Simultaneous mass and temperature determination using a single quartz wafer

Simultaneous mass and temperature determination using a

single quartz wafer

Citation for published version (APA):

van Ballegooijen, E. C. (1978). Simultaneous mass and temperature determination using a single quartz wafer. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR138061

DOI:

10.6100/IR138061

Document status and date: Published: 01/01/1978 Document Version:

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SIMULTANEOUS MASS

AND TEMPERATURE DETERMINATION

USING A SINGLE QUARTZ WAFER

SIMULTANEOUS MASS AND TEMPERATURE DETERMINATION

USING A SINGLE QUARTZ WAFER

SIMULTANEOUS MASS

AND TEMPERATURE DETERMINATION

USING A SINGLE QUARTZ WAFER

PROEFSCHRIFI'

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Eindhoven,op gezag van de rector magnificus,prof.dr. P.van der Leeden,voor een commissie aangewezen door het college van dekanen in het openbaar te verdedigen op

dinsdag 3 oktober 1978 te 16.00 uur

door

ERIK CORNELIUS VAN BALLEGOOIJEN

Dit proefschrift is goedgekeurd door de promotoren

Prof. Dr. P. van der Leeden en

TABLE OF CONTENTS

GENERAL OUTLINE

1. 1. Introduction 1

1.2. The pPesent investigation 2

REFERENCES

II THICKNESS-TWIST MODES IN ROTATED V-CUT QUARTZ CRYSTALS

2. 1. Introduction 6

2. 2. Rema:!'ks on the notations 7

2. 3. Fundamental. equations 7

2.4. Plane elastic waves in crystals 9

2. 5. Two-diiTI8nsionaZ. model. of an unbounded rotated :t-cut 11

quaPtZ 1iXZfeP

2. 6. Thickness-t1Jist modes

2. 7. The t::ux>-dimensional. vibrution pPobZ.em 2. 8. DetePmination of the Pesonant frequencies

REFERENCES/APPENDICES

Ill EXPERIMENTAL APPARATUS

3.1. Introduction

3. 2. DetePmination of the resonant frequencies

3. 2. 1. MeasU!'ing method 3.2.2. 11-net::ux>Pk

3.2.3. Measuring circuit

s.s.

Determination of the mass deposited on and the

temperuture of the crystal. under test 3.3.1. Determination of the mass

3. S. 2. Determination of the tempePature

3. 4. The crystal. hotdeP

REFERENCES/APPENDIX

IV ONE-DIMENSIONAL MODEL OF A COMPOSITE RESONATOR

4.1. Introduction

4. 2. "Rotated Y-cut quaPtz crystal. ~Aith t!Ao diffePent electrodes tPeated as a one-dimensional. acoustic composite resonator" (197?)

V THE ELECTRODE-TAB RESONATOR

14 18 21 30

so

32 34 38 40 40 46 46 5. 1. Introduction 48

5. 2. 11_{The infl-uence of mass l-oading outside the electrode area 48} on the Pesonant frequencies of a quaPtz crystal." (1977}

5. 3. "Influenae of the thiakness of tabs on the resonating 49 properties of a quartz arystal" (19??)

VI SIMULTANEOUS MASS AND TEMPERATURE DETERMINATION

6 .1. Introduation 50

6.2. 1_{~ppliaation of a quartz arystal with eleatrode-tab 50}

aonfiguration for simul-taneous mass and temperature detePmination" (19?8)

6.3. Resonant frequenaies of an eleatrode-gap-tab resonator 52

6. 4. "Simultaneous mass and temperature detePmination using 54

a single quartz wfer: An optimized arystal aut11 _(19?8)

6.5. Conalusions and remarks 55

VII REPRINTS/CONCEPT OF PUBLICATIONS

?.1. ''Rotated Y-aut quartz arystal with two different eleatrodes treated as a one-dimensional aaoustia aomposite resonator", F. Boersma and E. C. van

Ballegooijen, J. Acoust. Soc. Am. 62, 335 - 340 (1977)

59

?.2. 11_{The infiuenae of mass loading outside the eleatrode area 65} on the resonant frequenaies of a quartz arystal",

C. van der Steen, F. Boersma, and E. C. van Ba11egooijen, J. App1. Phys. 48, 3201 - 3205 (1977)

?. J. "Infiuenae of the thiakness of tabs on the resonating ?0

properties of a quartz arystal11_{, E.}

c.

_{van Ballegooijen,}

F. Boersma, and C. van der Steen, J. Acoust. Soc'. Am. 62, 1189 - 1195 (1977)

?.4. '~ppliaation of a quartz arystal with eleetrode-tab 7?

aonfiguration for simultaneous mass and temperature,

detePmination11_{, E. C. van Ballegooijen, F. Boersma, and}

C. van der Steen, J. Acoust. Soc. Am. 63, 806-814 (1978)

?.5. "Simultaneous mass and temperature detePmination using 86

a single quartz wafer: An optimized arystal aut",

E. C. van Bal1egooijen, accepted for publication in J. Appl. Phys., nov. 1978

SUMMARY

111

SAMENVATTING

113

DANKWOORD

115

Aan Meya

CHAPTER I GENERAL OUTLINE 1. 1. Intl'oduction

In 1957 Sauerbrey suggested that a quartz crystal oscillator could be used for the determination of the thickness of thin films,(l) From his experiments in 1959 he found that the frequency shift of a quartz resonator was linearly proportional to the amount of mass deposited on the crystal.(2) These investigations lead to the introduction of the quartz crystal as a microbalance with a possible sensitivity in the picogram/cm2 range.(3)

One of the fundamental complications when a quartz resonator is used as a microbalance is the fact that the resonant frequency of such a resonator depends - apart from the added mass - on many additional quantities, for· instance temperature and hydrostatic pressure, It is evident that the maximum sensitivity and accuracy of this technique can only be attained if the influence of these spurious effects is either properly taken into account or drastically minimized. Throughout this thesis we will confine ourselves to the most dominating effect, which is the influence of the temperature on the resonant frequency. Some of the methods which are most commonly used to reduce this effect are discussed below.

A rather straightforward method is the use of crystal plates, of which the temperature dependence of the resonant frequencies is very small. For this purpose the AT-cut (4) is commonly chosen, because its resonant frequencies are nearly temperature independent in the neigh-borhood of room temperature.(S) In order to prevent drifting of the temperature of the crystal the crystal holder is water-cooled in nearly all cases.(6-7) Note that this technique is used in most commercial film thickness monitors •. The mass determination, however, is then restricted to a relatively small temperature range around room temperature.

To avoid this disadvantage, other investigators used a second quartz crystal as a reference. From the known frequency-temperature dependence of this reference crystal the resonant frequency of the main crystal can be corrected for the temperature variations. This method requires

the knowledge of the temperature characteristics of the main crystal. (8-9) Another quite similar but more sophisticated method uses two identical resonators, of which only one is exposed to the mass loading. In this case the mass loading is derived from the beat frequency of two crystal controlled oscillator circuits containing the two resonators. (10-12) Apart from the fact that both crystals must be kept at the same environmental conditions, this technique requires that the frequency-temperature characteristics of both crystals are identical, which reduces the practical applicability and versatility of the method,

To meet these objections van Empel introduced a double resonator technique, in which two resonators are realized on a single quartz wafer.(l3-15) If the resonant frequency of the resonator, which is not exposed to the mass loading, is not influenced by the resonant frequency of the second resonator, the beat frequency of the two reso-nators is a measure for the mass increment.(16) By means of this technique mass determination proved to be possible in a wide range of temperatures,i.e., from room temperature up to 140°C,(l7-18)

In all aforementioned methods the influence of the temperature is reduced by choosing a particular crystal cut, by stabilization of the temperature, or by correcting the resonant frequency for the effect of temperature changes. However, no accurate information is obtained about the temperature itself. In contrast to these methods EerNisse introduced a multiple resonator technique to measure different quantities like temperature, mass, and stress, simultaneously.(l9-20) In this technique the number of crystals is chosen equal to the number

of quantities to be measured. However, the method still requ~res that

all crystals are kept at the same experimental conditions.

In the next section we will outline the present investigation, in which a method to determine mass and temperature simultaneously is developed, which meets the objections of the latter method.

1.2. The present investigation

In this thesis a method is presented, which may be used in practice to determine mass and temperature simultaneously using a single quartz

wafer. The method is based on the properties of a so-called

"eZeatrode-tab11 resonator. The electrode-tab resonator consists basically of a quartz wafer on which, apart from a pair of conventional electrodes,

an additional single electrode - the so-called "tab11

- is deposited

adjacent to one of the electrodes. At this point we like to note that the thesis can be distinguished in two parts, In the first part -chapters II and III - we will discuss the propagation of plane elastic waves in crystals and the experimental apparatus, The second part

-chapters IV, V, and VI - is mainly devoted to the method to determine mass and temperature simultaneously. Because this method has been the subject of a number of papers which were recently published by the author, the development of the method will be discussed in connection with these papers. Reprints of the papers are included in the final chapter VII of the thesis. In the chapters IV, V, and VI we will briefly review the topics that are discussed in much more detail in the corresponding papers.

Chapter II is devoted to a theoretical analysis of the propagation

.

of plane waves in crystals. After a short introduction (Sec. 2.1) and some remarks on the notations used in the thesis (Sec, 2.2), we will consider the fundamental equations describing the behavior of plane waves in piezoelectric crystals (Sec. 2.3, 2.4). Throughout this analysis we will confine ourselves to a two-dimensional model (cf, Sec. 2.1) of quartz wafers of the rotated Y-cut family of crystal cuts. It appears that for these cuts the only vibration modes of interest are the thickness-twist modes (Sec. 2.5). We will show that it is allowed to simplify the corresponding wave equation together with the appropriate boundary conditions by leaving out the piezo-electric properties of quartz (Sec. 2.6), However, even the solution of this simplified two-dimensional vibration problem can in general only be approximated, either analytically or numerically (Sec. 2,7), In order to obtain an analytical solution a further simplification of the vibration problem has to be made (Sec. 2.8).

Chapter III is devoted to a description of the experimental methods and apparatus.

Chapter IV deals with a theoretical analysis of a one-dimensional model of a quartz resonator coated with two different electrodes. This model represents one of the few exceptional cases in which we can obtain an analytical solution of the vibration problem without the simplifications mentioned above.

Chapter V gives an introduction to the physical principles on which the mechanism of an electrode-tab resonator is based, By means of a

two-dimensional model the resonating properties of the electrode-tab cesonator are-theoretically analyzed. The results from this analysis are compared with the results from a number of experiments. At the end of this chapter we will indicate how the resonant frequencies of an electrode-tab resonator might be used for simultaneous mass and temperature determination.

Chapter VI contains a detailed description of this method. A number of modifications which increase the performance of the method are discussed. In order to reach an optimal resolution and accuracy of the mass and temperature determination an analysis of the errors of the relevant parameters is presented. From this analysis an optimized crystal cut is inferred. Both these results and a number of experiments indicate that with the optimized method a very accurate determination of mass and temperature will be possible.

REFERENCES CHAPTER I

I. G. Sauerbrey, Phys, Verb.

!•

113 (1957). 2. G. Sauerbrey, z. Phys.

lli•

206 (1959).

3. A. W. Warner and c. D. Stockbridge, Vaouwn Mitn'oba'lanoe Teahniquee, edited R. F. Walker (Plenum, New York, 1962), Vol. 2, p. 71.

4. See for the definition of the crystal cut chapter 11, section 2.6. The AT-cut is defined by+~ +35°30'.

5. R. Bechmann, Proc. IRE 44, 1600 (1956).

6. K. H. Behrndt and R. w. Love, 'l'ran8. 7th Nat. Vac. Symp. (Pergamon Press, London, 1961), p. 87.

7. s. J. Lins snd R. s. Kukuk, ibid., p. 333.

8. H. L. Eschbach and E. W. Kruidhof, Vacuum Mitn'oba'lanoe Teehniquee, edit~ by K. H. Behrndt (Plenum, New York, 1966), Vol. 5, p. 207.

9. R. Niedermayer, N. Gladkich, and D. Hillecke, ibid., p. 217. 10, R.,M. Mueller and W. White, Rev. Sci. 1nstr,

2!•

291 (1968),

11, A. Langer snd T. J. Patton, Vaouwn MiCPobal.anae Tealmiquee, edited by

K. H. Behrndt (Plenum, New York, 1966), Vol. 5, p. 231.

12. A, W. Warner, UU;m MierotJJeight Dete1'1'11ination in ContvoUed Erwi1.'0n111ents.

edited by s. P. Wolsky and E. J •. Zadnuk (Interscience, New York, 1969), p. 137. 13, F. J, van Empel, E. C. van Ballegooijen, F. Boersma, J. A. Poulis, and

c. H. Massen, Thennat Anatym. P:!>oo. thim ICTA. lfavos (Birkhauser Verlag, Basel, 1971), Voi. 1, p. 583.

14. J. Ph. Termeulen, F. J. van Empel, J. J. Hardon, c. H. Massen, and

J. A. Poulis, l'l!ogPess in Vaouwn Mia:r'oba'lanoe Techniques. edited by T. Gast and E. Robens (Heyden, London, 1972), Vol. 1, p. 41.

15. F. Boersma and F. J. van Empel, Pl'ogroess in Mieroba'lanoe Teehniques, edited by C. Eyraud and M. Escoubes (Heyden, London, 1975), Vol. 3, p. 9,

16. F. J. van Empel, C. H. Massen, H. J. J. M. Arts, and J. A. Poulis, J. Acoust. Soc. Am. 1Q_, 1386 ( 1971).

17. F. J. van Empel, Ph.D. Thesis (Eindhoven University of Technology, Eindhoven, l 97S)(unpublis.hed).

18. R. W. Philips, L. U. Tolentino, and S, Feuerstein, J. Spacecraft & Rockets Ji, 501 (1977).

19. E. P. EerNisse, J, Appl. Phys. 43, 1330 (1972). 20. E. P. EerNisse, J. Vac. Sci. Technol.

l!•

564 (1975),

CHAPTER II

THICKNESS-TWIST !40DES IN ROTATED V-CUT QUARTZ CRYSTALS

2.1. Introduction

After an introduction of the fundamental theory on plane wave propaga-tion in crystals, this chapter will be devoted to a detailed study of the properties of thickness-twist modes in so-called "rotated Y-cut" quartz crystals. With the help of the fundamental equations, presented

in section 2.3, the "stiffened Christoffel" equation, describing the

behavior of plane waves in piezoelectric media, is derived in section 2.4. In section 2.5 the stiffened Christoffel equation is solved for plane waves propagating in an unbounded two-dimensional model of a quartz wafer. In our notation "two-dimensional" indicates that the direction of the wave vector is restricted to a plane, i.e., one of the components of the wave vector is always equal to zero, The resul-ting wave equations together with appropriate boundary conditions are in general extremely complicated, and hence these equations can only be solved approximately in terms of plane wave theory, For the discus-sions, however, we prefer an analytical solution, since from a physical point of view the latter usually can be interpreted more directly. In order to obtain such an analytical solution, it is necessary to sim-plify the vibration problem. Firstly, in section 2.6, we will show that the pertinent piezoelectric constants of quartz may be neglected

in the calculations of the resonant frequencies, suggesting ~n fact

that quartz is a purely elastic medium. Secondly, in section 2,7, we will suggest a further simplification of the wave equation and corres-ponding boundary conditions. Finally, in section 2.8, we will indicate to what extent the latter simplification is allowed in order to main-tain a realistic description of the original vibration problem. The simplifications mentioned above permit us to obtain an analytical solution, and hence they will be used throughout the considerations in this thesis. Furthermore, the concept of harmonic and anharmonic overtones is introduced, and the effect of the presence of electrodes is discussed.

2.2. Remarks on the notations

In this section we will present a survey of the notations used in this thesis. The following notations and conventions will be employed.

The Einstein summation convention is used for repeated tensor indices

(i,j ,k,l 1,2,3), i.e., when a letter occurs as a suffix twice in the same term, summation with respect to that suffix is to be automatically understood, e.g.,

Because the Einstein summation convention shall be repeatedly applied in a number of equations, we will indicate the order by which the sequence of summations has to be carried out by brackets,

Further we like to emphasize that in a number of equations the - much simpler - abbreviated matrix notation is used, in which a pair of interchangeable subscripts is replaced by one index (see section 2,5).

A subscript preceeded by a comma denotes differentiation with respect

to the space coordinate indicated by that subscript, e.g.,

ail>

E

=--k a~

We will denote the elastic, dielectric, and piezoelectric constants

referred to a rotated coordinate system x (a= 1,2,3) by a

super-a

scripted bar, e.g., E22•

Vectors are indicated by underlined symbols, e.g.,

y.

Occasionally we

will represent a vector by its magnitude and its direction, e.g.,

y = v~, where v is the magnitude and ~ is a unit vector in the direction

of

y.

2. 3. Fundamental equations

For W?Ve lengths large compared to the interatomic spacing the modes of vibration and the electrical properties of any piezoelectric

pertinent Haxwell equations, and the linear piezoelectric constitutive equations.(!) These equations do not include nonlinear effects nor losses due to internal friction or other effects which may give rise

to dissipation of energy. The so-called "statia fiel.d appro:dmation"

(2) is applied, in which the magnetic field is neglected, and the electric field is derived from a scalar potential. The equations governing the behavior of elastic waves in a piezoelectric medium can be summarized as follows:

Stress equations of motion:

T • • • l.J ,1. pu .• J

Charge equation of electrostatics:

D • •

=

0.

1.,1.

Strain/mechanical-displacement relations:

Electric field/electric potential relations:

Linear piezoelectric constitutive equations:

T •• l.J D. l. (2. I) (2,2) (2.3) (2,4) (2,5) (2.6)

In the Eqs.(2.1)-(2.6) u., D., and Ek are the components of mechanical

J l.

displacement, electric displacement, and electric field, respectively; T .. and Skl are the components of the symmetric stress and strain

. l.J • E S

tensor, respect1.vely; c •• kl' ek''' and €.k are the components of the

l.J l.J l.

elastic, piezoelectric, and dielectric tensor, respectively; p and ~

are the mass density and the electric potential, respectively. The superscripts E and S, denoting constant field and strain, respectively, will be dropped in the remaining part of the thesis,

The solution of Eqs.(2.1)-(2.6), together with appropriate boundary conditions, would - in principle -yield all the resonant frequencies, mode shapes, and all other information about the modes of vibration at low strain amplitudes, at least in materials where energy dissipation can be neglected. However, even for very simple sample geometries and nonpiezoelectric isotropic materials the solutions of these equations are extremely complex. In general, it is impossible to solve these equations in closed form. Therefore, in nearly all cases several sim-plifications have to be made. In section 2.6 and 2.8 we will examine to what extent these simplifications are allowed.

2.4. Plane elastic waves in crystals

First> we like to emphasize that the formalism used in this section was already developed for pure elastic media by E. Christoffel.(3) It was refined by A. Lawson who extended it for piezoelectric media.(4) To investigate the properties of plane elastic waves propagating within a crystal, we must return to the fundamental equations

presen-ted in section 2.3. This system of 22 equations can be reduced to four differential equations relating the four dependent variables u.

J

and ~. These equations read:

(2.7)

and (2.8)

In the following we will use a rectangular Cartesian coordinate system xa, a ; 1,2,3 (see next section). Let us consider the monochromatic plane waves given by

u. J and where A. exp[i(kg.E - wt)], J ~o exp[i(kg.r- wt)],

(sin6cos,, sin6sin,, cos6),

(2.9)

(2.10)

(2.11)

is a unit vector (5) in the direction of the wave vector ~; 6 and ' are the polar and azimuthal angle of this vector referred to the x1 ,

the circular frequency; ~o and A. are the amplitude of the electric

- J

potential and the components of the amplitude of the particle displa-cement, respectively; ! is the radius vector. These plane waves satisfy Eqs.(2.7) and (2.8) provided that

and

(2.12)

(2. 13)

where ni are the components city of the wave. Since the be transferred to the right

of n, and v = (w/k)n_ is the phase

velo-- -p

factor multiplying ~ is a scalar, it may hand side of Eq.(2,13), yielding the potential as a function of the particle displacement,

(nieiklnl)

(nie:ik~) uk,

Substitution of Eq.(2.14) into Eq.(2.12) yields

(2, 14)

(2. 15)

Eq.(2.15) is the so-called "stiffened Ch:t>istoffel11 _{equation. because}

it has exactly the same form as the original Christoffel equation (6),

except for the term cijkl being replaced by the expression within square brackets,

(2,16)

The constants cijkl are the "piezoeleat:riaaally stiffened elastia"

constants. It is common use to write Eq.(2.15), using the matrix notation (cf. Sec. 2.2), in the following much simpler form

(2.17)

where ojk is the Kronecker delta, and Ajk is a symmetric matrix given by

(2, 18) At this point we like to emphasize that Eq.(2,17) forms the basis of

crystals, and represents a set of three homogeneous equations of the first degree for the unknown variables u1, u2, and ug, This set of equations has nonzero solutions only if the determinant of coefficients vanishes, which yields the dispersion relations

o.

(2.19)

This cubic equation in v 2 generally yields three different real roots

p

v~(a)' a= 1,2,3. For a given wand g we obtain three values of k(a)' one for each v 2( )' Now each v_{. p a} 2_p ( ) determines an independent solution, _a which may be substituted into Eq.(2.17) to obtain the relative

amplitudes

A~a).

These three amplitude c.q. particle displacement

J

vectors are mutually orthogonal.(7) Note that once these relative particle displacements have been obtained, the electric potential ~

may easily be calculated from Eq.(2.14). In the next section the theory outlined above will be applied to a rotated Y-cut quartz wafer.

2.5. Two-dimensionaL modeL of an unbounded rotated Y-aut quartz crystal

Before starting the derivations we will briefly review some of the specific properties of quartz. There are two modifications of quartz, i.e., a- and ~-quartz, of which a-quartz is stable below the transi-tion temperature of 573°C. Above this transitransi-tion temperature a-quartz changes into ~-quartz, which shows hardly any piezoelectric properties, a-Quartz is trigonal and has symmetry class Ds (or 32), while B-quartz is hexagonal and has symmetry class D6 (or 622). It should be noted

that quartz occurs in optical right-hand and left-hand forms, i.e., crystals which rotate the plane of polarization of polarized light passing along the Z or optic axis counterclockwise (left-handed) or clockwise (right-handed) from the point of view of the observer facing the source of light. However, this distinction is of no importance in vibration problems. Since for quartz resonators always a-quartz is used the "a" is dropped throughout the thesis.

In our analysis we will confine ourselves to the so-called two-dimensional model of a rotated Y-cut quartz resonator. To define the crystal cut, indicated by ~. we will consider a quartz strip of infinite width, length 2~ and thickness 2d. The Cartesian coordinate system xa - the plate coordinate system - is chosen conform the plate

geometry. At this point we like to note that we will use "length" to

indicate the largest dimension of the aross-seation of the strip,

since all properties of the strip are independent of the width c.q. x 1-direction. In general the axes xa do not coincide with the

rectan-gular crystallographic coordinate system X, Y, and

z,

of which the

Z-axis coincides with the crystallographic a axis, and the X-axis

coincides with the a axis. As mentioned above the x1-axis is always

in the infinite direction of the strip, while the x2-axis is always in the thickness direction of the strip. Usually, the crystal is coated

with electrodes. These electrodes are deposited at the "front" (x 2

=

+d) and the "back" side (x2 = -d) of the quartz strip, The electrodes

are of finite length (ie ~ 2i) in the x3-direction, and have a profile

which does not depend on x1. The relation between the plate coordinate system and the crystallographic coordinate system is determined by the crystal cut. Any crystal cut may be characterized by three single

rotations about the x 1, x2, and x3 axes, For denoting the crystal cuts

we will apply the convention described by the "IRE Standards on

Piezo-electric Crystals".(8) For the special case of the rotated Y-cut

family (yxi, ~ - in standard IRE-notation) only one single rotation

about the x 1-axis is carried out (see Fig. 2,1). For a rotated Y-cut Fig. 2.1.

The relation ~etween the plate

coordinate system x and the a

arystallographia coordinate system X, Y, and Z for a rotated Y,-aut quartz plate. A single rotation is carried out about the x1-axis, The crystal aut is determined by ~.

quartz plate the x 1-axis corresponds with the X-axis, which is one of the three twofold c.q. digonal axes in quartz, The elastic, piezoelec-tric, and dielectric tensor components must refer to the plate coor-dinate system xa. Since these components are usually given with respect to the crystallographic coordinate system, a transformation is

neces-sary. In appendix A-1 at the end of this chapter the arrays (9) of the

elastic, piezoelectric, and dielectric tensor components referred to the crystallographic coordinate system are given, as well as their

numerical values.(lO) The components are given in compressed matrix notation (11), in which a pair of interchangeable subscripts is repla-ced by one index, e.g., c .. kl = c , and e.kl =e. , where i,j,k,l

1J pq 1 1q

1,2,3, and p,q = 1,2,3,4,5,6. The replacement of the subscripts is performed according to the formalism given in Table I. The tensor components referred to the plate coordinate system may be calculated straightforwardly from the usual tensor-transformation formulas.(12)

Table I. The comp~essed matrix notation fo~ tenso~ indices,

ij 11 22 33 23,32 13,31 12,21

p 2 3 4 5 6

We will now describe the tensor transformation for the special case of a rotated Y-cut quartz plate. The transformed components can be found by applying the following transformation,

(2.20)

where cijkl are the transformed tensor components. and where in our case B is the rotation matrix, describing a single rotation ~ about the x1-axis, given by

0

B

[:

_-sin~ cos~ _coscjl

.~.+

(2.21) At this point we will restrict ourselves to a two-dimensional model of the quartz strip, i.e., we will assume that the wave vector is confined to the x2-x3-plane,

(2.22)

where nz =sinS and n3 = cose, according to Eq.(2,11). Substituting Eq.(2.22) into Eqs.(2.16) and (2.18), and then using Eq,(2.20) and Eq.(2.21), and the arrays of elastic, piezoelectric, and dielectric tensor components given in appendix A-1, Eq.(2.19) reduces to

Au - v2 p 0 0 0 0 A22 - v2 p Azs = 0, (2.23) A23

A33

- v2 p

where

+ --- (2.24)

(2.25) (2.26)

and (2.27)

One of the roots of Eq.(2.23) is easily found, i.e.,

(2.28)

The corresponding direction of the particle displacement may be found by substitution of Eq(2.28) into Eq.(2.17),

A. - A(l, 0, 0).

J (2.29)

The other two roots are not of interest here, because the corresponding modes cannot directly be excited piezoelectrically (13) since

Azz,

Az3 , and A33 contain no elements of the piezoelectric tensor. One of

the important features of the first solution Eq.(2.28) is the fact that regardless of the direc.tion of the wave vector the particle displacement A. is polarized along the x1-axis, and that no coupling

J

occurs to the latter two modes, which represent a quasi-longitudinal and a quasi-transverse wave. One should bear in mind, however, that these features are a direct consequence of the assumption made in Eq.(2.22). The first wave is the so-called "thiak:ness-twist" mode -shortly denoted by TT3 - which is a pure mode, i.e., a pure

trans-verse wave. The existence of this mode may also be inferred directly from the crystallographic symmetry of quartz.(l4) The properties of thickness-twist modes will be discussed in the next section.

2.6. Thickness-twist modes

In the preceeding section we obtained the dispersion relation Eq,(2.28) related to the thickness-twist modes. This equation is still rather complicated because of the presence of the piezoelectric constants.

We will show below that it is allowable to simplify this equation by neglecting the piezoelectric constants. To this end Eq.(2.28) will be rewritten as

pv2

p

where k2 _{is the electromechaniaal coupling constant~} t

(2,30)

(2,31)

The relevant elastic, piezoelectric, and dielectric constants referred to the plate coordinate system are given as a function of the crystal cut ~ in appendix A-2 at the end of this chapter, The value of k~ is calculated as a function of the crystal cut for three values of the direction

e

of the wave vector, i.e.,

e

=

80°, 90° (x2-axis), 100°. The results are plotted in Fig. 2.2, together with the behavior of the piezoelectric constant ~26• Inspection of Fig. 2.2 shows that for 80° <

e

< 100° k2 will be less than 0.027. It appears that the maximum

t ---e26 - - - kt2, 8=80° -·-·.: gcf' 101t I I l 0.1 c(I(O) I I I I / 2.5 1.5~

"'

... 1.0 0.5

Pig. 2.2. The value of the electromechanical coupling constant k~ as a function of the ccystal out 4> for> th!'ee values of the direction of the wave vector>~

e

80°(- - -)~ 90°(-·-·-J~

and 100°( ••••• ). The

drawn

curve gives the value of the piezoelectric constant e26 (C/m2) as a function of~.

value k2 _tmax ~ 0.031 is reached for~= 0°, 6

=

70°, and hence we will

assume that k~ may be neglected in Eq.(2.30) without drastically

affec-ting the description of the vibration problem. In Fig. 2.2 we have restricted ourselves to those directions of the wave vector, which are oriented approximately along the x2-axis. As will be clarified in section 2.6, this assumption is allowable as long as the length/thick-ness ratio of the resonator plate is large. Although the elastic as well as the piezoelectric constants depend on the temperature the

value of k~ at higher/lower temperatures hardly differs from its value

at room temperature. The symbols AT (~ ~ 35°30') and BT (~ ~ -49°) in

Fig. 2.2 refer to two of the most widely used crystal cuts.

One should bear in mind that in the special case of a rotated Y-cut quartz plate the waves are generated by applying an alternating voltage across the surface electrodes. In this way an electric field is created,

which is directed along the x2-axis, i.e., E1

=

E3

= 0, E2

*

0. Because

of the fact that the mechanical deformation u 1 2 of the thickness-twist

•

mode is related to the electric field component E2 by the piezoelectric

constant ~26, it will be obvious that these modes can only be strongly

excited if ~26

*

0. From Fig. 2.2 it can be easily seen that it will

be hardly possible to excite strong resonances in the neighborhood of

~ = 90°(Z-cut), since ~26 ~ 0 in this region. The complications of

this effect are considered in chapter VI.

Although energy considerations are not the prime subject of this thesis, it may be useful to illustrate the well-known fact that the energy velocity does neither have the same direction nor the same magnitude as the phase velocity, as is the case in an isotropic medium. The energy velocity Ye is defined as the propagation velocity of the - average - power flow density related to an elastic wave, In the

absence of dissipation v is identical to the group velocity v (15),

-e -g

which is given by

(2.32)-The group velocity is calculated from the following two relations,

and V -g (2.33) (0 - lll'l/()k21 - ()Q/llk31 ) • antaw k • lll'l/<lw k • 3 .w 2 ,w (2.34)

From these expressions we obtain

V

-g (2.35)

For any crystal cut ~. the magnitude and the direction of Yg may be calculated from Eq.(2.35) as a function of the direction of the wave vector. In Fig. 2.3 the group velocity and the phase velocity (cf. Eq. 2.30) are plotted as a function of the direction of the wave vector, indicated by

e.

The curve which represents the phase velocity

I 5 I \ \ \

'

_'

'

_'

'

3

'

_'

A 2 2 3 ...

Fig. 2. 3. The group e. q. energy ve ~oei ty ( dx>a~;m eurve J and the phase Ve~oeity (dashed eUPVe) aa a function of the di~eetion 9 of the wave veato~ fo~ a p~e Y-eut (yz~, ~ = 0°). The ve~oai­ tiea a~e given in units of 1000 m/a.

is called the norma~ a~faae, the curve which represents the group c.q. energy velocity is called the Ray a~faae. From a theoretical analysis one may derive that there exists a straightforward relation between the normal- and the Ray surface.(l6) Using this relationship we may easily construct the group velocity for any given direction of the

phase velocity by drawing the normal to the phase velocity vector in the corresponding point of the normal surface (see Fig. 2,3, A). The group velocity is then given by the vector pointing from the origin to the point where this normal touches the Ray surface (see Fig. 2.3, B). From Fig. 2.3 it can be inferred that the phase and group velocity differ in general in magnitude as well as in direction.

2.7. The two-dimensionaZ vibration probZem

In the previous sections we discussed a two-dimensional model to des-cribe the propagation of plane waves in an unbounded rotated Y-cut quartz crystal. However, in order to obtain a resonance problem we have to consider quartz crystals of finite dimensions. Therefore we will return to the two-dimensional model of a quartz strip of infinite

width (xl-direction), length 2~ (xa-direction), and thickness 2d

(xz-direction), where E

=

d/~

<

I (see section 2,5).

We start the derivation from a more generalized form of the thick-ness-twist modes,

(2.36)

in which a harmonic time dependence is assumed, Since the particle displacement of the pure thickness-twist modes is directed along the

x1-axis [cf. Eq.(2.29)], we will drop the subscript 1 of u. Since we

showed above that the piezoelectric constants may be neglected in the

description of the vibration problem, the wave equation corr~sponding

to the thickness-twist modes may be found by substituting Eq,(2,3) and Eq.{2.5)- without the term ekijEk- into Eq.(2,1), which yields

(2,37)

In the case of plane elastic waves, where A(x2,x3) is given by

(2,38)

the dispersion relation

is obtained, which is equivalent to Eq.(2,33), The set of equations is completed with the stress-free boundary conditions (the effects of the presence of electrodes will be considered in section 2.8),

o,

at xz • ±d, (2.40)

0, at xs ±t, (2.41)

In general the set of Eqs.(2.37), (2,40), and (2,41) cannot be solved analytically in terms of plane wave theory, which is due to the fact that c₅₆ has a nonvanishing value. We will demonstrate this by means of a so-called "sloumess"-diagram. A slowness diagram is a plot of the reciprocal phase velocity 1/vp of a wave as a function of the direction of the wave vector. The reciprocal phase velocity is found from

Eq. (2.39),

(2.42)

Now let us consider, as an example, the reflection of an incident wave (k . ,k . )

2l. 3l.

reflected

at the boundary x2 • +d. The incident wave gives rise to a

wave (k ,k ). Since the components of the wave vectors of zr ar

these waves directed along the boundary of reflection must be equal (k _s i

=

k _ar ), it follows directly from Eq.(2.39) that the angles of the two wave vectors with respect to the surface normal are not equal. This fact may be easily demonstrated by means of the slowness diagram of aY-cut (yx!, ~ = 0°) quartz.wafer plotted in Fig, 2.4. The

direction of the reflected wave is found from the following construc-tion. First draw the normal to the xz c.q. kz/oo-axis from the point where the wave vector of the incident wave intersects the slowness curve (cf. Fig. 2.4, A). The direction of the.reflected wave is now given by the vector pointing towards the lower intersection point of the surface normal and the slowness curve (cf, Fig, 2.4, B). The

angle of the incident and reflected wave vector with the surface normal is different, which is due to the fact that the principal axes of the ellipse, representing the slowness curve. are rotated with respect to the plate coordinate system. To obtain phase resonance in a crystal plate it is necessary that - after a number of reflections at the different boundaries of the plate - a wave is obtained, which has a reduced phase identical to the phase of the original wave. Fig. 2.4

8

Fig. 2.4. Slowness diagram of aY-aut (y~l, $

=

0°) quartz crystal. The ellipse repr>esents the reaiproaal phase veloaity 1/v

p

as a funation of the direction

e

of the wave vector. For an explanation of the symbols~ see te~t.

illustrates that the slowness diagram is a rotated ellipse, ~ence it will generally be impossible to solve the resonance problem in terms of plane wave theory. The rotation angle ~ of the ellipse with respect to the x2- c.q. k2/w-axis is given by

(2.43) This angle is also indicated in Fig. 2.4 for two other crystal cuts,

i.e., $

=

-15°, +15°. Note that the slowness curves of the other crystal cuts cannot be obtained from the curve for $ = 0° by a simple rotation only.

As we indicated above the resonance problem cannot be solved analy-tically, since for rotated Y-cut quartz crystals C56 will generally

be unequal to zero. There are a few exceptional cases, however, in which we may obtain an analytical solution of the resonance problem:

1. The one-dimensional model of the quartz resonator~ in which the wave vectors have only one component unequal to zero.

2. The two-dimensional model of the quartz resonator. in which one of the dimensions of the plate is infinite.

3. The two-dimensional model of the quartz resonator of finite dimensions with css equal to zero.

In the next section we will consider some of the features of these cases in more detail. In particular we will demonstrate that - under certain conditions - for rotated Y-cut quartz crystals c56 may be

neglected without really affecting the solution of the resonance problem.

2.8. Determination of

the

Pesonant fPequenaies

In the preceeding section we indicated that only in a few cases an analytical solution of the two-dimensional resonance problem may be obtained in terms of plane wave theory, So far we did not account for the fact that in general the crystal plate is partially coated with electrodes. The presence of electrodes gives rise to additional com-plications in the description of the resonance problem, since we cannot apply the simple stress-free boundary conditions given by Eq.(2.40). To avoid the complications, which arise from the additional boundary conditions which appear due to the presence of the electrodes, it is common use to apply the so-called "smaU mass approximation"

(cf. Chap. VII, Sec. 7.1). In this approximation the presence of the electrodes is only taken into account by their inertia forces. Since these inertia forces should be in equilibrium with the tension force on the surface, we have to replace Eq,(2.40), for those parts of the crystal which are covered with electrodes, by

at x2 "' +d, (2.44)

at x2 -d, (2.45)

*

where p, 6df' and p, 6db are the specific density of the electrode material and the thickness of the front and back electrode,

respec-tively; u is the particle displacement, The small mass approximation

- 1

*

* ~

may be applied if w~df(p/css)2

_<

_I,

and ~db(p/c6s)

<

I, where c66

*

and c66 are the elastic constants of the electrode material of the

front and back electrode, respectively;

w

is the resonant frequency

of the resonator (cf. Chap. VII, Sec. 7.1). In all practical cases these conditions are satisfied as long as the frequency shift, resulting from the presence of an electrode (or mass load), is less than 2% of the resonant frequency of the unloaded resonator,

Throughout the thesis we will use the concept of harmonia and

anharmonia overtones. In order to illustrate the existence of these

overtones, we will consider a simple two-dimensional model of a quartz resonator, i.e., a quartz wafer which is not covered with electrodes.

We will also assume that

c

56 is equal to zero. The resonant frequencies

of this system, which may now easily be obtained from Eq.(2.37)-(2.41), are given by

f (2,46)

In Eq.(2.46) q and p denote the harmonic.and anharmonic overtone

number, respectively; q,p

=

0,1,2,3,., •• Since in practice only the

asymmetric modes in the x2-direction ~f the crystal can be excited

electrically we may restrict ourselves to odd q, q = 1,3,5,,,, As may

be inferred from Eq. (2.46), .each of the resonant frequencies is

characterized by two numbers (q,p), e.g., (1,0) is the so-called first harmonic or fundamental resonant frequency, (3,0) is the third

har-monic overtone, and (1,1) is the first anharmonic overtone r~lated to

the first harmonic. The waves corresponding to harmonic overtones, i.e., (q,O), are characterized by a wave vector which has only a

com-ponent along the x2-axis (thickness direction), while the anharmonic

overtones (q,p

*

0) also have a component of the wave vector along the

x3-axis (length direction). Because in all practical cases d/1

<

1,

the resonant frequencies of the lowest order anharmonic overtones are just slightly higher than the resonant frequency of the corresponding harmonic overtone. Further, we may confine ourselves to the lowest order anharmonic overtones, since in all practical cases only these anharmonic overtones are strongly excited, because of the principle of "trapped energy" which will be discussed below in more detail, In case of a more complicated resonance problem, in which the crystal is

partially coated with electrodes and cs6 is not equal to zero, the

characteristics of the frequency spectrum are largely similar, For that reason we will also use the introduced (q,p)-notation to denote the resonant frequencies of such systems,t

Next, we will discuss some of the features of the three cases, in which an analytical solution of the resonance problem may be obtained,

1. The one-dimensional model, in which the wave vector has only one component unequal to zero, is widely used to describe the resonating properties of crystal plates.(l7,18) Such one-dimensional models offer the possibility to calculate the resonant frequencies without any simplification of the mathematical description of the resonance problem, e.g., it is possible to include the piezoelectric proper-ties of quartz and a complete description of the electrodes. In chapter IV we will present an analysis of a quartz crystal resonator with two different electrodes, Although the one-dimensional model explains the existence of the higher order harmonic overtones, it will be obvious that it fails to explain the existence of anharmo-nic overtones.

2. The two-dimensional model, in which one of the dimensions - the xa-direction - of the crystal plate is assumed to be infinite, may be used to obtain a more general form of the dispersion relations, (19) Within this model it is possible to calculate the dispersion relations - even if css is not equal to zero - in two situations, i.e., when the crystal plate is completely covered with electrodes or is not covered at all. However, since within this model the component of the wave vector in the infinite x3-direction may

have any arbitrary magnitude, it also fails to explain the exis-tence of anharmonic overtones.

In case the crystal plate is partially covered with electrodes the model will be used to explain the principle of trapped energy,

(20-24) Because of this principle the "boundaries", which separate the coated and uncoated parts of the crystal plate, behave like real boundaries, i.e., boundaries formed by the finite dimensions of the crystal plate. Therefore, the solution of this resonance problem is largely similar to the solution of the case in which

the crystal plate is not coated, but has two finite dimensions. The latter will be discussed below,

3. When cs6 equals zero, one may simply solve the resonance problem

by separation of variables (see, for instance, the discussion about harmonic and anharmonic overtones given above), There are four

crys~al cuts where css is exactly equal to zero (see appendix A-2).

The well-known AT- and BT-cuts are in the immediate neighborhood of these cuts, and hence for the AT- and BT-cuts it is justified to neglect css· For other rotated Y-cut quartz crystals there are no physical arguments, which would justify such a simplification, To

investigate whether it is allowable to neglect c56 , without really

affecting the solution of the resonance problem, we used a numeri-cal approximation method, in which the small mass approximation is incorporated. This numerical method was developed by F,J, Jacobs

*

(Dept. of l~th) and implemented on the Burroughs B6700/7700

com-puter in cooperation with H. Willemsen (Dept, of l1ath,). In this

way we succeeded in calculating the resonant frequencies of the lowest order harmonic and some of the corresponding anharmonic overtones of a conventional trapped energy resonator, These reso-nant frequencies can be calculated as a function of a large number of parameters such as the temperature, the dimensions of the crystal plate and those of the electrodes, Since a description of the numerical method itself and a complete survey of the results following from the analysis are outside the scope of this thesis, they will be published in a separate paper.(25) To investigate the

influence of c5s on the resonance problem the resonant fr~quencies

of aY-cut (yx!, ~

=

0°) crystal for which

c

56

=

-1,794 N/m2 were

calculated for various lengths of the electrodes and various thick-nesses of the crystal, and compared with the corresponding cases

with

c

56 zero. In the Figs. 2.5 - 2.8 the results obtained from

the computations are plotted. In these graphs the resonant frequen--cies of the first harmonic and some of its corresponding anharmonic overtones, with respect to an arbitrarily chosen reference

frequen-cy f f' are plotted as a function of the thickness of the front

re

electrode adf. The resonant frequencies were calculated for silver

*present ad.dress: Kon./Shell Exploration Production Laboratory, Rijswijk, The Netherlands.

j 4 60 0 120 ~ 4 60 \ \

'

.

·'

'\

. ,ob .,ooi <Agl 2d .. 0.2mm 21 • 1Smm 2b • 10mm y • 2/J ~'56"'0 --+-+--CS6~ . - - Sao,Jerbrey Fig. \

'

\ \ \ \ 6<1J,•lOO~!Agl \ \ \ \ 2d "'0.15mm \ 2{ ""15mm \ 2b *'Smrn \ y •l/J \ ---<>-<>-<>-<so • 0 __._... c56 -.o - - - - Sauerbtey '\ \ 90 ~ \ ~ \ \ ., _'\ w _\ \ \ A<l(,•<OO.&!Agl _\ 2d *0.2mm \ 2! •lSmm \ 2b •7-Smm \ y .. ;,2 \ \ 30 \ Fig. 2.6.

90'-~

'\ '\ ., \ 60 \ _\

Adb • <OOJl fAg I \ _\ Zd • 0.2mm \ 21 •1Smm \. 2.b ><Smm \ V •l/J \

'

JO _\ -o-o-o-"so -o

'

_.._._...(~...0 ~--Sauerbrey

°

Fig. 2. 7. 2 Mt no3!.Agl

Figs. 2.5- 2.8. These gi'aphs sho!/J the effeat of

ass

on the I'esonant fi'equencies of the fii'st hamonia and some of the ao1.'I'esponding anhal'-monia ovei'tones of a Y-aut aryatal. The fi'equenaiea al'e plotted vs. the thickneaa of the fi'ont eleati'Ode ~df. The open (asG=OJ and alosed (asot 0) airulea I'epi'esent the aalaulated data pointa. Foi' e:cplanation of the symbols~ aee text.

(Ag,

p

10.5 g/cm3

) as electrode material and a crystal temperature

T

=

20°C. In Figs. 2.5- 2.7 the results are plotted for various

lengths of the electrodes (2b

=

10, 7.5, and 5 mm, respectively) and

a constant thickness of the crystal plate (2d • 0.2 mm), These figures

indicate quite clearly that with decreasing ratio of the thickness of

the crystal and length of the electrodes (d/b

=

0,040, 0,026, and

0.020, respectively) the influence of

c

5s on the values of the various

resonant frequencies decreases. This is due to the fact that for

increasing y

=

b/~ the direction of the wave vector is rotated towards

the x2-axis. For that reason we may expect that with increasing

anharmonic overtone number p the influence of cs6 on the corresponding

resonant frequencies becomes more pronounced, This tendency may also be inferred from the figures. We also may expect that with decreasing

thickness of the crystal plate the effect of Cs6 becomes less

pronoun-ced. This is demonstrated in Fig. 2.8, in which the calculations were

performed on a crystal with 2d • 0.15 mm. We wish to note that the

figures illustrate quite clearly some of the well-known properties of a trapped energy resonator, e.g., the decreasing frequency differences between equivalent anharmonic overtones with increasing length of the electrodes and decreasing thickness of the crystal. In the figures we also plotted the frequency-mass relationship according to the one-dimensional model of Sauerbrey (see Chap. VII, Sec. 7.1),

In view of the results presented above, we conclude that for quartz plates of which the thickness of the crystal plate is small compared to the length of the electrodes (or length of the crystal in case

there are no electrode coatings), e.g., d/b < 0,03, the infl~ence on

the mass dependence of the various resonant frequencies is very small, especially for the lowest order anharmonic overtones, From a similar calculation with the temperature as parameter it appeared that the influence of css on the temperature dependence of the various resonant

frequencies is also negligible. As in the experiments which are repor~

ted in this thesis only the lower order anharmonic overtones are con-sidered, we feel confident to state that in explaining our results,

REFERENCES CHAPTER 11

1. H. F. Tiers ten, LinBar Piezoelectric Plate Vibrations (Plenlllll, New York, 1969), Chap. 5, PP• 33-40.

2. H. F. Tiersten, ibid., Chap. 4, PP• 30-32.

3. E, Christoffel, Ann. Mat. Pura, Appl.

1•

193 (1877). 4. A. W. Lawson, Phys. Rev. 59, 838 (1941).

5. It is necessary that the propagating direction of the wave vector S and the stiffness constant,.. are referred to the same coordinate system. It is not

l.Jkl

necessary that this system corresponds to the crystallographic coordinate sys-tem.

6. E. I. Fedorov, Theory of elastic ~es in arystaZs (Plenum, New York, 1968), Chap, 3, p. 89.

7. B. A. Auld, Acoustic FieZda and Mxves in SoLids (Wiley-Interscience, New York, 1973), Vol. I, p. 219.

8. Anonymus, Proc. IRE. 11.• 1378 (1949).

9. see for instance Ref. 8, and

w. G. Cady, Piezoelectricity (Dover, New York, 1964), Vol. 1, p. 35, p. 191. 10. R. Bechmann, Phys. Rev, ~· 1060 (1958).

ll. H. F. Tiers ten, LinBar Piezoelectric nate Vibrations (Plenum, New York, 1969), Chap. 7, p. 51.

12. H. Jeffries, Cartesian tensors (Cambridge University Press, New York, 1931), Chap. I, pp. l-11.

13. I. Koga, Physics

l•

70 (1932).

14. B. A. Auld, Acoustic FieZda and Waves in Solids (Wiley Interscience, New York, 1973), Vol. I, Chap. 7, pp. 236-240.

15. B, A. Auld, ibid •• Vol. _{I '} Chap. 7, p. 229. 16. B. A. Auld, ibid., Vol. 1, Chap. 7, P• 223.

17. E. P. EerNisse, IEEE trans. Sanies & Ultrasonics

li•

59 (1967). 18, H. F. Tiersten, J •. Acoust. Soc. Am. 21• 53 (1963).

19. R. D. Mindlin, J. Acoust. Soc. Am.

±L•

969 (1967). 20. H, F. Tiersten, J. Acoust. Soc. Am. 21• 234 (1963).

21. W. Shockley, D. R, Currsn, and D. J. Konevsl, Proeeedings 17th Annual Sym-posium on Frequency Contrat. (U,S. Army Electronics COIIIllalld, Fort Momnouth, N.J,, 1963), P• 88.

23, D, R, Curran and D. J. Koneval, Proceedings 18th Annual Symposium on Fre-quency Control. (U.S, Army Electronics Command, Fort Monmouth, N.J., 1964),

P• 93.

23. D. R. Curran and D. J. Kotreval, Proceedings 19th Annual Symposium on Fre-quency Control (U.S. Army Electronics Command, Fort Monmouth, N.J., 1965),

P• 213.

24. W. Shockley, D. R. Curran, and D. J. Koneval, J. Acoust. Soc. Am. 41, 981 (1967).

APPENDIX A-I

SOME DATA OF LEFT-HANDED a-QUARTZ

The arrays of elastic, piezoelectric, and dielectric tensor components (cf. Ref. 8) with respect to the crystallographic coordinate system X,

Y, Z, given in compressed matrix notation (see Sec. 2.2),

c l l C12 Cl3 Cp, 0 0 C12 c11 C13 -cllt 0 0 C13 C13 C33 0 0 0 , where c66 Hcu - c12), c pq C14 -cl4 0 C44 0 0 0 0 0 0 c., .. C14 0 0 0 0 C14 css

[

·~'

-ell 0 e14 0

-·~·

J

n·

0 0

J

e. 0 0 0 -el.,

_•

_{and s:ik -} £11 0 •

l.q

0 0 0 0 0 EH

The values of these components are given in Table A-1 and Table A-2,

Table A-1. The values of the adiabatic elastic constants (1010 _N/m2 )

at constant field of quartz at 20°C, according to Bechmann (ref. 10). The differences between adiabatic and isothermal values, as well as the differences of the values at constant field and constant strain are neglected.

C!2

8.674 1.072 0.699 1.191 5.794 3.988 -1.794

Table A-2. The values of the piezoelectric constants (C/m2_{), and the}

dielectric constants (nF/m) at constant strain, according to Bechmann (Ref. 10). Note that for right-handed quartz the piezoelectric constants change sign (cf, Ref. 8), The differences between the values of the dielectric constants at constant strain. and constant stress are neglected.

APPENDIX A-2

TRANSFORMATION OF THE RELEVANT TENSOR COMPONENTS OF A ROTATED V-CUT

QUARTZ CRYSTAL

Transformation of some of the elements of the elastic, piezoelectric, and dielectric tensors with respect to a set of rotated axes - the plate coordinate system - corresponding to a rotated Y-cut, specified

by the rotation angle ~. The rotation is carried out about the x1-axis,

We will denote the crystal cut according to the convention as described

in the "IRE Standards on Piezoeteatria Crystals" (Ref, 8). Only the

elements which are relevant for the thickness-twist modes are given,

In the formulas S and C denote sin~ and cos~, respectively. The

transformed elements are indicated by the superscripted bar.

Elements of the elastic tensor:

CsG cs6C2 + 2SCcs6 + cssS2,

Css

=

cuS2 - 2SCcss + cssC2,

csG

=

(C2 - S2_{)cs6 +}

Elements of the piezoelectric tensor:

e:zs

=

ezsC2 - e2sSC,

ea s "' e2sS2 - e2sSC,

ezs e:u;C2 + e2sSC,

e35 =-e2sS2 - e2sSC,

Elements of the dielectric tensor:

E22 EzzC2 + E33S 2,

€23 • (€33 - E22)SC, Eaa

=

E:z2S2 + €33C2•

SC(css - css).

CHAPTER III

EXPERIMENTAL APPARATUS

3.1. Introduction

In this chapter a survey is given of the methods which are used to determine the resonant frequencies of the crystal under test, the mass deposited on the crystal, and the temperature of the crystal. In sec-tion 3.2 we will describe both the method to determine the resonant frequencies and the experimental apparatus. After a description of the electrical circuitry the influence of the length of the various connec-tion cables on the resonant frequencies of the crystal under test will be discussed. Section 3.3 deals with the methods to determine the mass deposited on the crystal and the temperature of the crystal. A descrip-tion of the crystal holder, which is used to determine the resonant frequencies during the process of mass evaporation, is presented in section 3.4.

3.2. Determination of the resonant frequencies of the crystal under test

3.2.1. Measuring method

The r•sonant frequencies of the crystal under test are determined with

a passive measurement method, based on a phase-sensitive ~-network.

This method is recommended by the Technical Committee 49 (TC 49) of the International Electrotechnical Commission (lEG) as a standard for mea-suring the resonant frequencies of piezoelectric devices.(1,2) Using this method, the resonant frequency is defined as the lower frequency, at which the argument of the crystal impedance is zero. This method has the advantage that the resonant frequencies are almost independent of

the external electronic circuitry. In Fig. 3.1 a block diagram of the

measuring circuit is given. The circuit consists basically of a very

stable digital frequency synthesizer (Af/f

<

3 x 10-8_{/24 hours), a}

network-analyzer, and a phase-sensitive TI-network. The coupler and mixer are necessary for proper connection of the synthesizer and the network-analyzer. The network-analyzer compares the phase and amplitude