Quasiparticle lifetime broadening in resonant x-ray scattering of ${\mathrm{NH}}_4{\mathrm{NO}}_3$

It has been previously shown that two effects cause dramatic changes in the x-ray absorption and emission spectra from the N $K$ edge of the insulating crystal ammonium nitrate. First, vibrational disorder causes major changes in the absorption spectrum, originating not only from the thermal population of phonons, but, significantly, from zero-point motion as well. Second, the anomalously large broadening $(\sim 4$ eV) of the emission originating from nitrate $\sigma$ states is due to the unusually short lifetimes of quasiparticles in an otherwise extremely narrow band. In this work, we investigate the coupling of these effects to core and valence excitons that are created as the initial x-ray excitation energy is progressively reduced toward the N edge. Using a $GW$/Bethe-Salpeter approach, we show the extent to which this anomalous broadening is captured by the $GW$ approximation. The data and calculations demonstrate the importance that the complex self-energies (finite lifetimes) of the valence bands have on the interpretation of emission spectra. We produce a scheme to explain why extreme lifetimes should appear in $\sigma$ states of other similar compounds.

Figure 1

The $GW$ band structure of ${\mathrm{NH}}_4{\mathrm{NO}}_3$. The coloring (inset) is by overlap with bright orange states being primarily $p$-type states on the nitrogens, which make up the ammonium, and the bright blue belonging to $p$-type states on the nitrate. Light grey bands either have $s$-type character or have little overlap with the nitrogen atoms.

Figure 2

(Top) The absolute values of the real (light-blue squares) and imaginary (red circles) components of the self-energy corrections as a function of the LDA energy for the valence and low-lying conduction bands of ammonium nitrate. (Bottom) The projected local densities of states $(p$-DOS) for $l=1$ around the nitrate-centered (orange, solid) and ammonium-centered (blue, dashed) nitrogen atoms. Only the NO $\sigma$ bonds near $-20$ eV show an imaginary self-energy that varies significantly from zero. The black vertical line marks the middle of the band gap.

Figure 3

The nitrogen $K$-edge absorption comparing theory and experiment. The effect of vibrational disorder is notable in going from a calculation using the “frozen” coordinates to one including disorder. This is especially true for the ${\mathrm{NH}}_4$ exciton around 407 eV in the absorption. The main ${\mathrm{NO}}_3$ peak at 405 eV is less affected by vibrational broadening.

Figure 4

Electron density plots of the ${\mathrm{NO}}_3$ (a) and (b) and ${\mathrm{NH}}_4$ (c) excitons are shown for select excitation energies and polarizations (plotted using VMD [41]). The nitrate exciton at 405 eV protrudes into the open space above and below the nitrate plane—whereas the ammonium exciton at 406.6 eV encompasses the hydrogen atoms and in turn is quite sensitive to the large vibrational displacements that the hydrogen atoms undergo. The higher energy ${\mathrm{NO}}_3{\sigma }^{*}$ exciton (412.8 eV) is also strongly in uenced by vibrational disorder.

Figure 5

Measured (blue/light blue, solid) and calculated (red/orange, dashed) RIXS of ${\mathrm{NH}}_4{\mathrm{NO}}_3$ (see text). Numbers along the left above each set of spectra denote the incident x-ray energy in eV. Spectra are offset vertically and scaled according to the numbers along the right. Measured spectra are normalized for beam intensity and count time and corrected for self-absorption. Calculated spectra are normalized to fit. At 405, 412, and 424 eV, the thin, dotted lines show the calculations without the complex-valued self-energy corrections. The black dashed line indicates a shift in the NO $\pi$ peak from 394.0 to 395.8 eV with increasing incident energy.