Question

A free neutron is unstable and decays into a proton with a half-life of \(10.4\) min. (a) What other particle forms? (b) Why don't neutrons in atomic nuclei decay at the same rate?

Short answer

(a) During neutron decay, an electron (\(e^{-}\)) and an electron antineutrino (\(\overline{\nu_e}\)) are formed. So, the other particle that forms is the electron antineutrino (\(\overline{\nu_e}\)). (b) The decay rate of neutrons in atomic nuclei differs from the decay rate of free neutrons due to two factors: the stabilizing effect of the nuclear binding energy provided by the strong nuclear force and the presence of protons in the nucleus, which affect the transformation process through their repulsive electrostatic force.

Step-by-step solution

Understand neutron decay

Neutron decay, also known as beta decay, is a process in which a neutron transforms into a proton by emitting an electron and an antineutrino. This process can be represented by the following equation: \( n \rightarrow p + e^{-} + \overline{\nu_e} \) where: - \(n\) is the neutron, - \(p\) is the proton, - \(e^{-}\) is the electron (also called a beta particle), - \(\overline{\nu_e}\) is the electron antineutrino.

Identify the other particle formed during neutron decay

Using the neutron decay equation from Step 1, we can now answer question (a): the other particle formed during neutron decay is the electron antineutrino (\(\overline{\nu_e}\)).

Understand the difference in decay rates for free neutrons and neutrons in atomic nuclei

To answer question (b), we need to discuss the factors that affect the decay rate of neutrons in atomic nuclei. There are two main factors: 1. Nuclear binding energy: Neutrons in an atomic nucleus are bound together by the strong nuclear force, which is a short-range attractive force that keeps protons and neutrons together in the nucleus. This binding energy provides a certain level of stability to the neutrons in the nucleus, making it more difficult for them to decay compared to free neutrons. Therefore, the decay rate of neutrons in atomic nuclei is generally slower than the decay rate of free neutrons. 2. The presence of protons: In an atomic nucleus, the protons and neutrons are in close proximity to each other. The presence of protons in the nucleus can affect the decay rate of neutrons because the repulsive electrostatic force between the protons and the newly formed protons during neutron decay works against the transformation process. In summary, the decay rate of neutrons in atomic nuclei is different from the decay rate of free neutrons because of the stabilizing effect of the nuclear binding energy and the presence of protons in the nucleus.

Key concepts

Beta Decay

In beta decay, a neutron turns into a proton. This transformation happens by expelling particles, specifically an electron and an electron antineutrino. The equation that represents this decay is \( n \rightarrow p + e^{-} + \overline{u_e} \).Let's break it down:

• Neutron ( \( n \)): Starts the process by decaying.
• Proton ( \( p \)): The newly formed particle.
• Electron ( \( e^{-} \)): Also called a beta particle, this is emitted during the decay.
• Electron antineutrino ( \( \overline{u_e} \)): Another particle released in this process.

For free neutrons, this reaction occurs with a half-life of about 10.4 minutes. This means that, for a batch of free neutrons, half will undergo beta decay within this time frame. Beta decay is essential to understanding nuclear reactions and processes.

Electron Antineutrino

The electron antineutrino is a fundamental particle released during beta decay. It carries no electric charge and only interacts weakly with other matter, making it extremely hard to detect. Here are some important points about it:

• Neutral Particle: The antineutrino has no charge, which distinguishes it from the electron.
• Weak Interaction: It interacts primarily through the weak nuclear force, one of the four fundamental forces of nature.
• Energy Conservation: In beta decay, the electron antineutrino helps conserve energy and momentum, balancing the decay equation.

Since antineutrinos are elusive, they provide exciting challenges for physicists studying fundamental particles and forces.

Nuclear Binding Energy

Nuclear binding energy is what keeps protons and neutrons together inside a nucleus. It's the energy needed to pull a nucleus apart into separate neutrons and protons. Key points to understand:

• Strong Force: The nuclear binding energy arises from the strong nuclear force, a powerful, yet short-range force that binds nucleons (protons and neutrons) together.
• Stability: The stronger the nuclear binding energy, the more stable the nucleus. Therefore, neutrons inside a nucleus do not decay as quickly as free neutrons.
• Energy Release: When a nucleus forms, this binding energy is released, often in the form of radiation or kinetic energy.

Nuclei with high binding energy are particularly stable and require more energy to break apart, thus affecting decay rates.

Neutron Stability

Neutron stability varies significantly depending on their location. Free neutrons are not stable and eventually undergo beta decay. Neutrons in a nucleus, however, behave differently due to being bound by nuclear forces. Here's how it works:

• Free Neutrons: These have a defined half-life of 10.4 minutes, after which they decay into protons, electrons, and antineutrinos.
• Bound Neutrons: Within a nucleus, neutrons are much more stable, as they are influenced by the nuclear binding energy and interactions with other protons and neutrons. This makes their decay rate slower compared to free neutrons.
• Decay Rate Variation: Nuclear binding energy and the electrostatic force among protons affect how quickly the bound neutrons decay. This is why they exhibit different decay rates depending on their environment.

Understanding neutron stability is crucial for predicting how atoms and nuclei behave under various conditions.