At the Stanford Linear Accelerator, electrons and positrons...

At the Stanford Linear Accelerator, electrons and positrons collide together at very high energies to create other elementary particles. Suppose an electron and a positron, each with rest energies of $0.511 \mathrm{MeV},$ collide to create a proton (rest energy $938 \mathrm{MeV}$ ), an electrically neutral kaon $(498 \mathrm{MeV}),$ and a negatively charged sigma baryon $(1197 \mathrm{MeV}) .$ The reaction can be written as: $$\mathrm{e}^{+}+\mathrm{e}^{-} \rightarrow \mathrm{p}^{+}+\mathrm{K}^{0}+\Sigma^{-}$$ (a) What is the minimum kinetic energy the electron and positron must have to make this reaction go? Assume they each have the same energy. (b) The sigma can decay in the reaction $\Sigma^{-} \rightarrow \mathrm{n}+\pi^{-}$ with rest energies of $940 \mathrm{MeV}$ (neutron) and $140 \mathrm{MeV}$ (pion). What is the kinetic energy of each decay particle if the sigma decays at rest?

At the Stanford Linear Accelerator, electrons and positrons collide together at very high energies to create other elementary particles. Suppose an electron and a positron, each with rest energies of $0.511 \mathrm{MeV},$ collide to create a proton (rest energy $938 \mathrm{MeV}$ ), an electrically neutral kaon $(498 \mathrm{MeV}),$ and a negatively charged sigma baryon $(1197 \mathrm{MeV}) .$ The reaction can be written as:
$$\mathrm{e}^{+}+\mathrm{e}^{-} \rightarrow \mathrm{p}^{+}+\mathrm{K}^{0}+\Sigma^{-}$$
(a) What is the minimum kinetic energy the electron and positron must have to make this reaction go? Assume they each have the same energy. (b) The sigma can decay in the reaction $\Sigma^{-} \rightarrow \mathrm{n}+\pi^{-}$ with rest energies of $940 \mathrm{MeV}$ (neutron) and $140 \mathrm{MeV}$ (pion). What is the kinetic energy of each decay particle if the sigma decays at rest?

Key Concepts

Rest Energy

The rest energy of a particle is the intrinsic energy associated with its mass, as given by Einstein's equation E = mc². It represents the minimum energy that a particle possesses even when it is at rest, and it is a crucial component in understanding particle creation and annihilation in high-energy physics.

Threshold Energy

Threshold energy is the minimum kinetic energy required for a reaction to occur, particularly in particle collisions. In high-energy physics, this concept is essential because the total energy (including rest energies and kinetic energies) of the incoming particles must be at least equal to the combined rest energies of the produced particles, ensuring the reaction is energetically possible.

Center-of-Mass Frame

The center-of-mass (COM) frame is a reference frame in which the total momentum of the interacting system is zero. This frame simplifies the analysis of collision processes because energy conservation equations become easier to handle, particularly when determining threshold energies and analyzing the kinematics of produced particles.

Conservation of Energy and Momentum

The principles of conservation of energy and momentum play a fundamental role in analyzing particle collisions and decays. These conservation laws ensure that the total energy and momentum before and after an interaction remain constant, allowing physicists to calculate the necessary kinetic energies for reactions and determine the energy distribution among decay products.

Two-Body Decay Kinematics

Two-body decay kinematics deals with the situation in which a particle at rest decays into two products. By applying conservation of energy and momentum, one can determine the energies and momenta of the decay products. This concept is especially relevant when analyzing decay processes where the initial particle is at rest, allowing for a straightforward calculation of the kinetic energies of the resultant particles.

Transcript

00:01 For the first of this question, we are given the collision of an electron and positron pair to produce a few particles which are listed here is protons, k -on and the sigma beryon.

00:21 And i've also given the relative, their rest and mass energies, their rest energies.

00:30 Return below directly below the particles and the first part of the question asks us what is the kinetic energy the minimum kinetic energy required to be possessed by each of the electrons right the electron the positron so to find out we need to use conservation of energy we know that the kinetic energy of our electrons right two times of that because we are given that they have the same kinetic energy.

01:06 So the kinetic energy plus the rest energies of the electrons must be equal to the final energies of our final products.

01:19 And since we are looking at the minimum kinetic energy, the final products must be produced at rest.

01:26 So this will give us the minimum energy that is required to be inputted into the system.

01:34 And so that will be just the rest energies of the proton, rest energies of the k -on, and rest energy of our sigma barrier.

01:45 All right, and all of these are already given.

01:50 So you can just substitute them in.

01:53 You should get to k to p .632g.

02:03 Finally, you can get the kinetic energy of each of the electron or the positron to be 1 .316 gv.

02:15 For the next part, we want to find what is the kinetic energy for a decay of our sigma barion.

02:26 So sigma baron actually decays into pion and also the neutron.

02:41 And their rest energies are given as such.

02:45 Neutron is 940, 40, 4 .5 is 140.

02:57 To find what is the kinetic energy being produced, we will take the difference in the rest energies from the initial sigma baron to the final products.

03:15 The change in the rest energies must be released in the form of kinetic energy.

03:19 And so change in the rest energy be 1197 minus 140 get 117 mvv so this must be the total kinetic energy let's label this as k right but this is but what we want to find is the constituent kinetic energies of the neutron and the ion right and and by conservation energy, the total must be equal to the sum of the kinetic energies.

03:59 Now in order to find what is the individual kinetic energy, we need to use the conservation of momentum.

04:05 That is, the momentum of the initial state of the sigma barion is zero.

04:16 So it means that the total momentum of our final system, including the neutron and pion, must be zero as well.

04:24 And what this tells us is that the magnitude of the momentum for neutron must be equal to the momentum of the pion.

04:34 They must be equal and opposite in order to cancel out and give us a total momentum of zero.

04:41 So that momentum is conserved.

04:47 Now we're going to use the equation pc square equals k square.

05:03 Plus e0...

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