Theoretical study on resonant x-ray scattering from the multipolar ordering phase

我们推导出了有用的表达式。基于局域电子图像的共振x射线散射（RXS）振幅。由于中间态的求和不使用所谓的快速碰撞近似，并且假设描述中间态的哈密顿量保持旋转不变性，因此表达式包含了光谱的精确能量依赖。这些表达式及其导数证明了f电子体系中RXS的光谱分析可以为阐明多极有序相的性质提供有力的支持。我们举一些例子来说明它们。首先，我们应用了多极有序相，如Ce_<1-x>La_xB_6中的八极有序相和其他一些多k有序相。他们很好地再现了实验结果，并预测了一些有趣的特征。其次，将该表达式推广到某些重稀土金属所表现出的螺旋和锥形磁相等长周期有序结构的情况。以钬为例，计算了其RXS光谱。结果表明，在某些情况下，光谱形状可以区分不同的相。我们还预测了在Ho的锥形相位的电偶极子（E1）跃迁中可以探测到第四个卫星斑点。第三，研究了磁铁矿中缺乏局部空间反演对称性的A位Fe离子的RXS谱。然后，当满足特定的实验条件时，奇偶过程，例如e1 -四极（E2）过程存在。我们计算了RXS和吸收光谱，后者显示了El-E2过程的清晰证据。最后，我们研究了La_2CuO_4共振非弹性x射线散射的双磁振子激发光谱，证明了实验观测到的光谱确实是由铜酸盐中的双磁振子激发引起的。

We have derived useful expressions. Of the resonant x-ray scattering (RXS) amplitude on the basis of the localized electron picture. The expressions include precise energy dependence of the spectra since the summation over the intermediate states are carried out without using the so-called fast collision approximation with the assumption that the Hamiltonian describing the intermediate states preserves the rotational invariance. The expressions and their derivatives have proved that the spectral analysis in RXS in the felectron system can lend a powerful support in elucidating the nature of the multipolar ordering phases. We present some examples to show them.First, we have applied to the multipolar ordering phases such as octupolar in Ce_<1-x>La_xB_6 and some other multi-k ordering phases. They reproduced the experimental results well and predicted some intriguing features.Second, the expression has been extended to the case of the long period ordered structure such as the helical and conical magnetic phases exhibited by some heavy rare earth metal. Taking holmium as an example, we have calculated the RXS spectra. The results indicate that the spectral shape can discriminate the different phases in some cases in this system. We also have predicted that the fourth satellite spot is detectable in the electric dipole (E1) transition in the conical phase of Ho.Third, RXS spectrum in the magnetite has been studied at the A site Fe ions, which lack local space inversion symmetry. Then, parity odd process, for instance, E1-quadrupole (E2) process survives when some specific experimental conditions are satisfied. We have calculated both the RXS and absorption spectra, the latter of which show clear evidence of the El-E2 process.Finally, we have investigated two-magnon excitation spectra of resonant inelastic x- ray scattering from La_2CuO_4, which proves that the experimentally observed spectra are indeed due to the two-magnon excitations in cuprates.