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Direct observation of the relaxation time of the symmetric molecular CO2-3 vibration of calcite
Volume 14, number 1
OPTICS COMMUNICATIONS
May 1915
DIRECT OBSERVATION OF THE RELAXATION TIME OF THE SYMMETRIC MOLECULAR CO;-
VIBRATION OF CALCITE
A, LAUBEREAU, G. WOCHNER and W. KAISER Physik-Department der Technischen Univem'tiit Miinchen, Munich, Fed. Rep. Germany Received 5 March 1975 The relaxation of the internal Arg-mode of the COI- group (Y = 1086 cm-‘) is investigated with picosecond pulses. The measured decay times are 4.4 psec and 8.7 psec at 295 K and 90 K, respectively. Our values are in agreement with spontaneous Raman data. Possible relaxation processes are discussed.
An extensive literature has accumulated during the past decades on the lattice dynamics of crystals. Most of this work was concerned with the frequency spectra of various absorption and scattering processes (see, for example, ref. [l]). A comparatively small number of papers exists on time-resolved investigations of acoustic modes at low temperature (ref. [2a] ; see also, for instance, ref. [2b]). The advent of picosecond pulses with defined pulse properties (for a short survey, see ref. [3]) was an important step towards direct measurements of fast relaxation phenomena in solids [4-61. For example, the lifetime of TO phonons (in the center of the Brillouin zone) was measured in diamond to agree with estimates obtained from spontaneous Raman linewidth data [4]. A more complicated situation was found for the polariton mode of ionic Gap, demonstrating that additional information on lattice dynamics can be obtained with the help of picosecond light pulses [S] . In this letter we report on the decay time of the internal Alg-mode of CaCO,. Free relaxation of this vibration was directly observed with a time constant r which agrees with spontaneous Raman linewidth data. Our experimental results differ from earlier reported data on CaC03 [6]. Our experimental technique has been described in previous publications [7,8]. Fig. 1 shows a schematic of the experimental system. A single, intense picosecond pulse is generated by a mode-locked Nd:glass laser system in conjunction with an electro-optic switch and
a following amplifier. Our pulse parameters are: frequency FL = 9455 cm- l, band width 3 cm-l, pulse duration tp = 6 psec and peak power w 500 MW. A second puise of green light is produced with the help of a KDP crystal (T2L = 18910 cm-l). The laser pulse serves as a pump pulse traversing a carefully oriented CaCO, crystal of high optical quality; the sample length is 3 cm. The symmetric molecular vibration at Tp = 1086 cm-l is excited via stimulated Raman scattering to a level many orders of magnitude above the thermal equilibrium value. The phonon modes, which are populated during the short pumping process, pos-
Fig. 1. Schematic of the experimental system; the pump pulse at h = 1.06 pm and the probe pulse at A = 0.53 nm interact in the CaCOs sample; glass rod for fiied delay FD, glass prisms for variable delay VD, filters F, photodetectors P, digital peak voltmeter DPV. Inset: wave vector diagram of the pump and probe scattering processes (schematic).
7.5
Volume 14, number 1
OPTICS COMMUNICATIONS
sess a definite wave vector k, close to the center of the Brillouin zone. In our experimental arrangement we calculate k, = 1 .l X lo4 cm-l. The green probing pulse is properly delayed with respect to the infrared laser pulse; it interrogates the instantaneous degree of excitation of the molecular vibration with wave vector k,. Coherent anti-Stokes Raman scattering of the probe pulse at frequency TAS = FZL +TP is observed by means of a grating spectrometer and a photomultiplier. The detected scattering signal represents a convolution of the vibrational excitation and the shape of the probe pulse. Phase matching of the various wave vectors is most important for the probe scattering experiment [9]. Careful crystal orientation and beam alignment are essential. The scattering geometry benefits from the optical birefringence of trigonal CaC03; the phase matching geometry is modified with respect to ref. 193 and shown by the inset of fig. 1. The pump pulse propagates as an ordinary beam kt through the specimen at an angle of 32.8” to the optic (trigonal) axis. The vibrational excitation is represented by wave vector k, = kf -kg, where kg is the wave vector of the Stokes pulse produced in the stimulated scattering. The color dispersion of CaCO, requires an extraordinary probe pulse kezLwith an angle of 5 8” with respect to the pump pulse in order to ensure phase matching of the probe scattering. The anti-Stokes emission is highly directional and occurs at 0.3’ with respect to the probe beam. Our experimental data on the decay time of the internal modes with wave vector k, are presented in fig. 2 for two temperatures. The coherent anti-Stokes scattering signal Scoh(tD) is plotted as a function of delay time tD between pump and probe pulse. tD = 0 marks the peak of the pumping laser pulse. The experimental points extend over five orders of magnitude. Each data point represents an average value over approximately ten individual measurements; the error bars indicate the observed r.m.s. deviation. The signal curves in the figure rise to a maximum at tD m 7 psec shortly after the peak of the pump pulse; i.e., time is required to build up the lattice excitation. For larger values of tD the excitation process rapidly terminates and the internal vibration relaxes freely according to the individual time constant 7. It is interesting to see in fig. 2 the exponential decay of the signal over several orders of ten. We are able to follow the material excitation with good accuracy for a period of 60 psec. 76
May 1975
I
I
I
D1086cm-’
,
I_
2; T 295K t ZL.Lf 0.3ps\ D
,_ I
0
20
LO
Delay
Time
4 Ii\
I
60
tD Cps3
Fig. 2. Anti-Stokes probe scattering signal versus delay 90 K (full points) and 295 K
two crystal temperatures,
cles); curves are calculated
time for (open
cir-
(see text).
From the slopes of the signal curves we find values of r of (4.4 kO.3) x 10-12 set and (8.7 + 0.7) X lo-l2 set at 295 K and 90 K, respectively. The curves in the figure are calculated from the existing theory of transient stimulated Raman scattering [lo] using our result on r and the parameters of our pumping pulse. The good agreement with the experimental data should be noted. The quality of the picosecond pulses strongly influences the reliability of time resolved studies. A complicated multiple pulse structure, for instance, destroys the free relaxation of the excited vibration and leads to apparently larger values of the relaxation time. The following points should be emphasized: (i) Our measurements were carried out with single picosecond pulses selected from an early position of clean mode-locked pulse trains. (ii) Our laser oscillator applies a thin contacted dye cell for the nonlinear absorber avoiding satellite pulses. (iii) The same decay rate was observed in different experimental runs with and without an external nonlinear absorption cell of 3% initial transmission positioned after
OPTICS COMMUNICATIONS
Volume 14, number 1 Table 1 Decay times of the internal Alg-mode of CaCOs
Decay time 7
(psec)
Spontaneous scattering a) [Ill
This work
3.8 to 4.8
4.4 * 0.3
7.8 f 0.9
8.7 f 0.7
Ref. [6]
8.5*2 19.1~~4a)
295 K 90K
a) at 100 K. the optical amplifier. This nonlinear filter discriminates against smaller satellite pulses and strongly reduces the general background. It is concluded that satellite pulses or background radiation do not disturb our measurements. In table 1 various experimental data of the relaxation of the Cog- internal mode in CaC03 are summarized. Spontaneous scattering was investigated by Park [ll].ARamanlinewidthof 1.1+0.08cm-1 and 1.4* 0.14 cm-l was reported for room temperature. At a temperature of 90 K a value of 0.68 f 0.08 cm-l was measured. Assuming a lorentzian line profile, these numbers correspond to phonon relaxation times of 7= 3.8 to 4.8 psec and T= 7.8 + 0.9 psec at 295 K and 90 K, respectively. Table 1 clearly shows that our data of 7 = 4.4 psec (295 K) and 7 = 8.7 psec (90 K) are in good agreement with the spontaneous measurements. In table 1, our results are compared with previous data of Alfano and Shapiro [6]. These authors used the same phase-matched probing technique [9] discussed in this paper. There is considerable disagreement between our experimental values and the data of ref. [6]. On account of our detailed study of our single picosecond pulses we are very certain that our data are correct. The following comment should be made. In related work on liquid N2, the authors of ref. [6] reported time constants again too long [ 121 in comparison with ref. [8] and to values deduced from spontaneous Raman data [ 131. The authors of ref. [6] used in their investigations [6,12,14] the whole mode-locked pulse train for the excitation and the probing of the molecular vibration. It has been well established that complicated pulse structures frequently occur around the peak and the later part of the pulse trains. We feel that well-defined single picosecond pulses are required to obtain quantitative and reliable relaxation data.
May 1975
Comparing our relaxation times and the values calculated from the spontaneous Raman results, the following points are of interest. The highly directional excitation process populates a small number of phonon modes to occupation numbers as high as lOlo, i.e., many orders of magnitude above the thermal equilibrium value. In spite of this high excitation per mode, the internal CO!- vibration decays with the same time constant estimated from the low excitation level of spontaneous Raman scattering. This result is not surprising if the total vibrational excitation is considered. During one pumping pulse, one CO;- ion out of approximately one thousand molecules accepts a vibrational quantum. Under these conditions the phonon relaxation process is expected to be independent of the degree of excitation. A phonon breakdown mechanism [ 151 has been discussed in the literature in context with the A,, mode of CaC03 [6,15]. This process requires efficient coupling of the internal vibration to TA phonons of the lowest acoustic phonon branch, which are expected to display very long lifetimes at low temperatures. However, available information on the phonon spectrum of CaCO, [ 161 rules out three phonon interactions of the type discussed in ref. [ 151. The excited optical mode at 1086 cm-l is far above the TA branches, which have energies 6 100 cm-l. Phonon breakdown for other decay routes is unlikely on account of the large number of terminating phonon modes [4]. The good agreement between our 7 values and the spontaneous Raman data suggest that parametric phonon decay processes do not affect our investigations. We turn now to the discussion of the physical proce!ses which contribute to the observed phonon decay time 7. Calcite has a very complicated phonon spectrum of 27 phonon branches [ 161. In the stimulated Raman process the incident pump pulse interacts with the internal A,, mode. Phonons of frequency ‘3 = 1086 cm-l and wave vector k, = 1 .l X lo4 cm- P are produced. These phonons are studied with our coherent probing technique. The measured time constant 7 represents the lifetime of the excited phonon modes with well defined frequency and wave vector. There are two kinds of prdcesses which are responsible for the disappearance of the generated phonons: a) decay to new phonons of different energy and b) change to new phonons of equal energy but different k-vector. 77
Volume 14, number 1
OPTICS COMMUNICATIONS
a) The decay of a high energy phonon has been considered frequently in experimental and theoretical papers in terms of three and four photon processes. For the Alg-mode of CaCO, , possible three phonon processes allowed by conservation laws and symmetry selection rules are: 1086(Alp)
+
712 (EJ + 376(EU),
1086 (Alp) + 1416 (Err) - 330 (EJ. The second decay route will be important at high temperature only. Other decay routes, especially of higher order, are possible. b) The dispersion of the internal Al, mode is very small. Phonons might “change” their k-vector along the Alg phonon branch without significant loss of energy. This process represents a change in the phase relation between the internal vibrations of the individual CO;- complexes, which are coherently excited in the stimulated Raman process. Dephasing between the molecular vibrations leads to a decay of the observed scattering signal. It is interesting to note that such phase relaxation processes have been studied in polyatomic liquids in recent years [ 171. In fact, the dephasing time and the energy relaxation time of a molecular vibration could be distinguished in the liquid state in a number of cases [ 171. An investigation of the energy relaxation time would be required for the internal mode of CaC03 to establish clearly the physical processes which determine the observed phonon decay time. In summary we wish to say that our experimental data at 295 K and 90 K give phonon relaxation times with an accuracy of better than 10%. Our values agree with line width data obtained from spontaneous Raman scattering. Nonlinear phonon decay is not observed for the Alg-mode in CaCO,. The authors gratefully acknowledge valuable discussions with Professor H. Schmidt and Professor S.F. Fischer.
78
May 1975
References R.W.H. Stevenson, ed., Phonons in perfect lattices and lattices with point imperfections (Oliver Boyd, Edinburth, 1966). [21 H.E. Bommel and K. Dransfield, Phys. Rev. Lett. 1 (1958) 234; W.P. Mason and R.N. Thurston, eds., Physical acoustics, vol. 8 (Academic Press, New York, 1971). [31 A Laubereau and W. Kaiser, Optoelectronics 6 (1974) 1. [41 A. Laubereau, D. von der Linde and W. Kaiser, Phys. Rev. Lett. 27 (1971) 802. 151 A. Laubereau, D. von der Linde and W. Kaiser, Opt. Commun. 7 (1973) 173. 161 R.R. Alfano and S.L. Shapiro, Phys. Rev. Lett. 26 (1971) 1247. [71 D. von der Linde, A. Laubereau and W. Kaiser, Phys. Rev. Lett. 26 (1971) 955. [81 A. Laubereau, Chem. Phys. Lett. 27 (1974) 600. [91 J.A. Giordmaine and W. Kaiser, Phys. Rev. 144 (1966) 676. 1101 S.A. Akhmanov, Mater. Res. Bull. 4 (1969) 455; R.L. Carman, F. Shimizu, C.S. Wang and N. Bloembergen, Phys. Rev. A2 (1970) 60. 1111 K. Park, Phys. Lett. 22 (1966) 39; 25A (1967) 490. 1121 R.R. Alfano and S.L. Shapiro, in: Phonons, ed. M.A. Nusimovici (Flammarion, Paris, 1971). 1131 W. Clements and B.P. Stoicheff, Appl. Phys. Lett. 12 (1968) 246; M. Scotto, J. Chem. Phys. 49 (1968) 5362. [I41 R.R. Alfano and S.L. Shapiro, Opt. Commun. 6 (1972) 98; see comments of: G. Mourou and M.M. Malley, Opt. Commun. 11 (1974) 282. [ISI R. Orbach, IEEE Trans. Sonics Ultrason. 14 (1967) 140; S.E. Bulgadaev and I.B. Levinson, JETP Lett. 19 (1974) 304. iI61 M. Plihal, Phys. Stat. Sol. (b) 56 (1973) 495; M. Plihal and G. Schaack, Phys. Stat. Sol. (b) 42 (1970) 485. iI71 A. Laubereau, D. von der Linde and W. Kaiser, Phys. Rev. Lett. 28 (1972) 1162; A. Laubereau, L. Kirschner and W. Kaiser, Opt. Commun. 9 (1973) 183.
ill
OPTICS COMMUNICATIONS
May 1915
DIRECT OBSERVATION OF THE RELAXATION TIME OF THE SYMMETRIC MOLECULAR CO;-
VIBRATION OF CALCITE
A, LAUBEREAU, G. WOCHNER and W. KAISER Physik-Department der Technischen Univem'tiit Miinchen, Munich, Fed. Rep. Germany Received 5 March 1975 The relaxation of the internal Arg-mode of the COI- group (Y = 1086 cm-‘) is investigated with picosecond pulses. The measured decay times are 4.4 psec and 8.7 psec at 295 K and 90 K, respectively. Our values are in agreement with spontaneous Raman data. Possible relaxation processes are discussed.
An extensive literature has accumulated during the past decades on the lattice dynamics of crystals. Most of this work was concerned with the frequency spectra of various absorption and scattering processes (see, for example, ref. [l]). A comparatively small number of papers exists on time-resolved investigations of acoustic modes at low temperature (ref. [2a] ; see also, for instance, ref. [2b]). The advent of picosecond pulses with defined pulse properties (for a short survey, see ref. [3]) was an important step towards direct measurements of fast relaxation phenomena in solids [4-61. For example, the lifetime of TO phonons (in the center of the Brillouin zone) was measured in diamond to agree with estimates obtained from spontaneous Raman linewidth data [4]. A more complicated situation was found for the polariton mode of ionic Gap, demonstrating that additional information on lattice dynamics can be obtained with the help of picosecond light pulses [S] . In this letter we report on the decay time of the internal Alg-mode of CaCO,. Free relaxation of this vibration was directly observed with a time constant r which agrees with spontaneous Raman linewidth data. Our experimental results differ from earlier reported data on CaC03 [6]. Our experimental technique has been described in previous publications [7,8]. Fig. 1 shows a schematic of the experimental system. A single, intense picosecond pulse is generated by a mode-locked Nd:glass laser system in conjunction with an electro-optic switch and
a following amplifier. Our pulse parameters are: frequency FL = 9455 cm- l, band width 3 cm-l, pulse duration tp = 6 psec and peak power w 500 MW. A second puise of green light is produced with the help of a KDP crystal (T2L = 18910 cm-l). The laser pulse serves as a pump pulse traversing a carefully oriented CaCO, crystal of high optical quality; the sample length is 3 cm. The symmetric molecular vibration at Tp = 1086 cm-l is excited via stimulated Raman scattering to a level many orders of magnitude above the thermal equilibrium value. The phonon modes, which are populated during the short pumping process, pos-
Fig. 1. Schematic of the experimental system; the pump pulse at h = 1.06 pm and the probe pulse at A = 0.53 nm interact in the CaCOs sample; glass rod for fiied delay FD, glass prisms for variable delay VD, filters F, photodetectors P, digital peak voltmeter DPV. Inset: wave vector diagram of the pump and probe scattering processes (schematic).
7.5
Volume 14, number 1
OPTICS COMMUNICATIONS
sess a definite wave vector k, close to the center of the Brillouin zone. In our experimental arrangement we calculate k, = 1 .l X lo4 cm-l. The green probing pulse is properly delayed with respect to the infrared laser pulse; it interrogates the instantaneous degree of excitation of the molecular vibration with wave vector k,. Coherent anti-Stokes Raman scattering of the probe pulse at frequency TAS = FZL +TP is observed by means of a grating spectrometer and a photomultiplier. The detected scattering signal represents a convolution of the vibrational excitation and the shape of the probe pulse. Phase matching of the various wave vectors is most important for the probe scattering experiment [9]. Careful crystal orientation and beam alignment are essential. The scattering geometry benefits from the optical birefringence of trigonal CaC03; the phase matching geometry is modified with respect to ref. 193 and shown by the inset of fig. 1. The pump pulse propagates as an ordinary beam kt through the specimen at an angle of 32.8” to the optic (trigonal) axis. The vibrational excitation is represented by wave vector k, = kf -kg, where kg is the wave vector of the Stokes pulse produced in the stimulated scattering. The color dispersion of CaCO, requires an extraordinary probe pulse kezLwith an angle of 5 8” with respect to the pump pulse in order to ensure phase matching of the probe scattering. The anti-Stokes emission is highly directional and occurs at 0.3’ with respect to the probe beam. Our experimental data on the decay time of the internal modes with wave vector k, are presented in fig. 2 for two temperatures. The coherent anti-Stokes scattering signal Scoh(tD) is plotted as a function of delay time tD between pump and probe pulse. tD = 0 marks the peak of the pumping laser pulse. The experimental points extend over five orders of magnitude. Each data point represents an average value over approximately ten individual measurements; the error bars indicate the observed r.m.s. deviation. The signal curves in the figure rise to a maximum at tD m 7 psec shortly after the peak of the pump pulse; i.e., time is required to build up the lattice excitation. For larger values of tD the excitation process rapidly terminates and the internal vibration relaxes freely according to the individual time constant 7. It is interesting to see in fig. 2 the exponential decay of the signal over several orders of ten. We are able to follow the material excitation with good accuracy for a period of 60 psec. 76
May 1975
I
I
I
D1086cm-’
,
I_
2; T 295K t ZL.Lf 0.3ps\ D
,_ I
0
20
LO
Delay
Time
4 Ii\
I
60
tD Cps3
Fig. 2. Anti-Stokes probe scattering signal versus delay 90 K (full points) and 295 K
two crystal temperatures,
cles); curves are calculated
time for (open
cir-
(see text).
From the slopes of the signal curves we find values of r of (4.4 kO.3) x 10-12 set and (8.7 + 0.7) X lo-l2 set at 295 K and 90 K, respectively. The curves in the figure are calculated from the existing theory of transient stimulated Raman scattering [lo] using our result on r and the parameters of our pumping pulse. The good agreement with the experimental data should be noted. The quality of the picosecond pulses strongly influences the reliability of time resolved studies. A complicated multiple pulse structure, for instance, destroys the free relaxation of the excited vibration and leads to apparently larger values of the relaxation time. The following points should be emphasized: (i) Our measurements were carried out with single picosecond pulses selected from an early position of clean mode-locked pulse trains. (ii) Our laser oscillator applies a thin contacted dye cell for the nonlinear absorber avoiding satellite pulses. (iii) The same decay rate was observed in different experimental runs with and without an external nonlinear absorption cell of 3% initial transmission positioned after
OPTICS COMMUNICATIONS
Volume 14, number 1 Table 1 Decay times of the internal Alg-mode of CaCOs
Decay time 7
(psec)
Spontaneous scattering a) [Ill
This work
3.8 to 4.8
4.4 * 0.3
7.8 f 0.9
8.7 f 0.7
Ref. [6]
8.5*2 19.1~~4a)
295 K 90K
a) at 100 K. the optical amplifier. This nonlinear filter discriminates against smaller satellite pulses and strongly reduces the general background. It is concluded that satellite pulses or background radiation do not disturb our measurements. In table 1 various experimental data of the relaxation of the Cog- internal mode in CaC03 are summarized. Spontaneous scattering was investigated by Park [ll].ARamanlinewidthof 1.1+0.08cm-1 and 1.4* 0.14 cm-l was reported for room temperature. At a temperature of 90 K a value of 0.68 f 0.08 cm-l was measured. Assuming a lorentzian line profile, these numbers correspond to phonon relaxation times of 7= 3.8 to 4.8 psec and T= 7.8 + 0.9 psec at 295 K and 90 K, respectively. Table 1 clearly shows that our data of 7 = 4.4 psec (295 K) and 7 = 8.7 psec (90 K) are in good agreement with the spontaneous measurements. In table 1, our results are compared with previous data of Alfano and Shapiro [6]. These authors used the same phase-matched probing technique [9] discussed in this paper. There is considerable disagreement between our experimental values and the data of ref. [6]. On account of our detailed study of our single picosecond pulses we are very certain that our data are correct. The following comment should be made. In related work on liquid N2, the authors of ref. [6] reported time constants again too long [ 121 in comparison with ref. [8] and to values deduced from spontaneous Raman data [ 131. The authors of ref. [6] used in their investigations [6,12,14] the whole mode-locked pulse train for the excitation and the probing of the molecular vibration. It has been well established that complicated pulse structures frequently occur around the peak and the later part of the pulse trains. We feel that well-defined single picosecond pulses are required to obtain quantitative and reliable relaxation data.
May 1975
Comparing our relaxation times and the values calculated from the spontaneous Raman results, the following points are of interest. The highly directional excitation process populates a small number of phonon modes to occupation numbers as high as lOlo, i.e., many orders of magnitude above the thermal equilibrium value. In spite of this high excitation per mode, the internal CO!- vibration decays with the same time constant estimated from the low excitation level of spontaneous Raman scattering. This result is not surprising if the total vibrational excitation is considered. During one pumping pulse, one CO;- ion out of approximately one thousand molecules accepts a vibrational quantum. Under these conditions the phonon relaxation process is expected to be independent of the degree of excitation. A phonon breakdown mechanism [ 151 has been discussed in the literature in context with the A,, mode of CaC03 [6,15]. This process requires efficient coupling of the internal vibration to TA phonons of the lowest acoustic phonon branch, which are expected to display very long lifetimes at low temperatures. However, available information on the phonon spectrum of CaCO, [ 161 rules out three phonon interactions of the type discussed in ref. [ 151. The excited optical mode at 1086 cm-l is far above the TA branches, which have energies 6 100 cm-l. Phonon breakdown for other decay routes is unlikely on account of the large number of terminating phonon modes [4]. The good agreement between our 7 values and the spontaneous Raman data suggest that parametric phonon decay processes do not affect our investigations. We turn now to the discussion of the physical proce!ses which contribute to the observed phonon decay time 7. Calcite has a very complicated phonon spectrum of 27 phonon branches [ 161. In the stimulated Raman process the incident pump pulse interacts with the internal A,, mode. Phonons of frequency ‘3 = 1086 cm-l and wave vector k, = 1 .l X lo4 cm- P are produced. These phonons are studied with our coherent probing technique. The measured time constant 7 represents the lifetime of the excited phonon modes with well defined frequency and wave vector. There are two kinds of prdcesses which are responsible for the disappearance of the generated phonons: a) decay to new phonons of different energy and b) change to new phonons of equal energy but different k-vector. 77
Volume 14, number 1
OPTICS COMMUNICATIONS
a) The decay of a high energy phonon has been considered frequently in experimental and theoretical papers in terms of three and four photon processes. For the Alg-mode of CaCO, , possible three phonon processes allowed by conservation laws and symmetry selection rules are: 1086(Alp)
+
712 (EJ + 376(EU),
1086 (Alp) + 1416 (Err) - 330 (EJ. The second decay route will be important at high temperature only. Other decay routes, especially of higher order, are possible. b) The dispersion of the internal Al, mode is very small. Phonons might “change” their k-vector along the Alg phonon branch without significant loss of energy. This process represents a change in the phase relation between the internal vibrations of the individual CO;- complexes, which are coherently excited in the stimulated Raman process. Dephasing between the molecular vibrations leads to a decay of the observed scattering signal. It is interesting to note that such phase relaxation processes have been studied in polyatomic liquids in recent years [ 171. In fact, the dephasing time and the energy relaxation time of a molecular vibration could be distinguished in the liquid state in a number of cases [ 171. An investigation of the energy relaxation time would be required for the internal mode of CaC03 to establish clearly the physical processes which determine the observed phonon decay time. In summary we wish to say that our experimental data at 295 K and 90 K give phonon relaxation times with an accuracy of better than 10%. Our values agree with line width data obtained from spontaneous Raman scattering. Nonlinear phonon decay is not observed for the Alg-mode in CaCO,. The authors gratefully acknowledge valuable discussions with Professor H. Schmidt and Professor S.F. Fischer.
78
May 1975
References R.W.H. Stevenson, ed., Phonons in perfect lattices and lattices with point imperfections (Oliver Boyd, Edinburth, 1966). [21 H.E. Bommel and K. Dransfield, Phys. Rev. Lett. 1 (1958) 234; W.P. Mason and R.N. Thurston, eds., Physical acoustics, vol. 8 (Academic Press, New York, 1971). [31 A Laubereau and W. Kaiser, Optoelectronics 6 (1974) 1. [41 A. Laubereau, D. von der Linde and W. Kaiser, Phys. Rev. Lett. 27 (1971) 802. 151 A. Laubereau, D. von der Linde and W. Kaiser, Opt. Commun. 7 (1973) 173. 161 R.R. Alfano and S.L. Shapiro, Phys. Rev. Lett. 26 (1971) 1247. [71 D. von der Linde, A. Laubereau and W. Kaiser, Phys. Rev. Lett. 26 (1971) 955. [81 A. Laubereau, Chem. Phys. Lett. 27 (1974) 600. [91 J.A. Giordmaine and W. Kaiser, Phys. Rev. 144 (1966) 676. 1101 S.A. Akhmanov, Mater. Res. Bull. 4 (1969) 455; R.L. Carman, F. Shimizu, C.S. Wang and N. Bloembergen, Phys. Rev. A2 (1970) 60. 1111 K. Park, Phys. Lett. 22 (1966) 39; 25A (1967) 490. 1121 R.R. Alfano and S.L. Shapiro, in: Phonons, ed. M.A. Nusimovici (Flammarion, Paris, 1971). 1131 W. Clements and B.P. Stoicheff, Appl. Phys. Lett. 12 (1968) 246; M. Scotto, J. Chem. Phys. 49 (1968) 5362. [I41 R.R. Alfano and S.L. Shapiro, Opt. Commun. 6 (1972) 98; see comments of: G. Mourou and M.M. Malley, Opt. Commun. 11 (1974) 282. [ISI R. Orbach, IEEE Trans. Sonics Ultrason. 14 (1967) 140; S.E. Bulgadaev and I.B. Levinson, JETP Lett. 19 (1974) 304. iI61 M. Plihal, Phys. Stat. Sol. (b) 56 (1973) 495; M. Plihal and G. Schaack, Phys. Stat. Sol. (b) 42 (1970) 485. iI71 A. Laubereau, D. von der Linde and W. Kaiser, Phys. Rev. Lett. 28 (1972) 1162; A. Laubereau, L. Kirschner and W. Kaiser, Opt. Commun. 9 (1973) 183.
ill