Commensurate to Incommensurate Charge Ordering and Its Real-Space Images in L${\mathrm{a}}_{0.5}$C${\mathrm{a}}_{0.5}$Mn${\mathrm{O}}_3$

The antiferromagnetic to ferromagnetic transition in L${\mathrm{a}}_{0.5}$C${\mathrm{a}}_{0.5}$Mn${\mathrm{O}}_3$ is accompanied by a commensurate to incommensurate charge-ordering transition. The presence of incommensurate charge correlation in the ferromagnetic phase is a remarkable interplay between charge and spin in double-exchange ferromagnets. Furthermore, a dark-field imaging technique has provided real-space images of charge-ordered domains revealing discommensurations inside the nearly commensurate domains. Local density fluctuations of discommensurations are responsible for the similar variations of the charge-ordering wave vector.