A particle of mass m1 is fired at a stationary particle of...

A particle of mass $m_{1}$ is fired at a stationary particle of mass $m_{2},$ and a reaction takes place in which new particles are created out of the incident kinetic energy. Taken together, the product particles have total mass $m_{3} .$ The minimum kinetic energy the bombarding particle must have so as to induce the reaction is called the threshold energy. At this energy, the kinetic energy of the products is a minimum, so the fraction of the incident kinetic energy that is available to create new particles is a maximum. This condition is met when all the product particles have the same velocity and the particles have no kinetic energy of motion relative to one another. (a) By using conservation of relativistic energy and momentum, and the relativistic energy-momentum relation, show that the threshold energy is $$K_{\min }=\frac{\left[m_{3}^{2}-\left(m_{1}+m_{2}\right)^{2}\right] c^{2}}{2 m_{2}}$$ Calculate the threshold energy for each of the following reactions: (b) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{p}+\overline{\mathrm{p}}$ (One of the initial protons is at rest. Antiprotons are produced.) (c) $\pi^{-}+$ $\mathrm{p} \rightarrow \mathrm{K}^{0}+\Lambda^{0}$ (The proton is at rest. Strange particles areproduced.) (d) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\pi^{0}$ (One of the initial protons is at rest. Pions are produced.) (e) $\mathrm{p}+\overline{\mathrm{p}} \rightarrow \mathrm{Z}^{0}$ (One of the initial particles is at rest. $\mathrm{Z}^{0}$ particles of mass 91.2 $\mathrm{GeV} / c^{2}$ are produced.)

A particle of mass $m_{1}$ is fired at a stationary particle of mass $m_{2},$ and a reaction takes place in which new particles are created out of the incident kinetic energy. Taken together, the product particles have total mass $m_{3} .$ The minimum kinetic energy the bombarding particle must have so as to induce the reaction is called the threshold energy. At this energy, the kinetic energy of the products is a minimum, so the fraction of the incident kinetic energy that is available to create new particles is a maximum. This condition is met when all the product particles have the same velocity and the particles have no kinetic energy of motion relative to one another. (a) By using conservation of relativistic energy and momentum, and the relativistic energy-momentum relation, show that the threshold energy is
$$K_{\min }=\frac{\left[m_{3}^{2}-\left(m_{1}+m_{2}\right)^{2}\right] c^{2}}{2 m_{2}}$$
Calculate the threshold energy for each of the following reactions: (b) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{p}+\overline{\mathrm{p}}$ (One of the initial
protons is at rest. Antiprotons are produced.) (c) $\pi^{-}+$ $\mathrm{p} \rightarrow \mathrm{K}^{0}+\Lambda^{0}$ (The proton is at rest. Strange particles areproduced.) (d) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\pi^{0}$ (One of the initial protons is at rest. Pions are produced.) (e) $\mathrm{p}+\overline{\mathrm{p}} \rightarrow \mathrm{Z}^{0}$ (One of the initial particles is at rest. $\mathrm{Z}^{0}$ particles of mass 91.2 $\mathrm{GeV} / c^{2}$ are produced.)

Key Concepts

Relativistic Energy-Momentum Relation

Expressed as E² = (pc)² + (mc²)², this relation links a particle’s energy, momentum, and mass in a frame-invariant way. It is fundamental for analyzing high-energy collisions because it allows one to connect the energy and momentum of particles, regardless of the reference frame, and is essential in deriving threshold conditions.

Threshold Energy Concept

The threshold energy is defined as the minimum kinetic energy that must be supplied to a reaction (via a projectile, for instance) so that it can produce new particles. At threshold, the produced particles have the same velocity (and thus minimal kinetic energy in their center-of-mass frame), ensuring that the energy is used most efficiently to create mass rather than impart kinetic energy.

Conservation of Relativistic Energy

In any particle reaction, the total energy—including both the rest energy (due to mass) and the kinetic energy—is conserved. This concept underpins the analysis of high?energy collisions where energy is transformed between kinetic and mass forms, ensuring that the total energy before the interaction equals the total energy after the reaction.

Conservation of Relativistic Momentum

Alongside energy, momentum is conserved in all particle interactions. In the relativistic regime, momentum conservation must include contributions from both moving mass and energy, and it plays a crucial role in determining the kinematics of the reaction, especially when using the energy-momentum relation.

Mass-Energy Equivalence

The principle that mass can be converted into energy and vice versa, encapsulated in the equation E=mc², is crucial in particle physics. In particle reactions, some of the kinetic energy of the initial particles is converted into the rest mass of new particles, and mass-energy equivalence provides the quantitative framework to calculate the required energy for these processes.

Transcript

00:01 To obtain the minimum kinetic energy, we're actually going to obtain the total minimum energy of the incoming particle.

00:08 So that is going to be denoted by e -min, and this once again is the total energy of the incoming particle, which then interacts with a particle at rest and produces a bunch of particles.

00:25 Now, we have, by the conservation of energy, first of all, e -min plus the rest energy of the second particle m2c squared, which is the particle at rest, equaling the relativistic energy of all the other particles that get produced.

00:46 Together, they move with the same velocity, they all do.

00:50 They basically act like a single particle of momentum p3 and mass m3.

00:58 And so we basically then just have this equation.

01:04 And the only particle initially moving is the incoming particle, which we denote its momentum by p1, and we denote the momentum, the overall momentum of all the produced particles as p3.

01:17 So by conservation momentum, p1 is equal to p3.

01:21 And so what we then do is we go on and we note that p3c squared is equal to p1c squared from conservation momentum, and by the relativistic energy equation for the first particle, p1c squared is going to equal e -min squared minus m -1 -c -squared squared.

01:45 So this equation, the relativistic energy equation for the first particle gives us this equaling this right here.

01:56 But from conservation momentum, we have these two momenta equaling each other.

02:02 So now what we can do is we can substitute this quantity right here for p3c squared in this equation right up here.

02:13 So we go ahead and we do that, and we get the following equation.

02:22 So we now have e -min plus m -2c -squared equals e -min squared minus m1, rather the square root of of emin squared minus m1 c squared squared plus m3 c squared squared.

02:43 Now we can square both sides of that equation, and that should be an e min squared.

02:51 But we're going to square, we square both sides of that equation, and this is what we get.

03:05 And the e min squared terms cancel, and we now have 2m2c squared times e min is equal to m3c squared squared, which is right here, minus m1c squared minus m2c squared squared squared.

03:28 And what i did, of course, was i moved this term m2c squared over.

03:34 I subtracted it over and it moved right here.

03:42 So now, when we divide by m2c squared on both sides, we get the following equation.

03:48 We have e -min equals m3 squared minus m1 squared minus m2 squared times c.

03:57 To the fourth divided by 2m2 c squared and this then can be simplified to m3 squared minus m1 squared minus m2 squared x squared over 2m2 to get the minimum kinetic energy we subtract the rest energy of the first particle the incoming particle so we subtract m1c squared from e min and to do that we get a common denominator we have 2m2 on the first this first quantity right here so we get 2m2 on the top and bottom of m1c squared.

04:43 So that gives us 2m2 on the bottom and 2m2 on the top right here.

04:52 We can then combine, and we get the following here in this form, and we can then factor.

05:06 There should be a square right here on m2, and this is then equal to m3 squared minus m1 plus m2 squared, divided by 2m2 times c squared.

05:25 So this is the equation that we want.

05:27 This right here for k -min.

05:30 Now we can apply it to some actual calculations for some reactions.

05:35 So for the first reaction, we have two protons interact and give us two proton, rather three protons plus an antiproton.

05:44 So m3 corresponds to the three protons and an antiproton.

05:50 M2, or m1 is this proton on the left right here, and m2 is this second proton on the left -hand side...

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