**3. Susceptibility** *χQ***(***ω***,** *<sup>T</sup>* **) for soft mode**

### **3.1. Intensity and line shape of Raman spectra**

Temperature and frequency dependence of Raman spectrum is represented by the imaginary part of the susceptibility *χQ*(*ω*, *T* ), which is the response of a phonon *Q* to its conjugate force *f <sup>Q</sup>*.

$$I\_{\mathbb{S}}\left(o\nu\_{\prime}T\right)\propto\left[n\left(o\nu\_{\prime}T\right)+1\right]\text{Im}\,\chi\_{\mathbb{Q}}\left(o\nu\_{\prime}T\right)\tag{4}$$

or

$$\operatorname{Im} \mathbb{X}\_Q \left( \alpha \nu, T \right) = \frac{I\_S \left( \alpha \nu, T \right)}{n \left( \alpha \nu, T \right) + 1} = \frac{I\_{AS} \left( \alpha \nu, T \right)}{n \left( \alpha \nu, T \right)} \tag{5}$$

where *IS* (*ω*, *T* ) and *I AS* (*ω*, *T* ) are intensity of Stokes and anti-Stokes side spectrum, respec‐ tively, and *n* (*ω*, *T* )= exp(ℏ*ω* / *kBT* )−1 <sup>−</sup><sup>1</sup> is the Bose factor. It should be noted that the approx‐ imations, ℏ*ω* / *kBT* > >1 or ℏ*ω* / *kBT* < <1, cannot be used for the soft mode.

In most cases, spectra are analyzed using either the so-called Damped Harmonic Oscillator (DHO) model for displacive-type transitions

$$\chi\_{\mathbb{Q}}(\alpha) = \chi\_{\mathbb{Q}}(0)\frac{\alpha \frac{\alpha}{\alpha\_0^2 - \alpha^2 - i\alpha\gamma}}{\alpha\_0^2 - \alpha^2 - i\alpha\gamma} \tag{6}$$

or the Debye model (DB) for the relaxational behavior of order–disorder-type transitions,

$$\chi\_{\mathbb{C}}(\alpha) = \chi\_{\mathbb{C}}(0)\frac{\chi'}{\chi'-i\alpha} \tag{7}$$

where in ferroelectrics *χQ*(0)≈*C* / (*T* −*Tc*) is proportional to the Curie constant *C*.

It seems rather strange, however, why these two formulas are exclusively used and what the meaning of the damping constant *γ* and *γ* ′ is. Since soft modes are often observed without any

1

apparent peaks, it is important to recognize the difference between the various forms of the susceptibilities. Thus, we discuss the more general formula of *χ* (*ω*).

### **3.2. Generalized susceptibility and the stability limit**

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6 Ferroelectric Materials – Synthesis and Characterization

**3. Susceptibility** *χQ***(***ω***,** *<sup>T</sup>* **) for soft mode**

**3.1. Intensity and line shape of Raman spectra**

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Temperature and frequency dependence of Raman spectrum is represented by the imaginary part of the susceptibility *χQ*(*ω*, *T* ), which is the response of a phonon *Q* to its conjugate force

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where *IS* (*ω*, *T* ) and *I AS* (*ω*, *T* ) are intensity of Stokes and anti-Stokes side spectrum, respec‐ tively, and *n* (*ω*, *T* )= exp(ℏ*ω* / *kBT* )−1 <sup>−</sup><sup>1</sup> is the Bose factor. It should be noted that the approx‐

In most cases, spectra are analyzed using either the so-called Damped Harmonic Oscillator

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or the Debye model (DB) for the relaxational behavior of order–disorder-type transitions,

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It seems rather strange, however, why these two formulas are exclusively used and what the

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Phase transitions occur when the system becomes unstable. In other words, as *T* →*Tc*, at least one of the poles of the *χ*(*ω*) approaches the origin in the complex **ω** plane. Typical traces are given in Fig. 2 for DHO, DB, and a more general case.

**Figure 2.** Motion of the pole of the susceptibility in the complex **ω**-plane as stability limit is approached. (a) Damped harmonic oscillator, (b) general case, (c) relaxation type (Debye).

Fig.2 Motion of the pole of the susceptibility in the complex ω-plane as stability limit

is approached. (a) Damped harmonic oscillator, (b) General case, (c) Relaxation type (Debye). As Van Vleck and Weisskopf showed in 1945 [8] for the case of microwave spectroscopy of gas, DHO and Debye types are qualitatively different in nature and DB cannot be obtained as a limit of vanishing characteristic frequency *ω*0. It is because the *<sup>γ</sup>* ′ in DB (Eq. (7)) means the inverse relaxation time *τ* of the order parameter, which is a stochastic variable. In contrast, the *γ* in DHO (Eq. (6)) is a dynamical variable representing the damping proportional to the velocity of the order parameter. Nevertheless, in the field of Raman spectra, especially for an overdamped soft mode associated with structural phase transitions, it is often misleadingly stated that DHO smoothly reduced to DB by decreasing the ratio *ω*<sup>0</sup> <sup>2</sup> / *γ* →0.

We note that the qualitative difference between DHO and DB originates from the fact that in DB the system is described for a finite time interval by neglecting the instantaneous change of the system. This "coarseness in the time domain" means that one cannot apply the Debye formula to the high frequency part of spectra. Therefore, the criteria, "high frequency tail of Raman spectra is proportional to *ω* <sup>−</sup><sup>3</sup> for DHO and *ω* <sup>−</sup><sup>1</sup> for DB" cannot be used to discriminate the displacive-type transitions from the order–disorder type.

Since a real system would not be represented by a single damping mechanism, we have proposed a more general form of susceptibility (GVWF) which includes two damping constants [9]. It is regarded as a generalization of the Van Vleck, Weisskopf, and Froehlich (VWF)-type susceptibility (*γ* =*γ* ′ ).

$$\chi(\alpha) = \chi(0)\frac{\alpha\_0^2 + \gamma\gamma' - i\gamma'\alpha}{\alpha\_0^2 + \gamma\gamma' - i\left(\gamma + \gamma'\right)\alpha - \alpha^2} \tag{8}$$

In general, this form can be described by the two poles Ω1,2 in the complex **ω** plane as follows:

$$\chi(\alpha) = \frac{\chi(0)}{2} \left\{ \frac{\Pi\_1}{\Omega\_1 - \alpha} + \frac{\Pi\_2}{\Omega\_2 - \alpha} \right\} \tag{9}$$

where

$$\begin{aligned} \Omega\_{1,2} &= -\frac{i\left(\boldsymbol{\gamma} + \boldsymbol{\gamma}'\right)}{2} \pm \sqrt{\boldsymbol{\alpha}\_{\boldsymbol{0}}^{2} - \left(\boldsymbol{\gamma} - \boldsymbol{\gamma}'\right)^{2}/4}, & \mathbf{a} \\ \Pi\_{1,2} &= -i\boldsymbol{\gamma}' \pm \left[\boldsymbol{\alpha}\_{\boldsymbol{0}}^{2} + \boldsymbol{\gamma}'\left(\boldsymbol{\gamma} - \boldsymbol{\gamma}'\right)/2\right] \frac{1}{\sqrt{\boldsymbol{\alpha}\_{\boldsymbol{0}}^{2} - \left(\boldsymbol{\gamma} - \boldsymbol{\gamma}'\right)^{2}/4}} & \mathbf{b} \end{aligned} \tag{10}$$

The behavior of the poles in the complex **ω** plane is shown in Fig. 3 for the case of the softening (*ω*0→0) with constant *<sup>γ</sup>* and *<sup>γ</sup>* ′ .


Behavior of various *χ*(*ω*) can be understood in terms of the three parameters (*ω*<sup>0</sup> , *γ*, *<sup>γ</sup>* ′ ) of GVWF. Figure 4 represents schematically the paths of the pole on approaching *Tc*. There are many paths leading to the divergence of *χ*(*ω*). The stability limit leading to the phase transition takes place when one of the poles moves toward the origin of complex **ω** plane (*γ* axis in Fig.4).

The pole of DHO moves in the DHO-plane (*ω*<sup>0</sup> , *γ*, *<sup>γ</sup>* ′ =0) as an ideal softening in displacivetype transitions. Ideal relaxation in order–disorder-type transitions (DB) corresponds to

2

**Figure 3.** Behavior of the poles of the GVWF in the complex *ω* plane for *ω*0→0 with fixed *<sup>γ</sup>* and *<sup>γ</sup>* ′ . (a) *γ* >> *γ*'=0, DHO; (b) *γ* > *γ*'≠0, GVWF; (c) *γ* = *γ*', VWF; (d) *γ* < *γ*', GVWF. Small white circles mean the po‐ sition of the pole for *ω*<sup>0</sup> =0 and the big black circles denote the DB relaxation mode.

*γ* ′ <sup>→</sup>0 regardless of *ω*<sup>0</sup> and *<sup>γ</sup>* . The pole of VWF moves in the (*ω*<sup>0</sup> , *<sup>γ</sup>* <sup>=</sup>*<sup>γ</sup>* ′ ) plane. The hatched plane indicates the position where the pole reaches the imaginary axis. Thus, GVWF is the most general approach to the instability, indicating that the application of DHO or DB is not trivial but based on the hidden assumptions. Fig.3 Behavior of the poles of the GVWF in the complex plane for 0 <sup>0</sup> with and . (a) ,0' DHO, (b) ,0' GVWF, (c) ,' VWF, (d) ' , GVWF. Small white circles mean the position of the pole for 0 <sup>0</sup> and the big black circles denote the DB relaxation mode.
