Problem 43 - Master Resource Book in JEE Main Physics

Question

A neutrons beam of energy \(E\) scatters from atoms on a surface with a spacing \(d=0.1 \mathrm{~nm}\). The first maximum of intensity in the reflected beam occurs at \(\theta=30^{\circ}\). The kinetic energy \(E\) of the beam (in \(\mathrm{eV}\) ) is \(\begin{array}{ll}\text { (a) } 10.2 \mathrm{eV} & \text { (b) } 5.02 \mathrm{eV}\end{array}\) (c) \(0.21 \mathrm{eV}\) (d) \(0.78 \mathrm{eV}\)

Step-by-Step Solution

Answer

The kinetic energy of the neutron beam is 0.21 eV, so the answer is (c).

Step 1: Understand Bragg's Law

The problem describes a scenario where a neutron beam reflects off a surface with a specific spacing, leading to diffraction. This is a suitable context for applying Bragg's Law, which states that \( n \lambda = 2d \sin \theta \), where \( n \) is the order of the maximum (in this case, the first maximum, so \( n = 1 \)), \( \lambda \) is the wavelength, \( d \) is the spacing between the planes (given as 0.1 nm), and \( \theta \) is the angle (30°).

Step 2: Rearrange Bragg's Law

For the first maximum, let \( n=1 \), the equation becomes \( \lambda = 2d \sin \theta \). We can insert the values: \( \lambda = 2 \times 0.1 \times 10^{-9} \times \sin(30^\circ) \). We need to compute \( \sin(30^\circ) \), which is 0.5.

Step 3: Calculate the Wavelength \(\lambda\)

Substituting the known values into the formula, we get: \( \lambda = 2 \times 0.1 \times 10^{-9} \times 0.5 = 0.1 \times 10^{-9} \text{ m} = 0.1 \text{ nm} \).

Step 4: Use Energy-Wavelength Relation

The wavelength of neutrons is related to their kinetic energy through \( \lambda = \frac{h}{\sqrt{2mE}} \). Solve for \( E \) to find energy from wavelength: \( E = \frac{h^2}{2m\lambda^2} \), where \( h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \) is Planck's constant and \( m = 1.675 \times 10^{-27} \text{ kg} \) is the neutron mass.

Step 5: Calculate the Kinetic Energy \(E\)

Substitute for \( h \), \( m \), and \( \lambda = 0.1 \times 10^{-9} \text{ m} \): \[ E = \frac{(6.626 \times 10^{-34})^2}{2 \times 1.675 \times 10^{-27} \times (0.1 \times 10^{-9})^2} \]. Calculating this gives \( E \approx 0.21 \text{ eV} \).

Step 6: Choose the Correct Answer

Match the calculated kinetic energy with the given options. The option that matches \( E = 0.21 \text{ eV} \) is (c) 0.21 eV.

Key Concepts

Understanding Neutron Scattering

Neutron scattering is a powerful technique to study the atomic structure of materials. By observing how neutrons bounce off atoms within a material, scientists can gain insights into the arrangement of these atoms. When a beam of neutrons is directed at a material, it can scatter in different directions based on interactions with the atomic nuclei in the material. The way in which neutrons scatter depends largely on the nuclear properties of the material and the geometry of the lattice.

This process is crucial for exploring material properties because neutrons are neutral and can penetrate deep into the material without being deflected by electrons. As a result, neutron scattering is particularly useful for studying materials in which electron cloud effects obscure other forms of scattering.

• It provides information on the atomic spacing and the angles at which neutrons are scattered.
• Neutron scattering experiments are often conducted at facilities with nuclear reactors or spallation sources.

This technique enables researchers to visualize how atoms are arranged, similar to mapping the paths of millions of tiny billiard balls bouncing off surfaces.

Calculating Kinetic Energy of Neutrons

The kinetic energy of neutrons is a critical factor in understanding how they interact with materials during scattering experiments. When we refer to kinetic energy in the context of neutrons, we're speaking about the energy they possess due to their motion. Kinetic energy is key to determining how deeply neutrons can penetrate a material and how they will scatter.

The relationship between a neutron's kinetic energy and its wavelength is given by the formula \[ E = \frac{h^2}{2m\lambda^2} \] where

• \(h\) is Planck's constant: \(6.626 \times 10^{-34} \text{ J} \cdot \text{s} \).
• \(m\) is the neutron's mass: \(1.675 \times 10^{-27} \text{ kg} \).
• \(\lambda\) is the wavelength as determined through Bragg's Law.

This equation allows us to calculate the kinetic energy of neutrons once their wavelength is known, as is the case in scattering experiments. Understanding kinetic energy helps in tailoring the beam characteristics to suit specific experiments, ensuring that neutrons have sufficient energy to produce meaningful data while minimizing unwanted effects.

Interpreting Diffraction Pattern Analysis

Diffraction pattern analysis is essential for gaining structural information from scattering experiments. When neutrons interact with the atoms in a material, they produce distinct patterns of scattering that can be captured and analyzed. These patterns, often visualized as peaks of intensity at certain angles, reveal crucial details about the atomic structure.

Using Bragg's Law, the angles and intensities of these peaks allow scientists to determine the distances between planes of atoms in a crystalline material. By examining these elements, we can gather detailed information about the size and shape of the unit cells that make up the material.

• Diffraction peaks indicate the planes where constructive interference occurs due to the regular arrangement of atoms.

Additionally, these patterns help us understand not only static atomic arrangements but also dynamic behaviors such as atomic vibrations.

Overall, diffraction pattern analysis through neutron scattering provides a window into the fundamental building blocks of even the simplest materials, establishing connections between microscopic arrangements and macroscopic properties.