  We show how the use of variational states to approximate the ground state of
a system can be employed to study a multi-mode Dicke model. One of the main
contributions of this work is the introduction of a not very commonly used
quantity, the cooperation number, and the study of its influence on the
behavior of the system, paying particular attention to the quantum phase
transitions and the accuracy of the used approximations. We also show how these
phase transitions affect the dependence of the expectation values of some of
the observables relevant to the system and the entropy of entanglement with
respect to the energy difference between atomic states and the coupling
strength between matter and radiation, thus characterizing the transitions in
different ways.
