Before You Begin

Question #43613

Difficulty: 8/10 | 🟢 Free

The Question

A hypothetical nucleus $A$ at rest undergoes a transition to a ground state nucleus $B$ , emitting a single photon. The rest mass energy of nucleus $A$ is $M_A c^2 = 500 \text{ MeV}$ and the rest mass energy of nucleus $B$ is $M_B c^2 = 300 \text{ MeV}$ .

The emitted photon is then incident on a free atom $C$ , initially at rest, with rest mass energy $M_C c^2 = 300 \text{ MeV}$ . The photon is completely absorbed, ejecting an electron and leaving behind an ion $C'$ with rest mass energy $M_{C'} c^2 = 360 \text{ MeV}$ .

Assume the ejected electron is ultra-relativistic and can be treated as effectively massless such that its kinetic energy $K_e$ and momentum $p_e$ are related by $K_e \approx p_e c$ . Assuming all motion occurs in one dimension, what is the maximum possible kinetic energy of the ejected electron?

Options

A.
$94 \text{ MeV}$
B.
$100 \text{ MeV}$
C.
$134 \text{ MeV}$
D.
$140 \text{ MeV}$

Check Your Answer

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

Einstein's Photoelectric Equation|Nuclear Fission Energy Balance|Photon Neutrality

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Special Relativity and Quantum Mechanics, specifically targeting the 'Photon Neutrality / Momentum Transfer' trap.

A common student mistake is to calculate the photon energy simply as the mass defect ( $\Delta m c^2 = 200 \text{ MeV}$ ) and the effective work function as the rest mass difference of the atom and ion ( $360 - 300 = 60 \text{ MeV}$ ), leading directly to the trap answer of $140 \text{ MeV}$ .

This fundamentally violates the conservation of momentum.

Because the photon carries momentum, the emitting nucleus must recoil, carrying away kinetic energy and red-shifting the emitted photon.

Similarly, during absorption by a free atom, the resulting ion must recoil to conserve momentum, which effectively increases the energy required to eject the electron.

Students must apply exact relativistic invariant mass equations to account for recoil in both the emission and absorption phases.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Einstein's Photoelectric Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free