Question

(a) 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? (b) If the alpha particles instead interact in a colliding-beam experiment, what must the energy of each beam be to produce the same available energy?

Short answer

(a) 17.0 GeV; (b) 11.727 GeV per beam.

Step-by-step solution

Understanding Available Energy in a Stationary Target Collision

When an alpha particle collides with a stationary helium target, the available energy is given by the formula: \( E_{avail} = \sqrt{s} - m_{1}c^2 - m_{2}c^2 \), where \( \sqrt{s} \) is the center-of-mass energy, and \( m_1 \) and \( m_2 \) are the rest masses of the alpha particle and helium nucleus, respectively. Here, \( m_1 = 3727.38 \text{MeV/c}^2 \) and \( m_2 = 3727.38 \text{MeV/c}^2 \) because both are essentially alpha particles.

Calculating the Total Energy of the Stationary Target Collision

The given available energy \( E_{avail} = 16.0 \text{GeV} \). Thus, the total beam energy required for a stationary target is \( E = m_0c^2( \sqrt{1 + 2\cdot u} - 1) \), where \( u = E_{avail}/(2m_0c^2) \), \( m_0 = 3727.38 \text{MeV/c}^2 \), substituting the value of \( E_{avail} \), \( u = 16,000/ (2 \times 3727.38) \), we get \( u \approx 2.146 \), and solving for \( E \), we find \( E \approx 17.0 \text{GeV} \). Therefore, each alpha particle must have a total energy of approximately 17.0 GeV.

Understanding Available Energy in a Colliding-Beam Experiment

In a colliding-beam setup, both beams have the same energy \( E \), and the available energy is given by \( E_{avail} = \sqrt{4E^2/c^4} - 2m_0c^2 \). Given \( E_{avail} = 16.0 \text{GeV} \), this simplifies to solving \( 2E = 16.0 \text{GeV} + 2m_0c^2 \).

Calculating the Beam Energy in a Colliding-Beam Experiment

Solving for \( E \), we have \( E = 8.0 \text{GeV} + m_0c^2 \). Since \( m_0c^2 = 3727.38 \text{MeV} \), \( E = 8.0 \text{GeV} + 3.72738 \text{GeV} \), we obtain \( E \approx 11.727\) GeV. Therefore, in a colliding beam experiment, each beam must contain an energy of approximately 11.727 GeV.

Key concepts

Alpha Particles

Alpha particles are fascinating objects in the world of particle physics. They are essentially helium nuclei, meaning they consist of two protons and two neutrons. These charged particles are a type of ionizing radiation and are often a product of radioactive decay. They have a relatively high mass compared to other particles like electrons, which makes them unique. When studying high-energy collisions, alpha particles provide a good experimental probe because of their mass and charge. Their nature helps physicists test various theoretical predictions by observing how they behave when accelerated to high energies. Alpha particles, due to their characteristics, can interact strongly with matter, making them useful for numerous applications, including medical and energy fields.

Available Energy

Available energy in particle physics is a critical concept, especially when dealing with high-energy collisions. It refers to the amount of energy that can be converted into other forms, like the production of new particles.In the context of colliding alpha particles with a stationary helium target or with another beam, the available energy dictates what reactions are possible. Not all the total energy of the system can be used to create new particles. Some of it will be tied up in the kinetic energy of the resulting particles or remain as mass (according to Einstein's famous equation, \( E=mc^2 \)).Understanding available energy in any collision is crucial because it helps predict what sort of interactions or particles can be observed. This can lead to the discovery of new particles or validate theoretical models within particle physics.

Center-of-Mass Energy

Center-of-Mass Energy is another pivotal concept in high-energy collisions. It represents the total energy available in the center-of-mass frame of the colliding particles. Think of it as an equivalent way to evaluate the energy interactions of particles. The center-of-mass energy is especially important because it provides a clean comparison of energy interactions without having to worry about the motion of the system as a whole. In the exercise context, when an alpha particle hits a stationary target, calculating the center-of-mass energy allows students to determine how much energy remains after accounting for the rest mass energies of the particles involved. In colliding beam experiments, the center-of-mass energy is maximized, as the particles move towards each other. This means more energy is available for the interesting stuff: particle creation and interactions!

Colliding-Beam Experiment

Colliding-beam experiments are a staple in experimental particle physics. In these setups, two beams of particles are accelerated towards each other and collide head-on. The advantage of this configuration is that it maximizes the available energy for particle interactions. In these experiments, unlike fixed target setups, almost all the kinetic energy in the beams can be used, as the frame of reference is the center of mass itself. This makes colliding-beam experiments particularly powerful for investigating high-energy phenomena and discovering new particles. Such experiments are employed in some of the world's most renowned accelerators, like the Large Hadron Collider (LHC). The goal is often to reach a high center-of-mass energy to uncover ever more fundamental layers of matter.