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The neutral pion (π0) is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of 8.4 × 1017 s before decaying into two gamma-ray photons. Using the relationship E=mc2 between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.

Short Answer

Expert verified
The uncertainty in the pion's mass as a fraction is about 2.4×1014.

Step by step solution

01

Identify Given Values

We are given the average lifetime of the neutral pion as Δt=8.4×1017s and its mass relative to the electron mπ0=264me. We need to find the uncertainty in the mass as a fraction of the mass.
02

Understand Energy-Mass Uncertainty Relationship

According to the energy-time uncertainty principle, ΔEΔt2, where ΔE is the energy uncertainty and is the reduced Planck's constant (=1.0545718×1034Js).
03

Relate Energy Uncertainty to Mass Uncertainty

Since energy is related to mass through E=mc2, the uncertainty in mass Δm can be expressed as ΔE=Δmc2. So, Δmc2Δt2.
04

Solve for Mass Uncertainty

Rearranging the energy-mass uncertainty relationship gives Δm2c2Δt. Substitute =1.0545718×1034Js, c=3×108m/s, and Δt=8.4×1017s.
05

Calculation Details

Δm1.0545718×10342(3×108)28.4×1017 Compute this value to find Δm.
06

Express as Fraction of Mass

Calculate the fraction Δmmπ0 where mπ0=264me. Since the electron mass me=9.10938356×1031kg, find mπ0=264×9.10938356×1031kg. Use this to find the fraction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Neutral Pion
The neutral pion, denoted as π0, is a type of meson, which is a particle made up of a quark and an antiquark. It is one of several types of pions, which are important players in particle physics, responsible for mediating the strong force that holds atomic nuclei together. Neutral pions have no electric charge, which distinguishes them from their charged counterparts. Despite their crucial role in physics, neutral pions are incredibly unstable. They are typically produced in high-energy particle collisions, such as those occurring in particle accelerators.

The mass of a π0 is about 264 times that of an electron. Electrons have a relatively small mass, known to be approximately 9.10938356×1031kg. At this mass, a π0 remains stable only for an exceedingly short amount of time, specifically about 8.4×1017s before it decays into lighter particles like gamma-ray photons. Learning about neutral pions helps us understand fundamental forces and interactions that occur in the subatomic world.
Mass-Energy Equivalence
Mass-energy equivalence is a profound principle in physics, famously encapsulated by Einstein's equation, E=mc2. This equation reveals that mass and energy are two forms of the same thing and can be converted into each other. E denotes energy, m represents mass, and c is the speed of light, which is approximately 3×108m/s. This principle is fundamental in understanding many phenomena in the universe.

In the context of the neutral pion, mass-energy equivalence allows us to connect an uncertainty in the particle's mass to its energy. When a particle like the π0 decays, its mass is converted into energy, emitting gamma-ray photons. The precise energy of these photons can vary, leading to an uncertainty in energy. Using the mass-energy equivalence concept, one can determine the uncertainty in mass from the uncertainty in energy.
Lifetime of Particles
The lifetime of particles like the neutral pion is an essential concept in particle physics. This lifetime, also known as the mean lifetime, is a measure of how long a particle exists before it decays into other particles. For a neutral pion, this lifetime is remarkably short, about 8.4×1017s. Such short lifetimes characterize many fundamental particles, especially those that are highly unstable.

The uncertainty principle, related to Heisenberg's Uncertainty Principle, tells us that the product of the uncertainty in energy ΔE and the uncertainty in time Δt is roughly equal to or greater than a constant value, 2, where is the reduced Planck's constant. This principle means that the shorter the lifetime of the particle (Δt), the greater the uncertainty in its energy ΔE.

This lifetime-uncertainty relationship is vital for understanding how experiments in particle physics measure energies and lifetimes of short-lived particles like the π0. By understanding these fleeting particles, scientists can probe the fundamental forces and rules that govern our universe.

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