We discuss how the crossovers in models like spin-boson model are changed by adding the coupling of the central spin to localised modes- the latter modelled as a 'spin bath'. These modes contain most of the environmental entropy and energy at low T in solid-state systems. We find that the low T crossover between oscillator bath and spin bath dominated decoherence, occurring as one reduces the energy scale of the central spin, is characterised by very low decoherence- we show how this works out in practise in magnetic insulators. We then reconsider the standard quantum-classical crossover in the dynamics of a tunneling system, including both spin and oscillator baths. It is found that the general effect of the spin bath is to broaden the crossover in temperature between the quantum and classical activated regimes. The example of tunneling nanomagnets is used to illustrate this.

13 Latex pages, 6 figures. Chem. Phys. 2003 (accepted)