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Alpha emission is a type of radioactive decay where an unstable nucleus releases an alpha particle. An alpha particle is essentially a helium nucleus, which includes 2 protons and 2 neutrons. This results in a decrease of the atomic number by 2 and a decrease in the mass number by 4. For instance, when Uranium-238 undergoes alpha emission, it releases an alpha particle (\(\text{He}_{2}^{4}\)), and transforms into Thorium-234. The process can be represented as:\[\text{U}_{92}^{238} \rightarrow \text{Th}_{90}^{234} + \text{He}_{2}^{4}\] Here, Uranium (\(\text{U}_{92}^{238}\)) loses 2 protons and 2 neutrons, transforming into Thorium (\(\text{Th}_{90}^{234}\)), thus becoming more stable. Alpha decay is prevalent in elements with very large nuclei in order to stabilize by decreasing their size.

Beta emission involves the emission of a beta particle, which is a high-energy, high-speed electron or positron. In beta minus (\(\beta^-\)) decay, a neutron changes into a proton, emitting an electron and an antineutrino. This increases the atomic number by 1, but the mass number remains unchanged.For example, with Aluminum-28, beta emission results in the creation of Silicon-28. The equation is:\[\text{Al}_{13}^{28} \rightarrow \text{Si}_{14}^{28} + e^-\]This transformation occurs because a neutron in Aluminum's nucleus is converted into a proton, resulting in an element with one more proton in the periodic table, therefore altering the original element's identity.Beta emission is crucial in processes where elements adjust the neutron-to-proton ratio to achieve greater stability.

Positron emission is a kind of beta decay where a proton is transformed into a neutron, releasing a positron and a neutrino in the process. The atomic number decreases by 1 as a result, but the mass number does not change.Taking Oxygen-15 as an example, it undergoes positron emission to become Nitrogen-15:\[\text{O}_{8}^{15} \rightarrow \text{N}_{7}^{15} + e^+\]Here, a proton from the oxygen nucleus is converted into a neutron, emitting a positron (\(e^+\)) and resulting in the formation of a different element, Nitrogen.Positron emission is often encountered in proton-rich nuclei seeking stabilization as they adjust the balance between protons and neutrons.

Electron capture happens when an inner electron is pulled into the nucleus, causing a proton to convert into a neutron. This process results in the decrease of the atomic number by 1 while leaving the mass number unchanged. In the case of Iron-55, electron capture leads to the transformation into Manganese-55:\[\text{Fe}_{26}^{55} + e^- \rightarrow \text{Mn}_{25}^{55}\]This process virtually "captures" the electron, reducing the number of protons by changing one into a neutron. As a consequence, the element shifts to another with one fewer proton in its nucleus.Electron capture is a mechanism used by some heavy elements to achieve a more stable neutron-to-proton ratio.