Attenuation of longitudinal ultrasonic wave near the diffuse phase transition in CdF2

Solid State lonics 15 (1985)65 69 North-Holland, Amsterdam

ATTENUATION OF LONGITUDINAL ULTRASONIC WAVE NEAR THE DIFFUSE PHASE TRANSITION IN CdF 2 M.O. MANASREH and D.O. PEDERSON Physics Department, University of Arkansas, Fayetteville, AR 72701, USA Received 13 August 1984

The attenuation of a longitudinal ultrasonic wave propagating in the [Iii] direction in CdF 2 is studied as a function of temperature from 300K to 103OK. An ultrasonic attenuation peak has been observed for the first time near 983K. This peak is used to define the diffuse transition temperature (T = 983K) in CdF~ which is well below its melting temperature of 1372K. The Arrhenius activation C . . energy of anlon mo~1on above T was obtained from the temperature dependence of the attenuation and c the theory of the dynamics of the coupled crystalline-cage-charged-liquid fluctuations. The elastic constant, (CII+2C12+4C44)/3, measured simultaneously with the ultrasonic attenuation displays a large decrease near 983K in addition to the nearly linear decrease in the elastic constant with temperature.

i. INTRODUCTION When an ultrasonic wave propagates through any medium, its amplitude and velocity change because of interactions due to many mechanisms. These mechanisms may be divided into two classes: those mechanisms that are of physical interest having to do with the fundamental physical properties of the material, such as thermoelastic dislocation, conduction electron, and phononphonon interactions, and those apparent mechanisms that arise as a consequence of the method by which such measurements are made, such as diffraction, wedging, nonparallelism, transducer, and bond effects. Ultrasonic attenuation and velocity near critical points in superionic conductors are complicated and may arise from a variety of processes. Superionic conductors themselves are divided into several classes according to the behavior of their ionic conductivity with temperature. Several classification schemes have been proposed [1,2,3]. The scheme proposed by Boyce and Hubermann [3] divides superionic conductors into three classes. In class I materials, the ionic conductivity jumps abruptly from a low temperature insulating value to a high conductivity value. The change in ionic conductivity is accompanied by a structural change in the crystal lattice. Typical class I examples are AgI and RbAg. I. In class II q D" materials, the conductivity increases gradually with temperature to a high conductivity value which tends to saturate at higher temperatures. These materials are said to undergo a diffuse

transition which is also known as the Faraday transition. Alkaline earth fluorides and PbF~ are typical examples of class II materials. ~n these fluoride crystals, the fluorine sublattice becomes disordered at the transition temperature while the complementary cation sublattice distorts negligibly to first approximation. Thus, there is no structural change in these materials The ionic conductivity of a class III material follows the Arrhenius equation and increases smoothly as the temperature is increased. The absence of cooperative effects in these materials is inferred from the lack of anomalies in conductivity data. Sodium beta-alumina is an example of class III superionic conductors. The purpose of this paper is to report on ultrasonic attenuation and an elastic constant of a CdF 2 single crystal. CdF 2 has the fluorite structure, and the defect formation is comparable to that of CaF 2 [4]. The Faraday transition has not been observed previously in CdF 2. This, in part, is due to the reactive nature of this material with oxides especially at high temperatures. The elastic constants of alkaline earth fluorides and PbF~ show anomalous behavior near their Faraday transltlon temperature [5,6]. Ultrasonic attenuation peaks near the Faraday transition in class II materials have not been previously reported.

2. THEORY The elastic constant,C, of CdF 9 measured simultaneously with the ultrasoni~ attenuation

M.O. Manasreh, D.O. Pederson

66

/ Attenuation o f longitudinal ultrasonic wave

was derived [7] using the thermal expansion coefficient to correct the crystal specimen's density and length and given by the expression T C = (4 ~ono2/t2)exp( (1)

J~(T)dT)

where Po and L ° are the mass density and the length of the specimen, respectively, measured at temperature To, t is the pulse round-trip time, and ~ ( T ) is the thermal expansion coefficient of the specimen. For the case of CdF~, the density /0o was taken to be [8] 6.386xI03 kg/m 3 at T = 30OK, and the thermal expanslon coefflclent was taken to be [9] ~,(T) = 1.3x10-5+2.7x10-ST. As mentioned earlier, the ultrasonic attenuation arises from a variety of processes, and it is necessary to separate and identify the mechanisms that cause the ultrasonic attenuation The apparent attenuations due to the nonparallelism and bond effects [10], and the diffraction effect [11] were identified and subtracted from the measured ultrasonic attenuation. The attenuation of ultrasonic wave interacting with the thermal phonons is dominant well below the transition temperature. Woodruff and Ehrenreich [12] and Bommel and Dransfield [13] have used a statistical transport model, which is based on the work of Akhieser [14], to derive an expression for the ultrasonic attenuation due to the lattice vibrations. They concluded that this attenuation is independent of temperature above the Debye temperature of the material. In the vicinity of the transition temperature the ultrasonic attenuation due to mobile ions (F- in CdF2) is dominant [15]. Huberman and Martin [16] visualized the coupling of the ionic liquid and the crystalline cage phonons as an effective frequency and wavevector dependent modulation of the chemical potential felt by the ions. Huberman and Martin's model predicts that the effect of local site fluctuations on the phonon dynamics dominates over other fluctuations. For the weak coupling and low-frequency limit, i.e. u2[~<< 1, the ultrasonic attenuation due to mobile ions can be written as •

O

.

~(pC)I/2(AL/L)2~2/(R(T)nokT)

(2)

where p i s the mass density of the material, C is the elastic constant, ( ~ L / L ) is a fractional change in dimensions of a crystalline cell in the presence of a mobile i o n , ~ K is the angular frequency of the propagating acoustic wave, R(T) is defined as the relaxation rate (relaxation frequency) that in general has Arrhenius behavior for simple defect formation and reflects the coupling of the pseudospin to the thermal reservoir (crystalline cage), n is the number of carriers per unit cell, k is °the Boltzmann constant, and T is the temperature. The relaxation rate can be written as R(T) = i/'~ = Ro(T ) exp(-E/kT)

(3)

where ~ is the relaxation time, R (T) is the attempt relaxation rate (attempt frequency), o expected to be of the order of an optical phonon frequency, 1012 to 1013 sec -I, and E is the activation energy for the hopping of mobile ions between available sites.

3. EXPERIMENT A 0.62 cm long crystal was cut from a 4 cm long and I cm diameter crystal of CdF. obtained from Optovac, Inc. [17] whose axis wa~ determined by back-reflection Laue photography to be oriented II ° from the [iii] direction. For the ii ° misalignment, the relative error ~v/v in the longitudinal velocity is [i0] +0.017. The end faces of this sample were parallel to within 4 parts in 104 . The electronic equipment has been described elsewhere [7,18]. Measurements of round trip transit time of the ultrasonic pulse were made using the pulse superposition method [19,20] for the correct cycle-for-cycle overlap of two pulse echoes assuming the thin-bond limit. DuPont 9770 [21], a platinum and silver conductor composition ordinarily used to provide thick film conductors for microcircuits, has been previously used as an effective acoustic bond at high temperature for compressiona] [22] and shear [23] waves. The transducer used for the longitudinal ultrasonic measurements was a i0 MHz 36 ° y-cut lithium niobate transducer. The temperature was measured using a type-K chromelalumel thermocouple with an Omega-CJ coldjunction compensator and a Keithley 171 digital multimeter. A Lindberg Hevi-duty furnace was used in this work. The measurements were performed in a vacuum of approximately 4xlO -3 Pa (3xlO -5 Torr). The estimated accuracy for the measured elastic constant including misorientation is approximately ± 4%. A precision of 0.5 dB in the ultrasonic attenuation could be achieved. The accuracy of the ultrasonic attenuation is calculated to be + 0.23 dB/cm from the standard deviation of the data about a smoothly increasing function.

4. RESULTS AND DISCUSSION The temperature dependence of the elastic constant, P V L 2 = (CII+2C12+4C44)/3 , and the longitudinal ultrasonic attenuation of CdF~ is z given in Fig. I. The nearly linear behavior of the elastic constant below the transition temperature is characteristic of the anharmonic contribution to the elastic constant [24]. The sharp decrease of the elastic constant near the transition temperature of 983K is due to the formation of a relatively low concentration of interstitial defects [5] and the hopping motion of defects [6]. The behavior of the elastic

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Fig. I. Temperature dependence of longitudinal ultrasonic attenuation,c~, (crosses) of CdF 2 is meas2 sured simultaneously with P v L = C = (CII+2C12+4C44)/3 (diamonds) at 9 MHz. The solid line represents ~ = 26.076JTc-T[-0"579. constant near the transition temperature agrees well with the theoretical prediction. Elcome [25] and Kleppmann [26] utilized the ionic potential model to derive analytical expressions for the elastic constants of a perfect fluorite lattice and concluded that the changes in the Coulomb and short-range forces with temperature yield canceling contributions to the elastic constants C.^ and C._ but additive contribu• Ig 4~ tzons to the elastic constant CII. Table I shows a comparison beEween the present measurements of (CII+2C12+4C44)/3 and calculations from previous measurements of elastic constants at room temperature. The agreement between the present measurements and Pederson and Brewer [18], Alterovitz and Gerlich [8], and Hart [27] is within experimental error. The nonparallelism, diffraction, and bond effects were estimated and subtracted from the measured ultrasonic attenuation. The result as shown in Fig. 1 displays a small variation around 870K and a large variation (anomalous behavior) around 983K. The variation of the ultrasonic attenuation near 870K is not well understood. The ultrasonic attenuation peak and the sharp decrease in the elastic constant around 983K resemble a diffuse transition type anomaly similar to those observed in this labo-

ratory in other solid electrolyte fluorites [28]. This leads to the assertion that CdF 9 undergoes a diffuse transition (Faraday transition) at T = 983K. c The transition temperature, T = 983K, is • . C defzned as follows assummng a power law temperature dependence, O(,= AITc-T ~. The ultrasonic attenuation in Fig. 1 was separated into c~ L for T < T c and ~H for T > Tc, assuming that 978K < T < 987K. By varying T in C ~ C this range of temperatures keeping IT -T I > i, • C the standard error of overlappmng o(L and ~ H can be calculated. The correct overlap of ln(~) vs. lnlT -T I corresponds to the minimum standard C . . error for a szngle fzt to both ~ L and ~ H as shown in Fig. 2 for T = 983K. The criticalC

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Present Work (298K) Ref. [8] (295K) Ref. [18] (295K) Ref. [27] room temperature

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Fig. 2. Attenuation vs. for CdF 2 using the ultrasonic attenuation data of Fig. 1 below T = 983K (diamonds) and above T (crosses) C C is f i t by t h e s o l i d l i n e u s i n g a f i r s t o r d e r l e a s t squares fit with the slope representing the critical-point exponent.

M.O. Manasreh, D.O. Pederson

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Fig. 4. The diamonds represent the calculated values of R(T)/R (T) from ultrasonic attenua• u tion data of Fig. I above the transition temperature, T = 983K. The solid line is the first C

variation of the critical-point exponent from the least squares fit for the choices of T is also shown, c

order least squares fit of the data with the slope of the solid line representing the Arrheniu= activation energy of the anion motion.

point exponent was calculated during the overlapping procedure and plotted along with the standard error in Fig. 3. The minimum standard error fit for T c =983K is for ~ = 26.0761Tc-T1-0"579

but the exact mechanism of the anion motion is difficult to identify [28]. The ultrasonic behavior of CdF 9 is consistent with other fluorites in that in the [100] direction near T there exists a sharp increase in the C . . longltudlnal ultrasonic attenuation but no sharp increase in the shear ultrasonic attenuation [29]. In terms of fluctuations of the local site population, this indicates a coupling of of the ultrasound to ions which are redistributed among energetically inequivalent sites [16]. While this argues against pure vacancy motion, there is no evidence in these measurements or in the literature that the ion motion may not be more complicated resulting from the interstitialcy mechanism or defect interactions rather than simple interstitial motion. Additional measurements from other kinds of experiment will be useful in understanding the mechanism of ion motion above T .

(see the solid line in Fig. I). Excluding data greater than InaT -T I = 6 yields a different C minimum standard error fit of T = 984K and ~ = C 15.931Tc-Ti -0"456.'' While there is no experimental reason to exclude such data, the fit indicates the confidence that can be placed in the values for T and the critical-point exponent. The C . . CdF 2 crltlcal-poznt exponent of -0.58+--0.12 is much smaller than the critical-point exponent of -1.7 in the attenuation of a longitudinal acoustic wave propagating in the [I00] direction in RbAg4I 5 reported by Nagao and Kaneda [15]. The temperature variation of the relaxation rate, R(T), of CdF 2 was obtained by analyzing the ultrasonic attenuation measurements above the transition temperature, which is defined in this work as T = 983K. Fig. 4 shows the plot of the naturalClog of R(T)/R (T) vs. i/kT. The solid line in Fig. 4 is °the first order least squares fit of the data. Eq. (3) predicts that the slope of the solid line in Fig. 4 is the activation energy for the hopping of mobile ions between available sites. The activation energy obtained from Fig. 4 is 1.73 ± 0.21 eV. There are two contributions to the estimated error of the activation energy: first, the estimated accuracy obtained from the standard error (± 0.II eV) of the data in Fig. 4, and secondly, the precision of + 0.5 dB in the ultrasonic attenuation yields an estimated error of ± 0.i0 eV in the activation energy measurements. The activation energy is thought to be associated with the onset of the transition in temperature region IIIB of the conductivity data

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5. CONCLUSION The present study has provided new measurements of the adiabatic elastic constant, (CII+ 2C12+4C44)/3, as well as the longitudinal ~ ultrasonic attenuation in CdFp up to I030K. The behavior of both the elastic constant and the ultrasonic attenuation is identical to that observed in the ultrasonic measurements in PbF 9 and BaF 2 near their Faraday transition [5,29]. ~ This leads to the conclusion that CdF 2 undergoes a diffuse phase transition at 983K. The sharp ultrasonic attenuation peak, which is used to define the transition temperature of T = • . . C 983K, is associated only wlth ion motlon rather than in combination with a crystallographic phase transition as in RbAg4I 5 [15].

M.O. Manasreh, D.O. Pederson /Attenuation of longitudinal ultrasonic wave The temperature dependence of ultrasonic attenuation measurements in CdF~ yield a z critical-point exponent much smaller than the critical-point exponent of RbAg4I~ reported D in the literature. An Arrhenius activation energy for anion motion above T of 1.73 + .c 0.21 eV was obtained by analyzzng the ultrasonic attenuation measurements utilizing the theory of fluctuations of local site populations derived by Huberman and Martin [16]. Additional work is required to understand the mechanism of this anion motion.

[10] R. Truell C. Elbaum, and B. B. Chick, [II] [12] [13] [14] [15] [16]

ACKNOWLEDGEMENTS [17] The authors are grateful to S. M. Day for loaning part of the electronic equipment.

[18]

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