L O W T E M P E R A T U R E C O E F F I C I E N T Q U A R T Z C R Y S T A L S 75
76 B E L L S Y S T E M T E C H N I C A L J O U R N A L
th e le n g th is c u t along th e X axis an d th e v ib ra tio n is ex cited b y fields ap p lied along th e le n g th of th e b a r. S ince a ro ta tio n a b o u t th e o p tic o r Z axis does n o t ch an g e th e p ro p erties of th e ela stic c o n s ta n ts in v o lv ed in th is v ib ra tio n , th is b a r should h av e a zero te m p e ra tu re coefficient a t a b o u t th e sam e ra tio of axes a s t h a t given a b o v e. T h e zero angle of o rie n ta tio n is, how ever, n o t th e m o st fa v o ra b le angle of o rie n ta tio n for th e fu n d a m e n ta l v ib ra tio n of a long b a r, for if th e le n g th of th e c ry s ta l lies a t a n angle of + 5° w ith re sp e c t to th e Y or m ech an ical axis, th e coefficient of a long b a r is n e a rly zero .5 T h e se
DEPTH OF O PTICAL AXES IN M IL L IM E T E R S
Fig. 1—T em perature coefficient of a perpendicularly cut crystal for varying ratios of w idth to length.
long b a r ty p e cry sta ls h a v e been used to a sm all e x te n t to co n tro l oscillators an d to stab ilize th e p ass b a n d s of filters. T h e ir sm all use is a ttr ib u ta b le to th e fa c t t h a t th e y v ib ra te a t low frequencies an d are difficult to excite in a n o scillato r c ircu it.
T h e A T a n d B T h ig h -freq u en cy sh e a r c ry sta ls a n d th e C T a n d D T low -frequency sh e ar c ry sta ls a re o th e r low te m p e ra tu re coefficient c ry sta ls a n d th e y are discussed in d e ta il in section I I . T h ese c ry sta ls are c u t w ith th e ir plan es a t specified angles w ith re sp e c t to th e c ry sta l- lographic axes an d all of th e m involve a single ro ta tio n a b o u t a n axis w hich is p a rallel or ap p ro x im a te ly p arallel to one of th e c ry sta llo g ra p h ic axes. I t is show n in section I I I t h a t such c ry sta ls a re n o t th e only zero coefficient c ry sta ls of th ese ty p e s t h a t can be o b ta in e d , for if w e allow th re e ro ta tio n s a b o u t th e c ry sta llo g ra p h ic axes a w hole surface of zero te m p e ra tu re coefficient c ry sta ls c an be found. T h ese c ry sta ls
6 M atsum ara and K ansaki, “ On th e T em perature Coefficient of Frequency of Y W aves in X C ut Q uartz P lates,” Reports of Radio Researches and Works in Japan, M arch 1932.
L O W T E M P E R A T U R E C O E F F I C I E N T Q U A R T Z C R Y S T A L S 77 are m ore difficult to c u t th a n th e s ta n d a rd cry stals a n d are m ore su b ject to couplings to o th e r m odes of m otion an d hence m ost of th em are p ro b a b ly of m ore th eo retical in te re st th a n of p ractical value.
All of th e zero coefficient cry sta ls described abo v e are zero coefficient a t a specified te m p e ra tu re only a n d for te m p e ra tu re s on eith e r side of
TEMPERATURECOEFFICIENTIN PARTSPERMILLIONPER°C
78 B E L L S Y S T E M T E C H N I C A L J O U R N A L
Fig- 2— D iagram illustrating angles used in expressing orientation of A T and B T plates within th e n atu ral crystal.
A N G L E OF R O T A T IO N A B O U T X A X IS IN D E G R E E S ( 6 )
Fig. 3—T em perature coefficient for thin plates plotted as a function of the angle of cut.
L O W T E M P E R A T U R E C O E F F I C I E N T Q U A R T Z C R Y S T A L S 79
(d ete rm in e d b y a com pression) is u p, th e valu es of I, m, a n d n are 1 = 0; m = cos 0; n = — sin 0. (4) In th is d efin itio n , a rig h t h a n d e d c ry sta l is ta k e n a s one w hich causes th e p la n e of p o la riz a tio n of lig h t tra v e lin g alo n g th e Z or o p tic axis to 80 B E L L S Y S T E M T E C H N I C A L J O U R N A L
- 9 0 - 80 - 7 0 - 60 - 5 0 - 4 0 - 3 0 - 2 0 -10 0 10 2 0 3 0 4 0 5 0 6 0 70 80 90 R O T A T IO N ON X A X IS - --- 0 ---► + R O T A T IO N AB O U T X A X IS
IN DEGREES IN DEGREES
Fig. 4— Frequency constant for thin plates plotted against angle of cut.
ro ta te in th e sense of a r ig h t h a n d e d screw . S u b s titu tin g th e v a lu es of (3) a n d (4) in (2) we find
An = ¿66 cos2 8 + c u sin2 8 — 2 cu sin 8 cos 8 = c6j', A23 = — C14 cos2 8 — (c44 + C23) sin 6 cos 6,
A22 = c22 cos2 8 + C44 sin 2 6 + 2ci4 sin 6 cos 6, (5) A33 = C44 cos2 8 + C33 sin2 8,
A12 = X13 - 0.
W ith th ese v a lu es of A, th e th re e so lu tio n s of e q u a tio n (1) are /ATi .
C l =
p
«■
- V s [ t + T =■= > / ( t - t ) ' + ^
7 7] ■
w here K 2 = A232/A22A33.
T h e freq u e n cy of a n y p la te w ith its edges free to m ove will be
(
6)
f = Yt (2n + i ) n = 0, 1,2, • - ■, (7)
L O W T E M P E R A T U R E C O E F F I C I E N T Q U A R T Z C R Y S T A L S 81
82 B E L L S Y S T E M T E C H N I C A L J O U R N A L
A T an d B T c ry sta ls can co n tro l co n sid erab ly m o re pow erful o scillato rs w ith o u t d a n g e r of th e c ry sta ls b re a k in g th a n can th e X or Y c u t c ry sta ls. T h e fre q u e n c y sp e c tru m of an A T c u t p la te g ro u n d dow n from a larg e ra tio of d im ensions to a sm aller one is show n on F ig. 5.
L E N G T H OF X A X IS IN M IL L IM E T E R S
Fig. 5—Frequency spectrum (dots) for an A T p late as a function of ratio of length to thickness. T he dashed lines represent calculated flexural vibrations. T he whole line is th e principal shear mode. T he dot dash lines are other shear modes.
M o st of th e p ro m in e n t freq u en cies can be id entified as sh e a r frequencies of th e ty p e discussed in a p re v io u s p a p e r 10 a n d h arm o n ics of flexure 10 “ Electrical W ave Filters Em ploying Q uartz C rystals as E lem ents,” W. P.
Mason, B. S. T. J., July 1934, page 446. T he verification was made by R. A. Sykes who kindly supplied Fig. 5.
LO W T E M P E R A T U R E C O E F F IC IE N T Q U A R T Z C R Y S T A L S 83
•
v ib ra tio n s. T h is figure show s clearly t h a t th e stro n g e st flexures e n te rin g are con tro lled b y th e len g th of th e X axis r a th e r th a n th e Z ' axis.
As show n b y eq u a tio n s (6), (7) an d (8) th e A T an d B T c u t cry sta ls hav e o dd h arm o n ic v ib ra tio n s w hich are con tro lled b y th e sam e elastic c o n sta n ts as th e fu n d a m e n ta l v ib ra tio n s. Since th e y are con tro lled b y th e sam e elastic c o n sta n ts, th e h arm o n ic v ib ra tio n s have th e sam e te m p e ra tu re coefficients as th e fu n d a m e n ta l m ode and hence will h av e n early zero coefficients. T h is p ro p e rty has been m ade use of in oscillators in con tro llin g h igh-frequency v ib ra tio n s w ith c ry stals whose thicknesses can be o b ta in e d com m ercially.
C T and D T Low-Frequency Zero Temperature Coefficient Crystals A n o th e r se t of zero te m p e ra tu re coefficient c ry sta ls w hich are p a rtic u la rly useful for low frequencies h as re c e n tly been described b y H ig h t an d W illa rd .11 T h e y are re la te d to th e A T a n d B T c u ts d is
cussed abo v e in t h a t th e y use th e sam e shearin g m otion to pro d u ce th e low coefficient. T h is relatio n is illu stra te d b y F ig. 6 w hich shows
11 “ Presented before the Institute of Radio Engineers, M arch 3, 1937. Published in I. R . E . Proc. M ay, 1937, p. 549. Similar crystals are also discussed in U. S.
P atents 2,111,383 and 2,111,384 issued to S. A. Bokovoy.
84 B E L L S Y S T E M T E C H N I C A L J O U R N A L
th e a p p ro x im a te o rie n ta tio n s of th e A T c u t an d th e D T c u t. In th e A T p la te th e x v' s tra in is p ro d u c ed b y a sh e a r m ode of v ib ra tio n as show n b y th e arro w s w hich rep re se n t in s ta n ta n e o u s d isp la c e m e n ts.
In th e D T p la te th e x y' s tra in is p ro d u ced b y a sh e a r m o d e of v ib ra tio n as show n a g ain b y th e arrow s. T w o d iag o n a lly o p p o site co rn ers m ove ra d ia lly o u tw a rd w hile th e o th e r tw o m ove ra d ia lly in w ard . T h e re la tiv e ly low freq u en c y of th e D T p la te re su lts from th e re la tiv e ly larg e freq u e n c y -d ete rm in in g d im ensions x an d y ' . T h e te m p e ra tu r e coefficient of freq u en c y of th e se p la te s m a y be m ad e zero, for th e p ro p e r angles of c u t, since it goes from a large p o sitiv e v a lu e a t one o rie n ta tio n to a larg e n e g a tiv e v a lu e for an o rie n ta tio n 90 deg rees from th e first. A c tu a lly th e angle of c u t of th e D T p la te is n o t e x a c tly 90 degrees from th e A T . T h is is d u e to th e fa c t t h a t th e fre q u e n c y
ORIENTATION ANGLE IN DEGREES (6 )
Fig. 7—Frequency constant for low-frequency shear crystal plotted against angle of cut.
for a sq u a re p la te involves th e s 86' c o n s ta n t r a th e r th a n th e c ^ ' c o n s ta n t w hich co n tro ls th e fre q u en cy of a th in p la te . S im ilarly w e find t h a t th e re is a c ry sta l a lm o st 90° from th e B T w hich h as a zero coefficient an d th is has been d e sig n ated th e C T .
F ig u re 6 show s t h a t th e electro d e faces of th e D T c ry sta l are placed on th e z 'x p lan e a n d hence th e sh e a r m ode g e n e ra te d w ould o rd in a rily be called th e z j m ode even th o u g h it is sim ila r to th e x y' sh e a r m ode in th e A T c ry sta l a t rig h t angles to it. T h e m e asu red freq u en cy c o n s ta n t of such a series of sq u a re p la te s is show n on F ig. 7% In th e ab sence of a co m p lete th e o re tic a l so lu tio n 12 ta k in g a c c o u n t of all th e e lastic couplings for a sq u a re p la te v ib ra tin g in sh e af, an em pirical
12 An approxim ate solution neglecting coupling was given in a form er paper
“ Electrical W ave Filters Em ploying Q uartz C rystals as E lem ents,” page 446. This solution is not complete enough, however, to allow calculations of tem perature coefficients with very great accuracy.
L O W T E M P E R A T U R E C O E F F I C I E N T Q U A R T Z C R Y S T A L S 85 form ula w as developed for th e frequency w hich is
<10>
w here d = x — z' if th e p la te is sq u a re a n d d = (x + z')/2 if only n early sq u are. T h e elastic c o n s ta n t 555' dep en d s on th e o rie n tatio n angle 9 according to th e eq u atio n ,
s55' = S44 cos2 9 - f 566 sin2 9 + 4i i4 sin 9 cos 9. (11) F igure 7 show s th e m easu red v alu es of freq u en cy an d th e valu es calcu lated from e q u a tio n s (10) an d (11). A g reem en t is ob tain ed w ith in 2 p er c e n t.
F ro m eq u a tio n s (10) an d (11) th e te m p e ra tu re coefficient of fre
q uen cy of a sh ear v ib ra tin g p la te should be for a sq u a re c ry stal T f = — (1/2) [ T x + 7 7 + T p
s u T Sii cos2 9 + SeeTsee sin2 9 + 4s u T eit sin 9 cos
+ ] • (1 2)
544 cos2 9 + 566 sin2 9 + 45x4 sin 9 cos 9
T h e te m p e ra tu re coefficient of length along th e o ptic axis is a b o u t 7.8 p a rts p er m illion (per degree cen tig rad e) while th a t p erp en d icu lar to th e o p tic axis is 14.3 p a rts p e r m illion. F o r an y o th e r directio n
Ti = 7.8 + 6.5 cos2 9, (13)
w here 9 is th e angle betw een th e length and th e op tic axis. H ence T x = 14.3; 7 7 = 7.8 + 6.5 cos2 9,
and
T p = — 36.4 p er degree C. (14)
T h e te m p e ra tu re coefficients of th e six elastic c o n sta n ts w ere e v alu ated in a form er p a p e r.13 Since th e n th e y hav e been slightly revised so t h a t th e b e s t valu es now are
T Sn = + 12, T Cn = —54.0,
T Slt = - 1,265, th is 7 7 , = - 2,350, 7 7 , = - 238, resu lts in T Cn = - 687,
77« = + 123, 7 7 , = + 96, (15)
Tsu = + 213, 7 7 , = - 251,
77« = + 189, 77« - - 160,
77« = - 133.5, 77« = + 161.
13 "E lectric W ave Filters Employing Q uartz Crystals as Elem ents,” W. P. Mason, B. S. T. J ., 13, p. 446, July 1934.
86 B E L L S Y S T E M T E C H N I C A L J O U R N A L
L O W T E M P E R A T U R E C O E F F I C I E N T Q U A R T Z C R Y S T A L S 87 P ra c tic a lly all th e w o rk done has been on sq u a re or n early sq u are p lates. Som e tim e ago B ech m an n 14 a n d K oga 16 published w o rk done on c ry sta ls w hich d e p a rte d from th e sq u a re sh ap e for w hich zero coeffi
cien ts w ere o b ta in e d a t som ew h at differen t angles an d d ifferen t fre q u e n cies th a n th o se given for th e C T an d D T c ry stals. T h is is due to th e fa c t t h a t w hen th e c ry sta l sh ap e d e p a rts from th e sq u are, th e freq u en cy ap p ro ach es m ore n ea rly th e re so n a n t freq u en cy of th e c ry sta l v ib ra tin g in its second flexure m ode an d th e increased coupling changes th e angle for w hich th e coefficient becom es zero. T h e sq u are c ry sta l is th e one w hich h as few er seco n d ary frequencies an d is th erefo re m ore desirable.
CL^ 64 001 ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
---> 4400 Oz
hi
---DO
l±j 40001______ ___ ___ ___ ___ ______ ______ ___ ___ ___ _________ ______
fr -90° -60° -30° 0° 30° 60° 90°
R O T A T IO N ABO U T X A X IS IN DEGR EES
Fig. 9—Frequency constant for E and F type vibrations.
I I I . Ze r o Te m p e r a t u r e Co e f f i c i e n t Cr y s t a l s f o r Mo r e Ge n e r a l Or i e n t a t i o n s
S h o rtly a fte r th e discovery of th e and B T cry sta ls it w as realized t h a t zero te m p e ra tu re coefficient cry sta ls could be o b tain ed a t a v a rie ty of angles provided tw o ro ta tio n s of th e cry sta l w ith resp ect to th e cry stallo g rap h ic axes w ere used. T h is w ould allow th e direction of th e shearin g axis to p o in t in a n y d irectio n w ith resp ect to th e c ry sta llo g rap h ic axes. U sing th e C66r c o n s ta n t as th e elastic c o n sta n t d ete rm in ing th e frequency, it w as found t h a t th e re w as a w hole series of zero
14 R. Bechmann, Hochfrequenstechnik u Elektroakustic, Vol. 44, No. 5, p. 145.
16 I. Koga, Report of Radio Research in Japan, Vol. IV, No. 2, 1934. See also P atents 2,111,383 and 2,111,384 issued to S. A. Bokovoy.
88 B E L L S Y S T E M T E C H N I C A L J O U R N A L 18 “ Researches on N atu ral Elastic V ibrations of Piezo-Electrically Excited Q uartz P lates,” R. Bechm ann, Z e it.f. Technisch Physik, Vol. 16, No. 12, 1935, pp. 525-528.
This m ultiple orientation of high-frequency shear crystals is also th e basis of the V cut crystal of Bokovoy and Baldwin discussed for example in B ritish P aten t No. 457,342 issued M ay 27, 1936.
L O W T E M P E R A T U R E C O E F F IC IE N T Q U A R T Z C R Y S T A L S 89 an d set
cos \l/ — p3/2 >
th e th re e solutions will be
. - ^ V P c o s f + ^ ’ + t f + g ) ,
. = y l ' - 2 V P c o s (
f
± r)
+ Ÿ Â ± T l ± M, (18)
F ro m th ese eq u a tio n s a n d e q u a tio n (2), th e frequencies and te m p e ra tu re coefficients of all th re e m odes of m otion have been calcu lated by B echm ann. B ased on th ese calcu latio n s th e angles of zero coefficient
Fig. 10—Angles of cut for zero tem perature coefficient high-frequency shear crystals for two rotations.
are show n on Fig. 10 for th e an g u la r p lacem ent of th e direction of p ro p ag atio n a d o p te d on Fig. 11.
U sing th e em pirical form ula (10) for th e low -frequency shear v ib ra tio n a surface of zero coefficient low -frequency sh ear v ib ra tin g cry stals c an be c a lc u la te d.17 F o r th is cry sta l th re e angles are required to
17 M ultiple orientation low- and high-frequency shear crystals are discussed in British P atent 491,407 issued to the writer on September 1, 1938.
90 B E L L S Y S T E M T E C H N I C A L J O U R N A L
RIG HT HA N D ED CRYSTAL
F i g . 11
X (+ b yc o m p r e s s i o n)
-A ngular system for locating th e axis of shear of high-frequency crystals with two rotations.
sp ecify th e position of th e p la te since, for a low -frequency sh e a r c ry sta l, r o ta tin g th e p la te a ro u n d its sh e arin g axis will c h an g e th e $55' c o n s ta n t a n d h ence th e fre q u en cy a n d te m p e ra tu re coefficient of th e p la te . If we le t th e po sitio n of th e p la te w ith re sp e c t to th e c ry sta llin e axes b e d e n o te d b y th e angles, 9, <p a n d 7, m easu red as show n on F ig. 12 it can be
z
Fig. 12—A ngular system for locating low-frequency shear crystals with three rotations.
show n t h a t th e £55' c o n s ta n t is given b y th e e q u a tio n
para-FREQUENCYCHANGEIN PARTSPERMILLIONVALUEOF 6INDEGREES
92 B E L L S Y S T E M T E C H N I C A L J O U R N A L
Fig. 13— Contour map of zero tem perature coefficient low-frequency shear crystals w ith three rotations.
Fig. 14— Frequency tem perature relations for zero tem perature coefficient crystals
L O W T E M P E R A T U R E C O E F F I C I E N T Q U A R T Z C R Y S T A L S 93 bolas. T h is is w h a t w ould be e x p ected for in general we can w rite th e frequency as a function of te m p e ra tu re b y th e series
/ = / 0[1 + a , { T - To) + a , { T - T 0)2 + a 3( T - T 0)3 + • • • ] > (22) w here To is an y a rb itra ry te m p e ra tu re . D iffe ren tiatin g f w ith respect to T we have
^ = / 0[ a i + 2 a , ( T - To) + 3a 3{ T - To)2 + • • • ] • (23)
F o r a zero coefficient c ry sta l th e change in frequency will pass th ro u g h zero a t som e te m p e ra tu re To- H ence a\ = 0, and th e frequency will th e n be
/ = / o [ l + a , ( T - To)2 + a 3( T - T 0)3 + • • ■ ] ■ (24) Since a , will o rd in a rily be m u ch larg er th a n succeeding term s, a parabolic cu rv e will be o b ta in e d . If a , is p ositive th e frequency will increase on e ith e r side of th e zero coefficient te m p e ra tu re To an d if neg ativ e it w ill decrease.
R e cen tly a new c ry sta l c u t, labeled th e GT, has been found for w hich b o th ai an d a, are zero. As a re su lt th e parabolic v a ria tio n w ith te m p e ra tu re is elim in ate d an d th e frequency rem ains c o n sta n t over a m uch w ider ran g e of te m p e ra tu re . T h e v a ria tio n o b tain ed is p lo tted on Fig. 14 b y th e cu rv e labeled GT, and, as can be seen, th e frequency does n o t v a ry over a p a r t in a m illion over a 100° C. change in te m p e ra tu re .
T h is c ry sta l, w hich will be described in a forthcom ing paper, has found considerable use in freq u en cy sta n d a rd s, in v e ry precise oscilla
tors, an d in filters s u b je c t to large te m p e ra tu re v a riatio n s. I t has given a co n stan cy of freq u en cy co nsiderably in excess of t h a t o b tain ed b y a n y o th e r c ry sta l.