J . Phys. Chem. 1990,94, 6653-6658
6653
FTIR Study of Vibrational Relaxation in KC104 Crystalt R. Bini, P. Foggi, P.R. Salvi, and V. Schettino* Laboratorio di Spettroscopia Molecolare. Dipartimento di Chimica, Universita' di Firenze. Via G.Capponi 9, 50121 Firenze, Italy (Received: January 18, 1990)
The infrared spectrum of KC10, single crystal has been studied in the region of weak infrared uI, u2, and v I + 2u2 vibrational excitons at low temperature. The u2 dispersion is found to be =10 c d . The infrared and Raman spectroscopy in the v I region, in close coincidence with the first overtone of v2, is discussed in terms of Fermi resonance in crystals. The dependence on temperature of the bandwidths of the u I and u I 2v2 modes has been measured in the 20-1 60 K temperature range. Our bandwidth data are interpreted on the basis of current theories on relaxation mechanisms. In particular, the main decay processes in the u, and uI + 2u2 case involve the cubic anharmonicity and are determined only by depopulation. The calculated decays compare well with experimental results. On the whole, our decay data are also in agreement with recently reported picosecond coherent anti-Stokes Raman spectroscopy (CARS) bandwidth measurements.
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Raman spectra. The study of their relaxation time as a function Introduction of temperature addresses the problem of dissipation of vibrational In recent years the vibrational relaxation in solids has been the object of experimentall-I2 and theoretical inve~tigation.'~-~* energy from different levels of the same vibron band for which not much is Relaxation dynamics is usually described in terms of depopulation processes, Le., energy flow from excited oscillators to a thermal The infrared spectrum of KC104 crystals has been discussed bath, and of dephasing processes, i.e., coherence loss among exby several a ~ t h o r s . ~ * - However, ~' the v 1 and 2v2 regions have citons.',2 Both mechanisms contribute to the total line width, and theoretically their relative importance is estimated as a function of crystal anharmonicity and density of states. ( I ) Laubereau, A,; Kaiser, W. Reo. Mod. Phys. 1978, 50, 607. Experimentally, the dynamics of a system excited in a given (2) Califano, S.; Schettino, V. Int. Reu. Phys. Chem. 1988, 7, 19. vibrational state can be followed by a variety of nonlinear tech(3) Geirnaert, M. L.; Gale, G . M.; Rytzanis, C. Phys. Reu. Lett. 1984,52, niques, including picosecond CARS,'" frequency-domain CARS? 815. and high-resolution Raman spectroscopy.lWl2 Frequency- and (4) Gale, G. M.; Guyot-Sionnest, P.; Zheng, W. Q.;Flytzanis, C. Phys. time-domain experiments give in principle the same information. Reo. Lett. 1985, 54, 823. For instance, the full width of a Lorentzian band, I?, and the decay (5) Dlott, D. D.; Schosser, C. L.; Chronister, E. L. Chem. Phys. Lett. 1982, constant of the temporal signal, r, are related as r = l / ( ~ c ~ ) . 90, 386. The experimental equivalence has been proved in several ~ a s e s . 2 ~ 9 ~ ~ (6) Duppen, K.; Hesp, B. M. M.; Wiersma, D. A. Chem. Phys. Letr. 1981, Resonant infrared absorption is another possible tool in the study 79, 399. of vibrational relaxation. Indeed, picosecond infrared pulses have (7) Ouillon, R.; Ranson, P. J . Chem. Phys. 1986, 85, 1295. been used to study population redistribution in the ground stateI9 (8) Velsko, S.; Trout, J.; Hochstrasser, R. M. J . Chem. Phys. 1983, 79, 21 14. or dephasing times in regime of nearly free induction decay.20 (9) Trout, J.; Velsko, S.; Bozio, R.; Decola, P.L.; Hochstrasser, R. M. J . However, the infrared technique has not been much used so far Chem. Phys. 1984,81,4146. to investigate vibrational relaxation in molecular solids. Recently, (IO) Ranson, P.; Ouillon, R.; Califano, S. Chem. Phys. 1984, 86, 115. high-resolution Fourier transform infrared ( F H R ) spectrometers (11) Ouillon, R.; Ranson, P.; Califano, S. Chem. Phys. 1984, 91, 119. have become available for research that make a systematic analysis (12) Becucci, M.; Castellucci, E. Chem. Phys. 1989, 135, 363. of infrared band shapes possible in conditions of instrumental (13) Maradudin, A. A.; Fein, A. L. Phys. Rev. 1962, 128, 2589. resolution well below the expected bandwidth. Here we wish to (14) Califano, S.; Schettino, V.; Neto, N. Lotrice dynamics of molecular report on bandwidth measurements of weakly active infrared crystals. Lecture Notes in Chemistry; Springer: Berlin, 1981; Vol. 26. modes in the molecular-ionic KCIO, crystal. (1 5) Procacci, P.; Righini, R.; Califano, S.Chem. Phys. 1987, 116, I7 I . In recent years measurements of vibrational lifetimes by pi(16) Jindal, V. K.; Righini, R.; Califano, S. Phys. Rev. B 1988, 38, 4259. cosecond CARS in molecular-ionic crystals have been carried out (17) Bogani, F.; Cardini, G . ;Schettino, V.; Tasselli, P. L. In Dynamics of in our l a b ~ r a t o r y . ~ ' -It~is~ of interest to compare infrared and moleculur crystals; Lascombe, J., Ed.; Elsevier: Amsterdam, 1987; p 99. Raman data on simple systems. In fact, IR experiments on (18) Bogani, F.; Cardini, G.; Schettino, V.; Tasselli, P. L. Phys. Reu. B, crystalline naphthalene and anthracene25 have been reported in press. showing that infrared line widths are in general much broader (19) Gottfried, N. H.; Seilmeier, A.; Kaiser, W. Chem. Phys. Lett. 1984, than found in Raman experiments. 1 1 1 , 326. An important point in infrared experiments on solids is the (20) Hartmann, H. J.; Bratengeier, K.; Laubereau, A. In Time-Resolued Vibrational Spectroscopy; Springer: Berlin, 1985. actual observation of Lorentzian band shapes. In general, the (21) Angeloni, L.; Righini, R.; Castellucci, E.; Foggi, P.; Califano, S. J . absorption coefficient does not show a Lorentzian profile around Phys. Chem. 1988, 92,983. the resonance frequency.26 The band asymmetry depends, among (22) Righini, R.; Angeloni, L.; Castellucci, E.; Foggi, P.; Califano, S . other factors, on the transition dipole moment. KCIO, is therefore Croat. Chem. Acta 1988, 61. 621. a particularly suitable crystal since two fundamental modes, v I (23) Righini, R.; Angeloni, L.; Foggi, P.; Castellucci, E.; Califano, S. and u2, are infrared inactive in the isolated ion and acquire intensity Chem. Phys. 1989, 131, 463. only through crystal effects. It may be expected that their crystal (24) Angeloni, L.; Righini, R. Chem. Phys. Lett. 1989, 154, 1 1 5. transition moment is quite small, and consequently the corre(25) Hill, J. R.; Chronister, E. L.; Chang, T. C.; Kim, H.; Postlewaite, J. sponding bands will have a Lorentzian shape. C.; Dlott, D. D. J . Chem. Phys. 1988, 88, 949. In addition, in KCIO, crystals more than one Davydov com(26) Loudon, R. The quanrum theory of light; Clarendon Press: Oxford, 1973. ponent of the internal modes can be observed in the infrared and 'This work was supported by the Italian Minister0 della Pubblica Istruzione and Consiglio Nazionale delle Ricerche. *To whom correspondence should be addressed.
0022-3654/90/2094-6653$02.50/0
(27) Ranson, P.; Ouillon, R.; Califano, S . J . Raman Spectrosc. 1986. 17, 155. (28) Cohn, H. J. Chem. Soc. 1952, 4282.
0 1990 American Chemical Society
Bini et ai.
6654 The Journal of Physical Chemistry, Vol. 94, No. 17, 1990 TABLE I: Correlation Diagram for CI0,- Vibrations factor group molecule 7, site C. 021
0.3
I,
468 . ' 462 WAVENUMBERS (cm-') Figure 1. Infrared spectrum of KCIO., single crystal ( a b face) in the u2 region at 20 K. 474
not been considered in great detail. In the present work the infrared spectra polarized along the three crystal axes have been obtained. From these experimental data the weak Fermi resonance between v1 and 2v2 is d i s c ~ s s e d .It~ is~ shown ~ ~ ~ that the Fermi interaction is greatly different in the various polarizations. The line width of the sharp peaks observed in the v1;2v2region has been measured in the 20-160 K temperature range. The broadening processes are discussed in terms of the energy level diagram of the KC104 and compared with previous results from coherent Raman experiments. It is found that IR and Raman line widths are comparable. Results are also discussed on the band shapes in the U , 2v2 region as a function of temperature.
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Experimental Section Large single crystals of KCI04 of good optical quality were grown from water and cut orthogonal to the principal optical directions, which were determined with the aid of a polarizing microscope. Room-temperature Raman spectra in various scattering geometries were coincident with those already reported," thus identifying the crystal axes. High-resolution infrared spectra were measured on a IFS 120 Bruker interferometer on thin samples (5100 p n ) cut on the ab and bc crystal faces. I n all circumstances, the spectra were obtained with an instrumental resolution at least 1 order of magnitude smaller than the bandwidth of the infrared bands under investigation, so that no band deconvolution was necessary. For high-resolution studies the MCT detector operating at liquid nitrogen has been used. Low-temperature measurements were taken with a closed circuit H e cryostat that was mechanically isolated from the interferometer with the aid of a special purging chamber.3s The temperature was monitored with a goldconstantane thermocouple above 100 K and with a carbon resistor below 100 K. Both thermocouples are believed to be accurate within 5 K. In the intermediate range (80-1 10 K) the accuracy of both thermocouples is lower and the uncertainty on the temperature can be slightly larger. The Raman spectra of KCIO, at low temperature (as well as those at room temperature) were obtained by using a standard experimental apparatus with the 5145-A line of an Ar laser and photon-counting detection. Vibrational Spectroscopy of v 1 and v 2 Modes KCI04 crystallizes in the ordered hase below 583 K, in the p6 with four formula units orthorhombic system, space group DZh, in the primitive unit The correlation diagram of the C104ion between ion ( T d ) ,site (Cs),and crystal (D2h) symmetry is illustrated in Table I. (29) Hezel, A.: Ross, S. D.Spectrochim. Acta 1966, 22, 1949. (30)Lutz, H.D.; Becker, R. A.; Eckers, W.; Holscher, B. G.;Berthold, H. J. Specrrochim. Acta, Part A 1983, 39, 7. (31) Jayasooriya, U. A.; Kettle, S. F. A.; Mahasuverachai, S. J. Chem. Phys. 1987,86, 3127. (32) Bogani, F.;Salvi, P. R.J . Chem. Phys. 1984, 81, 4991. (33) Cardini, G.; Salvi. P. R.;Schettino, V. Chem. Phys. 1987, 117, 341; 1988, 119, 241. (34) Toupry, N.; Poulet, H.; Le Postollec, M.; Pick, R. M.; Yvinec, M. J . Roman Spectrosc. 1983, 14, 166. (35) Bini, R.;Foggi, P.; Salvi, P. R.:Sirnone, R.;Schettino, V. J. Mol. Srruct. 1990, 219, 43. (36) Johansson, G . B.; Lindquist, 0. Acta Crystollogr. 1977, 833, 2819.
.
.
A 1.6 1.2
0.8 0.4 0.0
945
935 925 WAVENUMBERS (cm-') Figure 2. Infrared spectrum of KCI04 single crystal in the u,;2v2 region at 20 K: solid line, ab face; dashed line, bc face.
The symmetric stretching ( v I ) and bending ( v 2 ) vibrations belong to A, and E symmetry species, respectively, in the isolated ion and therefore are infrared inactive. According to Table I, these modes split in the crystal into four and eight components, respectively, two of which, of B,, and B,, symmetry, are infrared active for v , and three (Blu,B2,, and B3J for v2. All these components borrow intensity through crystal effects and should be much less intense than the infrared-allowed v3 and v4 modes. This is fully consistent with experiment^.^^ The low-temperature infrared spectra in the v2 region of a thin KC104 plate cut parallel to the ab and bc crystal faces have been measured. The spectrum of the ab face is shown in Figure 1. Two peaks are observed at 466 and 471.5 cm-I. The 466-cm-, peak is also observed in the infrared spectrum of the bc face, while that at 47 1.5 cm-I disappears. Therefore the first peak is assigned to BzUand the second to B,, symmetry, respectively. A second weak band is found in bc polarization at 468.5 cm-I. Also, the infrared spectrum of a KC104 plate not parallel to the main crystal faces shows three bands, at 466,468.5, and 471.5 cm-I. Therefore, the peak at 468.5 cm-' is assigned to the BI, component of v2. The Raman spectrum of a single crystal of KC104 at 20 K shows four peaks in the same region at 461.4,463.4,464.8, and 465.4 cm-I, which, on the basis of our scattering geometry, can be assigned as B,, A,, B,,, and B,,, respectively. The dispersion of the bending mode is therefore e 1 0 cm-I. Neglecting the internal anharmonicity of the C104- ion, the two-phonon band due to a simultaneous excitation of two crystal modes, u2(k) + u2(-k), will extend approximately in the range 923-943 cm-I. The infrared absorption of the KC104 ab and bc crystal faces in the 2v2 region reported in Figure 2 is in good agreement with this prediction. In fact, a broad structure is observed in the 920-940-cm-' range. A more detailed discussion of Figure 2 presents some difficulties. The two-phonon continuum falls near the fundamental u I ,giving rise to their Fermi resonance. The Fermi resonance in crystals between a fundamental and a two-phonon continuum has been discussed in several paper^.^^^^^ At low intramolecular anharmonicity the Fermi coupling produces only a distortion of the shape of the two-phonon continuum. At larger anharmonicities bound
Vibrational Relaxation in a KCIO4 Crystal
The Journal of Physical Chemistry, Vol. 94, No. 17, I990 6655
WAVENUMBERS (cm-’1 Figure 3. Polarized infrared spectra of KClO, single crystal in the vl;2u2 region at 20 K along the crystal directions a (lower),c (middle),and b (upper).
4.8001
! Po
carries most of the intensity. On the contrary, resonance states occurring on the continuum edge (quasi-bound states) have an intensity comparable to that of the continuum. Considering these general results and the intensity pattern observed in the (aa) Raman spectrum of Figure 4, it can be concluded that the 942.6and 921.8-cm-I peaks should most likely be assigned as quasibound states and will be denoted as Q+(A,) and Q-(A,), respectively. In these circumstances, the Fermi coupling changes the shape of the two-phonon continuum compared to the harmonic density of states. Two broad peaks are observed inside the continuum, one centered at -927 cm-’ and the second starting at -932 cm-’ and with a maximum at -936 cm-l. We notice that the frequency of these peaks coincides with twice the frequency of the A,, B2,, and B1, components of u2. This implies that the branches of the dispersion curves of u2 are relatively flat. (b) The B3,,(lla)and B,,,(IIc) (Figure 3) spectra are similar to each other. They show sharp peaks at 943.5 and 943 cm-I, respectively, which may be assigned as the Q+(B,,) and Q+(BI,) quasi-bound states due to Fermi interaction with the two-phonon continuum of appropriate symmetry. Combining these with Raman data, we notice that ul has a very small dispersion (C1 cm-I). On the contrary, no sharp peak is observed at lower frequency, meaning that the Fermi resonance interaction is not sufficientlystrong in this case to give rise to the second quasi-bound states, Q-(B3,) and o-(B,,,), respectively. This can be explained by remembering that the two-phonon density of states entering in the Fermi interaction is different for each symmetry species.32 If the profile of the density of states changes, the occurrence of quasi-bound states may require larger Fermi coupling constants. (c) In the B2,,(l16) spectrum (Figure 3) no sharp structure occurs due to absence of Fermi interaction. As was already pointed out,” the continuum may however be renormalized by the internal anharmonicity of the u2 mode, measured by the x22anharmonicity constant. If this were sufficiently strong, one bound state could split below the continuum. In our case, x22can be assumed to be very small and the observed B2,, profile is then related to the two-phonon density of states of B2, symmetry. Fermi resonance in crystals depends on the two-phonon v2 u2 density of states, on the position of the unperturbed uI level with respect to the continuum, and on the strength of the anharmonic interaction, W = -k,22/(2’/2).32 Limiting our attention to the A,(aa) Raman spectrum, a qualitative estimate of the Fermi parameters, namely, v I and W, may be obtained. For this we neglect crystal interactions and approximate the two-phonon continuum as a single level centered at the average frequency of -933 cm-’. In this case (the “molecular” case)38the treatment simplifies considerably and the equations to work with are
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Figure 4. Polarized Raman spectrum of KCIO, single crystal in the v1;2v2 region at 20 K in the aa scattering geometry.
two-phonon states are split above and below the continuum. The bound states appears as sharp peaks in the spectrum. At intermediate anharmonicities quasi-bound states (resonances) may occur and show in the spectrum as sharp peaks within the continuum close to the edges. Jn the present case it must be considered that not all the crystal components of v , and 2u2 are in Fermi resonance. In fact, assuming that the Fermi coupling has an intramolecular origin, it will occur only between the A , fundamental and the A, component of the 2u2overtone. Therefore, from the correlation diagram of Table I it can be seen that the Fermi resonance will be active for the A B2,, B,,, and B3, crystal components. On the other hand, t t e B2, ( v 2 v 2 ) continuum (coming only from the E component of 2v2) is not perturbed by Fermi interaction, and in the limit of vanishing anharmonicity of the bending mode, the two-phonon intensity profile will be proportional to the unperturbed density of states of the B2, two-phonon states3’ We have measured the infrared spectrum of the ul;2u2region using polarized light along a, 6, and c crystal axes. For completeness the A,(aa) Raman spectrum was also measured. These results are shown in Figures 3 and 4, respectively. The following points may be noticed: (a) The A,(aa) Raman spectrum displays (Figure 4) a continuum pattern extending within the dispersion limits of u2 + v2 and two peaks, one strong at 942.6 cm-l and the second, much weaker, at 921.8 cm-l. In previous ~ o r k sit~was ~ *pointed ~ ~ out that when a bound state splits off the two-phonon continuum, it
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( 3 7 ) Bogani, F. J . Phys. C1978, 11, 1283.
o+ + 0-= 2v2 + u ,
(la)
where 9+,0-are the observed upper and lower frequencies and vl, 2v2 the unperturbed levels. From eq 1 we find that Q+ = 942.6 cm-I, R- = 921.8 cm-I, and 2v2 = 933 cm-l yields v 1 = 931.4 cm-l and W = 10.4 cm-I. The energy bands of the KCIO., crystal, as they result from the IR spectrum, are schematically drawn in Figure 5. In particular, the polarized spectrum in the 2vl;vl 2v2 region is reported in Figure 6. The spectrum is completely polarized along the b crystal direction (B2, symmetry). The band shape is rather peculiar, and the overall spectrum can be deconvoluted into a continuum extending from 1865 to 1885 cm-l enclosed by two sharp peaks at 1867 and 1884 cm-I. The deconvoluted spectrum is shown in Figure 7. The continuum can be assigned as arising from V ( k ) v2(k’) v2(-k - k’) three-phonon states. Since, as discussed above, the dispersion of R+ is very small, the width of the continuum is the same as that of the v 2 + v 2 two-phonon manifold. The two sharp peaks of the deconvoluted spectrum, which will be denoted from now on as w+ and w-, can be interpreted as
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+
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(38) Cardini, G.; Salvi, P. R.; Schettino, V.; Jodl, H. J . J . Chem. Phys. 1989, 91, 3869.
Bini et al.
6656 The Journal of Physical Chemistry, Vol. 94, No. 17, I990 2200 7
-
32v2+v3 2ooo-i
VlfV3
vl-2 v2-
1600J
-
4
800
1
'2+"3)
v1+v4
i -
*
3
6oo 400
0
4
A
I
v- -
1875 cm-'
1865
'
1885
Deconvolution of the infrared v I + 2v2 band into the w+ and resonances and the continuum: a, experimental spectrum; b, continuum; c, calculated w- and ut band shapes (see text for details). The calculated spectrum is shown by squares on (a). Figure 7. w-
P
Energy level diagram for KCIOPvibrons. The diagram at the bottom is a schematic representation of the lattice phonon density of states. The length of vertical bars is equal to the lattice phonon band width. The infrared-active levels under investigation have been drawn by bold lines, and the main up- and down-conversion processes indicated by arrows. Figure 5.
i
I
4-
TABLE 11: Observed Infrared and Raman Components of KCIO, Crystal in the Region of the v, and vt Fundamentals and of the v, t Zv, Combination u,
cm-'
461.4 463.4 464.8 465.4 466 468.5 471.5
A
1.0
assignment u2
B28 A, Bl, B3, B2, B,, B3u
u,
assignment
cm-'
u,;2u2 921.8 Q-(A,) 942.6 Q+(A,) 943 Q+(BI,) 943.5 Qt(B,,) UI
1867 1885
+ 2u2 4B2J
W+(B2,)
If the coupling with radiation is of dipolar type and the N oscillators are damped, the real and imaginary part of n are expressed as a function of frequency asz6
0.5
WAVENUMBERS (cm-') Figure 6. Polarized infrared spectrum of KCIO, single crystal in the v I 2u2 region a t 20 K along b (solid line) and a (dashed line) crystal axes.
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resonant states produced by the intramolecular anharmonicity and falling at the edges of the continuum. The vibrational assignment is summarized in Table 11. Relaxation Dynamics of Infrared-Active Vibrons Absorption Line Shape. In the classical theory, the oscillator strength of N harmonic oscillators with frequency vi (cm-I) is given by39 Si = (4nN/V')[(no2
+ 2 ) / 3 ] 2 ( a ~ / a 9 i ) 2 / ( 4 . 2 c 2 u i 2 )( 2 )
where N / V is the number of oscillators per unit volume (cm3), a ~ / d is 9 ~the transition dipole moment (esu), and no is the fre-
quency-independent part of the refractive index. However, the refractive index depends on the radiation frequency u and must be defined more generally as a complex quantity, that is n=q+ik
where r i is the damping constant (cm-I) for the oscillator at frequency vi. If vi >> ri,as it is for most vibrational transitions, it is justified to replace u with vi in eq 4b everywhere except in the factor (v? - v 2 ) since 27k has significant values only when u ui. This gives
and therefore 7 k has a Lorentzian line shape with full width at half-maximum (fwhm) equal to ri. Since the experimentally observed absorption coefficient a is related to k a = 4nvk (6) the absorption profile will be distorted with respect to a Lorentzian shape, due to the fact that 7 also is frequency-dependent. However, the degree of distortion depends on Siui/ri,and if this quantity is small compared to unity, Siui/ri
