J. Phys. Chem. 1984, 88, 5899-5902
5899
Nonllnear Electroacoustic Phenomena: Phonon Echo in &Tartaric Acid and Its Salts Jacek Swiatkiewiczt and Paras N. Prasad* Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214 (Received: June 21, 1984)
The phonon echo, also called polarization or electroacoustic echo, is investigated for d-tartaric acid and its diammonium salt as well as for the Rochelle salt by using the two-pulse technique. The dependence of the echo amplitude on the delay time is described by both the anharmonic oscillator model and the model of coupled micromotions of the grains recently proposed by Pouget and Maugin. The echo under the experimental conditions behaves according to the small signal limit. The dependence of the damping factor on the pressure of the surrounding gas and the temperature is used respectively to measure an average speed of sound and to derive the mechanism of damping. Under vacuum, the damping factor is found to depend on the square of the radio frequency (rf) as is expected from an intrinsic loss mechanism. A plot of the temperature dependence of the damping factor for d-tartaric acid exhibits a maximum. This maximum is explained by a thermally activated acoustic relaxation within the powder particles. The observed deuteration effect both on the shape and the position of the maximum points out the importance of hydrogen motions for the acoustic relaxation in these hydrogen-bonded solids.
Introduction The area of nonlinear interaction in organic solids is highly active. The investigation in the past has focused on nonlinear optical effects such as second harmonic generation. The increased interest stems from the recognition that organic solids, rich in ?r electrons, possess large nonlinear electronic susceptibilities. Although electrostrictive and elasticity tensors are known for many of organic materials, the study of nonlinear interactions in the radio-microwave-frequency range has not been reported so far. Such studies are highly relevant in order to determine any potential application of organic solids as phase shifters or nonlinear mixing devices in the radio-microwave region. One manifestation of the nonlinear electroacoustic interactions in piezoelectric solids is the phonon echo phenomenon. The phonon echo, also called polarization or electroacoustic echo, is a coherent rf pulse emitted at time delay 27, as a response of the system to the pulses of electromagnetic radiation applied at times 0 and T . One or both of the rf pulses applied may be substituted by an acoustic pulse of appropriate frequency. Since the phonon echo cannot be produced by the linear response alone, it is a representative of nonlinear electroacoustic interactions. The phonon echo can be observed in both single crystals and powdery materials. It has been extensively investigated for inorganic piezoelectric and ferroelectric systems. A comprehensive review of the literature can be found in ref 1 and 2. The investigation of the echo amplitude dependence on the pulse time delay has established separate models for single crystal and powder samples: the parametric field-mode interaction model for the former and the anharmonic oscillator model for the latter kind of sample. These two models have been thoroughly discussed by Fossheim and H o l t 2 Although the echo phenomenon involves nonlinear effects, any estimation of the respective parameters from the experimental data can be made only after a proper separation of the time-dependent relaxation terms. Therefore, a better understanding of the acoustic relaxation process in an organic piezoelectric powder plays a key role in the study of nonlinear electroacoustic effects in organic solids. Recently, we reported the observation of the phonon echo in an organic solid, specifically the d-tartaric acid powdere3 In the present paper, an extensive study of the phonon echo in the powders of d-tartaric acid and its diammonium salt as well as the Rochelle salt is presented. The emphasis of this work is on the interpretation of the echo relaxation time (damping factor). The effect of temperature and pressure of the surrounding gas medium on the time evolution of the echo amplitude is investigated to derive information on the acoustic properties of the material. ‘On leave from the Institute of Organic and Physical Chemistry, Technical University of Wroclaw, Poland.
0022-3654/84/2088-5899$01.50/0
Experimental Section Reagent grade d-tartaric acid and its diammonium salt were crystallized from water and dried. Deuterated d-tartaric acid crystals were prepared by repeated crystallization from D20. The Rochelle salt was made by adding equimolar amounts of KOH and NaOH, in the proper stoichiometric ratio, to a d-tartaric acid solution in water. Dry crystals were ground in a mortar and sieved in order to separate the appropriate size particles (75-150 Mm), The powders underwent an etching procedure which involved a mild sonification of the powder in a solvent. During this treatment, smaller particles which adhere to the major crop of the microcrystals were separated out. Simultaneous etching and recrystallization remove some surface damage induced by grinding the crystals into the powder form. The best results were obtained by using a mixture of methanol and hexane as the solvent for dtartaric acid. But pure methanol was found to be more effective for the etching of the salt powders. Dried powders were sieved again to thoroughly separate particles smaller than 75 bm. A weighted amount of the powder, which approximately contained 1O6 particles, was placed between the disks of the capacitor. The sample cell was evacuated for several hours at room temperature. The measurements were performed under vacuum or in a cell filled with an inert gas. The spectrometer used has been described previ~usly.~In experiments reported here, we used two rf pulses of equal amplitude, identical frequency, and constant pulse width (2 11s). Special attention was paid to minimize the signal distortion in the external circuitry. A phase-sensitive detection was used. The preamplifier stage was eliminated for this study. The signal amplitude was kept constant by a scaled attenuator. Therefore, no saturation effect of the receiver was observed. The capacitor containing the sample powder was mounted in a glass container, which allowed us to evacuate the system prior to the measurements. The glass container was immersed in a thermostat bath. The CLTS-2 temperature sensor was located a t the opposite side of the grounded electrode of the capacitor. This arrangement minimizes any temperature lag between the powder and the temperature probe during the cooling and heating cycles. Results and Discussion Time Dependence of the Echo Amplitude. The observed dependence of the echo amplitude on the variable delay T between the two applied rf pulses is shown in Figure 1 for d-tartaric acid, its diammonium salt, and the Rochelle salt. In each case, a fast ~
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(1) Pouget, J.; Maugin, G. A. J. Acoust. SOC.Am. 1983, 74, 941. (2) Fossheim, K.; Holt, R. M. ’Physical Acoustics”; Mason, W. P., Thurston, R. N., Eds.; Academic Press: New York, 1982; Vol. 16. (3) Swiatkiewicz, J.; Talapatra, G. B.; Kurland, R. J.; Prasad, P. N. J. Chem. Phys. 1983, 78, 7500.
0 1984 American Chemical Society
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The Journal of Physical Chemistry, Vol. 88, No. 24, 1984 21
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2-r [PSI Figure 1. Time evolution of the phonon echo amplitude for d-tartaric acid (top), the Rochelle salt (middle), and d-diammonium tartarate (bottom), recorded at 77 K.
buildup followed by a relatively longer decay is observed. The decay is almost linear on the semilogarithmic plot. The two models, widely used to interpret the phonon echo behavior, are (a) the field-mode intraction model and (b) the anharmonic oscillator model? In the field-mode interaction, the echo is generated by parametric mixing of the propagating sound wave (generated by the first pulse) with the electric rf field imposed during the second pulse. This model, which has been shown to be applicable to single crystals, predicts a monotonic exponential decay of the echo amplitude. In the anharmonic oscillator model, the first pulse generates the acoustic oscillations of a standing wave nature in small particles. These vibrations undergo dephasing during the time 0 to 7. The second pulse also generates another set of acoustic oscillations which anharmonically interacts with the acoustic oscillations produced by the first pulse. A coherent rephasing starts at time 7, resulting in an echo emitted at time 27. The echo amplitude first builds up to a maximum before its decay is observed. Clearly, our result relates to the anharmonic oscillator model because a buildup of the echo amplitude is initially observed. Recently, Pouget and Maugin1a4 have proposed a different theory of polarization echo in piezoelectric powders. This theory involves a coupling between the piezoelectric oscillations of the powder particles with their librational oscillations. This model, like the anharmonic oscillator model, predicts an initial buildup of the echo amplitude followed by an exponential decay for longer times. The expression derived for the echo amplitude e(2r) can be cast, for both models, in a similar form: e(27) = KE3 exp(-r12r)(1 - exp(-r2(2r - 2A)))
the damping of the piezoelectric oscillations of the powder particles, while r2 refers to the damping of their librational oscillations (microgyration). The applicability of eq 1 also requires that the experimental conditions fulfill the small signal limit; i.e., the rf pulses are not strong enough to drive higher order nonlinearities. The most common criterion for the small signal limit is the power dependence of the echo signal. According to (l), the echo amplitude is expected to be proportional to E3. Indeed, we observed this relation for d-tartaric acid. However, for the two salts, this relationship does not hold for the entire range of applied voltages. Another manifestation of the small signal limit is that the damping factors are not dependent on the rf power. This is found to be the case for all three powders (d-tartaric acid and two salts) for the applied rf voltage in the range 50-800 V. In a previous study3 we had reported that, for d-tartaric acid, the relaxation time was dependent on the rf power. This particular experiment was conducted on a powder that was not etched before experiment. Since then, we have found the relaxation time (reciprocal of the damping factor) to be highly dependent on the sample preparation, being almost an order of magnitude longer for properly etched materials. In the case of untreated powders, we frequently observed that the echo decay was dependent on the rf power used, but no such dependence was noticed for etched samples. Next, we analyze the time dependence of the echo amplitude in order to evaluate the relaxation times (damping factors). The minimum number of parameters in eq 1 needed to fit the observed decay is two: the preexponential factor and a common damping factor I' = rl = r,. This assumption refers to the anharmonic oscillator model. This assumption, however, did not provide any acceptable fit to the observed data points. A significant improvement was achieved, by combining two expressions like eq 1 and, therefore, by using four independent parameters. A similar assumption was made by Fossheim et al. for fitting the data on the AsGa p ~ w d e r . ~In the anharmonic oscillator model, these two damping factors, estimated by the fitting procedure, may represent two effective limits of the distribution of damping factors for independent oscillators (particles). On the other hand, one can fit the data points equally well, if the parameters rl and r2 in one expression, like eq 1, are allowed to be independent. This situation refers to the theory of Pouget and Maugin. The curves drawn in Figure 1 were calculated by using three independently adjustable parameters in eq 1. For this model, the best fit was obtained with I'l and I', as 4.35 X lo3 and 16.7 X lo3 s-l for d-tartaric acid, 7.69 X lo3 and 8.93 X lo3 s-l for the Rochelle salt, and 2.08 X lo3and 8.3 X lo3 s-l for d-diammonium tartarate. Because both fit procedures provide acceptable set of two damping factors, it is not possible at this moment to conclude the discussion in favor of any of the two models. Acoustic Parameters and Damping Factors. The damping factor, which determines the slow decay, refers to the rate of energy loss by the oscillating modes of the particle. There are two main channels of energy loss: the loss due to intrinsic process and the loss due to energy exchange between the particles and the surrounding gaseous medium. According to Fossheim et al.? the external and the internal energy loss components are additive. If the interaction between the particles is neglected, the effective damping factor can be expressed as
(1)
This expression is applicable when the applied rf amplitudes El and E , are the same, i.e. E , = E , = E , and the respective pulse widths A1 = A, = A. The term K represents, collectively, various time-independent constants, The terms rl and are damping factors. The two models differ in the physical interpretation of these damping factors. In the anharmonic oscillator model I'l = I', = I' which is the damping factor for the oscillation of individual particles. In the theory of Pouget and Maugin, these parameters are different in their physical meaning. describes (4) Pouget, J.; Maugin, G. A. J. Acous?. SOC.Am. 1983, 74, 925.
Swiatkiewicz and Prasad
In the above equation, w is the frequency of the rf pulses and Zg and Z, are the acoustic impedances of the surrounding gas and the solid particles, respectively. Both Z, and the constant A describe the acoustic properties of the particles. The average acoustic impedance of the particles was determined by measuring the damping factor at different pressures of the surrounding gas (N,) but keeping w the same for each measurement. The result of this study on d-tartaric acid is shown in ( 5 ) Fossheim, K.; Kajimura, K.; Kazyaka, T. G.; Melcher, R. L.;Shiren, N. S.Phys. Rev. B Solid State 1978, 17, 964.
Phonon Echo in d-Tartaric Acid and Its Salts
The Journal of Physical Chemistry, Vol. 88, No. 24, 1984 5901
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Figure 2. The effective damping factor plotted as a function of the nitrogen gas pressure in the sample cell.
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T [KI Figure 4. Temperature dependence of the damping factor as measured from exponential time decay of the phonon echo amplitude: top, nondeuterated normal d-tartaric acid; bottom, deuterated d-tartaric acid.
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Figure 3. The damping factor plotted as a function of frequency. ,’I was chosen as a measured value for the lowest frequency wo = 16 MHz.
conditions, the effective damping parameter, evaluated from the slow decay of the echo amplitude, is a measure of the energy loss rate due to intraparticle processes. Therefore, we can relate the observed temperature dependence of the effective damping factor to the thermal behavior of the intrinsic dissipation process as it occurs in a crystalline material. The temperature dependence of the damping factor, I?, for the deuterated and nondeuterated d-tartaric acids is shown in Figure 4. For each system (protonated or deuterated acid), the temperature dependence curve exhibits a broad maximum. Since the general shape of the observed maximum is reproducible through the reversible heating and cooling cycles, it cannot be associated with the kinetics of thermal changes. No structural phase transition has been reported for d-tartaric acid. We, therefore, rule out the possibility of the observed maximum to be associated with any acoustic anomaly driven by a structural transformation. The exponential decay of the echo amplitude has been noticed in experiments performed in the entire temperature range, including the temperature at which the peak of the damping parameter is observed. The observed peak may be explained by assuming that the powder sample behaves as a heterogeneous system with different subgroups of particles exhibiting different thermal damping characteristics. In such a case, the effective damping will be determined by a superposition of several expressions of form (1). However, this superposition predicts an overall nonexponential decay of the echo amplitude, which is contrary to what is observed. The parameter r can, therefore, be referred to the attenuation of the acoustic waves in the particles, as it is normally done for crystals.2 The acoustic attentuation, caused by internal friction loss, is described in terms of various relaxation processes. The usual relaxation equation yields the following form for the damping parameter r:
Figure 2. The term Z,, calculated from the slope, is -2.7 X lo5 g om-* sd. The average speed of sound, given by Z,lp with p being the density of the crystal, was found to vary in the range 1.6 X 105-0.9 X lo5 cm s-l for samples prepared differently. In comparison, the speed of sound, calculated from the density and the compliance tensor data6 of d-tartaric acid, is of the same order of magnitude, but 2-3 times larger. Under vacuum, the second term of eq 2 is dominant. Therefore, a quadratic dependence of r on w is predicted by eq 2. This dependence was investigated for d-tartaric acid. The result is shown on a log-log plot in Figure 3. It can be seen that the slope is -2, which supports the validity of eq 2. The measurement of the damping factor under vacuum provides an estimate for the intrinsic loss. This value is equivalent to the one determined from Figure 2 when the line is extrapolated to p = 0. When the sample chamber was filled with helium gas under low pressure ( lo4 Pa), the measured damping factor was about the same as that measured under vacuum. Therefore, the damping factor in the presence of the helium gas is primarily derived from the internal loss mechanism. This method has proved useful when collecting data on the temperature dependence of the damping term due to the internal loss. The helium gas at low pressure helped maintain the thermal equilibrium in the sample cell. This study is described in the next subsection. Temperature Dependence of the Damping Factor. It has been discussed in the previous section that, under properly chosen
The parameter 7,is a correlation time or, in classical approach, a phenomenological relaxation time. The physical meaning of this parameter is specified according to the relaxation mechanism chosen. The term C contains various acoustic constants. There are several relaxation mechanisms which result in the absorption and dispersion of ultrasonic waves in solids. The effect of the impurities and the attenuation due to phonon-phonon interactions have been discussed by Mason.’ His phonon-phonon interaction theory, based on the Akhieser effect, has recently been used for
(6) Kitaigorodsky, A. I. “Molecular Crystals and Molecules”; Loebl, E. M., Ed.; Academic Press: New York, 1973.
(7) Mason, W. P. “Physical Acoustics”; Mason, W. P., Ed.; Academic Press: New York, 1966; Vol. 3B.
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The Journal of Physical Chemistry, Vol. 88, No. 24, 1984
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Figure 5. Temperature dependence of the damping factor of the Phonon echo amplitude: above, Rochelle salt; below, d-diammonium tartarate.
the temperature study of the nonlinear parameters in pure silicon crystals.* Since many acoustic parameters needed to fit this model are not known for d-tartaric acid, we are not able to compare our result with this theory. Another relaxation mechanism involves thermally activated processes. The relaxation time (correlation time) satisfies an Arrhenius-type equation: T~ = T , , ~exp(U/kT). The parameter Uis an activation energy for the relaxation process. This equation is used for different kinds of thermally activated processes, such as diffusion or energy exchange between two states. Thermally activated relaxation has been reported by Smolenskii et. a1 in the phonon echo and in the N Q R studies on the bismuth germanate powder at low t e m p e r a t ~ r e . ~ J ~ We have used the thermally activated relaxation mechanism in an attempt to explain our results. In order to fit our data, it was found necessary to use an appropriately weighted sum of two expressions like eq 3. One of these described the slowly rising background in the plot of the damping factor vs. temperature. The other term described the observed peak in this plot. Here we discuss, in detail, the parameters which are related to this second term (the one describing the peak). The correlation times, T ~ found by the fitting of the data of Figure 4, are rC= 2 X lo-” exp(2600/T) for the protonated acid and T, = 7.5 X lo-” exp(990/T) for the deuterated acid. Therefore; our result indicates the following pronounced deuteration effects: (a) the activation energy, obtained by the fit, decreases by more than 2 times (which (8) Rajagopalan, S.;Joharapurkar, D. N.; Shende, P. R. J . Appl. Phys. 1984, 55, 275. (9) Smolenskii, G. A,; Krainik, N. N.; Baisa, D. F.; Tarakanov, E. A.; Skorbun, A. D.; Popov, S.N. Sou. Phys.-Solid Srate (Engl. Transl.) 1980, 22, 853. (10) Smolenskii, G . A.;Krainik, N. N.; Tarakanov, E. A,; Grekhova, T. I.; Popov, S.N. Sou. Phys.-Solid Stare (Engl. Transl.) 1982, 24, 1377.
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Swiatkiewicz and Prasad is observed as a broadening of the peak in Figure 4), and (b) the peak shifts by almost 30 K toward lower temperature in going from the protonated acid to the deuterated acid. The observed isotopic effect strongly suggests that proton motions must play a significant role in the acoustic attenuation mechanism. Although, deuteration does not cause any substantial change in the crystal structure, its influence on the dynamics of the hydrogen bonds is paramount.” The hydrogen bonds in d-tartaric acid make a three-dimensional network which can be expected to play an important role in determining the elastic properties of the crystal. we infer that the acoustic relaxation process observed in d-tartaric acid involves flutuations related to the hydrogen bond. The influence of the molecular and crystal structure on the temperature characteristics of the damping factor is even more evident when the result obtained for d-tartaric acid is compared with those for its salts. The observed temperature dependence of the damping constant for d-diammonium tartarate and for the Rochelle salt is shown in Figure 5. In contrast to curves reported for d-tartaric acid, no maximum is observed for the salts. The main structural differences lie in the molecular and ionic type of structures for the acid and the salts, respectively. Moreover, despite the fact that the structure for the tartaric acid molecule and that for the tartrate ions are geometrically similar,** the hydrogen bond network in these systems differ substantially. These circumstances may not only have a quantitative influence on parameters of a given model, but such structural modifications may also change the predominant mechanism of acoustic attenuation. In conclusion, the phonon echo experiments on powders of the molecular systems appear to be a sensitive probe for molecular and crystal structure. However, in order for this technique to be used as a probe, more studies of molecular systems are needed to correlate the structure with the acoustic attenuation observed by the phonon echo technique. Acknowledgment. This work was supported in part (P.N.P.) by Air Force Office of Scientific Research Grant No. AFOSR82-01 18 and in part (J.S.) by N S F Grant No. DMR-818298. Registry No. &Tartaric Acid, 87-69-4; diammonium d-tartarate, 3164-29-2; Rochelle salt, 304-59-6. (1 1) Hamilton, W. C.; Ibers, J. A. “Hydrogen Bonding in Solids”;W. A. Benjamin: New York, 1968. (12) Wyckoff, W. G . “Crystal Structures”;Interscience: New York, 1966; Vol. 5.
