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Physics LibreTexts

3.5: Stability of Nuclei

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In Figure 3.5.1 we have color coded the nuclei of a given mass A=N+Z by their mass, red for those of lowest mass through to magenta for those of highest mass. We can see that typically the nuclei that are most stable for fixed A have more neutrons than protons, more so for large A increases than for low A. This is the “neutron excess”.

mass_tab.png
Figure 3.5.1: The valley of stability

β decay

If we look at the mass of nuclides with fixed nucleon number A (i.e., roughly perpendicular cuts through the valley of stability in Figure 3.5.1), we can see that the masses vary strongly,

1.png
Figure 3.5.2: The negative of binding energy per nucleon for nuclides with fixed A: (left) A=56 and (right) A=150. The profile of binding energy across the valley of stability is roughly a parabola (e.g., Iron-56 is stable, while Vandium-56 is unstable to β^− decay.

It is known that a free neutron is not a stable particle, it actually decays by emission of an electron and an antineutrino,

n \rightarrow p + e^-+\bar{\nu}_e. \nonumber

The reason that this reaction can take place is that it is endothermic,

m_n c^2 > m_p c^2 + m_e c^2. \nonumber

Here we assume that the neutrino has no mass.

The degree of allowance of such a reaction is usually expressed in a Q value, the amount of energy released in such a reaction,

\begin{align} Q &=m_n c^2 - m_p c^2 - m_e c^2 \\[5pt] &=939.6-938.3-0.5=0.8\text{ MeV}. \end{align} \nonumber

Generically it is found that two reaction may take place, depending on the balance of masses. Either a neutron “\beta decays” as sketched above, or we have the inverse reaction

p \rightarrow n + e^++{\nu}_e. \nonumber

For historical reason the electron or positron emitted in such a process is called a \beta particle. Thus in \beta^- decay of a nucleus, a nucleus of Z protons and N neutrons turns into one of Z+1 protons and N-1 neutrons (moving towards the right in Figure \PageIndex{2A}. In \beta^+ decay the nucleus moves to the left. Since in that figure I am using atomic masses, the Q factor is

\begin{aligned} Q_{\beta^-}&= M(A,Z)c^2-M(A,Z+1)c^2,\nonumber\\ Q_{\beta^-}&= M(A,Z)c^2-M(A,Z-1)c^2-2m_e c^2.\end{aligned} \nonumber

The double electron mass contribution in this last equation because the atom looses one electron, as well as emits a positron with has the same mass as the electron.

In similar ways we can study the fact whether reactions where a single nucleon (neutron or proton) is emitted, as well as those where more complicated objects, such as Helium nuclei (\alpha particles) are emitted. I shall return to such processed later, but let us note the Q values,

\begin{aligned} \text{neutron emission} && Q&=[M(A,Z)-M(A-1,Z)-m_n]c^2, \nonumber\\ \text{proton emission} && Q&=[M(A,Z)-M(A-1,Z-1)-M(1,1)]c^2, \nonumber\\ \text{$\alpha$ emission} && Q&=[M(A,Z)-M(A-4,Z-2)-M(4,2)]c^2, \nonumber\\ \text{break up} && Q&=[M(A,Z)-M(A-A_1,Z-Z_1)-M(A_1,Z_1)]c^2.\end{aligned} \nonumber


This page titled 3.5: Stability of Nuclei is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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