This thesis investigates the role of local inhomogeneities and mesoscale features to understand material behavior not possible to realize with measurements on the average. It leverages a novel Dark Field X-ray Microscopy (DFXM) experimental technique to image mesoscale structures in a bulk-sensitive manner with hard X-rays at synchrotron facilities. DFXM utilizes an X-ray objective to magnify an individual diffraction peak and capture spatially resolved intensity distributions that highlight local variations from the nanometer to tens of micron length scales. However, challenges remain in the application of this technique to examine low-temperature phenomena and subtle periodic structures in quantum materials, which exhibit complex phases at cryogenic temperatures.

One such quantum material is NaMnO2, which demonstrates frustrated antiferromagnetism (AFM) at low temperatures and a high reversible capacity suitable for use in rechargeable battery applications. Its magnetic and electronic properties are further complicated by a series of crystal defects (including twinning, stacking faults, polymorphism and Mn3O4 intergrowths), whose size and spatial distribution remain unresolved. Furthermore, magneto-elastic coupling has been hypothesized to result in the formation of nanoscale triclinic domains below the AFM transition at 45 K, but this effect has yet to be experimentally validated. DFXM is well suited to characterize the heterogeneous structure of NaMnO2, provided that the challenges in performing the experiment at low temperature are overcome. DFXM studies performed on NaMnO2 reveal linear micro-structures that span hundreds and tens of microns in the longitudinal and transverse directions, respectively. Increased DFXM magnification exposes the linear domains are further broken up into a myriad of nanoscale structures, which are likely the root of enigmatic experimental data. A first-of-its-kind DFXM experiment performed below liquid‑helium temperature displays significant microscale intensity variations through the AFM transition. Although the formation of triclinic domains were not directly observed, DFXM images reveal that temperature-dependent local actors are responsible for mimicking a bulk structural distortion in X-ray measurements of the average structure.

Another non-trivial electronic material is the recently discovered superconducting kagome, CsV3Sb5, which hosts a charge-density-wave (CDW) instability below 94 K. Three-dimensional in nature, the CDW phase in CsV3Sb5 is accompanied by in-plane structural distortions that form Star of David (SoD) or Tri-Hexagonal (TrH) patterns, but inconsistencies in the out-of-plane component have left the stacking order ambiguous. Peak splitting in conventional X-ray diffraction data indicates the formation of 2 x 2 x 2 and 2 x 2 x 4 supercell structures below T_CDW. The primary question addressed in this thesis is whether these two CDW types compete during formation, resulting in phase separation, or coexist within the same crystallographic domain, comprised of two Fourier components. CsV3Sb5 is further complicated by a reduction from six-fold hexagonal symmetry to two-fold orthorhombic symmetry following the CDW transition. Elucidating the origin of the CDW instability is a critical step toward understanding the novel phenomena in this complex material. Real-space DFXM images of CsV3Sb5, collected across multiple length scales, provide evidence for a rich microstructure at low temperatures. DFXM images of the (1/2, 1/2, 1/2) and (1/2, 1/2, 1/4) CDW peaks, collected over hundreds of microns, are presented for evaluation. Comparison of each CDW peak reveals a significant spatial separation in three-dimensions and supports a theory of two distinct crystal domains being responsible for the split CDW signal. The experiments detailed in this thesis further demonstrate two novel implementations of the DFXM methodology, including in situ cryogenic temperature variation and full-field imaging of a bulk charge-density wave using hard X-rays and direct detection. Extending DFXM to low temperatures through the use of a low-vibration cryostat creates an avenue for experimental investigations of other complex physical phenomena. Additionally, DFXM imaging of charge-order Bragg peaks demonstrates the potential of this methodology to be extended to other weakly scattered structures (e.g., magnetic, elemental, etc.).