X-ray Thomson scattering in warm dense matter at low frequencies

The low-frequency portion of the x-ray Thomson scattering spectrum is determined by electrons that follow the slow ion motion. This ion motion is characterized by the ion-ion dynamic structure factor, which contains a wealth of information about the ions, including structure and collective modes. The frequency-integrated (diffraction) contribution is considered first. An effective dressed-particle description of warm dense matter is derived from the quantum Ornstein-Zernike equations, and this is used to identify a Yukawa model for warm dense matter. The efficacy of this approach is validated by comparing a predicted structure with data from the extreme case of a liquid metal; good agreement is found. A Thomas-Fermi model is then introduced to allow the separation of bound and free states at finite temperatures, and issues with the definition of the ionization state in warm dense matter are discussed. For applications, analytic structure factors are given on either side of the Kirkwood line. Finally, several models are constructed for describing the slow dynamics of warm dense matter. Two classes of models are introduced that both satisfy the basic sum rules. One class of models is the “plasmon-pole”-like class, which yields the dispersion of ion-acoustic waves. Damping is then included via generalized hydrodynamics models that incorporate viscous contributions.

Figure 1

(Color online) The $\mathcal{W}$ parameter is shown for Be (left panel) and Al (right panel) over wide ranges of temperature and density. The contours are at levels of $\mathcal{W}=0.1,0.3,0.5,0.7,0.9$. Note that the WDM regime captures the center of the phase diagram, away from regions where simplifying assumptions can be made. Also, note that the WDM regime is shifted to slightly higher densities and lower temperatures for higher-$Z$ elements.

Figure 2

(Color online) The ionization states of Be (left panel) and Al (right panel) are shown versus temperature and density, as computed with a finite-temperature Thomas-Fermi model. The contours are at ionization states of $⟨Z⟩=1.0,2.0,3.0$ for Be and $⟨Z⟩=1.0,3.0,5.0$ for Al. Comparing with Fig. 1, we see that WDM corresponds to matter in which ionization is moderate, thus highlighting the importance of finite-temperature atomic physics. More interestingly, note that WDM occurs along the “knees” of the iso-ionization curves, where temperature and pressure ionization have about equal contributions.

Figure 3

(Color online) The ion-ion Coulomb coupling parameters for Be (left panel) and Al (right panel). The contours are at levels $\Gamma =3.0,5.0,10.0$. Note that the level of ionic coupling varies from element to element in the WDM regime; here, in the centers of their respective WDM regimes, Al ions are more strongly coupled than Be ions.

Figure 4

(Color online) The predicted ionization state of Be is shown versus temperature for $\rho =1.69\text{g}/{\text{cm}}^3$ (liquid density) using the two definitions of ionization discussed in the text. The upper (blue) curve corresponds to the definition $Z^{\ast }$, whereas the lower (green) curve corresponds to $⟨Z⟩$. Also in the left panel, the thin blue lines with squares correspond to results obtained from a fully quantum mechanical calculation using the $Z^{\ast }$ definition of the ionization state. The upper(diamonds)/lower(squares) thin blue curves neglect/include exchange correlation contributions, so that the relative effects of orbitals and exchange correlation can be disentangled. Note that there is good agreement between the fully quantal results and the TF model over a wide range of temperatures, indicating the reasonable efficacy of TF models for describing WDM. The right panel shows $P_{bound}\left(r\right)=4\pi r^2{n}_{bound}\left(r\right)$ (arb. units) for Be at temperatures $T=1\text{eV}$ (upper, purple) and $T=20\text{eV}$ (lower, cyan), which shows how the bound density varies with increasing temperature and, therefore, ionization state.

Figure 5

(Color online) The Yukawa structure factor, using the HNCB method, for Al at just above the melting point ($\rho =2.27\text{g}/{\text{cm}}^{-3}$ and $T=0.114\text{eV}$) is shown as a solid green line with no symbols. The purple curve with diamonds is the experimentally measured result.

Figure 6

(Color online) The Kirkwood lines for Be (left panel) and Al (right panel) are shown versus temperature and density. The figures show modified ion-ion Coulomb couplings with 1/2 added if the material is above the Kirkwood line and $-1/2$ added if it is below. Thus, the discontinuity in intensity/color is the boundary between monotonic and oscillatory decay of the pair correlation functions. Note that the location of the Kirkwood line is quite different for the different materials.

Figure 7

(Color online) Comparison of the analytic structure factor with the hypernetted chain plus bridge result for two cases. The lower, more weakly coupled cases have $\Gamma =10$ and $\kappa =1$ (blue curves), whereas the upper, more strongly coupled cases have $\Gamma =50$ and $\kappa =1$ (green curves). The solid (no symbols) lines are HNCB results, and the lines with open circles are from the analytical model. In general, the analytical model tends to slightly overestimate the peak heights. This reveals the extent to which an effective coupling parameter model is useful.

Figure 8

(Color online) Comparison of the analytic structure factor with HNCB results (blue line with no symbols), as in the previous figure, but for the case $\Gamma =50$ and $\kappa =3$. In this case, the actual system is closer to a real HS system (red line with circles), and the resulting agreement is improved. Thus, for strongly screened WDM, most of the ionic structure can be accounted for by simple HS repulsion; the remainder is due to details of the electronic structure.