An electron strongly correlated in a solid has three degrees of freedom, charge, spin and orbital. The electronic orbital represents the shape of the electron cloud around an atom. The orbital degree of freedom often plays an important role in the electric and magnetic properties through strong coupling with the charge, spin, and lattice degrees of freedom. However, experimental techniques to observe the orbital state have been limited to date. We have developed one of the techniques using the resonant xray scattering, and quantitatively discussed the wave function of the orbitally ordered state in YTiO3.
Near the xray absorption energy of the target ion, the atomic scattering factor (ASF) sensitively reflects the local symmetry around the ion absorbed xray. Then, the ASF is written by a tensor instead of a scalar. We can discuss the orbitally ordered state through the tensor of the ASF. For example, when the orbital elongates in the zdirection as shown in the left panel of Fig. 1, the atomic scattering factor becomes anisotropic between the x, ydirection and the zdirection. Then, we can write the atomic scattering tensor as shown in the right panel of Fig. 1.
Figure 1 : (left) A orbitally ordered state, 3z²r².
(right) The tensor of atomic scattering factor
reflecting the orbital ordering, 3z²r².

Using a diffraction technique near the absorption energy,
we can observe a Bragg peak
reflecting the periodicity of the anisotropic ASF,
which is a kind of resonant xray scattering (RXS).
The RXS has three characteristics
different from a normal Bragg scattering.
(1) The RXS is observed only near the absorption edge energy
of the target ion.
(2) The polarization of the scattered xray changes from that of the
incident xray, though the xray polarization does not
change in normal xray scattering.
(3) The intensity oscillates depending on the azimuthal angle
which is the rotating angle around the scattering vector.
The polarization, azimuthal angle, and Qposition dependences
provide direct information on the atomic scattering tensor.
From these experimental results, we can know the periodicity and the
correlation length of the orbitally ordered state, and also discuss the
wave function.
A perovskitetype transition metal oxide, YTiO3, shows a ferromagnetic(FM)insulator phase below TC ∼30 K. The origin of the FM ordering has been theoretically considered to be a ferromagnetic superexchange interaction caused by the orbital ordering of Ti3+ t2gelectron with s=1/2. The orbitally ordered state below TC was studied by the polarized neutron scattering and NMR. However, the techniques are limited to the observation in the magnetic ordered phase. Using the RXS technique, therefore, we have studied the orbitally ordered state in YTiO3 above TC and quantitatively discussed the wave function of the Ti ion.
The RXS experiment was performed at beamline
X22C of Brookhaven National Laboratory's National
Synchrotron Light Source.
Figure 2 :
(a) Fluorescence at (0,0,1.5) near Ti Kedge energy.
(b)(d) Energy dependence of RXS intensities at forbidden reflections.
The scattering components, σ→σ' and
σ→π', are shown by open circles and filled circles,
respectively.
[The polarization vector, σ (π),
is perpendicular(parallel) to the scattering plane.]

Next, the tensor of atomic scattering factor is quantitatively determined on the basis of the polarization, the azimuthal angle, and the Qposition dependences. Finally, we could obtain the orbitally ordered image of YTiO3 by comparing these results with the orbital model as shown in Fig. 3.
Figure 3 : The orbitally ordered image in YTiO3,
which is determined by the RXS technique.

This work illustrates the possibility to quantitatively determine the orbitally ordered state through the anisotropic ASF. The application of this RXS technique is expanding to other materials orbitally ordered. However, the origin of the anisotropic atomic scattering factor still is controversial problem; The detailed report is given in the reference.