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Three solvable models are set out in some detail in reviewing different typesof phase transitions. Two of these relate directly to emergent criticalphenomena, viz. melting and magnetic transitions in heavy rare-earth metals,and secondly, via the $3d$ Ising model, to critical behaviour in an insulatingferromagnet such as CrBr$_3$. The final `transition', however, concernsionization of an electron in an isoelectronic series with $N$ electrons as theatomic number $Z$ is reduced below that of the neutral atom. These solvablemodels are, throughout, brought into contact either with experiment, or withvery precise numerical modelling on real materials.

Insights gained from solvable models into a variety of phase transitions, including emergent assemblies plus isoelectronic series of atomic ions