistic while those in lighter nuclei are strictly non-relativistic
(D)
Heavier nuclei possess zero centrifugal barrier
Step-by-Step KaTeX Derivation & Detailed Explanation
### Step-by-Step Derivation & Verification: 1. **WKB Barrier Transmission**: $T \approx e^{-2G}, \quad G = \frac{\pi Z_1 Z_2 e^2}{4\pi\varepsilon_0 \hbar v} = \frac{K Z}{\sqrt{E_\alpha}}$ 2. **Exponential Amplification**: Because $G$ sits in the exponent of $e^{-2G}$: $\ln T_{1/2} \approx A - \frac{B Z}{\sqrt{E_\alpha}}$ A small increase in $E_\alpha$ substantially reduces $G$, creating an astronomical exponential increase in barrier transmission $T$, thereby shortening the half-life from billions of years down to microseconds. 3. **Conclusion**: Option (A) is correct.