Collective acoustic modes as renormalized damped oscillators: Unified description of neutron and x-ray scattering data from classical fluids

In the $Q$ range where inelastic x-ray and neutron scattering are applied to the study of acoustic collective excitations in fluids, various models of the dynamic structure factor $S(Q,\omega )$ generalize in different ways the results obtained from linearized-hydrodynamics theory in the $Q\to 0$ limit. Here we show that the models most commonly fitted to experimental $S(Q,\omega )$ spectra can be given a unified formulation. In this way, direct comparisons among the results obtained by fitting different models become now possible to a much larger extent than ever. We also show that a consistent determination of the dispersion curve and of the propagation $Q$ range of the excitations is possible, whichever model is used. We derive an exact formula which describes in all cases the dispersion curve and allows for the first quantitative understanding of its shape, by assigning specific and distinct roles to the various structural, thermal, and damping effects that determine the $Q$ dependence of the mode frequencies. The emerging picture describes the acoustic modes as $Q$-dependent harmonic oscillators whose characteristic frequency is explicitly renormalized in an exact way by the relaxation processes, which also determine, through the widths of both the inelastic and the elastic lines, the whole shape of collective-excitation spectra.

Figure 1

Results of the viscoelastic analysis of liquid-${\mathrm{CD}}_4$ simulated spectra [8]. (a) Fitted $S\left(Q\right)$ (circles; error bars are smaller than the symbols); the line marks the value of $S\left(0\right)$. (b) Dispersion curve. Fitted ${\omega }_s$ (dots with error bars); the other symbols refer to quantities calculated from the fit parameters using (71, 76, 77, 78): $\sqrt{⟨{\omega }_Q^2⟩}$ (circles), $\sqrt{\gamma \left(Q\right)⟨{\omega }_Q^2⟩}$ (stars), and $\Omega \left(Q\right)$ (squares). The straight lines are $\sqrt{⟨{\omega }_0^2⟩}=c_{s}Q∕\sqrt{{\gamma }_0}$ (dashed) and $c_{s}Q$ (solid). (c) $\gamma \left(Q\right)$ (circles) derived from fit parameters using (71e); the line marks the value of ${\gamma }_0$. (d) $r\left(Q\right)$ (circles) and $\sqrt{1-z_s^2∕{\Omega }^2\left(Q\right)}$ (stars), from (78).

Figure 2

Dispersion curve of the RB triplet calculated with the thermophysical coefficients of liquid ${\mathrm{CD}}_4$ [8]. (a) $\sqrt{\gamma \left(Q\right)⟨{\omega }_Q^2⟩}=\sqrt{{\gamma }_0⟨{\omega }_0^2⟩}=c_{s}Q$ (solid line); $\Omega \left(Q\right)$ (upper dashed line); ${\omega }_s$ from solution of (31) (lower dashed line). (b) $r\left(Q\right)$ (dashed line) and $\sqrt{1-z_s^2∕{\Omega }^2\left(Q\right)}$ (solid line).

Figure 3

Results of fitting the generalized hydrodynamic and viscoelastic models to the liquid-${\mathrm{CD}}_4$ simulated spectra [8]. $⟨{\omega }_Q^2⟩$, calculated from fit parameters using (49) (dots) and (71) (circles), respectively. Theoretical values calculated from (6) are shown by stars (solid line as a guide for the eye).

Figure 4

Results of fitting the generalized hydrodynamic (dots) and viscoelastic (other symbols) models to the liquid-${\mathrm{CD}}_4$ simulated spectra [8]. Error bars are indicated for quantities obtained directly from the fit unless smaller than the symbol size. (a) Half-width at half-maximum of central Lorentzian lines: $z_h$ and, for viscoelastic model, $z_h$ (circles) and $z_2$ (squares). Lines show calculation of ${\Gamma }_T\left(Q\right)∕\gamma \left(Q\right)$ [approximation (B2) to $z_h$, solid line] and of $1∕\tau \left(Q\right)-\tau \left(Q\right){\Delta }_L^2\left(Q\right)$ [approximation (B3) to $z_2$, dashed line]. (b) Amplitudes of central Lorentzian lines. $I_h$ and, for viscoelastic model, $I_h$ (circles), and calculated $I_2$ (diamonds) and $I_h+I_2$ (squares). (c) $\gamma \left(Q\right)$; (d) ${\Gamma }_T\left(Q\right)$.

Figure 5

Results of fitting the generalized hydrodynamic (dots) and viscoelastic (other symbols) models to the liquid-${\mathrm{CD}}_4$ simulated spectra [8]. Error bars are indicated for quantities obtained directly from the fit unless smaller than the symbol size. (a) Dispersion curve ${\omega }_s$. (b) Half-width at half-maximum $z_s$ of Brillouin lines. At $Q=15{\mathrm{nm}}^{-1}$, $-z_A$ and $-z_B$ (stars) replace the viscoelastic parameter $z_s$ (circles). Lines show $\Omega \left(Q\right)$ (generalized hydrodynamic model, solid line; viscoelastic model, dashed line). (c) $r\left(Q\right)$. For viscoelastic model, the total (circles), and the two factors $z_h\gamma \left(Q\right)∕{\Gamma }_T\left(Q\right)$ (squares) and $z_2\tau \left(Q\right)$ (diamonds) are separately shown.

Figure 6

Results of fitting the generalized hydrodynamic (dashed line) and viscoelastic (other symbols) models to the liquid-${\mathrm{CD}}_4$ simulated spectra [8]. Differences between the simulated and the fitted spectra at $Q=15{\mathrm{nm}}^{-1}$ and $\omega ⩾0$. Viscoelastic fit with complex roots, dots; with all real roots, solid line. For comparison, the peak value is $S(Q,0)=95.8\times {10}^{-3}\mathrm{ps}$.