Question

A gamma-ray photon changes into a proton-antiproton pair. Ignoring momentum conservation, what must have been the wavelength of the photon (a) if the pair is stationary after creation, and (b) if each moves off at \(0.6 c\) perpendicular to the motion of the photon? (c) Assume that these interactions occur as the photon encounters a lead plate and that a lead nucleus participates in momentum conservation. In each case, what fraction of the photon's energy must be absorbed by a lead nucleus?

Short answer

For the scenario when the created pair is stationary, the wavelength of the gamma-ray photon should be \(1.32 * 10^{-15} m\) and the fraction of photon’s energy absorbed by the lead nucleus is 0. For the scenario when both move off at \(0.6 c\) perpendicular to the motion of the gamma-ray photon, the wavelength should be \(8.43 * 10^{-16} m\) and the fraction of photon’s energy absorbed by the lead nucleus is about 0.5.

Step-by-step solution

Calculate the energy of a photon

First, calculate the energy of a photon that transforms into a proton-antiproton pair using the following equation, \(E = 2m_{proton}c^2\). Here, \(m_{proton}\) represents the mass of a proton and equals \(1.67*10^{-27} kg\), \(c\) represents the speed of light and equals \(3*10^8 m/s\).

Calculate the wavelength of a photon

Now, input the photon’s energy calculated in step 1 into the Planck-Einstein equation \(E = h c / \lambda\) to calculate the wavelength of a photon for both scenarios. Here, ‘h’ represents the Planck constant and equals \(6.63*10^{-34} Js\).

Calculate the fraction of absorbed photon's energy

To calculate the fraction of a photon’s energy absorbed by a lead nucleus, identify the total initial energy before interaction (Energy of a photon) and the total final energy after interaction (Energy of proton-antiproton pair and energy of lead nucleus, which equals 0 in scenario 1), and then use the equation \((E_{initial}- E_{final}) / E_{initial}\).

Key concepts

Energy Conservation in Particle Physics

Understanding energy conservation in particle physics is essential for comprehending how particles interact and transform. The law of energy conservation states that the total energy in an isolated system remains constant over time. In the context of a photon transforming into a proton-antiproton pair, this principle dictates that the photon's energy before the transformation must equal the total energy of the newly created particles.

When considering the exercise scenario, a gamma-ray photon, which possesses a certain amount of energy characterized by its wavelength, converts into a proton and an antiproton. If these particles are stationary after creation (scenario a), all the photon's energy is utilized in creating the rest mass energy of the proton-antiproton pair. For the situation where the particles move at a speed (scenario b), the total energy also includes the kinetic energy of the particles. This scenario demonstrates a beautiful aspect of energy conservation; not only must the rest mass energy be accounted for, but also any additional kinetic energy imparted to the particles post-creation.

In practice, the conservation of energy in particle physics allows us to perform critical calculations. By knowing the proton's mass and recognizing that the total initial energy (from the photon) must equal the total energy of the produced particles plus any absorbed energy by other objects (like the lead nucleus), students can understand how to calculate the photon's wavelength. This concept forms the bedrock of energy calculations in particle physics and ensures that, regardless of the complex processes involved, energy is neither created nor lost; it merely changes form.

Proton-Antiproton Pair Creation

Proton-antiproton pair creation is a process inherent to high-energy physics, showcasing particles coming into existence from pure energy, a phenomenon predicted by Einstein's famous equation, E=mc². In our exercise, this concept is illustrated by the conversion of a gamma-ray photon into a proton and an antiproton. It’s important to understand that in the universe of quantum mechanics and particle physics, particles and antiparticles are regularly created and annihilated.

For a proton-antiproton pair to be created, the incoming gamma-ray photon must possess enough energy to account for the mass of the two particles. This requirement is because protons and their antiparticles have a property called 'rest mass', and creating them from a photon - which is massless - requires an equivalent energy conversion. The rest mass energy of a single proton is given by the expression: \(E = m_{proton}c^2\). Because we are creating a pair, we must consider twice this energy for the process – hence the equation: \(E = 2m_{proton}c^2\) used in our solution.

The creation of particles from photons is a critical area of study in high-energy astrophysics and can occur naturally, such as in cosmic rays or artificially in particle accelerators. Learning about this process helps students appreciate the profound connections between light (photons), matter (protons), and antimatter (antiprotons), deepening their understanding of particle interactions and the fabric of the universe.

Planck-Einstein Equation

The Planck-Einstein equation is a cornerstone in the field of quantum mechanics, linking a photon's energy to its wavelength. This equation is expressed as \(E = h c / \lambda\), where:\(E\) is the energy of the photon,
\(h\) is the Planck constant (\(6.63*10^{-34} Js\)),
\(c\) is the speed of light (\(3*10^8 m/s\)), and
\(\lambda\) is the wavelength of the photon.

This equation is integral in solving the textbook example because it allows us to back-calculate the wavelength of the photon that transformed into the proton-antiproton pair. By applying the principle that the energy of a photon corresponds to the energy required for particle creation, we can see the direct relationship between wavelength and energy: a shorter wavelength means a higher energy photon and vice versa.

The use of this equation enables students to work with real values associated with subatomic particles and photons. By understanding how wavelength affects energy, students can grasp why certain photons have enough energy to create particle pairs while others do not. It also illustrates the fundamental concept that light has both wave-like and particle-like properties, a phenomenon essential to the dual nature of light accepted in modern physics.