Beta+ Decay or Electron Capture?

[rwooduk]

Homework Statement

Homework Equations

See below.

The Attempt at a Solution

We have β⁺ decay $X_{Z}^{A} \rightarrow Y_{Z-1}^{A}+ e^{+}+\upsilon _{e}$ which leads to the mass condition $M(A,Z)>m(A,Z-1)+2m_{e}$.

We have electron capture $X_{Z}^{A} + e^{-}\rightarrow Y_{Z-1}^{A}+ \upsilon _{e}$ which leads to the mass condition $M(A,Z)>m(A,Z-1)+\frac{\varepsilon }{c^{2}}$.

Those are the mass conditions. From my notes it says $\frac{\varepsilon }{c^{2}}\ll m(A,Z-1)$ therefore electron capture can happen whenever β⁺ decay does, but β⁺ decay is more likely. I am unsure about this statement or how it helps distinguish which one would happen for certain conditions.

Any ideas would really be appreciated.

 

[mfb]

You can check the inequalities to see what is possible.

I am unsure about this statement or how it helps distinguish which one would happen for certain conditions.

If beta+ decay is possible (and if the inequality is not too close to an equality), it is more likely.

 

[rwooduk]

You can check the inequalities to see what is possible.
If beta+ decay is possible (and if the inequality is not too close to an equality), it is more likely.

the inequalities are $\frac{\varepsilon }{c^{2}}$ and $2m_{e}$ , sorry it should have been if $\frac{\varepsilon }{c^{2}}\ll 2m_{e}$ in the original post. So if $\frac{\varepsilon }{c^{2}}\ll 2m_{e}$ then it's beta decay, if $\frac{\varepsilon }{c^{2}}\gg 2m_{e}$ it's electron capture and if they are equal it is equally likely? Is that what you mean?

Many thanks for the reply

 

[mfb]

the inequalities are $\frac{\varepsilon }{c^{2}}$ and $2m_{e}$ , sorry it should have been if $\frac{\varepsilon }{c^{2}}\ll 2m_{e}$ in the original post. So if $\frac{\varepsilon }{c^{2}}\ll 2m_{e}$ then it's beta decay, if $\frac{\varepsilon }{c^{2}}\gg 2m_{e}$ it's electron capture and if they are equal it is equally likely? Is that what you mean?

That does not make sense.

If both inequalities are satisfied, both processes are possible. Then beta+ decay is more likely, with an exception:
If beta+ is "just" allowed (the left side of the beta+ inequality condition is just a tiny bit larger than the right), then beta+ decay is not very likely.

 

[rwooduk]

That does not make sense.

If both inequalities are satisfied, both processes are possible. Then beta+ decay is more likely, with an exception:
If beta+ is "just" allowed (the left side of the beta+ inequality condition is just a tiny bit larger than the right), then beta+ decay is not very likely.

Ahh so by stating the mass condition for each I've answered the question, excellent, I think I got confused by trying to link the two mass conditions somehow as independent entities when they can both be satisfied at the same time. thanks again.