What is the maximum charge limit for an electron-positron pair creation?

[rubertoda]

Calculate the energy is takes to create a positron-electron pair and place the electron
in the lowest energy state of a nucleus of charge 푍푒 and the positron infinitely far
away. Show that for a sufficiently high charge the creation of the electron-positron
pair gives a net release of energy and that, therefore, there is an upper limit for the
1magnitude of a free charge and find an approximate number for this charge. help me...

 

[TSny]

Hi, rubertoda.

It would be helpful for us if you indicated your line of thought regarding how to approach this problem.

Do you know how to calculate the energy required to create a electron-positron pair in free space (that is, without worrying about the presence of a nucleus)?

If you can answer that question, then you can think about how to include the additional energy associated with the electron being bound to the nucleus in the lowest energy state.

 

[rubertoda]

yes, i have calculated the necessary photon energy (511 Mev + 511 Mev - 13.6 ev + 13.6 ev)
But i need to understand the second part of the question; at what Z is there a net energy released in the pair production

 

[TSny]

OK. mc^2 for an electron is 0.511 MeV (rather than 511 MeV). So to create an electron positron pair in free space requires 2(.511 Mev) = 1.02 MeV.

It's only the electron that will get trapped by the nucleus. So, you just need to consider the additional energy corresponding to the lowest energy for an electron bound do a nucleus of charge Ze, where Z is the number of protons in the nucleus and e is the elementary charge of a proton.

If you were to use the Bohr model, what would be the lowest energy of an electron bound to a nucleus with Z protons?

The important thing is that the energy will be negative and will become more negative as Z increases. At some value of Z, the negative energy of the atom will offset the 1.02 MeV of the pair creation. At that value of Z, no input energy would be required to create the pair! That is, such a nucleus just sitting alone in space would cause electron-positron creation!

Once you have found Z, then the charge of the nucleus would be Ze.

Now, in actual fact, for large Z the Bohr model will not be accurate due to relativistic effects. So, your answer will not be very accurate. One would have to use relativistic quantum electrodynamics, etc. But that's a thesis not a homework problem! So, hopefully using the energy levels according to the Bohr model is all that you are expected to use.

 

[rubertoda]

Thanks!Now i know