unwanted mode

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The analysis of unwanted-mode vibration patterns in AT-cut quartz oscillator crystals,

revealed by X-ray diffraction topography. II. A partial theoretical description of the unwantedmode

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1975 J. Phys. D: Appl. Phys. 8 1843

(http://iopscience.iop.org/0022-3727/8/15/016)

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J. Phys. D: Appl. Phys.,Vol. 8, 1975. Printed in Great Britain.0 975

The analysis of unwanted-mode vibration patterns in AT-cutquartz oscillator crystals, revealed by x-ray diffractiontopography: 11. A partial theoretical description of theunwanted mode

F N Goodall? and C A Wallace$The General ElectricC OLtd., C entral Research Laboratories, H irst Research Centre,Wembley

Received 16 September1974, in final form 17 March 1975

Abstract. The characterization of an unwanted m ode in AT-CUt qu art z oscillator crystals,from vibration patterns observed in x-ray diffraction topographs, has been describedin part I. From experimental measurements of the frequency and wavelength of theperiodic component in theX direction, it is concluded that it is not a well-knowncoupling mode. A theoretical description is presented, of a mode periodic alongXvibrating at twice the fundamental resonance frequency, which is in close agreementwith experimental observations. The unw anted mod e is believed to be coupled t o thefundamental m ode by virtueof non-linear elastic behaviour.

1. Introduction

In part I of the present paper (Isherwood a nd W allace 1975), a inode of ultrasonic vibra-tion in AT-cut quartz crystal plates, which causes unwanted temperature-dependentdeviations in the frequency and activity of crystal oscillator units in the frequency range5-20 MHz by coupling to the thickness-shear mode, has been characterized by meansof x-ray diffraction topography. The x-ray diffraction topographs record the nodaldistribution of standing wave patterns in the plane of the plate( X ,Z) with respect tothe component of particle displacement perpendicular to the selected Bragg plane. Thus,

some indication of the relative phases of th e displacement components in different direc-tions can be deduced froma comparison of topographs obtained with different Braggplanes. The directionsX , Y , Z are defined by the IR E Convention (S tandards on Piezo-electric Crystals 1949) an d a re shown in figure1.

Although the vibration patterns of the coupled mode generated in crystal units inwhich the geometry and mass distribution of the gold electrodes is irregular can beextremely complex,it is possible as a first approximation to resolve the spatial distribu-tion of amplitude in theX , Z plane into two components, one a mode periodic inxwith a period of the same order as the crystal thickness, the other more slowly varyingwith x and 2 . The latter is termed the modulating mode because it causes an apparentamplitude and wavelength modulationof the nodal distribution. By suitable designoft Now at Radio and Space and R esearch Station, Ditto n Park, Slough, Bucks.$ Now at M arconi-Elliott Avionic Systems Ltd., Boreham wood, Herts,W D 6 1RX.

1843


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1844 F N Goodall and C A Wallace

'

XJFigure 1. Principal directions inthe AT-CUt of quartz ( Y X I35' 1 5 ' ) .

the electrode geometry it has proved possible to excite unwanted-m ode vibration patternsin which substan tial areas consist of the periodic component in relative isolation. Thewavelength in theX direction Ax can be measured directly from the x-ray diffractiontopographs, after allowing fo r the residual effects of the superimposed modulating m ode.

The frequency of the unwanted m ode has also been determined. Inspection of thetopographical patterns from different Bragg planes has indicated that the unwantedmode belongs to the 'point-time' symmetry group designated2/n1 and its frequency,when coupled, has been demonstrated experimentally to equal twice the resonancefrequency 2 The mode can thus be clearly distinguished from the more familiarcoupling modes, face-shear and thickness-flexure, both of which belong to the point-time group 2t/mt and couple at the fundamental frequency.

From these measurements of wavelength and frequency it is possible to attemptRtheoretical description of the periodic component of the unwanted mode,in which thewaveform of the particle displacement vectors(x, y ' ) may be established.

2. Review of the experimental data

Theoretically, the types of coupled mode excited in a crystal and the conditions ofcoupling are invariant under a uniform change of scaleof all the linear dimensions of the

crystal, electrodes and resultant wavelengths generated, such tha t their produc ts with thefrequency are constants. However, provided that in the observed unwanted-modepatterns the XZ' dimensions of the electrodes and the extent of weakly-modulatedvibration are large compared withAx and the crystal thicknessT and the electrode thick-ness is negligible compared withT , the fundamental frequency is principally governedby T while that of the unwanted mode is controlled byAx and T. It follows that theproduct Xxf at the point of coupling should be practically constant, with only residualdependence on the crystal and electrode dimensions in theXZ' plane and the electrodethickness, although the actual incidence of coupling will be critically affected by theseparameters.

In order to determine the productA x f ,15 vibration patterns were selected exhibitinglarge areas of weakly-modulated periodic vibration extending over7-21 periods. Thewavelengths were measured following the procedureof part I, $4.2. The total spread of


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The analysisof unwanted-mode vibration patterns 1845

results amounted to about6 % , and within this range it was not possible to discern anysystematic correlation with frequency (between9.0 and 11 5 MHz), electrode dimensionalong X between 0.3 and 1.0 times the crystal length) o r the frequency decrem ent dueto electrode thickness (between1 and 4 % ) . This spread is probably associated with theirregularities observed to some extent in all the vibration patterns recorded and prom-inently revealed in the modulating mode amplitudes, which may result from residualinhomogeneitiesin the crystal units (see part I,$3.3) . Using the natural frequency of theunplated crystalfy the average value and standard deviation of the productX,fx werefound to be

h 5 f x= 4 5 3 x lo3t .08 x lo 3 (SI units). ( 1 )A phase velocity for the periodic comp onent a t the point of coupling can now be definedas

V r = 2 h , f ~ = 9 . 0 6 x1 0 3 + 0 * 1 6 xo 3 (SI units). ( 2 )YTcan be related to the elastic constants of quartz and the density (taken as2.65 x lo3SI units). Th e relevant elastic constants fo r the AT-cut have been given by M indlin(1961)and are as follows (in units of109 SI):

Cl , = 86.74; C,,=8*26; C14= -3.65

It is interestingto com pare the observed value ofh5 with those found experimentallyfor thickness-flexure and face-shear modes, propagating alongX, oupled to the funda-mental thickness-shear mode. For flexureXzf=2*677x I O 3 (SI units) and for face-shearh,f = (C55/p)1/2 5.08 x lo3 (SI units) (Sykes 1946). Extensional modes propagatingalong X can be coupled to the fundamental mode provided that they too belong to the2/m group and have twice the resonance frequency. Th e extensional phase velocityV,is given theoretically by

V, = [(C,, - C,2L/C22)/p]1/2 5.70 x l o 3 (SI units) ( 4 )(Mindlin and Spencer1967)and is not comparable toVz. It is concluded tha t the periodiccomponent of the unwanted mode cannot be identified with simple flexural, face-shearor extensional modes of vibration.

3. Solution of the equations of motion

If the vibration is considered to be effectively unbounded in theX and Z directions,plane periodic solutions to the equations of motion, belonging to the space-time group2 /mand with frequency2A have particle displacementss(u , U , w ) f the following form :

u ( x , y , t ) = CdUi cos (2nk ly lT)sin ( 2 n h x / T )cos ( 4 n f r )

v (x , , t ) = &Vi in (2nk ty lT)cos (2nlzxlT)cos ( 4 n f t ) ( 5 )w(x, y , t ) = CiWi sin (2nk ty lT)cos ( 2 n k x / T )cos ( 4 n f t ) .

It is convenient to normalize with respect to the crystal thickness by writing forT / h , and T/X,f the wave numbersh and kt respectively. Th e method of solution is givenin detail in the Appendix a ndis summ arized briefly here. Substitution from equation(5)


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1846 F N Goodalland C A Wallace

into the wave equations yieldsa secular equation with three roots(kt ) or any given valueof h , with corresponding amplitudesof displacement(Ui, i, Wi).

Th e relative contributions fr om each roo t (or Fourier component) a re determined bythe need t o satisfy the boundary conditions, which require tha t the surfaces of the crystalshould be free from all external stresses. It is assumed that the gold electrodes aresufficiently thin for their inertia to be ignored, and the piezoelectric stiffening effect isalso ignored. An exact solution is possible only fora specific value ofh, and may befou nd by varyingh until a combination of values ofki, Ui, Vi and Wt is yielded whichsatisfy all the boundary conditions. The discrepancy betweenh and the experimental value,derived from the measured value ofVz (see equation A4) is a measure of the e rrors in-volved. An alternative procedu re is t o use the experimentally determined value ofhand find the combination of Fourier amplitudes which represents the closest approachto a solution of the boundary conditions. Although b oth procedures were adopted inthis study, only the latter is presented here.

Tablel. Solutions to hesecularequationforh=0.365

Index (i) koc kl uti v W,I Vi

1 0.775 0.794 -8.36 - .502 0.440 0.448 0.269 0.1073 0,715 0.720 4.41 -20.1

Table 2. Fourier amplitude components for the minimum-error waveform puttingv;=1*00

Index(i) U4 VI w,,1 - 1.00 0.120 -0,1802 0.269 1.00 1.073 -0,0457 -0.0103 0.208

From the experimental phase velocityV , of equation (2) we have h=0.365. Thecorresponding values ofki, Utivi, and Wit/ Vi or the three rootsof the secular equationare given in table1. Also tabulated are the approximate values ofkt (denoted bykoz)calculated by ignoring the non-diagonal terms in the secular equation. These terms are

relatively small because the displacement in each component is polarized predominantlyalong one of the axes of the plate,so that the koi provide a good first approximation.Finally, from the closest fit to the b oundary conditions,a complete set of Fourier ampli-tudes has been derived, as shown in table2 , normalized with respect toVZ. The totalerror E , as defined by equation (A12), was found to be0.023, representing a very smalldiscrepancy between theory and experiment.

Putting V Z = , one finds that the total amplitudes of displacement aty=O andy = &T/2are

UO= 0,779 sin ( 2 ~ r h x l T ) ; V O =Wo=O) ( 6 )Us=O.884COS ( 2 7 ~ h ~ l T )Vs= 1.041C O S( 27~hx l T )Ws=0.156sin (2nhxlT) .

(7)


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The analysisof unwanted-mode vibration pat tern s 1847

- , -igure 2. Spatial amplitude of displacement in the XY'-plane.The general form of the spatial amplitude of displacement in theXY' plane is showndiagrammatically in figure2 over a single wavelengthA, Each period contains six nodesat the following positions:

x = o ; y'=O

X = 1.37T; J I ' = O

X = k0.68T; y ' = t0-28T.

4. Discussion

Th e small discrepancy factor,E = 2.3 %, indicates a fairly close agreement between theoryand experiment. Th e error is of the same order of magnitude a s the observed variationin VZ nd may be attributed to the same factors, viz. the finite dimensions of the electrodesin the X 2'-plane, coupling to the modu lating mode, an d the finite mass of the electrodeswhich has to be balanced by non -zero surface stresses in the boundary equations, as wellas to experimental erro rs in the physical constantsCi' nd p, and inhomogeneity in thecrystal units. This solution can therefore be accepted asa partial theoretical description

of the unwanted mode, confined to the periodic component only vibrating in a quasi-infinite p late.

Coupling between the fundamental thickness-shear mode and the unwanted mode(both periodic and modulating components) at twice the fundamental frequency ispossible by virtue of non-linear phenomena within the crystal unit and/or its associatedoperating circuitry. Fo r example, harmonic m odes may be activated by non-linearelastic behaviour represented by third-order coefficients of the formCijk'. Alternatively,harmonics may be genera ted by non-linear effects within the oscillator circuit. How ever,comparison of topographs obtained at similar amplitudes of lattice strain when thecrystal was operated ina test oscillator and ina transmission 7-network (usinga Rhodeand Schwarz frequency synthesizer atf ~ ) howed very similar coupled-mode vibrationpatterns. It is therefore believed that the origin of the coupling is more likely to be thenon-linear elastic forces within t he crystal (W ood and Seed1967).

144


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1848 F N Goodall and C A Wallace

The condition of time inversion symmetry requires that the displacement maximaof the thickness-shear mode, for whichf t = O , 4, nd the zeroes( f t = $ , a),correspondto opposite phases of the unwanted-mode amplitude.

Two branches of the coupled mode may be distinguished, according to whether thethickness-shear displacement maxima coincide with the positive configurationof theunwanted mode, shown in figure2, or with the negative or reversed-phase configuration,The zeros of the thickness-shear displacement then coincide respectively with the nega-tive or positive configuration of the unwanted mode. Th e frequency difference betweenthe two branches can be observed as an apparent splitting of the frequency responsepeak of the fundam ental mode a t the temperatureof coupling, when the crystalis drivenin a transmission n--network. When the crystal is driven in a test oscillator there is asimilar sha rp oscillation in the frequency as the temperature is varied, accompanied by adip in activity. Th e magnitude of all these effects has been observedto increase withamplitude of vibration, a s is consistent with their origin in non-linear phenomena (Woodand Seed1967).

Because of the complexity of its amplitude distribution in the X2-plane, no attempthas been made in this paper to describe the modulating component of the unwantedmode . It is known tha t its frequency is2fR, and the observation (see partI, 96) thatdisplacements in the modulating and periodic modes tend either to reinforce or opposeeach other simultaneously in theX and Y directions implies that the forms ofs ( y ) neach componen t are not greatly dissimilar. It is worth noting, however, that ina crystalplate unbounded in the X2-plane with electrodesof negligible inertia, the thickness-shear harmonic of frequency2 R and wavelengthAy= T , having the waveform

21= UCOS (27rylT)cos (4n-f?) (c=w=O), (9)is a solution of the equations of motion and boundary conditions, as may be verified byputting h = 0 in equations(AS) and (A7). Related harm onics belonging to the point-timegroup 23m with amplitudes U ( x ) varying very slowly withx have frequencies closeto2fR and small but finite amplitudes ofL and M. As in the fundam ental thickness-shearmode, the actual formof U ( x ) is critically dependent on the mass distribution in theelectrodes. It seems most likely tha t the modulating modes belong to this groupofthickness-shear harmon ics.

A complete theoretical description of the coupling phenomenon and its dependenceon temperature and amplitude of vibration requires a more detailed descriptionof theamplitude distribution in each of the modes and knowledge of the third-order elasticconstants CijkJ.

Acknowledgments

This work was initiated by Salford Electrical Instruments Limited, pa rt ofGEC ElectricalComponents, through a research programme being carried out in the Quartz CrystalsGroup at the Hirst Research Centre.

Appendix

Formulationand solution of the wave equationsLet us express the particle displacements( U , U, w) n the periodic component as the


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The analysiso unwanted-mode vibration patterns 1849

following Fourier su m :

u ( x ,y , t ) = CzUicos (2nkty lT)sin (27rhxlT)cos (47rft)

v(x,y , t ) = CiVisin (27rkay/T)cos (27rhxlT)cos (47rft) ( A l l

w(x,y , t ) = C,W,l sin (2xk ty lT )cos (27rhxlT)cos (47rft).Since there is assumed to be no wave motion in the2 direction, all derivatives in

2 can be ignored, while the symmetry of the AT-Cut demands that the elastic constants

The equations of motion then take the form

Th e fundam ental frequency for thickness-shear is given by

After substitution from equations(AI) and (A3) (and also dropping the summation,

dividing by the sino and cosine factors, and multiplying by(T/27r)2),he equations ofmotion reduce to the form

[cl, +c~,, k 2 - I ) ] Uf(C12+C6e)hkV+(Cl , +C56)hkW=0(CI2+CGO) kU+[C66(h2- 1)+c22k2]V+(C,6h2+C24 k) w=o ( 4 5 )(Cl4+Cje) hk U + (C66h2+CZ4 k 2 )V + Csj h2+C44 k 2- C66) W = 0.Th e condition tha t the amplitudesU , V ,W should have finite values is tha t the deter-minan t of their coefficients be zero. This yields a secular equation w hich is cubic ink2,and thus has three rootskf for any given value ofh, with corresponding ratios of the

componentsU ; , Vi and Wi.The general solution to the wave equation is now a linear combinationof the threeroots of the secular equa tion and the ratios of the coefficients (i.e. the F ourier amplitudesof equations(A l ) ) , eing determined by the boundary conditions. These are tha t, if theinertia of electrodesis ignored, the stresses acting on the surface planes0 2 , a4 and (56 bezero, i.e.


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1850 F N Goodall and C A Wulluce

After substitution from equations (A l) and the use o f y= i- /2 , he boundary conditionsare

.Ci(CI2 U,+ Cz2 iVi+C24 i w i ) OS nki=O

Ci(CBGi i+Cos 7 Vi+CjS h W,) sin nk, = 0.Ci(C1l h V i CZ4 ,V i+Cd4 iW,)COS nki = 0 (A71

Substitution of the app rop riat e values ofUi/Vi and Wi/ Vi for each root, correspondingto a particular value ofh, yields a set of simultaneous questions in the three variablesVi.

Let us abbreviate the coefficients of the variablesVi n the boundary equations by thematrix elementsuji , wherej (= 2, 4, 6) is the subscriptof the corresponding componentof stress U and i ( = 1, 2 , 3) is the index num ber of the solution to equations (A5). (Bydefinition solutions are indexed so that fori = 1, U is predominant, etc). The co-factorof each matrix element is denotedA j i . A combination ofVir exactly satisfying the bound-ary conditions can be foun d at the particular value ofh for which the determinantA ofthe coefficientsutj is zero. Ifh diKers from this value,A#O and IAl is a direct measureof the discrepancy. A solution may then be sought by a variational method.Sincethe three u j are not all zero simultaneously let us writeoj=ejV2 as an error term inequation (A7) equivalent to an unbalanced stress at the crystal surface. The choiceofvariable used for normalization is entirely arbitrary and there is no special reason forchoosing VZ.The boundary equations may now be written as

u21 VI +a22 V Z a23V3 = 2 772a41 VI+a42 VZ+a43 V3= e4VZ (AS)0 6 1 vl +a62 VZ +a63 V3= e6VZ.

Following the principles of least-squared errors, we can find the solution of equations(A8) for which the sumEjej2is a minimum, i.e. stationary with respect to variationof

V I/ VZ nd V3/ V2.The conditionsd (Cjej2)/d(VI/VZ) d (Cjej2)/d( V3/ Vl) = 0 (A91

yield two additional linear equations

Equations (A8) and (A10) constitute five relations between the five unknownsVljVz,Vl/V2, e2, Vs/Vz, e4 and eG. Th e complete solution is as follows:

The total err or may be normalized in the following expression:

References

Isherwood BJ and Wallace CA 1975J . Phys . D: Appl . Phys. 8 1827-42Mindlin R D 1961 Quart. Appl . Math . 19 51Mindlin R D and SpencerW J 1967J . Acoust. Soc. A m . 42 1268Standards on Piezoelectric Crystals 1949Proc. Inst . Radio Engrs, New York 37 1378Sykes R A 1946 Quartz C rystals fo r Electrical Circuits ed R A Heising (NewYork: Van Nostrand)

Wood AFB and SeedA 1967Proc. 21st A . Frequency Control Symp. (Fort Monmouth, NJ:US Armypp 205-48

Electronics Command) p 420

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