Nuclear properties

Nuclei consist of positively charged protons and electrically neutral neutrons packed closely together.  Protons and neutrons are each about 1.4*10^–15 m in diameter,  and the size of a nucleus is essentially the size of a ball of these particles.

The total number of nucleons in a nucleus is usually denoted by the mass number A, where A = Z + N, Z protons and N neutrons.  The chemical properties of an atom are determined by the number of electrons, the same as the number of protons Z.  This is called the atomic number.  Nuclei can have the same atomic number, but different numbers of neutrons.  These nuclei are called isotopes.

The nuclear force which holds the nucleus together is attractive, strong, and short ranged.  Its range is about equal to 4 proton diameters.  The nuclear force treats protons and neutrons equally and does not act on electrons. 

Iron 56 has the largest binding energy per nucleon, no other nucleus is more tightly bound.  The reason Iron 56 is most strongly bound is because the diameter of an iron nucleus is about equal to the range of the nuclear force.  Iron 56 is the largest nucleus in which every nucleon attracts every other nucleon.  The diameter of the Iron 56 nucleus is the distance over which the attractive nuclear force is stronger than the repulsive electrostatic force for each particle pair.

Atomic and nuclear data tables often list the mass of the neutral atom (not that of the nucleus) in atomic mass units (u).  
In terms of the nuclear masses, we write for the binding energy B(Z,N) of a nucleus with Z protons and N neutrons

B(Z,N) = c²(Z*m_p + N*m_n - M_nuc(Z,N)).

In terms of the atomic masses, we write

B(Z,N) = c²(Z*m_H + N*m_n - M_atom(Z,N)).

Nuclear decay

The decay of an unstable nucleus is a random (probabilistic) process.  The rate at which nuclei decay, dN/dt, is proportional to the number N of nuclei present.

dN/dt = -λN,    N(t) = N₀exp(-λt).

Different radionuclides decay at different rates, each having its own decay constant λ.   Their mean life is τ = 1/λ.
The half-life of a nucleus is given by t_1/2 = τ ln2 = ln2/λ.

Alpha decay is a form of radioactive decay in which an atomic nucleus characterized by mass number A and atomic number Z ejects an alpha particle and transforms into a nucleus with mass number A - 4 and and atomic number Z - 2 .  
Example: ²³⁸U --> ²³⁴Th + α
Alpha decay is a quantum-mechanical tunneling process.  The quickest decays correspond to the most energetic α-particles.

Beta decay is a type of radioactive decay in which a β-particle is emitted.
In β⁻ decay, the weak nuclear interaction converts a neutron into a proton while emitting an electron and an anti-neutrino.

In β⁺ decay, a proton is converted into a neutron, a positron and a neutrino.

If the proton and neutron are part of an atomic nucleus the decay processes transmute one chemical element into another.

 γ-decay is the emission of a photon by an exited state of a nucleus, as it returns to its ground state.