Static and Dynamic Behavior of Quartz Resonators

zyxwvutsr zyxwvu zyxwv zyxwvut zyxw IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-26, NO. 4, JULY 1979 299 sonified region, so as long asy, is less than the minitank width, there will be no reflection from the minitank walls. For the case of dissimilar minitank and main tank fluids, W is simplyreplacedby theterm 2 p C intheexpressionfor P U C , +PlCI. [ 21 P. Pille, “Real Time Liquid Surface Acoustic Holography,” Master’s Thesis, University of British Columbia (October 1972). (31 P. Pille, B. P. Hildebrand, “Rigorous Analysis of the Liquid-Surface Acoustical Holography System,”Acoustical Holography, Vol. 5, Plenum Press, New York,N.Y.(July1973),pp.335-371. [ 4 ] Z . A. Goldberg, “Acoustic Radiation Pressure,” High Intensity Ultrasonic Fields, Plenum Press, New York, N.Y. (1971). [S] I. A . Viktorov, Rayleigh and Lamb Waves, Plenum Press, New REFERENCES York, N.Y. (1967) Chapter 11. [ l ] P. S. Green, Lockhced Internal Report No. 6-77-67-42 (September [S] B. P. Hildebrand, B. B. Brendcn,An Introduction to Acoustical 1967). Holography, Plenum Press,141-142. New York,pp. N.Y. (1972), Static and Dynamic Behavior of Quartz Resonators zyxwvutsrqpo ARTHUR BALLATO, SENIOR MEMBER, IEEE Abstract-The frequency-temperature cf-T) behavior of a crystalrestors of special orientations. This development fortunately onator depends upon temperature,and its spatial and temporal gradients. coincides with stringent frequency control requirementsnewly f-T curve can beused to deterFor quasi-isothermal changes the static imposed by the latest generations of communication systems mine frequency shifts that occur,e.g., in oven-controlled units. The and systems for navigation and position location. frequency then depends uponthe parameters of the staticf-T curve, In order to identify an important present contribution to the temperature range overwhich the oven cycles, and upon the oven setting point. The maximum frequency excursion has been computed frequency instability and to separate out its effects, itis confor theA T and SC cuts of quartz in terms of these parameters as a funcvenient to discuss first the static frequency-temperature (f- T) tion of the orientation angle. When thermal-transient-compensated characteristic of a quartz resonator. Staticf-T curves are of cuts are not utilized, oven cyclings or other temperature perturbations introduce an additional nonnegligible component of the frequency shift. interest in their own right in connection with crystals operating without ovens over broad temperature ranges, such as in This effect is quantified by means of a simple mathematical model. The model is capable of predicting the thermal transient effects for AT temperature compensated crystal oscillators [2] . Maximum cuts appearing in the literature. Simulations, using the model, disclose frequency deviations, obtained from the static f-T curves, are that sinusoidal temperature variations with periods of hours can readily shown to be simply represented in normalized form for such lead to frequency instabilities much larger than would be expected usapplications. They are subsequently used for comparison with ing the staticf-T curve for theA T cut. This effect shouldbe greatly the dynamic behaviorof resonators. diminished in the vicinity of theSC cut. zyx zyxwvuts For crystal resonators subjected to wide temperature variations, the angle of cut is ordinarily chosen to minimize the frequency excursions over the entire temperature range. High INTRODUCTION precision oscillators, on the other hand,utilize resonators N THEIR sixty years quartz resonators have shown themmaintained within an oven system that closely regulates the selves capable of dramatic improvements in frequency stabilthermal environment. In this latter case it is important to ity: on the average, one order of magnitude per decade. Beknow how the unavoidable oven instabilities affect the resonacause of a long history compared with other solid-state devices, tor frequency and how the oven parameters interact with the it is easy to look uponthis component as the product of a maf-Tcurve as a function of the crystal cut. ture technology and to presume that theleveling-off plateau Use of the staticf-T curve in conjunction with the known has been reached; that the last bit of stability has been,or characteristics of commercial ovens leadsto the conclusion shortly is to be, wrung out of thevenerable vibrator. that frequency stabilities ordersof magnitude better than Such a conclusion would be entirely wrong. Prospects are those observed ought to be possible withA T-cut resonators. exceedingly good that the near-term future will see a further The discrepancy is due to theneglect of the dynamicf-T behundredfold improvement in the stability of high precision oshavior of the crystal. By making use of various types of dycillators, brought about by using doubly rotated quartz resonanamic f-T data from the literature, a phenomenological parameter can be extracted that satisfactorily explains the effects Manuscript received December 29,1978. This work is based on a found in practice. When applied to simulations of oven cypaper given at the 32nd Annual Frequency Control Symposium[ l ] . cling, it shows the dominance of dynamic effects in high stabilThe author is with the U.S. Army Electronics Technology and Deity situations whenA T cuts are used. vices Laboratory, USAERADCOM, Fort Monmouth, NJ 07703. I U.S. Government work not protected by US.copyright. zyx z IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-26, NO. 4, JULY 1979 300 t . zyxwvu OVEN CYCLING R A N G E , , zyxwvutsrqponmlkjihg 1 zyxwvutsr , l l l l E Y PlE A A I U Rl f l l l l l 1 Fig. 1. Definition of oven characteristics on static frequencytemperature resonator curve. Cycling range is symmetric about setting point. p 2 'OVEN OFFSET OVEN CYCLE RkNGE Ae MINUTES OF ARC zyxwvu Fig. 2. Normalized frequency excursion versus orientation angle difference for A T-cut quartz as function of oven parameters. Abscissa reference angle is that for which the slope at the inflection point of the static frequency-temperature curve is zero. Angle varies somewhat with resonator design and construction. At the SC cut the parameter linking dynamic thermalbehavior to frequency becomes zero, and for this doubly rotated quartz cut thef-T behavior predicted from the static curveis capable of being realized. In the first part of this paper the staticf - T behavior of A T and SC cuts is examined. The dynamic behavior of resonators is considered in the second portion. STATICBEHAVIOR For most types of quartz resonators the frequencytemperature behavior takes the form of the cubic curve depicted in Fig. l . When the resonator is operated in an oven there will be, in general, an oven offset error A that measures the amount by which the oven setting point misses the desired . reference point where the first-order temperature coefficient vanishes. In addition, the oven will have a cycling range &AT, about the setting point, asseen in Fig. 1. As the oven cycles, the frequency will vary, but in the case shown the maximum frequency excursion will be from the reference point to the frequency at the point A + AT,. As A and AT, are allowed to vary, the maximum frequency excursionwill of course vary and will be a function of A , A T , ,and the shapeof the cubic curve. The cubic curve varies with the angle of the cut. For the ATand SCcutsof orientations (Yx7)O = 35.2", (YXwZ)@ 21.9"/6' 2: 33.9", the reference angle is taken as that for which turnover temperatures coincide with the inflection temperature. In terms of deviations A0 from this angle, the maximum frequency excursions are shown for AT-cut quartz inFig. 2. zyx zyx zyxwvuts zyxwvutsrqp zyxwv zyxwvutsrqponmlkjihgfed BALLATO: BEHAVIOR OF QUARTZ RESONATORS 301 TABLE I FREQUENCY D E V I A I - I O N S FOR AT-CUTQUARTZ, W I T H AB = 5' A N D TRANSIENT EFFECTS OMITTED I I zyxwvutsr zyxwvutsrqponmlkj SC CUI W A R T 2 A B - 5' (K1 O CV YE C N L RI N A G N G E zyxwvuts zyxw 0 1 T 1.Ex10-8 t 0.01 4.6~10-g 0.001 2 . OVEN OFFSET a 4 . 6 ~ 1 0 - ~ ~ 4.6~10-l~ p -- OVEN C Y C L E RANGE l =la I C t zyxwvutsrq 30 1 IC 5 "0 I I5 I l 20 25 30 A 0 MINUlES O F kRC I Fig. 3. Normalized frequency excursion versus orientation angle difference for SC-cut quartz as function of oven parameters. Curves are weak functions of azimuthal angle 0. TABLE 11 FREQUENCY DEVIATIONS FOR AT-CUTQUARTZ, WITH A8 = 0.5' AND TRANSIENT EFFECTSOMITTED ... __. . . ne = __ O C V Y E N C L RI N A G N G E .5' TABLE 111 FREQUENCY DEVIATIONS FOR SC-CUI QUARTZ, W I T H AB = 5' A@= ( K ) (K) O CV YE C N L RI N A G N G E 5' 0.1 j 0.001 0.01 2.1x10-10 Z,1X10_9 2.1x10-11 *,lxlol, tW (0 H I I I I l I I I L L 0 0 II t z W > 0 zyxwvu zyxwvutsrqpo l TABLE IV FREQUENCY DEVIA-~IONS FOR SC-CUTQUARTZ, W I T H A@ = 5' The angle difference is in minutes of arc, and the ordinateis the quantity ___.____- A$= IAf/fIrnaxI{(ATo)' . (1 Rho is defined as + IPI>'}. p = A/ATo = 2 . offset/cycle range. For IpI 1, a single curve suffices. The curves imated by the relation 0 (K, + K 2 A@). (1) /l 5' ( K ) O CV YE CN L RI A N N GG E ' 0 1 I 0.001 0.01 I I (2) may be approx- (31 where 0 is the ordinate expressed in (1) and K , , K , are simple functions of IpI for a given cut. Tables I and I1 provide values of IAf/fl,,, for AT-cut quartz for AO = 5 and 0.5 minutes of arc, respectively. The doubly rotatedSC cut of orientation (YXwl)@= 21.9"/0 +33.9' has two angles to vary. Fig. 3 gives the variation in 0 versus AO. Tables I11 and IV present values of lAf/flrnaXfor A@and A@= 5', respectively, while Tables V and VI are for A0 and A@= 0.5', alsorespectively.This With reference t o Table 11, one sees that an A T cut operated in a good oven with a stability of a few millikelvins should produce frequency stabilities in the order to in fact is farfromtheactual values obtained. zyxwvutsr zyxw zyxwvutsr zyxwvut zyxwvutsrqponmlkjihgfedc IEEE TRANSACTIONS ON 302 TABLE V FREQUENCY ___ DEVIATIONS FOR SC-CUTQUARTZ, .___ A 9 = .5' O V E N WITH M 3 = 0.5' ~~ C Y C L I N G R A N G E ( K ) SONICS AND ULTRASONICS, VOL. SU-26, NO. 4, JULY 1979 lem. The traditional mannerof expressing the static frequencytemperature behavior of quartz resonators is due to Bechmann [28] : zyxw Af/f = a, AT + bo ATZ + c o AT3. (4) In (4) AT = T - T o ,where Tis the present temperature and To is the specified reference temperature, often taken as 25°C. The parameters a,, b o ,and c. depend on material, cut,geometry, electroding, etc., but do not depend on time, for a given resonator design. Our assumption consists of the inclusion of the term a". T in the expression for the effective firsta(t), which now dependson time order temperature coefficient implicitly through the temporal behavior of the temperature: c L Af(t)/f = a ( t ) . AT([) + bo . AT(t)' + c o . AT(t)3,(5) zyxwvuts with FREQUENCY TABLE VI DEVIATIOM FOR SC-CUTQUARTZ, a ( t ) = a. WITH ~. A$=.5' A+ = 0.5' (K1 OC VY EC N L RI N AG N G E f 0 001 1.7~10-~~ (6) The quantity C? is a function of the same entities as a o , b o , and co, as well as several others such as the method of setting up thermal gradients (thickness gradient, azimuth-dependent lateral gradients), thermal conductivity of supports and electrodes, etc., but is a constant for a given design. It is treated simply as a phenomenological quantity to be evaluated separately for each crystal design. It is obvious that the assumption concerning a ( t ) could be extended to the coefficients bo and c. by introducing similar constants b" and F ; the results are qualitatively the same, and theywill be omitted here. zyxwvutsrqpo 2.cx10-13 2.cx10-;4 + Z . T(t). 2.~0-15 5.C ~ 1 0 - l ~ L DYNAMIC BEHAVIOR That the resonance frequency of a crystal vibrator depends on temperature-rate and gradient effects, aswell as temperature itself, has been known for some time [3] -[26] . Experimental evidence exists in a number of forms. Bistline [ 131 showed that thef-T cubic curve appeared to exhibit cutangles differing in A0 by several minutes of arc for different temperature sweep rates. Kusters [20] showed how the upper turning point of an A T c u tshifts in frequency with temperature rate. In both cases the temperature-time graphs were ramps, so that T = dT/dt is a constant. The influence of a "step function" of temperature on ATcutswas shown by Warner (71, [ l l ] and Munn [14]. For BT-cut and rotated-X-cut crystals [27], the frequency spike is reversed in sign. Based upon the observed effects, a number of modelshave been used to characterize the thermal transient, or dynamic thermal, behavior of quartz resonators [ 9 ] , [16], [19], [21], [24]. We propose here a simple model that explains the results of a variety of experiments. More careful experiments, coupled with a thorough analysis involving nonlinear elasticity theory, are required to provide a firm foundation for understanding dynamic effects in resonators. It is hoped that the results presented here will stimulate such an approach to the prob- PUBLISHED EXPERIMENTS For the case of a ramp temperature-time profile, where T i s constant, a ( t ) will reduce to a constant different froma,. The f-T curve simply appears asif it had changed its apparent orientation angle 0 . In the case of Bistline's graphs, the value of a" required toaccount for thedata is + l .7 X s/K2. This value is arrived at using the known quantity &/de = - 5.08 X 10-6/K, deg B for the ATcut [2], [23]. Applying the model to Kusters' curves [20] , straightforward algebra yields the relation for the normalized frequency shift of the turning point S=(T,-To).a".T, (7) where T, is the upper turnover temperature and To is the reference temperature. Using the parameters of the AT-cut crystal discussed by Kusters [20] , the value of a", for the crystal used by him, is zyxwvu Z = -2.27 X IO-' s/KZ. (8) The Warner curves [7] , [ 1 l ] , [l41 may be simulated closely by making the assumption that the thermal "step function" is of the form (1 - e-'') and involves a single system thermal time constant r . The values of Z are larger in magnitude than for the Bistline and Kusters experiments. Thesign ofzchanges according to the electrode metallization pattern, being negative for thickness excitation and positive for electrodes with a gap for lateral excitation [ 1l ] . It may be expected that other factors will influence the sign and size of a" and that a proper zyxwvutsrqpon zyxwvut BALLATO: BEHAVIOR OF QUARTZ RESONATORS 303 choice of resonator and electrode design, electrode material(s), and method of depositionwill produce resonators with small a" values. The possibility of controllinga" by the use of doubly rotated plates is discussed below. The presence of a frequency spike, as foundby Warner in response to a thermal step function applied to an AT-cut bulk wave resonator, has been recentlyverified for surface acoustic wave (SAW) resonators of ST-cut orientation by Parker[29] . SINUSOIDAL TEMPERATURE VARIATION zyxwvut Based upon the simple model adopted it is possible to simulate the effect of sweeping the temperature in a sinusoidal manner. It is known that this leads experimentally to hysteresis effects. The result of a wide temperature range sweep is shown in Fig.4. The heavyline represents the static f-Tcurve. When the temperature is swept as sin wr by 60K about the 30°C point, the resulting curve exhibits an hysteresis thatvaries with the rate W as shown. The effect is more pronounced at the upper turning point (UTP); this is due to thesign of a". The orbits are counterclockwise,also because of the sign of a", at the UTP. The general features of the curves in Fig. 4, representing dynamic thermal cyclings, have been verified in the main by experiments performed by Hughes [30] using sinusoidal temperature cyclings. In particular, the much greater deviations at the UTP (implying a negativea" in his experiment), compared with the lower turning point, were unequivocally established for the resonator examined. Fig. 5 shows the hysteresis effects expected for k10K excursions about the UTP. For asweep with a 10-min period the frequency change is roughly three times that expected on the basis of the static curve.In Fig. 6 hysteresis orbitsfor ATo = 5K, offset from the UTP at T,, by +5K and by - 1 OK, are shown. The frequency scale is magnified in Fig. 7 to parts in lo-''. Fortemperaturedeviations AT, of k5 X K, thestatic f-T curve is nearly a horizontal line,while the orbits for the dynamic frequency behavior are elliptical, with amplitudes greatly exceeding the static behavior when the period of the temperature cycle is shorter than about 1 h. The constant a" from (8) has been used in the computations. In Fig. 8 the frequency scale has been further magnified to the IO-'' range, with a temperature scale in millikelvins about the UTP. Again we can see the dominance of the dynamicbehavior compared to the static,even when the temperature variations take place over periods of the order of a day. Table VI1 provides values for the fractional frequency deviadue to the dynamic thermal effect in AT-cut tion lAf/fl,,, quartz, as a function of both oven cycling range and cycling time, assuming sinusoidal temperature variation andthe a" determined from Kusters' experiment [20]. The oven offset has been taken as zero. The entriesare to be compared to those in Tables I and 11. The dramatic influence of the dynamic component of the frequency shift be canseenfrom the comparison. Gagnepain [31] has confirmed the hysteresis orbits shown, e.g., in Figs. 5-8, for cyclings in a small temperature range near a turnover point; heuses a model slightly different from that of ( 5 ) and (6), where the a" term is not multiplied byAT(r) in (5); this model has been used by Anderson and Merrill(9) and zyxw zyxwvuts DEGREESCELSIUS F i g . 4. Simulation of frequency hysteresisarising from sinusoidal temperature cycling. Sixty-kelvin sweep about inflection temperature. Orbital eriods are 628,1257,and 6283 S for W = lo-', 5 X lo-', and lo-?, respectively. I I I I I I AT CUT QUARTZ , / I l zyxwvutsrqpon zyxwvutsrqpo zyxwvut I 50 I I 60 I I 10 I I I Bo DEGREES CELSIUS Fig. 5. Simulation o f frequency hysteresis arising from sinusoidal temperature cycling. Ten kelvin sweep about upper turnover point. Counterclockwise orbits stem from sign of a". Koehler er al. [21] , [24]. For small temperature sweeps both models give equivalent results, but for wide sweeps( 5 ) predicts an apparent orientationangle shift for temperature ramp excitation, whereas the modified model predicts a constant frequency offset. Experimental results for ramp functions of temperature have been obtained byLagasse [32] ,whose curves show very interesting behavior. Data were supplied on two groups of A T-cut resonators having thicknesses very nearly in the ratio of 2 : 1 . For the thinner plates,60%showed only a constant frequency offset,while the remainder showed both an offset as well as a rotation in the apparent angle. The thicker plates showed the frequency offset alone in only10% of the cases, while 90%had both an offset and an apparent angle change. Each resonator in a group was of identical construction, but between the groups the designs were similarbut not identical. zyxw zy zy zyxwvutsrqponm IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-26,NO. 4, JULY 1979 304 l 1 I I I l I A T CUTQUARTZ / l I I I I AT CUT QUARTZ- I 1 zyxwvutsrqponmlkjihgfed zyxwvutsrqpon zyx l l l 1 I m 60 50 DEEREES CELSIUS l I W J Fig. 6. Hysteresis loops for sinusoidal temperature cycles offset from upper turnover point. Cycling range is five kelvins; offsets are -10 and +5 K . Orbital periods are 2094 S for outer loops and 6283 S for inner loops, corresponding to 1 2 W =3X I i? - - 2 27 1. IO" S/Kz -I x10-' and respectively. I xO I -l0 A T CUT QUARTZ CCW O R B I T S I I l 0 I x IO" (1- T p l K E L V I N S Fig. 8. Elliptical hysteresis orbits for sinusoidal temperature cycling. Temperature sweep ranges 0.5 and 1 millikelvin. Orbital periods are and lo-', respectively. It 6.28 x lo4 and 6.28 X l o 5 S, for W = is seen that even diurnal rhythms contribute significantly to frequency instability. TABLE VI1 FREQUEUCV DEVIATIONS D U E TO DYVAMIC THERMAL. BEHAVIOR OF A T-CUT af f 0 I Fig. 7 . Elliptical hysteresis orbits for sinusoidal temperature cycling. Temperature sweep range 5 millikelvins. This figure is drawn for Z= -2.27 X s/K2. Orbital periods are 3142,6283, and 62832 S, corresponding to W = 2 X and lo4, respectively. locus [35] theapproximate value for ;is +1.8 X s/K2 for EerNisse's design. For a given vibrator design, a plot of -a"versus 4 along the upper zero temperature coefficient locus in quartz would look very much like Fig. 3 in [33] ;the derivaThe simulations seen in Figs. 4-8 are for sinusoidal variations tive aa"/a@is zero at theAT cut and reaches its maximum positive value at the rotated-X-cut. in temperature. In practice, the fluctuations in the thermal The SCcut permits the possibility of a revolutionary change field will have a certain spectral distribution. The result of the in the stabilities of crystal oscillators, particularly in the coupling term a" wdl be to produce a corresponding frequency intermediate-to-long-term regimes. With the coefficienta" respectral distribution, with a resulting magnified frequency duced to a negligible level, and with state-of-the-art ovens in instability when compared to what would be expected in the the millikelvinrange,stabilities in the range to static case. What are the prospects for reducing r?? Fortunately, it apfor 128-S ought to beachievable. A stabilityof 6 X pears that by design even A T-cut resonators can be produced sampling times has recently been achieved withan SC-cut cryswith smaller values of ii. However, by using doubly rotated tal of special design, utilizing an airgap fixture and special crystals cut to orientationsin the vicinity of the SCcut the mountings [36] , [37] . This stability also depended upon a value of a" can be reduced to a negligible value. These cuts are special circuit configuration in addition to the SC-cut resonacompensated for a variety of other effects aswell [23], [33], tor [38] . The ability to achieve medium-term stabilities on [34]. For rotated-X-cuts on the zero temperature coefficient this order would make doubly rotated quartz resonators come zyxwvutsrqp zyxwvuts BALLATO: BEHAVIOR OF QUARTZ RESONATORS zyxwvutsrqp zyxwv 305 parable to rubidium atomic frequency standards in precision applications. Medium precision stabilities will be available for fast warmup oscillators, and they represent a second important application area for this latest representativeof a long lineof quartz vibrators. CONCLUSION The actual frequency instabilitiesobserved in quartz crystal oscillators for medium- and high-precision applications are greater than can be accounted for by the static frequencytemperature characteristic of the vibrator and the noise characteristics of the active devices and ancillary circuitry. One mechanism contributingto theinstability is the dynamic f-T coupling. This paper has used available experimental data with a simple mathematical model to explore some of the consequences of the dynamic effect on the characteristicsof the crystal resonance frequency. Doubly rotated quartz resonators that are compensated for the dynamic thermal effect offer promisein the near-term future of making available very small, light weight, and cheaposcillators with stabilities approaching those of rubidium frequency standards. [9] T. C. Anderson and F. G. Merrill, “Crystal-controlled primary frequency standards: Latest advances for long-term stability,” IRE Trans. Instrumentation, vol. 1-9, pp. 136-140, Sept. 1960. [ 101 W. J. Spencer and W. L. Smith, “Precision crystal frequency standards,”in Proc. 15th AFCS, May-June 1961, pp. 139-155. [ 1l ] A. W. Warner, “Use of parallel-field excitation in thedesign of quartz crystal units,” in Proc 17th AFCS, May 1963, pp. 248266. [ 121 W. L. Smith and W . J. Spencer, “Quartz crystal thermometer for measuring temperaturedeviations in theto 10”OC range,” Rev. Sci. Instr., vol. 34, pp. 268-270, Mar. 1963. [ 131 G. Bistline, Jr., “Temperature testing tight tolerance crystal units,” in Proc. 17th AFCS, May 1963, pp. 314-315. [ 141 R. J. Munn, ‘Warm-up characteristics of oscillators employing3. M.C. fundamental crystals in HC-27/U enclosures,” inProc. 19th AFCS, Apr. 1965, pp. 658-668. 151 H. F. Pustarfi, “An improved 5 MHz reference oscillator for time and frequency standard applications,” IEEE Trans. Instrum. Meas., vol. IM-15, pp. 196-202, Dec. 1966. 161 L. E. Schnurr, “The transient thermal characteristics of quartz resonators and their relation to temperature-frequencycurve distortion,” in Proc. 21stAFCS. Apr. 1967, pp. 200-210. 171 E. F. Hatman and J.C. King, “Calculation of transient thermal imbalance within crystal units following exposureto pulse irradiation,”in Proc. 27th AFCS, June 1973, pp. 124-127. 181 R. Holland, “Nonuniformly heated anisotropic plates: I. Mechanical distortion and relaxation,” IEEE Trans. Sonics Ultrason., vol. SU-21, pp. 171-178, July 1974. 191 -, “Nonuniformly heated anisotropic plates: 11. Frequency transients in AT and BT quartz plates,” in Proc. IEEE Ultrason. S y m p . , Nov. 1974, pp. 592-598. 201 J. A. Kusters, “Transient thermal compensation for quartz resonators,” IEEE Trans. Sonics Ultrason., vol. SU-23, pp. 273-276, July 1976. 211 D. R. Koehler, T. J. Young, and R. A. Adams, “Radiation induced transient thermal effectsin 5 MHz AT-cut resonators,” in Proc. IEEE Ultrasonics S v m p . . Oct. 1977. PP. 877-881. J. A. Kusters and J. G. Lkach,’“Further experimental data on stress and thermal gradient compensated crystals,”Proc. IEEE, vol. 65, Feb. 1977, pp.282-284. A. Ballato, “Doubly rotated thickness mode plate vibrators,”in Physical Acoustics: Principles and Methods, vol. 13, W.P. Mason and R. N. Thurston, Eds. New York: Academic, 1977, Ch. 5 , pp. 115-181. T. J. Young, D. R. Koehler, and R. A. Adams, “Radiation ininduced frequency and resistance changes in electrolyzed high purity quartz resonators,” in Proc. 32nd AFCS, May-June 1978, pp. 34-42. J. A. Kusters, M. C. Fischer, and J. G. Leach, “Dual mode operation of temperature and stress compensated crystals,” in Proc. 32nd AFCS, May-June 1978, pp. 389-397. F. Euler, P. Ligor, A. Kahan, P. Pellegrini, T. M. Flanagan, and T. F. Wrobel, “Steady state radiation effects in precision quartz resonators,” in Proc. 32nd AFCS, May-June 1978, pp. 24-33. E. P. EerNisse, Sandia Laboratories, Albuquerque, NM 87185, private communication, June 1978. R . Bechmann, “Frequency-temperature-angle characteristics of AT-type resonators made of natural and synthetic quartz,” Proc. IRE, vo1.44,Nov. 1956,pp. 1600-1607. T. Parker, Raytheon Research Division, Waltham, M A 02154, private communication, Dec. 1978. S . J. Hughes, Royal Aircraft Establishment, Farnborough, private communication, June 1978. J. J. Gagnepain, C.N.R.S., BesanGon, private communication, Nov. 1978. G. Lagasse, M d o y Electronics Co., Mount Holly Springs,PA 17065, private communication, June 1978. E. P. EerNisse, “Quartz resonator frequency shifts arising from electrode stress,” in Proc. 29th AFCS, May 1975, pp. 1-4. E. P. EerNisse, “Calculations on the stress compensated (SC-cut) quartz resonator,” in Proc. 30th AFCS, June 1976, pp. 8-1 1. E. P. EerNisse, Rotated X-cut quartz resonators for high temperature applications,” in Proc. 32nd AFCS, May-June 1978, pp. 255-259. R. J. Besson, E.N.S.C.M.B., Besanqon,and F. L. Walls, N.B.S., Boulder, CO 80303, private communications, May 1978, S . R. Stein, C. M. Manney, Jr., F. L. Walls, J. E. Gray, and R . J. zyxwvut zyxwvut zyxwvutsrqpo ACKNOWLEDGMENT The author wishes to thank T. J. Lukaszek of USAERADCOM, Ft. Monmouth for general discussions; E.P. EerNisse, Sandia Laboratories, for rotated-X- cut data;D. R. Koehler of Sandia Laboratories, for references and discussions;A. W.Warner of Frequency Electronics for discussions and a copy of [7] ;J . A. Kusters of Hewlett Packard for discussions anddata; J. H. Sherman, Jr., of General Electric for calling attention to [S] ;and G. Lagasse of McCoy Electronics, S. J. Hughes of R.A.E., and J. J. Gagnepain of C.N.R.S. for kindly supplying data and information. REFERENCES Note: Many of the references were presented at the Annual Frequency Control Symposium, U.S. Army Electronics R&D Command, Fort Monmouth, NJ 07703. They are cited here as AFCS for brevity. zyxwvutsrqpo [ l ] A. Ballato and J. R. Vig, “Static and dynamic frequencytemperature behavior ofsingly and doubly rotated, ovencontrolled quartz resonators,” inProc. 32nd AFCS, May-June 1978, pp. 180-188. [ 21 A. Ballato, “Frequency-temperature-load capacitance behavior of resonators for TCXO application,” IEEE Trans. Sonics Ultrason., vol. SU-25, pp. 185-191, July 1978. [3] J. S . Lukesh and D. G . McCaa, “An anomalous thermal effect in quartz oscillator-plates,”Am. Mineral., vol. 32, pp. 137-140, Mar.-Apr. 1947. [ 4 ] V. E. Bottom, “Note on the anomalous thermal effect in quartz oscillator plates,” Am. Mineral., vol. 32, pp. 590-591, Sept.Oct. 1947. [S] I. E. Fair, “Design data on crystal controlled oscillators,” in “Information bulletin on quartz crystal units,”ASESA 52-9, Fort Monmouth, NJ, Appendix 11, Aug. 1952. [ 6 ] A. W. Warner, “Ultra-precise quartz crystal frequency standards,” IRE Pans. Instrumentation, vol. 1-7, pp. 185-188, Dec. 1958. [7] A. W. Warner and D. L. White, “An ultra-precise standard of frequency,” Eleventh Interim Rep. on Contract DA 36-039 SC73078 to U.S. Army Signal R&D Lab., Fort Monmouth, NJ, July 1959,42 pp. [8] A. W. Warner, “Design and performance of ultraprecise 2.5-mc quartz crystal units,” Bell Syst. Tech. J . , vol. 39, pp. 1193-1217, Sept. 1960. zyxwvutsrqpo 306 zyxwvutsrqponml zyx zyxwvutsrqp zyxwvutsr zyxwvu zyxwvuts IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. Besson, “A systems approachto high performance oscillators,” in Proc. 32nd AFCS, May-June 1978, pp. 527-530. [38] F. L. Walls and S. Stein, “A frequency-lock system for improved quartz crystal oscillator performance,”IEEE Trans. Instrum. Mas., vol. IM-27, Sept. 1978, pp. 249-252. [ 391 D. R. Koehler, “Radiation induced frequency transientsin AT, BT, and SC cut quartz resonators,”to appear in Proc. 33rd AFCS, May-June 1979. SU-26, NO. 4, JULY 1979 [40] Y.Teramachi, M. Horie, H. Kataoka, and T. Musha, “Frequency response of a quartz oscillator to temperature fluctuation,”to appear inProc. 33rd AFCS, May-June 1979. [41] G. Thiobald, G. Marianneau, R. Pritot, and J. J. Gagnepain, “Dynamic thermal behavior of quartz resonators,” to appear in Proc. 33rd AFCS, May-June 1979. [42] T. E. Parker and D. L. Lee, “Stability of phase shift on quartz SAW devices,” to appear in Proc. 33rd AFCS, May-June 1979. Correspondence zyxw The purpose of this paperis an investigation of further conditions on the existenceof a leaky Rayleigh wave. The condiNEALG.BROWER,DOUGLAS E. HIMBERGER, tions involve restrictions on thefollowing ratios which describe A N D WALTERG.MAYER the liquid/elastic solid system: p L / p s , V L / V D ,V,/Vs, where p~ and ps are the densities of the liquid and the solid, respecAbsfruct-The Rayleigh wave, an inhomogeneous surface wave, exists tively, and V,, V D ,and Vs are the sound velocityin the liqwave velocities in the for all isotropic elastic solid infinite half-spaces., When the free surface uid, the longitudinal and shear bulk of the solidis bounded by a liquid a leaky Rayleigh wave does not nec-solid. essarily exist forall liquid/isotropic solid systems. The well-known conLimits o n these ratios are derived from numerical solutions dition for the existence of a leaky Rayleigh wave, the sound velocityin of the secular equation for leaky Rayleighwaves and specific the liquid must be less than the shear wave velocity in the solid, is examples are given of liquid/elastic solid combinations which shown to be a necessary but not a sufficient condition. Additional con- exceed the limits. ditions on density ratios and velocity ratios are given. Examples are listed showing liquid/sdid combinations which satisfy the liquid-shear 11. THEORETICAL BACKGROUND wave velocity condition but not the additional restrictions and thus do Consider an unbounded isotropic bulk medium. In this case not support a leakyRayleigh wave. two and only two differentwaves can be propagated. If the medium is bounded, otherwaves may exist. I. INTRODUCTION A . The Rayleigh Wave The Rayleigh wave is a plane inhomogeneous surfacewave For an isotropic elastic solid with a plane boundary, a surwhich can propagate undiminished along the surfaceof an elasface wave always exists. The wave is known as a Rayleigh tic half-space. The wave amplitude decreases exponentially wave. Briefly, one seeks a solution to the equationsof motion normal to the surface. The existence of such surface waves, for an elastic solid where the displacement amplitude decays which can be supported by all elastic media, was shown by exponentially perpendicular to the surface. At the boundary, Lord Rayleigh [ 11. all stress components must vanish. This leads to a secular There also exists surfacewaves for the case of a liquid/elastic equation for the velocity of the surface wave V . The secular solid system. One surface wave, the leaky Rayieigh wave, is an equation is given by inhomogeneous wave which is damped along the interface 4(V,/V)2(1 - (V,/V)211/2[(Vs/VD)2 - (V,/V)21’/2 formed by the adjoining two media. The damping is a manifestation of radiation into the liquid along the path of propa+ [ 1 - 2( V,/V)212 = 0 (1) gation. In the limit where the density of the liquid tends to where V D and Vs are the longitudinal and shear bulkwave zero, the leakyRayleigh wave becomes a free Rayleigh wave. velocities, respectively. There are three roots to the secular Unlike the free Rayleighwave, the leaky Rayleighwave does not exist forall real combinations of liquid/elasticsolid media. equation (1). The range of the Poisson ratio U determines the nature of the roots. The Poisson ratio may be written in terms A well-known example is an elastic solid in which the bulk of the bulk velocities as shear wave velocity is less than the sonic velocity in the adjoining liquid. U = (V:, - 2 V 3 / 2 (V:, - v;>. (2) Restrictions on the Existence of Leaky Rayleigh Waves zyx If the Poisson ratio is in the range where Manuscript received April 2, 1979. This work was supported by the Office of Naval Research, U.S. Navy. N. G . Brower was with the Physics Department, Georgetown University, Washington, DC 20057. He is now with the Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20810. D. E. Himberger and W. G . Mayer are with t h e Physics Department, Georgetown University, Washington,DC 20057. CJ < 0.263 . . . (3) then there exist three real roots t o (1). However, if the Poisson ratio falls in the range where a > 0.263 . .. 0018-9537/79/0700-0306$00.75 0 1979 IEEE (4) z