Method of angle correcting doubly rotated crystal resonators

Doubly rotated quartz crystal blanks such as the SC, IT and FC cuts have ir apparent angles of cut and hence their frequency vs. temperature characteristics modified by changing the contours of one or both of the convex sides thereof, or by changing the electrode areas thereof; or a combination of these two changes.

BACKGROUND OF THE INVENTION 
This invention relates in general to a method of treatment of quartz 
crystal resonators for varying the frequency vs. temperature (f vs. T) 
characteristics thereof in a desired manner. More particularly, the 
invention relates to a method of treating SC-cut quartz resonator blanks, 
or other doubly rotated cuts, in such a way that a desired f vs. T 
characteristic can be achieved. Unmodified resonator blanks of these types 
will inherently have a certain variation in f vs. T characteristic due to 
the inherent errors in the cutting machinery and due to variations 
introduced during lapping. The novel method of the invention permits all 
of these randomly varying blanks to be converted to the same 
characteristic, within an acceptable tolerance, or each to different 
desired characteristics. 
The invention is based on the discovery that the f vs. T characteristic of 
the c-mode of SC-cut quartz resonators and of other doubly rotated cuts is 
a function of the curvature or diopter value of the convex side thereof 
and of the electrode area. By changing this diopter value and/or the 
electrode area, the slope of the f vs. T characteristic at its point of 
inflection, as well as the turning points thereof can be controlled in a 
predictable manner. These changes in diopter value and/or electrode area 
thus effectively change the effective angles of cut of the crystals and 
thus provides a relatively easy and inexpensive way of increasing the 
useability of crystal blanks which might otherwise have to be discarded or 
modified by other more costly methods. The diopter value, D, is a measure 
of the curvature of the convex side of these resonator blanks and is equal 
to the reciprocal of the radius of curvature in meters. The units of all 
of the slopes referred to herein is ppm/.degree.C. 
SUMMARY OF THE INVENTION 
The invention comprises a method for modifying the f vs. T characteristic 
of a plano-convex, fundamental mode, SC-cut quartz crystal resonator to 
have a new slope, df/dT.vertline..sub.D.sbsb.f at its point of inflection, 
comprising the steps of; measuring the f vs. T characteristic of such a 
resonator having an initial diopter value D.sub.i, determining from said f 
vs. T characteristic the slope, df/dT.vertline..sub.D.sbsb.i, at its point 
of inflection, determining from the following formula the diopter value, 
D.sub.f, required to yield said new slope; 
##EQU1## 
and then re-contouring the convex side of said resonator to the calculated 
new diopter, D.sub.f. 
For third and higher overtone blanks of this type, and for biconvex blanks, 
and for other doubly rotated cuts, the procedure is similar but the 
coefficient 0.216 in the formula is different for the different blank 
types. 
In a variation of this concept, the characteristics of such blanks can be 
modified by means of the aforementioned diopter value change to achieve an 
approximation of the desired characteristic, and then the electrode area 
of the blank can also be changed to fine tune the f vs. T characteristic 
to the desired value, or if the initial diopter value is fairly close to 
the desired one, electrode area change alone can be used.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT 
A doubly rotated cut resonator blank is one which is cut at an angle .phi. 
rotated around the Z-axis of the mother crystal and at an angle .theta. 
around the X'-axis thereof. Doubly rotated crystal resonators, such as the 
SC-cut, have significant advantages over singly rotated resonators such as 
the AT-cut. One of these advantages is the lack of a thermal transient 
effect, which means that the warmup time of oven-controlled SC-cut 
crystals can be much shorter than with singly rotated crystals in which 
thermal gradients during warmup cause long warmup times. Also, SC-cut 
resonators can exhibit low drive level effect, and little frequency error 
due to such things as electrode stress relief, vibration and gravitational 
forces. 
In order to exhibit the aforementioned advantages, doubly rotated crystal 
blanks must be accurately cut at the proper angles of .phi. and .theta.. 
Inherent tolerances of the mechanical cutting machinery and of lapping 
processes make this impossible to achieve in practice except with a very 
small yield. Techniques are known for economically correcting inaccurate 
angles of cut of singly rotated blanks such as the AT-cut. This involves 
such things as regrinding the planar surfaces of the blanks to physically 
change the angle of cut to a desired value. This is feasible because with 
singly rotated cuts, generally available x-ray diffraction equipment can 
be used to measure the actual angle of cut and thus the amount of angular 
error is easily determined. Further, only one angle, the angle .theta., 
needs to be corrected. High precision x-ray diffraction techniques are not 
as yet economically feasible for the doubly rotated cuts. Moreover, for 
the doubly rotated cuts, both the angles .phi. and .theta. must be 
accurately controlled. The degree of control, especially of the angle 
.theta., must be much tighter, particularly when making crystals for 
oven-controlled oscillators. The tight angle tolerances, and the lack of 
an inexpensive technique for correcting angles of cut of such resonators 
has limited the application of these cuts. Without an angle correction 
technique, it is impossible to tailor the frequency vs. temperature 
characteristics of the doubly rotated resonators to a desired value. 
Prior art techniques for correcting actual or apparent angles of cut are 
shown in, for example, U.S. Pat. No. 4,070,502, issued May 5, 1976, which 
describes a method for changing the apparent angle of cut of singly 
rotated resonators by depositing an adherent film thereon. Temperature 
changes cause the film to exert stresses on the resonator which affects 
the f vs. T characteristic in a desired manner. An article by Husgen et al 
entitled "A Method of Angle Correction" in the "1976 Proceedings of the 
30th Annual Symposium on Frequency Control" describes a method of actual 
angle change including etching away selected portions of a resonator blank 
followed by lapping to a final configuration. U.S. Pat. No. 3,803,774, 
issued on Dec. 22, 1972 shows another method of changing actual angles of 
cut involving forming one or more mesas on an inaccurately cut resonator, 
followed by double face lapping until the desired angle is reached. 
The frequency vs. temperature characteristic of most crystal resonators 
follows a third order law, as shown in FIG. 3, in which the abscissa is T 
and the ordinate is the change in frequency from some nominal value, 
.DELTA.f. The maxima and minima of the curves are the lower and upper 
turning points, respectively. These points of zero slope are preferred as 
the operating points of the crystal ovens of oven-controlled oscillators. 
Approximately halfway between the upper and lower turning points is the 
point of inflection. The point of inflection of SC-cut crystals is 
approximately 100.degree. C., the exact value depending on the overtone. 
With the method of the present invention, f vs. T characteristics of 
SC-cut and other doubly rotated crystals can be changed simply by 
re-contouring the convex side thereof to a new diopter value while leaving 
the flat side unchanged. This new value will shift both the upper and 
lower turning points and change the slope at the point of inflection. In 
fact the slope at the point of inflection has been found to be a linear 
function of the change in diopter value. With the technique of the present 
invention it is possible to obtain a zero slope, or any other desired 
slope at the point of inflection. A zero slope is desirable because then 
there is a large temperate range about this point in which a temperature 
excursion results in a nearly zero change in frequency. Such an operating 
point is advantageous for many applications, particularly for 
oven-controlled oscillators. Further, the operating temperature of an 
oven-controlled crystal must always be above the ambient temperature so 
that the crystal always loses heat to its surroundings. The present 
invention makes it possible to raise the lower turning point of a crystal 
to near the inflection point for such applications. 
Furthermore, adjusting the contour of an SC-cut crystal resonator can 
produce a significant shift in f vs. T characteristic, while producing 
only insignificant shifts in the other resonator parameters. For example, 
for 14 mm diameter fundamental mode plano-convex 5 MHz resonators, 
changing the contour from 1 diopter to 3 diopter changed the slope at the 
inflection temperature by 0.432 ppm per .degree.C. The resonator Q's 
remained in the range of 1 million, no activity dips were observed, the 
capacitance ratios increased from about 1200 at 1 diopter to about 1800 at 
3 diopter, the anharmonic modes between the b-mode and c-mode remained at 
a much higher resistance than the c-mode resistance, and the closest to 
the c-mode anharmonic mode moved from being 120 kHz from the c-mode at 1 
diopter to about 190 kHz from the c-mode at 3 diopter. 
The mounted resonator blank 7 of FIG. 1 is of the plano-convex type which 
is generally of disc shape with one side thereof flat and the other side 
convex. The cross sectional view of FIG. 2 illustrates the shape of the 
sides. Electrodes, which usually are gold, are attached to opposite sides 
of the resonator for making electrical connection thereto. The electrode 
17 is applied to the flat resonator side and electrode 19 to the convex 
one. The electrode 17 makes contact with conductive contact 15 and 
mounting clip 9 and the electrode 19 with contact 13 and clip 11 at 
diametrically opposite points on the resonator blank. The clips connect 
with pins 23 which are supported by base 21. 
FIG. 3 shows how the present invention was used to modify the frequency vs. 
temperature characteristics of a fundamental mode, plano-convex SC-cut 
resonator blank having nominal angles of cut of; .phi.=21.degree.56' and 
.theta.=33.degree.44' by simply re-contouring the convex face to a new 
diopter value. This blank was originally cut with a diopter value of 1.0 
and the measured f vs. T characteristic of this blank is labelled as D=1.0 
in FIG. 1. It can be seen that this curve has a lower turning point (LTP) 
of approximately 64.degree. C., and an upper turning point (UTP) of 
approximately 137.degree. C. This same blank was then re-contoured to have 
diopter values of D=1.37 and D=1.62 by the simple expedient of grinding 
the convex surfaces thereof to the new curvature corresponding to these 
diopter values. After each such re-contouring, the f vs. T characteristic 
was re-measured. The results are plotted in FIG. 3 by means of the curves 
labelled D=1.37 and D=1.62. It can be seen that the point of inflection, 
T.sub.i, of all of these curves is the same and is about 100.degree. C. 
Also, as the diopter value increases, the lower turning point is lowered 
and the upper turning point is raised. Also, the slope at the point of 
inflection becomes stepper, i.e., more negative, as the diopter value 
increases. 
FIG. 4 shows the characteristics of a similar blank which had an initial 
diopter value of D=1.0. The nominal angles of cut of this blank were 
.phi.=21.degree.56' and .theta.=33.degree.38'. This blank showed no 
turning points but when it was re-contoured to D=1.37 it showed definite 
turning points at about 80.degree. C. and 120.degree. C., as shown. 
It has been found empirically that the following linear relationship exists 
between the diopter values and the inflection point slopes for the c-mode 
of fundamental mode, SC-cut, plano-convex resonator blanks with nominal 
frequencies of approximately 5.0 MHz; 
##EQU2## 
wherein df/dT.vertline..sub.D.sbsb.i is the inflection point slope at the 
initial diopter value, D.sub.i, and df/dT.vertline..sub.D.sbsb.f is the 
inflection point slope at the final diopter value, D.sub.f. The units of 
the slopes in this equation are ppm per .degree.C., i.e., for D.sub.f 
-D.sub.i =1.0 diopter, df/dT.vertline..sub.D.sbsb.f will be 0.216 ppm per 
.degree.C. lower than df/dT.vertline..sub.D.sbsb.i. If it is desired to 
determine the final diopter value, D.sub.f, required to yield a desired 
inflection point slope, Eq. (1) can be solved for D.sub.f, as follows: 
The curves of FIG. 5 show how this technique can be used to obtain a zero 
slope at the point of inflection, with the advantages explained above. The 
curve labelled D=1.37 therein represents the curve of a resonator 
initially cut with a diopter value of D.sub.i of 1.37. The measured slope 
at the inflection point of this curve was found to be 0.124 
ppm/C..degree.. It was desired to shift this slope to zero and Eq. (2) was 
utilized to yield the required final diopter value D.sub.f. This was found 
to be D.sub.f =1.94. A standard 2.0 diopter cup was used to re-contour 
this blank. The re-contoured blank showed the f vs. T curve labelled as 
D=2.0 in FIG. 5. It can be seen that this curve has a nearly zero slope at 
T.sub.i, and remains at near zero slope over a wide temperature range 
about T.sub.i. 
It has also been found that the c-modes of 5 MHz, third overtone SC-cut 
resonators have a linear relationship between diopter value and inflection 
point slope. This relationship is as follows: 
##EQU3## 
This equation was derived by measuring the f vs. T characteristics of a 
group of six 5 MHz, third overtone SC-cut resonators, all initially cut to 
4.0 diopters, plano-convex, with 14 mm diameters. The blanks were all then 
re-contoured to higher and lower diopters, in 1.0 diopter steps. The 
measured change in the inflection point slope at these steps were used to 
derive Eq. (3). Eq. (3) can be solved for the final diopter value, 
D.sub.f, as follows: 
##EQU4## 
In a modification of this method, the f vs. T characteristic can be 
modified by a combination of the aforementioned re-contouring of the blank 
and modifying the electrode area of the blank. It has been found that the 
f vs. T characteristic of quartz resonator blanks can be affected to a 
certain degree by changing the electrode area. This provides and 
alternative way of fine tuning crystal characteristics. The diopter cups 
used in re-contouring can be purchased in one eighth diopter increments, 
and electrode area modification provides a convenient way of fine tuning 
which would otherwise require the use of non-standard diopter cups. For 
example, it has been found that when the 7 mm diameter circular electrodes 
of a 14 mm diameter SC-cut fundamental mode resonator were replaced with 4 
mm electrodes, the slope of the f vs. T characteristic at the inflection 
point changed from +1.07.times.10.sup.-8 per .degree.C. to 
-1.25.times.10.sup.-8 per .degree.C., i.e., the change in slope was 
slightly less than the change that would have been produced by a one 
eighth diopter increase in contour. According to Eq. (1), a one eighth 
diopter change in contour produces a 2.7.times.10.sup.-8 per .degree.C. 
change in slope vs. a 2.3.times.10.sup.-8 per .degree.C. change produced 
by decreasing the electrode diameter from 7 mm to 4 mm. Of course, to 
produce a smaller change in slope, one can make a smaller change in 
electrode dimensions. The change in electrode dimensions can be used alone 
to produce relatively small changes in f vs. T characteristics, or the 
method can be used in combination with changing the contour and thereby 
produce large but precisely controlled changes in f vs. T characteristics. 
It has also been found that the method can be applied to biconvex blanks. 
The f vs. T characteristic of a biconvex resonator can be shifted by 
re-contouring either one or both sides of the blank. The method can also 
be applied to other doubly rotated cuts, such as the IT and FC-cuts, 
however, the proportionality constant in equation (1) will be different 
for different types of resonators. The proportionality constant for any 
desired cut or any particular doubly rotated design can be conveniently 
determined by performing a simple experiment similar to that described 
above following equation (3) for 5 MHz, third overtone SC-cut resonators. 
While the invention has been described in connection with specific and 
illustrative embodiments, obvious variations will occur to those skilled 
in the art, accordingly the invention should be limited only by the scope 
of the appended claims.