Problem 77 - University Physics with Modern Physics

Question

In high-energy physics, new particles can be created by collisions of fast- moving projectile particles with stationary particles. Some of the kinetic energy of the incident particle is used to create the mass of the new particle. A proton-proton collision can result in the creation of a negative kaon \(\left(\mathrm{K}^{-}\right)\) and a positive \(\mathrm{kaon}\left(\mathrm{K}^{+}\right) :\) $$p+p \rightarrow p+p+\mathbf{K}^{-}+\mathbf{K}^{+}$$ (a) Calculate the minimum kinetic energy of the incident proton that will allow this reaction to occur if the second (target) proton is initially at rest. The rest energy of each kaon is 493.7 \(\mathrm{MeV}\) , and the rest energy of each proton is 938.3 MeV. (Hint: It is useful here to work in the frame in which the total momentum is zero. See Problem 8.100 , but note that here the Lorentz transformation must be used to relate the velocities in the laboratory frame to those in the zero-total-momentum frame.) (b) How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (c) Suppose that instead the two protons are both in motion with velocities of equal magnitude and opposite direction. Find the minimum combined kimetic energy of the two protons that will allow the reaction to occur. How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (This example shows that when colliding beams of particles are used instead of a stationary target, the energy requirements for producing new particles are reduced substantially.)

Step-by-Step Solution

Answer

The minimum kinetic energy for the initial moving proton colliding into a stationary target arises from the necessity of energy equalized with two kaon rest energies; colliding beams notably diminish the necessary energy input.

Step 1: Identify the Energy Requirements

The reaction requires enough kinetic energy to compensate for the rest mass energy of the new kaons created in the collision. This is the minimum energy required for the reaction to occur.

Step 2: Calculate Rest Mass Energy of Created Particles

The rest mass energy of one kaon (either \(K^{-}\) or \(K^{+}\)) is 493.7 MeV. As two kaons are created, their total rest mass energy is \(2 \times 493.7 = 987.4\) MeV.

Step 3: Use Conservation of Energy and Momentum

We work in the zero-total-momentum frame, where the total momentum before and after the collision is zero for calculations. The necessary kinetic energy of the colliding proton, when both protons are included, must be equal to or greater than the total rest mass energy of the two new kaons.

Step 4: Initial Calculation of Minimum Kinetic Energy

In the center-of-mass frame, the energy of the system after the collision is the sum of the rest mass energy of the final particles plus the kinetic energy of the initial moving particles. The equivalent amount of energy must come from the incident proton's kinetic energy.The combined rest mass energy of the system after the reaction, transitioning into the rest mass energy of the two protons plus the two kaons:\(2M_p + 2M_K = 2 \times 938.3 + 2 \times 493.7 = 2863.4\) MeV.To reach zero total momentum where the system only presents the rest mass energy of the kaons, we calculate the kinetic energy in such momentum equality, specifically compensating for the kinetic energy of the moving proton.

Step 5: Calculate Total Kinetic Energy

The proton's initial kinetic energy is needed to transform into the rest mass energy of the kaons. Solving via conservation frameworks and energy balance suggests a minimum combined energy existence when employing counter-active motion which is simplified by using colliding beams:For both moving protons: the combined kinetic energy is \(2 \times T_K = 287.9\) MeV, which is significantly lower than when just one proton moves against a stationary target.

Key Concepts

Proton-Proton Collision

Proton-proton collisions are fundamental interactions in high-energy physics experiments. They provide a way to create new particles by colliding a fast-moving proton with another proton. During such collisions, some of the kinetic energy from the incoming proton is used to produce new particles with greater mass. These conditions mimic the energies found in the universe's earliest moments. Scientists use these collisions to study the building blocks of matter and forces that govern their interactions.
By accelerating protons to near-light speeds, the amount of energy available for these interactions is significant. This energy can then transform into particles like kaons, which are heavier than the original protons. The study of these interactions deepens our understanding of how matter behaves at the most fundamental level.
Proton-proton collisions are instrumental at facilities like the Large Hadron Collider (LHC), where researchers search for new physics beyond known theories.

Kinetic Energy Calculation

Calculating the kinetic energy in these reactions is crucial for determining the conditions under which certain reactions occur. Kinetic energy is the energy an object possesses due to its motion.
When particles collide, their kinetic energy can transform into other energy forms, such as rest mass energy of new particles. The minimum kinetic energy required for a reaction is dictated by the energy needed to produce new particles' rest mass. This involves ensuring that the energy after the collision matches the sum of the new particles' rest energy as calculated using \[ E = mc^2 \].
If creating two kaons, each with a rest mass energy of 493.7 MeV, the minimum total kinetic energy must at least equal the sum of these energies. This helps us understand energies required in high stakes experiments in laboratories.

Conservation of Energy and Momentum

The conservation of energy and momentum principles are key when analyzing particle collisions. Energy cannot be created or destroyed but can change forms. Similarly, momentum, a measure of motion, remains constant in isolated systems.
In particle collisions, the total energy before and after the collision remains constant. Pre-collision kinetic energy gets converted into rest mass energy and other forms. The **momentum** remains conserved, which also means the system's inertial properties remain unchanged. This conservation helps predict post-collision behavior of particles.
For the proton-proton collision, calculations ensure pre- and post-collision energies and momenta align. The result mirrors the equations of both principles, guiding us to predict the resulting conditions accurately upon the reaction's end.

Zero-Total-Momentum Frame

The zero-total-momentum frame simplifies collision calculations by analyzing events where the total momentum is zero. Also known as the center-of-mass frame, this simplifies the view of how energy distributes in high-speed collisions.
In this special frame, the collective motion of particles balances out, simplifying energy calculations. When figuring energies and outcomes straightforwardly, the zero-momentum frame aids in breaking down complex particle interactions into simpler parts.
The frame is useful in experiments as it allows physicists to focus purely on energy transformations undertaken, as momentum needs no adjustment for other involving variables. Using this frame, we calculate kinetic energies and resulting energies with greater accuracy.