  Neutron and x-ray diffraction techniques have been used to study the
competing long and short-range polar order in the relaxor ferroelectric
Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$ (PMN) under a [111] applied electric field.
Despite reports of a structural transition from a cubic phase to a rhombohedral
phase for fields E $>$ 1.7 kV/cm, we find that the bulk unit cell remains cubic
(within a sensitivity of 90$^{\circ}$-$\alpha$ =0.03$^{\circ}$)for fields up to
8 kV/cm. Furthermore, we observe a structural transition confined to the near
surface volume or `skin' of the crystal where the cubic cell is transformed to
a rhombohedral unit cell at T$_{c}$=210 K for E $>$ 4 kV/cm, for which
90$^{\circ}$-$\alpha$=0.08 $\pm$ 0.03$^{\circ}$ below 50 K. While the bulk unit
cell remains cubic, a suppression of the diffuse scattering and concomitant
enhancement of the Bragg peak intensity is observed below T$_{c}$=210 K,
indicating a more ordered structure with increasing electric field yet an
absence of a long-range ferroelectric ground state in the bulk. The electric
field strength has little effect on the diffuse scattering above T$_{c}$,
however below T$_{c}$ the diffuse scattering is reduced in intensity and adopts
an asymmetric lineshape in reciprocal space. The absence of hysteresis in our
neutron measurements (on the bulk) and the presence of two distinct temperature
scales suggests that the ground state of PMN is not a frozen glassy phase as
suggested by some theories but is better understood in terms of random fields
introduced through the presence of structural disorder. Based on these results,
we also suggest that PMN represents an extreme example of the two-length scale
problem, and that the presence of a distinct skin maybe necessary for a relaxor
ground state.
