Problem 45 - University Physics with Modern Physics

Question

In the LHC, each proton will be accelerated to a kinetic energy of 7.0 \(\mathrm{TeV}\) (a) In the colliding beams, what is the available energy \(E_{\mathrm{a}}\) in a collision? (b) In a fixed-target experiment in which a beam of protons is incident on a stationary proton target, what must the total energy (in TeV) of the particles in the beam be to produce the same available energy as in part (a)?

Step-by-Step Solution

Answer

(a) 14 TeV, (b) approximately 105 TeV.

Step 1: Understanding Part (a)

In part (a), the protons in the Large Hadron Collider (LHC) are moving towards each other, each with a kinetic energy of 7 TeV. Since they are head-on colliding, the center of mass energy is simply the sum of their energies. Therefore, the available energy \( E_a \) in a collision is given by:\[ E_a = E_p + E_p = 2E_p \]where \( E_p = 7 \text{ TeV} \).

Step 2: Calculating Available Energy in Part (a)

Substitute the value of \( E_p \) into the equation from Step 1:\[ E_a = 2 \times 7 \text{ TeV} = 14 \text{ TeV} \]

Step 3: Understanding Part (b)

In a fixed-target experiment, a moving proton beam collides with a stationary proton. To find the beam energy necessary to achieve the same available energy of 14 TeV from part (a), we use the invariant mass equation for a system of two particles. The invariant mass \( s \) is given by:\[ s = (E_{beam} + m_p c^2)^2 - (p_{beam} c)^2 \]where \( m_p \) is the rest mass energy of the proton (0.938 GeV) and \( p_{beam} \) is the momentum of the moving proton, and \(c\) is the speed of light.

Step 4: Relate Invariant Mass to Available Energy

The available energy \( E_a \) in a collision is related to the invariant mass \( s \) as follows:\[ E_a = \sqrt{s} \]We require \( \sqrt{s} = 14 \text{ TeV} \).

Step 5: Simplify Equation for Total Energy of Beam

Using the mass-energy-momentum relation, we know:\[ s = (E_{beam} + m_p c^2)^2 - (p_{beam} c)^2 = 2 m_p c^2 E_{beam} + 2 (m_p c^2)^2 \]Substituting \( m_p c^2 = 0.938 \text{ GeV} \) and \( \sqrt{s} = 14 \text{ TeV} \), solve for \( E_{beam} \):\[ 14^2 = 2 \times 0.938 \times E_{beam} + 2 \times (0.938)^2 \]

Step 6: Solve Equation for Beam Energy

We need to solve the equation obtained in Step 5:\[ 196 = 2 \times 0.938 \times E_{beam} + 2 \times (0.938)^2 \approx 2 \times 0.938 \times E_{beam} \]Solving for \( E_{beam} \), we get:\[ E_{beam} = \frac{196}{2 \times 0.938} \approx 104.6 \text{ TeV} \]

Key Concepts

Center of Mass Energy

In a particle collision, the center of mass energy is a crucial concept that represents the total energy available to create new particles. In the context of the Large Hadron Collider (LHC), protons are accelerated to extremely high energies and collide head-on.
This configuration maximizes the available energy due to the symmetry and the efficient use of the kinetic energy possessed by all colliding particles.

• For protons each with 7 TeV of kinetic energy, the center of mass energy is the sum of these energies.
• This results in a total available energy of 14 TeV.

This energy is used in collision experiments to look for new particles and to understand the fundamental forces of nature.

Fixed-Target Experiment

Unlike colliding-beam configurations, a fixed-target experiment involves a beam of particles incident on a stationary target. This setup is less energy-efficient compared to a head-on collision in colliders like the LHC.
To match the 14 TeV available energy seen in part (a) with this method, the beam must have a substantially higher energy.

• The colliding beam allows equal sharing of energy and momentum.
• In fixed-target setups, the stationary target limits the energy transfer efficiency.

This inefficiency arises because a significant portion of the projectile's energy is used to propel the stationary mass rather than converting into the usable energy of the interaction.

Available Energy in Collisions

Available energy is the portion of the total energy in a collision that can be used to produce new particles. It depends heavily on the collision configuration, particularly distinguishing between collisions at the LHC and fixed-target experiments.
For LHC-style collisions:

• The available energy is nearly equal to the center of mass energy since both particles are moving.
• All kinetic energy is effectively utilized for particle creation.

In contrast, for fixed-target experiments, only a fraction of the total energy appears as available energy due to the stationary nature of one mass. The equation for the invariant mass, \( s = (E_{beam} + m_p c^2)^2 - (p_{beam} c)^2 \\), helps show how energy is utilized and needs to be calculated carefully to achieve desired collision outcomes.

Proton-Proton Collisions

Proton-proton collisions are a focal point in high-energy physics, especially in experiments conducted at the LHC. These collisions are utilized to explore the behavior of fundamental particles under extreme conditions.
The high energy involved enables interactions that can reveal new physics beyond the Standard Model, such as the discovery of new particles.

• The collision energy permits probing deep into the structure of protons.
• Complex interactions within these collisions often produce rare particles.

Understanding proton-proton collisions helps explore unexplained phenomena in particle physics, leading to potential groundbreaking discoveries.