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Question #43812

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The Question

In a realistic model of a heavy nucleus of mass number $A$ , the nucleon number density $\rho(r)$ is not perfectly uniform. It is modeled as being constant at $\rho_0$ from the center up to a radial distance $R - t$ , and then decreasing linearly to zero at the geometric surface radius $R$ . Here, $t$ is a constant 'skin thickness' ( $t \ll R$ ) arising from the finite range of the strong nuclear force, and is independent of $A$ . Let $R_0$ be the standard nuclear radius parameter defined by the relation $\rho_0 = \left(\frac{4}{3}\pi R_0^3\right)^{-1}$ . Which of the following expressions best represents the geometric surface radius $R$ of the nucleus as a function of $A$ , to the first order in the small parameter $(t/R)$ ?

Options

A.
$R = R_0 A^{1/3}$
B.
$R = R_0 A^{1/3} + \frac{t}{2}$
C.
$R = R_0 A^{1/3} + t$
D.
$R = R_0 A^{1/3} - \frac{t}{2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Nuclear Radius Scaling|Nuclear Force Range and Saturation|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to move beyond the empirical $R = R_0 A^{1/3}$ formula and understand its origin from the assumption of uniform nuclear density.

The primary trap is that students commonly misapply the uniform density formula by either assuming the core itself is $R_0 A^{1/3}$ and adding the skin thickness $t$ directly (leading to Option C), or by completely ignoring the density gradient and assuming the geometric radius remains $R_0 A^{1/3}$ (Option A).

By forcing an integration of the density profile, the problem highlights how the 'missing' nucleons in the linearly decreasing skin must be compensated by a slight expansion of the overall geometric radius by $t/2$ to conserve the total nucleon number $A$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Radius Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free