Premjeet Singh
email: premsingh@iitkgp.ac.in

Goutam Saha
email: gsaha@ece.iitkgp.ac.in

Md Sahidullah
email: md.sahidullah@inria.fr

Deep scattering network for speech emotion recognition

Keywords: Deep convolutional networks, Deep scattering transform, EmoDB, IEMOCAP, RAVDESS, Shift invariance, Speech emotion recognition

This paper introduces scattering transform for speech emotion recognition (SER). Scattering transform generates feature representations which remain stable to deformations and shifting in time and frequency without much loss of information. In speech, the emotion cues are spread across time and localised in frequency. The time and frequency invariance characteristic of scattering coefficients provides a representation robust against emotion irrelevant variations e.g., different speakers, language, gender etc. while preserving the variations caused by emotion cues. Hence, such a representation captures the emotion information more efficiently from speech. We perform experiments to compare scattering coefficients with standard melfrequency cepstral coefficients (MFCCs) over different databases. It is observed that frequency scattering performs better than time-domain scattering and MFCCs. We also investigate layerwise scattering coefficients to analyse the importance of time shift and deformation stable scalogram and modulation spectrum coefficients for SER. We observe that layer-wise coefficients taken independently also perform better than MFCCs.

I. INTRODUCTION

The introduction of deep learning has caused rapid development in different speech signal processing tasks. One such domain is speech emotion recognition (SER) which finds applications in health-care systems, sentiment analysis, and many other human-computer interaction applications [START_REF] Shah Fahad | A survey of speech emotion recognition in natural environment[END_REF]- [START_REF] Ayadi | Survey on speech emotion recognition: Features, classification schemes, and databases[END_REF]. Even after two decades of research, SER is still considered as a challenging task [START_REF] Akc | Speech emotion recognition: Emotional models, databases, features, preprocessing methods, supporting modalities, and classifiers[END_REF]. One of the major challenges is the interpersonal and intrapersonal variability in emotion expression, e.g., different speaking styles, cultural background, language, context, speaker's mood etc [START_REF] Shah Fahad | A survey of speech emotion recognition in natural environment[END_REF], [START_REF] Ayadi | Survey on speech emotion recognition: Features, classification schemes, and databases[END_REF]. This causes SER systems to face difficulty in generalization across different emotion speech samples, especially when the train and test data samples belong to different domains [START_REF] Parry | Analysis of deep learning architectures for cross-corpus speech emotion recognition[END_REF]. This downside motivates the search for a robust feature that can extract emotion-specific information irrespective of different variabilities.

The deep learning based SER approaches generally addresses the issue of variabilities by using deep convolutional neural networks (CNN), which are considered efficient feature extractors in terms of invariances learned against signal shifts [START_REF] Abdel-Hamid | Convolutional neural networks for speech recognition[END_REF], [START_REF] Mao | Learning salient features for speech emotion recognition using convolutional neural networks[END_REF]. However, the complexity and lack of explainability of deep CNNs come as an added disadvantage to automatic feature extraction ability [START_REF] Lipton | The mythos of model interpretability[END_REF].

To obtain an end-to-end framework, some SER studies use 1D CNNs to learn the relevant features directly from raw speech [START_REF] Trigeorgis | Adieu features? End-to-end speech emotion recognition using a deep convolutional recurrent network[END_REF]. Such studies observe that deep networks learn representations similar to handcrafted features [START_REF] Muckenhirn | Understanding and visualizing raw waveform-based CNNs[END_REF]. However, due to inherent instability and requirement of large data samples for training, suitability of deep neural networks becomes uncertain [START_REF] Rolnick | Deep learning is robust to massive label noise[END_REF].

Following similar lines, we propose the use of scattering transform for SER. Scattering transform was introduced in [START_REF] Mallat | Recursive interferometric representation[END_REF], [START_REF]Group invariant scattering[END_REF] as a deep convolutional network involving convolution with predefined kernel instead of automatic kernel learning. It also introduces stability to both temporal shift and deformations in the feature representation of signals. As emotion cues spread temporally in speech, such characteristics of scattering transform should provide a representation robust to both time-shifting of cues and emotion irrelevant temporal variations. In [START_REF] Andén | Deep scattering spectrum[END_REF], authors compute scattering coefficients across the log-frequency domain for frequency transposition invariance to obtain speaker-independent representation for speech recognition. Such characteristics may also help in learning speaker-independent emotion cues.

Scattering transform is used in both 1-D and 2-D data processing. Regarding 1-D signals, authors in [START_REF] Andén | Joint time-frequency scattering[END_REF] introduce joint time-frequency scattering that incorporates multiscale frequency energy distribution over time-invariant representation for various audio classification frameworks. Authors in [START_REF] Ghezaiel | Hybrid network for end-to-end text-independent speaker identification[END_REF] use two-layer scattering coefficients with CNN layers to obtain a stable descriptor of speaker information from raw speech. In [START_REF] Bruna | Audio texture synthesis with scattering moments[END_REF], authors compute different moments of scattering coefficients layers and found that such moments contain enough information for decent voice synthesis quality. Authors in [START_REF] Salamon | Feature learning with deep scattering for urban sound analysis[END_REF] used scattering transform for urban sound classification and report a marginal performance improvement but with reduced training data. They attribute their finding to better background information learning ability of temporal modulations. Along similar lines, [START_REF] Baugé | Representing environmental sounds using the separable scattering transform[END_REF] also applies to scatter coefficients for environmental sound classification. In [START_REF] Lostanlen | Time-frequency scattering accurately models auditory similarities between instrumental playing techniques[END_REF] a joint time-frequency based scattering representation is used to analyze the timbral similarity between different acoustic instruments. Our work is the first to apply scattering transform for SER. We conduct experiments on three different SER datasets using time and frequency scattering coefficients and achieve noticeable improvement over standard mel-frequency cepstral coefficients (MFCCs). The main contributions of this work are summarized below: 1) Optimization of scattering transform parameters and analysing its performance for SER, 2) Analysis of time-domain and frequency-domain scattering coefficients over different databases, and 3) Layer-wise analysis of scattering coefficients for SER.

II. SCATTERING TRANSFORM

For 1D signals, the idea behind scattering transform is to obtain a robust feature that remains invariant to the location of information cues (translation invariance) and time warping (diffeomorphism) of information across different instances of the signal.

A. Notion of Lipschitz continuity and information loss in MFCC

Let the signal x(t) be translated by c. Then we have, x c (t) = x(t -c). Taking its Fourier transform, we obtain, X c (ω) = e -jωc X(ω). Taking modulus on both sides, we obtain, |X c (ω)| = |X(ω)|. In short time Fourier transform (STFT), translation c is localized to the time frame duration T . If |c| T , the STFT representation is already invariant to such translation. However, a robust feature representation also requires stability to time deformations appearing in the signal. Let x(t) is warped (deformed) in time by a factor τ , i.e., x τ (t) = x(t -τ (t)). A transformation φ of x(t) is said to be stable to deformation τ if, ||φ(x) -φ(x τ )|| ≤ Csup t |τ (t)|||x||, i.e., the distance between the non-deformed (φ(x)) and deformed (φ(x τ )) feature in transformation space φ should be less than a factor C of maximum amplitude/size of deformation (sup t |τ (t)|). This is also known as Lipschitz continuity condition. Applying this to Fourier transform, we have,

|||X(ω)|-|X τ (ω)||| ≤ C ||x||, where, time deformation τ (t) = t. Now the Fourier transform of x τ (t) = x(t -t) is given as X τ (ω) = 1 1-X( ω 1
-). Hence, the frequency components at ω are shifted to 1 1-ω . This effect is more prominent at high frequencies. Hence, modulus of Fourier transform is not Lipschitz continuous. In MFCC, filters with non-linearly varying bandwidths are applied over the STFT. As output of every mel-filter is averaged to get a single coefficient, filter application can be viewed as averaging performed over frequency domain which counters the effect of instability. The mel-filters applied around high frequency regions have higher bandwidth which reduces the effect of deformation instability. However, this averaging in frequency also leads to information loss, especially at higher frequencies. Now, consider x t (u) to be the signal frame at time t. Then x t (u) = x(u)φ(u-t) where φ is a window of length T . The application of mel-filters over Fourier transform of x t (u), can be written in frequency as,

M x(t, λ) = 1 2π |x t (ω)| 2 |ψ λ (ω)| 2 dω (1)
where x t (ω) is the frequency response of x t (u) and ψ λ (ω) is the mel-filter with support λ. Since multiplication in frequency becomes convolution in time, Eq. 1 becomes,

M x(t, λ) = |x t * ψ λ (v)| 2 dv = x(u)φ(u -t)ψ λ (v -u)du 2 dv (2)
If T is much greater than the filter support λ in time, φ(t) is constant over λ. Hence we can write, φ(u

-t)ψ λ (v -u) ≈ φ(v -t)ψ λ (v -u), which gives, M x (t, λ) ≈ x(u)ψ λ (v -u)du 2 |φ(v -t)| 2 dv = |x * ψ λ | 2 * |φ| 2 (t) (3)
Eq. 3 shows that averaging in frequency domain is equivalent to time-domain averaging of |x * ψ λ | 2 with window duration T and becomes the basis for scattering transform. Hence, M x (t, λ) remains invariant to time shifts smaller than T . In MFCC, T generally equals 20ms which limits the capturing of long time scale information. If T is increased, loss of high frequency information takes place due to averaging [START_REF] Andén | Deep scattering spectrum[END_REF].

B. Layer-wise scattering coefficients

To prevent the information loss appearing in MFCC while, at the same time, capturing high frequency information, scattering transform computes second layer coefficients or the modulation spectrogram of signal. This includes computing the frequency response of the time-series of scalogram frequency bins. This is followed by averaging of both scalogram and modulation spectrum coefficients for stability against deformation. Scattering transform uses constant quality factor wavelet filters over complete frequency range to compute scalogram and modulation spectrogram. If Q defines the number of wavelets per octave, the center frequencies of wavelets becomes λ = 2 k Q , where k is an integer and the bandwidth is proportional to 1/Q. Every wavelet has a support of bandwidth λ/Q around center frequency λ. The same wavelet in time provide a spread of 2πQ λ . To make sure that this spread is less than T (because T is the time window over which time averaging will be computed for stable feature representation) wavelets are only defined for λ ≥ 2πQ T . For frequencies less than 2πQ T the frequency interval is covered with wavelets of equal bandwidth given by 2π T , hence covering complete frequency range. The mel-filter operation and filtering due to time window (T ) in MFCC is replaced with wavelet-based filters in scattering transform.

An issue with using wavelets is that they "commute with translation" [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. However, invariance to translation can generally be introduced by averaging. Following this, x(t) * ψ λ (t)dt should be translation invariant. But due to symmetric nature of wavelet, x(t) * ψ λ (t)dt = 0. Hence, a modulus nonlinear operator is applied to prevent this vanishing of integral, i.e. |x(t) * ψ λ (t)|dt. The reason behind choosing modulus non-linearity is because modulus operation is contractive and preserves the norm of the vector to which it is applied [START_REF] Andén | Joint time-frequency scattering[END_REF]. The modulus of Fourier transform coefficients are then lowpass filtered with filter φ to obtain feature representation robust against time-warping induced deformation instability. These features are then invariant to translations smaller than 2 J , where J is the scale of the low-pass wavelet filter (φ). Therefore, any mth order scattering coefficients are given as,

S Jm x(u) = U m x * φ 2 J (u) = U m x(v)φ 2 J (u -v)dv (4)
where,

U m x = U [λ m ]...U [λ 2 ]U [λ 1 ]x = |||x * ψ λ1 | * ψ λ2 |...| * ψ λm |
To obtain invariance to frequency transpositions, wavelet transform is first computed over the log-frequency axis (e.g., log(λ 1 )) of time-domain scattering coefficients followed by wavelet averaging. This is similar to the DCT operation in MFCC [START_REF] Andén | Deep scattering spectrum[END_REF]. Such averaging introduces invariance to shifts in frequency defined by the scale of the averaging wavelet. The frequency scattering coefficients are cascaded with time scattering coefficients to generate a feature representation of the signal.

III. EXPERIMENTAL DETAILS A. Dataset description

Three different speech corpora used in the experiments are described in Table I. The speech samples are downsampled at 16 kHz sampling frequency when required. EmoDB database is a German-language database whereas, both IEMOCAP and RAVDESS contain speech samples in the English language. For IEMOCAP, we use only four emotions (Happy, Angry, Sad, and Neutral) following other works with this dataset [START_REF] Parry | Analysis of deep learning architectures for cross-corpus speech emotion recognition[END_REF], [START_REF] Luo | Investigation on joint representation learning for robust feature extraction in speech emotion recognition[END_REF].

B. Experimental evaluation & Methodology

We first optimize the scattering transform parameters over the EmoDB database. The optimized parameter set is then used for SER evaluation over other databases. We perform only twolevel decomposition as higher level coefficients contain a very small fraction of signal energy [START_REF] Andén | Joint time-frequency scattering[END_REF]. We optimize the number of wavelets per octave (Q) and maximal wavelet length or averaging scale (T ) while using an empirically chosen fixed value of input signal duration (N ). For frequency scattering, we select the maximal wavelet length as 32. We choose this to obtain invariance over five octaves, found optimum for EmoDB database. Also, following [START_REF] Andén | Deep scattering spectrum[END_REF], for the first layer timescattering coefficients, we compute only the frequency-domain wavelet decomposition. The required optimum frequency averaging is computed and applied by the classifier itself [START_REF] Andén | Deep scattering spectrum[END_REF]. We use Morlet wavelet for both time and frequency scattering coefficients computation [START_REF] Andén | Deep scattering spectrum[END_REF]. For implementation, we use the ScatNet toolkit 1 .

For classification purposes, we use a radial basis function (RBF) kernel-based support vector machine (SVM), as the feature representations obtained are utterance-level representations. We use the leave-one-speaker-out (LOSO) crossvalidation strategy and report average performance obtained over different train-test pairs. For every experiment, we keep speech samples of one speaker in validation, one in test and remaining in training set. We optimize the hyperparameters of the SVM classifier on the validation set.

For comparison, we also compute MFCCs for speech utterances of duration N and evaluate the performance with the same LOSO cross-validation strategy. We choose the MFCC feature because of its characteristic of mimicking human sound perception and its versatility in various speech processing domains. For performance analysis, we use both accuracy and unweighted average recall (UAR) performance metrics. The UAR is defined as,

UAR = 1 K K i=1 A ii K j=1 A ij . ( 5
)
Here A is the contingency matrix, A ij is the number of samples in class i classified into class j and K is the number of classes. UAR is better suited as a performance metric in class imbalance situations as compared to standard accuracy [START_REF] Rosenberg | Classifying skewed data: Importance weighting to optimize average recall[END_REF]. The optimum value of signal duration (N ) is empirically at 51000 samples (or 3.18 seconds at 16 kHz sampling rate). We start with the lowest value of T to be 4096 samples, which correspond to 256 ms at 16KHz, a duration considered optimum to convey sufficient emotion cues [START_REF] Zhang | Speech emotion recognition using deep convolutional neural network and discriminant temporal pyramid matching[END_REF]. T is then increased by a factor of 2 until its value is below N . Interestingly, T = 4096 shows poor performance, whereas, T = 32768 shows better performance over different values of Q. In terms of parameter Q, we observed a general increase in performance for lower values of wavelets per octave. This indicates the importance of coarse-level frequency domain information for SER. This result is also comparable to the experiments done using constant-Q transform (CQT) for SER [START_REF] Singh | Non-linear frequency warping using constant-q transformation for speech emotion recognition[END_REF]. We select T = 16384 and Q = 5 as the optimum value of parameters for further experiments. Figure 3 shows the confusion matrices for scattering coefficients and MFCCs over EmoDB database. Scattering coefficients can better classify speech samples of different emotion classes in EmoDB as compared to MFCC.

IV. RESULTS

A. Parameter optimization over EmoDB

B. Evaluation on other datasets

Table II shows the performance obtained with the optimised scattering parameters (ScatNet) over different databases and its comparison with standard MFCC. We compute 13 MFCCs over window length of 20 ms, 10 ms hop and 512 frequency bins over same utterance duration N (3.18 seconds). Mean and standard deviation of MFCCs is computed over all the frames to obtain a vector representation for every utterance. The scattering coefficients are observed to outperform MFCCs over every database. We also compare performance with frequency domain scattering transform (F-ScatNet) with similar optimized parameters (Q, T and N ). The frequency transposition invariance obtained from frequency domain scattering introduces speaker invariance hence improving the performance. However, such improvement is not very prominent in the EmoDB database. One probable reason for such observation could be increased redundancy in frequency scattering coefficients because of small database size and low number of speakers.

C. Layer-wise analysis

To analyse emotion relevance of different layers of scattering transform, we compare SER performance by using the first and second layer coefficients separately in Table III. The first layer coefficients are averaged scalogram coefficients similar to MFCC. However, the averaging provided by filter φ introduces time shift invariance which results in better performance than MFCC. The second layer coefficients further perform better than first layer coefficients showing the relevance of modulation spectrum coefficients from SER perspective. We observe that the scattering transform coefficients, with optimized parameters, provide feature representations that are better suited for SER. The optimized wavelet averaging scale provides sufficient invariance and stability to irrelevant temporal variations to capture the emotion cues in the time domain. The low value of wavelets per octave generates coarse-level frequency domain information improving SER performance. Invariance to frequency transposition in frequency scattering further reduces the speaker-dependent variations enhancing the system performance. The performed layer-wise analysis shows the importance of time-domain averaging over the typical scalogram/mel-spectrogram coefficients. Results obtained with second layer scattering coefficients indicate the relevance of amplitude modulation of time series of different scalogram frequency bins.

We conclude that deep convolutional scattering coefficients capture more relevant emotion-related information than the standard mel-filterbank based feature. However, the performance with optimized parameter set on EmoDB varies noticeably across databases, indicating the lack of generalization. We also observe that the system faces difficulty in classifying emotions which have similar arousal characteristics (e.g., Happy & Angry). This hints towards the requirement of further analysis on improving the scattering coefficient based feature representation for SER. Our work is a preliminary study that introduces scattering transform for SER. In future, we will explore other back-end classifiers like multi-layer perceptron, gaussian mixture models etc. with scattering coefficients. To further evaluate the generalisation across samples from different domains, cross-corpus analysis for SER can also be evaluated.

Fig. 1 .

 1 Fig. 1. Graphical representation of scattering transform. Here, x is the input signal and φ is the low-pass filter used for stability to deformations. ψ 1 and ψ 2 are wavelet filter banks corresponding to 1st and 2nd layer of scattering transform. The figure describes only 2 layer decomposition of signal x.

  Angry, Sad and Neutral)

Figure 2

 2 Figure2shows the results obtained by varying Q and T parameters of scattering transform over EmoDB database.

Fig. 2 .

 2 Fig. 2. Comparison of various time-domain scattering transform parameters over EmoDB database. Q is number of wavelets per octaves and T is the averaging scale in samples. Improved performance over lower values of Q shows the importance of coarse level frequency-domain information for SER.

FFig. 3 .

 3 Fig. 3. Confusion Matrices of time-domain scattering coefficients and MFCC features over EmoDB database. Although scattering coefficients show higher classification rate for every emotion class as compared to MFCC, confusion across emotion classes of similar arousal characteristics, such as, Happy & Angry, Neutral & Boredom and Fear & Happy can be observed.

TABLE I SUMMARY

 I OF THE SPEECH CORPORA USED IN THE EXPERIMENTS.

	Databases	Speakers	Emotions
	Berlin Emotion Database
	(EmoDB) [22]

(F=FEMALE, M=MALE)

TABLE III SER

 III PERFORMANCE COMPARISON WITH FIRST AND SECOND LAYER TIME-DOMAIN SCATTERING COEFFICIENTS TAKEN INDEPENDENTLY. GIVEN VALUES ARE IN PERCENTAGES.

	Database	First layer	Second layer
		Accuracy	UAR	Accuracy	UAR
	EmoDB	62.18	57.94	68.64	61.22
	RAVDESS	38.06	36.72	48.61	47.20
	IEMOCAP	56.48	47.30	58.93	48.80
	V. DISCUSSION & CONCLUSION


David Chapron

Francesco Cordero

Marc D Fontana

Characterization of oxygen vacancies in SrTiO 3 by means of anelastic and Raman spectroscopy

Keywords: strontium titanate, oxygen vacancies, Raman spectroscopy, PCA

Oxygen vacancies in reduced SrTiO 3 are investigated by Raman spectroscopy and dynamic elastic modulus measurements. The anelastic spectrum is used to evaluate the amount and state of the O vacancies, which result to be almost completely paired and otherwies aggregated at a concentration of 0.5 mol%, with only 1/6 of them isolated at room temperature. Raman spectra recorded as a function of temperature in as-grown and reduced samples are examined regarding the different processes which can contribute to them. Chemometric analysis is used to highlight the specific influence of oxygen vacancies on the Raman spectra. It is shown that oxygen vacancies induce a relaxation of selection rules leading to the activation of first-order lines even in the cubic phase. Furthermore, regions around oxygen vacancies are demonstrated to be responsible for the asymmetric broadening of Raman line.

I. INTRODUCTION

Among the ABO 3 perovskite compounds SrTiO 3 (STO) shows peculiar interest for fundamental research and applications as well. On the one hand it is generally considered as a textbook example of quantum paraelectric, since it remains paraelectric down to 0 K 1 . On the other hand, due to the high dielectric permittivity, it is an attractive material for microwave or THz devices, and dynamic random access memories [START_REF] Gevorgidan | Ferroelectrics in microwave devices, circuits and systems. physics, modelling, fabrication and measurements[END_REF] . Point defects play an important role, in particular for electrical and electrochemical applications. Oxygen vacancies (V O ) govern the transport properties and can induce a superconducting state, which is still under study [START_REF] Rischau | A ferroelectric quantum phase transition inside the superconducting dome of sr1xcaxtio3[END_REF] . In addition, they may induce a state of giant permittivity, interesting for capacitors and energy storage devices [START_REF] Wang | Giant permittivity and low dielectric loss of srtio3 ceramics sintered in nitrogen atmosphere[END_REF] and participate in filamentary resistive switching [START_REF] Lee | Resistive switching phenomena: A review of statistical physics approaches[END_REF] . Furthermore, V O are considered to be responsible for electrical breakdown and damage in microelectronic devices.

Notwithstanding the importance of V O in determining the electrical properties of STO, and their use for this purpose, the literature is confusing regarding their mobility, and very little is actually known on their aggregation state. There is a large spread of data for the diffusivity of V O in STO, but they are usually considered as uniformly dispersed and diffusing with an activation energy of about 1 eV. The only techniques that can probe the hopping of isolated V O in perovskites are anelastic [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] and NMR [START_REF] Hackmann | Nmr investigation of defect properties in single crystal srtio3[END_REF] relaxation, and agree on a barrier of 0.6 eV both for isolated V O free in the lattice [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] and trapped by acceptor Fe [START_REF] Hackmann | Nmr investigation of defect properties in single crystal srtio3[END_REF] . This is a much faster mobility than generally a) Electronic mail: david.chapron@univ-lorraine.fr found, but the anelastic experiments also show that most V O aggregate in pairs and larger clusters already at concentrations as low as 0.5 mol% below 800 K; the activation energy for the motion of the aggregated V O is 1 eV [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] , explaining why this is the value generally found. While the more extended clusters of V O have been assumed to be chains for the purpose of fitting the anelastic spectra, their nature is actually unknown. Yet, the nature of the aggregations of the V O is certainly of great interest in connection with the relatively stable confinement in near surface regions [START_REF] Szot | Localized metallic conductivity and self-healing during thermal reduction of srtio 3[END_REF] or filamentary defects responsible for resistive switching [START_REF] Lee | Resistive switching phenomena: A review of statistical physics approaches[END_REF] .

Raman spectroscopy is known to be an efficient tool for studying vibrational modes and structural phase transitions. It can also provide information on point defects introduced in the lattice, by studying the change of the spectra [START_REF] Fontana | Microstructure and defects probed by raman spectroscopy in lithium niobate crystals and devices[END_REF] . Therefore, the question of the ability of Raman spectroscopy to probe oxygen vacancies is of primary interest.

Ten years ago, Tenne et al. have investigated the effect of reduction on the Raman spectra (RS) of STO [START_REF] Tenne | Raman study of oxygen reduced and re-oxidized strontium titanate[END_REF] . They found two effects: the appearance of first order peaks corresponding to optical phonons of the STO lattice and new lines attributed to local vibrational modes related to the oxygen vacancies. Nevertheless, this interpretation is debatable owing to the complexity of the RS and the various processes which can contribute to them. Thus the RS in cubic STO includes the possible coexistence of strong and broad second order scattering and first order peaks of different origins [START_REF] Perry | Temperature dependence of the raman spectrum of srtio3 and ktao3[END_REF][START_REF] Nilsen | Raman spectrum of strontium titanate[END_REF][START_REF] Petzelt | Dielectric, infrared, and raman response of undoped srtio 3 ceramics: Evidence of polar grain boundaries[END_REF] . The first-order RS is in principle forbidden in a centrosymmetric phase, so that only second-order scattering should be detected. In fact first-order peaks can be observed in the cubic phase due to the disappearance of inversion symmetry, which can be caused by a deviation from stoichiometry [START_REF] Jang | Ferroelectricity in strain-free srtio 3 thin films[END_REF] , the presence of impurities or dopant ions [START_REF] Ouillon | A raman scattering study of the phase transitions in srtio 3 and in the mixed system (sr 1x ca x )tio 3 at ambient pressure from t = 300 k down to 8 k[END_REF] , the grains [START_REF] Ostapchuk | Origin of soft-mode stiffening and reduced dielectric response in srtio 3 thin films[END_REF] or to the existence of strains, particularly in thin films [START_REF] Sirenko | Observation of the first-order raman scattering in srtio 3 thin films[END_REF] . Furthermore, some additional lines appear below the antiferrodistortive (AFD) transition at 105 K, and are due to unit cell doubling and attributed to phonons at the Brillouin zone boundary point R [START_REF] Petzelt | Dielectric, infrared, and raman response of undoped srtio 3 ceramics: Evidence of polar grain boundaries[END_REF] . All these peaks show their own dependence on temperature (T ). This renders difficult the analysis of the effect of the different processes.

Therefore, we have undertaken a new Raman investigation of V O in STO with the comparison of the spectra in as-grown and reduced single crystals, by means of principal components analysis (PCA), which is able to evidence the variability (even very small) between spectra (or data) due to any external parameter change. The reduced sample was firstly studied by means of anelastic measurements in order to reliably characterize the concentration and state of the oxygen vacancies.

The samples were cut from a wafer of STO 0.5 mm thick from M.T.I. Corporation. One sample was left in the virgin transparent state, while the other, was cut as a bar approximately 28 x 2.6 x 0.5 mm 3 , with the edges parallel to the (100) directions, for the anelastic measurement. In order to reduce it, it was inserted in an envelope of Pt foil open at both ends, sitting inside a water cooled quartz tube and heated by induction, as described in [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] . The sample was heated to approximately 1150 • C for 3 h in a flux of 0.9Ar + 0.1CO maintained at 920 mbar, followed by 1 h homogenization at 800 • C in the same flux. The O deficiency deduced from the mass loss was δ = 0.0043 in SrTiO 3-δ , but there is some uncertainty on an estimate only based on the mass loss; for example, it is impossible to assess if the sample picked up some material from the heated Pt foil. In order to have a better characterization of the concentration and state of the O vacancies, the complex dynamic Young's modulus E = E + iE of the reduced sample was measured between 90 and 650 K. The sample was suspended on thin thermocouple wires in correspondence to the nodal lines of the first flexural resonance (5.0 kHz) and electrostatically excited in high vacuum or <0.1 mbar He, as described in ref [START_REF] Cordero | An insert for anelastic spectroscopy measurements from 80 k to 1100 k[END_REF] . The Young's modulus is obtained from the resonance frequency, E ∝ f 2 , and the elastic energy loss, Q -1 = E /E , from the decay of the free oscillations, after switching off the excitation. The measurements were made during both heating and cooling.

II. RESULTS

Anelastic spectra

Figure 1 shows the real part E of the Young's modulus, E = 1/s 11 normalized to its maximum value, and the elastic energy loss Q -1 of the reduced sample measured at 5 and 27 kHz during the same cooling and heating runs. The third flexural mode at higher frequency could be measured reliably only in limited temperature inter-vals, but enough to see the temperature dependence of the intensity of peak P3. In the present context we are interested in the Q -1 (T ) curves, which contain peaks associated with various anelastic relaxation processes from hopping of V O and rearrangements of their complexes under the influence of the applied stress at frequency ω/2π. Consider the case of an isolated V O hopping with characteristic time τ = τ 0 exp(W/kT ). Each jump is accompanied by a reorientation of 90 • of the direction of the nearest neighbour Ti atoms, and hence of the associated local distortion, or elastic dipole tensor λ. The populations of the three possible types of sites for the V O with Ti along x, y and z, otherwise identical, are perturbed by the external stress and the resulting anelastic relaxation produces a Debye peak in the losses 6 :

Q -1 (T ) ∝ c(∆λ) 2 T ωτ 1 + (ωτ ) 2 , ( 1
)
where c is the (temperature dependent) volume concentration of the isolated V O , ∆λ the change of the elastic dipole after a jump, projected in the direction of the applied stress, and τ is the defect hopping time, apart from a factor close to unity. The maximum is at the temperature T m such that ωτ (T m ) ∼ = 1, so that the same process measured at higher frequency is shifted to higher temperature. Jumps of isolated V O , reorientations of pairs and aggregation/dissolution of complexes occur with different characteristic times and produce different peaks in the anelastic spectrum Q -1 (ω, T ).

The virgin sample has a flat Q -1 (T ) curve above 110 K, with none of the peaks found in the reduced samples, and labeled according to [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] (see curve 1 in Fig. 2). The peaks certainly associated with the hopping of V O are P1 and P3. P3 is due to the jumps of isolated V O , while P1 to the reorientation of pairs, but the formation of larger and more stable aggregates of V O , possibly longer chains, has been deduced from the dependence of the intensity of P1 on temperature and annealing [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] . Peak P5 is probably due to small polarons introduced by the V O . The sharp rise of dissipation and drop of modulus below 100 K (Fig. 1) is due to the AFD transition. Its shift to lower temperature with respect to the undoped case (105 K) allows the O deficiency to be assessed independently of the mass loss [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] , but a full comparison of the anelastic spectrum with those measured under various level of reduction from Ref. [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] (Fig. 2) is useful to discuss the state, in addition to the concentration of the V O in the presently reduced sample. The continuous lines in Fig. 2 and the present data (empty symbols) are close to the curve of δ = 0.0049. According to this comparison, the present δ is slightly greater than 0.0049, but certainly smaller than 0.0066, so that we may estimate the O deficiency as δ = 0.0050 ± 0.0005, not far from the estimate from the mass loss.

Figure 3 shows the calculated fractions of V O that are free, paired and aggregated (including paired), assuming δ = 0.005 and the approximations and parameters described in detail in Ref. [START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] for fitting the Q -1 (T ) curves. The approximation consists in dividing the crystal into non overlapping regions, for each of which a grand partition function can be constructed [START_REF] Cordero | Anelastic (dielectric) relaxation of point defects at any concentration, with blocking effects and formation of complexes[END_REF] including V O that are free or aggregated into pairs or chains of length up to four. The following parameters have been used, setting the energy of an isolated V O as E f = 0: E p /k B = -2140 K for paired V O and E c /k B = -3000 K for V O inside chains of 3 to 4 vacancies, the V O at the ends being considered as paired. From Fig. 3 it appears that at room temperature the majority of V O is aggregated and less than 1/6 of them are isolated; the fraction is further reduced at lower temperature. That the crossover region between aggregated and free V O is slightly above room temperature can be qualitatively checked by observing the temperature dependence of the intensity of P3, due to free V O . The intensities of the Debye relaxation peaks of defects with constant concentrations decrease as 1/T , but P3 increases instead with T (compare the curves measured at different frequencies in Fig. 1), due to the thermal dissolution from the aggregated to the free state, which increases the fraction of free V O 6 . The Raman data described below are aimed at providing: (i) spectral signatures of V O by comparing measurements on as-grown and reduced samples (ii) possibly information on the aggregation state of V O .

Raman spectra

Raman measurements were carried out by means of Aramis Horiba Jobin-Yvon spectrometer with a 532 nm laser at a power of 50 mW as the exciting line. The resolution was 1.2 cm -1 . Measurements were performed as function of temperature from room T down to 93 K, i.e. just below the phase transition temperature (105 K) by placing the sample on a Linkam THMS600 stage.

Figure 4 shows the Raman spectrum recorded for different temperatures in the untreated crystal. At room temperature the spectrum exhibits several broad bands labelled 1 to 9, well attributed to 2 nd order scattering [START_REF] Perry | Temperature dependence of the raman spectrum of srtio3 and ktao3[END_REF][START_REF] Nilsen | Raman spectrum of strontium titanate[END_REF][START_REF] Petzelt | Dielectric, infrared, and raman response of undoped srtio 3 ceramics: Evidence of polar grain boundaries[END_REF] , which is solely expected by Raman selection rules for the stoichiometric crystal in the cubic phase. At 93 K, i.e. below the AFD transition, two new lines denoted R and lying at 143 and 449 cm -1 occur in the tetragonal phase. They are due to a folding of the Brillouin zone (BZ) resulting from cell doubling, so that the phonon modes at the zone boundary point R appear at the BZ center in the tetragonal phase and are therefore detected by Raman spectroscopy. The phonons R are associated with antiphase tilting of the Ti-O 6 octahedra. It is to be noted that no additional lines, that could be assigned to impurities, are detected, proving the stoichiometric and undoped nature of the crystal under study. Therefore, the comparison with spectra on reduced sample, prepared from the same crystal, should be more reliable, since any difference between them can be related to the treatment only.

The spectra of the reduced sample, shown in figure 5, displays the second-order bands and R peaks located at the same positions as in the untreated sample, but new lines labelled A, B and C are detected around 175, 549, and 860 cm -1 . On cooling, these peaks become narrower and increase in intensity so that they more and more emerge from broad 2 nd order scattering.

No significant shift of the positions of the lines A, B and C is detected with change of T . It is to be noted that the peaks A and B are asymmetric. In addition to the appearance of new lines A, B and C, the spectra in the reduced sample differ from that in the untreated sample by the occurrence of a broad signal below 150 cm -1 . Its origin is not clear, since we did not perform measurements at lower frequency owing to the use of edge filter for suppressing the Rayleigh line.

In figure 6 are compared the spectra recorded at the same T (93 K) in both crystals, pointing out the new lines A, B and C in the reduced crystal. These bands can be straightforwardly related to the presence of oxygen vacancies in this crystal. This feature is corroborated below by PCA.

III. ANALYSIS 1. Generalities on Chemometric analysis and PCA

Generally, chemometric analysis concerns statistical and mathematical methods for extracting information from large numbers of data [START_REF] Wold | Chemometrics; what do we mean with it, and what do we want from it?[END_REF][START_REF] Wold | Chemometrics, present and future success[END_REF] . It is widely used in IR spectroscopy [START_REF] Muehlethaler | The application of chemometrics on infrared and raman spectra as a tool for the forensic analysis of paints[END_REF] and more and more in Raman studies [START_REF] Svensson | Reaction monitoring using raman spectroscopy and chemometrics[END_REF] . Several methods can be used to correlate linearly spectral changes with experimental parameters or to extract each component signal from a mixture. Here is shown how Principal Components Analysis (PCA) can be efficiently used to extract more information from spectra recorded in STO, which are very complicated and/or display very weak changes. The principle of a PCA is to reduce the number of spectral variables using an orthogonal transformation and turn them into uncorrelated variables called principal components (PC) ranked in order of largest possible variance [START_REF] Kim | An efficient model order selection for pca mixture model[END_REF] . Raman spectra have spectral variables which are reduced by PCA to scores and loadings. Scores are the weights of the spectra in the different dimensions (PC) and loadings are the representation of these dimensions according to the wavenumbers. The distribution of these scores in the space of dimensions is used to group or rank individuals (spectra) according to common principal components and thus to link them to a physical parameter. Signal processing, PCA and regression were performed according to a self-made program using the R language and the FactoMineR library 25 .

Results

As shown above the Raman spectra in STO are rather complex and involve lines with different origins. The PCA is used here to decorrelate in the changes of Raman spectra the pure effect of temperature from the influence of reduction which is here of our interest. The PCA was done on 12 Raman spectra grouping the two sets obtained on untreated and reduced samples, from 100 to 1200 cm -1 .

The analysis was done after applying a standard normal variate (SNV) pretreatment on the spectra. The transformation is applied to each spectrum individually by subtracting the spectrum mean and scaling with the spectrum standard deviation, in order to normalize the intensity and to correct the baseline of all spectra.

In Fig. 7 are plotted the scores of the two first components PC1 and PC2 derived from PCA analysis. The scores for each sample can be deduced by projection on axis for both components.

The total variance associated to PC1 and PC2 are equal to 96% so that all the variance in spectra can be nearly explained by both components. A clear separation is obtained between both samples along the axis PC2. The opposite signs reflect the opposite variations in intensities with respect of the mean spectrum between the untreated and reduced crystals. The influence of temperature on the spectra is clearly seen on the PC1 axis for both samples. Indeed, projecting the scores of both samples on the axis PC1 it is shown that the spectra are classified according to T change.

These results provide distinct effects of the two experimental parameters causing changes of the Raman intensities: the sample treatment (on PC2) and the temperature (on PC1). The discrepancy between spectra at the same T is enlarging for lowest T : 133 K and 93 K, indicating a strengthening of the reducing treatment effect on these spectra, consistently with the direct comparison of spectra (Figs. 4, 5, 6). The loadings of the corresponding components are presented in Fig. 8. They represent the value of each component at every point of the spectrum and therefore provide the influence of each component on the spectrum. We observe the variations of Raman intensities vs wavenumber due to the main components. As shown in PC1, the part of the spectrum between 300 and 600 cm -1 is very sensitive to T and the intensity increases with T for both samples. In PC2 the positive changes in several wavenumber ranges reflect the difference of Raman spectra between both samples and thus the reduction effect. It is to be underlined that some among these fea-tures correspond to the peak positions A, B and C of the Raman spectra on the treated sample. We receive confirmation that these peaks are linearly correlated, thus coming from the same physical process induced by the reduction treatment, likely related to oxygen vacancies. Furthermore, PC2 points out below 150 cm -1 a positive contribution which could result from V O as well as the peaks A, B and C.

IV. INTERPRETATION AND DISCUSSION

We have shown above that the Raman spectra on the reduced sample exhibit lines A, B and C, which are attributed to V O from the direct comparison between untreated and treated samples and PCA applied to all data. Nevertheless our spectra in the reduced crystal are significantly different from those reported by Tenne et al [START_REF] Tenne | Raman study of oxygen reduced and re-oxidized strontium titanate[END_REF] . In particular, in our studies we did not observe the lines (lying around 505, 630, 700 cm -1 ) attributed by Tenne et al. to local vibrationnal modes associated to V O . The origin of the discrepancy between previous data and our results is not very clear.

Nevertheless we note that Tenne et al. have performed measurements at much lower temperature (10K). In our studies the lines A, B and C are detected in cubic phase as in the tetragonal phase, in both sides of the AFD transition (110K). The additional lines detected by Tenne et al. do not correspond to cubic stoichiometric STO lattice and, to our opinion, could be due to a process different from this induced by V O only. Indeed, we can note that their spectra appear relatively close to those reported in Fe-doped STO samples. In these samples, the Raman spectra on the Fe-doped samples exhibit a strong peak around 690-730 cm -126-28 , which appears very similar to the highly intense peak attributed by Tenne to be due to V O . This peak occurs in nanoparticles [START_REF] Van Minh | Dopant effects on the structural, low temperature raman scattering and electrical transport properties in srti1xfexo3 nanoparticles synthesized by sol-gel method[END_REF][START_REF] Van Minh | Srti1xfexo3 nanoparticle: a study of structural, optical, impedance and magnetic properties[END_REF] and in single crystals [START_REF] Lenser | Spectroscopic study of the electric field induced valence change of fe-defect centers in srtio3[END_REF] as well. We can therefore assume that the additional lines reported by Tenne et al., is due to local vibrations including both Fe and V O .

In cubic undoped STO all first-order Raman lines are symmetry forbidden and optical phonons can be obtained by hyper-Raman measurements only. It is remarkable that the lines A, B and C of the reduced sample correspond to the positions if 1 st order lines TO 2 +LO 1 , TO 4 and LO 4 of the pure stoichiometric STO lattice, respectively, as detected by hyper-Raman [START_REF] Vogt | Hyper-raman tensors of the zone-center optical phonons in srtio 3 and ktao 3[END_REF][START_REF] Inoue | Hyper-raman scattering study of ferroelectric phase transitions in srtio3 and batio3[END_REF] .

The new peaks are therefore due to the activation by local symmetry lowering, due to oxygen vacancies, and are assigned to zone center optical phonons of STO lattice. The strong scattering at low frequency (below 150 cm -1 ) in the reduced sample likely arises from the same phenomenon by the activation of the soft mode TO 1 .

According to lattice dynamics calculations [START_REF] Stirling | Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models[END_REF][START_REF] Cowley | Lattice dynamics and phase transitions of strontium titanate[END_REF] the TO 2 phonon corresponds to the vibration of Sr ions against TiO 6 octahedra while TO 3 involves Ti-O stretching. Besides the TO 1 ferroelectric soft mode, they are the TO components of the three F 1u modes of the cubic phase. No indication of the TO 3 phonon expected at 264 cm -1 , corresponding to the silent and non-polar F 2u mode is seen in our spectra.

The detection of forbidden 1 st order lines was previously reported in nominally pure single crystals [START_REF] Jang | Ferroelectricity in strain-free srtio 3 thin films[END_REF] , thin films 17 , ceramics [START_REF] Ostapchuk | Origin of soft-mode stiffening and reduced dielectric response in srtio 3 thin films[END_REF] or nanocubes [START_REF] Banerjee | Observation of fano asymmetry in raman spectra of srtio3 and caxsr1xtio3 perovskite nanocubes[END_REF] . In fact, grains in ceramics, strain in thin films, or Sr deficiency, likewise O vacancies in our investigation, cause a breakdown of selection rules leading to the detection of 1st order peaks.

An unified explanation can be therefore deduced from these observations, whatever is the nature of the STO material (ceramic, thin film, and single crystal) and its intrinsic point defects related to non-stoichiometry.

The second peculiar feature of the Raman spectra of the reduced SrTiO 3 sample is the asymmetric line shape shown by the activated lines A and B. This asymmetry results from the breakdown of the translation symmetry of the crystal, and is thus an additional consequence of the presence of O vacancies. Indeed, in a perfect crystal only optical phonons with a wave vector q = 0 (or close to it) can participate to the Raman scattering. The corresponding line is symmetric and centered on the phonon frequency (if its damping or line width is not so large). By contrast, in defected material, O vacancies (in relatively large concentration) or any kind of point defect break the symmetry of the STO lattice, creating locally distorted regions. This induces a relaxation of Raman selection rules, leading to the detection of phonons with a wave vector q which can differ from zero. This gives an asymmetric broadening of the Raman line. Earlier Sirenko et al. [START_REF] Sirenko | Observation of the first-order raman scattering in srtio 3 thin films[END_REF] have pointed out this asymmetry for the line at 175 cm -1 and interpreted it as a Fano resonance. Fano line shapes have been described as the interaction of a phonon and a continuum, arising from rapid polarization fluctuations in polar nano-regions induced by impurities. We invoke here a different interpretation of the line asymmetry, which was at first proposed by Richter et al. [START_REF] Richter | The one phonon raman spectrum in microcrystalline silicon[END_REF] in the framework of the phonon confinement model, and which is now more and more used for nano-structures [START_REF] Adu | Raman scattering as a probe of phonon confinement and surface optical modes in semiconducting nanowires[END_REF][START_REF] Kumar | Qualitative evolution of asymmetric raman line-shape for nanostructures[END_REF] .

Accordingly, in our model we assume the reduced SrTiO 3 compound sample as a sum of spherical regions around each V O pair with a radius L (the mean size of regions). Assuming that all V O in the reduced sample are paired, there are 0.25% pairs, each occupying a region of average size L = 4.6a (a is the cell parameter).

We express the 1 st induced Raman line shape as a summation of all phonons into the BZ between q = 0 and q max = π/L, supposing an isotropic phonon dispersion:

S(ω) = qmax 0 C(q)dq ((ω -ω(q)) 2 + (γ/2) 2 , ( 2
)
where γ is the linewidth in the stoichiometric material, q is the wave vector in units 2π/a, ω(q) is the phonon dispersion and C(q), considering a Gaussian distribution, is the weight of each phonon: C(q) = exp (-q 2 π 2 /4).

(

The spectral response S(ω) or line shape of the Raman line therefore depends on the dispersion curve of the phonon ω(q), and varies with q max and then L. Results of the calculations performed for different values of L (as a multiple of the cell parameter a) are shown in figure 9. We observe that if L is very large (i.e. q max small, case of perfect stoichiometric crystal) the line is symmetric and centered on the frequency ω 0 which corresponds to the frequency at the Brillouin zone center (see the dispersion branch in the insert). By contrast, for small values of L (large concentration of defects), we obtain a broadening of lines, with an asymmetric shape in the high frequency side.

According to this analysis, it can be concluded that no new modes specifically associated with isolated or aggregated V O appear in the Raman spectra, but the distortions induced by these defects activate scattering from the main transverse optic modes, otherwise forbidden in the cubic lattice. In addition, the broadening of the new peaks is well accounted for by the confinement of phonons within the volumes defined by the defects, if almost all the V O are assumed to be paired, as indicated by the anelastic spectra.

V. CONCLUSION

The Raman spectra of SrTiO 3 can exhibit lines due to different processes: second-order scattering and first-order scattering induced by defects or by the AFD transition. The Principal Component Analysis of data recorded in both as-grown and reduced samples, highlights lines directly related to V O , whose concentration and aggregation state have been characterized by anelastic relaxation measurements. These bands in fact correspond to the activation of Brillouin zone center optical phonons which are in principle forbidden in stoichiometric material but are related to the breakdown of selection rules caused by the presence of V O . These peaks, denoted A (phonon TO 2 +LO 1 ) and B (phonon TO 4 ), are asymmetric in the high frequency side. This asymmetric line shape results from V O as well, which form distorted regions breaking the translational symmetry, so that phonons out from the Brillouin zone center take part to the scattering. The asymmetric broadening is relatively well reproduced by our model based upon clusters of size inversely proportional to the V O concentration, assumed to be completely paired below room temperature. Our study underlines the ability of Raman spectroscopy to probe oxygen deficiency, even in a relatively small content. Furthermore, the asymmetric line broadening can be used to estimate the size of distorted regions around V O .

FIG. 1 .

 1 FIG. 1. Elastic energy loss and normalized Young's modulus of the reduced sample. The arrows indicate cooling and heating.

FIG. 2 .

 2 FIG.2. Anelastic spectrum of the reduced sample (curve 4), plotted together with other spectra at various O deficiencies δ taken from Ref.[START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] .

FIG. 3 .

 3 FIG.3. Fractions of free, paired and total aggregated fraction of the VO, assuming δ = 0.005 and the approximations and parameters of Ref.[START_REF] Cordero | Hopping and clustering of oxygen vacancies in SrTio 3 by anelastic relaxation[END_REF] .

1 FIG. 4 .

 14 FIG. 4. Raman spectra of the untreated sample as function of T . The second order bands are labelled 1 to 9. R denote the lines linked to the AFD transition.

1 FIG. 5 .

 15 FIG. 5. Raman spectra of the reduced sample as function of T . The second order bands are labelled 1 to 9. R denote the lines linked to the AFD transition.

FIG. 6 .

 6 FIG.6. Comparison of Raman spectra between untreated and reduced samples at T = 93 K (tetragonal phase). The second order bands are labelled 1 to 9. R denote the lines linked to the AFD transition. The differences between spectra are pointed by A, B and C.

FIG. 8 .

 8 FIG.8. Principal components analysis : loadings of PC1 and PC2 (lower part) compared to the experimental Raman spectrum of the reduced sample at 93K (upper part).

FIG. 9 .

 9 FIG. 9. Raman line shape calculations derived from Eq. 2 with L = 20a (nearly stoichiometric crystal) and then L = 4.6a (crystal with defects) with γ = 8 cm -1 compared to Raman spectrum of the reduced sample at 93 K corrected by the Bose-Einstein thermal population factor. The dispersion branch of the TO2 phonon 31 used in the calculations is reported in the inset.