In the rare decay π^+ →e^++νe, what is the kinetic energy of...

Key Concepts

Two-Body Decay Kinematics

In a two-body decay, a parent particle at rest decays into two daughter particles. The kinematics of such a process is uniquely determined by the conservation of energy and momentum. This concept provides a framework for calculating the energies and momenta of the decay products based solely on the masses involved, as the constraints force the daughter particles to share the available energy in a predictable way.

Conservation of Energy and Momentum

The principles of conservation of energy and momentum are fundamental in analyzing particle decay processes. In any decay, the total energy (including rest mass energy) and the total momentum before the decay must equal the corresponding totals after the decay. This allows one to set up equations that link the kinetic energies and momenta of the decay products, a critical step in determining the energy distribution among them.

Rest Frame Kinematics

When a particle decays from rest, the analysis is simplified because the initial momentum is zero, which means the momenta of the daughter particles are equal in magnitude and opposite in direction. This situation makes it easier to apply conservation laws and derive explicit expressions for the energies of the decay products, as the absence of initial momentum reduces the number of variables in the problem.

Relativistic Energy-Momentum Relationship

In particle physics, the relation between energy, momentum, and mass is governed by the relativistic energy-momentum equation. This relationship is essential when dealing with decays where the kinetic energies of the particles are comparable to their rest mass energies. It ensures that both the rest mass and the motion of the particles are correctly accounted for, enabling accurate calculations of the kinetic energy released in the decay process.

Transcript

00:01 So in this problem, we are looking at the decay of the pion into a positron and electron neutrino.

00:07 And it wants us to find the kinetic energy of the positron from the energy release from the decay process.

00:16 So the q value is equal to the connect energy of the positron plus the connect energy of the neutrino.

00:23 The neutrino rest mass is approximately zero.

00:26 So q is defined as the mass.

00:31 For the pion subtracted the positron plus the mass the neutrino which that is sorry so that's approximately 0 times c squared so these are mc square terms so that can be equivalently written as 139 that 6 minus 0 .511 minus 0 m .v.

01:15 C squared times c squared.

01:19 So that cancels.

01:23 And we end up with 139 .089 m .e .v.

01:33 So q is approximately 139 .1m .m .v.

01:41 So the two connect guarantees will end up having this sum up to that value of q.

01:46 So then we note that we're given that the pion began at rest, so there's no extra kinetic energy in the system, no extra momentum, so p total equals zero, and also is the momentum of the positron plus the momentum of the electron neutrino.

02:20 So then we can say, you either as magnitude you could write as squared, so there's some i'll write it squared, positron momentum, neutrino momentum.

02:40 And so that's p squared.

02:42 They're the same value, so we'll just replace it with p.

02:45 So in the kinetic energy equations, where you have the p -c -squared term for the different particles, we can just write, replace those with the lower p, and it'll be equivalent.

02:58 So then the kinetic energy of the positron becomes pc squared plus 0 .511 mep squared minus 0 .511 mep and k of the neutrino is pc squared plus 0 .11 .m...

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