  We investigate the entanglement properties of an ensemble of atoms
interacting with a single bosonic field mode via the Dicke (superradiance)
Hamiltonian. The model exhibits a quantum phase transition and a
well-understood thermodynamic limit, allowing the identification of both
quantum and semi-classical many-body features in the behaviour of the
entanglement. We consider the entanglement between the atoms and the field, an
investigation initiated in [N. Lambert, C. Emary and T. Brandes, Phys. Rev.
Lett. {\bf 92}, %073602 (2004)]. In the thermodynamic limit, we give exact
results for all entanglement partitions and observe a logarithmic divergence of
the atom-field entanglement, and discontinuities in the average linear entropy.
