Annihilating a proton or electron

[Curl]

So if the total energy of a particle is E² = m²c⁴ + c²p²

what about a charged particle, when you annihilate it you also destroy its electric field, so you get the field energy by collapsing it.

how does this factor in?

 

[Drakkith]

Just to throw out a guess, I'd say that it's already included in the mass, or that it is ignored. Not sure honestly.

 

[Curl]

It would make sense, but then why is the neutron mass larger than that of a protons' ?

 

[IsometricPion]

what about a charged particle, when you annihilate it you also destroy its electric field, so you get the field energy by collapsing it.

$$\sqrt{m^{2}c^{4}+|\vec{p}|^{2}c^{2}}$$ is the mechanical energy of a particle with rest mass, m, and (relativistic 3-)momentum, p. This energy does not include any other form of energy (such as potential or that stored in an associated field). The total energy of a charged particle is $$\sqrt{|\vec{p}|^{2}c^{2}+m^{2}c^{4}}+q\phi$$ where q is the charge of the particle, and $$\phi$$ is the electrical potential acting on the particle (not including the potential due to the particle itself) (wiki: Hamiltonian mechanics).

how does this factor in?

Since $$\phi{q}$$ is negative when the particles have opposite charges, the expression for the energy given above implies that the total energy of the gamma rays produced in the annihilation is less than the total of the initial kinetic and rest mass energies (of the particle-antiparticle pair).

 

[Drakkith]

It would make sense, but then why is the neutron mass larger than that of a protons' ?

A proton is composed of two Up Quarks and 1 Down quark, while the Neutron is composed of 1 UP Quark and 2 Down Quarks. Down quarks are slightly more massive than Up Quarks, which makes Neutrons slightly more massive than Protons.

Up quarks have an electric charge of +2/3 while Down Quarks have a charge of -1/3.

 

[Phrak]

A proton is composed of two Up Quarks and 1 Down quark, while the Neutron is composed of 1 UP Quark and 2 Down Quarks. Down quarks are slightly more massive than Up Quarks, which makes Neutrons slightly more massive than Protons.

Yeah, maybe, but the answer is not that simple.

The mass/energy of a nucleon consists of its quarks and the binding energy of the gluons. The gluons contribute the majority of the mass. The quarks clock in at only 2-15 Mev each. The masses of the quarks are not known with enough accuracy (to my knowledge) give yield an unambiguous answer.

In addition, there's at least one reason to claim that partitioning the mass between particles makes no sense.

 

[Drakkith]

Yeah, maybe, but the answer is not that simple.

The mass/energy of a nucleon consists of its quarks and the binding energy of the gluons. The gluons contribute the majority of the mass. The quarks clock in at only 2-15 Mev each. The masses of the quarks are not known with enough accuracy (to my knowledge) give yield an unambiguous answer.

In addition, there's at least one reason to claim that partitioning the mass between particles makes no sense.

Got a link to somewhere i could find that information?

 

[Phrak]

There's so much. Which parts do you want? I'm not a particle physicists with a proper appreciation of the standard model (or where it fails). But with some luck, one will come by who can tell you all about the knowns and unknowns within the Context of the standard model.

Wikipedia, if you look up 'quark', will give you exact mass/energy values for up and down quarks without error bars. You could go with that and make your claim work, but I think the wiki article is mickey-mouse. Other wiki articles contradict it, by the way.

 

[cortiver]

Contrary to what IsometricPion said, it is true that the field energy of the electron is already included in its mass. This is the point of Einstein's E = mc²; any energy which the electron has in its rest frame translates into inertia (i.e. mass). (Actually, the problem in classical physics is that the field energy of a point particle is infinite, so the non-electromagnetic part of the mass of the electron has to be negative infinity to compensate! Apparently this resolved somehow in quantum field theory).

 

[Curl]

Is it really infinite? Doing a triple integral over all space for the energy due to a point charge won't give you infinity, it converges. I don't get what you're saying.

 

[cortiver]

Is it really infinite? Doing a triple integral over all space for the energy due to a point charge won't give you infinity, it converges. I don't get what you're saying.

Really? The electric field of a point charge is
$$E \propto \frac{1}{r^2}$$

So the electrostatic energy density is
$$u \propto E^2 \propto \frac{1}{r^4}$$

So the total energy is
$$U = \int_0^\infty 4\pi r^2 u(r) dr \propto \int_0^\infty \frac{1}{r^2} dr$$

This integral diverges at the zero end.

 

[Dickfore]

So if the total energy of a particle is E² = m²c⁴ + c²p²

This equation is only true for free particles.

what about a charged particle, when you annihilate it you also destroy its electric field, so you get the field energy by collapsing it.

how does this factor in?

To annihilate a particle, you also need an antiparticle. If these are charged, they interact.

 

[cortiver]

To annihilate a particle, you also need an antiparticle. If these are charged, they interact.

However, you could consider a neutral pion, which is its own antiparticle (and therefore can decay without interacting with any other particle), yet still has electric potential energy since it consists of charged quarks.

 

[IsometricPion]

Contrary to what IsometricPion said, it is true that the field energy of the electron is already included in its mass

You are correct (I was wrong), the field energy associated with the electron's own field (and its interaction therewith) is included in the rest mass. However, the $$\phi$$ term in the second energy equation I gave was the energy due to an external electric field, such as that produced by a positron when considering the annihilation of an electron with a positron, and must be included in addition to the self-energy due to the particle's electromagnetic field (which is included in the "mechanical" energy formula). The wikipedia article on renormalization covers this topic (the divergence of the electron's self-energy and how (conceptually) a finite mass is attained in Quantum Field Theory).

 

[Dale]

yet still has electric potential energy since it consists of charged quarks.

That potential energy is part of the mass of the particle.