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 that the bombarding particle must have 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 situation occurs when all the product particles have the same velocity; then the particles have no kinetic energy of motion relative to one another. (a) By using conservation of relativistic energy and momentum, and the relativistic energymomentum relation, show that the threshold energy is given by $$K_{\min }=\frac{\left[m_{3}^{2}-\left(m_{1}+m_{2}\right)^{2}\right] c^{2}}{2 m_{2}}$$ Calculate the threshold energy for each of the following reactions: (b) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{p}+\overline{\mathrm{p}}$ (c) $\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\Lambda^{0}$ (The proton is at rest. Strange particles are produced.) $(d) p+p \rightarrow p+p+\pi^{0}$ (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 (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 that the bombarding particle must have 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 situation occurs when all the product particles have the same velocity; then the particles have no kinetic energy of motion relative to one another. (a) By using conservation of relativistic energy and momentum, and the relativistic energymomentum relation, show that the threshold energy is given by
$$K_{\min }=\frac{\left[m_{3}^{2}-\left(m_{1}+m_{2}\right)^{2}\right] c^{2}}{2 m_{2}}$$
Calculate the threshold energy for each of the following reactions:
(b) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{p}+\overline{\mathrm{p}}$
(c) $\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\Lambda^{0}$
(The proton is at rest. Strange particles are produced.)
$(d) p+p \rightarrow p+p+\pi^{0}$
(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 (mass 91.2 $\mathrm{GeV} / c^{2}$ ) are produced.

Key Concepts

Relativistic Energy?Momentum Relation

This principle unifies energy, momentum, and mass in the expression E² = (pc)² + (mc²)². It is essential for analyzing high?speed particle interactions, as it allows us to relate kinetic energy and momentum with the rest masses of particles. This relation is pivotal when determining the energy thresholds necessary for particle creation in reactions.

Conservation Laws in Relativistic Collisions

In relativistic physics, both energy and momentum are conserved in particle collisions. These conservation laws are applied to both the initial and final states of a reaction, enabling the derivation of key relationships such as the threshold energy. They ensure that the total four-momentum before and after the reaction remains constant, regardless of the reference frame.

Threshold Energy

Threshold energy is the minimum kinetic energy required by an incident particle to initiate a reaction that produces new particles. At this energy, the produced particles have the same velocity (i.e., they are at rest relative to one another in the center-of-mass frame), meaning all available energy goes into creating the new rest mass. This concept is crucial in analyzing and predicting the outcomes of particle reactions.

Center-of-Mass Frame and Invariant Mass

The center-of-mass (COM) frame is where the total momentum of the system is zero, simplifying the analysis of collision problems. In this frame, the invariant mass of the system, which is derived from the energy and momentum of all the particles, remains constant and provides a natural basis for identifying the minimum energy needed for a reaction to occur. This concept aids in understanding how energy is partitioned between kinetic energy and the creation of new mass.

Transcript

00:01 In this question we have a scenario where a mass m1 moving at a certain velocity is heading towards a stationary particle m2 and colliding with it.

00:18 And subsequently it breaks into its products and forms many products which are moving at the same below.

00:32 With respect to each other, moving at a certain velocity with a total mass of m3.

00:44 Now we are given that this is the scenario where the kinetic energy required at the start would be a minimum.

00:53 The question argued that this is the case, and so we're going to try to find what is the equation governing the minimum.

01:04 Energy.

01:08 Now by conservation of energy we know that initial total energy is just the energy of our of our m1, right, minimum, plus the energy of m2 which is just the rest energy.

01:30 So this is the total initial energy.

01:34 We create to the final energy, final energy of the system.

01:41 Which we use the energy mass relation.

01:45 E square goes to pc square plus m .c.

01:56 Which would mean that it is square root of momentum times c square plus m3c square square.

02:12 Now we can write this down because we have the final products.

02:17 They are moving together that are equivalent to a single particle right if they are moving together with the same velocity then we can kind of attribute them to be a single particle all right they are moving together with a momentum p3 and a mass of m3 now by conservation of momentum the initial momentum of this system which is just from our mass one.

02:52 So just p1 is our initial momentum must be equals to the final momentum of the system which is just p3.

03:05 And so this implies that p1c squared must be equal to p3 c squared.

03:20 We know from the energy mass relation that p1c squared is equal to p3 c squared.

03:28 To e of minimum square minus the rest energy of m1.

03:42 And we're going to substitute this, sorry, there should be a bracket over here, going to substitute this expression into this expression over here.

04:28 Now to remove the square root, we will square both sides of the equation.

04:56 We see that we can cancel out the minimum energy square term and we are left with only one unknown which is the energy minimum.

05:17 All we have to do next is to just rearrange the equation to find what is the minimum energy.

05:56 And from here we can find the kinetic energy, right, the minimum kinetic energy.

06:02 Since total energy is equals to kinetic energy plus the rest energy.

06:13 So the kinetic energy must be just the total energy minus the rest energy...

Please give Ace some feedback