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Question #43805

Difficulty: 8/10 | 🟢 Free

The Question

A hypothetical atomic nucleus is modeled as a uniform solid sphere of rest mass $M$ and radius $R$ . Initially, the nucleus is moving with a relativistic velocity $v = \beta c$ in the $+x$ -direction in the laboratory frame. A gamma-ray photon of energy $E$ , traveling in the $-x$ -direction, is absorbed by the nucleus at grazing incidence (effective impact parameter $b = R$ ). This absorption excites the nucleus to an internal energy state $\Delta E$ above its ground state. Assuming the recoil and rotational velocities in the nucleus's initial rest frame are non-relativistic ( $\Delta E \ll Mc^2$ ), which of the following expressions gives the required incident photon energy $E$ in the laboratory frame? (Keep terms up to first order in $\frac{\Delta E}{Mc^2}$ ).

Options

A.
$\left( \Delta E + \frac{(\Delta E)^2}{2Mc^2} \right) \sqrt{\frac{1-\beta}{1+\beta}}$
B.
$\left( \Delta E + \frac{7(\Delta E)^2}{4Mc^2} \right) \sqrt{\frac{1-\beta}{1+\beta}}$
C.
$\left( \Delta E + \frac{7(\Delta E)^2}{4Mc^2} \right) \sqrt{\frac{1+\beta}{1-\beta}}$
D.
$\left( \Delta E + \frac{5(\Delta E)^2}{4Mc^2} \right) \sqrt{\frac{1-\beta}{1+\beta}}$

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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

Emission vs Absorption|Impact Parameter Geometry|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the relativistic Doppler effect with conservation of both linear and angular momentum during photon absorption.

A common trap for JEE Advanced students is to blindly apply the standard linear recoil formula ( $E = \Delta E + \frac{(\Delta E)^2}{2Mc^2}$ ) while ignoring the angular momentum carried by a photon absorbed at an impact parameter.

By specifying grazing incidence, the photon imparts significant angular momentum, forcing the nucleus to acquire rotational kinetic energy.

Students must correctly model the nucleus as a rigid rotor, calculate the rotational recoil, and then correctly apply the relativistic Doppler shift transformation between the laboratory and the nucleus's rest frame to find the observed photon energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Emission vs Absorption
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free