Relativistic electrons

1. Apr 28, 2005

IntuitioN

Just a general q: at what kinetic energy do electrons speeds become relativistic? I heard some people mention from 20keV onwards... is that correct?

2. Apr 28, 2005

eNathan

I dont understand your question. If you are talking about relavistic speed, it is always relative (to something else). Even if its going .001 meter per second, it has more KE than a body going 0 meters per second.

Are you asking at what speed does relavistic mass (hence energy) become noticable?

3. Apr 28, 2005

Ich

"relativistic" usually means v> 0.1 c, where SR corrections are more than 1%. That would mean about 50 keV for an electron.

4. Apr 28, 2005

Miserable

He means when do you need to use relativistic mechanics rather than classical mechanics to describe it.

Obviously there's no definite cut-off point, but as a general rule of thumb a particle is considered to become relativistic when its kinetic energy is roughly equal to its rest mass energy, so for an electron that would be ~500keV.

5. Apr 28, 2005

A_I_

KE = 1/2 m v^2

m = 9.1*10^-31 kg

and to be a relativist electron its speed should exceed 0.1v where v is the speed of light; thus exceeds 30000 km/s

the minimum kinetic energy required for an electron to be relativist is proportional to a minimum of 30000 km/s as speed.
do the calculations and u will find the energy required in Joules unit

if you need to convert to eV
energy (J) / 1.6*10^-19 = energy (eV)

Hope this helps to get a clearer view

6. Apr 28, 2005

robphy

This reminds me of a neat calculation:
What is the speed of an electron in a (Bohr model) hydrogen atom? (Express your answer as a fraction of the speed of light.) [You don't need a calculator.]

7. Apr 28, 2005

eNathan

Well, I guess that all depends on how presise you want your calculations to be. Relavistic effects always take place from a stationary reference frame (even tho its small) when v > 0.

8. Apr 28, 2005

mathlete

0 in the frame of the electron!

9. Apr 29, 2005

yogi

ropbhy - first Bohr orbit it is apha = (1/137)c

10. Apr 29, 2005

Ich

ups, that means 2.5 keV of course (1/2 %).

11. Apr 29, 2005

robphy

Yes! So, one could use this as a rule of thumb to determine if a non-relativistic treatment is good enough.

It's interesting how important energies and lengths in atomic/quantum physics are related by powers of alpha.

12. Apr 29, 2005

Meir Achuz

Coulombs law has no intrinisc dimension if written in appropriate units as V=e^2/r.
Any physical result for the hydrogen atom must either be dimensionless or depend on some power of m and powers of alpha (which is dimensionless).

13. May 1, 2005

Creator

Here's another:
An electron has a spin angular momentum of $\frac{\sqrt{3}}{2}{\hbar}$ and a classical radius $$R_0=\frac{\alpha{\hbar}}{mc}$$.
Classically, what is the maximum surface speed of the electron,
assuming the electron has spherical shape and its mass is uniformly distributed.

Creator

14. May 2, 2005

Creator

What; no takers?

15. May 4, 2005

Meir Achuz

NR, I would get 5\sqrt{3}/(4*alpha), which means it is a relativistic question. Is that what you meant? This becomes relativistic, with a bit of a messy integral involved.

Last edited: May 5, 2005
16. May 8, 2005

Creator

Yes; thanks for the attempt, Meir. But its more than simply 'relativistic', it becomes an 'impossibility'.

Start with the classical spin angular momentum L = I$\omega$.
Then substitute in values for electron spin angular momentum, solid spherical moment of inertia I, and angular velocity $\omega=\frac{v}{R}$, and R=classical electron radius as previously given.
Then solve for v (surface velocity) in terms of c.
What do you get?

Creator

Last edited: May 8, 2005
17. May 9, 2005

Meir Achuz

The model is wrong, but not impossible. Your simple formula L=I\omega is wrong in SR.
Actually angular momentum can get quite complicated in SR.
If the model had any interest, I would plow through the messy integration.

18. May 9, 2005

pmb_phy

It depends on the precision of your equipment/experiment. Take protons as an example. There have been experiments in which the mass of low speed protons has been detectable. I think the experiment was called a "Penning Trap."

Pete

19. May 9, 2005

Creator

That's the whole point, Meir! Duh! Double Duh! You obviously missed the entire purpose of the exercise!

The whole point of my post to Robphy was in response to his question about trying to figure the 'orbital velocity' of an electron. Which actually one cannot use as 'correct' model either.
Well, my question was about the analogous situation involving the velocity of 'spin'.

Neither of us was trying to produce a 'correct ' model; it was merely an exercise.
The whole point of my exercise was that (if one uses the classical radius) then in order to account for the electron angular momentum classicallythen the surface velocity of the electron would have to be around 300 c !!

And yes, that is an impossibility, (whether you care to agree or not).
Had you done the simple calculation I suggested you would have realized that!..MAYBE. :tongue2:
I'm sure you'll want to argue about it...Just forget it ...it's not worth trying to explain.

Creator

Last edited: May 9, 2005
20. May 10, 2005

Meir Achuz

L=I\omega is wrong. Wrong is wrong. Using the right formula would give v<c.