Which Oxygen Isotope Undergoes Mirror Nuclei Decay?

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Homework Help Overview

The discussion revolves around identifying which isotope of oxygen undergoes decay into mirror nuclei, specifically focusing on the types of beta decay, β+ or β-. The original poster attempts to apply the semi-empirical mass formula to analyze the decay process.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to compare the masses on both sides of the decay equation and suggest selecting known isotopes of oxygen to determine the appropriate value of A. There is uncertainty about how to proceed without a specific value for A.

Discussion Status

The conversation is ongoing, with participants exploring the relationship between mass and decay types. Some guidance has been provided regarding the selection of isotopes, but no consensus has been reached on the specific approach to take.

Contextual Notes

Participants note that common isotopes of oxygen include 14, 15, 16, 17, 18, 19, and 20, with some being stable and others having short half-lives. There is an acknowledgment of the need to guess and check for the correct isotope.

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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