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?

Verified step by step guidance

1. 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.
2. 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.
3. 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 \( 1 \; \text{u} = 931.5 \; \text{MeV}/c^2 \).
4. 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.
5. 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.

Key Concepts

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.