  We present a neutron triple-axis and resonant spin-echo spectroscopy study of
the spin correlations in untwinned YBCO crystals with x= 0.3, 0.35, and 0.45 as
a function of temperature and magnetic field. As the temperature T approaches
0, all samples exhibit static incommensurate magnetic order with propagation
vector along the a-direction in the CuO2 planes. The incommensurability delta
increases monotonically with hole concentration, as it does in LSCO. However,
delta is generally smaller than in LSCO at the same doping level. The intensity
of the incommensurate Bragg reflections increases with magnetic field for
YBCO(6.45) (superconducting Tc = 35 K), whereas it is field-independent for
YBCO(6.35) (Tc = 10 K). These results suggest that YBCO samples with x ~ 0.5
exhibit incommensurate magnetic order in the high fields used for the recent
quantum oscillation experiments on this system, which likely induces a
reconstruction of the Fermi surface. We present neutron spin-echo measurements
(with energy resolution ~ 1 micro-eV) for T > 0 that demonstrate a continuous
thermal broadening of the incommensurate magnetic Bragg reflections into a
quasielastic peak centered at excitation energy E = 0, consistent with the
zero-temperature transition expected for a two-dimensional spin system with
full spin-rotation symmetry. Measurements on YBCO(6.45) with a triple-axis
spectrometer (with energy resolution ~ 100 micro-eV) yield a crossover
temperature T_SDW ~ 30 K for the onset of quasi-static magnetic order. Upon
further heating, the wavevector characterizing low-energy spin excitations
approaches the commensurate antiferromagnetic wave vector, and the
incommensurability vanishes in an order-parameter-like fashion at an
"electronic liquid-crystal" onset temperature T_ELC ~ 150 K. Both T_SDW and
T_ELC increase continuously as the Mott-insulating phase is approached with
decreasing doping level.
