Composition dependence of the diffuse scattering in relaxor (1-$x$)Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$-$x$PbTiO$_{3}$ ($0\leq x\leq0.40$)

We have studied composition dependence of diffuse scattering in the relaxor system (1-$x$)Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$-$x$PbTiO$_{3}$ (PMN-$x$PT) with $x=0$, 10, 20, 30, and 40\% by neutron diffraction. The addition of ferroelectric PbTiO$_{3}$ (PT) modifies the ``butterfly" and ``ellipsoidal" diffuse scattering patterns observed in pure PMN ($x=0$), which are associated with the presence of randomly oriented, polar nanoregions (PNR). The spatial correlation length $\xi$ derived from the width of the diffuse scattering increases from 12.6~\AA\, for PMN ($x=0$) to 350~\AA\, for PMN-20\%PT, corresponding to an enlargement of the PNR. The integrated diffuse scattering intensity, which is proportional to $\chi''$, grows and reaches a maximum at $x = 20$\%. Beyond $x =$30\%\,PT, a concentration very close to the morphotropic phase boundary (MPB), no diffuse scattering is observed below $T_{C}$, and well-defined critical behavior is observed. By contrast, the diffuse scattering for $x \leq 20$\% persists to low temperatures, where the system retains an average cubic structure ($T_{C}=0$). We can simulate the wave vector dependence of the diffuse scattering by assuming that it arises from the condensation of a soft transverse-optic (TO) phonon.