# Element 129 superluminal? zαc

1. Oct 31, 2013

### Superposed_Cat

I found the equation for the speed of an electron today. atomic number*fine-structure constant*speed of light. rearranging we get z= v/σc. if you set v to c then you find that at element 137 the electrons become superluminal. This can be possible right? a retrocausal atom! Could someone please explain why this wont happen. Thanks for any help.

sorry, for the typo in the title.

Last edited by a moderator: Oct 31, 2013
2. Oct 31, 2013

### The_Duck

This formula is a non-relativistic approximation, valid only for Z much smaller than 137. It is not true for large Z. (It is also a huge oversimplification that ignores things like the fact that different electrons in an atom have different velocities and the fact that that no electron has a definite velocity; nevertheless it does give the right order of magnitude for the electron velocity, as long as Z is much less than 137).

3. Oct 31, 2013

### Superposed_Cat

what's the relativistic equation?

4. Oct 31, 2013

### Bill_K

There's more involved than just "relativistic". For the nonrelativistic atom the energy of the ground state is

E = - mc22 Z2/2)

The corresponding relativistic formula (exact solution of the Dirac equation) is

E = mc2 √(1 - α2Z2)

For αZ << 1 it reduces to the rest mass mc2 plus the nonrelativistic value. However for αZ = 1 it goes to zero. This is a warning that the single-particle Dirac solution is no longer valid, since the negative energy solutions can no longer be ignored. In fact for αZ > 1 the energy becomes imaginary.

5. Oct 31, 2013

thanks