One of the most fundamental properties of an interacting electron system is its frequency- and wave-vector-dependent density response function, $chi({bf q},omega)$. The imaginary part, $chi({bf q},omega)$, defines the fundamental bosonic charge excitations of the system, exhibiting peaks wherever collective modes are present. $chi$ quantifies the electronic compressibility of a material, its response to external fields, its ability to screen charge, and its tendency to form charge density waves. Unfortunately, there has never been a fully momentum-resolved means to measure $chi({bf q},omega)$ at the meV energy scale relevant to modern elecronic materials. Here, we demonstrate a way to measure $chi$ with quantitative momentum resolution by applying alignment techniques from x-ray and neutron scattering to surface high-resolution electron energy-loss spectroscopy (HR-EELS). This approach, which we refer to here as M-EELS, allows direct measurement of $chi({bf q},omega)$ with meV resolution while controlling the momentum with an accuracy better than a percent of a typical Brillouin zone. We apply this technique to finite-q excitations in the optimally-doped high temperature superconductor, Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (Bi2212), which exhibits several phonons potentially relevant to dispersion anomalies observed in ARPES and STM experiments. Our study defines a path to studying the long-sought collective charge modes in quantum materials at the meV scale and with full momentum control.
Download