How much energy is required to produce the electron-positron pair

[vst98]

Homework Statement

How much energy is required to produce the electron-positron pair in the filed of nucleus (Z=20) if electron is captured to the ground 1s state of the system.

The Attempt at a Solution

I can calculate the energy of the captured electron from

E = - εZ²/(2n²) where ε=me⁴/hbar²≈27.2eV

E = - 5442eV

Now I have to add this energy to the energy of created positron to have total required energy, right ? But I do not know how to find energy of a positron.

 

[tiny-tim]

hi vst9!

… But I do not know how to find energy of a positron.

won't it be the same as the energy of the electron?

 

[vst98]

hi tiny-tim !

Yes I think so, then it is the energy and momentum conservation that places a restriction
on the positron to have the same energy as electron , right ?

I was a little bit confused by the "... in the filed of nucleus" part of the question, then
this filed of a nucleus does not change much (except maybe the needed energy).

So, the needed energy for pair creation is E=2*5442=10.88keV

 

[tiny-tim]

hi vst98!

right!

i'm not sure why they mentioned the field (not "filed" ) … it doesn't seem to add anything to the rest of the question

 

[vst98]

Ok, thanks !

(yes it is "field" and not "filed" in the original question, my flip error :) )