Nuclear Decays

$\alpha,\beta$ and $\gamma$ radiation can be created either by nuclei decay or through fusion and fission reactions

$\beta$ decay

$\beta$ -decay: Nuclear reaction where a nucleon changes type by emitting either:

An electron and an electron anti-neutrino
A positron (anti-electron) and an electron neutrino

🗒️ Note: the “$\beta$ particle” is the electron without the neutrino due to Rutherford

Means for a nucleus to increase its binding energy (move towards stability)

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At the level of nucleons we can write reaction 1. and 2. as

$$ \begin{aligned} n&\to p+e^{-} +\overline \nu _e & \quad \beta^{-}&\;\text{decay ($\beta$ decay)}\\ p&\to n+e^{+} + \nu _e & \quad \beta^{+}&\;\text{decay (positron decay/emission)}

\end{aligned} $$
The energy release is the difference in nuclear mass between initial and final, for $\beta^{-}$ decay

$$ ^A_Z{\rm X}\to {Z+1}^{\;\;\;\,A}{\hspace*{-0.3em} \rm Y} +e^{-}+\overline \nu e \\ Q=\left [ M\text{nuc.} (^A_Z{\rm X}) -M{\text{nuc.}} ({Z+1}^{\;\;\;\,A}{ \rm Y})-m_e-m\nu\right ]c^2 $$

where $M_\text{nuc.}$ refers to the nuclear mass. We will approximate $m_\nu \approx 0$

If we re-write the equation in terms of atomic masses $M_\text{at.}(Z,A)=M_\text{nuc.} (Z,A)+Zm_e$

$$ \begin{aligned} Q&=\left [ M_\text{at.} (^A_Z{\rm X}) -Z m_e -(M_{\text{at.}} ({Z+1}^{\;\;\;\,A}{ \rm Y}) -(Z+1)m_e)-m_e\right ]c^2 \\ &=\left [ M\text{at.} (^A_Z{\rm X}) -M_{\text{at.}} (_{Z+1}^{\;\;\;\,A}{ \rm Y}) \right ]c^2 \end{aligned} $$

🗒️ Note: we ignored the electron binding energies as they will largely cancel out
For a $\beta ^+$ the reaction is $^A_Z {\rm X} \rightarrow _{Z-1}^{\;\;\;\,A}{\hspace*{-0.3em} \rm Y} + e^+ + \nu_e$ using the same method we find

$$ \begin{aligned} Q&=\left [ M_\text{nuc.} (^A_Z{\rm X}) -M_{\text{nuc.}} ({Z-1}^{\;\;\;\,A}{ \rm Y})-m_e-m\nu\right ]c^2 \\ &=\left [ M_\text{at.} (^A_Z{\rm X}) -Z m_e -(M_{\text{at.}} ({Z-1}^{\;\;\;\,A}{ \rm Y}) -(Z-1)m_e)-m_e\right ]c^2 \\ &=\left [ M\text{at.} (^A_Z{\rm X}) -M_{\text{at.}} (_{Z-1}^{\;\;\;\,A}{ \rm Y})-2m_e \right ]c^2

\end{aligned} $$

🗒️ Note: this leads to an electron deficit in the final atom
An electron can be absorbed by a proton through the following electron capture process

Due to the asymmetry term, nuclei are more stable if the Fermi energies for neutrons and protons are similar

If we study an isobar (constant $A$) with SEMF (curve) we see for odd $A$ we get a good correlation with lab data (dots)

🗒️ Note: the arrows are $\beta^\pm /EC$ decays

For even term we have the paring term $a_p$ thus we get 2 parabolas with good agreement between theory and data

🗒️ Note: in the second diagram $^{92}{42} {\rm Mo}$ could double decay to $^{92}{40}{\rm Zr}$ bypassing $^{92}_{41}{\rm Nb}$ completely. however this process is very unlikely with a half life of $10^{21}$ years

Fusion

For $A<56$ the binding energy per nucleon $E_B/A$ increases with $A$ which means energy is released when light nuclei fuse

💼 Case: stellar nucleosynthetic

$$ \small\begin{aligned} p+p&\to ^2_1 \hspace*{-0.2em}{\rm D} + e^+ + \nu &Q&=0.4\, \text{Mev} \\ p+^2_1\hspace*{-0.2em} {\rm D}&\to ^3_2 \hspace*{-0.2em}{\rm He}+\gamma &Q&=5.5\,\text{Mev} \\ ^3_2{\rm He}+^3_1 \hspace*{-0.2em}{\rm He}&\to ^4_2 \hspace*{-0.2em}{\rm He}+2p &Q&=12.9\,\text{Mev}

\end{aligned} $$

where $^2_1\rm D$ is deuteron

🗒️ Note: the process end $\sim^{56}{26} \hspace*{-0.2em}\rm Fe$ explaining the presence of ${28}\rm Ni$ and $_{26}\rm Fe$ in the core of planets

$s$ and $r$ processes

Slow neutron capture process ($s$)

📖 Definition: if the neutron flux is such that the process is more likely to decay before it absorbs another neutron

💃 Example: The next isotopes of $^{56}{26}{\rm Fe}$ are $^{57}{26}\rm Fe$ and $^{58}{26}\rm Fe$ but the third $^{59}{26}\rm Fe$ has a half life of $\sim 45$ days.

If the likelihood of absorbing an additional neutron is less than 45 days then it will $\beta$ decay to $^{59}_{27}\rm Co$

🗒️ Note: this process could continue until $^{209}_{83}\rm Bi$ at which point there are no isotopes stable enough. It follows a zig zag path along the line of stability

Rapid neutron capture process $(r)$

📖 Definition: When nuclei are bombarded with neutrons faster than they can $\beta$-decay

This process happens in neutron star mergers

🗒️ Note: this process continues until the isotope is so unstable that it decays faster than it can absorb an additional neutron. This phenomena leads to straight horizontal and vertical lines along the neutron drip lines

🗒️ Note: the nuclear drip line is the boundary beyond which atomic nuclei are unbound with respect to the emission of a proton or neutron

$\alpha$ decay

For $A>56$ the binding energy per nucleon $E_B/A$ decreases with $A$ which means energy is released when nuclei split into light nuclei

Cluster decay: are fission processes that a more reliable like $\alpha$ -decay.

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🗒️ Note: the $\alpha$ particle is $^4_2\rm He$, a light strongly bond nuclei

$\alpha$-decay reaction have the form

$$ ^A_Z {\rm X} \to ^{A-4}_{Z-2}\hspace*{-0.2em} {\rm Y}+^4_2\hspace*{-0.2em}{\rm He} $$
We can write the energy released as

$$ \begin{aligned} Q&=(M_\text{nuc.}(^A_Z {\rm X})-M_\text{nuc.} (^{A-4}{Z-2} {\rm Y})-M\text{nuc.}(^4_2{\rm He}))c^2 \\ &=(M_\text{at.}(^A_Z{\rm X})-Zm_e-(M_\text{at.} (^{A-4}{Z-2} {\rm Y})-(Z-2)m_e)-(M\text{at.}(^4_2{\rm He})-2m_e))c^2 \\ &=(M_\text{at.}(^A_Z{\rm X})-M_\text{at.}(^{A-4}{Z-2} {\rm Y} )-M\text{at.}(^4_2{\rm He}))c^2

\end{aligned} $$