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Question #43750

Difficulty: 8/10 | 🟢 Free

The Question

A stationary nucleus $X$ (nuclear rest mass $M_X$ ) undergoes $\beta^-$ decay to an excited state of nucleus $Y$ . The emitted $\beta^-$ particles are observed to have a continuous kinetic energy spectrum with a maximum value $K_{max}$ . The excited nucleus subsequently decays to its ground state $Y$ (nuclear rest mass $M_Y$ ) by emitting a single $\gamma$ -photon of energy $E_\gamma$ . Assuming the antineutrino is massless, neglecting the recoil kinetic energy of the nucleus during the $\beta^-$ decay, and using the non-relativistic approximation for the recoil during $\gamma$ emission ( $E_\gamma \ll M_Y c^2$ ), which of the following is the correct expression for the rest mass energy of the initial nucleus $X$ ( $M_X c^2$ )? Let $m_e$ be the rest mass of the electron and $c$ be the speed of light in vacuum.

Options

A.
$M_Y c^2 + m_e c^2 + K_{max} + E_\gamma + \frac{E_\gamma^2}{2 M_Y c^2}$
B.
$M_Y c^2 + m_e c^2 + K_{max} + E_\gamma$
C.
$M_Y c^2 + K_{max} + E_\gamma + \frac{E_\gamma^2}{2 M_Y c^2}$
D.
$M_Y c^2 + m_e c^2 + K_{max} + E_\gamma - \frac{E_\gamma^2}{2 M_Y c^2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Photon Neutrality|Mass Defect vs. Binding Energy|Energy Distribution in Beta Decay

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly couple conservation laws across two distinct nuclear decay mechanisms.

The primary trap exploits the common misconception that a gamma photon carries away the entirety of the nuclear transition energy; students often forget to account for the recoil kinetic energy of the nucleus ( $E_\gamma^2 / 2Mc^2$ ) required by momentum conservation.

A secondary conceptual filter involves the continuous beta spectrum: students must recognize that the maximum kinetic energy of the beta particle corresponds to a state where the undetected antineutrino has zero energy, allowing for an exact mass-energy balance.

Finally, by explicitly specifying nuclear rest masses, the problem prevents the accidental cancellation of the electron mass term, catching students who blindly apply atomic mass formulas without understanding the underlying physics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Photon Neutrality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free