Light scattering in a phonon gas

Light-scattering spectrum in dielectric crystals is studied with extended thermodynamics (ET) for a phonon gas, covering from hydrodynamic to ballistic (collisionless) regimes. The ET equation is solved to obtain the power spectrum of the energy density in a phonon-gas mixture, which consists of interacting phonon gases of longitudinal and transverse acoustic (LA and TA) phonon modes. In the hydrodynamic regime, where phonon collisions take place frequently, it is found that the light-scattering spectrum consists of two components, which can be interpreted as the two normal modes formed by the two second-sound modes defined in each of the LA and TA phonon gases. Out of the two normal-mode spectra, the low-frequency component arises from thermal fluctuations and gives rise to a narrow quasielastic spectrum, which corresponds to the well-known thermal Rayleigh scattering due to thermal diffusion, i.e., due to overdamped second sound. With sufficient momentum-conserving phonon collisions (normal phonon scattering), the narrow quasielastic component is demonstrated to develop, from the diffusive central peak, into a pair of shifted inelastic peaks due to propagation of underdamped second sound. The other spectrum component gives rise to a much broader quasielastic scattering, whose wing extends out to the Brillouin-scattering lines of the LA and TA phonons (first sounds). The broader quasielastic spectrum has a linewidth equal to the phonon-collision rate, which suggests that this component originates from a nonequilibrium process in the phonon gas. As the ballistic regime is approached, the line shapes and linewidths of the two normal-mode spectra approach each other, and the two components finally coincide in the limit of the ballistic regime, which is in good agreement with the reported behavior for these spectra that were experimentally observed in many crystals. In the ballistic limit, the light-scattering mechanism in the present model is found to become formally equivalent to the previously proposed microscopic framework, i.e., the second-order difference Raman scattering (two-phonon difference light scattering) on a same phonon dispersion. The derived spectral formula is fitted to the spectra previously observed in the experiments for rutile $\left({\text{TiO}}_2\right)$ and strontium titanate $\left({\text{SrTiO}}_3\right)$. The fits are quite successful in wide ranges of frequency and temperature, i.e., regardless of degree of nonequilibrium, owing to the ET analysis. The relaxation times for the normal and resistive phonon collisions (${\tau }_{\text{N}}$ and ${\tau }_{\text{R}}$) are determined through the analysis. The temperature dependences of ${\tau }_{\text{N}}$ and ${\tau }_{\text{R}}$ indicate that the origin of the broad shifted peaks (the “broad doublet”), which were observed in ${\text{SrTiO}}_3$ at around 30 K, is likely due to underdamped second sound at least in a narrow temperature range around 30 K.

Figure 1

Two coupled (a) mechanical oscillators and (b) second sounds (temperature waves). In both figures, Mode 1 and 2 are the “in-phase” and “out-of-phase” normal modes, respectively. In (b), $\delta T$ and $e$ represent the temperature fluctuation and the energy-density fluctuation, respectively. Note that the interaction between the two oscillators is much more effective in Mode 2 than in Mode 1.

Figure 2

Phonon regimes and phonon Knudsen numbers. ${\text{Kn}}_{\text{R}}$ and ${\text{Kn}}_{\text{N}}$ are defined in Eqs. (25, 26), respectively. The hatched region corresponds to the “second-sound regime,” where a wavelike propagation of heat is expected to exist in solids.

Figure 3

${\text{Kn}}_{\text{R}}$ dependence of the spectra of $S_1$ and $S_2$. The vertical axes are scaled in arbitrary units. The horizontal axes are measured in units of $cq$, which is the averaged Brillouin frequency.

Figure 4

Spectrum in the ballistic limit without the asymptotic termination. For ${\text{Kn}}_{\text{R}}=0.1$, there is only one peak (central peak) due to thermal diffusion. For larger values of ${\text{Kn}}_{\text{R}}$, splittings of the spectra are seen.

Figure 5

(Color online) A chart of light-scattering spectra from a phonon gas for combinations of $\left({\text{Kn}}_{\text{R}},{\text{Kn}}_{\text{N}}\right)$. All the spectra are plotted on log-log scales. We set $n_{\text{max}}=5$ and ${\gamma }_6={\gamma }_{\infty }$ in calculating these spectra.

Figure 6

Shifted-peak structure in the collisionless regime $\left({\text{Kn}}_{\text{R}},{\text{Kn}}_{\text{N}}\right)=\left(100,10\right)$. Both axes are on linear scales.

Figure 7

(Color online) ${\text{Kn}}_{\text{R}}$ dependence of the normalized linewidths. ${\gamma }_1$ and ${\gamma }_2$ are the normalized half widths at half maximum of the spectra of $S_1$ and $S_2$, respectively. (a) Experimentally measured ${\text{Kn}}_{\text{R}}$ dependence of the normalized linewidths reported in Ref. 7. The BS and ISTS represent backscattering and impulsive stimulated thermal scattering, respectively. (b) Calculated linewidths of $S_1$ and $S_2$.

Figure 8

(Color online) Temperature dependence of the quasielastic light-scattering spectrum observed in rutile $\left(\mathit{q}\parallel \left[110\right]\right)$. “LA” and “TA” denote longitudinal and transverse acoustic Brillouin lines, respectively. The inset shows the spectra at high temperatures $\left(\mathit{q}\parallel \left[001\right]\right)$. The solid lines are the fit of Eq. (54).

Figure 9

(Color online) Temperature dependence of the low-frequency light-scattering spectra in ${\text{SrTiO}}_3$ from 6 to 295 K (log-log plot). The LA and TA denotes longitudinal and transverse acoustic phonons’ Brillouin lines, respectively. The inset is a semi-log plot for the spectra below 72 K. Each spectrum is appropriately scaled for visual clarity. The solid lines are fits of Eq. (54).

Figure 10

(Color online) Temperate dependences of the normal $\left({\tau }_{\text{N}}^{-1}\right)$ and the resistive $\left({\tau }_{\text{R}}^{-1}\right)$ phonon relaxation rates in ${\text{SrTiO}}_3$. Legends are shown in the figure. All the lines between the symbols are guides to the eyes. The dashed horizontal line at $\sim 2\times {10}^{11}\text{rad}/\text{s}$ represents a constant $q{v}_{\text{D}}/\sqrt{3}\approx {\omega }_0$. The shaded area indicates the “window” for second-sound propagation.

Figure 11

Phonon Knudsen numbers measured in ${\text{SrTiO}}_3$ (the closed squares). See Figs. 2, 5 for qualitative comparison.

Figure 12

(Color online) Determination of the frequency shifts and the envelope function for the peaks of the round-shaped comb spectrum when $\text{Kn}⪢1$. Here $N=7$ is assumed for visual clarity although the number is actually too small for the approximation presented in the text to be valid.