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Vibrational Dephasing of PHD2 in liquid and solid PD3
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
15 November 1979
VIBRATIONAL DEPHASING OF PHD 2 IN LIQUID AND SOLID PD 3 Richard E. WILDE Department o f Chemistry, Texas Tech University, Lubbock, Texas 79409, USA Received 3 August 1979
Vibrational correlation functions obtained from the isotropic Raman scattering of the P-H stretching vibration of PHD2 isolated as an isotopic impurity in liquid and solid PD3 have been modeled using a memory-function appraoch. The modulation is fast at all temperatures. It is postulated that rotation-translation coupling is responsible for the vibrational relaxation. Information about the molecular dynamics has been obtained out to 7 ps.
1. Introduction The memory-function approach [1 ] to correlationfunction modeling is proving to be very informative as far as the molecular dynamics affecting vibrational relaxation are concerned. The approach can be carried to second or higher order in the memory function depending upon the mechanism of the relaxation process. There are many processes that can influence the relaxation including pure dephasing, resonant vibrational energy transfer, population relaxation, and vibration-rotation coupling. For small molecules in condensed phases, pure dephasing and resonant vibrational energy transfer appear to be the dominant relaxation processes on the picosecond time scale. Pure dephasing arises from quasielastic molecular collisions that modulate the vibrational frequency.The mechanism of this process is thought to involve the rotational and translational motions of the molecules. These motions form a heat bath, and the stochastic nature of the heat bath provides the time dependence for the interaction between the molecular vibrations and the heat bath. The vibrational energy exchange can be viewed either as an oscillator-bath interaction or as a collective oscillator-oscillator interaction depending on the formalism employed. Spontaneous Raman linewidths are a measure of molecular processes on the picosecond time scale. This technique is simple and has the potential of
yielding considerable information about molecular dynamics provided that the isotropic and anisotropic spectra can be analyzed. The method of analysis [2] involves the Fourier transform of the spectral band shapes to obtain vibrational and reorientational timecorrelation functions. In this letter we discuss the mechanism of vibrational dephasing of an asymmetrictop molecule, PHD2, isolated as an impurity molecule in the liquid and plastic crystalline phases of PD 3. Nice features of this system are that the P - H stretching band is not overlapped by any bands of PHD 2 or PD 3 and that resonant vibrational energy exchange interactions are minimized.
2. Experimental methods The experimental techniques employed have been discussed in detail elsewhere [3]. Therefore, only a brief description is given here. Spontaneous Raman scattering experiments were done using a Jarrell-Ash double monochromator capable of a resolution of better than 1 c m - l . The source was a Coherent Radiation model 54 argon-ion laser. The signal was analyzed using photon-counting electronics. The PHD 2 sample was condensed into an 8 mm o.d. Pyrex tube mounted on the tail section of an Air Products Heli-Tran refrigerator. Right-angle scattering was observed with an S-5 response PMT. The PHD 2 was present in PD 3 555
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
to the extent of 1 2 - 1 5 tool %. The preparation of the sample has been described previously [4].
15 November 1979
~°I. *
PHD 2
08
T=153
K
* "+
3. Method of analysis
06
The isotropic part of the 2313 c m - 1 P - H stretching band of PHD 2 was Fourier transformed to provide the vibrational time-correlation function; This function was then modeled [1 ] beginning with a "frozenlattice" gaussian function which reflects the inhomogeneous broadening at short times. Then successively higher-order frozen-lattice memory functions were calculated via coupled Volterra equations. In the present case the memory functions were truncated at the second-order memory function K2FL(t), and the frozenlattice memory function was related to the actual second-order memory function by the relation
/%(0 -- 2K
L(t)e
,
(1)
where n 2 is related to the second and fourth spectral moments, M 2 and M 4, of the isotropic Raman band, and r is an adjustable parameter. The Volterra equations are then used in a reverse manner to give the correlation function C(t) of the real system. The parameter K2 is related to the speed of modulation through the relation K 2 : ½r/+
I,
The correlation functions and normalized firstorder memory function are shown in fig. 1 for the liquid at 153 K. The correlation and memory functions of the liquid at 143 K just above the freezing point (f.p. 139 K), the plastic crystalline phase (phase I) at 130 K, and phase I at 95 K just above the I - I I transition temperature (T c = 93.5 K) are very similar to those of the liquid at 153 K. The correlation functions are not reliable beyond 6 - 7 ps. The wide slit width (2.0 cm - 1 ) necessary at 95 K makes this corre556
*+
÷.+
-+ o+l+
02
o
"*. o
+"%+e
o o
+°+e*°+o+o+~, ~ o
O.O
o o
o
÷°+°+*÷°+~.÷,+°+e÷ oooooo
°ooOOOooOooooooooo
o
o
-0.2
o % I
-04
2
I
I
3
4
I
5
I
6
7
t(ps)
Fig. 1. Experimental (o) and calculated (+) vibrational correlation functions and the calculated memory function (0) of PHD2 at 153 K in the liquid phase. lation function unreliable beyond about 5.5 ps. The resuits of the correlation-function modeling analysis are given in table 1. Before discussing these results, it should be pointed out that the intensity of the Raman Table 1 Summary of vibrational dephasing analysis Temperature (K)
Mexptl M~Xptl
(3)
4. Correlation and memory functions
o+
0.4
(2)
where ,/-1/2 < 1 for fast modulation and r/-1/2 >> 1 for slow modulation [5]. The speed of modulation is ! characterized by a correlation time r c such that 72= 1/M2(r'c) 2 .
.*
M2 (cm -2) •2 r (ps) M4 (cm -4) n r~ (ps)
rl-1/2
153
143
130
112 99053 110 3.3 1.0 91960 4.6 0.24 0.466
88 59056 85 3.0 0.9 50575 4.0 0.29 0.500
116 90 109320 61418 120 90 3.7 3.4 1.2 0.6 120960 63180 5.4 4.8 0.21 0.26 0.430 0.456
gaussian a) fwhh (cm -1) 24.7 fwhh (cm- 1) fwhh (cm -1) 10.0 exptl c) fwhh (cm -1 ) 6.4
95
21.7
25.8
22.3
9.3
9.5
8.8
6.5
5.5
6.6
a) 2(2 In 2)1]2M1]2. b) 41rcM2./o c) No correction for finite slit width.
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
scattering by the P - H stretching mode of PHD 2 is necessarily weak because the PHD 2 is present at small concentration. This is a disadvantage because the spectrometer slit width must be fairly wide (1.5 cm -1 for the liquid and 1.7-2.0 cm -1 for phase I) and the background noise is high due to the increased gain of the photon counter. The background noise level was about 3% compared with 0.3% for the PD 3 bands [3]. This gives almost twice the experimental errors reported in ref. [3]. Thus, for example, the errors in the second moments reported in table 1 are about -+20% with a truncation error on the band wings of -+6 cm -1 . All correlation functions were corrected for finite slit width. The data of table 1 indicate that the second moments of the liquid and solid are on the order of I00 cm - 2 and that the vibrational modulation is fast. Upon comparing the spectra of PHD 2 with that of PD 3 previously reported [3], one is struck by the much broader bandwidth of PHD 2. This leads to faster relaxingcorrelation and memory functions as reflected t ! in the r c values of table 1. The r c values are very close to the relaxation times of the first-order memory functions. Apparently, the PHD 2 molecule in the PHD 2 PD 3 mixture is relaxing much faster than either the PD 3 or PH 3 molecules [6]. As pointed out in refs. [3, 6], the evidence suggests that the rotational motions dominate the dephasing process. In the present case, however, the modulation is faster than one would expect from rotations alone, since the PH 3 molecule has the smallest moments of inertia of the three molecules. The fairly large percentage of 1H ( 4 - 5 % ) in the sample undoubtedly leads to some band broadening through interaction between the 1H atoms. Part of this is an inhomogeneous effect which is reflected in the second moment. Resonant vibrational energy exchange is negligible in the cases [3,6] of PD 3 and PH3, and we see no reason for this effect to be of importance in an isotopically dilute sample. In the case of the ordered solid it was concluded [4] that the 1 H -
15 November 1979
1H interaction was negligible. Therefore, we do not believe that P H D 2 - P H D 2 interactions are an important source of band broadening. There is another effect which is present in PHD 2 but not in the symmetric-top molecules that must be considered in order to explain the fast relaxation. This is the r o t a t i o n translation coupling that arises because the center of interaction and the center of mass do not coincide [7]. This coupling allows the translational as well as the rotational degrees of freedom to be effective in modulating the vibrational frequency. It appears, therefore, that the rotation-translation coupling must dominate the dephasing process in the case of PHD 2. The fact that it was not necessary to include higher-order memory functions in the analysis suggests that this is the only mechanism effectively modulating the vibrational frequency.
Acknowledgement This research was supported b y a grant from the Robert A. Welch Foundation. Computer time was provided by Texas Tech University.
References [1] S.S. Cohen and R.E. Wilde, J. Chem. Phys. 68 (1978) 1138. [2] L.A. Nafie and W.L. Peticolas, J. Chem. Phys. 57 (1972) 3145. [3] R.E. Wilde and S.S. Cohen, J. Chem. Phys. 70 (1979)4557. [4] R.E. Wilde and B.C. Covington, J. Phys. Chem. Solids 36 (1975) 1225. [5] R. Kubo, in: Fluctuation, relaxation and resonance in magnetic systems, ed. D. ter Haar (Oliver and Boyd, Edinburgh, 1962) p. 23. [6] R.E. Wilde and T.-C. Chang, J. Chem. Phys., submitted for publication. [7] H. Friedmann and S. Kimel, J. Chem. Phys. 47 (1967) 3589.
557
CHEMICAL PHYSICS LETTERS
15 November 1979
VIBRATIONAL DEPHASING OF PHD 2 IN LIQUID AND SOLID PD 3 Richard E. WILDE Department o f Chemistry, Texas Tech University, Lubbock, Texas 79409, USA Received 3 August 1979
Vibrational correlation functions obtained from the isotropic Raman scattering of the P-H stretching vibration of PHD2 isolated as an isotopic impurity in liquid and solid PD3 have been modeled using a memory-function appraoch. The modulation is fast at all temperatures. It is postulated that rotation-translation coupling is responsible for the vibrational relaxation. Information about the molecular dynamics has been obtained out to 7 ps.
1. Introduction The memory-function approach [1 ] to correlationfunction modeling is proving to be very informative as far as the molecular dynamics affecting vibrational relaxation are concerned. The approach can be carried to second or higher order in the memory function depending upon the mechanism of the relaxation process. There are many processes that can influence the relaxation including pure dephasing, resonant vibrational energy transfer, population relaxation, and vibration-rotation coupling. For small molecules in condensed phases, pure dephasing and resonant vibrational energy transfer appear to be the dominant relaxation processes on the picosecond time scale. Pure dephasing arises from quasielastic molecular collisions that modulate the vibrational frequency.The mechanism of this process is thought to involve the rotational and translational motions of the molecules. These motions form a heat bath, and the stochastic nature of the heat bath provides the time dependence for the interaction between the molecular vibrations and the heat bath. The vibrational energy exchange can be viewed either as an oscillator-bath interaction or as a collective oscillator-oscillator interaction depending on the formalism employed. Spontaneous Raman linewidths are a measure of molecular processes on the picosecond time scale. This technique is simple and has the potential of
yielding considerable information about molecular dynamics provided that the isotropic and anisotropic spectra can be analyzed. The method of analysis [2] involves the Fourier transform of the spectral band shapes to obtain vibrational and reorientational timecorrelation functions. In this letter we discuss the mechanism of vibrational dephasing of an asymmetrictop molecule, PHD2, isolated as an impurity molecule in the liquid and plastic crystalline phases of PD 3. Nice features of this system are that the P - H stretching band is not overlapped by any bands of PHD 2 or PD 3 and that resonant vibrational energy exchange interactions are minimized.
2. Experimental methods The experimental techniques employed have been discussed in detail elsewhere [3]. Therefore, only a brief description is given here. Spontaneous Raman scattering experiments were done using a Jarrell-Ash double monochromator capable of a resolution of better than 1 c m - l . The source was a Coherent Radiation model 54 argon-ion laser. The signal was analyzed using photon-counting electronics. The PHD 2 sample was condensed into an 8 mm o.d. Pyrex tube mounted on the tail section of an Air Products Heli-Tran refrigerator. Right-angle scattering was observed with an S-5 response PMT. The PHD 2 was present in PD 3 555
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
to the extent of 1 2 - 1 5 tool %. The preparation of the sample has been described previously [4].
15 November 1979
~°I. *
PHD 2
08
T=153
K
* "+
3. Method of analysis
06
The isotropic part of the 2313 c m - 1 P - H stretching band of PHD 2 was Fourier transformed to provide the vibrational time-correlation function; This function was then modeled [1 ] beginning with a "frozenlattice" gaussian function which reflects the inhomogeneous broadening at short times. Then successively higher-order frozen-lattice memory functions were calculated via coupled Volterra equations. In the present case the memory functions were truncated at the second-order memory function K2FL(t), and the frozenlattice memory function was related to the actual second-order memory function by the relation
/%(0 -- 2K
L(t)e
,
(1)
where n 2 is related to the second and fourth spectral moments, M 2 and M 4, of the isotropic Raman band, and r is an adjustable parameter. The Volterra equations are then used in a reverse manner to give the correlation function C(t) of the real system. The parameter K2 is related to the speed of modulation through the relation K 2 : ½r/+
I,
The correlation functions and normalized firstorder memory function are shown in fig. 1 for the liquid at 153 K. The correlation and memory functions of the liquid at 143 K just above the freezing point (f.p. 139 K), the plastic crystalline phase (phase I) at 130 K, and phase I at 95 K just above the I - I I transition temperature (T c = 93.5 K) are very similar to those of the liquid at 153 K. The correlation functions are not reliable beyond 6 - 7 ps. The wide slit width (2.0 cm - 1 ) necessary at 95 K makes this corre556
*+
÷.+
-+ o+l+
02
o
"*. o
+"%+e
o o
+°+e*°+o+o+~, ~ o
O.O
o o
o
÷°+°+*÷°+~.÷,+°+e÷ oooooo
°ooOOOooOooooooooo
o
o
-0.2
o % I
-04
2
I
I
3
4
I
5
I
6
7
t(ps)
Fig. 1. Experimental (o) and calculated (+) vibrational correlation functions and the calculated memory function (0) of PHD2 at 153 K in the liquid phase. lation function unreliable beyond about 5.5 ps. The resuits of the correlation-function modeling analysis are given in table 1. Before discussing these results, it should be pointed out that the intensity of the Raman Table 1 Summary of vibrational dephasing analysis Temperature (K)
Mexptl M~Xptl
(3)
4. Correlation and memory functions
o+
0.4
(2)
where ,/-1/2 < 1 for fast modulation and r/-1/2 >> 1 for slow modulation [5]. The speed of modulation is ! characterized by a correlation time r c such that 72= 1/M2(r'c) 2 .
.*
M2 (cm -2) •2 r (ps) M4 (cm -4) n r~ (ps)
rl-1/2
153
143
130
112 99053 110 3.3 1.0 91960 4.6 0.24 0.466
88 59056 85 3.0 0.9 50575 4.0 0.29 0.500
116 90 109320 61418 120 90 3.7 3.4 1.2 0.6 120960 63180 5.4 4.8 0.21 0.26 0.430 0.456
gaussian a) fwhh (cm -1) 24.7 fwhh (cm- 1) fwhh (cm -1) 10.0 exptl c) fwhh (cm -1 ) 6.4
95
21.7
25.8
22.3
9.3
9.5
8.8
6.5
5.5
6.6
a) 2(2 In 2)1]2M1]2. b) 41rcM2./o c) No correction for finite slit width.
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
scattering by the P - H stretching mode of PHD 2 is necessarily weak because the PHD 2 is present at small concentration. This is a disadvantage because the spectrometer slit width must be fairly wide (1.5 cm -1 for the liquid and 1.7-2.0 cm -1 for phase I) and the background noise is high due to the increased gain of the photon counter. The background noise level was about 3% compared with 0.3% for the PD 3 bands [3]. This gives almost twice the experimental errors reported in ref. [3]. Thus, for example, the errors in the second moments reported in table 1 are about -+20% with a truncation error on the band wings of -+6 cm -1 . All correlation functions were corrected for finite slit width. The data of table 1 indicate that the second moments of the liquid and solid are on the order of I00 cm - 2 and that the vibrational modulation is fast. Upon comparing the spectra of PHD 2 with that of PD 3 previously reported [3], one is struck by the much broader bandwidth of PHD 2. This leads to faster relaxingcorrelation and memory functions as reflected t ! in the r c values of table 1. The r c values are very close to the relaxation times of the first-order memory functions. Apparently, the PHD 2 molecule in the PHD 2 PD 3 mixture is relaxing much faster than either the PD 3 or PH 3 molecules [6]. As pointed out in refs. [3, 6], the evidence suggests that the rotational motions dominate the dephasing process. In the present case, however, the modulation is faster than one would expect from rotations alone, since the PH 3 molecule has the smallest moments of inertia of the three molecules. The fairly large percentage of 1H ( 4 - 5 % ) in the sample undoubtedly leads to some band broadening through interaction between the 1H atoms. Part of this is an inhomogeneous effect which is reflected in the second moment. Resonant vibrational energy exchange is negligible in the cases [3,6] of PD 3 and PH3, and we see no reason for this effect to be of importance in an isotopically dilute sample. In the case of the ordered solid it was concluded [4] that the 1 H -
15 November 1979
1H interaction was negligible. Therefore, we do not believe that P H D 2 - P H D 2 interactions are an important source of band broadening. There is another effect which is present in PHD 2 but not in the symmetric-top molecules that must be considered in order to explain the fast relaxation. This is the r o t a t i o n translation coupling that arises because the center of interaction and the center of mass do not coincide [7]. This coupling allows the translational as well as the rotational degrees of freedom to be effective in modulating the vibrational frequency. It appears, therefore, that the rotation-translation coupling must dominate the dephasing process in the case of PHD 2. The fact that it was not necessary to include higher-order memory functions in the analysis suggests that this is the only mechanism effectively modulating the vibrational frequency.
Acknowledgement This research was supported b y a grant from the Robert A. Welch Foundation. Computer time was provided by Texas Tech University.
References [1] S.S. Cohen and R.E. Wilde, J. Chem. Phys. 68 (1978) 1138. [2] L.A. Nafie and W.L. Peticolas, J. Chem. Phys. 57 (1972) 3145. [3] R.E. Wilde and S.S. Cohen, J. Chem. Phys. 70 (1979)4557. [4] R.E. Wilde and B.C. Covington, J. Phys. Chem. Solids 36 (1975) 1225. [5] R. Kubo, in: Fluctuation, relaxation and resonance in magnetic systems, ed. D. ter Haar (Oliver and Boyd, Edinburgh, 1962) p. 23. [6] R.E. Wilde and T.-C. Chang, J. Chem. Phys., submitted for publication. [7] H. Friedmann and S. Kimel, J. Chem. Phys. 47 (1967) 3589.
557