Happy Rads! (Part 2 of 2)

[6_bleen_7]

[Continued from this post.]

It occurred to me many times, while listening to the Geiger counter chattering away beside me, to wonder just how fast the beta particles were traveling once their parent nuclei had spat them out. And I looked all over for the answer, but nobody, it seems, was willing to part with such information. The maximum decay

Comments

[jedibl]

OK, it's late, so I'm not thinking too clearly. I think you have arrived at correct conclusions, but your formulation is technically faulty.

Mass is the magnitude of the energy-momentum 4-vector. It does not increase when kinetic energy increases, whatever the popular texts on special relativity may say. It is invariant, which means that regardless of how fast a particle is moving relative to you, its mass stays the same. The energy and momentum may increase, but the magnitude of the 4-vector, mass, stays the same.

So I think the way you *should* have formulated the discussion goes something like this: The relativistic energy of something is equal to gamma*m*c^2 (but most physicists would use units where c = 1), where gamma = 1/Sqrt(1-v^2/c^2). the total energy is also equal to rest energy (mc^2) plus kinetic energy. So given your 529.6 KeV for tritium, we can set that equal to gamma*m*c^2 to solve for gamma and thence for mass. Which I think will end up being, ultimately, the same calculation you did. Just for slightly different reasons.

For a great discussion of special relativity (and an explanation of where these formulas I used actually come from), see Taylor and Wheeler, _Spacetime Physics_.

[6_bleen_7]

Thanks-I was hoping you'd check my work, so to speak. By "4-vector," do you mean a vector of length 4? I don't remember that from my college physics class, but we barely scratched the surface of relativity. I was a freshman at the time, which will give you an idea of how advanced the class was. (On the other hand, it was Oberlin, so we probably did more than the average first-semester class does. And I'm happy to have remembered so much of it after 24 years!)

Yes, your formulation seems to be equivalent to mine. Sounds like I had the right idea, but I'm using the terms mass and energy improperly.

Thanks also for the recommendation! Taylor and Wheeler sounds exactly like what I am interested in. I'll put it on my Amazon wish list and see if anyone feels generous this Christmas. : )

[jedibl]

A four-vector is a four-dimensional vector. The components of the energy-momentum 4-vector are (Energy, x-momentum, y-momentum, and z-momentum). The key difference between a 4-D spacetime vector and a traditional 3-D vector is that to find the magnitude (squared), you square the first component and then subtract the squares of the other three components. (Rather than just adding the squares of all components as you would in traditional Euclidean geometry.)

[6_bleen_7]

That's pretty strange-even stranger than quaternions. I think I see how the magnitude is equal to the mass, however.

[6_bleen_7]

Indeed, Kathy did give me Spacetime Physics for Christmas! I may have to put it on hold 'til I finish teaching my class, though; it doesn't look like light reading. Still, if I could tackle Clayton's Principles of Stellar Evolution and Nucleosynthesis, I shouldn't have too much trouble with Taylor & Wheeler.