Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Retroreflector wikipedia , lookup
Ultraviolet–visible spectroscopy wikipedia , lookup
Birefringence wikipedia , lookup
Anti-reflective coating wikipedia , lookup
Vibrational analysis with scanning probe microscopy wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Atmospheric optics wikipedia , lookup
Raman spectroscopy wikipedia , lookup
Resonance Raman spectroscopy wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
JOURNAL OF RAMAN SPECTROSCOPY J. Raman Spectrosc. 2003; 34: 633–637 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jrs.1031 Brillouin scattering at high pressure: an overview Alain Polian∗ Physique des Milieux Condensés, CNRS–UMR 7602, Université Pierre et Marie Curie, B77, 4 Place Jussieu, 75252 Paris Cedex 05, France Received 21 March 2003; Accepted 8 April 2003 Brillouin scattering allows the determination of acoustic velocities and adiabatic elastic moduli in matter. These data are crucial in many areas of science, such as fundamental physics, geosciences and technology, especially when measured as a function of the density (pressure). In this paper, we present a review of the work performed on Brillouin scattering under high pressure in diamond anvil cells, emphasizing the most recent results. Copyright 2003 John Wiley & Sons, Ltd. KEYWORDS: Brillouin scattering—High pressure—High temperature—Elastic properties INTRODUCTION Sound waves propagate in matter, independently of its structure. Brillouin scattering consists in the inelastic scattering of light by the sound waves, or thermal excitation in a material, in particular acoustic phonons in a crystal. The energy of light is therefore modified, increased in the case of the annihilation of excitation, or decreased in the case of creation of excitation. The measurement of this energy difference gives information on the energy of the phonons and therefore on the interatomic potentials of the material. The theory of Brillouin scattering has been extensively developed in the literature,1 – 4 so only the main ideas will be covered here. In the interaction process between a photon and thermal excitation, the energy and momentum are conserved: h̄ D šh̄ωi ωs 1 qE D šEki Eks 2 where ω and k are the wavenumber and the wavevector of the photon, the subscripts i and s refer to the incident and scattered beams and and q are the wavenumber and the wavevector of the phonon, respectively, the plus sign corresponds to the creation of a phonon (Stokes) and the minus sign to annihilation (anti-Stokes). It should be noted that the equations of conservation hold for both Brillouin scattering and Raman scattering because the physical processes are identical. From the experimental point of view, the differences are large, coming from the difference in the energies involved in the processes: the wavenumbers of Ł Correspondence to: Alain Polian, Physique des Milieux Condensés, CNRS–UMR 7602, Université Pierre et Marie Curie, B77, 4 Place Jussieu, 75252 Paris Cedex 05, France. E-mail: [email protected] the optical phonons giving rise to Raman scattering are of the order of 10–1000 cm1 (1.2–120 meV), whereas the Brillouin wavenumbers are of the order of 0.1–6 cm1 . Because of this proximity with the elastically scattered Rayleigh line, apparatus of much higher resolution has to be used to discriminate the Brillouin scattered light compared with Raman scattering where a grating spectrometer is well suited. Such a high-resolution apparatus is the Fabry-Pérot (FP) interferometer. Recent advances in Fabry-Pérot spectroscopy have been reviewed5 and will be described here only briefly. A Fabry-Pérot system consists basically of two partially transmitting plane mirrors, separated by a distance d. A parallel beam of light is incident along the mirror axis. Owing to interferences between the two parallel mirrors, the light is transmitted only when nd D m 2 3 where n is the refractive index of the medium between the two mirrors, is the wavelength of the light and m is an integer. Therefore, the wavelength of the transmitted light is selected by varying the optical path nd of the light, by changing either the distance between the mirrors or the refractive index of the inter-mirror medium. Two parameters describe the quality of an FP system: the first is the contrast (C), defined as the ratio of the maximum intensity transmission to the minimum intensity transmission, and this quantifies the ability of the set-up to measure signals that are small in comparison with the Rayleigh scattering. The second is the finesse (F), defined as the ratio of the distance between two successive transmission peaks to the full width at half-maximum of a transmission peak, and provides the capability of measuring peaks close to the Rayleigh peak. The most important parameter is the contrast. For a good quality FP system, one typically obtains Copyright 2003 John Wiley & Sons, Ltd. 634 A. Polian contrasts of the order of a few hundred, which is far from sufficient to measure Brillouin scattering in most materials. The solution to increase considerably the contrast is the multipass FP system, which was introduced by Sandercock6 and yields measured contrasts above 1010 . One measurement is not sufficient to determine the wavenumber of a phonon: this quantity is only determined modulo one ‘free spectral range’ (FSR), i.e. a wavenumber equal to the distance between two successive transmission peaks of the FP. The solution was again given by Sandercock,7 with the tandem FP interferometer, where two interferometers with different spacings are scanned synchronously. BRILLOUIN SCATTERING Brillouin scattering provides information about the acoustic branches at a given point of the dispersion curves of the material under study. The measured wavenumber shift of the radiation is equal to that of the phonon under consideration [Eqn (1)], and its wavevector is deduced from Eqn (2), so the sound velocity may be calculated by V D /q 4 This relation may also be written as cm1 D 2nV sin c 2 5 where is the measured wavenumber shift (in cm1 , n is the refractive index at , the wavelength of the exciting light, c is the speed of light in vacuum and is the angle between ki and ks . The sound velocity is related to the elastic constants by X D V 2 6 where X is a combination of elastic constants depending on the direction of propagation of the phonon with respect to the crystallographic axis and is the density. For example, in a cubic crystal, a longitudinal phonon propagating along the [110] direction is related to X D C11 C C12 C 2C44 /2 and the two non-degenerate transverse phonons are related to X D C11 C12 /2 and X D C44 . Therefore, by correctly choosing the geometry of the experiment, it is possible to determine the complete set of elastic constants. Figure 1 shows scattering geometries commonly used in order to determine the elastic constants. In Fig. 1(a), the p modulus of the phonon wavevector is equal to nk 2, where k ³ ki ³ ks . In Fig. 1(b), jqj D 2nk. This is the geometry that produces the largest wavenumber shift. Figure 1(c) shows an interesting geometry for Brillouin scattering under pressure (provided that an ad hoc cell is available), because measurement of the refractive index is not required. Indeed, in the crystal the modulus of the wavevector of light is equal to nk. Owing to the incidence angle, jqj D 2nksin(r), where Copyright 2003 John Wiley & Sons, Ltd. ki ki kd kd q i q q i r kd (a) ki (b) (c) Figure 1. Most commonly used geometries in Brillouin scattering, (a) 90° scattering, jqj D 21/2 nk; (b) backscattering geometry, jqj D 2nk; (c) platelet geometry, jqj D 2nksin i D 2ksin r. r is the angle of the refracted photon inside the crystal with respect to the normal to the surface. Using the Descartes law of refraction, one obtains jqj D 2ksin(i), where i is the known angle of incidence. The knowledge of the relative orientation of ki and ks with respect to the crystallographic axes allows the determination of that of q, and the selections rules2 allow one to deduce the related combination of elastic constants. BRILLOUIN SCATTERING IN A DIAMOND ANVIL CELL With a ‘classical’ diamond anvil cell (DAC), i.e. with only optical access along the axis of the diamond, it is basically only possible to use the backscattering geometry (only one experiment has been performed in the platelet geometry with this type of cell).8 Unfortunately in the backscattering geometry, the selection rules favor the longitudinal mode, so most of the experiments performed in this geometry only give information on these phonons. Another problem which occurs is due to the diamond anvils which have also acoustic phonons, and the difference in volume between the anvils and the sample make the intensity of the Brillouin of the former much more intense than that of the latter. The problem of the longitudinal modes of diamonds may be avoided by using a tandem FP set-up. The ‘Swiss cheese’ DAC9 enables one to measure in the platelet geometry, and then gives information on the transverse modes. Nevertheless, the maximum pressure is achieved, on average, in a backscattering geometry.10 Once the spectrum has been recorded, the sound velocity can be computed, which means that [Eqn (5)] the refractive index at the wavelength of the laser and at the desired density should be measured or computed. The measurement of the pressure dependence of the refractive index of compounds is not easy. Various techniques have been proposed and applied in a few cases for H2 and D2 ,11,12 GaS,13 xenon,14 argon15 and helium.16 In case where n is not measured with pressure, a good approximation is given by the Clausius–Mossotti equation, provided that the equation of state is known. Once the sound velocity has been determined, Eqn (6) gives the effective elastic modulus corresponding to the J. Raman Spectrosc. 2003; 34: 633–637 Brillouin scattering at high pressure direction of propagation of the measured phonon. In order to determine the linear combination of elastic constants involved, one has to know the crystallographic orientations of the crystal inside the DAC. This may be done when the sample is solid under ambient conditions. When the sample is loaded as a fluid in the DAC, there is no control on the crystal orientations, and the determination of individual elastic constants is not straightforward using the classical methods. The first attempt17 was made in backscattering geometry. In this geometry, only the longitudinal acoustic (LA) modes are detected. The pressure dependence of the velocity was measured for unknown directions, without identifying the crystal orientation. The envelope for all experimental points corresponds to maximum LA velocities, v2 D [C11 C C12 C 4C44 ]/ along the h111i direction and to minimum velocities along h100i, v2 D C11 /. For the determination of all the elastic moduli, the isothermal bulk modulus B D C11 C 2C12 /3 is used. Two problems may arise by using this technique. First, it is assumed that the maximum and minimum values of the sound velocities are really reached, and second, the ratio of the specific heats, cp /cv , is supposed to be constant and equal to 1. The problem was solved for cubic crystals by Shimizu and Sasaki18,19 using the platelet geometry, a large aperture cell and a method first proposed by Every.20 Rotating a crystal with an unknown crystallographic orientation in the platelet geometry results in the determination of a set of curves of the angular dependence of the speed of sound. In an arbitrary direction, the sound velocity is related to the elastic moduli and the direction cosines n1 , n2 and n3 of the wavevector relative to the axes of the cubic crystal by v2j D fj C11 , C12 , C44 , n1 , n2 , n3 7 where is the density of the crystal. The wavevector direction may be expressed as a function of the Euler angles of the crystal, , ϕ, and hence: vj D gj C11 /, C12 /, C44 /, , ϕ, 8 The rotation angle ϕ is varied, and vj is then determined by a least-squares fit between the calculated gj ϕ and the experimental elastic moduli. Using this technique, the orientation of the crystal is also precisely obtained. A similar, but somewhat simpler, procedure is now commonly used when working on compounds which are crystalline under ambient conditions, i.e. in platelet geometry, and rotating the cell around the load axis. The orientation of the sample is known from the beginning, and in that case the full set of elastic moduli may be obtained for samples of lower symmetry. Nevertheless, it should be noted that the highest pressures are obtained in the backscattering geometry.29,34 Table 1 presents an extensive review of the results obtained at ambient temperature in a DAC. It is clearly Copyright 2003 John Wiley & Sons, Ltd. Table 1. Summary of the compounds studied by Brillouin scattering in a DAC Material Scattering geometry NaCl H2 , D2 a-SiO2 a-SiO2 a-SiO2 GaS Mg2 SiO4 Mg2 SiO4 H2 O, D2 O H2 O-VII a-B2 O3 a-GeO2 a-As2 S3 TiO2 N2 Ar Ar HF He CH4 CH4 CS2 Kr Kr BaTiO3 Platelet Platelet Backscattering Platelet Platelet Backscattering Backscattering Platelet C rotation Backscattering Platelet C rotation Backscattering Backscattering Backscattering Backscattering Backscattering Backscattering Combined Platelet Backscattering Backscattering Platelet C rotation Platelet Backscattering Platelet C rotation Backscattering and platelet Backscattering and platelet Backscattering Platelet C rotation Backscattering Platelet Platelet Backscattering Platelet Platelet C rotation Backscattering Platelet C rotation Platelet C rotation Platelet C rotation Backscattering Platelet C rotation Platelet C rotation Platelet C rotation Platelet C rotation Platelet C rotation Platelet C rotation Platelet C rotation SrTiO3 NH3 NH3 C6 H6 CH3 OH ZnO Xe Pentane/isopentane H2 S NaCl, KCl CO2 H2 N2 O 1-Propanol (MgFe)SiO4 HCl MgO MgO HBr Quartz SiO2 CaF2 Max. pressure (GPa) Ref. 4 20 17 14 57.8 18 4 16 74 8 16 20 4 14 20 35 70 12 20 32 5 7 33 8 11 and 8 21 22 23 24 25 26 27 28 29 30 31 33 33 32 33 17 34 8 35 36 37 38 39 40 41 36 and 9 42 20 3.5 30 8.4 4 31 12 7 17 6 24 4.5 8.3 32 8 20 50 7 20 9.3 43 44 45 46 47 10 48 49 50 51 52 53 54 55 56 57 58 59 60 61 J. Raman Spectrosc. 2003; 34: 633–637 635 A. Polian seen that in recent years, most of the experiments have been performed in the platelet geometry, so that the complete set of elastic moduli is obtained up to the highest pressure. Most of the results obtained to date at high pressure in a DAC were obtained at ambient temperature. Nevertheless, some preliminary results at high temperatures have been published on liquid H2 O62 and low-temperature data were obtained on H2 S.63 Figure 2 represents the Brillouin spectrum obtained at high pressure and high temperature in water. This study was undertaken up to 450 ° C and 9 GPa (F. Datchi and A. Polian, to be published). One application of the determination of the full elastic moduli set is the establishment of a thermodynamic pressure scale. The measurement of both the volume and the elastic constants as a function of the same secondary pressure scale 8 ×104 H2O P=0.9 GPa, T=450 °C Intensity/counts 6×104 R2 B1 B2 0 -1.5 -1.0 1.4 -0.5 0.0 0.5 1.5 1.6 1.0 1.7 1.5 ∆σ/cm-1 Figure 2. Brillouin scattering of liquid H2 O at 0.9 GPa and 450 ° C. The peaks labeled Ri are the Rayleigh lines and Bi are the Stokes and anti-Stokes lines of water (F. Datchi and A. Polian, to be published). 1400 1200 1000 800 600 MgO at 54.7 GPa, phonon direction [100] R R R VL(s) VL(s) D D 400 VT(s) 200 0 -1.5 -1.0 VT(s) -0.5 0.0 0.5 1.0 1.5 Frequency/cm-1 Figure 3. Brillouin scattering spectrum of MgO at 54.7 GPa, from Ref. 55. VL and VT are the longitudinal and transversal sound velocities, respectively, from MgO, R the Rayleigh lines and D the Brillouin lines from the diamond. Copyright 2003 John Wiley & Sons, Ltd. B A 1 PGPa D B 0 9 hold with A D 1904 GPa and B D 7.715, in good agreement with the previous determination with A D 1904 GPa and B D 7.665. CONCLUSION REFERENCES R3 R1 -1.7 -1.6 -1.5 -1.4 enables one to eliminate this scale and to establish a primary pressure scale. This has been done using x-ray diffraction and Brillouin scattering on MgO up to 55 GPa by Zha et al.55 (Fig. 3). The primary pressure scale determined in such a way, expressed in the ‘classical’ form We have briefly reviewed the results obtained to date with Brillouin scattering under high pressure in a diamond anvil cell. It should be noted that even with the limitation that it is necessary to work with transparent samples, it is a powerful technique, one of the few that can determine the properties of amorphous compounds or fluids. 4×104 2×104 Counts 636 1. Born M, Huang K. Dynamical Theory of Crystal Lattices. Oxford University Press: London, 1954. 2. Cummins HZ, Schoen PE. In Laser Handbook, Arecchi FT, SchulzDubois EO (eds). North-Holland: Amsterdam, 1972; 1029. 3. Hayes W, Loudon R. Scattering of Light by Crystals. Wiley: New York, 1978. 4. Dil JG. Rep. Prog. Phys. 1982; 45: 285. 5. Nizzoli F, Sandercock JR. In Dynamical Properties od Solids, Horton GK, Maradudin AA (eds) North-Holland: Amsterdam, 1990; 281. 6. Sandercock JR. In Light Scattering in Solids, Balkanski M (ed). Flammarion: Paris, 1971; 9. 7. Sandercock JR. In Light Scattering in Solids III, Cardona M, Güntherodt G (eds). Springer: Berlin, 1982; 173. 8. Lee SA, Pinnick DA, Lindsay SM, Hanson R. Phys. Rev. B 1986; 34: 2799. 9. Bassett WA, Wilburn DR, Hrubec JA, Brody EM. In High Pressure Science and Technology, vol. 2, Timmerhaus KD, Barber MS (eds). Plenum Press: New York, 1979; 75. 10. Polian A. In Frontiers of High Pressure Research, Hochheimer HD, Etters RD (eds). Plenum Press: New York, 1991; 181. 11. Shimizu H, Brody EM, Mao HK, Bell PM. Phys. Rev. Lett. 1981; 47: 128. 12. Shimizu H, Brody EM, Mao HK, Bell PM. In High Pressure Research in Geophysics, Akimoto S, Manghnani MH (eds). Center for Academic Publishing Japan: Tokyo, 1982; 135. 13. Polian A, Besson JM, Grimsditch M, Vogt H. Phys. Rev. B 1982; 25: 2767. 14. Itié JP, Le Toullec R. J. Phys. (Paris) Colloq. 1984 45: C8–53. 15. Grimsditch M, Le Toullec R, Polian A, Gauthier M. J. Appl. Phys. 1986; 60: 3479. 16. Le Toullec R, Loubeyre P, Pinceaux JP. Phys. Rev. B 1989; 40: 2368. 17. Grimsditch M, Loubeyre P, Polian A. Phys. Rev. B 1986; 33: 7192. 18. Shimizu H, Sasaki S. Jpn. J. Appl. Phys. 1992; 32(Suppl. 32-1): 355. 19. Shimizu H, Sasaki S. Science 1992; 257: 514. 20. Every AG. Phys. Rev. Lett. 1979; 42: 1065. J. Raman Spectrosc. 2003; 34: 633–637 Brillouin scattering at high pressure 21. Whitefield CH, Brody EM, Bassett WA. Rev. Sci. Instrum. 1976; 47: 942. 22. Shimizu H, Brody EM, Mao HK, Bell PM. Phys. Rev. Lett. 1981; 47: 128; Shimizu H, Brody EM, Mao HK, Bell PM. In High Pressure Research in Geophysics, Akimoto S, Manghnani MH (eds). Center for Academic Publishing Japan: Tokyo, 1982; 135; Shimizu H, Kumazawa T, Brody EM, Mao HK, Bell PM. Jpn. J. Appl. Phys. 1983; 22: 52. 23. Grimsditch M. Phys. Rev. Lett. 1984; 52: 1312; Grimsditch M. Phys. Rev. B 1986; 34: 4372. 24. Schroeder J, Dunn KJ, Bundy FP. In Proceedings of VII AIRAPT International Conference, Backman CM, Johanisson T, Tegner L (eds). University of Uppsala: Uppsala, 1989; 259; Schroeder J, Bilodeau TG, Zhao XS. High Press. Res. 1990; 4: 531. 25. Zha CS, Hemley RJ, Mao HK, Duffy TS, Meade C. Phys. Rev. B 1994; 50: 13 105. 26. Polian A, Besson JM, Grimsditch M, Vogt H. Appl. Phys. Lett. 1981; 53: 620; Polian A, Besson JM, Grimsditch M, Vogt H. Phys. Rev. B 1982; 25: 2767. 27. Shimizu H, Bassett WA, Brody EM. J. Appl. Phys. 1982; 53: 620. 28. Duffy TS, Zha CS, Downs RT, Mao HK, Hemley RJ. Nature (London) 1995; 378: 170. 29. Polian A, Grimsditch M. Phys. Rev. B 1984; 27: 6409; Polian A, Grimsditch M. Phys. Rev. Lett. 1984; 52: 1312; Polian A, Grimsditch M. Phys. Rev. B 1984; 34: 6362. 30. Shimizu H, Ohnishi M, Sasaki S. Phys. Rev. Lett. 1995; 74: 2820. 31. Grimsditch M, Bhadra R, Meng Y. Phys. Rev. B 1988; 38: 7836. 32. Polian A, Grimsditch M. J. Phys. Lett. (Paris) 45: L1131 1984. 33. Grimsditch M. Phys. Rev. B 1985; 32: 514. 34. Shimizu H, Tashiro H, Kume T, Sasaki S. Phys. Rev. Lett. 2001; 86: 4568. 35. Polian A, Grimsditch M. Europhys. Lett. 1986; 2: 849. 36. Hebert P, Polian A, Loubeyre P, LeToullec R. Phys. Rev. B 1987; 36: 9196. 37. Shimizu H, Nakashima N, Sasaki S. Phys. Rev. B 1995; 53: 111. 38. Shimizu H, Sasaki S, Ishidate T. J. Chem. Phys. 1987; 86: 7189. 39. Polian A, Grimsditch M, Besson JM, Grosshans WA. Phys. Rev B 1989; 39: 1332. 40. Shimizu H, Saitoh N, Sasaki S. Phys. Rev. B 1998; 57: 230. 41. Fischer M, Polian A. Phase Trans. 1987; 9: 205; Ishidate T, Sasaki S. Phys. Rev. Lett. 1989; 62: 67. 42. Fischer M, Polian A. In High Pressure Geosciences and Materials Synthesis, Vollstädt H (eds). Akademie Verlag: Berlin, 1988; Copyright 2003 John Wiley & Sons, Ltd. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 156; Fischer M, Bonello B, Polian A, Léger JM. In Perovskite: a Structure of great Interest in Geophysics and Materials Science. AGU Monographs No. 45, American Geophysical Union: Washington, 1989; 125; Ishidate T, Sasaki S, Inoue K. High Press. Res. 1988; 1: 53. Gauthier M, Pruzan Ph, Chervin JC, Polian A. Solid State Commun. 1988; 68: 149. Kume T, Daimon M, Sasaki S, Shimizu H. Phys. Rev. B 1998; 57: 15 347. Gauthier M, Polian A, Chervin JC, Pruzan Ph. High Press. Res. 1991; 7: 130. Lee SA, Anderson A, Lindsay SM, Hanson RC. High Press. Res. 1990; 3: 230. Ishidate T, Yamamoto K, Inoue K, Shibuya M. In Solid State Physics Under Pressure: Recent Advances with Diamond Anvil Devices, Minomura S (ed). KTK Scientific Publishers: Tokyo, 1985; 99. Oliver WF, Herbst CA, Lindsay SM, Wolf GH. Phys. Rev. Lett. 1991; 67: 2795. Shimizu H, Sasaki S. Science 1992; 257: 514. Campbell AJ, Heinz DL. Science 1992; 257: 66. Shimizu H, Kitagawa T, Sasaki S. Phys. Rev. B 1993; 47: 11 567. Zha CS, Duffy TS, Mao HK, Hemley RJ. Phys. Rev. B 1993; 48: 9246. Shimizu H, Satoh H, Sasaki S. J. Phys. Chem. 1994; 98: 670. Takagi Y, Ahart M, Yano T, Kojima S. J. Phys. Condens. Matter 1997; 9: 6995. Zha CS, Duffy TS, Downs RT, Mao HK, Hemley RJ. Earth Planet. Sci. Lett. 1998; 159: 25. Kume T, Tsuji T, Kamabuchi K, Sasaki S, Shimizu H. Ferroelectrics 1998; 219: 101. Sinogeikin SV, Bass JD. Phys. Rev. B 1999; 59: 14 141. Zha CS, Mao HK, Hemley RJ. Proc. Natl. Acad. Sci. USA 2000; 97: 13 494. Shimizu H, Kanazawa M, Kume T, Sasaki S. J. Chem. Phys. 1999; 111: 10 617. Gregoryanz E, Hemley RJ, Mao HK, Gillet P. Phys. Rev. Lett. 2000; 84: 2117. Speziale S, Duffy TS. Phys. Chem. Miner. 2002; 29: 465. Grimsditch M, Popova S, Polian A. J. Chem. Phys. 1996; 105: 8801. Murase S, Yanagisawa M, Sasaki S, Kume T, Shimizu H. J. Phys. Condens. Matter 2002; 14: 11 537. J. Raman Spectrosc. 2003; 34: 633–637 637