Nuclear energy (AQA A-Level Physics): Revision Notes

Nuclear instability

Understanding nuclides and nuclear stability

A nuclide is a type of nucleus with a specific number of protons, a specific number of neutrons, and a specific energy state. This precise specification is what distinguishes one nuclide from another, even among isotopes of the same element.

In nature, we find that some nuclides are inherently stable and will never undergo radioactive decay, while others are unstable and will spontaneously transform into different nuclides through radioactive processes. Of the naturally occurring nuclides, a significant proportion have remained unchanged since the Earth's formation approximately 4.6 billion years ago. These are the stable nuclides that form the stable matter we observe today.

When an unstable nuclide undergoes radioactive decay, it may transform into another unstable nuclide, which will itself decay. This process can continue through multiple stages, creating what is known as a decay series. A decay series continues until a stable nuclide is finally produced.

The neutron-to-proton ratio and nuclear stability

The stability of a nucleus depends critically on the balance between the number of neutrons and the number of protons it contains. When we plot neutron number (N) against proton number (Z) for all stable nuclides, a clear pattern emerges that reveals the requirements for nuclear stability.

For light nuclei with proton number 20 or less, the pattern is straightforward: stable nuclei follow approximately a straight line where N ≈ Z. This means light stable nuclei contain roughly equal numbers of neutrons and protons. For instance, carbon-12 has 6 protons and 6 neutrons, giving a perfect 1:1 ratio.

However, as we move to heavier nuclei with proton numbers greater than 20, a different pattern appears. The ratio N/Z becomes greater than 1 and progressively increases for heavier stable nuclei. In other words, heavier stable nuclei require increasingly more neutrons than protons. The heaviest stable nuclide is bismuth-209, which has 83 protons and 126 neutrons, giving a ratio N/Z = 1.52. For the very heaviest nuclei to have any chance of stability, this ratio must reach approximately 1.6.

Alpha decay

Alpha decay is a radioactive process in which an unstable nucleus emits an alpha particle. An alpha particle consists of 2 protons and 2 neutrons bound together, which is identical to a helium-4 nucleus. The alpha particle can be represented by the symbol $^4_2\text{He}$ or sometimes simply as $\alpha$.

When a nucleus undergoes alpha decay, the resulting nucleus (called the daughter nucleus) has 4 fewer nucleons than the original (the parent nucleus) and 2 fewer protons. To illustrate this process, consider the naturally occurring uranium-238 nucleus, which is unstable and decays by alpha emission with a half-life of 4.5 billion years.

The decay of uranium-238 can be represented by the equation:

$^{238}_{92}\text{U} \rightarrow ^{234}_{90}\text{Th} + ^4_2\text{He}$

In this equation, we can verify that both the nucleon numbers (mass numbers) and the proton numbers balance on both sides. The uranium-238 has 238 nucleons and 92 protons. After decay, the thorium daughter has 234 nucleons and 90 protons, while the alpha particle contributes 4 nucleons and 2 protons, giving a total of 238 and 92 respectively, which matches the parent nucleus.

An important characteristic of alpha decay is that the alpha particles emitted from a sample of a particular radioactive isotope all carry the same distinct kinetic energy. This is described as being monoenergetic. The decay energy is the kinetic energy of the daughter nucleus and the alpha particle, shared in a fixed ratio. Since the parent and daughter in alpha decay are always the same species, and the decay energy is always released in this same fixed ratio, all alpha particles from that isotope have identical kinetic energy.

Beta-minus decay

Beta-minus decay occurs in nuclei that have too many neutrons relative to protons for stability. In this process, a neutron within the nucleus transforms into a proton, and the nucleus emits an electron (the beta-minus particle) and an electron antineutrino.

The transformation that occurs within the nucleus during beta-minus decay can be represented as:

$\text{n} \rightarrow \text{p} + \text{e}^- + \bar{\nu}_e$

where n represents the neutron, p represents the proton, $\text{e}^-$ represents the electron, and $\bar{\nu}_e$ represents the electron antineutrino. The electron symbol $^0_{-1}\text{e}$ is sometimes written as $^0_{-1}\beta$. The proton remains within the nucleus after this transformation, but the electron and antineutrino are emitted.

As an example, consider thorium-234, which is the daughter nucleus produced from uranium-238 alpha decay. Thorium-234 is itself unstable and decays by beta-minus emission with a half-life of 24.1 days. The decay equation is:

$^{234}_{90}\text{Th} \rightarrow ^{234}_{91}\text{Pa} + ^0_{-1}\text{e} + ^0\bar{\nu}_e$

The daughter nucleus, protactinium-234, has the same nucleon number (234) as the parent thorium-234, but its proton number has increased by 1 (from 90 to 91). This makes sense because a neutron has been converted to a proton inside the nucleus.

Beta-plus decay

Nuclei that are over-rich in protons relative to neutrons can decay by beta-plus emission. In this process, a proton within the nucleus transforms into a neutron, and the nucleus emits a positron (the antiparticle of the electron, also called a beta-plus particle) and an electron neutrino.

The transformation occurring within the nucleus is:

$\text{p} \rightarrow \text{n} + \text{e}^+ + \nu_e$

where p represents the proton, n represents the neutron, $\text{e}^+$ represents the positron, and $\nu_e$ represents the electron neutrino. The positron can be represented by the symbol $^0_{+1}\text{e}$ or $^0_{+1}\beta$. The neutron remains in the nucleus following this transformation, while the positron and neutrino are emitted.

An example of beta-plus decay is oxygen-15, which is frequently used in PET (positron emission tomography) imaging. Oxygen-15 can be produced artificially by bombarding nitrogen-14 with high-energy deuterons using a cyclotron. It decays with a half-life of about 2 hours. The decay equation is:

$^{15}_8\text{O} \rightarrow ^{15}_7\text{N} + ^0_{+1}\text{e} + ^0\nu_e$

The daughter nucleus, nitrogen-15, has the same nucleon number as oxygen-15 but has one fewer proton (decreasing from 8 to 7), which corresponds to the conversion of a proton to a neutron.

Similar to beta-minus decay, the decay energy in beta-plus emission is shared between the daughter nucleus, the positron, and the electron neutrino. Consequently, the emitted positrons have a continuous energy spectrum up to a specific maximum value characteristic of the parent nucleus, as in beta-minus decay. The change of a proton to a neutron, a positron, and an electron neutrino only occurs inside an unstable nucleus; a free proton is a stable particle and does not undergo this transformation.

Electron capture

An alternative decay mechanism for nuclei that are over-rich in protons is electron capture, sometimes referred to as K-capture. This process is relatively rare in nature but can be produced artificially. The most well-known naturally occurring example is beryllium-7, produced in the atmosphere by cosmic ray interactions, which decays by electron capture with a half-life of 53 days.

In electron capture, the nucleus absorbs one of its inner orbital electrons, typically from the innermost (K) shell. This captured electron combines with a proton in the nucleus to form a neutron, with the emission of an electron neutrino. The transformation within the nucleus can be written as:

$\text{p} + \text{e}^- \rightarrow \text{n} + \nu_e$

The decay of beryllium-7 by electron capture is represented as:

$^7_4\text{Be} + ^0_{-1}\text{e} \rightarrow ^7_3\text{Li} + ^0\nu_e$

The daughter nucleus, lithium-7, has the same nucleon number as beryllium-7 but has one fewer proton (decreasing from 4 to 3), corresponding to the proton-to-neutron conversion.

The general equation for electron capture is:

$^A_Z\text{X} + ^0_{-1}\text{e} \rightarrow ^A_{Z-1}\text{Y} + ^0\nu_e$

Electron capture produces the same change in the nucleus as beta-plus decay: both reduce the proton number by 1 while keeping the nucleon number constant. The choice between these competing processes depends on the specific nuclear energy levels and binding energies involved.

Other decay mechanisms

Nuclear stability and the strong nuclear force

To understand why the neutron-to-proton ratio must increase for heavier nuclei, we need to consider the forces acting within the nucleus. The strong nuclear force is responsible for binding protons and neutrons together to form stable atomic nuclei. This force is attractive between nucleons when they are separated by distances of approximately 1 femtometre (1 fm = $10^{-15}$ m) and can overcome the electrostatic (Coulomb) repulsion between positively charged protons.

However, at distances around 3 fm, the strong nuclear force between two protons becomes negligible, while the Coulomb repulsion between them remains significant and continues to act over much larger distances. This creates a fundamental challenge for nuclear stability: as more protons are added to create heavier nuclei, the Coulomb repulsion between all the protons becomes increasingly problematic.

Predicting decay modes from nuclear composition

The position of an unstable nuclide on a graph of neutron number versus proton number provides valuable information about which decay mode it is likely to undergo. Stable nuclei occupy a narrow region called the stability region or "valley of stability" on this graph.

Unstable nuclei decaying by alpha emission tend to be heavy nuclei with corresponding points on the N versus Z graph lying mostly beyond the stability region in the heavy nucleus area. Alpha decay is effective at reducing both the neutron and proton numbers significantly, moving the nucleus toward lighter, more stable configurations.

Points corresponding to beta-minus emitters tend to lie to the left of the stability region. These nuclei are neutron-rich relative to the stability requirement for their mass. Beta-minus decay converts a neutron to a proton, increasing Z while keeping A constant, which moves the nucleus rightward on the graph toward the stability region.

Conversely, points corresponding to nuclei decaying by beta-plus emission or by electron capture tend to lie to the right of the stability region. These nuclei are proton-rich relative to what is required for stability. Both beta-plus decay and electron capture convert a proton to a neutron, decreasing Z while keeping A constant, moving the nucleus leftward toward the stability region.