# $\beta^-$ and $\beta^+$ $Q$ value issue

It's unclear to me why the following doesn't work when I calculate the $$Q$$ value for $$\beta^-$$ and $$\beta^+$$ decay. If we start with a parent nucleus $$P_z$$ with $$Z$$ electrons decaying through $$\beta^-$$ producing a daughter nucleus $$D_{z+1}$$ with $$Z+1$$ electrons:

$$Q = (P_z + Z*m_e) - (D_{Z+1} + (Z+1)*m_e + m_e).$$

Assuming that the Daughter is charge neutral and $$m_e$$ is the mass of an electron I find:

$$Q = P_z - D_{Z+1} -2m_e$$

I was expecting to get: $$Q = P_z - D_{Z+1}$$

For $$\beta^+$$ decay I have a similar problem where I get: $$Q = P_z - D_{Z-1}$$ when I expect to get $$Q = P_z - D_{Z-1} -2m_e$$

Can someone please point out where I have gone wrong?

• Does this answer your question? When to involve electron mass in energy calculation of beta decay May 20, 2020 at 22:20
• Not really in my case $P_Z$ and $D_Z$ are atomic nuclei (I shall update my question to reflect this). Hence I don't think the electrons are included in this. I guess my question is more related to this en.wikipedia.org/wiki/Beta_decay#Energy_release. Why do we need to convert from the mass of the nucleus to atomic masses.. May 20, 2020 at 22:29
• But you have electrons included in your calculations. Compare what is in that question with yours - it shows how to include the creation of the relevant beta. May 20, 2020 at 22:31
• Yeah but the question suggests that I have double-counted electrons and I don't think I have. May 20, 2020 at 22:32
• If you prefer, try physics.stackexchange.com/q/186897, or search for 'beta decay q value' and find other Q/A. May 20, 2020 at 23:25

Therefore, to get the nuclear mass you should subtract the electron masses from the tabulated atomic masses. In your case for $$\beta^-$$ decay you should have $$Q = (P_Z - Z*m_e) - (D_{Z+1} - (Z+1)*m_e + m_e),$$ where $$P_Z$$ and $$D_{Z+1}$$ will be the tabulated neutral atomic masses. I think you'll get what you expected, both in this case and the $$\beta^+$$ case.