Stokes to Anti-Stokes Intensity Ratio in the Raman Scattering Spectra of Rutile TiO2
+ AffiliationsGraduate School of Frontier Biosciences, Osaka University, Suita, Osaka 565-0871, Japan
Received March 31, 2019; Accepted July 8, 2019; Published August 8, 2019
We investigate the Stokes to anti-Stokes intensity ratio in the phonon Raman spectra of rutile titanium dioxide (TiO2). The frequency dependence of the ratio for first-order scattering shows a plateau where the ratio does not vary with frequency near the peak frequency of Eg mode, while the second-order component satisfies the relation of the Boltzman factor. The frequency region of the plateau broadens with temperature, and it is suggested that the irreversible relaxation that broadens the phonon energy plays an important role.
©2019 The Physical Society of Japan
1. IntroductionFor a Raman scattering spectrum under non-resonant excitation, the Stokes [\(I_{\text{S}}(\Omega)\)] to anti-Stokes [\(I_{\text{AS}}(\Omega)\)] intensity ratio at a frequency shift Ω is given by the Boltzmann factor:1–3) \begin{equation} I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) = \mathrm{e}^{\hbar \Omega/k_{\text{B}}T}. \end{equation} (1) This relation is a result of the time-reversal symmetry of spontaneous Raman scattering under a canonical distribution of non-magnetic material and is often used in experiments to accurately determine the sample temperature T. However, it is important to determine whether Eq. (1) still holds at each frequency shift Ω for a Raman peak with homogeneous broadening. When the energy \(\hbar\Omega_{0}\) of an elementary excitation that scatters light broadens with \(\Delta E\sim \hbar/\tau_{\text{c}}\) due to an irreversible relaxation process with a coherence time \(\tau_{\text{c}}\), its energy has no definite value within \(\Delta E\). In this frequency region of \(\Delta E/\hbar\), \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) may not vary with Ω according to the Boltzman factor. If a deviation from the Boltzman factor is observed in the Raman spectrum, the ratio analysis gives useful information about the relaxation process of elementary excitation.
Focusing on the dissipative nature of the relaxational modes, which appear as a central peak in Raman spectra, the ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) is observed and found to deviate from Eq. (1), taking a symmetric spectrum with \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\approx 1\) around \(\Omega = 0\) in liquid carbon disulfide,4) amorphous solids and supercooled liquids such as As2O3 and ZnCl2,5) and potassium dihydrogen phosphate (KDP) near the ferroelectric-phase-transition temperature.6) The relation between the symmetric spectrum of relaxational modes around \(\Omega = 0\) and their exponential decay in the long-time region of the time-response function has been discussed from the perspective of Markovian nature of their dynamics.7)
In this paper, we study the intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) of phonon Raman spectra. According to the theory of first-order phonon Raman scattering, the broadening of the spectrum has been analysed based on self-energy of the mode due to its anharmonic interaction with other phonon modes.8–10) As reviewed later, these theories predict that \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) also varies with Ω within the spectral broadening according to the Boltzman factor, which has not yet been experimentally clarified. The Stokes to anti-Stokes intensity ratio is usually evaluated to determine the sample temperature by using the peak intensity or frequency(Ω)-integrated intensity of the Raman line. Here, we investigate the intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) of phonon Raman spectra in a rutile TiO2 for each Ω. The rutile TiO2 crystal shows strong phonon Raman lines with wide homogeneous broadening, which have been studied in terms of phonon anharmonicity.11–18) We briefly reported a deviation of the ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) from the Boltzman factor for \(E_{g}\) mode (\(\Omega_{0}\approx 450\) cm−1) of TiO2 at 150 K.19) Because the spectral broadening due to phonon anharmonicity increases with temperature, the present analysis is conducted over a high temperature range including 473 K and a wide spectral region (\(10\leq \Omega\leq 600\) cm−1) including the second-order component of Raman scattering is investigated. We found that the Ω dependence of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) for \(E_{g}\) phonon mode presents a plateau where \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) does not vary with Ω near the phonon peak frequency, while the second-order component satisfies Eq. (1) over the spectral region. The frequency region of the plateau broadens with temperature, suggesting that the irreversible relaxation that broadens the phonon energy plays an essential role, which is not usually considered in the analysis of phonon Raman scattering.
2. Theory of Phonon Raman ScatteringAt first, we briefly summarize the theoretical background of Eq. (1) and the standard way of analyzing the first-order phonon Raman spectrum. The differential cross section of spontaneous Raman scattering under non-resonant excitation is given, apart from unimportant factors, by2,3,7) \begin{equation} \frac{d^{2}\sigma}{d\Omega_{\text{s}}\,d\omega_{2}} = \omega_{1} \omega_{2}^{3} I(\omega_{1}, \omega_{2}), \end{equation} (2) where \(I(\omega_{1},\omega_{2})\) is the Raman spectrum, \(\omega_{1}\) and \(\omega_{2}\) are the frequencies of the excitation and scattered photons, respectively, and \(\Omega_{\text{s}}\) is the solid angle. From the perspective of fluctuation-dissipation theorem, the Stokes and anti-Stokes Raman spectra are expressed as2,3) \begin{align} I_{\text{S}}(\Omega) &= I(\omega_{1}, \omega_{1} -\Omega) = [n(\Omega)+1] \text{Im} R(\Omega), \end{align} (3) \begin{align} I_{\text{AS}}(\Omega) &= I(\omega_{1}, \omega_{1} +\Omega) = n(\Omega) \text{Im} R(\Omega), \end{align} (4) where \(\Omega=|\omega_{1} -\omega_{2} |\), \(n(\Omega)\) is the Bose–Einstein distribution function, and the imaginary part of the response function \(\text{Im} R(\Omega)\) is expressed by a set of δ-functions, each located at \(\Omega =\omega_{m}-\omega_{n}\) with the energy \(\hbar\omega_{n}\) of the stationary state \(| n\rangle\) of the relevant system.20,21) Thus, the Stokes and anti-Stokes Raman spectra with spectral broadenings are expressed by a set of δ-functions, each satisfying \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) = [n(\Omega)+1]/n(\Omega)=e^{\hbar\Omega/k_{\text{B}}T}\). In the case of first-order phonon Raman scattering, the imaginary part of the response function \(\text{Im} R(\Omega)\) can be expressed, apart from unimportant factors, as8–10) \begin{equation} \text{Im} R(\Omega) = \frac{\Gamma(\Omega)}{\{\Omega-\Omega_{0}-\Delta(\Omega) \}^{2}+\Gamma(\Omega)^{2}}, \end{equation} (5) where the frequency shift \(\Delta(\Omega)\) and broadening \(\Gamma(\Omega)\) of the phonon mode with harmonic frequency \(\Omega_{0}\) are caused by its anharmonic interaction with other phonon modes and are given by the real and imaginary parts of the phonon self-energy, respectively. In Eq. (5), it is assumed that \(\Delta(\Omega)\) and \(\Gamma(\Omega)\) are much smaller than \(\Omega_{0}\). \(\Gamma(\Omega)\), which is determined by the two-phonon density of states and the coupling strength between the phonon modes, is given by a set of δ-functions associated with the energy conservation among the interacting phonons and the incident and scattered photons.8–10) The spectral shape of \(\text{Im} R(\Omega)\) reduces to a simple Lorentzian if both \(\Delta(\Omega)\) and \(\Gamma(\Omega)\) do not depend on the frequency Ω. However, it generally deviates from Lorentzian owing to the frequency dependence of the phonon self-energy.
3. ExperimentalIn the following, we investigate \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) in the Raman spectra of rutile TiO2. As an excitation source, we employed a cw-Ar–Kr ion laser (Spectra Physics 2020) operated at 514.5 nm with power 120.0–150.0 mW under a fluctuation less than \(\pm 0.5\) mW. The scattering geometry was \(y(x,z)x\) or \(y(z,z)x\) for the observation of the \(\alpha_{zx}\) or \(\alpha_{zz}\) component of a scattering tensor α, respectively, where x, y, and z are the crystallographic axes. The \(E_{g}\) phonon mode is observed in the scattering geometry of \(y(x,z)x\) but not in \(y(z,z)x\) owing to its scattering tensor.22) To change the geometry from \(y(x,z)x\) to \(y(z,z)x\), the polarization of excitation laser light was rotated by inserting a half-wave plate. For the measurement at temperature \(T\leq 273\) K, a TiO2 crystal with size \(9\times 9\times 9\) mm3 was maintained in liquid isopentane in glass cell in an atmosphere of cooled N2 gas from a liquid nitrogen vessel. The temperature of the sample was measured using a thermocouple in isopentane and controlled with a stability of 0.1 K. For the measurement at 299 and 473 K, the TiO2 crystal was maintained in silicone oil heated to the required temperature. Because the fundamental absorption band of rutile TiO223) lies in a much higher energy region than the present spectral range, the non-resonant excitation condition assumed in Eq. (1) is satisfied. This is also confirmed by the result that the broad component of second-order scatterings satisfies Eq. (1) as shown later.
The scattered light was analyzed by a polarizer (z-direction) and a double monochromator (Jobin Yvon U-1000) with a spectral resolution of 2.0–3.0 cm−1. The double monochromator was scanned with a step of 0.5 cm−1 and the photon number of scattered light was counted using a photomultiplier tube (Hamamatsu R464 operated with a supply voltage of 1150 V between anode and cathode) with an accumulation time of 1 s for each channel.
In the present analysis, it is quite important that the detection system is operated in its dynamic range of photon counting. To confirm it, we measured elastic scattering spectrum from the crystal around \(\omega_{2} = 0\). The obtained spectrum with maximum count of \(1\times 10^{6}\) s−1 is well fitted by a Gaussian function with a spectral width corresponding to a slit width of the double monochromator. This indicates that the linearity of output count is preserved at least under the count of \(1\times 10^{6}\) s−1. Because the photon number detected in the spectral region around \(E_{g}\) mode in the present Raman spectra lied within \(10^{2}\)–\(10^{5}\) s−1 with dark count of about 10 s−1, we consider that the linearity of the detection system is preserved in the following Raman spectra.
After correcting the measured spectra by the \(\omega_{2}^{3}\) factor in Eq. (2), we carefully corrected the wavelength-dependent sensitivity of the monochromator/detection system including the polarizer using a measured spectrum of light from a standard lamp.
4. Results and DiscussionIn Fig. 1, we show the Raman spectra \(I(\omega_{1},\omega_{2})\) of TiO2 measured for the \(\alpha_{zx}\) and \(\alpha_{zz}\) components as a function of \(\omega =\omega_{1} -\omega_{2}\). In the spectrum of the \(\alpha_{zx}\) component, a strong line of \(E_{g}\) phonon mode and a broad component lying over the spectral range are observed. The half-width at half-maximum (\(\mathit{HWHM}\)) of the \(E_{g}\) mode at high energy side of the Stokes spectrum is obtained as \(\mathit{HWHM}= 10.0\), 15.7, and 23.3 cm−1 for \(T=170\), 299, and 473 K, respectively. The origin of the broad component has been understood as a set of continuously distributed two-phonon scattering peaks with combinations of various phonon modes (combination bands).14–17) Contrastingly, in the spectrum of the \(\alpha_{zz}\) component, only the combination bands are observed. The qualitative features of these spectra are the same as those of the reported ones for Stokes scattering.11)
Figure 1. Raman scattering spectra \(I(\omega_{1},\omega_{2})\) of rutile titanium dioxide (TiO2) measured for \(\alpha_{zx}\) and \(\alpha_{zz}\) components as a function of \(\omega =\omega_{1} -\omega_{2}\).
The intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) is calculated as a function of \(\Omega = |\omega |\) for each spectrum. In the analysis, we found that the ratio for Ω around \(E_{g}\) mode is sensitive to the determination of the spectral origin (\(\omega=0\)), as discussed in the case of central peak.4,6,7,24) In this analysis, the spectral origin of each spectrum is determined from the center frequency of elastic scattering peak, which has a Gaussian shape that reflects the instrumental function of the monochromator. Here, we make a linear interpolation of the spectrum measured with 0.5 cm−1 step of \(\omega_{2}\) when the spectral origin exists between the data points. As the elastic scattering spectrum can be affected by the scattered light from the sample surface or impurity in the scattering volume, we confirmed the validity of the origin determined at \(T=299\) and 473 K by comparing the results of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) obtained from different positions of scattering volume in the sample crystal.
The intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) thus obtained is shown for temperature \(T=170\), 299, and 473 K for the \(\alpha_{zx}\) component in Fig. 2(a), and \(T=298\) and 473 K for the \(\alpha_{zz}\) component in Fig. 2(b). For each temperature and polarizability component, the results of the two scans are plotted by squares and triangles to show the reproducibility of the ratio. The solid lines indicate the relation \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) = e^{\hbar\Omega/k_{\text{B}}T}\). As shown in Figs. 2(a) and 2(b), the intensity ratio almost satisfies the Boltzman factor in the range of \(10<\Omega<600\) cm−1 at all temperatures for the components \(\alpha_{zx}\) and \(\alpha_{zz}\).
Figure 2. (Color online) The intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) for (a) \(\alpha_{zx}\) and (b) \(\alpha_{zz}\) components. The solid line indicates the relation \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) = e^{\hbar\Omega/k_{\text{B}}T}\).
The broad component observed in TiO2 was analyzed based on a model of continuously distributed second-order scattering peaks with various combinations of phonon modes.14–17) However, there has been no experimental study on the intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) of these combination bands. For a peak of second-order scattering centered at frequency shift Ω, two phonons with frequencies of \(\Omega_{1}\) and \(\Omega_{2}\) are involved, which satisfy the relation \(\Omega =\Omega_{1} +\Omega_{2}\) or \(\Omega =\Omega_{1} -\Omega_{2}\) (\(\Omega_{1} >\Omega_{2}\)). In these cases, on neglecting the spectral broadening of each phonon, the Stokes to anti-Stokes intensity ratio for frequency shift Ω are given by25,26) \begin{align} I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) &= [\{n(\Omega_{1})+1\}\{n(\Omega_{2})+1\}]/[n(\Omega_{1})n(\Omega_{2})]\notag\\ &= e^{\hbar (\Omega_{1} + \Omega_{2})/k_{\text{B}}T}, \end{align} (6) or \begin{align} I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) &= [\{n(\Omega_{1})+1\}n(\Omega_{2})]/[n(\Omega_{1})\{n(\Omega_{2})+1\}]\notag\\ &= e^{\hbar (\Omega_{1} - \Omega_{2})/k_{\text{B}}T}, \end{align} (7) respectively. As a result, the broad component, which is considered as a set of second-order scattering peaks, is also expected to satisfy Eq. (1). This is consistent with the results of broad components shown in Figs. 2(a) and 2(b). To our knowledge, this is the first report that experimentally proves that the broadband of continuously distributed second-order scattering peaks in Raman spectrum satisfies Eq. (1).
Next, in Fig. 3, we plot the Ω dependence of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) around the peak frequency \(\Omega_{E_{g}}\) of \(E_{g}\) mode for \(\alpha_{zx}\) component. It is found that the ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) shows a plateau where \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) does not vary with Ω and takes a constant value of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\sim e^{\hbar\Omega_{E_{g}}/k_{\text{B}}T}\) near \(\Omega =\Omega_{E_{g}}\). The frequency range of the plateau is indicated by arrows in Figs. 3(a)–3(c) with 9, 12, and 17 cm−1 for \(T=170\), 299, and 473 K, respectively, which are much larger than the spectral resolutions of 2.0 (170 K) and 3.0 (299 and 473 K) cm−1.
Figure 3. (Color online) The intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) and \(I_{\text{S}}(\Omega)\) for \(\alpha_{zx}\) around the \(E_{g}\) mode at (a) 170 K, (b) 299 K, and (c) 473 K. The arrow indicates the frequency range where \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) does not vary with Ω. The solid line indicates the relation \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) = e^{\hbar\Omega/k_{\text{B}}T}\).
Figure 4 shows the spectral width \(\mathit{HWHM}\) of \(E_{g}\) mode and frequency range of plateau \(\Delta E_{\text{p}}\) at high-energy side \(\Omega\geq \Omega_{E_{g}}\) obtained at several temperatures. Here, we plot \(\mathit{HWHM}\) and \(\Delta E_{\text{p}}\) at \(\Omega\geq \Omega_{E_{g}}\) because a shoulder-like component resulting from second-order scattering located at the tail of low-energy side \(\Omega\leq \Omega_{E_{g}}\) (Fig. 1) can affect the spectral shape and \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\). As seen in Fig. 4, \(\Delta E_{\text{p}}\) increases with temperature along with \(\mathit{HWHM}\).
Figure 4. (Color online) The spectral width \(\mathit{HWHM}\) of \(E_{g}\) mode (circles) and the frequency range of plateau \(\Delta E_{\text{p}}\) (squares) at high-energy side \(\Omega\geq \Omega_{E_{g}}\) as defined in the inset. The values at 150 K are obtained from the spectrum in Ref. 19.
We consider that the plateau originates from an irreversible relaxation in the phonon scattering process, which is not usually considered in the analysis of phonon Raman spectra. As seen in Eqs. (3)–(5), the Raman spectrum of \(E_{g}\) mode can be regarded as a set of δ-function-like spectrum associated with energy conservation among interacting phonons and photons. In the case of TiO2, the principal contribution to the spectral broadening of \(E_{g}\) mode is derived from the cubic anharmonicity associated with three-phonon processes including \(E_{g}\) mode and two other modes with frequencies \(\omega(\mathbf{k},j_{1})\) and \(\omega(-\mathbf{k},j_{2})\) satisfying \(\omega(\mathbf{k},j_{1})\pm \omega(-\mathbf{k},j_{2})\sim \Omega_{E_{g}}\) with wave vectors \(\mathbf{k}\) and \(-\mathbf{k}\) in the phonon branch \(j_{1}\) and \(j_{2}\), respectively.18) If the quantum coherence between these interacting phonons disappears by time \(\tau_{\text{c}}\), their energy becomes uncertain by \(\Delta E\sim \hbar/\tau_{\text{c}}\) according to the energy-time uncertainty relation.27) Thus, each δ-function-like spectrum associated with these phonons has a spectral width of \(\Delta E\), which results in the deviation of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) from Eq. (1).
To confirm whether the above interpretation qualitatively reproduces the experimental results, we calculated \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) under the following assumptions. The two-phonon density of states and coupling strength in \(\Delta(\Omega)\) and \(\Gamma(\Omega)\) in Eq. (5) are independent of Ω. A set of δ-function associated with \(\text{Im} R(\Omega)\) has a spectral shape of Lorentzian around center frequency \(\Omega_{0}\) with width \(\mathit{HWHM}=\gamma\). Each δ-function at \(\Omega =\Omega'\) broadens to Gaussian spectrum with \(\mathit{HWHM}=\Delta E\) that is same for all δ-functions and \(n(\Omega)\) for the Gaussian is given by \(n(\Omega')\). Based on these assumptions we calculated following spectra: \begin{align} I_{\text{S}}(\Omega) &= \sum_{\Omega'} [n(\Omega')+1] \frac{1}{(\Omega'-\Omega_{0})^{2}+\gamma^{2}} \mathrm{e}^{-(\ln 2/\Delta E^{2}) (\Omega-\Omega')^{2}}, \end{align} (8) \begin{align} I_{\text{AS}}(\Omega) &= \sum_{\Omega'} n(\Omega') \frac{1}{(\Omega'-\Omega_{0})^{2}+\gamma^{2}} \mathrm{e}^{-(\ln 2/\Delta E^{2}) (\Omega-\Omega')^{2}}. \end{align} (9) In Fig. 5, we show \(R = \{I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) \}/\mathrm{e}^{\hbar\Omega_{0}/k_{\text{B}}T}\) and \(I_{\text{S}}(\Omega)\) as a function of \(\Delta\Omega =\Omega -\Omega_{0}\) calculated with \(\gamma = 3\) cm−1, \(\Delta E = 6.0\), 7.5, and 9.0 cm−1 and \(T = 170\) K. As seen in the figure, \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) shows a plateau-like behavior around \(\Delta\Omega = 0\). The frequency region of the plateau-like behavior broadens with \(\Delta E\) along with the spectral width. Thus, the results of calculation qualitatively reproduce the experimental results of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\). For a detailed discussion, we need to consider the assymmetric nature of the spectral shape of \(E_{g}\) mode that reflects the frequency dependence of the phonon self-energy.
Figure 5. (Color online) \(R = \{I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) \}/\mathrm{e}^{\hbar\Omega_{0}/k_{\text{B}}T}\) and \(I_{\text{S}}(\Omega)\) as a function of \(\Delta\Omega =\Omega -\Omega_{0}\) calculated with \(\gamma = 3\) cm−1, \(\Delta E = 6.0\) (triangles), 7.5 (circles), and 9.0 (squares) cm−1 and \(T = 170\) K. The solid line indicates the relation \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega) = e^{\hbar\Omega/k_{\text{B}}T}\).
As seen in Fig. 4, the ratio \(\Delta E_{\text{p}}/\mathit{HWHM}\) becomes 0.6–0.7 in the measured temperature range, implying that \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) shows a plateau only around the center frequency of the Raman peak. If we assume that the behavior of \(\Delta E_{\text{p}}/\mathit{HWHM}\) is almost the same for other phonon modes as that of \(E_{g}\) mode, a peak of second-order scattering by two phonon modes with spectral broadening is also expected to show a plateau around its center frequency in Ω dependence of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\). However, the continuous distribution of second-order peaks with spectral overlap makes it difficult to detect the plateau, which may be the reason why broad components almost satisfy the relation of Boltzman factor in the range of \(10<\Omega<600\) cm−1 as seen in Figs. 2(a) and 2(b).
Finally, the observed spectrum \(I_{\text{S}}(\Omega)\) [or \(I_{\text{AS}}(\Omega)\)], in principle, has all information about the relaxation process related to the interacting phonon modes. However, it is difficult to show if the thermal factor of observed spectrum varies with Ω as \(n(\Omega)+1\) [or \(n(\Omega)\)] from the spectral shape of \(I_{\text{S}}(\Omega)\) [or \(I_{\text{AS}}(\Omega)\)] because the spectral shape of the response function \(\text{Im} R(\Omega)\) is unknown and is not generally expressed by a simple function such as Lorentzian. However, the ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) depends only on the thermal factors, not on the spectral shape of \(\text{Im} R(\Omega)\). Thus, the analysis of \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) is effective for investigating the relaxation process related to the phonon Raman modes.
5. SummaryWe investigated the Stokes to anti-Stokes intensity ratio \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\) for the phonon Raman spectra of TiO2. The ratio for broad components of the second-order scattering satisfies the Boltzman factor. In contrast, the ratio for the first-order scattering near peak frequency \(\Omega_{E_{g}}\) of \(E_{g}\) mode shows a plateau with \(I_{\text{S}}(\Omega)/I_{\text{AS}}(\Omega)\simeq e^{\hbar\Omega_{E_{g}}/k_{\text{B}}T}\). The frequency range of plateau broadens with temperature along with spectral width. It is suggested that the irreversible relaxation that broadens the energy of the phonon plays an important role in the Raman scattering process.
Acknowledgment
The author would like to thank Professor S. Kinoshita for many fruitful discussions.
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