In beta minus (β⁻) decay, a neutron-rich parent nucleus \( P \) undergoes a transformation in which one of its neutrons is converted into a proton, an electron (\( e^- \)), and an electron antineutrino (\( \bar{\nu}_e \)) are emitted. This process results in a change in the atomic number of the nucleus, which increases by one, while the atomic mass number remains unchanged. The equation for β⁻ decay is as follows:
\[ _{Z}^A P \rightarrow _{Z+1}^A D + e^- + \bar{\nu}_e \]
Where:
Beta minus decay is a type of weak interaction, meaning that it occurs due to the weak nuclear force, which is responsible for changing the flavor of quarks (a neutron consists of one up quark and two down quarks, while a proton consists of two up quarks and one down quark). In β⁻ decay, a neutron in the parent nucleus \( P \) is converted into a proton, emitting an electron and an electron antineutrino in the process. The newly formed proton becomes part of the daughter nucleus \( D \), which has an atomic number one unit higher than the parent nucleus.
This process is energetically favorable because the transformation of a neutron to a proton results in the emission of energy, which is carried away by the emitted electron and the antineutrino. The energy released in β⁻ decay is referred to as the "decay energy" or \( Q \)-value.
A typical example of β⁻ decay is the decay of \( \ce{^{60}Co} \), a radioactive isotope with a half-life of 5.26 years. In this decay, the \( \ce{^{60}Co} \) nucleus undergoes β⁻ decay to form an excited state of \( \ce{^{60}Ni} \). The process proceeds as follows:
\[ _{27}^{60} \text{Co} \xrightarrow{\beta^-} _{28}^{60} \text{Ni}^* \xrightarrow{\gamma} _{28}^{60} \text{Ni} \]
Where:
In this case, a neutron in the \( \ce{^{60}Co} \) nucleus is converted into a proton, emitting an electron (\( e^- \)) and an electron antineutrino (\( \bar{\nu}_e \)) in the process. The daughter nucleus formed is an excited state of \( \ce{^{60}Ni} \) (\( \ce{^{60}Ni^*} \)), which then de-excites by emitting gamma radiation (\( \gamma \)) to reach the stable ground state of \( \ce{^{60}Ni} \).
Another well-known example of β⁻ decay is the decay of \( \ce{^{14}C} \) (carbon-14), which is used in radiocarbon dating. Carbon-14 has a half-life of about 5730 years and decays into nitrogen-14 (\( \ce{^{14}N} \)) through β⁻ decay:
\[ _{6}^{14} \text{C} \rightarrow _{7}^{14} \text{N} + e^- + \bar{\nu}_e \]
Where:
The decay of \( \ce{^{14}C} \) is used in the dating of organic materials by measuring the ratio of \( \ce{^{14}C} \) to stable carbon isotopes in a sample. As \( \ce{^{14}C} \) decays over time, the amount of \( \ce{^{14}C} \) decreases, and by knowing the half-life of \( \ce{^{14}C} \), scientists can estimate the age of the sample.
β⁻ decay is crucial in many medical applications, particularly in nuclear medicine. Radioactive isotopes that undergo β⁻ decay are commonly used in imaging techniques such as Positron Emission Tomography (PET). For example, \( \ce{^{18}F} \), a fluorine isotope used in PET scans, decays via β⁻ decay. The emitted positron interacts with an electron, resulting in the annihilation of both particles and the emission of two gamma photons. These gamma photons are detected to produce high-resolution images of metabolic activity in the body.
In addition to medical applications, β⁻ decay also plays a role in environmental studies, such as carbon dating, where the decay of \( \ce{^{14}C} \) in organic materials helps estimate the age of fossils and archaeological finds. The understanding of β⁻ decay and the use of carbon dating has revolutionized paleontology, archaeology, and other fields.
The energy released during β⁻ decay is referred to as the decay energy, also known as the Q-value. This energy is shared between the emitted electron (\( e^- \)), the electron antineutrino (\( \bar{\nu}_e \)), and the daughter nucleus.
The decay energy \( Q \) can be calculated using the following equation:
\[ Q = \left[ M(P) - M(D) - m(e^-) \right] c^2 \]
Where:
The decay energy is typically on the order of several hundred keV (kiloelectron volts), depending on the specific isotopes involved. This energy is shared between the daughter nucleus and the emitted particles, and the daughter nucleus typically carries away the smallest share of the energy.