Spin and energy in nuclear processes

[jjr]

Hello

1. I am having some trouble with getting the correct answers when trying to solve problems that involve spin (Problems from textbook Physics for scientists and engineers, Serway & Jewett). I'll give two examples to illustrate:
The process p + p -> p + p + n (protons and a neutron). It's not allowed because it violates the conservation of the baryon number, but that is irrelevant to my problem. Here's my logic when trying to see if spin is conserved: Both p and n has spin 1/2, so it should go something like this: 1/2 + 1/2 ≠ 1/2 + 1/2 + 1/2, and therefore angular momentum is not conserved. (The task is to find ALL the broken laws of conservation which are broken in this process. The answer in the book only mentions the violation of the conservation of baryon number.)
Another example: p + p -> p + pion(+)
From my logic: 1/2 + 1/2 -> 1/2 + 0 => 1 ≠ 1/2, and therefore conservation of angular momentum is violated. Book says the only thing violated is the conservation of energy.
I'm obviously dead wrong, so how should I approach the problem?
(I can provide a few more examples where I got the wrong answer, if needed)

2. Now a second question related to the first nuclear process mentioned above (p + p = p + p + n). The book says that energy is conserved in this process, even though the rest mass of the right side of the equation clearly is higher than the rest mass of the left side. I'm thinking that it has something to do with the kinetic energy of the two protons on the left side being transformed to the extra mass we see on the right side? If this is the case, I would love to get a few pointers on how to do some calculations on this kinetic energy, to see which processes that potentially could be well within the limits of conservation of energy when taking into account this kinetic energy. For example how to find a maximum value for this kinetic energy, and how it relates to the maximum allowed difference in rest energies for the particles on the left side and the particles on the right side.

Hopefully you can make some sense of that. Any answers appreciated!
JJR

 

[vela]

1. I am having some trouble with getting the correct answers when trying to solve problems that involve spin (Problems from textbook Physics for scientists and engineers, Serway & Jewett). I'll give two examples to illustrate:
The process p + p -> p + p + n (protons and a neutron). It's not allowed because it violates the conservation of the baryon number, but that is irrelevant to my problem. Here's my logic when trying to see if spin is conserved: Both p and n has spin 1/2, so it should go something like this: 1/2 + 1/2 ≠ 1/2 + 1/2 + 1/2, and therefore angular momentum is not conserved. (The task is to find ALL the broken laws of conservation which are broken in this process. The answer in the book only mentions the violation of the conservation of baryon number.)
Another example: p + p -> p + pion(+)
From my logic: 1/2 + 1/2 -> 1/2 + 0 => 1 ≠ 1/2, and therefore conservation of angular momentum is violated. Book says the only thing violated is the conservation of energy.
I'm obviously dead wrong, so how should I approach the problem?
(I can provide a few more examples where I got the wrong answer, if needed)

The book's answers apparently aren't correct in that they're not complete. In the second process, for example, baryon number isn't conserved either. I wouldn't worry that you're missing something.

There is, however, a problem with your logic about angular momentum, simply because the addition of angular momentum is a bit more complicated. (Consider, for instance, e⁺e^- → γγ, which does happen.) But your basic idea is right. Angular momentum can't be conserved in those two processes above.

2. Now a second question related to the first nuclear process mentioned above (p + p = p + p + n). The book says that energy is conserved in this process, even though the rest mass of the right side of the equation clearly is higher than the rest mass of the left side. I'm thinking that it has something to do with the kinetic energy of the two protons on the left side being transformed to the extra mass we see on the right side? If this is the case, I would love to get a few pointers on how to do some calculations on this kinetic energy, to see which processes that potentially could be well within the limits of conservation of energy when taking into account this kinetic energy. For example how to find a maximum value for this kinetic energy, and how it relates to the maximum allowed difference in rest energies for the particles on the left side and the particles on the right side.

That's right. The kinetic energy is transformed into mass. These sorts of calculations are straightforward applications of special relativity. You might want to read the chapter on special relativity in David Griffith's Introduction to Elementary Particles. He does a great job, in my opinion, of showing how to do these types of calculations.