Question about electron capture and mass difference in the decay

Question

I’m a radiation oncologist resident, and physics is part of my daily studies. Recently, while studying electron capture, I came across the following passage on Wikipedia:

"Around the elements in the middle of the periodic table, isotopes that are lighter than stable isotopes of the same element tend to decay through electron capture, while isotopes heavier than the stable ones decay by electron emission."

I thought I had a good understanding of the topic, but this sentence confused me. Let me explain my question:

As I understand it, in electron capture, the nucleus of an atom captures an electron from its inner orbital. This electron combines with a proton in the nucleus, forming a neutron. As a result, the atomic number decreases by 1, while the mass number remains the same.

For example: An element with mass number and atomic number decays via electron capture, producing a new element with atomic number − 1 and the same mass number , along with a neutrino and energy .

Here’s where my doubt comes in: for the energy to be released, doesn’t the parent element necessarily have to be more massive than the daughter element ? After all, the energy released should correspond to the difference in mass between the two, right?

If that’s the case, why does the Wikipedia passage suggest that isotopes lighter than the stable ones tend to decay via electron capture? This seems contradictory to me, as I would expect electron capture to occur when the parent nucleus (being heavier) could release enough energy to form a more stable daughter nucleus.

I’d appreciate it if someone could clarify this for me!

Answer 1

This is because the masses of different isotopes of the same element depend primarily on their neutron number $(N)$, and binding energy per nucleon $(B/A)$ vs $N$ is roughly parabolic. This $\beta^-$ decays possible for lighter isotopes and $\beta^+$ or Electron Capture (EC) decays possible for heavier isotopes.

Beta decays change neutron number and proton/atomic number $(Z)$, but conserve nucleon/mass number $(A=N+Z)$.

• for $\beta^-$ decays: $Z\rightarrow Z+1$, $N\rightarrow N-1$
• for $\beta^+$ decays: $Z\rightarrow Z-1$, $N\rightarrow N+1$ (including Electron Capture)

Lighter isotopes of a mid-range element with proton/atomic number (Z) are typically less bound than their corresponding $(Z+1,N-1)$ isobar and hence can decay via $\beta^-$ emission. Similarly heavier isotopes are less bound than their corresponding $(Z-1,N+1)$ isobar and hence can decay via $\beta^+$ emission or Electron Capture (EC).

An example of this is illustrated in the plot below of $B/A$ vs $N$ for $Z=39, 40, 41$. (The binding energy has been estimated from the semi-empirical mass formula since this was faster than getting the actual experimental values.) The lighter isotopes of Zirconium $(Z=40)$ on the left-hand side of the plot are less bound their their corresponding Yttrium $(Z_Y=Z-1=39)$ isobars, but more bound than their corresponding Niobium $(Z_{Nb}=Z+1=41)$ isobars, so $\beta^-$ decays are possible but $\beta^+$/EC decays are not. The opposite is true for the heavier Zirconium isotopes on the right-hand side of the plot.