Maximum entropy deconvolution of resonant inelastic x-ray scattering spectra

Resonant inelastic x-ray scattering (RIXS) has become a powerful tool in the study of the electronic structure of condensed matter. Although the linewidths of many RIXS features are narrow, the experimental broadening can often hamper the identification of spectral features. Here we show that the maximum entropy technique can successfully be applied in the deconvolution of RIXS spectra, improving the interpretation of the loss features without a severe increase in the noise ratio.

Figure 1

Comparison between raw (second order) Zn x-ray emission spectra recorded at 0.78 eV spectrometer resolution and the results of the MaxEnt deconvolution procedure. For comparison, a spectrum recorded at 0.28 eV spectrometer resolution is also shown. (a) Integrated spectra, (b) raw 2D spectrum for $\Gamma =0.78$ eV, (c) MaxEnt deconvoluted 2D spectrum for $\Gamma =0.78$ eV, and (d) raw 2D spectrum for $\Gamma =0.28$ eV. The raw 2D spectra (before correcting for the curvature of the image) are shown on a logarithmic false color scale.

Figure 2

Co $L$-edge RIXS data of Co${}_3$V${}_2$O${}_8$. (a) Raw spectra and (b) MaxEnt deconvoluted spectra, in which the raw spectra are reproduced in light gray. Solid lines are a guide to the eye and represent the data with a binomial smoothing of 0.15 eV. Vertical lines show the elastic line at 0 eV and loss features identified in the raw data. The symmetry of the excitations is labeled in (a).

Figure 3

V $L$-edge RIXS data of NdVO${}_3$. (a) The absorption spectrum and excitation energies used for RIXS. Raw RIXS data are shown on (b) an emission-energy and (c) a loss-energy scale, for which the individual MaxEnt spectra are also shown. (d) Summed RIXS spectra before and after MaxEnt deconvolution. The symmetry of the excitations are labeled in (d); the dotted line represents the MaxEnt summed data with one spectrum missing (see text).

Figure 4

The results of deconvoluting $M=100$ simulated noisy spectra with the MaxEnt procedure. (a) Input spectra and (b) deconvoluted spectra. The individual spectra $P_i\left(E\right)$ are shown in light gray, with the mean spectra $\overline{P}\left(E\right)$ in blue (dark gray). The error bars shown represent the standard deviation $\sigma \left[P\right(E\left)\right]$ of the $M$ spectra for each energy. The inset shows the distribution of each simulated spectrum from its mean, normalized to its standard deviation. In (b), the initial test spectrum $T\left(E\right)$ is shown by the dashed line. (c) Comparison between the mean of the input spectra and their variance ${\sigma }^2$. (d) Comparison between the mean of the deconvoluted spectra and ${\sigma }^{1.54}$. For (c) and (d) the solid line represents $\overline{P}={\sigma }^2$ and $\overline{P}={\sigma }^{1.54}$, respectively.

Figure 5

The (a) raw and (b) MaxEnt deconvoluted RIXS spectra shown in Fig. 2, reproduced here with error bars. In (a), the statistical error is shown ($\sigma =\sqrt{N}$), whereas in (b) the empirical error $\sigma =N^{1/1.54}$ is used (see text).