  We present time-of-flight inelastic neutron scattering measurements of
$La_{1.965}Ba_{0.035}CuO_{4}$ (LBCO), a lightly doped member of the high
temperature superconducting La-based cuprate family. By using time-of-flight
neutron instrumentation coupled with single crystal sample rotation we obtain a
four-dimensional data set (three {\bf Q} and one energy) that is both
comprehensive and spans a large region of reciprocal space. Our measurements
identify rich structure in the energy dependence of the highly dispersive spin
excitations, which are centered at equivalent ($\frac{1}{2},\frac{1}{2}, L$)
wave-vectors. These structures correlate strongly with several crossings of the
spin excitations with the lightly dispersive phonons found in this system.
These effects are significant and account for on the order of 25$\%$ of the
total inelastic scattering for energies between $\sim$5 and 40 meV at low
$|{\bf Q}|$. Interestingly, this scattering also presents little or no
$L$-dependence. As the phonons and dispersive spin excitations centered at
equivalent ($\frac{1}{2},\frac{1}{2}, L$) wave-vectors are common to all
members of La-based 214 copper oxides, we conclude such strong quasi-two
dimensional scattering enhancements are likely to occur in all such 214
families of materials, including those concentrations corresponding to
superconducting ground states. Such a phenomenon appears to be a fundamental
characteristic of these materials and is potentially related to superconducting
pairing.
