  Bipartite and global entanglement are analyzed for the ground state of a
system of $N$ spin 1/2 particles interacting via a collective spin-spin
coupling described by the Lipkin-Meshkov-Glick (LMG) Hamiltonian. Under certain
conditions which includes the special case of a super-symmetry, the ground
state can be constructed analytically. In the case of an anti-ferromagnetic
coupling and for an even number of particles this state undergoes a smooth
crossover as a function of the continuous anisotropy parameter $\gamma $ from a
separable ($\gamma =\infty $) to a maximally entangled many-particle state
($\gamma =0$). From the analytic expression for the ground state, bipartite and
global entanglement are calculated. In the thermodynamic limit a discontinuous
change of the scaling behavior of the bipartite entanglement is found at the
isotropy point $\gamma =0$. For $% \gamma =0$ the entanglement grows
logarithmically with the system size with no upper bound, for $\gamma \neq 0$
it saturates at a level only depending on $\gamma $. For finite systems with
total spin $J=N/2$ the scaling behavior changes at $\gamma =\gamma
_{\mathrm{crit}}=1/J$.
