If neutrons exert only attractive force, why don’t we have a nucleus containing neutrons alone?

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By definition, a nucleus is the central core of an elemental atom. Atoms consist of a nucleus surrounded by a cloud of electrons. If the nucleus contains only neutrons, that are electrically neutral, then the electron cloud will collapse due to repulsive Coulomb’s force between the electrons. Thus protons are required in a nucleus to hold the electron cloud bound to the nucleus of the atom.

The hypothetical substance that consists purely of neutrons is called ‘Neutronium‘. Neutronium is stable only in the case of neutron stars where the neutrons are tightly held by immense gravitational pressure.

In a regular stable nucleus, there are protons and neutrons. Neutrons are slightly heavier than protons and just like protons, are made up of quarks. A neutron on its own is not stable. With a half-life of about 10.2 minutes, an isolated neutron tends to undergo beta minus decay to transform into a lighter proton.

n \rightarrow p+e+ \bar{\nu_e} (where \bar{\nu_e} is an electron antineutrino)

Inside a regular stable nucleus where there are many protons and neutrons, the electron released from a beta minus decay is captured by a proton and a neutron is produced, thus keeping the proton and neutron numbers constant. But in the case of Neutronium, this equilibrium does not hold and the nuclear reaction tends to favour the production of protons through beta minus decays. The only exceptions, in this case, are the neutron stars that have high gravitational pressure where even beta decays are not spontaneous.