  We study the ground-state Wigner bilayers of pointlike particles with Yukawa
pairwise interactions, confined to the surface of two parallel hard walls at
dimensionless distance $\eta$. The model involves as limiting cases the
unscreened Coulomb potential and hard spheres. The phase diagram of Yukawa
particles, studied numerically by Messina and L\"owen [Phys. Rev. Lett. 91
(2003) 146101], exhibits five different staggered phases as $\eta$ varies from
0 to intermediate values. We present a lattice summation method using the
generalized Misra functions which permits us to calculate the energy per
particle of the phases with a precision much higher than usual in computer
simulations. This allows us to address some tiny details of the phase diagram.
Going from the hexagonal phase I to phase II is shown to occur at $\eta=0$,
which resolves a longtime controversy. We find a tricritical point where
Messina and L\"owen suggested a coexistence domain of several phases which was
suggested to divide the staggered rhombic phase into two separate regions. Our
calculations reveal one continuous region for this rhombic phase with a very
narrow connecting channel. Further we show that all second-order phase
transitions are of mean-field type. We also derive the asymptotic shape of
critical lines close to the Coulomb and hard-spheres limits. In and close to
the hard-spheres limit, the dependence of the internal parameters of the
present phases on $\eta$ is determined exactly.
