# (SP. RELATIVITY) Annihiliation of Electrons into Photons

• Kunhee
In summary: Is the frame moving in the +x direction with v, or the -x direction with v? This won't change your answer - but I think it is worth a mention here.In summary, an electron and positron annihilate to produce two photons. Momentum must be conserved after annihilation, so two photons are created with equal and opposite momentum. In the rest frame, the energies and momenta of the photons are determined by the rest mass of the particles. In another frame moving with (v/c)^2 = 4/5 with respect to the rest frame, the photon energies and momenta are affected by the Lorentz transformation, with the momentum being multiplied by the Lorentz factor and the energy being multiplied by
Kunhee

## Homework Statement

[/B]
An electron e- and positron e+ annihilate to produce two photons.

a_ Why are two photons produced rather than one?

b_ Assume that the e- and e+ are at rest just before they annihilate. In their rest frame, what are the energies and momenta of the photons? Define the +x axis to be the direction of motion of one of the photons.

c_ What are the energies and momenta of the photons in another frame that movies with (v/c)^2 = 4/5 with respect to the rest frame?

## Homework Equations

p = y m0 u
E total = y m0 c^2
E rest = m0 c^2
E kinetic = (1-y)(m0 c^2)

## The Attempt at a Solution

a _ I think this is a simple answer that reads "momentum should be conserved after annihilation so there must be two photons created with equal and opposite momentum.

b _ Because the electron and positron are at rest before annihilation, there is no velocity transformation or Lorentz transformation required so should the momentum of the photon be "p = m0 c" and energy be "E =m0 c^2" ?

c _ This time the frame with photons is moving with (v/c)^2 = .8. I am guessing the electron and positron are still at rest before annihilation so there is no velocity transformation necessary. So should the momentum of the photon apply just the Lorentz transformation, making "p = y m0 c" and energy be "E = y m0 c^2" where y is the Lorentz factor with (v/c)^2 = .8 ?

Thanks!

(a) you appear to have assumed that the electron and positron are at rest when the annihilation has occurred - total momentum after annihilation must be zero, but photons cannot be at rest. That how you are thinking?
What about the case where they have equal and opposite velocity in the x direction, but equal positive velocity in the z direction?
Wouldn't energy and momentum be satisfied with a single photon traveling in the +z direction?
Would it be possible, even in the "annihilation at rest" case, to satisfy conservation of momentum with 3 or 4 photons?

(b) The assumption of "at rest" is explicitly stated here - so is that a good assumption for part (a)? However, your reasoning is good for this part - but why does the question tell you to consider the +x axis as the direction of one photon? (hint: momentum is a vector...)

(c) That looks good - a way to check your understanding here is to consider what happens to the photons in this moving frame compared with the rest frame of the particles. (hint: the photons do not go any faster or slower, so what changes, and in what way?) Is your answer. above, consistent with this?
Aside: I see the direction of the relative velocity is not given above - and the photons are in all inertial frames - so there is a slight ambiguity.

Kunhee

## 1. What is the process of annihilation of electrons into photons?

The annihilation of electrons into photons is a process in which an electron and its antiparticle, a positron, collide and produce two photons. This process is governed by the laws of special relativity and can only occur if the total energy of the collision is equal to or greater than the rest mass energy of the electron and positron combined.

## 2. Why is this process important in the study of special relativity?

The process of annihilation of electrons into photons is important in the study of special relativity because it provides evidence for the principle of mass-energy equivalence. This principle states that energy and mass are interconnected and can be converted into one another. The annihilation process also demonstrates the conservation of energy and momentum, which are fundamental principles in special relativity.

## 3. Can this process occur in reverse, with photons turning into electrons?

Yes, this process can occur in reverse. It is known as pair production, where two photons with enough energy can create an electron and a positron. This process is also governed by the laws of special relativity and follows the principles of conservation of energy and momentum.

## 4. How is the energy of the photons produced in this process calculated?

The energy of the photons produced in the annihilation of electrons into photons can be calculated using the equation E=mc^2, where E is the energy, m is the rest mass of the electron or positron, and c is the speed of light. The total energy of the photons produced is equal to the total rest mass energy of the electron and positron before the annihilation.

## 5. What are the potential applications of this process in science and technology?

The annihilation of electrons into photons has potential applications in medical imaging, such as positron emission tomography (PET) scans. It is also used in particle accelerators to produce high-energy photons for research purposes. Additionally, this process is being studied for potential use in energy generation through matter-antimatter reactions. However, harnessing this energy is currently not feasible due to the high energy requirements and technological limitations.

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