Why doesn't Beta Decay violate the laws of physics?

In Beta decay, a neutron decays into a proton, and "throws out" an electron at high speed. However, this, to me, suggests that the law of conservation of mass is not being kept here.

Neutrons have a mass of "1",

Protons have a Mass of "1",

Electrons have a mass of "1/1840"

This means that, before the decay, the total mass is "1", but after the decay has occurred, the total mass in the system is "1841/1840", we have gained an electron's worth of mass from somewhere.

DISCLAIMER: I have been taught at a GCSE level, so much dumbing down has occurred in terms of what I have been taught. If I say something wrong, its because I've been taught that at school. Sorry!

This means that, either energy is being converted to mass here, or the "mass" values I have been taught are wrong, or the reason could be completely different. Which is it?

TL;DR: In beta decay, we seemingly gain one electron's worth of mass. Where has it come from?

• Conservation of mass is not a physical law. What are you asking about? – ACuriousMind May 22 '15 at 22:56
• where did the extra mass of the electron come from if protons and neutrons have the same mass? – Deep May 22 '15 at 23:07
• 1. Protons and neutrons do not have exactly the same mass. 2. See, for example, when matter and antimatter annihilate, only photons come out - mass is gone! Mass can vanish, and it can appear, there simply is no conservation law for mass. – ACuriousMind May 22 '15 at 23:09
• @Deep $E = mc^2$ – Jimmy360 May 22 '15 at 23:18
• @Deep: "In beta decay, we seemingly gain one electron's worth of mass. Where has it come from?" Well, that's still small potatoes. In the meantime, Beta decay is believed to produce an intermediary particle, the W boson, whose mass (80 GeV/c2) is 80 times that of the original particle (less than 1 MeV/c2) it was created from. Now, isn't that something?! – bright magus May 23 '15 at 9:07

The mass of a free neutron is 939.566 MeV/c$^2$ (almost 1 GeV/c$^2$, so that's probably where your instructor got the "1" value), and the mass of a free proton is 938.272 MeV/c$^2$. A free neutron will decay into a free proton, free electron ($\beta^-$), and an anti-neutrino, $\bar{\nu}$. The mass of the electron is 0.511 MeV/c$^2$, and of the anti-neutrino, practically zero.

In the center-of-mass (CoM) reference frame (the rest-frame of the neutron), the total energy to start with is the mass energy of the neutron: 939.566 MeV.

After the decay, the mass energy of the products is 938.783 Mev, so there remains 0.783 MeV of energy to be shared as kinetic energy between the proton and the anti-neutrino. The net momentum in the CoM frame must be zero, but with three particles involved, the energy is not split uniquely among the three. The $\beta^-$ and $\bar{\nu}$ carry most of the kinetic energy, but again, not uniquely split.

The non-unique energy and momentum of the $\beta^-$ is what led physicists to consider the existence of the third particle in the decay.

• For a bit of perspective, the amount of energy converted from mass in beta decay per nucleon (i.e per combined number of protons and neutrons involved) is much, much larger than the amount of energy created from chemical reactions that break a single chemical bond (e.g. burning hydrogen gas) . Converting mass to energy is a very energy dense process. – ohwilleke Nov 3 '16 at 4:24

There is no conservation of mass. There is conservation of mass/energy. Proton mass does not equal neutron mass.

• why are they not equal? What are they defined as? – Deep May 22 '15 at 23:11
• @Deep Proton mass is a fraction of neutron mass. – Jimmy360 May 22 '15 at 23:13
• @Deep: They are not defined, they are measured. The proton's mass is measured to be $938.2\ \text{MeV}/c^2$, while the neutron's mass is $939.5\ \text{MeV}/c^2$. They are different particles, so there is no a priori reason why they should have the same mass. – Javier May 22 '15 at 23:13
• and, I assume, the electron therefore has a mass of 1.3 MeV/c2? So it's simply a misunderstanding because of what I've been taught? – Deep May 22 '15 at 23:15
• @Deep Yes, they must have rounded. – Jimmy360 May 22 '15 at 23:16