Elements with fewer protons, such as the ones near the top of the periodic table, are stable if they have the same number of neutrons and protons. For example carbon (carbon-12) is stable and has six protons and six neutrons.
However, as the number of protons increases, more neutrons are needed to keep the nucleus stable. For example lead (lead-206) has 82 protons and has 124 neutrons.
If the nucleus has too few neutrons, it will emit a 'package' of two protons and two neutrons called an alpha particle.
An alpha particle is also a helium-4 nucleus, so it is written as and is also sometimes written as
A beta particle has a relative mass of zero, so its mass number is zero. As the beta particle is an electron, it can be written as , and sometimes as .
The beta particle is an electron but it has come from the nucleus, not the outside of the atom.
Electrons are not normally expected to be found in the nucleus but neutrons can split into a positive proton (same mass but positive charge) and an electron (which has a negative charge to balance the positive charge) which is then ejected at high speed and carries away a lot of energy.
Beta decay causes the atomic number of the nucleus to increase by one while the mass number remains the same.
After emitting an alpha or beta particle, the nucleus will often still be too 'hot' and will lose energy in a similar way to how a hot gas cools down. A hot gas cools by emitting infrared radiation, which is an electromagnetic wave.
High energy particles will emit energy as they drop to lower energy levels. Since energy levels in the nucleus are much higher than those in the gas, the nucleus will cool down by emitting a more energetic electromagnetic wave called a gamma ray.
Gamma ray emission causes no change in the number of particles in the nucleus, meaning both the atomic number and mass number remain the same.
Occasionally it is possible for a neutron to be emitted by radioactive decay. This can occur naturally, ie absorption of cosmic rays high up in the atmosphere can result in neutron emission, although this is rare at the Earth's surface. Or it can occur artificially, eg firing alpha particles at beryllium results in neutrons being emitted from that.
A further example of neutron emission is in nuclear fission reactions, where neutrons are released from the parent nucleus as it splits.