- Email: [email protected]
Vibrational relaxation of three-phonon bound states in crystal CO2
Journal of Molecular Structure, 266
(1992) 165-170 Elsevier Science Publishers B.V., Amsterdam
165
Vibrational relaxation of three-phonon bound states in crystal C@ R.Binia, P.R.Salvia, V.Schettinoa , H.-J.Jodlb and N.Orlicc aLaboratorio di Spettroscopia Molecolare, Dipartimento di Chimica, Universita’ di Firenze, via G.Capponi 9, 50121 Firenze (ITALY) and Laboratorio Europeo di Spettroscopie non Lineari, Dipartimento di Fisica, Universita’ di Firenze, Largo E.Fermi 2, 50135 Firenze (ITALY) bFB-Physik, Universitat Kaiserslautem, (GERMANY)
E.Schrodinger
Strasse, 6750 Kaiserslautem
cPedagoski Facultet, University of Rijeka, Omladinska 14, Rijeka (YUGOSLAVIA)
Abstract Vibrational lifetimes of three-phonon bound states (triphonons) observed in the vibrational region 3q;(ol+q) of crystal CO2 have been measured in the temperature range lo-200 K. The relaxation is interpreted in terms of depopulation of excited levels and dephasing by low-frequency phonons. A comparison with results for biphonons in the Fermi resonance region 2q;wl is made and similarities are discussed.
1. zNTRoDucIlON Phonon relaxation in molecular crystals has been a subject of current interest in solid state physics in the last few years [ 11. Lifetime and linewidth measurements have been performed by CARS and Raman techniques on Raman active phonons and their temperature dependence has given important information on the mechanisms contributing to vibrational decay [I]. High-resolution FTIR spectroscopy is an additional experimental technique for bandwidth measurements of infrared active phonons 121. Not much is known about the latter due to the experimental difficulties in preparing samples of convenient thickness. Also, the infrared lineshape is lorentzian only for transitions of small transition moment. Therefore the true band shape can be directly obtained only for weakly absorbing phonons. An interesting application of high-resolution FTIR is the study of multiphonon bound states in molecular crystals. Vibrational complexes such as biphonons and triphonons have been discussed in several papers [3-51. They occur as sharp states split off the multiphonon continuum whenever the molecular anharmonicity is sufficiently high with
0022-2860/92/$05.000 1992 Elsevier Science Publishers B.V. All rights reserved
166
respect to the phonon dispersion. A convenient source of high anharmonic coupling is Fermi resonance in simple triatomic molecules such as CO2, N20 and CS2. Very recently we reported on experimental evidence of triphonons in crystal CO2 in the infrared region 39;(01 +q) [6]. Here we wish to report on their relaxation properties as a function of temperature and to give a preliminary interpretation of the decay mechanisms responsible of the observed behaviour. Our results are compared with those relative to biphonon relaxation in the 2~2;01 Raman active region of crystal CO2 [A.
2. EixPmAL Single crystals of CO2 were grown from liquid in a specially designed infrared cell equipped with sapphire windows. The high- resolution spectra were taken on a Bruker FTIR spectrometer mod. IFS 120 with resolution less than .005 cm-l.
3. CRYSTAL TRIPHONONS In the harmonic approximation phonons do not interact with each other and accordingly move as independent particles in the crystal. Intermolecular interactions spread their energy over a band of width A . Intramolecular anharmonicity is a residual attractive interaction among harmonic phonons. If anharmonicity is large with respect to A, phonons may correlate their motion and, as a consequence, bound states with phonon excitations travelling on the same molecule and energy lower than the multiphonon continuum are formed. Fermi resonance in the wl;2q vibrational region of crystal CO2 is a well-known case of two-phonon bound states (biphonons) [6]. In the free molecule the unperturbed levels 01 and 2~2 are split apart as far as - 100 cm-1 by the anharmonic Fermi constant W=-kl22M2=
-52.8 cm-t [8] . The observed levels are denoted
conventionally as Q+
and O- . In the crystal the optical phonon wl(0) interacts with all members of the twophonon set q(k)+w2(-k). Fermi coupling is specified by W and by the position, width and shape of the q+w2 continuum. As a result, the observed spectral structure consists of two sharp peaks (biphonons), fI+ and O-, and of an intermediate broad continuum of dissociated two-phonon states. The Fermi coupling depends on the vibrational quantum numbers of the interacting levels. For the higher order 3w2;(wl+o2) resonance the anharmonic constant is larger than for 20991 and equal to w\/2. The molecular levels, o+ and w-, are distant - 150 cm-l. The increase of the anharmonic constant has important consequences on the crystal spectrum. Considering the crystal as a simple collection of N non interacting molecules the 3~7;(01+~2) resonance may be obtained either assuming that (i) the two excitations 3~2 and (wl +q) are on the same molecule or that (ii) 2~2 and wl sit on one molecule and the
167
third ~2 excitation on another. Therefore both anharmonic constants, W and w\/2, be considered for the crystal and four levels, w+, w-, fI++q and O-+q from the 3q;(wl+q) interaction. A straightforward anharmonic calculation
must
will arise assuming
(wl+q)=3q gives f&+w 2=3-W and &=302_42 W. Intermolecular interactions broaden the energy spectrum of the ~2 phonons into a dispersion band of width A. The tU+q states, consisting of a biphonon and a phonon (BP+P), are dispersed by A while the set of free three-phonon states q+q+q (P+P+P) have an energy band equal to 36. On the contrary, the w+ and w- levels remain single even in the presence of an intermolecular potential. They are called triphonons (TP). The spectral structure of the infrared spectrum of a CO2 single crystal in the 3q;(wl +q) region, shown in Fig. 1, is in full agreement with these expectations. In fact the following features are observed: a) a multiphonon continuum of free (P+P+P) excitations, all on different sites, extending from 1960 to 2034 cm-l (MP in Fig. 1); b) two narrower bands corresponding to localization of 2q and 01 excitations on the same site and of third ~2 on another (BP+P in Fig. 1) which have the same width as the dispersion of the q mode; c) two sharp bound states due to 3~2 and
(WI+%)
excitations on the same site (TP
in
Fig. 1) occurring at 2066 and 1914 cm-l.
MPxlO
*
Fig. 1. Infrared spectrum of the CO2 crystal in the 3q;(wl +q) line) .
region at 50 K (*, isotope
168 4. VIBRATIONAL RELAXATION OF TRIPHONONS
By means of high resolution FTIR we have studied the bandwidths 2I’ (FWHM) of the triphonons at 2066 and 1914 cm-l in the temperature range lo-200 K with a spectral resolution of 0.005 cm-l, well below the observed 2p. Being our CO2 sample a single crystal, we expect that TP’s exhibit a lorentzian band profile if inhomogeneous broadening is absent, given the small value of their transition moment. The lorentzian or gaussian character of the observed bands has been calculated by a fitting procedure with a non-linear least-squares program. Since the spectral resolution is much lower than the observed widths, deconvolution with instrumental bandwidth was not necessary. We have found that both TP’s have a perfectly lorentzian lineshape at all temperatures. This confirms that the co+ and w- bands must be assigned to a single phonon complex. In Fig. 2 and 3 the 2I’ values are plotted as a function of temperature between 10 and 200 K.
01 0
1
cl
O.Sj
I
I
50
100
I
150
I
200
_
Temperature (K) Fig. 2. Temperature dependence of the lower TP (w-) bandwidth in the range lo-200 K. Different symbols refer to different series of measurements.
The temperature dependence of bandwidth provides important clues on the vibrational decay of excited phonons. In the high-temperature limit (i.e kT > > hq, where OL is the frequency of a lattice phonon) a linear dependence of 2I’ on T indicates that third-order depopulation processes, either o-- > O’+q_ (down-conversion) or o+q--- > o’
169 (up-conversion), are responsible for decay. On the other hand, a quadratic behaviour means, in the same approximation, that also fourth-order depopulation and/or dephasing processes, involving two phonons, contribute to relaxation. Our results of Fig. 2 and 3 show for both TP’s a quadratic dependence. Therefore, in our case the preferred relaxation or mechanisms may be dissociation of the TP into component particles (TP-->P+P+P TP-- > BP+P) or dephasing, with assistance of two lattice phonons . In addition, the lower
Temperature (K) Fig. 3. Temperature dependence of the upper TP (w+) bandwidth in the range lo-200 K. Different symbols refer to different series of measurements.
TP has a smaller bandwidth than the upper at all temperatures. These results may be compared with those about the biphonon relaxation in the Raman active 2~2;wl region of crystal co2 [7]. Also in that case it was found that n- has a smaller bandwidth than W and relaxes with a quadratic dependence on T . This was discussed in terms of dephasing by lattice phonons [9]. It is reasonable to assume that also the lower TP (w-), showing the same T dependence, relaxes through a dephasing mechanism involving lattice phonons. As a further evidence of it, we have measured at 10 K the bandwidth of the lower isotopic TP’s of CO2 crystal occurring at 1874 cm-l and 1885 cm-l and found a substantial independence of these values on the isotopic species, as it should be for a relaxation through dephasing . The bandwidth of the upper TP (o+) has a quadratic dependence on T, in contrast with the s2+ results [7j. It has been already calculated that the W relaxation at zero pressure has a small quadratic contribution 191. In addition, for a CO2 crystal under pressure the relaxation is quadratic and this arises from depopulation and dephasing contributions by lattice and low-frequency phonons [lo]. Therefore, assuming that quadratic
170
contributions become more important not only with pressure but also increasing the vibrational quantum number, the different behaviour of w+ and W with T is plausible. 5. REFERENCES 1. SCalifano and VSchettino, Int.Rev.Phys.Chem. 7 (1988) 19. 2. R.Bini, P.Foggi, P.R.Salvi and V.Schettino,J.Phys.Chem. 7(1990) 6653. 3. FBogani, J.Phys.C 11 (1978) 1283. 4. FBogani and P.R.&X, J.Chem.Phys. 81 (1984) 4991. 5. V.M.Agranovich and O.A.Dubovski, in Optical Properties of Mixed Crystals, chp .6, edited by R.J.Elliott and I.P.Ipatova (Elsevier, Amsterdam, 1988). 6. R.Bini, P.R.Salvi, V.Schettino and H.-J. Jodl, Phys.Lett.A, in press. 7. PRanson, R.Ouillon and S.Califano, J&man Spectrosc. 17 (1986) 155. 8. I.Suzuki, J.Mol.Spectry 25 (1968) 479. 9. FBogani, G.Cardini and V.Schettino, J.Chem.Phys., in press. lO.M.Baggen and ALagendijk, Chem.Phys.Lett. 177 (1991) 361.
(1992) 165-170 Elsevier Science Publishers B.V., Amsterdam
165
Vibrational relaxation of three-phonon bound states in crystal C@ R.Binia, P.R.Salvia, V.Schettinoa , H.-J.Jodlb and N.Orlicc aLaboratorio di Spettroscopia Molecolare, Dipartimento di Chimica, Universita’ di Firenze, via G.Capponi 9, 50121 Firenze (ITALY) and Laboratorio Europeo di Spettroscopie non Lineari, Dipartimento di Fisica, Universita’ di Firenze, Largo E.Fermi 2, 50135 Firenze (ITALY) bFB-Physik, Universitat Kaiserslautem, (GERMANY)
E.Schrodinger
Strasse, 6750 Kaiserslautem
cPedagoski Facultet, University of Rijeka, Omladinska 14, Rijeka (YUGOSLAVIA)
Abstract Vibrational lifetimes of three-phonon bound states (triphonons) observed in the vibrational region 3q;(ol+q) of crystal CO2 have been measured in the temperature range lo-200 K. The relaxation is interpreted in terms of depopulation of excited levels and dephasing by low-frequency phonons. A comparison with results for biphonons in the Fermi resonance region 2q;wl is made and similarities are discussed.
1. zNTRoDucIlON Phonon relaxation in molecular crystals has been a subject of current interest in solid state physics in the last few years [ 11. Lifetime and linewidth measurements have been performed by CARS and Raman techniques on Raman active phonons and their temperature dependence has given important information on the mechanisms contributing to vibrational decay [I]. High-resolution FTIR spectroscopy is an additional experimental technique for bandwidth measurements of infrared active phonons 121. Not much is known about the latter due to the experimental difficulties in preparing samples of convenient thickness. Also, the infrared lineshape is lorentzian only for transitions of small transition moment. Therefore the true band shape can be directly obtained only for weakly absorbing phonons. An interesting application of high-resolution FTIR is the study of multiphonon bound states in molecular crystals. Vibrational complexes such as biphonons and triphonons have been discussed in several papers [3-51. They occur as sharp states split off the multiphonon continuum whenever the molecular anharmonicity is sufficiently high with
0022-2860/92/$05.000 1992 Elsevier Science Publishers B.V. All rights reserved
166
respect to the phonon dispersion. A convenient source of high anharmonic coupling is Fermi resonance in simple triatomic molecules such as CO2, N20 and CS2. Very recently we reported on experimental evidence of triphonons in crystal CO2 in the infrared region 39;(01 +q) [6]. Here we wish to report on their relaxation properties as a function of temperature and to give a preliminary interpretation of the decay mechanisms responsible of the observed behaviour. Our results are compared with those relative to biphonon relaxation in the 2~2;01 Raman active region of crystal CO2 [A.
2. EixPmAL Single crystals of CO2 were grown from liquid in a specially designed infrared cell equipped with sapphire windows. The high- resolution spectra were taken on a Bruker FTIR spectrometer mod. IFS 120 with resolution less than .005 cm-l.
3. CRYSTAL TRIPHONONS In the harmonic approximation phonons do not interact with each other and accordingly move as independent particles in the crystal. Intermolecular interactions spread their energy over a band of width A . Intramolecular anharmonicity is a residual attractive interaction among harmonic phonons. If anharmonicity is large with respect to A, phonons may correlate their motion and, as a consequence, bound states with phonon excitations travelling on the same molecule and energy lower than the multiphonon continuum are formed. Fermi resonance in the wl;2q vibrational region of crystal CO2 is a well-known case of two-phonon bound states (biphonons) [6]. In the free molecule the unperturbed levels 01 and 2~2 are split apart as far as - 100 cm-1 by the anharmonic Fermi constant W=-kl22M2=
-52.8 cm-t [8] . The observed levels are denoted
conventionally as Q+
and O- . In the crystal the optical phonon wl(0) interacts with all members of the twophonon set q(k)+w2(-k). Fermi coupling is specified by W and by the position, width and shape of the q+w2 continuum. As a result, the observed spectral structure consists of two sharp peaks (biphonons), fI+ and O-, and of an intermediate broad continuum of dissociated two-phonon states. The Fermi coupling depends on the vibrational quantum numbers of the interacting levels. For the higher order 3w2;(wl+o2) resonance the anharmonic constant is larger than for 20991 and equal to w\/2. The molecular levels, o+ and w-, are distant - 150 cm-l. The increase of the anharmonic constant has important consequences on the crystal spectrum. Considering the crystal as a simple collection of N non interacting molecules the 3~7;(01+~2) resonance may be obtained either assuming that (i) the two excitations 3~2 and (wl +q) are on the same molecule or that (ii) 2~2 and wl sit on one molecule and the
167
third ~2 excitation on another. Therefore both anharmonic constants, W and w\/2, be considered for the crystal and four levels, w+, w-, fI++q and O-+q from the 3q;(wl+q) interaction. A straightforward anharmonic calculation
must
will arise assuming
(wl+q)=3q gives f&+w 2=3-W and &=302_42 W. Intermolecular interactions broaden the energy spectrum of the ~2 phonons into a dispersion band of width A. The tU+q states, consisting of a biphonon and a phonon (BP+P), are dispersed by A while the set of free three-phonon states q+q+q (P+P+P) have an energy band equal to 36. On the contrary, the w+ and w- levels remain single even in the presence of an intermolecular potential. They are called triphonons (TP). The spectral structure of the infrared spectrum of a CO2 single crystal in the 3q;(wl +q) region, shown in Fig. 1, is in full agreement with these expectations. In fact the following features are observed: a) a multiphonon continuum of free (P+P+P) excitations, all on different sites, extending from 1960 to 2034 cm-l (MP in Fig. 1); b) two narrower bands corresponding to localization of 2q and 01 excitations on the same site and of third ~2 on another (BP+P in Fig. 1) which have the same width as the dispersion of the q mode; c) two sharp bound states due to 3~2 and
(WI+%)
excitations on the same site (TP
in
Fig. 1) occurring at 2066 and 1914 cm-l.
MPxlO
*
Fig. 1. Infrared spectrum of the CO2 crystal in the 3q;(wl +q) line) .
region at 50 K (*, isotope
168 4. VIBRATIONAL RELAXATION OF TRIPHONONS
By means of high resolution FTIR we have studied the bandwidths 2I’ (FWHM) of the triphonons at 2066 and 1914 cm-l in the temperature range lo-200 K with a spectral resolution of 0.005 cm-l, well below the observed 2p. Being our CO2 sample a single crystal, we expect that TP’s exhibit a lorentzian band profile if inhomogeneous broadening is absent, given the small value of their transition moment. The lorentzian or gaussian character of the observed bands has been calculated by a fitting procedure with a non-linear least-squares program. Since the spectral resolution is much lower than the observed widths, deconvolution with instrumental bandwidth was not necessary. We have found that both TP’s have a perfectly lorentzian lineshape at all temperatures. This confirms that the co+ and w- bands must be assigned to a single phonon complex. In Fig. 2 and 3 the 2I’ values are plotted as a function of temperature between 10 and 200 K.
01 0
1
cl
O.Sj
I
I
50
100
I
150
I
200
_
Temperature (K) Fig. 2. Temperature dependence of the lower TP (w-) bandwidth in the range lo-200 K. Different symbols refer to different series of measurements.
The temperature dependence of bandwidth provides important clues on the vibrational decay of excited phonons. In the high-temperature limit (i.e kT > > hq, where OL is the frequency of a lattice phonon) a linear dependence of 2I’ on T indicates that third-order depopulation processes, either o-- > O’+q_ (down-conversion) or o+q--- > o’
169 (up-conversion), are responsible for decay. On the other hand, a quadratic behaviour means, in the same approximation, that also fourth-order depopulation and/or dephasing processes, involving two phonons, contribute to relaxation. Our results of Fig. 2 and 3 show for both TP’s a quadratic dependence. Therefore, in our case the preferred relaxation or mechanisms may be dissociation of the TP into component particles (TP-->P+P+P TP-- > BP+P) or dephasing, with assistance of two lattice phonons . In addition, the lower
Temperature (K) Fig. 3. Temperature dependence of the upper TP (w+) bandwidth in the range lo-200 K. Different symbols refer to different series of measurements.
TP has a smaller bandwidth than the upper at all temperatures. These results may be compared with those about the biphonon relaxation in the Raman active 2~2;wl region of crystal co2 [7]. Also in that case it was found that n- has a smaller bandwidth than W and relaxes with a quadratic dependence on T . This was discussed in terms of dephasing by lattice phonons [9]. It is reasonable to assume that also the lower TP (w-), showing the same T dependence, relaxes through a dephasing mechanism involving lattice phonons. As a further evidence of it, we have measured at 10 K the bandwidth of the lower isotopic TP’s of CO2 crystal occurring at 1874 cm-l and 1885 cm-l and found a substantial independence of these values on the isotopic species, as it should be for a relaxation through dephasing . The bandwidth of the upper TP (o+) has a quadratic dependence on T, in contrast with the s2+ results [7j. It has been already calculated that the W relaxation at zero pressure has a small quadratic contribution 191. In addition, for a CO2 crystal under pressure the relaxation is quadratic and this arises from depopulation and dephasing contributions by lattice and low-frequency phonons [lo]. Therefore, assuming that quadratic
170
contributions become more important not only with pressure but also increasing the vibrational quantum number, the different behaviour of w+ and W with T is plausible. 5. REFERENCES 1. SCalifano and VSchettino, Int.Rev.Phys.Chem. 7 (1988) 19. 2. R.Bini, P.Foggi, P.R.Salvi and V.Schettino,J.Phys.Chem. 7(1990) 6653. 3. FBogani, J.Phys.C 11 (1978) 1283. 4. FBogani and P.R.&X, J.Chem.Phys. 81 (1984) 4991. 5. V.M.Agranovich and O.A.Dubovski, in Optical Properties of Mixed Crystals, chp .6, edited by R.J.Elliott and I.P.Ipatova (Elsevier, Amsterdam, 1988). 6. R.Bini, P.R.Salvi, V.Schettino and H.-J. Jodl, Phys.Lett.A, in press. 7. PRanson, R.Ouillon and S.Califano, J&man Spectrosc. 17 (1986) 155. 8. I.Suzuki, J.Mol.Spectry 25 (1968) 479. 9. FBogani, G.Cardini and V.Schettino, J.Chem.Phys., in press. lO.M.Baggen and ALagendijk, Chem.Phys.Lett. 177 (1991) 361.