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Question #43654

Difficulty: 8/10 | 🟢 Free

The Question

A hydrogen-like ion consists of a single electron of rest mass $m_e$ and a nucleus. The sum of the rest masses of the isolated nucleons constituting the nucleus is $M_0$ , and the total nuclear binding energy is $E_b$ . Let $\nu_0$ be the frequency of a photon emitted during a specific electronic transition, calculated by accounting for nuclear recoil but incorrectly assuming the nuclear mass is $M_0$ . Let $\nu$ be the true frequency of the photon for the same transition, correctly accounting for both nuclear recoil and the nuclear mass defect. Assuming $m_e \ll M_0$ and $E_b \ll M_0 c^2$ , which of the following best approximates the fractional shift in frequency, $\frac{\nu - \nu_0}{\nu_0}$ ? ( $c$ is the speed of light in vacuum)

Options

A.
$-\frac{m_e E_b}{M_0^2 c^2}$
B.
$+\frac{m_e E_b}{M_0^2 c^2}$
C.
$-\frac{E_b}{M_0 c^2}$
D.
$-\frac{m_e^2 E_b}{M_0^3 c^2}$

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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

Rydberg Constant Sensitivity|Center of Mass (COM) Frame|Mass Defect vs. Binding Energy

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of atomic spectra and nuclear mass defect.

Students often compartmentalize these topics, applying the reduced mass correction $\mu = \frac{mM}{m+M}$ using the sum of nucleon masses ( $M_0$ ) while forgetting that nuclear binding energy ( $E_b$ ) reduces the effective inertial mass of the nucleus to $M = M_0 - E_b/c^2$ .

The trap lies in assuming the mass defect only affects nuclear reaction Q-values, whereas it fundamentally alters the center-of-mass frame dynamics of the atom, leading to a measurable shift in the Rydberg constant and spectral lines.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Rydberg Constant Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free