Stokes to anti-Stokes intensity ratio in Raman spectra of the soft mode in KH 2 PO 4 near the phase transition temperature J. Watanabe * , R. Yoshida, S. Iwane, S. Kinoshita Graduate School of Frontier Biosciences, Osaka University, Suita 565-0871, Japan Available online 30 October 2007 Abstract We investigate the Stokes to anti-Stokes intensity ratio for the Raman spectrum of a central peak in a potassium dihydrogen phos- phate (KDP) crystal. The ratio deviates from the Boltzmann factor as the temperature approaches the phase transition temperature. The response functions evaluated by three ways assuming the fluctuation–dissipation theorem do not agree in a frequency region below 30 cm 1 . This means that the usual relation based on the fluctuation–dissipation theorem are not directly applicable to the spectral analysis of the soft mode. Further, we compare the result of KDP with that of the cooperative rotational dynamics in liquid carbon disulfide. Ó 2007 Elsevier B.V. All rights reserved. PACS: 77.84.Fa; 78.30.Hv; 63.20.e Keywords: Raman scattering; Optical spectroscopy; Raman spectroscopy; Fluctuations 1. Introduction The Stokes (I S ) to anti-Stokes (I AS ) intensity ratio for light scattering at a frequency shift x is known to be given by the Boltzmann factor I S =I AS ¼ e  hx=k B T . This relation is a result of the time-reversal symmetry of the scattering pro- cess under a canonical distribution of the system [1], and it is often used to accurately determine the sample temper- ature T. Recently, the deviation of the intensity ratio from the Boltzmann factor has been reported for the spectra of central peaks in liquids and amorphous solids. In liquid carbon disulfide [2], I S =I AS for cooperative rotational dynamics deviates from the Boltzmann factor and becomes unity in a frequency region below several wavenumbers, while the above relation holds for low-frequency phonon modes. It has also been reported for the quasi-elastic light scattering in amorphous solids and supercooled liquids such as As 2 O 3 and ZnCl 2 [3]. It has been discussed that the spectrum can be divided into a symmetric quasi-elastic line and Boson peak that satisfies the Boltzmann factor. Despite the importance of understanding these low- frequency dynamics, the standard formulation of light scattering based on the quantum-mechanical fluctuation– dissipation theorem [4,5] cannot be applied to the spectral analysis of these central peaks, since these experimental data show a symmetric spectrum with I S =I AS ¼ 1, which will only be satisfied in the high-temperature or the low-fre- quency limit. In general, the differential cross section of spontaneous light scattering under non-resonant excitation is given by [5,6] d 2 r dXdx 2 / x 1 x 3 2 I ðx 2  x 1 Þ; ð1Þ with I ðx 2  x 1 Þ¼ Z 1 1 dtha y aðtÞie iðx 2 x 1 Þt ; ð2Þ where x 1 and x 2 are the angular frequencies of the excita- tion and scattered photons, respectively; X is the solid 0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.08.078 * Corresponding author. Tel./fax: +81 6 6879 4601. E-mail address: junw@fbs.osaka-u.ac.jp (J. Watanabe). www.elsevier.com/locate/jnoncrysol Available online at www.sciencedirect.com Journal of Non-Crystalline Solids 354 (2008) 112–116