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Question #43703

Difficulty: 8/10 | 🟢 Free

The Question

A stationary parent nucleus undergoes beta decay into a daughter nucleus of mass $M$ , an electron of mass $m$ , and a massless antineutrino. The total energy released in the decay (Q-value) is $Q$ . In a specific decay event, the electron is emitted with a kinetic energy $K$ (where $0 < K < Q$ ). Assume that the daughter nucleus and the electron are non-relativistic ( $Q \ll mc^2$ ), and that $M$ is sufficiently large such that the recoil kinetic energy of the daughter nucleus is negligible in the energy conservation equation. Which of the following expressions gives the minimum possible recoil kinetic energy of the daughter nucleus? (Let $c$ be the speed of light in vacuum).

Options

A.
$\frac{1}{2M} \left( \sqrt{2mK} - \frac{Q-K}{c} \right)^2$
B.
$\frac{m}{M} K$
C.
$\frac{1}{2M} \left( \sqrt{2mK} + \frac{Q-K}{c} \right)^2$
D.
$\frac{(Q-K)^2}{2Mc^2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Energy Distribution in Beta Decay|Reduced Mass Effect|Impulse-Momentum Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply conservation laws to a 3-body decay system, specifically targeting the misconception that beta decay can be treated as a simple 2-body problem.

Students often incorrectly assume that the nucleus only recoils against the electron (ignoring the undetected antineutrino's momentum), which leads directly to the trap Option B ( $K_Y = p_e^2/2M = mK/M$ ).

By asking for the minimum recoil energy for a given continuous electron energy, the problem forces students to recognize the phase-space constraints: they must apply the triangle inequality to the momentum vectors ( $\vec{p}_Y = -\vec{p}_e - \vec{p}_\nu$ ) and realize that the minimum recoil occurs when the electron and antineutrino are emitted in opposite directions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Energy Distribution in Beta Decay
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free