Thermal excitation signals in the inhomogeneous warm dense electron gas

We investigate the emergence of electronic excitations from the inhomogeneous electronic structure at warm dense matter parameters based on first-principles calculations. The emerging modes are controlled by the imposed perturbation amplitude. They include satellite signals around the standard plasmon feature, transformation of plasmons to optical modes, and double-plasmon modes. These modes exhibit a pronounced dependence on the temperature. This makes them potentially invaluable for the diagnostics of plasma parameters in the warm dense matter regime. We demonstrate that these modes can be probed with present experimental techniques.

In prior diagnostics efforts of WDM, a homogeneous electronic structure of the samples has commonly been assumed, particularly regarding the modeling aspects. Under this assumption, the electronic temperature of the induced WDM states has been determined from X-Ray Thomson scattering signals, for instance, in warm dense aluminum either by dielectric models 59 or KS-DFT calculations 57,58,60 . Despite the combined efforts of experiment and simulation, determining the electronic temperature is still subject to significant uncertainties 61 . Assuming a homogeneous WDM sample is justified when relaxation effects are considered. Their time scale is typically on the order of several picoseconds 22,62 . Moreover, this assumption underpins the uniform electron gas 42,45 as a paradigm for modeling phenomena in WDM.
In this Report, we break away from the common assumption of a homogeneous electronic structure in WDM diagnostics. We (1) show the emergence of yet unexplored modes of electronic excitations in inhomogeneous WDM, (2) illustrate the potential utility of these modes for the temperature diagnostics of WDM, and (3) numerically demonstrate the emergence of these modes in the inhomogeneous electronic structure imposed on warm dense aluminum.
We show that features in the new modes are absent in the spectrum of homogeneous WDM states which is dominated by the well-known plasmon mode. The variety of features we observe depends on the degree of the density deviation from the homogeneous state. They include satellite signals around the standard plasmon feature, transformation of plasmons to optical modes, and double-plasmon modes. We demonstrate that both the realization of the required perturbation amplitudes and the observation of the proposed features are feasible using present and upcoming experimental facilities 25,[63][64][65][66] . For example, the dynamics of emerging electronic structures was recently shown in laser-driven WDM samples where inhomogeneities were imposed by a periodic grating structure 65 . Furthermore, spatially modulating electronic structures can also be triggered by free electron 25 , VUV 67 , and THz lasers 66 . Therefore, we discuss the temperature dependence of excitation signals in the inhomogeneous warm dense electron gas that can be utilized for WDM diagnostics.

Results
Our investigation begins with imposing spatial modulations on the warm dense uniform electron gas (UEG) in terms of the Hamiltonian where N is the total number of electrons and H UEG the standard Hamiltonian of the interacting UEG. Note that we adopt Hartree atomic units, where the length is expressed in Bohr and the energy in Hartree. This elementary Hamiltonian captures the fundamental physics that leads to the proposed modes. Likewise, it is motivated by the fact that electronic oscillations in WDM are well described by the UEG due to the relatively weak electron-ion coupling 46,68,69 . In Eq. (1), the degree of inhomogeneity is determined by the perturbation amplitude U 0 . The length scale of the imposed modulations is set by the wave vector Q . This Hamiltonian has been used extensively in prior work to study response properties of ambient and warm dense electrons with respect to local field corrections 43,70 and response functions 71,72 .
We utilize the electron energy-loss function (EELF) to illustrate the emergence of electronic excitations in inhomogeneous WDM which is defined in terms of the macroscopic dielectric function ε(q, ω) 73 . Electronic excitations are identified by distinct features located at the maxima of the EELF 74 .
The EELF is proportional to the double-differential cross section in inelastic electron scattering. It is measured directly in electron energy-loss spectroscopy 75 . Likewise, the X-Ray Thomson scattering signal is proportional to the EELF via the dynamical structure factor 36 obtained from the fluctuation-dissipation theorem 76 . The electronic excitations we predict are computed using first-principles calculations of the EELF based on the Hamiltonian in Eq. (1). We use time-dependent KS-DFT 53 as implemented in the GPAW code [77][78][79][80] and employ the adiabatic local density approximation 81 . We corroborate the accuracy of our results based on the electronic density of the initial state associated with Eq. (1). For this purpose, we compare our KS-DFT calculations with quantum Monte-Carlo calculations. This comparison is meaningful, as the KS states that comprise the total electronic density are used to compute the EELF of the perturbed system. We provide all computational details in the "Methods" section. Furthermore, we also rule out a source of systematic errors by providing an extensive analysis of finite size effects 82 .
In the following, we consider the typical range of WDM conditions 3 with a density parameter r s = 2 (corresponding to an electron number density n ≃ 2 × 10 23 cm −3 ) and a degeneracy parameter θ in the range from 0.1 to 1.0.
The choice of amplitude and wave number of the perturbation is grounded in our interest of inhomogeneities on excitations which become relevant when q/q F ≪ 1 83 . We therefore consider U 0 in the range from 0.1 to 1.0 and Q ≃ 0.84q F = 1.5 Å −1 . Furthermore, these perturbation parameters are viable in current experimental setups 65,72 . For example, available THz lasers with an intensity of 600 kV/cm 66 correspond to a perturbation amplitude U 0 ≃ 0.3 72 . Likewise, free electron lasers with intensities of up to 10 22 W/cm 225 provide perturbation www.nature.com/scientificreports/ amplitudes up to U 0 ≈ 2 . Alternatively, periodically inhomogeneous electronic structures can be generated using laser irradiation of pre-designed grating targets 65 .
Our central result is illustrated in Fig. 1. It displays a new mode of electronic excitation (solid blue and solid green) that emerges in inhomogeneous WDM due to the perturbation in Eq. (1). Additionally, the figure displays the pronounced temperature dependence of the new mode (solid blue versus solid green). We highlight the generality of these new modes by deliberately focusing on the interacting electron gas. Furthermore, we also demonstrate the emergence of these modes in isochorically heated Aluminum towards the end.
In Fig. 1, the EELF is shown for q = 0.431 Å −1 . The predicted mode emerges in the direction along the wave vector Q . It is shown at two temperatures corresponding to θ = 0.5 (solid blue) and θ = 0.1 (solid green). It exhibits a rich structure distinctly different from the well-known plasmon mode (dashed blue and dashed green). The new mode exhibits a damped structure close to the position of the plasmon peak and shows several satellite peaks. More importantly, the new mode has a strong temperature dependence. The broadened feature in the new mode is shifted by about 0.64 eV when the temperature is increased, while the plasmon mode is shifted by about 0.45 eV . The shift in the satellite peaks ( A and B ) is stronger, roughly E A ≈ 0.75 eV and E B ≈ 0.75 eV . Besides their energy shift, we expect the emergence of satellite peaks to be potentially helpful for diagnostics.
Next in Fig. 2, we predict additional modes and features that are observed along the wave vector Q for various values of q . These features emerge when we assess a wider range of perturbation amplitudes. We consider both a weak perturbation amplitude U 0 = 0.1 and a strong amplitude U 0 = 1.0 . We simultaneously probe the temperature dependence of these new modes for increasing values of the degeneracy parameter, θ = 0.1 (temperature T ≃ 1.25 eV , solid blue), θ = 0.5 ( T ≃ 6.27 eV , solid green), and θ = 1.0 ( T ≃ 12.54 eV , solid red).
First (left panel of Fig. 2), we assess the emerging mode when the perturbation is weak ( U 0 = 0.1 ). At low temperature ( θ = 0.1 ), the mode remains centered around the same frequency, but its satellite features become more separated with increasing q . With increasing temperature the mode peak intensity also rises and shifts to higher energies. Conversely, the satellite peaks become increasingly suppressed. This is due to thermal excitations that decrease the effect of the inhomogeneity.
Next in the right panel of Fig. 2, we further increase the perturbation amplitude (to U 0 = 1.0 ). This leads to a transformation of the plasmon mode to an optical mode. The physics behind this transformation is a localization effect. Due to the strong perturbation, a large number of electrons becomes localized in the strong density enhancement regions. This is a manifestation of electron density oscillations around the center of mass of these localized regions. We observe that the energy of the optical mode strongly depends on temperature. For example, at the smallest considered wave number q = 0.048 Å , the energy of the optical mode shifts by 0.4 eV from 32 eV at θ = 0.1 to 31.6 eV at θ = 0.5 . A further increase of the degeneracy parameter to θ = 1.0 results in a significant shift of the optical mode to 31 eV.
Furthermore, we provide further supporting evidence on the observation of the modes and features predicted in Fig. 2. To that end, we provide a detailed analysis of the underlying electronic structure in Fig. 3. There, we display the density distribution of orbital densities n i (r) = |φ i (r)| 2 (solid grey) and the total density distribution n(r) = i f i |φ i (r)| 2 (solid black) along Q at θ = 1.0 , where f i denotes the occupation numbers, here a Fermi-Dirac distribution. The left panel shows the results for a weak ( U 0 = 0.1 ), the right panel for a strong ( U 0 = 1.0 ) perturbation amplitude. While the KS orbitals φ i (r) are, strictly speaking, auxiliary quantities, they can nevertheless be used to explain the observed behavior of the EELF qualitatively. Furthermore, we confirm the accuracy of our KS-DFT calculations by comparing them to accurate quantum Monte-Carlo calculations  Fig. 3. At U 0 = 1.0 , most of the orbitals are localized at the central region. This causes the formation of an optical mode due to the collective oscillation of these orbitals around the center of the localization region. Further increase in U 0 leads to a stronger signal from optical modes.
So far, we discussed emerging modes along the direction of Q . However, in the transpose direction to Q , the EELF of the inhomogeneous system exhibits a mode similar to the standard plasmon 82 . Next, in Fig. 4, we demonstrate the formation of a double-plasmon mode at an angle of 45° to Q . The EELF is plotted for a perturbation amplitude of U 0 = 1.0 at various temperatures. The double plasmon signal appears at well separated energies. For example, at q = 0.136 Å one signal emerges around 10.7 eV and the other in the range from 29.1 eV to 30.1 eV . The plasmon excitation at larger energies is particularly sensitive to thermal effects. It shifts by about 0.9 eV when θ increases from 0.1 to 1.0.  www.nature.com/scientificreports/ We conclude this Report by demonstrating the emergence of these novel modes in isochorically heated aluminum. To this end, we impose an inhomogeneous electronic structure by applying the same perturbation as in Eq. (1). The final results of our analysis are shown in Figs. 5 and 6. Figure 5 illustrates the EELF in aluminum without any perturbation (dashed blue) and when the perturbation in Eq. (1) is imposed (solid blue). This is compared against the uniform (dashed black) and the perturbed electron gas (solid black) as defined in Eq. (1). In Fig. 6, we illustrate the temperature dependence of the predicted features in aluminum. The peak of the feature in aluminum shifts by about 0.4 eV when the temperature increases from 1.25 eV (solid blue) to 6.27 eV (solid green). The results for an unperturbed electronic structure in aluminum are provided as a reference (dashed blue and green). The comparison between solid and dashed curves further illustrates the effect a perturbed electronic structure has-while the peak position in the unperturbed case (dashed blue vs. dashed green curves) does not change significantly with temperature, a clear shift is observed in the perturbed case (solid blue vs. solid green curves).

Experimental feasibility
We demonstrate that the resolution achieved in present experiments is sufficient to measure the predicted electronic excitations that emerge in inhomogeneous WDM. To that end, in Fig. 7 we show the EELF at the wave-number at which the plasmon signal was detected in a recent experiment on Aluminum 84 . We observe that, first of all, our calculations in the homogeneous system reproduce the plasmon position (black curve) measured in the experiment (purple data point). Secondly, our predicted features in inhomogeneous WDM (blue curve) can be resolved with the current experimental accuracy, i.e., the uncertainty in the experimentally measured plasmon energy (dashed purple) is sufficiently small. Importantly, this example is as an illustration and,  www.nature.com/scientificreports/ in general, other (smaller) values of the wave numbers can be probed. We note that recently an energy resolution better than 0.1 eV for diagnostics using inelastic X-ray scattering at the high energy density instrument of the European XFEL has been reported 85 . This enables probing various new features in the electronic excitation spectrum discussed in this work.

Conclusion
We have demonstrated the emergence of novel electronic excitations in the inhomogeneous warm dense electron gas. From a physical perspective, the well-known plasmon is transformed into a more complicated mode with multiple peaks. Its details can be readily controlled via the imposed perturbation amplitude. Ultimately, an increasing degree of inhomogeneity results in both an optical mode and a double-plasmon feature. These physical effects are interesting in their own right, and the required perturbation amplitudes can be realized in experimental facilities. We also show that the energy resolution of present experiments 85 is sufficient to resolve the emerging modes and features. From a practical point of view, the presented EELF spectra of inhomogeneous WDM exhibit a substantially stronger dependence on the electronic temperature than the standard plasmon in homogeneous WDM. Furthermore, their directional dependence and richer structure provide additional constraints on theory and simulation. This can be used to solve the long-standing problem of temperature diagnostics in WDM, which is rather unconstrained at present.
The inhomogeneous electron structures that we investigate in this work can be realized using various already existing experimental setups. For example, sufficient perturbation amplitudes (see Ref. 72 for details) can be realized at LCLS 25 , the European X-FEL 28 , and FLASH 86 . A particularly promising experimental setup capable of  www.nature.com/scientificreports/ measuring dynamics following intense terahertz excitation of 600 kV/cm 66 (corresponding to a perturbation amplitude U 0 ≃ 0.3 ) has been developed at SLAC. This setup combines a picosecond terahertz pulse with an electron beam diagnostics operating on the femtosecond time scale. Finally, it is also a viable option to impose an inhomogeneity in the electronic structure by target design. Grating structures 87 can be used to fabricate targets with a spatial modulation that will impose an inhomogeneity in the electronic structure. Here, the required perturbation amplitude can be controlled in the manufacturing of the target. While the presented results assume a harmonic perturbation, we expect that other forms of electronic perturbations-characterized by amplitudes and wave numbers comparable to those considered here, will lead to similar results. In particular, these will lead to deviations in the electronic excitation spectrum from what is expected in the unperturbed case. Therefore, we speculate that other forms of density perturbations, if realizable in experiments, can also be used for improved WDM diagnostics.
We expect that the interplay of experiment and simulation in probing inhomogeneous WDM will potentially become an invaluable new method of diagnostics enabling the reliable inference of important plasma parameters. Prospective work to confirm our predictions in WDM experiments is planned.

Methods
The time-dependent density functional theory (TDDFT) calculations were performed with the GPAW code [77][78][79][80] . We used the Perdew-Zunger LDA functional 81 in the adiabatic approximation, periodic boundary conditions, and 14 unpolarized electrons in the simulation cell. The results are computed using 40 bands at θ = 0.1 , 95 bands at θ = 0.5 , and 180 bands at θ = 1.0 . The momentum transfer parameter q of the EELF spectrum is restricted to the difference between two k points from the equilibrium state calculation. In the longitudinal and transverse directions with respect to Q , the results where computed using 32 k-points along q and 8 k-points in the other directions. For example, for computing the EELF spectrum along Q (with Q chosen along the x axis), 32 × 8 × 8 k-points were used.
In the case of isochorically heated Aluminum, we compute changes in the electronic structure due to both the imposed external perturbation and the electron-ion interaction. Here, the face-centered-cubic crystal structure with a lattice constant a = 4.025 Å was used. Correspondingly, the perturbation wave number was set to 4π/a . The results where obtained using 32 k-points along q and 8 k-points in the other directions. The plane-wave mode was used with a cutoff energy 600 eV . We used the Perdew-Zunger LDA functional 81 . The number of bands was set to 60 at 1.25 eV and was set to 95 at 6.27 eV . Not that at 1.25 eV , the number of partially occupied states is 6. At 6.27 eV , the number of partially occupied states is 35. Remaining 54 (at 1.25 eV ) and 60 (at 6.27 eV ) unoccupied bands provide converged and accurate results for the EELF at considered wave numbers.
The QMC data shown in Fig. 3 have been obtained via the direct path integral Monte Carlo (PIMC) method (see, e.g., Refs. [88][89][90] for details) without any nodal restrictions. Furthermore, we employ the primitive factorization of the density matrix using P = 200 imaginary-time slices. The convergence has been checked carefully. Therefore, the PIMC reference data are exact within the given error bars and constitute an unassailable benchmark for the KS-DFT results. An analysis of finite-size effects is provided as Supplementary Information