9.1 Types of radioactive decay and nuclear equations

Types of Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei emit particles or energy to reach a more stable configuration. The type of decay a nucleus undergoes depends on its specific imbalance of protons and neutrons, and each decay type changes the nucleus in a predictable way. This makes it possible to write balanced nuclear equations and predict the products of any decay.

Types of Radioactive Decay

Alpha Decay

Alpha decay occurs in heavy nuclei that have too many protons and neutrons to remain stable. The nucleus ejects an alpha particle, which is a helium-4 nucleus ($^4_2He$) containing 2 protons and 2 neutrons. This decreases the atomic number by 2 and the mass number by 4.

Alpha decay is common in elements with Z > 83 (heavier than bismuth). For example, uranium-238 undergoes alpha decay to become thorium-234:

$^{238}_{92}U \rightarrow ^{234}_{90}Th + ^4_2He$

Beta Decay

Beta decay occurs in nuclei with an imbalanced neutron-to-proton ratio. There are two forms:

• Beta minus ($\beta^-$) decay: A neutron converts into a proton, emitting an electron and an antineutrino. The atomic number increases by 1, and the mass number stays the same. This happens in nuclei with too many neutrons. Example: carbon-14 decays into nitrogen-14.
• Beta plus ($\beta^+$) decay (positron emission): A proton converts into a neutron, emitting a positron and a neutrino. The atomic number decreases by 1, and the mass number stays the same. This happens in nuclei with too many protons (but not heavy enough for alpha decay). Example: fluorine-18 decays into oxygen-18.

Electron Capture

Electron capture is an alternative to positron emission for proton-rich nuclei. Instead of emitting a positron, the nucleus captures an inner-shell electron, which combines with a proton to form a neutron and a neutrino. The result is the same as $\beta^+$ decay: atomic number decreases by 1, mass number stays the same.

$^A_ZX + ^{\ 0}_{-1}e \rightarrow ^A_{Z-1}Y + \nu$

Electron capture tends to occur in heavier proton-rich nuclei where positron emission is energetically unfavorable. A common example is iron-55 capturing an electron to become manganese-55.

Gamma Radiation

Gamma ($\gamma$) radiation is the emission of high-energy photons when a nucleus transitions from an excited state to a lower energy state. Gamma emission changes neither the atomic number nor the mass number. It often accompanies alpha or beta decay, since the daughter nucleus is frequently left in an excited state. Cobalt-60 is a well-known gamma emitter used in medical applications.

Balanced Nuclear Equations

Every nuclear equation must be balanced: the total mass numbers (A) and total atomic numbers (Z) must be equal on both sides.

• Alpha decay: $^A_ZX \rightarrow ^{A-4}_{Z-2}Y + ^4_2He$
• Beta minus decay: $^A_ZX \rightarrow ^A_{Z+1}Y + ^{\ 0}_{-1}e + \bar{\nu}$
• Beta plus decay: $^A_ZX \rightarrow ^A_{Z-1}Y + ^{\ 0}_{+1}e + \nu$
• Electron capture: $^A_ZX + ^{\ 0}_{-1}e \rightarrow ^A_{Z-1}Y + \nu$
• Gamma radiation: $^A_ZX^* \rightarrow ^A_ZX + \gamma$

The asterisk ($^*$) indicates an excited nuclear state. The antineutrino ($\bar{\nu}$) and neutrino ($\nu$) carry away energy but have negligible mass and zero charge, so they don't affect the mass or atomic number balance.

Predicting Products of Radioactive Decay

To identify the daughter nuclide from a decay, follow these steps:

1. Write down the atomic number (Z) and mass number (A) of the parent nuclide.

2. Apply the changes based on the decay type:

   • Alpha decay: $Z_{daughter} = Z_{parent} - 2$, $A_{daughter} = A_{parent} - 4$
   • Beta minus decay: $Z_{daughter} = Z_{parent} + 1$, $A_{daughter} = A_{parent}$
   • Beta plus decay or electron capture: $Z_{daughter} = Z_{parent} - 1$, $A_{daughter} = A_{parent}$

3. Use the periodic table to identify the element that matches the new atomic number.

4. Write the full balanced equation, confirming that both A and Z totals match on each side.

For example, uranium-238 ($^{238}_{92}U$) undergoes alpha decay: $Z = 92 - 2 = 90$ and $A = 238 - 4 = 234$. Element 90 is thorium, so the daughter nuclide is $^{234}_{90}Th$.

A common mistake is forgetting to look up the new element after calculating Z. The atomic number defines the element, so changing Z always means you have a different element (except in gamma decay, where Z doesn't change).

Nuclear Stability and Radioactive Decay

The Band of Stability

Nuclear stability depends on the ratio of neutrons to protons in the nucleus. The band of stability is a plot of stable nuclei graphed by their number of protons (Z) versus their number of neutrons (N). Stable nuclei fall within a narrow region on this graph:

• For light nuclei (Z ≤ 20), the stable ratio is roughly 1:1 (equal protons and neutrons).
• For heavier nuclei, more neutrons are needed to offset the growing electrostatic repulsion between protons. The ratio shifts toward about 1.5 neutrons per proton for the heaviest stable nuclei.

Where a nucleus sits relative to the band of stability predicts which type of decay it will undergo:

• Above the band (neutron-rich): too many neutrons, so the nucleus undergoes $\beta^-$ decay to convert a neutron into a proton. Example: $^{14}_6C \rightarrow ^{14}_7N + ^{\ 0}_{-1}e + \bar{\nu}$
• Below the band (proton-rich): too many protons, so the nucleus undergoes $\beta^+$ decay or electron capture to convert a proton into a neutron. Example: $^{18}_9F \rightarrow ^{18}_8O + ^{\ 0}_{+1}e + \nu$
• Heavy nuclei beyond the band (Z > 83): no stable configuration exists, so these nuclei undergo alpha decay to shed both protons and neutrons. Example: $^{238}_{92}U \rightarrow ^{234}_{90}Th + ^4_2He$

Think of the band of stability as a target zone. Unstable nuclei "aim" for it by undergoing whichever decay moves them closer. Neutron-rich nuclei need fewer neutrons ($\beta^-$ converts a neutron to a proton). Proton-rich nuclei need fewer protons ($\beta^+$ or electron capture converts a proton to a neutron). Very heavy nuclei need to lose mass overall (alpha decay removes 4 nucleons at once).

Forces Inside the Nucleus

Two competing forces govern nuclear stability:

• The strong nuclear force acts over very short distances and binds protons and neutrons (collectively called nucleons) together. It's the strongest of the fundamental forces but only operates within about $10^{-15}$ m, roughly the diameter of a nucleon.
• Electrostatic (Coulomb) repulsion pushes protons apart because they all carry positive charge. Unlike the strong force, this repulsion acts over longer distances. As the number of protons increases, repulsion grows, which is why heavier nuclei need proportionally more neutrons to remain stable.

When electrostatic repulsion overwhelms the strong force, the nucleus becomes unstable and decays. This is also why no element with Z > 83 has any stable isotopes. Uranium-238, for instance, undergoes a long decay series of multiple alpha and beta decays (14 steps total) before finally reaching stable lead-206 ($^{206}_{82}Pb$).