Ch 44: Particle Physics and Cosmology

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 44: Particle Physics and Cosmology

Problem 11a

Chapter 44, 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 $16.0$ 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.

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.

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 $p + p \to p + p + η^{0}$ reaction. The rest energy of the $eta to the 0 power$ is $547.3$ MeV (see Table $44.3$ ).

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Textbook Question

In Example $44.3$ , it was shown that a proton beam with an $800$ -GeV beam energy gives an available energy of $38.7$ GeV for collisions with a stationary proton target. In a colliding-beam experiment, what total energy of each beam is needed to give an available energy of $2 open paren 38.7$ GeV $close paren equals 77.4$ GeV?

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Textbook Question

Deuterons in a cyclotron travel in a circle with radius $32.0$ cm just before emerging from the dees. The frequency of the applied alternating voltage is $9.00$ MHz. Find the magnetic field.

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Textbook Question

What is the speed of a proton that has total energy $1000$ GeV?

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Textbook Question

The magnetic field in a cyclotron that accelerates protons is $1.70$ T. How many times per second should the potential across the dees reverse? (This is twice the frequency of the circulating protons.)

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Textbook Question

An electron with a total energy of $30.0$ GeV collides with a stationary positron. What is the available energy?

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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?

Verified video answer for a similar problem:

Key Concepts

Alpha Particles

Conservation of Energy

Center of Mass Energy

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 $16.0$ GeV?