  We report an experimental investigation of fractional quantum Hall effect
(FQHE) at the even-denominator Landau level filling factor $\nu$ = 1/2 in very
high quality wide GaAs quantum wells, and at very high magnetic fields up to 45
T. The quasi-two-dimensional electron systems we study are confined to GaAs
quantum wells with widths $W$ ranging from 41 to 96 nm and have variable
densities in the range of $\simeq 4 \times 10^{11}$ to $\simeq 4 \times
10^{10}$ cm$^{-2}$. We present several experimental phase diagrams for the
stability of the $\nu=1/2$ FQHE in these quantum wells. In general, for a given
$W$, the 1/2 FQHE is stable in a limited range of intermediate densities where
it has a bilayer-like charge distribution; it makes a transition to a
compressible phase at low densities and to an insulating phase at high
densities. The densities at which the $\nu=1/2$ FQHE is stable are larger for
narrower quantum wells. Moreover, even a slight charge distribution asymmetry
destabilizes the $\nu=1/2$ FQHE and turns the electron system into a
compressible state. We also present a plot of the symmetric-to-antisymmetric
subband separation ($\Delta_{SAS}$), which characterizes the inter-layer
tunneling, vs density for various $W$. This plot reveals that $\Delta_{SAS}$ at
the boundary between the compressible and FQHE phases increases
\textit{linearly} with density for all the samples. Finally, we summarize the
experimental data in a diagram that takes into account the relative strengths
of the inter-layer and intra-layer Coulomb interactions and $\Delta_{SAS}$. We
conclude that, consistent with the conclusions of some of the previous studies,
the $\nu=1/2$ FQHE observed in wide GaAs quantum wells with symmetric charge
distribution is stabilized by a delicate balance between the inter-layer and
intra-layer interactions, and is very likely described by a two-component
($\Psi_{311}$) state.
