We have studied (1) systems with orbital degrees of freedom which are strongly coupled to the other degrees of freedom - 3d transition metal oxides and (2) systems with orbital degrees of freedom which spontaneously order - 4f rare-earth metal compounds. As for (1), we have focused on layered Mn oxide La_<2-2x>Sr_<1+2x>Mn_2O_7. In 0.5 【less than or equal】 x 【less than or equal】 0.6, we have succeeded in growing single crystals with the FZ method, and found that an antiferromagnetic (AF) metallic state becomes stable with increasing the doping rate x. Our neutron scattering studies have revealed that the A-type AF metallic state and the CE-type AF charge-ordered state competitively coexist in an electronically phase separated state, which drastically changes with temperature and x. We have also employed other techniques such as resonant x-ray scattering, polarized neutron diffraction, magnetic Compton profile, spin wave measurement and analysis, and clarified quantitatively the orbital state and its effect to the magnetism and transport in 0.3 【less than or equal】 x 【less than or equal】 0.5.
As for (2), we have scrutinized the antiferro-quadrupole (AFQ) ordering process and structure in DyB_2C_2 by resonant x-ray scattering, which is the first example of resonant x-ray scattering measurement of AFQ state. We have also confirmed that HoB_2C_2 undergoes AFQ and AFM ordering below T_Q = 5.0 K, and found that there remains short-range magnetic correlations, well above T_Q, with an exotic incommensurate periodicity different from that of the low temperature state. We also revealed the details of AFQ ordering process and state hi CeB_6, known for a prototypical AFQ material for a long time, by resonant x-ray scattering.