,RD-A42 456 STATIC FREQUENCY-TEMPERATURE CHARACTERISTICS OF DOUBLY i/iROTATED QURRTZ(U) ROME AIR DEVELOPMENT CENTER GRIFFISSRFB NV A KAHAN DEC 83 RRDC-TR-83-285

UNCLASSIFIED F/G 20/2 NL

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RADC-TR-83-285In-House ReportDecember 1983

S TA TIC FREOUENC Y- TEMPERA TURECHARACTERISTICS OF DOUBI YROTATED O UARTZ

LO

Alfred Kahan

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITEO

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In-house reportSTATIC FREQUENCY-TEMPERATURIE CHAR- July 82 - August 83ACTERISTICS OF DOUBLY ROTATED QUARTZ 6 PERFORMING DIG REPORT NIMSR

* 7 AUTMORIS, 8 COTRC O RANT NUMBER.A

Alfred Kahan

9 PERFORMING ORGANI~ZATION NAEAND A~PS SPORME ~'POESolid State Sciences Division (RADC/ES)ARAN*RI NMS '-

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QuartzS .Crystal resonatorsFrequency standards

20 ABSTRACT 'ConI,.,, 1n -. 1e-..0 d. It n.-I...Y -d ,d.nify b, block -b.,e)

hThe static frequency characteristics of crystallographically doubly rotatedquartz as a function of temperature, f(T), are described. Curves showing thesensitivity of the normalized frequency offset to small misorientations in angleare developed. Three different sets of currently accepted temperature coef -ficients of elastic constants are applied to calculate f(T). We snow that forspecific orientations of practical interest, the three sets predict significantdifferences in the inflection and turnover temperatures. The crystallographicorientations showing this angular sensitivity are the ones to be exploited

O 1473 EDITION OF I NOV 6S IS OBSOLETE Ucasfe

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A. 20. Abstract -Contd.

- experimentally to establish the regions in which the different set~ are valid.Detailed computations are performed for orientations yielding 80% C turnovertemperatures and for the SC-cut. L

-7.

-7 K27 1 U-N -7 1 17 7 - 7 -

Preac

NTI

D.r TI

J.. S t f.

By-.

d~ e--s

Dist 6 i-.

* DO

Distr3

. -% . -. -.C . r ' * . ." • • -.• "

'.a-. . '." '.

°% a--.

Contents

S1. INTRODUCTION 9

2. COORDINATE SYSTEM 11

3. TEMPERATURE DEPENDENCE OF FREQUENCY 12

4. MATHEMATICAL PROPERTIES OF CUBIC CURVES 13,.

a,.5. ANGULAR ROTATIONS FOR ao 0 AND T1 = T to 180

'7. FREQUENCY OFFSETS FOR Tt= 80 "C 29

8. THE SC-CUT 3 6

REFERENCES 391

Illustrations

-a.. ,, . .

1. Coordinate System for a Doubly Rotated Crystal Plate 12H 2. Normalized Frequency Offset as a Function of Temperatureand 0-angle Orientation for the AT-cut, =00 15 '

3.Normalized Frequency Offset as a Function of Temperatureand 0-angle Orientation for the BT-cut, 0 = 600 15

s14 4. Sensitivity of Normalized Frequency Offset to Small e-angleVariation for the AT-cut Around 80 OC Turnover Temperature 16U 5

. .- " . ." .

2. COORDINATE SYSTM I*1 ..'.. ... .

.....- o

Illustrations

5. Sensitivity of Normalized Frequency Offset to Small o-angleVariation for the BT-cut Around 80 OC Turnover Temperature 17

% 6. Comparison of Normalized Frequency Offsets as a Function

of Temperature for 0 = 0° Oriented for "Zero-Angle"(T i = Ttc), and for Zero First Order Temperature Coef- 3 "ficient(a = 0) 1

7. Loci in q and 0 of Zero First Order Temperature (oef-ficients of Frequency, ao = 0, Based on BBL(1P(i2)and BBL(1963) LSF Determined Temperature Coef-ficients of Elastic Constants 20"

8. Loci in 0 and 0 of T- = Tto Based on Three Different Sets ofTemperature Coefficients of Elastic Constants 2 1 - '

9. Loci of T. T as a Function of Rotation Angle 0 221 to

10. First Order Temperature Coefficients of Frequency of thec-mode at T = Tto as a Function of Rotation Angle 23

11. Second and Third Order Temperature Coefficients of Fre-quency of the c-mode at T =Tto as a Function ofRotation Angle 9 24

12. Frequency Offsets as a Function of Temperature for T i = " .

T and Selected 0-angles Normalized to the RespectiveInfection Temperatures 24 .5 -

13. Normalized Frequency Offsets as a Function of Tempera-ture for 9 = 20.710 o2 .

14. Turnover Temperatures for the c-mode as a Function of

Rotation Angle 0 for Selected 9-angles Between 0 and 260 2';

15. Turnover Temperatures for the c-mode as a Functionof Rotation Angle € for Selected 0-angles Between34.5 and 35.50 27

16. Loci in 6 and 0 of Constant Turnover Temperatures -.Between 60 and 100 oc 27

17. Loci in 6 and 0 of 80 OC Turnover Temperatures Basedon Various Sets of Temperature Coefficients ofElastic Constants 28

18. Normalized Frequency Offsets as a Function of Temper-ature for Selected 0-angles and 80 °C TurnoverTemperatures 29

19. Frequency Offsets as a Function of Temperature forSelected 0-angles and 80 °C Turnover Temperatures 31)

20. Frequency Offsets as a Function of Temperature forT - 80 °C, Normalized to Respective Frequenciesat80 OC 30

0-o21. Minimum Frequency Offsets Between 79 and 81 C asa Function of Rotation Angle 0 31 -.. ".

22. Minimum Frequency Offset Between 79 and 81 °C forBetween 19.4 and 220 32

6..#!:', -4O

,:* :.,., :

.4% %

- .- .

IllustrationsA -.

*- "Oo° .4

23. Normalized Frequency Offsets Between 79 and 81 C

as a Function of Temperature for Selected 0-angles 33

24. Interaction Between 4 and n0 Misorientations 34 "'"

25. Separation Between b- and c-mode Frequencies (Left "Scale), and Electromechanical Coupling Factors(Right-hand Scale) as a Function of Rotation Angle0 for 80 °C Turnover Temperatures 3 G

26. DO Adjustment as a Function of Operating (Turnover)Temperature for SC-cut, = 2 1. 930 38

p.

* 4-.'° °

IPI

° .

• V°°w°°"'

,:

0 *'O

C...:: .-. .

0

Static Frequency-Temperature -.

Characteristics of Doubly Rotated Quartz

1. INTRODUCTION

Single crystal a-quartz is the most commonly used material for piezoelec-

tric resonators. It is chosen because of its superior mechanical, physical, and

chemical properties. The most important performance parameter of a crystal

resonator is the static temperature -dependent frequency characteristic, f(T). of *

the device. The temperature dependence of the frequency is determined by the

elastic, piezoelectric, and dielectric constants of the resonating crystal, their

temperature dependencies, and the crystal expansion coefficients. In the tem-

4perature range from -200 to +200 0 C, f(T) of the thickness mode vibrations of

quartz is described well by a third order polynomial. Explicit algebraic equa-

tions relating the three coefficients of the polynomial to the material constants

and their temperature coefficients are summarized in References 2 and 3.

Quartz belongs to the trigonal crystal system, and material properties are

crystal-orientation dependent. Orientations exist that give relatively flat f(T)

(Received for publication 2 December 1983)

4%1. Bechmann, R. , Ballato, A.D. , and Lukaszek, T. J. (1962) Higher -ordertemperature coefficients of elastic stiffnesses and compliances of a-quartz,Proc. IRE 50:1812-1822.

2. Kahan, A. (1982) Elastic Constants of Quartz, HAD(-TP-82-117, AD) A121672.

3. Kahan, A. (1982) Temperature Coefficients of the Elastic Constants of Quartz,RADC-TIR-82-224, AD A 125709.

90

a. ° .%

PRFV1C!11US PAGE*aq ISBLANK

a................

.,.. . . . . . .

..O7-..7i-.7

over a wide temperature range. Typical exairpies i:: I>: .I . : r,.ot, ti

AT- and BT-cuts, and the doubly rotated SC'-cut.

The frequency may exhibit extremes as a function of temrnperatui c. Th-

temperatures corresponding to the frequency nax imum and minimum are desig-

nated as the turnover temperatures, To. For high stability oscillator applia-

tions, the resonator is enclosed in a precisely controlled ov(n and imaintainked at

one of the Tto points. Most military applications spt cify maximum temperatures "

to which the oscillator may be exposed to be bet\%cen 1; and 1I,1 (,7 he crystal

orientation is then selected to generate one Tto several degreecs a)oVe this uppe....

environmental temperature. Because T is sensitive to crvst:illwraphii o icl- "-"-totation, the tolerances on cutting the crystal are narrow, of te order of minutc S

or seconds of arc, and these tolerances must be mnaintained throughout the

resonator fabrication process. Both reiluirentents, prcise- ovn -ontrol and

accurate crystal orientation, affect the oscillator costs considrai,,'.

The singly rotated AT- and BT-cuts are the ('rystal orientations most w ilev -

used for high-stability applications. In the temperature range of practical

interest, the doubly rotated cuts, for example, the SC-cut, can have consider-4ably smaller frequency variation. In principle, one, can obtain several orders

of magnitude decrease in temperature sensitivity, but, in practice, this decrcasc

is coupled with considerably narrower angular toleran'es. )oublv rotated

orientations of interest include, in addition to the SC-cut, the IT-, iF-, BT-,

and LC-cuts. This topic has been reviewed extensivvly bv Rallato. , .

There are several ways to define the temperature-istsitivi orientations.

" The most common one, exemplified by Ballato, is th( an.ular o)rietntation cor-

S..' responding to zero values of the first order coefficient of temperature: a -''-

Cuts situated along ao =0 angles are attractive because only the second and third

order coefficients b and co contribute to the frequency offset, and V r) becomes0 0

a slowly varying function of temperature. Resonator parameters of interest

include the positions of the inflection and turnover temperatures, T. and TSto' .-

and the magnitude of frequency variation around T These parameters are

determined by all three coefficients and are not necessarily optimized at the

angles corresponding to a 0. Another description references the angular

positions of the temperature-insensitive cuts to the orientations determined by"_-

i to'

4. Kusters, J.A., Adams, C.A., Yashida, H., and Leach, J. G. (1977)TTC's - Further developmental results, 31st Annual Symposium on Frequency Control, 3-7.

5. Ballato, A. D. (1977) Doubly rotated thickness mode plate vibrators,Physical Acoustics, Vol. XIII, W. P. Mason and T. N. Thurston, lds., -Academic Press, New York, pp. lV-181.".-

II : -.-: -'.55.

U,. ." "i;2-21*.0.J

.'."5---

F.W.,~-y- 7-.* -7 -7. *

There are. three sets of temperatureL koe)ffic'ientS Of elastic (-,I ta flts-

T n(c ) that can be applied equally well for c-alculating f(T). In ,, iic5

these sets are deLSignated as 13131.19f12), 13 BL(l1 (;3), and Adais (,t il All f(ll

Calculations reported in the literature are based on 13 1L( I9;2 ). In IL~t fci&( ri; cv- 2

and 3 it was shown that B1BI.(V'(;2) may not prvdic t the f(T) charactc rt-tik -;o

arbitrary doubly rotated orientations well, and instead, I3BL.( IN3' LSI-V r

chiosen as the working set of material constants. In this report, the threef

T (c ) sets are applied to illustrate f(T), and show that for specific L~'n pqrotated orientations, the three sets predict significant angular dfeec5j

the positions of T. = Tt or a = 0. These crystallographic orientatio.ns ,lhoak -i to 0

ing this angular sensitivity can be exploited to determn~ e T (c )n p)q(

2. COORD)INATE SYSTEM

Figure 1 depicts the Y-cut plate geometry and crystallographic coordinate

system used '*n this investigation. The initial position of the plate is aligned

w ith the x-, y-, and z-axes. The position of the rotated plate, x', y, and 7',

is described by angles (0, f)), where (o and o are rotations around the z- and

x-axes, respectively. Owing to crystal symmeitries applicable to quartz, all

rotations are mnappedi into

0 6a(0o and 0 e 9 Oo

In this nomenclature, the approximate values of (p and o for the AT-, BT-, and000

SC-cuts are (0, 35) , (60, 49)0, and (22, 34)c), respectively. Another common-

ly used mapping region is

0 -s 1 Lc300 and -90 :5 ' !S+900

*CFor 0 > 30~ the two descriptions are related by the transformation

* 0'=600 -'6 and 0'=-

In the W.~t W) nomenclature, the 13T-cut becomes (0, -49)O and, for example,

the combination (40541, 1r, 34') transforms to (1~9 o) -1 l634').

LIN O -

,,,,,,€',,, .-. .. '

C-,, -. ~*-'.-

Rotated:i Crsa Plat

* N

qurzrsoaosigthe 1.eCoriate ange fr 0 touy 000-'.Te-hav

I ~ ~ ~ hw evalat the nrie frequency offseuntso of th tepransvre modaeviristins .O

. s£ ~are described well by a cubic equation of the form-".'

3

f f - Io""-',f(T) f f(T, o, o) = = =0 fn(, (T -To)0 0"

o o n=1I,...

a a(' -T o + bo( 'r--o ) + c ( - o3 ,'2.[ - "

".,.JO

0 0. xc ( O

where f is the frequency at temperature T, f the frequency at the reference

temperature T , and 1n(a0) are the first, second, and third order temperature

coefficients evaluated at T o . In most applications the series is expanded using,.

6. Bechmann, r. , Ballato, A. D.a, and Lukaszek, T.J. (1963) igher-OrderTemperature Coefficients of Elastac Stiffnesses and ompaters of .

r-Quartz , rSAELRDL TR 2261. r f

12-- "

s h n e u f t am b

are dscribd wel by cubc equtionof"th for'p • .

"-" '-, :,' ." "." : " :, -': "." ." - -" ", " f-f' 3'.i.? -': - . - . - - : , ' . -'- . "- - ." '. ~ -. - .

-;W~ .7.

.- .7° . . .

7P 7. •7,7

the reference temperature T = 25 0(. For ease of notation we designatefl(, 0), f2(0,0), and f 3 (O, 0) as a o , b o and c o , respectively..

For quartz, fn(P, o) are calculated from the six elastic, two dielectric, and

two piezoelectric constants, the temperature dependencies of these constants,

and the two expansion coefficients of the material. In References 2 and 3 we

discuss the validity of currently accepted material constant values of quartz, -2 .m.

and we list three alternate sets of T (c ). The first, designated as I3l,(1962),n pq-

0 is the one listed in Bechmann, Ballato, and Lukaszek, and is derived from data"reported in that publication. The second set, designated as BL,(1!;3) LSF', is

derived in Reference 3, and is based on the least squares fit of the data published

by Bechmann, Ballato, and Lukaszek. The third set, designated as Adams et al,

is given by Adams, Enslow, Kusters, and Ward. Unless noted other ise, the

numerical results of this report are based on BII(1(,;3) ISF'. For illustration

purposes, the particular Tn ( pq) set used to calculate Ti, Tto' or f(T) is not

critical. All sets give similar results, and no conclusions (an be drawn regard-ing the superiority of a particular set. However, the three T sets Pq o sho-

n pqsubstantial numerical differences in predicting f(T) characteristics for specific ..

(0, 0) orientations, for example, the SC-cut. There is now extensive experi-

mental data available on the SC-cut, and, in turn, this data can he utilized to

rederive T (cp) values.n pq

-. N-ATIIEI-ATIC.\L PROPERTIES OF CB|C CURVES

Be chmnn des cribed the mathemoatical properties of Eq. (1), %%hich

gives the variation of the vibration frequency of a quartz plate as a function of

temperature.

By definition, all Af/f as a function of T curves have a root at T . In

addition, two additional roots, Tr, are at

[ ,~~~/2] ""'''1 T) b 2 4 c ) (2)Fr, s = uF 2 cLo ± (b

0 - 4ao ( : )():".'O

0 0 . .

7. Adams, (A. , I:nslow , G. M., Kusters, J. A. , and Ward, R. W. (1970)Selected topics in quartz crystal research, 24th Annual Symposium onYrequency (ontrol, M5-:3, AI) 74r;210.

8. Bechmann, 1-. W, T;) ltrequency-temperature -angle charac'teristics ofAT-type resonators made of natural and synthetic quartz, Proc. ME31K44: 1 ;0)()- 1f;(7.

13S

il

,.-

".~% -- o

The temperature positions of the maxima in f(T), designated as the Tuinv"-

temperatures Tto' are symmetric around the inflection temarature T', and a re

-- V obtained by differentiating Eq. (1),

/ b 2 1/2T-"= F ° T° ± - (b -3a cto-"0Ttc 3 - ...

2/ o 1 2= i± [u - '- a 3 ] U:'. ....

,"-'.'i = r iT . (T . T° - 1 a° 3 c° .... ,

where

bST

° T.U-

0 j 1 0 3c

All f(T) curves have an inflection temperature, but the existence of T andtoS.1' three real roots depends %hether the radical terms in . qs . (2) "tnd (:1 ;ii- , I,.

or imaginary. The existence of three roots automaticai 1 implies the. xist( 1,"

of Tto, but the existence of Tto does not necessarily insure that ,fif =.

, satisfied at three points.

The general behavior of f(T) based on 13BL.(163) L.SF is illustratd in I< -

"" .ures 2 and 3 for -- inode AT-cut (0 = O0 ) and b-mode BF-cut (Q = ! u) alt'

plates, respectively. In the -G0 to +10) C range, tht slopes of th( AT-, ut

curves decrease N% ith increasing O-values; at (0, 35. 25) the radical term of

Eq. (3) vanishes and T = T. 11.86 C. For (0, 35. 5) " increases toto I

T. = 16.75 0 C, the Tt are at -47.00 and 8!). 51 C, and the roots are at -r7. 57,

25, and 122.82 °C, respectively. Similar considerations apply to the BT-(ut.

At (00, 50. 81)0 the radical term of Eq. (3) vanishes and Tto T. f,-. 3 . . -to I

'-4 The normalized frequency offset at T. is

= (b!27c:) (2 -oa)

and at Tt

-f1)7 c2 (2 b 3 , )u (b2 c 2 4

f = (o ) -.

. .b) 2 0 3a o )I

- -2 c ( -to o -T .)S14o0 .•-.

N..

•1 .%

4 . . . . . . . . . 4.... . ... .

.4 ".* *".. . .

150

150 BBL(1963) 1SF c-MODEm - .1

8 34 5' 347

35100

50 -

10

00

100

() 0TEMPERATURE (T60 801)

-e Figure 2. Normalized Frequency Offset as a Function of Tern-*4perature and o-angle Orientation for the AT-cut, 9 00

250 . - -~~b-MODE

200-530 4

150446

0

o0.5

10 4

-150

-200 -

-250 L. . L.... -L..

-60 -40 -20 0 20 40 60 .s0 100.11\IPERATURE ('C)

Figure 3. Normalized Frequency Offset as a Function of Tern-%4 p erature and n-angle Orientation for the BT-cut, 6 Go

15. -4

t o tl j( jj'l V O lS t 1 iii!1 1iill tiol? I ti :I-

ott,

11'sjoctivel .\tTF the t'ilJ VItW 'd j l((l(( ff--II1(,fl

tit' _k F-Lot is 2' 1) -an ld for' tilt, 13'T-i-ut it is fioi / 11) /0 (. fh figure'(s

alSO sho%% til tn 'litiv 'it\ of 4I/ to small changes in (I . Shifts occLUr both in tia-

-r to positions anfd inl )f 'if . In fabricating resonators, the f 0 translation duti to

-anlgle. ILL isortenta tion is Lomnpensated by adjusting the plate thiLckness, and theL

major pra(ALal effect of iniisorientation is a shift in T Fo r T between 5oto, to

and 10 OCi C, the t shifts bv 2. 6 0(*/m mute of arc for the AT-cut, and o. :J5

0 m-/IIiute- of ar~c for the( LIT-cut.

1% 1[SF

00 AB- -2'

CL- 000

C,-4

-60 A-U

60 70 8o 90 100

TEMPERATURE (OC)

Figure 4. Sensitivity of Normalized Fre-quency Offset to Small 0-angle Variationfor the AT-cut Around 80 0 C TurnoverTemperature

Inmn xeimns ()dtai nviabea 50Cadtenr

0

malized frequency is fitted with a cubic equation

=/f a (T -T1

+ b 1

(T-T1

+ c (T(7) -

referenccd to temperature T 1with coefficients al, b1 , and~ c 1 ,. h transforma-

tion to re-ference( tempe.rature T is given by.0

a [a 1 + 2b (T -rT + 3 c (T 0 Tl)1 (8)

1% 0 0 p-

170B BL Q %3)LS F 0- 600

0.

ISO

14060 70 s0 90 100

TEMPERATURE (.

Figure 5. Sensitivity of Normalized Fre-quency Offset to Small 0-angle Variationfor the BT-cut Around 80 0 C TurnoverTemperature

b = [b + 3c I(T - T 1)]/p()

c c c/p (10)

2 1

16 - L2" ]" '-

p I + a(T 0-TI + b (T 0-T I)+ c(T 0-T) 1I

1 0

Of particular interest is the normalized frequency offset referenced to T. This-* o Ifunction, anti-symmetric with respect to Tis given by

T....

(f f f.)/f. [ i + c.i(T T Ti 2] (T -T. (12)

where

a, -(b7 3a8 0 c /3cq (13) e

Figure 5.Sniivt £Nom/zdq;r -i'"'

c C / (14)

and"d

s,. .". .-"S.-:

179%.

T ' 1 3

S'' ::

p = + a (T° - Tl + l(T° - + l(T - ( I '' .:,4 fp ri u a n e e ti h o m l z d f e u n y of e e e e c d t i h s''< i:"-'f n to ,an.s m e r e wih r s e tt T4i i e y " S::: :.

-4(f " fil/fi~ - = [a * + *............ i •(1) -.

" ,% .W .7

f. b( 2b2 - 9ac)q = 1+ 0 2 (15)

0* 4, V r% .'

The turnover temperatures, symmetric with respect to Ti, are given by

a. 1/2

Tto T i 3c-.4:.:, . .

* 5. ANGULAR ROTATIONS FOR ao 0 AND Ti Tto

- Ballato a illustrates many f(T) characteristics of doubly rotated crystal cuts

by considering the loci of the angular orientations (€, 0) corresponding to the

special case of a = 0. In this report, we prefer to reference characteristics

:'. to the condition of the two turnover temperatures coalescing with the inflection

temperature, Ti = Tto. The corresponding angles are also known as the "zero-to*

angles", or the "ideal" angles, and f(T) variations around T. = T remain small9 1 to

over a wide temperature range. 9 Mathematically this condition is satisfied when

b 2 - 3a c = 0. Figure 6 illustrates the difference between the two formulations

0 0 0

for the c-mode at = The T = Tto curve is the one depicted in Figure 2,

= 35.249 ° , and it corresponds to the lower limit of 0-angles with real maxima. k' 900

• ""The t0-angle for a.= 0 is 0 =35. 259 . The 0. 6 minute of are difference between"'..,-.-'

the two 0-angles is sufficient to cause parts per million offsets in f(T).

An inspection of Eq. (3) shows that for a = 0 the Tto are real, and that theyare located at T and at 2T. - T . The special case of a = 0 describes the con- -',

0 1 0 0dition of one Tto restricted to T0 For the a = 0 curve shown in Figure 6,

T. = 12.07 °C, Tto = -0.87 and 25 °C, whereas the temperature corresponding . ,.-I t

to the "zero-angle" is T i = T = 11.86 0 C. Figure 7 shows the c-mode (0, #)it~ 0

loci for a = 0, between 6 = 0 and 0 = 30 based on BBL(1962) and BBL(1963)

LSF. The figure also includes a graphic representation of the straight line

formula for a = 0 given by Ballato in Reference 5, Eq. (85). We note that at A

*.:.: b = 220, near the SC-cut, the 0-angle based on BBL(10963) LSF is 34. 500, the ".. ..

o. 6-angle based on BBL(1962) is 34.000, and the straight line approximation gives

33. 91° . The corresponding 0-angle computed from Adams et al is 33. 86 ° . At

higher -angles the divergence increases, and, for example, at Q = 300 the dif-

ference in 0 based on the two T (c ) sets is more than 10. Considering the fact_______n pq

9. Vig, J. , Washington, J. W. , and Filler, R. L. (1981) Adjusting the frequency ,vs. temperature characteristics of SC-cut resonators by contouring, 35th -.Annual Symposium on Frequency Control, 104-109.%

* 18-9

4%

.4 :?. ..'p. ?•

: * ~ .-.

.

BBL(1963 LSF

000 0

00

-20 0 20 40 60TEMPERATURE (0C) 0 7

Figure 6. Comparison ofNormalized Frequency Off-sets as a Function of Tern-perature for ¢ = 00 Orientedfor "Zero-Angle" (Ti = Tto),and for Zero First OrderTemperature Coefficient(a 0 = 0)

- ..--.. .:

that for any €, a0 = 0 rigorously defines T = 25°C, the substantial differencestoin 0-angles predicted by the T (c ) sets are easily accessible to experimental '.n pqverifications.

Figure 8 shows the c-mode (0, O) loci for T i = Tt based Adams et al,

BBL(1962), and BBL(1963) LSF parameters, respectively. At higher 0-angles,

similar to a = 0, the curves diverge. Figure 9 shows the corresponding T i .

T For all cases, T i = T increase with 0, but there are substantial differ- PTtoi toences in numerical values. Figures 10 and 11 show the first, second, and third

order temperature coefficients ao , b , and c , for T i = Tto based on the0 O 0 ' to"'.%". .

BBL(163) LSF parameters. At 6 = 10. 56, a° = b° = 0, and T = T = T :25 0 C.0-i

Figure 12 shows c-mode (f - f) 'f. as a function of temperature for T i Tto "r.:

and several 6-angles normalized to their respective frequencies at T i = Tto.

Consistent with the fact that T. increases "~ith 0, the "flat" region of the curve

shifts to higher temperature, and at the same time it becomes broader. This -.-

can be seen in a clearer manner by noting that in Lqs. (12) - (15), for T. : r 1 ,a. = 0, q = 1 - (a b /9c o ) I, and the normalized frequency offset referenced to I l

1 00 0

T. reduces to

d - . . .

. 4 q% % ~ % . . 4 '-. .. . . - . .~.. . , o , . o . . ... .. -, -% ;.," • " " '" "' " " "° " "w " '" °

." "- " " " • "- °,- ". . " - m " " ." ". .' - . .. ". " - ",,. ". "% .- ". " ",

*-.:.- - - -.

3A5.5*A 35.25 -(11/180).

L. - BBL62)

C - BBL(%3) LSF

4- .-.

, . .

35.0

G C

W0

*1 . -.- w-D.~34.5 B

A

34.0

c- MODE"%."..33.75 I . -

0 5 10 15 20 25 30ROTATION ANGLE 0 (DEG) ' U

Figure 7. Loci in 0 and 0 of Zero First Order Tern- -

perature Coefficients of Frequency, a = 0, Basedon BBL(1962) and BBL(1963) LSF Determined Tem-perature Coefficients of Elastic Constants

(f-f.)fT' = c° (T T )3 (17)

Figure 11 showed that c0 decreases with increasing @. Thus, for the same tem-

perature difference, (f - f )/f i , proportional to co , will also decrease with in-

creasing .

The frequency offset AF i. for a temperature range AT = ±(T - T.) aroundT T is1

I to ' ,...

AF. = c.(T - T.) 3 = 2c.(AT)3 2(co/q)(T 2c (AT) 3 (18)i 1 0 0q)""")

20

*° °°, "

°- .. 4--:.-9...%..-.

35.5 ', .

... 5.

" -"34.5

CMC

. 34.04-o

z A -ADAMS et al33.5

o ~~B -BBL(I%2)":"--"

;!s CCB(93 S

33.0

32.5

c -MDE32.0 f

0 5 10 15 20 25 28ROTATION ANGLE * (DEG)

Figure 8. Loci in @ and 0 of T i = Tto Based on Three Dif-

ferent Sets of Temperature Coefficients of Elastic Constants

Figure 13 shows 6F between 79 and 81 °C for T i Tto :80 °C. For 6T to C.

0 1T - T. = ±1 C, F. = 11.2 x 1011. The f(T) curve corresponding to T i = to

does not define the absolute frequency minimum between two temperatures. It '

is generally true (Reference 6) that for any temperature range ±6T, depicted

in the top scale of Figure 13, the minimum frequency offset is obtained by choos-

ing to, 0) so that T. is halfway between -6T and +6T and Tto are situated at

±6T/2. The frequency offset for the minimum condition [Eq. (6)] is given by

3 3

6F = 2c 6 ) + 2c0( "T =-0-(6T) 3 (19)

and the (f - fi)'f values are equal at -6T and 6T12 (the upper Tto), and at +6TI I- ,'

and -6T/2 (the lower Tto). Figure 13 also shows f(T) corresponding to 6Fmin"

21

'CC,

-.,....C-.:C-.-,.C,-CC.,",C *C:.. C'CC-C C -.. -..C . -

, # , , .. .. . ..,..* C . '. C. -.

175

A- ADAMS et al SB -BBL(1962)

150 C -BBL(19t3) [SF

125 0

0

75

50

A O~tO AN- *(OEG)

Fi5r 9.Lc fT=T 0 a unto fRtto nl%i

whereas T .detrmne ofTi e differenci n he RFtadtFion -angle is

to 0veryh smallie T. 8007',d bu this is sufficientto reptectfreuny ofhet nor-

mae eryqiencal Equaetion (18)qua and (9) show th5Cad for 79. bot and and

0. 0tatd is, for the teprue rang from0 to unCtirque ncy offsets cor-2.7

repnin to T.to 0 i F .2vl~*a atro agrta o1,whereas teminimu .Thdfreuenc i otaied by choos6Fin thO-anglei

buc as ator positio Te at 75 vand85 C.rrsThnisg gitetuves of =2.84 e 130facre ofarly ldnae thaos (1) but a1 facor ohfosar Tbth 6~F..an

22 inres as(T n ht6 6 . o xml oT±0

i~ to

0, 8

suha opstonTa 5ad8 C. Thi gie 6FC 2. 84 y0**'t min *..* . . . . . . . 7.~.. . ' .

3 ~ ~ * S

"' BBU1963) LSF

10-7*

4.~ V / 4

-c 1OD10 -0L

0 5 10 15 20 25 30ROTATION ANGi[ O(DEG)

Figure 10. First Order Temperature Coefficientsof Frequency of the c-mode at T i = T to as a Func-tion of Rotation Angle 0

onsequently, the "zero-angle" is not the "ideal" angle to get a minimum vani-%

ation of the static f(T).

The following example illustrates the recipe for selecting the optimum

doubly rotated (0, n) orientation for the minimum frequency offset, 6F m in' in

an%, temperature range t6T. Assume that the temperature range of interest

extends from -55 to +85 0 C. This defines 6T = ±70 0 C, T. 13 0 C, and Tt0

T. :t 6V2, at -20 and +50 0 C. For T = T = 13 0 C, Figure 9, curve C, givesi i to

,Pz . Fo r 0=4 , the 40 ,-0 To positions define the 0-angle as 0 = 35.285 . From

Figure :11, o 40 c =103.3 5 1012 and, from Eq. (19), we obtain 6Fi3 0 6mi

0. 5 c(7 0) =17. 8/1000

23 IV..

%4

,, .P--o*. .*.* ... o. o .- * .'-'. . . .. . . . .• , o ..- ... ,.-. . * - . . .2 1 . .,- ,

-..

... .:. -::

5 -.-b I- BEBLLt%)) LSF" '

i ,S

0.

J100 o "5 - u - So " "

--5 -- -

-- ----

, o . " .

%0

%

%°

%

5 10 15 0 25 0 J

ROTA1 IN Gll (DEG)

Figure 11. Second and Third Order Temperature . 'Coefficients of Frequency of the c-mode at Ti =

Tto as a Function of Rotation Angle "

%. %5=-.-

15 1 9 2 -3. -

C25

Q.r

_o 5.... .: r *::

*,:. .:: -: .:

/--J .

-i/ %M." - a

A e I

Figure 12. Frequenc'y Offsets as a lIuntion of "r,-" .p e r a tU r e fo r i -- T to a n d 0(I , t~ -a n l s N oi') a l-

- . .- - ..

ized to the liespectiv( Infle(tion leine(,ratures

24

%. **=f* *.- /' *.T .-..... ...... ,-;,.#, S°

-AT -&T/2 0 &T/2 aT6'-- II 11

=20.7 1' A'

Sb 2 AF 1 ...

* 0

"--2 AFmin -T

=2071I /

.-- - -" 'C-MODE

.- 9 00 2 Fi /

TEMPERATURE (*C)

Figure 13. Normalized Frequency Offsets as a Function of Te""- -.

perature for 2 = 0. 710. The curve depit ted as 2 I'j correspondsto T= Tt= 80 0 C. The curve designated as - is the m-

-~~~~ 0 .. :,

imum obtainable frequency offset between 79 and

6. LOCI OF TURNOVER TEMPERATURESM T .

In previous sections we have discussed the special situations arising from

a i0 and a. 10. For high precision oscillator applications, the resonator is .'O0 1

enclosed in an oven and operated at T t, designed to be several degrees above".-

the highest anticipated environmental temperature. For military applications,

T m tlrange between 60 and 100 C. The static f(T) characteristic for a specified

% ~T is determined by all f (0, 0) coefficients, and a judiciouIs Selection Of a doulyNto n%rotated angular orientation may result in improved performance, regardless of .- /'

-.,

*the conditions defined by a = 0 or a. = 0..0- 01

Figure 14 shows (0, 0) loci of Tt for, c-mode vibrations between 0 and[

160 0 C for selected 9p-angles between C) and 26. These curvsaerpoue

Sfrom Reference 10. For each -angle the curves are continuous and -ultivalucd.

The horizontal markers on the curves indicate T = T values, that is, thero 1-

0-angle limits for which Tt exist. The curve segments above the markers

define the upper T hereas the curve segments below the markers denote the .

9to-

lower turnover temperatures. T = T increases with and for a specified Ti to tu

10. Kahan, A. (1982) Turnover temperatures for, doubly rotated quartz, .thAnnual Symposium on Frequency Control, 170-181).

25

of

__

the ondtios dfine bya ° ora i . ........

Figre14shws(@ 0 lciofTt fr -. 5ode. virtin be S¢c - n .O

War." iml"° . .- °-

%--00

_'2 .. .2

• - .j

tt/.

40

%0

v .-, . . .

% 33 5 34 34 5 35 35.5 36 36.5

ROTATION ANGLE 0 (DEG) I .

Figure 14. Turnover Temperatures for the c-mode as a .Function of Rotation Angle 0 for Selected 0-angles Be-

.- ¢, 5tween 0 and 200. The horizontal bars indicate T i = TtoPositions

-. _-..

one may be situated either at the upper or at the lower Tto. For a given 0, the

* slope indicates 0-angle sensitivity, and the figure shows that at T. Tt small

0-angle misorientations will drastically shift Tt. For (P 0 the standardto*,tesadrAT-cut, the most sensitive region is at 12 °C, whereas for 0 200, the sensi-

tive region has shifted to 73 °C. For the SC-cut, O = 21. 930, a 60 = 28" mis-

orientation will shift Tt from 80 °C to 75 o(.

.e % Figure 15 shows the same type of information, Tto as a function of 0, for

'Yri selected 0 in the -100 to +200 range. For a given 0 Tt not necessarily

a continuous function. For example, for 0 35. 50, the upper and lower Tto ,'". .'.O

branches are separated, and for specific 0-angles there are defined temperature

- . ranges where Tt do not exist. A clearer illustration may be obtained by pre-% .to°

senting this data as (0, ) loci for constant Tt0 . Figure 16 shows the loci for

T between 60 and 100 0 C, in 10 C intervals. Consistent with the fact thatto

T i = T shifts to higher temperatures, T curves are well separated in 0 forto StoC

small 6-angles, coalesce in the 6 = 15 ° to 6 = 250 region, and again separate O

between 6 = 25 and 6 = 45 . For a specific Tto, 0 decreases with increasing

6 up to 6 z 250 increases between 6 250 and 6 370 , and again decreases

*:. .. with increasing 6. We note that a continuous (6, 0) region exists that gives Tto

26 . .- -.

% 4

,.0.. °

b'A. i ..

200

Li35.50 / e34 50 ~100 -

- 35 0

35 25 34 75 7,.

o -7z

6:35 50

0 10 20 30 40

ROTATION ANGLE O(DEG)

V iguitI . ral-2er Tempc'a turcs for the -mod<aI'u fl t io 11 of otation A ngle 0 for Se( ted ',-aI ules 13(t,..e n 34. and :3-. :0O

000

5 30 45

*Figure 16. Loci in 0 and 0 of ConstantTurnover Temperatures Between 60and 1 00 0 C

27

'4 0 0

-4-O

values between 60 and 100 °C. This allows us the flexibility to select an orienta-

tion that also optimizes the resonator with respect to some other performance 0 0

parameter.

The shapes of the curves illustrated in Figure 16 for (0, 0) loci of Tto are

not peculiar to BBL(1963) LSF. Computations based on 1BI(1962) or Adams

et al result in similar shapes. Figure 17 shows the (, 0) loci for c-mode T =80 0 C based on the three Tn(cpq) sets. The actual o-angles differ, but the O -

general pattern is the same. Similar to Ti = Tto, the higher 0-angles serve as

sensitive indicators to determine the validity of the various T (c ) sets. Inn pq

this report we are interested in illustrating f(T) behavior, and we omit the Tt.

computations for the angular region between 0 = 450 and 0 = 600, shown in

Reference 10. The results for this angular region are very complicated, are O O0

influenced by the b- and c-mode crossover near 0 z 50 , and are very sensitiveto T (c ). In principle, this would be the ideal angular region from which to

n pqderive an accurate set of elastic constants and their temperature coefficients.

Similarly, in the T i = T and a = 0 computations we do not show results for - -i to 0

> 3 0° . This is because calculated temperatures for these O-angles exceed t- 0

• z36

35

2 34

433 A-ADAMASet al

32 S ,

31 1c-MODE0 1 5 i 30 45- . ...'" ""

ROTATION ANGLE . -EG)

Figure 17. Loci in 0 and o of 80 °C TurnoverTemperatures Based on Various Sets of Tem-

4 perature Coefficients of Elastic Constants

28

" - .:- .-' "

......................................................................

040

200 °C, the experimental limit of the data from which Tn(cpq) were derived.

Also, the results are complicated by interferences of the AK-cut and other S

branches. 10

7. FREQUENCY OFFSETS FOR T10 80 °C

We extend the discussions of the previous sections by considering the staticf(T) characteristics at T = 80 C. Figure 18 shows the normalized frequency

tooffsets as a function of temperature for several 0-angles for Tto = 80 0C. We

note that with increasing ¢ the slope at Tt becomes less concave, reflecting

the fact that T i increases with 4. At 4 20 0, on the scale plotted on this figure, *- ;O

the curve at 80 C is "flat". For larger e-angles, T i increases above 80

T becomes the lower turnover, and the curves become convex. Figure 19to

shows, on an expanded scale, the normalized frequency offsets between 79 and

4' 81 °C for Tto = 80 C for selected 0-angles. The frequency offsets are normal-

ized to the respective frequencies at 80 0 C. The temperature dependence of

frequency decreases with increasing 4', and, between 4 = 200 and 4 = 220 the

curves change from concave to convex. This would imply that there is a unique0

"" for which the frequency offset between 79 and 81 C' is zero. As illustrated on

" an expanded scale in Figure 20, this does not happen. The T i = Tto = 80 C- . -

condition occurs at 4 = 20. 71. For smi;lle' ;ingles, T 80 C is s;tisrieI by 0t 0 ( o ltiq[i lb

00

02

C " 100

CL". .

03

30 . ,"-,20 430 4

"-- ~~~-100 "-" '

-100 0 so 200TEMPERATURE (00)

- Figure 18. Normalized Frequenc!y Offsetsas a Function of Temperature for Selected_-angles and 80 1 (' Turnover Temperatur(,s.. --

29 -'..I

4. O

4 - -:-.>,V.,-; 0 ,:-,< .:.<) .:i.;.::::.,..:,...)...:;?.,..)..... :, :

-. * *.,,, * * .*'-.- ** . ... W.7, ... *. ,,. .7 .S .,-,., . 7-W % . . .. .o.-. 17 1.77 - .,

20

110

o=Oo /

0 - " -

'" / // //15 .

g. . . .-''.0

20

o 22

4025

79 80 81

'.. TEMPERATURE (°C)

Figure 19. Frequency Offsets as a Function of Temperature forSelected 4-angles and 80 °C Turnover Temperatures. Offsetsnormalized to respective frequencies at 80 °C

b=20.65°

20.680 20.710 20.740 20.77'1. 5

." /. .,

.1 ~0.5L -%

- .6

-0.5

~1. - L /.. "

STt = 80 C, Normalized to Respective Frequencies at 80 0---4 20. 710 corresponds to T1 Tt = 0 c

' 30

-:.., . 5. C * 'q e q........-.

- ", - - - -. - . - . --, • -. r..r'. -. •

0S

the upper Tto. 0\s the 0-angle approaches 20.71 , the lower Tto (convex) also

approaches 80 C, and the temperature separation between the upper and lower I .

T decreases. At Q = 20. 680, the two turnover temperatures are within 0.C5%to0

of each other, and at 20. 71 , the Tto coalesce "ith T. (The 0 = 2). 710 curve.

is identical with the AF. curve of Figure 13.) For larger 6-values, the Tto

exchange roles, and Tto = 80 °C is now given by the convex lower Ttc. Conse -

queritly, with one of the Tto values restricted to 80 °C the frequency offset b- "

tween 7 P and 81 0oC is finite, occurs at 0 = 2 0. 710, and has a value of%Io

1. 12 / 10I In Figure 13, the curve marked gFmin' depicts f(T) correspond-

ing to the minimum frequency offset, with T not restricted to 80 C'.to

Doubly rotated cuts allow improvements in the static f(T) characteristics of

the resonator. Figure 2 1 shows the minimum frequency offsets obtained between " S79 and 81 oC as a function of 0, with the turnover temperatures not necessarily

0 0

restricted to TC = 80 C(. Figure 22 expands the region betwveen 9 = 19. 4 and._{-",

22. 00. At O = 00 the frequency offset is 20 / 10 - 9, decreases at 0 = 20.71

to a minimum of 2.8 , 10 - 1 increases at O = 300 to 12 / 10 - , and between

9 = 30 0 and p = 45 ° it oscillates between 10 and 12 / 10 - 9 . The top scale of ne .

Figure 22 gives Do, the 9-angle referenced to9 = 20.710.

We have shown that for a resonator operating between 79 and 81 °C, the

normalized frequency offset for doubly rotated orientations can be reduced by

almost 3 orders of magnitude over the singly rotated cut. This decrease in

-.-.. 4%:? T"I20 ,.

j 16'

12- - '"

- ". "" ." ",.

00 10 20 30 40

ROTATION ANGLE O(DEG)

Figure 21. Minimum Frequency Offsets Between I "79 and 81 °C as a Function of Rotation Angle 4.Turnover temperature is not necessarily at 80 °C.The 0 value is chosen to give minimum offset ateach

.

31

3. ....1

,- ." .- Oo

5= SD:,

D+ (minutes)-75 -50 -25 0 25 50 75

2.0J

/ I

0 1.2-

4 2 2. 2

Figure 22. Minimum Fre-quency Offset Between79 and 81 uC for 0 Between 19. 4 and 220. Topscale references 0-angle to 0. 20. 710, the an-gular position of the minimum

temperature sensitivity is at the expense of a decrease in allowable crystal-

lographic tolerances on (0, 0) throughout the resonator fabrication process.

Numerically, the following example illustrates the problem of crystallographic

tolerances. Assume that for the doubly rotated cut operating between 79 and

81 "C we are satisfied to improve resonator performance by 1 order of magni-

tude over the singly rotated AT-cut, a reduction from 120 y 10- to 12 Y' l0-IFigure 22 showed that this offset can be satisfied in the continuous angular region

between *=1..47- and 2 21.9030, a range of 2. 4 6. We carried out computa-

tions to determine the corresponding 0-angle range, and the tolerances on 0 or 0g

Swhich have to be maintained to achieve this improvement..

Figure 23 shows, for selected 0-angles, examples of 12 y' 10 I frequency

,- .-. ',,..."..0 . . °°o >

offsets between 79 and 81 C. For each 0-angle, we also include the curves.

def ining the m inim um off set. For example, for 0 = 2 1. 00, the minimum offset

-9 0 . . . .

is 0.148 10 . and the Tt are at 79.94 and 85.82 C. The two curves with

_12 X10'91 offsets have T at 81.29 and 84.47 .C, and at 79.05 and 86.73 "C,

4.4 respectively. For all three curves T. = 82. 89 dC. This is also true for all

%- other sets as well, that is, T. is the same for the set, is determined by the% 0-agland he 1 x 1- 9 1 1

0-Ngeicandy the 2fo0ow limnits are obtained by changing the 0-angle.

Figure 24 shows the allowable Do and Do deviations and ±0 and ±60 toler-

ances, refreenced to 20.71 that have to be maintained to be within 12>)e 10

frequency offset in the 79 to 81 C range. We reference angular deviations to the -.-

32

*.bten =1.4°

ad€=2I 3°

ag f246.W are u cmua ""-"

,%"ofst ewen7 nd8 C Frec '.,-ang'e,4we als include the-crves.'" -'.-'

,.. '" *- ,

*21.5 01

t

-I "-~ 0 -

20.0

*. '.- °O

20.71

00

• ", ~~~20. 7[i-• ,, ,

imum, an th te wocre.epc h

-0'0

( 0 .0 ....(.

set ialw tinl2010 " oL.o-- --

.9 80 81"- .%t .

TEMPERA TURE (°C)|I"":, .t,I.-

Figure2(a Figuroes 23. enveormawized Freqency ±6 offsanetso ahD n

Do deviationfrom(Between 79 and 81 °C as a Function of Tern- 24 " and24() Fo(1)perature for Selected bc-angles. For each ",-angles. The C-angle, one curve corresponds to the ar--."-"-• q? ~~imum, and the other two curves depict the"- "."'

f2/10- 9 12 offsets """ "

-_* . angles corresponding to the absolute minimum frequency offset, that is, (@oc' o) -'-"-,

",', , (20.7 1, 34. 332950). Figure 24(a) gives the (D, DO) combinations for the mini-"-'-'--

~~~~mum frequency offset curve of Figure 22. It shows that to achieve the minimum7-..','

frequency offset attainable at that particular 4-angle, each minute of arc mis- I',ti'UO

orientation in 4 must be compensated by -4.3" of arc in e. Ifthe frequency off-set is allowed to increase from the local minimum to ]27"10'9I, the line of -'.."

Figure 24(a) becomes an envelope, with ± 4) and +A0 tolerances on each D4) and ,-.'..

DO deviation from (4)o, 00). These tolerances are plotted in Figures 24(b) and ,-....,

24(c). For (4)o.0o) the corrected angles become 4) = 4o ± A4) = 4o ± 19. 2" and . -Si

0 o ± o = 0o ±1.0.M ". This points out the sensitivity differences in the 4) and .'i'.,._,,€

-angles. The o-angle is approximately 14 times more sensitive to crystal- ":." ""

lographlc msrettosthan the corresponding 4)-angle.-'. ""

33 gqI, ." 4 "

464

4'.,-%.-..,... .

C.

% 0.5-

%4

• °%•0."0

-6 -4 -

20

15~

=---10- - _.

+1

. , - ...

- . - , .~ " • ",

: o.." "Delinutes)

4-

-41r

-75 -50 -25 0 25 50 75 " "

o (mhin utes)

Figure 24. Interaction Between 0 and 0 Misorien-tations. (a) Do adjustment for DO' m-isorientationfor minimum frequency offsets between 79a~nd81 0 C, (b) ±6O tolerance on DO' misorientation for12 / 109 frequency offset, ad (c) ±6 toleranceon DO adjustment for 12>Y 10- 1 frequency offset

5._%- • .. "

- We illustrate the application of Figure 24 by several additional examples.

Assume that the crystal was intended to be oriented to (0 In) but measuring it . '

one finds that the -angle is misoriented by D = 3. 18'. The frequency offset-7

corresponding to the misoriented angle is 3. 51 o/ 10 . To bring the crystal

within the desired frequency offset limit, one can reorient either the 0- or the

%0-angle. From the relative 0 and o sensitivities it seems reasonable that one .~

will always try to reorient the 0p-angle. Figure 24(a) shows that the 0-angle

adjustment for the optimum frequency offset is Do -42.6' (0 20.00). Fig- S.

ure 22 showed that the combination ( + DOP, 0 + DO) 4 ( -42. 6', 00 + 3. 18')-9 000

gives a frequency deviation of 1. 15 10 For DO = -42. 6 Figure 24b gives

Assum 9. 90" as the tolerance that has to be maintained to be within 12 / 10.

4'34

.,n-angle. Frm the relaive and sestvte itsesraoa- ta n j'.'.

silawy r oroin h -nl.Fgr 4a hw htte0age"'''"

adutetfrteotmmfrqecbfsti w=-2 f'( 00) i-v O

* Consequently, the corrected 4-angle, 4) = + (DO) 60~), for an angular mis-0

orientation of Do 3. 18' that still maintains < 12 x 10-91 frequency offset between 04 0 i79and81 C, is -42. 6 ±9. 90".

0Consider another example. One orients the crystal and finds that the 0-angle

, is misoriented by DO = 17.4' (0 = 21.00). For extraneous reasons it is decided

that it is more convenient to accept the misoriented 4-angle and optimize the

design and adjust the 0-angle. Figure 24(a) shows that the corresponding a-angle

adjustment is DO = -1.23'. Figure 22 showed that the minimum frequency offset

for the adjusted angles is 0.48 y 10 9. For the given DR value, Figure 24(c) -

gives the tolerance, ±60 = 1. 1", which must be maintained to be within 2 .' 10-9.

Consequently, the corrected 0-angle, a = 00 + i10 ± Lo), for a D4) = 17.4' is-

orientation that still is within 12 y 10"91 frequency offset between 79 and 81 °C, -0 .

is 0 = 0o - 1.23' ± 1. 1". Incidentally, the maximum allowable tolerances do not

occur at 20.71 , the position of the absolute minimum frequency offset. They .-

are located near 4 21.00, the position when one T is near 79 °C and the otherto

one near 81 °C. ""

The static f(T) characteristics for Tto = 80 °C define 4 = 20.710 as the :j- 1 ..

*. optimum doubly rotated orientation. The frequency difference between Tto ± 1 °C

is not the only criterion to be considered for selecting doubly rotated cuts. The .o--

frequency separation between the b- and c-mode of vibrations, and the magnitude

of the electromechanical coupling factors, are also important parameters. Fig- -

ure 25 shows these parameters for Tto = 80 0 C. We note that the b- to c-mode t*r- ":g

frequency separation decreases from 12.6 percent at 0 to 9.5 percent at

20.71". At the same time the c-mode electromechanical coupling factor de-

creases from 8.7 percent at 0 =00 to 5.4 percent at 20.71 ° , whereas the b-mode

coupling factor increases from 0 percent to 4.4 percent. Consequently, the

improved frequency-temperature characteristics that can be obtained from

crystallographically doubly rotated cuts are at the expense of tighter angular

* tolerances in the fabrication process, a relatively weaker c-mode, and a more

complicated oscillator circuit to suppress or isolate the nearby b-mode.

354

' %

°°* S

"xJ

13 12

I 10

-2. C J

At

-%:. - ...0.~ 1010

n I

94

8-

7~ 0

-.-...- .

'.ROTATION ANGLE (EG

Figure 25. Separation Between b- and c-mode ~Frequencies (Left Scale), and Electromechani-cal Coupling Factors (Right-hand Scale) as aFunction of Rotation Angle for 80 OC Turn-over Temperatures

8. THE SC-CUT .-

The advantages and developments of the SC-cut have been reviewed by many

inyestigators. 1-4Specifically the SC-cut yields improvements in stress in-

sensitivity and compensation of thermal transients. The static f(T) of the SC-cut

t11. Kusters, J.A., and Adams, C.A. (1980) Production statistics of SC (or TT12. acrystals. 34th Annual Symposium on Frequency Control, 167-174.

12. Kusters, J.A. (1981) The SC-cut - An overview, 1981 Ultrasonics Sympo-sium, (IEEE 81CH1689. 9), 402-409.

13. Vig, J.R., Filler, R.L., andKosinski, J.A. (1982) SC-cut resonators for

the temperature compensated oscillators, 36th Annual Symposium onFrequency Control, 181-186.

14. Warner, A.W., Goldfrank, B., and Tsaclas, J. (1982) Further develop-ments in SC cut crystal resonator technology, 36th Annual Symposiumon Frequency Control, 208-214.

36 -O

_"-0 "% .

A;:"°"

'C" : -.

* *... ,... ... '....... ,.. - .. .. '..'.....-... .. '. .,,.. ... ',.. .'",. .. '. -. .:. ,.. .. "

°-77 7 .. W.

,,,q.

is no better than other optimally designed doubly rotated cuts. For our purposes

it can be defined as the cut oriented at 0 2 1. 930 Its behavior is very similar- 0

to the description we gave for 0 = 20. 710, the optimal angle for operating at080 C.

For the SC-cut, the optimum reference angles corresponding to the absoluteminimum frequency offset are calculated as (o , o)sc = (21. 93, 34. 24463)0. At

these angles T i = Tto = 92. 92 0°C. Figures 8 ind 9 show that t o 21. 930 , -

BBL(1962) based computations give 0 = 33.67120 and Ti = T = 102.92, andto

Adams et al based coefficients yield 0 = 33. 3941 ° , and T = T = 113. 96 C.* i to

Based on BBL(1963) LSF, the optimum operating temperature for the SC-cutis z 93 0 C. The frequency offset for a range of 6T = ± 1 0 C around Tto ri is

F. = 1. 02 / 10 1 , whereas the minimum frequency offset that can be obtained .1 0 1in the ± 1 C range around 92.92 C is AFmin 2. 58 y 10 -

. The angular dif-

ference between the 6F. and 6F m. curves is A = 0. 043, similar to the valuefo = 8C The a 0 condition is defined by (21.93 34434)0 , atto 0

which angles T. = 97.44 0 C, and the T are at 25 and 169.9 0 C, respectively.--toThe experimental T sensitivity to small variations in 0 and 0 around (21.03,to 134. 1151 ° has been published by Kusters and Adams 11 and is reproduced in

Vig et al. We obtain similar curves based on BBL(1963) LSF. %.--

In many applications it is desirable to operate the SC-cut at other than the

optimal 93 C. The right hand scale of Figure 26 shows the minimum frequency

offsets 6F as a function of operating temperature T , between 60 and 100 0C,

that can be achieved for 0 = 21. 93 and variable o-angles, and a range of T-

± 1 0C around To. For example, the minimum offset for a SC-cut plate opti-

mized for T 60 ( and operating in the temperature range of 59 to 61 C ismizedfor op_9 ,.-..-

6F = 5.2", 10". A minimum 6F of 2.6Y 10 1 is reached at T = 92.92 °C,opand increases at T = 100 °C to 6F = 1. 07 >' 10 The left hand scale of Fig

ure 26 shows the correction angle Do, referenced to (0)sc, that is required to

shift the minimum frequency offset from 92. 92 °C to some other operating tem-

perature. At a particular temperature, the minimum frequency offset occurswhen Top coincides with T to. The DO curve of this figure corresponds to T to .

variation with 0, and it shows, for example, that for DO = 200" the Tto is shifted .

from 93 to 60 °C. Consistent with results of Figure 21, we note that the mini-

mum frequency offset of 12 >e 10 91 is reached at T = 80 C. Consequently,

one cannot operate an SC-cut crystal below 80 0 C and still obtain a frequency

offset less than 2 y 10 9 1 in the ±1 °C range of T .opFigure 24 showed sensitivity curves DO ± 60 and DO ± A0 around 0 = 20.71

One can construct the same type of curves, with very similar results, for 4) =0 0

21.93 °. For example, for T 80 C, the 0-angle adjustment is DO -3.46"op:o

37

,..-,€,,- ,- ¢,.-...,-..-,.. .....,.. . . . . . . . ................ .-..--- ,.. .... ..-.... ..... , .,- .'.

~r 6

-0=2 1.9 3"~S

200 AT= 1 'C 5

.. 4" "lbO 4

160

0"120 3 . :

40

01 0.

0 70 .0 90 100

...--

!90 080g 100

OPERATING TEMPERATURE (°C)

Figure 26. DO Adjustment as a Function of Operating(Turnover) Temperature for SC-cut, o = 21. 930 .

Right-hand scale gives the minimum frequency offset

in a ± 1 °C range around the operating temperature

per minute of Do misorientation. For T = 80 0°C, the minimum frequency off-.,., op-_.

set is not symmetric around = 21. 93 °, but follows the curve of Figure 22. lor

0-. 9.- = 21. 93 ° and Do = -20', the minimum offset is 1.45 / 10 ", whereas for Do =

+20', the offset is 2.54 / 10 - 9 .

From a practical point of view, 0 and 0 are then selected to give the best

compromise for the desired operating temperature, between stress insensitivity,

".' ,'. thermal transient response, b- to c-mode separation, elcctromechanical coupling. . -

static f(T) characteristics, and ease of fabrication.

38..-:::::::.

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,,..: .:-.:" :" :

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0

References

1. Bechmann, R., Balato, A.D. and Lukaszek, T.J. (1962) Higher-order% temperature coefficients of elastic stiffnesses and compliances of a-quartz,

Proc. IRE 50:1812-1822.

2. Kahan, A. (1982) Elastic Constants of Quartz, BADC-TB-82-117, AD A 121672.

3. Kahan, A. (1982) Temperature Coefficients of the Elastic Constants of Quartz,RADC-TR-82-224, AD A12 5709.

4. Kusters, J. A. , Adams, C. A., Yashida, H. , and Leach, J. G. (1977) TTC's-Further developmental results, 31st Annual Symposium on FrequencyControl, 3-7.

.* . ~ 5. Ballato, A. D. (1977) Doubly rotated thickness mode plate vibrators, PhysicalAcoustics, Vol. XIII, W.P. Mason and T.N. Thurston, Eds., Acaem.Press, New York, pp. 115-181.

6. Bechmann, R., Ballato, A.D. , and Lukaszek, T. J. (1963) Higher-OrderTemperature Coefficients of Elastic Stiffnesses and Compliances ofa-Quartz, USAELBDL TBR 2261.

7. Adams, C.A., Enslow, G.M., Kusters, J.A., and Ward, B.W. (1970)* Selected topics in quartz crystal research, 24th Annual Symposium onY Frequency Control, 55-63, AD 746210.

8. Bechmann, R. (1956) Frequency-temperature-angle characteristics ofAT-type resonators made of natural and synthetic quartz, Proc. IRE44:1600-1607.

P. Vig, J. , Washington, J. W. , and Filler, B. L, (1981) Adjusting the frequency.vs. temperature characteristics of SC-cut resonators by contouring, 35thAnnual Symposium on Frequency Control, 104-109.

10. Kahan, A. (1082) Turnover temperatures for doubly rotated quartz, 36thAnnual Symposium on Frequency Control, 170-180.

11. Kusters, J.A. , and Adams, C. A. (1980) Production statistics of SC* (or TTC)crystals, 34th Annual Symposium on Frequency Control, 167-174.

'. \

S .-

39'

... .- .. . . .

12. Kuster~s, J. A. (1981) The SC-cut - An over-view, 181 Ultirasonics Sympo-sium, (IEEE 810111689. 9), 402-409.

13. Vig, J. R. , Filler, 1R. L. , and Kosinski, J. A. (1982) SC-(ut r (sonatoi-s uibrthe temper'ature' compensated oscillator-s, 36th AnnualSymposium onFr-equency Control, 181-18G.

14. Warner, A. W. , Goldfrank, B., and Tsaclas, J. (1982) Furthel- develop-% ments in SC cut crystal resonator technology. 36th Annual Symposium

on Frequency Control, 208-214.

400

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