Pair production and photon

In summary, the minimum energy required for a photon to create an electron-positron pair in the presence of a stationary nucleus of mass M is 2mc^2[1 + (m/M)], where m is the electron rest mass. This can be obtained by considering the conservation of energy and momentum in the center of mass frame, where the three particles (electron, positron, and nucleus) are stationary. The necessary initial energy and momentum of the photon in the center of mass frame can be calculated using Lorentz transformations from the zero momentum frame.
  • #1
babblingsia
9
0

Homework Statement



To show that the minimum energy a photon must have to create an electron-positron pair in the presence of a stationary nucleus of mass M is 2mc^2[1 + (m/M)], m is the electron rest mass.

Homework Equations



Conservation of energy and momentum.Also the minimum energy a photon should have is 1.02MeV to form an electron and a positron ( won't be needing the figures here though)


The Attempt at a Solution



I tried doing it in the center of mass frame ( coz the CM frame happens to give the minimum energy) but I'm getting nowhere. Should I be using the lorentz invariance E^2 - p^2c^2 in CM and lab frame?
 
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  • #2
Your thinking is along the correct lines. The CofM frame should include the nucleus. Thus the question is really asking what the minimum energy in the lab frame is. You should be able to convince yourself that the minimum energy case correspond to three stationary particles in the CofM frame: electron, positron and nucleus. From this you should be able to deduce the necessary initial energy and momentum for the photon in the CofM frame. Lorentz transform and it should give the correct answer.
 
  • #3
Okay is the energy of initial photon energy in Cm frame 2mc^2 + Mc^2?(I haven't really many solved problems in center of mass frames so I'm finding it a lil difficult )but then how do I get momentum of photon? the momentum of all three particles are zero in CM frame.
 
  • #4
babblingsia said:
Should I be using the lorentz invariance E^2 - p^2c^2 in CM and lab frame?
Yes. The threshold cm energy is given (2m+M)^2=(k+M)^2-k^2, where k is the photon lab energy. (c=1).
 
  • #5
I'm getting the photon momentum in CM frame to be zero( cos Center of mass is nucleus and its at rest in CM frame?)hows that possible!?
 
  • #6
(In natural units) the 4-momentum of the nucleus is (sqrt(M^2 + p^2), p). The CofM frame is possibly a bad name, because really we want the zero momentum frame (ZMF) --- which is the appropriate relativistic concept; in non-relativistic cases the two are the same. So the momentum of the photon must be -p, and so its 4-momentum is (p, -p). So for pair creation to *just about* happen, you want it to be the case that (sqrt(M^2+p^2)+p)=(M+2m). Now, the motion of the ZMF relative to the lab is given by p=(gamma)Mv, so you just need to Lorentz transform the 4-momentum of the photon to the lab frame.
 

1. What is pair production?

Pair production is a phenomenon in which a photon (a particle of light) interacts with an atomic nucleus, producing a particle and its antiparticle. This process requires a minimum energy of 1.02 MeV and can only occur in the presence of a strong electric field, such as near an atomic nucleus.

2. How does pair production occur?

Pair production occurs when a high-energy photon interacts with the electric field of an atomic nucleus. This interaction allows the photon to convert its energy into mass, creating a particle and its antiparticle. The most common particles produced through this process are electrons and positrons.

3. What is the role of energy in pair production?

Energy is a crucial factor in pair production, as it is required to create a particle and its antiparticle. The minimum energy needed for pair production is 1.02 MeV, which is equivalent to the rest mass of an electron and positron. If the photon has less energy than this minimum, pair production cannot occur.

4. Can pair production occur in a vacuum?

Yes, pair production can occur in a vacuum. In fact, it is more likely to occur in a vacuum than in a dense material, as there are fewer particles to interact with the photon. However, pair production still requires a strong electric field, which can be found near atomic nuclei or in the presence of high-energy particles.

5. What is the significance of pair production in the study of particle physics?

Pair production is a crucial process in particle physics, as it allows us to understand the relationship between mass and energy. It also provides evidence for the existence of antimatter, as the particles produced through this process are antiparticles. Pair production is also an important tool for studying high-energy particles and their interactions with matter.

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