For the reaction p+p →3 p+p̅, where one of the initial...

Key Concepts

Center-of-Mass Frame

The center-of-mass frame is a reference frame in which the total momentum of a system is zero. Analyzing particle reactions in this frame often simplifies the calculations because the energy available for particle production is directly related to the invariant mass of the system. At threshold, all produced particles move collectively, allowing the entire available energy to be used to create their rest mass.

Energy and Momentum Conservation

Energy and momentum conservation are fundamental principles in physics stating that the total energy and total momentum of an isolated system remain constant over time. In the context of relativistic particle reactions, these quantities are treated as components of the four-momentum, which combines energy and momentum into a single entity that is conserved in all inertial reference frames.

Reaction Q-value

The Q-value of a reaction represents the net change in the rest mass energy between the initial and final states. It indicates whether energy is absorbed (endothermic) or released (exothermic) during the reaction. In threshold reactions, the Q-value is directly related to the energy required to produce additional particles, making it a critical parameter in determining the threshold energy necessary for the reaction to occur.

Relativistic Kinematics

Relativistic kinematics is the framework used to describe the motion and energy of particles when they travel at speeds close to the speed of light. It replaces classical mechanics by incorporating Einstein’s theory of special relativity, ensuring that calculations include the mass-energy equivalence (E=mc^2) and properly account for time dilation and length contraction effects. This is crucial in high-energy particle reactions where rest mass and kinetic energies are comparable.

Threshold Energy

Threshold energy refers to the minimum amount of energy that the incoming particles must have in order to produce a specific reaction, ensuring that all conservation laws (such as energy and momentum conservation) are satisfied. In particle physics, it is the energy at which the reaction can occur with the final particles just being produced, often assumed to have no kinetic energy relative to one another in a chosen reference frame.

Transcript

00:01 Okay, so problem number 60, we're given the reaction that a proton collides with a proton to form three protons plus a anti -proton.

00:14 Now, it says one of these protons are at rest, and we want to show that the energy threshold is equal to six times the mass energy of the proton, where the energy threshold is basically going to.

00:30 Going to, that's just kind of the minimum energy that this proton needs to form that.

00:37 So let's go ahead and start.

00:40 We're given, again, from like in the previous problem, we're given that all final particles have the same velocity.

00:48 So from that, we can treat all of these four particles as one giant mass.

00:56 We'll set as big m, which is equal to 4 times the mass of the proton, since an antiproton is same mass as a regular proton.

01:08 So now we want to figure out the q, the q value.

01:13 So we have, i'll come over here, so i give myself some more room.

01:16 So q is going to be equal to 2 times the mass of the proton c squared, minus 4 times the mass of the proton c squared, which is going to be equal to minus 2.

01:33 I'm not going to do that yet.

01:34 I'm going to do it in terms of big m.

01:37 So this is going to be equal to, sorry, two times mass of proton c squared minus big m, c squared, which is going to be equal to minus two times the mass of proton c squared.

01:57 All right.

01:58 So that's the minimum energy, basically, total energy.

02:03 So now we want to find the threshold energy.

02:10 So we're going to give the threshold energy of the proton that we're going to give it the term, so we're going to say k -t -h is the threshold energy of the proton, and now the total energy of the proton is going to be equal to k -th plus the mass of the proton c -square.

02:34 So this is the total energy of the proton.

02:38 So we're going to use conservation of energy to show we need, or using conservation energy, it basically states that the total energy on this side has to equal the total energy on that side.

02:48 So that gives us trying to save room here because there's kind of a bit of map in this.

02:57 So i'll put it here.

02:58 So the threshold energy plus mpc squared plus mpc squared is equal to the total, the kinetic energy of this mass, told the big m plus mc squared.

03:20 And then, so also from conservation and momentum, momentum has to be equal on both sides.

03:28 So the momentum is being carried by the proton that's doing, that has the energy, the threshold energy.

03:34 So that's going to be equal to the total momentum of the particles on the other side.

03:37 So we can state that p, the momentum of the proton, is equal to the momentum of big m.

03:53 And then in putting this in form of, we can then state that p, c, so the proton momentum c squared is equal to the big mass momentum c squared, which using the form of the relativistic.

04:14 Kinetic energy, eralificic energy, e squared is equal to p c squared plus m0c squared c fourth.

04:23 We can put that into this.

04:27 So that's what p c squared is equal to e squared minus m1 squared c fourth.

04:36 So we can plug that in knowing that we have ep squared here.

04:45 The total energy of the proton is given by this, and the total energy of the mass is given by this.

04:56 So we can plug that in on both sides, and we can state that here.

05:03 We have, excuse me, k -t -h plus m -p -c -squared -squared, minus mpc squared, squared to the fourth, sorry.

05:30 So the fourth is equal to plug in this term over here, the kinetic energy of the mass plus its mass energy, squared minus big m squared, squared, c to the fourth.

05:47 All right.

05:51 Now we have to expand both sides, so we get k, 2k -t -h squared plus 2k -t -h mpc squared plus mp to the fourth mp squared c to the fourth minus mp squared c to the fourth is equal to total momentum of the combined mass squared plus 2 kmn c squared plus m squared plus m squared c4th minus m squared c4th okay so quick simplification those go away we end up with k t h squared plus two k t h m mass of the proton, c squared, is equal to the total mass, the total kinetic energy, m squared, plus 2km, mc squared.

07:04 Okay, so now we are going to put this in another form here.

07:16 So give me a second here to get my bearings with the math...

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