Problem 77 - Master Resource Book in JEE Main Physics

Question

After absorbing a slowly moving neutron of mass \(n_{N}\) (momentum \(\sim 0\) ) a nucleus of mass \(M\) breaks into two nuclei of masses \(m_{1}\) and \(5 m_{1}\left(6 m_{1}=M+m_{N}\right)\), respectively. If the de-Broglie wavelength of the nucleus with mass \(m_{1}\) is \(\lambda\), then de-Broglie wavelength of the other nucleus will be (a) \(25 \lambda\) (b) \(5 \lambda\) (c) \(\frac{\lambda}{5}\) (d) \(\lambda\)

Step-by-Step Solution

Answer

The de-Broglie wavelength of the other nucleus is (b) \( 5 \lambda \).

Step 1: Understand the Conservation of Momentum

When the initial nucleus absorbs the neutron, the combined system has zero momentum (since the neutron has negligible momentum). After breaking apart, the conservation of momentum implies that the resulting nuclei momentum must cancel each other out. Thus, \( m_1 \cdot v_1 = 5m_1 \cdot v_2 \), where \( v_1 \) is the velocity of the nucleus with mass \( m_1 \) and \( v_2 \) is the velocity of the nucleus with mass \( 5m_1 \).

Step 2: Relate Velocities Using Conservation

From \( m_1 \cdot v_1 = 5m_1 \cdot v_2 \), solve for \( v_1 \) and \( v_2 \): \( v_2 = \frac{v_1}{5} \). This relationship between the velocities of the two nuclei is essential for comparing their de-Broglie wavelengths.

Step 3: Apply the de-Broglie Wavelength Formula

The de-Broglie wavelength \( \lambda \) is given by \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant, \( m \) is the mass, and \( v \) is the velocity. For the nucleus with mass \( m_1 \) and wavelength \( \lambda \), we have \( \lambda = \frac{h}{m_1 v_1} \).

Step 4: Calculate de-Broglie Wavelength of 5m1 Nucleus

For the nucleus with mass \( 5m_1 \) and using \( v_2 = \frac{v_1}{5} \), its de-Broglie wavelength \( \lambda_2 \) is \( \lambda_2 = \frac{h}{5m_1 v_2} = \frac{h}{5m_1 \frac{v_1}{5}} = \frac{5h}{5m_1 v_1} = \frac{h}{m_1 v_1/5} = 5 \lambda \).

Step 5: Choose the Correct Answer

Based on our calculation, the de-Broglie wavelength of the other nucleus is \( 5 \lambda \). Therefore, the answer is (b) \( 5 \lambda \).

Key Concepts

Conservation of Momentum

The concept of momentum conservation is fundamental in understanding the behavior of particles in nuclear physics.
In our exercise, we encounter a nucleus absorbing a neutron and then splitting into two smaller nuclei. Despite the complexity of this transformation, momentum conservation provides the key to understanding what happens with particle movement.

• Initially, the nucleus and the absorbed neutron form a system with very little, or practically zero, momentum, due to the neutron's low speed.
• As they collectively decay, the momentum must be conserved, meaning the total momentum of the resulting nuclei equals the initial item's momentum.

This requires that for every action there is an equal and opposite reaction, mathematically described for our case as:\[ m_1 \cdot v_1 = 5m_1 \cdot v_2 \]Thus, the velocity of these fragments are related inversely through their masses, allowing us to determine one if the other is known.
Momentum conservation not only helps in particle physics calculations but also supports the principle that energy cannot be created or destroyed in a closed system.

Nuclear Physics

Nuclear physics delves into the components and behaviors of atomic nuclei. This field is key to understanding the exercise at hand, where we see the nucleus interacting with a neutron and breaking apart. These interactions reflect fundamental reactions known as nuclear fission and fusion.
Nuclear fission, which is relevant here, involves splitting a heavy nucleus into smaller nuclei, often resulting in gamma-rays and free neutrons. When a neutron is absorbed by a nucleus, it can destabilize the nucleus, making it split.

• Neutron absorption increases the nuclear mass momentarily before the split.
• The stability of a nucleus is influenced by the delicate balance of forces within, primarily the strong nuclear force and electromagnetic forces, both active over very short ranges.

This exercise simplifies many real-world considerations to focus on the essential principles, providing a meaningful application of nuclear physics in problem-solving.

Particle Velocity

Understanding particle velocity is central to predicting the behavior of subatomic particles like nuclei and neutrons. Velocity in this context relates to both the speed and direction of particle movement.
In nuclear interactions such as the one in this exercise, calculating velocities post-reaction is essential for further determining properties like de-Broglie wavelength. Since \[ v_2 = \frac{v_1}{5} \] is derived from our momentum conservation principle, we can confidently predict the velocity of the nucleus with mass \(5m_1\).

• Velocity can often influence a particle's properties, including its observable wavelength according to de Broglie's equation: \( \lambda = \frac{h}{mv} \).
• Adjustments in particle speeds due to nuclear interactions lead to differences in wavelengths, as observed in our solution.

Through understanding both momentum conservation and particle velocity, we bridge theoretical concepts with lived physics, linking observable phenomena to mathematical calculations.