Effective operator for $d–d$ transitions in nonresonant inelastic x-ray scattering

Recent experiments by Larson et al. [Phys. Rev. Lett. 99, 026401 (2007)] demonstrate the feasibility of measuring local $d–d$ excitations using nonresonant inelastic x-ray scattering (IXS). We establish a general framework for the interpretation where the $d–d$ transitions created in the scattering process are expressed in effective one-particle operators that follow a simple selection rule. The different operators can be selectively probed by employing their different dependence on the direction and magnitude of the transferred momentum. We use the operators to explain the presence of nodal directions and the nonresonant IXS in specific directions and planes. We demonstrate how nonresonant IXS can be used to extract valuable ground state information for orbiton excitations in manganite.

Figure 1

(Color online) Inelastic scattering from $e\to e^{\ast }$ in the manganites. (a) The top part shows the lowest $e$ orbitals for $|e_{+}⟩=\alpha |2⟩+\beta |0⟩$ for ${\alpha }^2=1$, 0.9, 0.75, and 0.5. Excitations are made into the orbital $|e_{+}^{\ast }⟩=\alpha |0⟩-\beta |2⟩$. The angular dependence for ${\alpha }^2=0,0.1,0.25,0.5$ is the same as those for $1-{\alpha }^2$. The inversion of the relative energy positions of the orbitals does not affect the nonresonant IXS, which is proportional to ${\left|⟨e_{\pm }^{\ast }|{\rho }_{\mathbf{q}}|e_{\pm }⟩\right|}^2$. (b) The angular distribution of the nonresonant IXS intensity in the quadrupolar region ($A_2=1$ and $A_4=1-A_2=0$). The intensity in a certain direction is proportional to the distance to the origin. The insets show the intensity in the $xy$ plane. The intensities give the sum over both orientations of the $e_{\pm }$ orbitals; (c) same as (b) for $A_2=0.67$; (d) idem for $A_2=0$.

Figure 2

(Color online) Nonresonant inelastic scattering for the manganites with an electron in the $|e⟩$ states as in Fig. 1a, but now for scattering $t_2\to e^{\ast }$ for ${\alpha }^2=1$, 0.9, 0.75, 0.5, 0.25, 0.1, and 0. Note that now ${\alpha }^2$ and $1-{\alpha }^2$ are inequivalent. (a) Nonresonant IXS in the quadrupolar region ($A_2=1$ and $A_4=0$); (b) idem in the hexadecapolar region ($A_2=0$ and $A_4=1$). The insets show the scattering intensity in the $xy$ plane.