What atomic particle determined specific isotopes?

An isotope of a number has a different number of neutrons than the normal atom has. Carbon-13 has a weight of 13 amu. Carbon has 6 protons because it has the atomic number of 6, and it would normally have the same number of neutrons as well. But since amu is protons plus neutrons, carbon-13 has 7 neutrons. Thus, the change in the number of neutrons makes it an isotope.