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Ch 44: Particle Physics and Cosmology
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 44: Particle Physics and CosmologyProblem 11a
Chapter 43, Problem 11a

A high-energy beam of alpha particles collides with a stationary helium gas target. What must the total energy of a beam particle be if the available energy in the collision is 16.016.0 GeV?

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Understand the concept of available energy in a collision: The available energy is the energy in the center-of-mass (COM) frame that can be used for particle interactions. For a collision between a moving particle and a stationary target, the relationship between the total energy in the lab frame and the available energy in the COM frame is key.
Use the formula for available energy in the COM frame: The available energy \( E_{\text{available}} \) is given by \( E_{\text{available}} = \sqrt{2m_1c^2E_1 + (m_1c^2)^2 + (m_2c^2)^2} \), where \( m_1 \) and \( m_2 \) are the masses of the two particles, \( E_1 \) is the total energy of the moving particle in the lab frame, and \( c \) is the speed of light.
Identify the masses of the particles: The alpha particle has a mass \( m_1 = 4.0026 \, \text{u} \) (atomic mass units), and the helium nucleus (stationary target) has a mass \( m_2 = 4.0026 \; \text{u} \). Convert these masses to energy units using \( 1 \; \text{u} = 931.5 \; \text{MeV}/c^2 \).
Rearrange the formula to solve for \( E_1 \): Substitute \( E_{\text{available}} = 16.0 \; \text{GeV} \) and the converted masses into the formula. Rearrange to isolate \( E_1 \), the total energy of the alpha particle in the lab frame.
Perform the necessary algebraic manipulations: Simplify the equation step by step, ensuring all terms are correctly handled. This will yield the required total energy \( E_1 \) of the alpha particle in the lab frame.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Alpha Particles

Alpha particles are positively charged particles consisting of two protons and two neutrons, essentially a helium nucleus. They are emitted during radioactive decay and have significant mass compared to other subatomic particles. Understanding their properties, such as charge and mass, is crucial for analyzing their interactions with other matter, particularly in high-energy collisions.
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Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant over time. In the context of particle collisions, this means that the kinetic energy of the incoming particles and any potential energy must equal the energy of the products after the collision. This principle is essential for calculating the required energy of the alpha particles to achieve a specific outcome in the collision.
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Center of Mass Energy

Center of mass energy is the total energy available for particle interactions in the center of mass frame, where the total momentum is zero. It is a critical concept in high-energy physics, as it determines the threshold energy needed for reactions to occur. In collisions involving stationary targets, the energy of the incoming particles must be sufficient to account for the rest mass energy of the target particles and any additional energy required for the reaction.
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Related Practice
Textbook Question

Calculate the minimum beam energy in a proton-proton collider to initiate the p+p→p+p+η0p + p → p + p + η^0 reaction. The rest energy of the η0\(\eta\)^0 is 547.3547.3 MeV (see Table 44.344.3).

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Textbook Question

Deuterons in a cyclotron travel in a circle with radius 32.032.0 cm just before emerging from the dees. The frequency of the applied alternating voltage is 9.009.00 MHz. Find the magnetic field.

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Textbook Question

You work for a start-up company that is planning to use antiproton annihilation to produce radioactive isotopes for medical applications. One way to produce antiprotons is by the reaction p+p→p+p+p+pˉp + p → p + p + p + p̄ in proton-proton collisions. You first consider a colliding-beam experiment in which the two proton beams have equal kinetic energies. To produce an antiproton via this reaction, what is the required minimum kinetic energy of the protons in each beam?

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Textbook Question

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon if the pp and p‾\(\overline{p}\) collide head-on, each with an initial kinetic energy of 620620 MeV.

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Textbook Question

What is the speed of a proton that has total energy 10001000 GeV?

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Textbook Question

An electron with a total energy of 30.030.0 GeV collides with a stationary positron. What is the available energy?

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