Implications of mass energy not being conserved for ##\beta^+## decay

[JD_PM]

Homework Statement

Beta + decay

Relevant Equations

Beta + decay

$\beta^{+}$ decay (a proton decays to a neutron + positron + electron neutrino) is a decay in which mass energy is not conserved.

I've been taught that if that ought to be the case for a decay/reaction, it would be forbidden (please see solution manual below):

But $\beta^{+}$ decay is observed to happen.

What am I missing here?

 

[Orodruin]

Homework Statement: Beta + decay
Homework Equations: Beta + decay

What am I missing here?

Kinetic energy.

 

[JD_PM]

Kinetic energy.

To get the maximum kinetic energy for the positron (for instance) we have to assume that the neutron has no kinetic energy so that momentum is conserved like:

$$p_{ntr} = p_e$$

Where ntr means neutrino.

Was your comment pointing at this assumption?

 

[Orodruin]

To get the maximum kinetic energy for the positron (for instance) we have to assume that the neutron has no kinetic energy so that momentum is conserved like:

$$p_{ntr} = p_e$$

Where ntr means neutrino.

Was your comment pointing at this assumption?

No. There is no such thing as ”mass conservation” in the sense of needing the sum of the masses of individual particles to be the same before and after. The appropriate conservation law is energy and this has contributions from both the masses and the kinetic energies of the particles. So when you are just comparing the sums of the individual masses, you are missing kinetic energy.

 

[JD_PM]

OK I see it now thanks. Then this reaction is not forbidden, because energy (mass+ kinetic energies) is conserved for Beta + decay, right?

 

[Orodruin]

Yes