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Electron capture is a type of nuclear decay in which an atomic nucleus absorbs an inner-shell electron from its own electron cloud. This process results in the transformation of a proton into a neutron and simultaneously emits a neutrino, which is a nearly massless particle.

During electron capture, the mass number (A) of the isotope remains the same because the nucleus is simply reconfiguring its particles without adding or losing nucleons. However, the atomic number (Z) decreases by one because one proton (positively charged particle) is converted into one neutron (a neutral particle).

The notation for electron capture involves adding an electron (\(e^-\)) on the reactant side of the nuclear equation. For instance, when Chromium-51 (\( ^{51}_{24}Cr \)) undergoes electron capture, the equation is written as \( ^{51}_{24}Cr + e^- \rightarrow ^{51}_{23}V \), where Vanadium-51 (\( ^{51}_{23}V \) ) is the resulting nuclide.

Beta decay represents a form of radioactive decay where a beta particle, which is essentially an electron or a positron, is emitted from an atomic nucleus. For beta-minus decay (\( \beta^- \) decay), a neutron is converted into a proton with the emission of an electron and an antineutrino. This causes the atomic number (Z) to increase by one while the mass number (A) remains constant.

In beta-plus decay (\( \beta^+ \) decay), the opposite occurs where a proton is transformed into a neutron with the emission of a positron and a neutrino, causing the atomic number to decrease by one. The general formula for beta-minus decay is \( ^A_ZX \rightarrow ^A_{Z+1}Y + \beta^- \).

As an example, Iodine-131 (\( ^{131}_{53}I \) ) decaying into Xenon-131 (\( ^{131}_{54}Xe \) ) with the emission of a beta particle can be represented as \( ^{131}_{53}I \rightarrow ^{131}_{54}Xe + \beta^- \). Similarly, Phosphorus-32 (\( ^{32}_{15}P \) ) undergoes beta decay to form Sulfur-32 (\( ^{32}_{16}S \) ), which is expressed as \( ^{32}_{15}P \rightarrow ^{32}_{16}S + \beta^- \).

Radioisotopes have wide-ranging applications in the field of medicine, industry, and scientific research due to their unique radioactive properties. In medical applications, radioactive tracers can help diagnose and treat various health conditions. For example, Chromium-51, with its ability to target the spleen, is utilized in red blood cell studies to understand and diagnose blood disorders.

Another common use is in the treatment of thyroid conditions, where Iodine-131 is administered to patients to manage hyperthyroidism. The beta particles emitted by the decaying Iodine-131 can destroy overactive thyroid cells, thus treating the condition.

In the research domain, radioisotopes such as Phosphorus-32 are used to trace the assimilation of nutrients in organisms. This isotope accumulates in the liver, allowing scientists to monitor liver function and health. These examples demonstrate the critical role radioisotopes play in enhancing our ability to detect and remedy medical challenges, as well as to facilitate advancements in biological research.