Two Dimensional Incommensurate and Three Dimensional Commensurate Magnetic Order and Fluctuations in $La_{2-x}Ba_{x}CuO_{4}$

We present neutron scattering measurements on single crystals of $La_{2-x}Ba_{x}CuO_{4}$, with $0 \leq x \leq$ 0.035. These experiments reveal the evolution of the magnetism: from a three dimensional (3D) commensurate (C) antiferromagnet, with a relatively high T$_{N}$, to a two dimensional (2D) incommensurate (IC) antiferromagnet with finite range static correlations, with relatively low effective T$_{N}$s. At low temperatures, the 2D IC magnetism co-exists with the 3D C magnetism for Ba concentrations as low as x = 0.0125. We find that 3D C magnetism disappears by x = 0.025; consistent with the limit of x $\sim$ 0.02 observed in the sister family of doped Mott insulators $La_{2-x}Sr_{x}CuO_{4}$. We construct a phase diagram based on magnetic order parameter measurements, which displays much of the complexity of standard high temperature superconductivity phase diagrams discussed in the literature. Analysis of high energy-resolution inelastic neutron scattering shows the low energy dynamic susceptibility to fall off with temperature on a scale much higher than the effective 2D IC T$_{N}$s in the sample. This effect is such that appreciable dynamic 2D IC magnetic fluctuations inhabit much of the ``pseudogap'' regime of the phase diagram.