Calculation of the damping constant and the relaxation time of the LA mode in the incommensurate phase of quartz

References

• W. G. Caddy, Piezoelectricity (McGraw-Hill, New York, 1946).
   Google Scholar
• W. P. Mason, Crystal Physics of Interaction Processes (Academic Press, New York and London, 1966).
   Google Scholar
• J. P. Bachheimer, An anomaly in the β phase near the α-β transition of quartz, J. Phyique. Lett. 41 (23), 559 (1980). DOI: 10.1051/jphyslet:019800041023055900.
   Google Scholar
• G. van Tendeloo, J. van Landuyt, and S. Amelinckx, The α → β phase transition in quartz and AlPO4 as studied by electron microscopy and diffraction, Phys. Stat. Sol. (A) 33 (2), 723 (1976). DOI: 10.1002/pssa.2210330233.
   Google Scholar
• K. Gouhara, Y. H. Li, and N. Kato, Observation of satellite reflections in the intermediate phase of quartz, J. Phys. Soc. Jpn. 52 (11), 3697 (1983). DOI: 10.1143/JPSJ.52.3697.
   Google Scholar
• G. Dolino et al., Incommensurate phase of quartz: I. Elastic neutron scattering, J. Phys. France. 45 (2), 361 (1984). DOI: 10.1051/jphys:01984004502036100.
   Google Scholar
• J. Van Landuyt et al., Interpretation of Dauphiné-twin-domain configurations resulting from the α - β phase transition in quartz and aluminum phosphate, Phys. Rev. B 31 (5), 2986 (1985). DOI: 10.1103/PhysRevB.31.2986.
   Web of Science ®Google Scholar
• K. Abe et al., Effect of Uniaxial Stress on Central Peaks in the 1q and 3q Incommensurate Phase of Quartz, J. Phys. Soc. Jpn. 60 (2), 404 (1991). DOI: 10.1143/JPSJ.60.404.
   Google Scholar
• B. Berge et al., Incommensurate phase of quartz. - II : Brillouin scattering studies, J. Phys. France. 45 (4), 715 (1984). DOI: 10.1051/jphys:01984004504071500.
   Google Scholar
• K. Abe et al., Lineshape of brillouin spectra in the incommensurate phase of quartz (SiO 2 ): Analysis by fano's formula, Ferroelectrics 203 (1), 153 (1997). DOI: 10.1080/00150199708012842.
   Web of Science ®Google Scholar
• E. J. K. B. Banda et al., αβ transition in quartz: Classical versus critical behavior, Solid State Commun 17 (1), 11 (1975). DOI: 10.1016/0038-1098(75)90323-3.
   Google Scholar
• S. Ishibashi et al., Acoustic phonon dispersion of SiO2 (quartz) near the incommensurate phase transition, Physica B 219-220 (220), 593 (1996). DOI: 10.1016/0921-4526(95)00822-5.
   Google Scholar
• T. Ochiai et al., Piezoelectric properties in α-, incommensurate- and β-phases of quartz SiO 2, Jpn. J. Appl. Phys. 37 (Part 1, No. 5A), 2606 (1998). DOI: 10.1143/JJAP.37.2606.
   Web of Science ®Google Scholar
• H. Yurtseven, and S. Ateş, Resonant frequency shifts related to the elastic constants near the α-β transition in quartz, J. Mol. Struct. 1179, 421 (2019). DOI: 10.1016/j.molstruc.2018.11.044.
   Web of Science ®Google Scholar
• J. F. Scott, Evidence of coupling between one- and two-phonon excitations in quartz, Phys. Rev. Lett. 21 (13), 907 (1968). DOI: 10.1103/PhysRevLett.21.907.
   Web of Science ®Google Scholar
• S. M. Shapiro, and H. Z. Cummins, Critical opalescence in quartz, Phys. Rev. Lett. 21 (23), 1578 (1968). DOI: 10.1103/PhysRevLett.21.1578.
   Web of Science ®Google Scholar
• J. D. Axe, and G. Shirane, Study of the α − β quartz phase transformation by inelastic neutron scattering, Phys. Rev. B 1 (1), 342 (1970). DOI: 10.1103/PhysRevB.1.342.
   Web of Science ®Google Scholar
• U. Höchli, and J. F. Scott, Displacement parameter, soft-mode frequency, and fluctuations in quartz below its α − β phase transition, Phys. Rev. Lett. 26 (26), 1627 (1971). DOI: 10.1103/PhysRevLett.26.1627.
   Web of Science ®Google Scholar
• H. Yurtseven, and O. Tari, Calculation of the soft-mode frequency for the alpha – beta transition in quartz, Optik 127 (10), 4470 (2016). DOI: 10.1016/j.ijleo.2016.01.167.
   Web of Science ®Google Scholar
• G. Dolino, The α-inc-β transitions of quartz: A century of research on displacive phase transitions, Phase Trans 21 (1), 59 (1990). DOI: 10.1080/01411599008206882.
   Web of Science ®Google Scholar
• G. Dolino, and P. Bastie, Phase transitions of quartz, Key Eng. Mater. 101–102, 285 (1995). DOI: 10.4028/www.scientific.net/KEM.101-102.285.
   Google Scholar
• M. C. Lider, and H. Yurtseven, Correlations between the resonant frequency shifts and the thermodynamic quantities for the α-β transition in quartz, J. Mol. Struct. 1159, 1 (2018). DOI: 10.1016/j.molstruc.2018.01.053.
   Web of Science ®Google Scholar
• H. Gibhardt, G. Eckold, and H. Mitlacher, The influence of periodic mechanical stresses on the incommensurate phase of quartz, Physica B 234–236, 149 (1997). DOI: 10.1016/S0921-4526(96)00932-5.
   Web of Science ®Google Scholar
• B. Berge, M. Vallade, and G. Martinez, Raman scattering investigation of the α-β transition and of the incommensurate phase in quartz, J. Phys. C: Solid State Phys. 17 (6), L167 (1984). DOI: 10.1088/0022-3719/17/6/002.
   Google Scholar
• T. Shigenari et al., Light scattering from E-symmetry amplitudon in the incommensurate phase of quartz, Solid State Commun. 64 (3), 367 (1987). DOI: 10.1016/0038-1098(87)90984-7.
   Web of Science ®Google Scholar
• G. Lahajnar, R. Blinc, and S. Zumer, Proton spin-lattice relaxation by critical polarization fluctuations in KH2PO4, Phys. Cond. Matter 18 (4), 301 (1974). DOI: 10.1007/BF01464399.
   Web of Science ®Google Scholar
• G. Schaack, and V. Winterfeldt, Temperature behaviour of optical phonons near Tc in triglycine sulphate and triglycine selenate, Ferroelectrics 15 (1), 35 (1977). DOI: 10.1080/00150197708236718.
   Web of Science ®Google Scholar
• I. Laulicht, and N. Lucnar, Internal-mode line-broadening by proton jumps in KH2PO4, Chem. Phys. Lett. 47 (2), 237 (1977). DOI: 10.1016/0009-2614(77)80008-0.
   Web of Science ®Google Scholar
• I. Laulicht, On the drastic temperature broadening of hard mode Raman lines of ferroelectric KDP type crystals near Tc, J. Phys. Chem. Solids 39 (8), 901 (1978). DOI: 10.1016/0022-3697(78)90153-1.
   Web of Science ®Google Scholar
• A. Hushur et al., Ferroelectric phase transition of stoichiometric lithium tantalate studied by Raman, Brillouin, and neutron scattering, Phys. Rev. B 76 (6), 064104 (2007). DOI: 10.1103/PhysRevB.76.064104.
   Web of Science ®Google Scholar
• M. Mączka et al., Temperature- and pressure-induced phase transitions in the metal formate framework of [ND][Zn(DCOO)] and [NH][Zn(HCOO)], Inorg. Chem. 53 (18), 9615 (2014). DOI: 10.1021/ic501074x.
   PubMed Web of Science ®Google Scholar
• H. Mori, Collective motion of particles at finite temperatures, Prog. Theor. Phys. 28 (5), 763 (1962). DOI: 10.1143/PTP.28.763.
   Google Scholar
• C. Carabatos-Nedelac, and P. Becker, Order-disorder and structural phase transitions in solid-state materials by Raman scattering analysis, J. Raman Spectrosc. 28 (9), 663 (1997). DOI: 10.1002/(SICI)1097-4555(199709)28:9 < 663::AID-JRS157 > 3.0.CO;2-L.
   Web of Science ®Google Scholar