Question

Thermal neutrons are neutrons that move at speeds comparable to those of air molecules at room temperature. These neutrons are most effective in initiating a nuclear chain reaction among \({ }^{235} \mathrm{U}\) isotopes. Calculate the wavelength (in \(\mathrm{nm}\) ) associated with a beam of neutrons moving at \(7.00 \times 10^{2} \mathrm{~m} / \mathrm{s}\) (mass of a neutron \(=1.675 \times 10^{-27} \mathrm{~kg}\) ).

Short answer

The wavelength is approximately 0.565 nm.

Step-by-step solution

Understanding the Problem

We need to calculate the wavelength associated with neutrons moving at a given speed using the de Broglie wavelength formula. The neutron's speed and mass are given, and we aim to find the wavelength in nanometers.

De Broglie Wavelength Formula

The de Broglie wavelength is given by the equation: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s}) \), \( m \) is the mass of the neutron, and \( v \) is the speed of the neutron.

Substitute Values into Formula

Substitute the given values into the de Broglie wavelength formula: \( \lambda = \frac{6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s}}{1.675 \times 10^{-27} \text{ kg} \times 7.00 \times 10^{2} \text{ m/s}} \).

Calculate the Wavelength

Perform the calculation: \( \lambda = \frac{6.626 \times 10^{-34}}{1.1725 \times 10^{-24}} = 5.65 \times 10^{-10} \text{ meters} \).

Convert Meters to Nanometers

Since 1 meter is equal to \( 10^9 \) nanometers, multiply the wavelength in meters by \( 10^9 \) to convert it to nanometers: \( 5.65 \times 10^{-10} \text{ m} \times 10^9 = 0.565 \text{ nm} \).

Key concepts

Thermal Neutrons

Thermal neutrons are neutrons that have a kinetic energy similar to thermal motion of particles at room temperature, which typically means speeds around a few hundred meters per second. These neutrons are called "thermal" because their speed distribution is similar to that of gas molecules at room temperature. Thermal neutrons play a critical role in nuclear reactions, especially in nuclear reactors. They are essential for sustaining a nuclear chain reaction, particularly with isotope \({}^{235}U\), due to their ability to be easily captured by the nucleus. This characteristic makes thermal neutrons valuable in nuclear fission, where the nucleus of an atom splits into two or more smaller nuclei, releasing a large amount of energy. Such reactions are fundamental to the operation of nuclear power plants.

Nuclear Chain Reaction

A nuclear chain reaction occurs when a single nuclear reaction causes an average of one or more subsequent nuclear reactions. This process leads to a series of self-sustaining reactions. In the context of fission, when a nucleus like \({}^{235}U\) absorbs a thermal neutron, it becomes unstable and splits into two smaller nuclei while releasing additional neutrons and a significant amount of energy. The released neutrons can further initiate more fissions, leading to a chain reaction.

• Fission releases energy due to the conversion of mass into energy, described by Einstein's equation \(E=mc^2\).
• The moderation and control of neutron speeds and quantities are crucial to managing the rate of the chain reaction.

Controlled nuclear chain reactions are the basis for electricity generation in nuclear power plants, while uncontrolled chain reactions are used in nuclear weapons.

Planck's Constant

Planck's constant, denoted by \(h\), is a fundamental constant in quantum mechanics characterized by its crucial role in the relationship between the energy of a photon and the frequency of its associated electromagnetic wave. The formula \(E = hf\) signifies this relationship, where \(E\) is the energy, \(f\) is the frequency, and \(h\) is Planck's constant. Planck's constant has a precise value of \(6.62607015 \times 10^{-34}\) \( ext{m}^2 \text{kg/s}\). It is a pivotal element in the de Broglie wavelength equation \(\lambda = \frac{h}{mv}\). This equation connects particle properties, like mass and velocity, to wave-like properties, such as wavelength. The discovery of Planck's constant revolutionized our understanding of microscopic processes, bridging the gap between wave and particle behaviors.

Neutron Speed

Neutron speed is a critical factor in nuclear physics and reactor design. It directly affects the capability of neutrons to induce fission in materials like \(^{235}U\). The kinetic energy of neutrons, dictated by their speed, can vary from very slow (thermal) to very fast (fast neutrons).

• Thermal neutrons have lower speeds, around hundreds of meters per second, and are most effective in causing fission in certain isotopes due to their high probability of being captured.
• Fast neutrons move at much higher speeds and are typically slowed down by a moderator in reactors to increase the probability of fission.

Understanding and controlling neutron speed is essential for maintaining safe and efficient nuclear reactions. This is why in reactor design, materials called moderators, like water or graphite, are used to slow fast neutrons to thermal speeds.