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Question #43815

Difficulty: 8/10 | 🟢 Free

The Question

In a theoretical liquid-drop model, the binding energy per nucleon $B$ of a nucleus with mass number $A$ depends on a dimensionless deformation parameter $x$ as $B(x) = B_0 - \alpha x^2 + \beta x^3$ where $B_0$ , $\alpha$ , and $\beta$ are positive constants in $\text{MeV}$ . The state $x=0$ corresponds to the stable spherical equilibrium of the nucleus. To induce nuclear fission, the nucleus must be deformed beyond its restoring stability limit. What is the minimum total kinetic energy $K_{\text{min}}$ that must be imparted to the deformation mode of the nucleus at $x=0$ to cause it to undergo fission?

Options

A.
$\frac{4 A \alpha^3}{27 \beta^2}$
B.
$\frac{4 \alpha^3}{27 \beta^2}$
C.
$\frac{A \alpha^3}{\beta^2}$
D.
$\frac{2 A \alpha^3}{9 \beta^2}$

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

Binding Energy per Nucleon Curve|Equilibrium Stability|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map the binding energy per nucleon to a macroscopic potential energy landscape for the entire nucleus.

The primary trap is twofold: first, students often forget to multiply the binding energy per nucleon by the mass number $A$ to get the total energy of the system, leading to an answer off by a factor of $A$ (Option B).

Second, students accustomed to simple harmonic oscillators might incorrectly assume the restoring force is purely linear and fail to recognize that the cubic anharmonicity ( $\beta x^3$ ) creates a local maximum in the potential energy landscape, which defines the physical threshold for fission.

Misidentifying the peak of the potential barrier by setting $U(x) = U(0)$ instead of $dU/dx = 0$ is another common lethal mistake that leads to Option C.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Binding Energy per Nucleon Curve
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free