Question

The Relativistic Heavy Ion Collider (RHIC) can produce colliding beams of gold nuclei with beam kinetic energy of \(A \cdot 100 .\) GeV each in the center-of- mass frame, where \(A\) is the number of nucleons in gold (197). You can approximate the mass energy of a nucleon as approximately \(1.00 \mathrm{GeV}\). What is the equivalent fixed-target beam energy in this case?

Short answer

Answer: The equivalent fixed-target beam energy for this case is 9849.5 GeV.

Step-by-step solution

Determine the total energy in the center-of-mass frame

In this step, we will determine the total energy of each colliding beam in the center-of-mass frame. We know that each colliding beam has a kinetic energy of \(A \cdot 100\) GeV and the mass energy of a nucleon is \(1.00\) GeV. Total energy of each colliding beam in the center-of-mass frame = Kinetic energy + Mass energy = \(A \cdot 100 \, GeV + 1.00 \, GeV\) Total energy of each colliding beam = \(197 \cdot 100 \, GeV + 1.00 \, GeV = 19700 \, GeV + 1.00 \, GeV = 19701 \, GeV\)

Calculate the energy in the fixed-target frame

Now, we need to convert this to the energy in the fixed-target frame. In order to do this, we have to take into account that in the fixed-target frame one of the nuclei will be at rest while the other moves with the beam's energy. We know that the total energy of each colliding beam in the center-of-mass frame is double the total energy in the fixed-target frame, as both beams contribute equally. Total energy in the fixed-target frame = \(\frac{1}{2} \cdot\) Total energy in the center-of-mass frame Total energy in the fixed-target frame = \(\frac{1}{2} \cdot 19701 \, GeV = 9850.5 \, GeV\)

Separate the fixed-target frame energy into kinetic and mass components

Now, we have to separate the fixed-target frame energy back into its kinetic and mass components. Remember that the mass energy of a nucleon is \(1.00 \, GeV\). Equivalent fixed-target beam energy = Total energy in the fixed-target frame - Mass energy Equivalent fixed-target beam energy = 9850.5 GeV - 1.00 GeV = 9849.5 GeV In conclusion, the equivalent fixed-target beam energy in this case is \(9849.5 \, GeV\).

Key concepts

Center-of-Mass Frame

Understanding the center-of-mass frame is pivotal when studying high-energy collisions, such as those in the Relativistic Heavy Ion Collider (RHIC). Simply put, the center-of-mass frame is a reference frame where the total momentum of a system is zero. When two particles collide, this frame allows us to analyze the collision symmetrically, because both particles have equal and opposite momentum.

In the context of the RHIC and similar colliders, the center-of-mass frame is often more useful than the laboratory frame because it removes the complication of dealing with differing momenta of individual particles. This symmetric view simplifies calculations and is essential for understanding the dynamics of the collision. It's equivalent to transforming the problem into one where two objects collide while sliding toward each other on an ice rink, with no net movement of the system's 'center'.

The concept is crucial for physicists, as it helps them to determine kinetic energies, collision angles, and post-collision particle paths in a way that would be enormously complicated without this frame of reference.

Fixed-Target Beam Energy

Fixed-target experiments differ from colliding beam setups, such as those at RHIC, in that one beam - usually of particles such as protons or ions - is accelerated towards a stationary target. The fixed-target beam energy is critical because it represents the amount of energy available to create new particles and initiate reactions in the target material.

In fixed-target experiments, the beam's total energy comprises its kinetic energy - the energy due to its motion - and the mass energy of the particles that make up the beam. Fixed-target beam energy is thus crucial for designing experiments and interpreting results, as well as for comparing the efficiencies of fixed-target setups against colliding beam setups like those at RHIC. It's important to note that if you're presented with center-of-mass energy, converting to the equivalent fixed-target beam energy involves accounting for the mass energy of the stationary particles and requires a deep understanding of relativistic physics.

For example, when gold nuclei are collided in RHIC, the energy in the center-of-mass frame is different from that in a fixed-target configuration. Transforming to the fixed-target energy requires knowing how to balance energy contributions in terms of both the kinetic and mass energy components.

Mass Energy of a Nucleon

The concept of mass energy is drawn from Einstein's famous equation, E=mc^2, which states that energy (E) and mass (m) are directly proportional, with the speed of light (c) as the constant of proportionality. For a nucleon, which can be a proton or neutron, the mass energy is the amount of energy contained in its mass.

It's a fundamental principle in nuclear physics and plays a crucial role in high-energy particle experiments where particles are accelerated to near-light speeds. In these cases, the mass energy becomes a significant part of the total energy of the particles. For instance, a gold nucleus at RHIC, with its approximately 197 nucleons, carries a mass energy of 197 GeV if we approximate the mass energy of each nucleon to be around 1 GeV.

Understanding nucleon mass energy is not only about knowing a static number but also about comprehending its implications in larger scale processes, such as in the energy considerations in particle accelerators. It's the reason particle collisions at high speeds in facilities like RHIC can lead to the formation of new particles and the study of states of matter under extreme conditions, like the quark-gluon plasma.