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Basics
Inside of a nucleus, any combination of protons and neutrons can be present. The most simple nucleus is that of hydrogen, which contains just one proton, although some nuclei contain over 300. The protons and neutrons contained in a nucleus are referred to as nucleons, which are held together by a force known as binding energy.
Binding Energy
Binding energy within the nucleus increases in strength as the mass of the nucleus increases. Binding energy can vary, which causes a less stable nucleus or shorter half-life. Protons are particles that carry one positive charge, while neutrons are neutral particles. Protons can be further dissected into quarks and leptons, which are believed to be the source of all matter (along with their corresponding antimatter particles).
Leptons and Quarks
Leptons can be neutral or positively charged -- positive leptons carry the charge equivalent to one electron, which is why they are noted as a lower case e in equations. Quarks on the other hand, are known as point-like particles, and have one-third or two-thirds the charge of an electron.
Standard Model
The mapping of sub-nucleus particles like quarks and leptons is a major development in the world of physics, and is known as the Standard Model of Fundamental Interactions. This model was developed over two to three decades, and is constantly refined as new discoveries are made. In this model, leptons and quarks are categorized into three families. Two families have little impact on the world, as the particles in these categories are short lived, but the main family is known as up-and-down quarks and leptons.