Bubble Chamber Lesson 5 Interaction Creating Additional Particles
Teacher Introduction: This lesson introduced the relationship E = mc^{2} using a bubble chamber interaction in which four particles emerge from an interaction between a K^{} and a proton.
Objectives:
Students build on understanding of collisions and decays in a bubble chamber and apply physics principles to a more complex interaction.
Students use E = mc^{2 }to explain the appearance of additional particles in an interaction; students apply conservation laws to determine which interactions are allowed.
The Interaction: link to 2156
This is the same interaction used in Lesson 3. This shows the following:
K^{ }p K^{ }p π^{ }π^{+}
Introduction: Begin with a drawing (see above) based on the photo of the reaction. Ask students to tell the story that this shows. They should recognize that a collision takes place and one of the resulting particles decays into a charged plus a neutral particle. But now we have a new phenomenon, the production of additional particles. Is this allowed?
How could an interaction between two particles produce four particles?
Teaching Strategy:
If conservation laws are not discussed, the students should be asked whether charge, energy and momentum appear to be conserved.

Are there any conservations laws or other physics principles violated in the initial collision?
This can be a test for misconceptions students have about what limits conservations place on these reactions. No conservation law is obviously violated (at least not that we can tell from the picture).
Write: E = mc^{2}
Explain what this means. Students are probably aware that in a nuclear bomb or nuclear power plant, this relationship tells us that a small amount of mass can be converted into a large amount of energy.
So – mass is not conserved it can be converted into energy! This is not consistent with the Newtonian model of mass and motion. In fact, the Newtonian model applies only to objects travelling at speeds that are a fraction of the speed of light. Virtually everything we encounter in normal activities meet this criterion – but beam particles shot in to a bubble chamber travel at speeds near c.
Does it work the other way? (Yes, that is what the “=” means) So an amount of energy, E, can be converted to an amount of mass, m = E/ c^{2}. Or, more strictly, mc^{2 }is an amount of energy.
We include mc^{2} for each particle as part of the energy available in each interaction. This need only be considered for objects travelling at speeds approaching the speed of light (we refer to these as “relativistic speeds”)
The relevant conservation laws are, therefore, conservation of energy, momentum and charge.

How is this “mass energy” included in the conservation of energy calculation?
The total energy of a kaon beam particle travelling at relativistic speeds consists of its kinetic energy plus its mc^{2} energy where m is the mass of the kaon. (Really the kaon at rest in its reference frame but let’s not get into that here.)
For the kaon, mc^{2} = 0.497 GeV. The total energy of kaon beam is 4.2 GeV.
So, what is the kinetic energy of the kaon in the beam? What percentage is this of the total energy?
In the interaction shown, a 4.2 GeV K^{} strikes a proton. For the proton,
mc^{2 }= 0.938 GeV.
What is the total energy available for the resulting particles? Ans: 4.2 GeV + 0.938 GeV = 5.138 GeV.
Additional particles can be created as long as the total energy, including mc^{2} equals the total energy of the interacting particles.

Is it possible for all the energy to be converted to mc^{2} energy in an interaction?
Consider conservation of momentum. The momentum of the original beam particle must be conserved. So, in an interaction with a stationary proton, the resulting particles must have some momentum; therefore the resulting particles must have a finite velocity.
Conclusion: What does the discussion of E = mc^{2}add to your understanding of the interaction shown? How is this type of interaction possible? Could these interactions occur K^{ }p K^{ }p K^{ }K^{+ }
K^{ }p K^{ }p π^{ }π^{+ }
Explain why or why not. What could you say about the kinetic energy of the resulting particles compared to the intial K^{ }in each case?
Compare the interaction given
Exercises:
1. For the interaction shown in the photograph:

find the point where a charged particle decays

What charge does the decaying particle have?

What is the charge of the particle emitted from the decay?

Was there another charge emitted from the decay? How can you tell?

Draw an arrow showing the approximate direction of the missing particle – justify your answer.

The following interaction is observed to take place in a hydrogen bubble chamber:
K^{ }p K^{ }p π^{ }π^{+}
The mc^{2 }for the π^{ }and the π^{+} is 0.134 GeV.

What is the total mc^{2 }energy of the resulting particles?

What is the total kinetic energy of the resulting particles?

Draw a diagram of the bubble chamber interaction:
a. K^{ }p four particles
Explain your reasoning for the paths shown and indicate (qualitatively) how momentum, energy and charge conservation are obeyed. State the assumed direction of the magnetic field.

Assume that any one of the resulting particles in your diagram decays into one
neutral and one charged particle  show the resulting path on your diagram.
