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Electron capture is a fascinating nuclear decay process in which the nucleus of an atom captures an inner orbital electron. This captured electron merges with a proton, transforming it into a neutron and releasing a neutrino. As a result, the atomic number of the element decreases by one, while the mass number remains unchanged. This type of decay generally occurs in proton-rich, heavy nuclei.

For example, when Gallium-67 undergoes electron capture, it transforms into Zinc-67. The reaction can be written as:
\[ ^{67}_{31}\text{Ga} + e^- \rightarrow ^{67}_{30}\text{Zn} + \text{neutrino} \]

It's important to remember that the overall charge and nucleon number are conserved during this process. This transformation helps unstable isotopes move towards a more stable state.

Positron emission, symbolized as \(\beta^+\), involves a proton in the nucleus transforming into a neutron. During this transformation, a positron (the antimatter counterpart of an electron) and a neutrino are emitted. As a result, the atomic number decreases by one, but the mass number stays the same.

This process is common in nuclei that have an excess of protons compared to neutrons. For Potassium-38, the representation of positron emission is:
\[ ^{38}_{19}\text{K} \rightarrow ^{38}_{18}\text{Ar} + \beta^+ + \text{neutrino} \]

By emitting a positron, the nucleus moves towards a more balanced and stable state, with the reduction in the overall positive charge aiding in this stabilization process.

Unlike other types of decay, gamma (\( \gamma \) ) emission involves the release of energy rather than particles. It occurs when a nucleus in an excited state releases energy to move to a lower energy state. This release of energy takes place in the form of gamma rays, which are high-energy photons.

In this process, neither the atomic number nor the mass number changes, which means the identity of the element remains the same.

For Technetium-99m, the process can be expressed as:
\[ ^{99m}_{43}\text{Tc} \rightarrow ^{99}_{43}\text{Tc} + \gamma \]

Gamma emission is crucial in medical imaging and other applications, as it allows for the transition of atomic states without altering the elemental composition.

Beta emission, or \(\beta^-\) decay, involves a neutron turning into a proton, accompanied by the emission of an electron (known as a beta particle) and an antineutrino. This conversion increases the atomic number by one while the mass number stays constant.

This process is frequent in neutron-rich isotopes, helping them achieve a more stable and balanced nuclear configuration. For Manganese-56, the beta decay can be represented as follows:
\[ ^{56}_{25}\text{Mn} \rightarrow ^{56}_{26}\text{Fe} + \beta^- + \text{antineutrino} \]

Through beta emission, the nucleus releases an excess negative charge and transitions into a slightly more positively charged state, shifting towards stability.