Which Oxygen Isotope Undergoes Mirror Nuclei Decay?

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The discussion revolves around determining which oxygen isotope undergoes mirror nuclei decay, specifically whether it is through beta plus (β+) or beta minus (β-) decay. Participants suggest using the semi-empirical mass formula to calculate the energy balance for different isotopes of oxygen, emphasizing the need to compare the masses on both sides of the decay equation. It is noted that common isotopes of oxygen include 14, 15, 16, 17, 18, 19, and 20, with only a few being stable. The approach involves selecting various isotopes and calculating to see which decay mode is energetically favorable. Ultimately, the solution requires guessing the value of A and testing known isotopes to find the correct one.
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Homework Statement


Using semi empirical mass formula calculate which isotope of oxygen (##_8O##) decays into mirror nuclei. Is it ##\beta +## or ##\beta -##?

Homework Equations


$$W(A,Z)=w_0A-w_1A^{2/3}-w_2\frac{Z^2}{A^{1/3}}-w_3\frac{(A-2Z)^2}{A}-w_4A^{-3/4}\delta(A,Z)$$ where $$w_0=15.8MeV$$ $$w_1=17.8MeV$$ $$w_2=0.71MeV$$ $$w_3=23.7MeV$$ and $$w_4=11.2MeV$$

The Attempt at a Solution


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Hmmm, firstly: $$\beta ^+ : \qquad ^A_8O \rightarrow ^A_7X+e^++\nu _e$$ and $$\beta ^- : \qquad ^A_8O \rightarrow ^A_9Y+e^-+\bar{\nu _e}$$ I don't really know what to do here but... My idea was that maybe the energy balance would give me the answer?...

Something like:
For ##\beta ^+##: $$M_Oc^2=M_xc^2+m_ec^2$$
$$8m_pc^2+(A-8)m_nc^2-W(A,8)-(7m_pc^2+(A-7)m_nc^2-W(A,7)+m_ec^2)=C$$ Now my idea was if that ##C>0## than the decay is not possible. But... I amnot sure this is the right way, since I don't have the number ##A##.

Does anybody have a hint on what to do here? :/
 
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You want to show (as you stated) that the mass of the left side of the decay equation is greater than the mass on the right side. You must choose the value of A for known isotopes of oxygen and try each one to see what happens.
 
gleem said:
You want to show (as you stated) that the mass of the left side of the decay equation is greater than the mass on the right side. You must choose the value of A for known isotopes of oxygen and try each one to see what happens.

Aha, so there isn't any mysterious way to find that A. I just have to guess it and look for the right one?
 
Yep. Common isotopes of oxygen are 14,15,16,17,18,19, 20 .three are stable. There are a few more with Half-Lifes less than 1 sec