US2212139A - Piezoelectric quartz element - Google Patents

Description

Aug. 20, 1940.

.C. F.'BALDW|N ET AL PIE-ZOELECTRIC QUARTZ ELEMENT Filed March 18, 1936 4 Sheets-Sheet l Z. m. F i@ G 1 F1,

I/vvE/vfrons SamueZHBoovo Charles EBaZdwL' Allg 20,. 1940- c. F. BALDWIN ET An. l PIEZOELECTRIC QUARTZ ELEMENT 4 sheets-sheet 2 /R a mmm y 1MM MP5 c fsf Twlfu/ wwwa caw W5 my@ 00N mi oma,

Arx N non n Charles EBaZcZw Aug. 20, 1940. c` F. BALDWIN E'r A1. 2,212.13'9

PIEZOELECTRIC QUARTZ ELEMENT Filed March 18. 1936 4 Sheets-Sheet 3 Aug 20 1940- c. F. BALDWIN ET AL PIEZOELECTRIC QUARTZ ELEMENT Filed March 18, 1956.

4 Sheets-Sheet 4 ATTORNEY Patented Aug. 20, 1940 UNITED STATES PIEZOELECTRIC QUARTZ ELEMENT Charles F. Baldwin, Schenectady, N. Y., and Samuel A. Bokovoy, Audubon, N. J., assignors to Radio Corporation of America, a corporation of Delaware Application March 18,

21 Claims.

This invention relates to the piezoelectric art, particularly to the cutting of quartz piezoelectric elements and has special reference tok the manufacture of piezoelectric quartz elements of the type possessing a natural mode of vibration which is a function of thickness. It is a continuation in part of application Serial Number 721,675 to Charles F. Baldwin and Samuel A. Bokovoy, led April 2l, 1934, and assigned to the same assignee as the instant case.

It is known to those skilled in the art that an X-cut quartz piezoelectric element (that is to say, one cut in the form of a rectangular plate having its electrode faces so orientated withI respect to the mother crystal that its thickness dimension is along an electric axis and its length or width along either the optic axis or a mechanical axis), will exhibit variations in frequency when subjected to temperature changes. The same phenomenon obtains if the crystal is Y-cut, i. e., if the thickness dimension of the element is along a mechanical axis and its length or width along either an electric axis or the optic axis.

In order to compensate for variations in frequency resulting from variations in temperature, all sorts of auxiliary frequency control apparatus have heretofore been employed. Obviously a piezoelectric-element possessing a zero temperature coefficient of frequency will not require an elaborate thermostatically controlled constant temperature cabinet to maintain a constant frequency of oscillation.

An object, therefore, of the present invention is to provide a quartz piezoelectric element which shall exhibit a zero temperature coefficient of frequency.

Another object of the invention is to provide a quartz-piezoelectric element which shall exhibit a zero or some other desired low temperature co-v efficient of frequency when vibrated at a frequency which is a function of its thickness dimension. v

Another object of the invention is to provide a process for so cutting a piezoelectric element that its fundamental frequency will be high relative to its thickness.

Another object of the invention is to provide a simple, accurate and efficient mode of pro- 50 cedure in the cutting of quartz crystals to eliminate as far as possible any uncertainties with regard to the frequency, the temperature coefficient of frequency and other operating characteristics of the finished element.

Still another object of the invention is to provide a process for so cutting piezoelectric crystals that there will be minimum wastage due to the production of non-usable elements.

Other objects and advantages will be apparent and the invention itself will be best understood 1936, serial No. 69,49

by reference to the following specification and to the accompanying drawings, wherein:

Figure lis a plan view of a natural or mother quartz crystal, the optic (Z) axis of which is perpendicular to the plane of projection, the

relative location of the apex, major and minor apex surfaces, an electric (X), a mechanical (Y) axis, a reference W axis and a reference Y-l-H axis being here illustrated as an aid to a clear understanding of the system of orientation followed in producing quartz piezoelectric elements in accordance with the invention.

Fig. 2 shows vin outline and in perspective a piece of natural quartz having a section cut and divided to provide a rough bar having top and bottom surfaces lying in planes which are normal to the Z-axis and its long edges parallel to a W-axis which is 20 removed from an X-axis.

Fig. 3 is an elevational View looking in the direction of the arrow of Fig. 2 showing the position of two blanks cut from the bar of Fig. 2, one of these blanks is tilted towards parallelism with the plane of a major apex surface of the mother-crystal and the other toward parallelism with the plane of a minor apex surface of the mother crystal.

Fig. 4 shows the left hand blank of Fig. 3 removed and trimmed, the plane of projection being perpendicular to the plane of the paper.

Fig. 5 shows the right-hand blank of Fig. 3, removed and trimmed.

Fig. 6 is a cross-sectional view taken on the line Y-Y of Fig, 1 showing the rotation of the blanks about a Y+0 reference axis of a mother crystal and with respect to the major and minor apex surfaces thereof.

Fig. 7 is a chart showing the correlation between certain angles Wn-Vn, and WHL-Vm, required to achieve a thickness-mode oscillator having a zero or other low temperature coefficient of frequency.

Fig. 8 is a chart of the frequency constants (K1n and Km) peculiar to zero temperature coefficient thickness-mode crystals.

Fig. 9 is a chart indicative of the characteristics of quartz plates rotated throughout the entire 360 scale about the Z-axis.

As above indicated the present invention contemplates and its practice provides a quartz piezoelectric element which exhibits a zero temperature coefficient of frequency, or a temperature coeiicient of either sign and of a desired low value, when vibrated at a frequency which is a function of its thickness dimension.

This desired operating characteristic obtains, in accordance with the invention, by reason of a rotation or inclination of the principal electrode faces of the element about any of certain Y-iaxes. Each of these Y|6 axes is normal to a W-axis, which W-axes lie in certain later speciv jection in Fig. 1.

.electrode faces of the element are directed and also upon the exact temperature coefcient of frequency desired.

Certain frequency constants (K) are given for each W-axis or Vangle so that it is possible to know in advance the substantially exact thickness dimension required to achieve a desired thickness-mode frequency. The temperature col eicients are also given for each angle of rotation in a standard frequency range.

Since the present invention involves a system of orientation in which (a) the major and minor apex surfaces of the mother-crystal are employed as reference planes and (b) certain W-axes and (c) certain Y-l-0 axes are employed as reference axes,` it is first necessary to identify thesev planes and axes.

As to (a), referring to Figs. 1 and 2 of the drawings and having in mind vthat all unbroken quartz crystals are hexagonal bi-pyramids, it will be seen that certain of the terminal surfaces of the quartz extend to the apex of the pyramid. These surfaces are designated M and are the major'apex surfaces. lThose terminal surfaces which do not touch the apex are designated N and are the minor apex surfaces of the mothercrystal.V Occasionally a mother crystal will be found in which more than three of the apex or cap faces extend to the tip of the pyramid; other crystals may have their pyramid ends broken off. No confusion, however, need exist as to the virtual location of the major and minor apex faces of a broken or otherwise abnormal crystal providing that the side faces m and n, or one of them, is intact for it will be apparentf'from an inspection of Fig. 2 that those side edges of the mother crystal which approach each other in the direction of its ends terminate in a major apex face, while 'those which diverge in this direction terminate in a minor apex face.' This is so in the case of both left-hand and righthand quartz.

As to (b), Fig. l is marked to show an electric, X-'axis' and an adjacent mechanical or crystallographic, Y-axis. The optic or Z-axis, marked in Fig'. v2, is perpendicular to the plane of pro- The W-axes lie between an X-axis and an adjacent Y-axis in the XfY plane,

i. e. in a plane normal' to the Z-axis. There are a substantially infinite number of such W-axes; however, the present invention contemplates the use as reference axes only those W-axes which in intersecting'anyreference X-axis form a W- angle therewith of from substantially 13 to substantially 29 or from substantially 15 to substantially 29, 4depending upon the direction of tilt. In Fig'. 1, but one W-axis is marked, it forms a W-anglei of with that X-axis which is designated X-X.

As to (c), there is a Y|0 axis for each W-axis,

each is normal to its W-axis and normal to the .Z-axis. It is about a Y-l-0 axis that the element is rotated or tilted to achieve a desired temperaturelcoefficient of frequency and frequency constant.

It may here be noted that a W-axis 30 removed from an X-axis will coincide with a Y- axis, hence its Y|0 axis will coincide with another one of the'X-axes; When.a.crystal blank is rotated about such a Y-l-H axis it is being rotated about this another one of the X-axis. Such orientation is not withi-n the purview of the present invention but forms the subject matter of vcopen'ding application Serial Number 62,- 300, filed February 4, 1936, in the name of the present applicants and assigned to the same assignee'as the instant case.

In order to produce a quartz piezoelectric element or plate having a predetermined lowtemperature coefficient, asuitable angle of orientation (angle W) with respect to an X-axis, and a coordinated angle representing the inclination of the electrode (i. e. tcp and bottom) `faces must be chosen. This latter angle will be referred to generally as the V-angle and more specifically as the Vm angle'or the Vn angle as determined by the direction of rotation, i. e., whether towards parallelism with the plane of a major apex surface (m) or a minor apex surface (n) of the mother crystal.

Referring now to Fig. 7 which shows the 'cor-Y relation between the temperature coecient and the angles W and V. In this chart an X-axis and an adjacent Y-axis are indicated by the vertical, correspondingly desig-nated, linesand the space or region therebetween is calibrated in degrees from the X-axis, the calibrations being along the horizontal line which intersects the centrally marked optic axis, Z. In agreement with the previously given definition a line extending at right angles at any point along thisline willcoineide with a W-axis. Horizontal line n--n extends from appoint substantially 13 from the X-axis to within 1 of the Y-axis (i. e. 29 from the X-axis) and horizontal line m-m extends from a point substantially 15 to 29 from the X-axis, both lines n n and m-m marked preferred limits within which a reference W-axis may be selected.

It will be noted that there are two scales marked along the line representing the X-axis. That scale which reads in an upward direction from 0 represents a rotation of the electrode faces of the element about its Y-i-H axis in a direction away from parallelism with the Z-'axisand t0- wards parallelism with the plane of a minor apex surface of the mother-crystal, the calibrations Vn being-in degrees from parallelism (0) with the Z-axis. That scale which reads in a downward direction from-0 represents a rotation of the electrode faces of the element about its Y-f-G axis in a direction away from parallelism with the Z-axis and towards parallelism Awith the plane of a major apex surface of the mothercrystal; the calibrations Vm like those-f Vn being inv degrees from parallelism `with the optic axis.

Now, having in mind that there is a Y-laxis for, and normal to, each JI-axis, then, from an examination of this Fig. 7 it will be seen that for each W-axis which forms an angle (W) with an X-axis within the 13-29 n-n range, yor 15-29 m-m range, there is a definite angle of rotation a minor apex surfa-ce of the mother-crystal. Now, selecting any W-angle within the limits 11-1L, say 20 (read along the line which intersects the Z-axis), then, the correlated Vn angle of rotation of the electrode surfaces of the element about the Y+0 axis is substantially 36.5, this latter reading being obtained by observing 'at what point on the upper left hand scale a line will fall projected at right angles to a 20 Weaxis from a point (p) at which the said W-axis crosses the zero temperature-coefficient line.

An element exhibiting a zero temperature coefficient of frequency may likewise be achieved by tilting the electrode surfaces about a Y-iaxis towards parallelism with the plane of a major apex surface. Thus, assuming as in the above example that a 20 W-axis is selected, then, reading the lower left-hand scale at the point (p') Where a 20 W-axis crosses the zero temperature coefficient line it will be seen that the correlated Vm angle of rotation is substantially 43.

The curves of Fig. '7 show clearly that the invention is not limited in its scope to the cutting of piezoelectric plates having an exactly zero ternperature coeflicient of frequency. For certain uses the characteristic to be sought is a definite positive or negative temperature coefcient. Such crystal elements have great utility in oscillator circuits which inherently have certain temperature coefficients of frequency irrespective of the crystal frequency control therefor. Thus it becomes advantageous at times to use a crystal control element having a temperature coefficient of frequency which is of opposite sign to, and substantially compensates for, the natural temperature coefficient of frequency of an .oscillator network per se.

To determine the Vn 'or the Vm angle of rotation required to achieve a piezoelectric element having a particular temperature coefficient of frequency the left-hand scale of Fig. 7 should be read `at the point where the selected W-axis crosses the line specific to the temperature coefficient desired. But three temperature coefficient curves are marked in ea-ch of the charts of Fig. '7. There is a zero temperature coefficient curve (the center curves, each marked i 0) for each direction of tilt. The curves on opposite sides of the zero line of the upper chart ernbrace the Wn and V angles required to achieve an element exhibiting a temperature coefcient of frequency within +15 cycles and l5 cycles, respectively, per million, per degree centigrade. The curves on the opposite sides of the zero line of the lower chart embrace the Wm and Vm angles required to achieve an element exhibiting a temperature coeicient of frequency within -5 cycles and +5 cycles, respectively, per million, per degree centigrade. Since all of the curves of the respective charts of Fig. 7 are substantially parallel one with another, it is obvious that other curves representative of other temperature coeiicients of frequency may be derived from the ones illustrated.

Before proceeding with the cutting of the blanks it is first preferable to determine of what thickness the finished element must be to achieve a desired fundamental thickness-frequency. This is simply accomplished in accordance with the formula where,

t, is the required thickness in mils of an inch Kn is a frequency constant which varies for each W-a-xis, when the direction of tilt of the element (about its Y-l-H axis) is towards parallelisin with the plane of a minor apex surface of the mother-crystal.

Km is a frequency constant which varies for each W-axis, when the direction of tilt of the element about its Y-}-0 axis is towards parallelism with the plane of a major apex surface of the mother-crystal, and

,f is the desired fundamental thickness-frequency,

expressed in megacycles.

he frequency ccnstants K and Km are shown in the chart of Fig. 8; constants Kn appear on the upper left-hand scale and constants Km on the lower left-hand scale. As in Fig. 7 the horizontal line which intersects the Z-axis is calibrated in degrees from an X-axis, and the lines n--n and nee-m niark the preferred limits within which a reference W -axis may be selected. The frequency constants Kn vary from substantially 68.2 for a 13 Whangle to substantially 66.6 for a 29 Wfl-angle. The frequency constants Km vary from substantially 88 for a 15 Win-angle to substantially .1.00.6 for a 29 Wm-angle.

The above substantially exact K values obtain only for quartz piezoelectric elements designed to exhibit a zero temperature coefficient of frequency. The range of frequency constants (Km and K11) for crystal-elements designed to exhibit temperature coefeients of frequency other than zero are substantially as follows:

Corresponding freq. constant ran gc Temp. cocllicient in cycles par l s r million per degree C. angles Km= to 99. Km= t0 103. 10:63h to 66.9. i Ifn=70-2 t0 68.6.

lt will be noted from the chart of Fig, 8 that the constant Km for any WE1-angle is greater than the constant Kn for corresponding WD-angle. Thus, for the 20 W-angle, constant Kn is substantially 67.5 whereas the constant Km is substantially 95. Since the fundamenta-l frequency of crystal elements of the type described is a function of the thickness dimension, it follows, that the thickness dimension required to achieve a given frequency will be greater` when the electrede faces of the element are tilted in a direction towards parallelism with a major apex surface than if these faces are tilted towards parallelism with the plane of a minor apex surface.

Accordingly, and although a finished element having substantially any thickness-frequency and temperature coefficient may be obtained in a blank tilted in either direction, it is recommended that for the relatively higher frequencies (say, above two megacycles) the direction of tilt or rotation about the Y-laxis should be in a direction toward parallelism with the plane of a major apex surface, for when so orientated practical difliculties attendant the cutting and the use of very thin quartz plates are substantially cbviated.

Summerizing the data thus far set forth, it will be apparent that before the actual cutting of the element from the mother-crystal commences there are three factors to be considered, to wit:

(l) The thickness-mode frequency at which the finished element is to respond. This factor rection of orientation of the electrode faces of the elementwith respect to the plane of a major or minor apex surface of the mother-crystal, As

above recommended, for. relatively high fre-v quencies the electrode surfaces of the blank should preferably, but not necessarily, be rotated (about a Y-laxis) in a direction toward parallelism with the plane of a major apex surface of the mother-crystal,

(2) The temperature coemcient of frequency to be exhibited by the finished element. This factor, like that of frequency, will of course be determined by the use to which the element is to be put. The temperature coefficient of frequency is a function of the angle (V) of rotation about a Y|6 axis so that it dictates one (the V) of the two angles (the V and W-angles) necessary to be known.

r(3), The relative location of the W-axis with respect to a reference X-axis, i. e. the angle (W) formed by the intersection of a W-axis with a reference X-axis. As previously set forth any1 W-angleror axis may be selected within the ii-ii 13-29 range or the m-m 15-29 range. (If angles lower than 13 and 15, respectively, are chosen the nished element may respond to extra and usually undesired spurious frequencies.) As the frequency constant, especially Km, increases, the W-angle increases. The selection of the W-angle is important only insofar as it is desirable to avoid the use of relatively thin crystal elements. As a matter of manufacturing convenience, it is often desirable to select standard W-angles from the substantially infinite number of AVil-angles available. Recommended standard W-angles and their correlated V- angles and frequency constants (Km, Kn) required to achieve a substantially zero and other low temperature coeflicient of frequency crystals are:

Tilt towards parallelism with a minor apex face Temp. cocf=0 Temp. coef.=-l5 WJ [vn Kn v1l K vn K Degrees Degrees 67. 9 29. 5 68. 2 40 70 67. 7 31 68 4l 69. 7 67. 5 31A 5 67. 8 4l. 5 69. 5 67 30. S 67. 3 40. 8 69 66. 6 80 66. 9 40 68. 6

Tilt towards parallelism with a major ape face Temp. coef.= -5

Temp. cocf.= Temp. cocf.=l5 Wm Vm Km Vm Km Vm Km Degrees Degrees Among the factors' not necessary to baconsidered is whether the mother-crystal is of so called left-hand quartz or right-hand quartz. The system for orientation of the present invention is applicable (without change) to quartz possessing crystalline structure of either type.

Any X-axis may be selected as the reference axis from which the W-axis is measured. This is brought out in the chart of Fig. 9 which shows the characteristics of quartz plates rotated throughout the entire 360 scale about the Z- ams. This chart shows, at a single glance, all of the information required in the planning of a quarts `piezoelectric element which shall exhibit a substantially zero temperature coefficient of `'13 to 29 or from substantially 15 to 29 are employed as reference W-axes in carrying the invention into effect.

Each Y-.axis bisects four curved lines, desigrespectively Vm, Km, K11, V11. Wherever a 'l -axis touches one of the curved lines V111 or V11, lere, the second or il-angle ofl rotation reuired to be known in the cutting of zero ternerature coenicient quartz crystal elements. The scale for reading V-angles is marked in degrees in two directions along that X-axis which divides the main 360 scale, horizontally. The exponents m and n are indicative of the direction of tilt with respect to the plane of a major (m) and minor (n) apex surface of the mother crystal. v

The other two curves, i. e. Km and Kn are frequency constant curves. The scale for reading these curves is marked on the lower half of the Y-axis which extends from the 90 main-scale @king through the Z-axis to the 270 marking. ./-ls with the case of the Vm and Vn curves the exponents m and n are indicative of the direction of tilt with respect to the plane of a major (m) and minor (n) apex surface of the mother crystal.

A technician in using the chart of Fig. 9 will rst select any W angle which intersects the curved V and K lines. Assuming, as in the prior example, that the W-axis selected is 20 from'an X-axis, then, toward the center of the chart (i. e. toward the Z-axis) it will be seen that this line intersects line V at a, Kn at b, Vm at c and Km at d, Points a and "c are read on the central horizontal scale and give the precise V-angle of rotation required for eachl direction of tilt; i. e. towards a minor apex face (a) or towards a major ap X face ("c). Projecting points a and c upon this central scale, it will be seen that if the blank is to be tilted towards a minor apex wardly from the center or Z-axis of the chart.-

The frequency constant Kn for a 20 'WH 36.5 Vn blank when read on this scale is seen to be 67.5

following the 20 line inwardlyl and the frequency constant Km for a 2O Wm, 43 Vm blank is 95.

As illustrative of a preferred manner of cutting piezoelectric elements having a zero or some other low temperature coefficient of frequency reference is had to Figs. 1 to 6 inclusive.

Referring first to Fig. 2, a section 2, say one inch thick, is first sliced from the body of the mother-crystal; the thickness dimension of this section is parallel with, and the length-breadth dimensions normal to the Z-axis. A bar 3 is then cut from this section 2, preferably at substantially the exact W-angle selected as a reference axis. In the illustrated embodiment the long dimensions of the bar 3 coincide with a W- axis which is 20 removed from that X-axis (and hence removed from that Y-axis) marked in Fig. 1.

Referring to Figs. 3 and 4, the blanks i and from which the finished elements are formed are then sliced from' bar E at either a Vm angle or Vn angle dictated by the temperature coefficient desired. The electrode faces of blank l, as shown, have been rotated substantially 36.5 (the Vn angle) about its Y-laxis in a direction away from parallelism with the Z-axis toward parallelism with the plane of that minor apex surface of the mother-crystal which is designated N1 in Figs. l and 4. (In Fig. 2, N1 is opposite the major apex surface designated M1.) This 36.5 V1"- angle is shown in Figs. 7 and 9 to be the precise angle required to achieve a substantially zero temperature coefficient of frequency-when the selected W-axis forms a W-angle with an X-axis of 26.

The electrode faces of blank of Figs. 3 and i have been rotated substantially 43 (the Vm angle) about its Y-laxis in a direction away from parallelism with the Z-axis towards parallelism with the plane of that major apex surface of the mother crystal which is designated MI in Figs. 1, 2 and 4.

Assuming, for purposes of illustration, that the finished element to be ground or lapped from blank i is to respond to a thickness-mode frequency of, say 2 megacycles, then since K" (frequency constant-l,

t (thlckngss) :f- (frequency in rnegacycles) and Kn (as shown in Figs. 8 and 9) for a 26 W-angle is equal to substantially 67.5. Therefore,

or, solved, t=substantially 33.75 thousandths of an inch.

Assuming, for purposes of illustration, that the finished element to be ground or lapped from blank 5 is to respond to a thickness-inode frequency of, say 8 megacycles. Applying the same formula,

which, solved, shows the thickness dimension required to achieve a 8 megacycle, zero temperature coefficient, Vm type crystal is substantially 11.87+ thousandths of an inch.

These dimensions having been determined it remains only to grind and lap the elements to the thickness dimensions set forth and to trim the edges (e, e4 Fig. Li, e e5 Fig. 5) so that the thickness dimension W-l-V, is normal to that one of the greater dimensions of the finished element which is designated Z-l-V. It is preferable to bevel the edges and round the corners of the otherwise finished elements of Figs. Ll and 5 in order to remove minute irregularities in the cutting and to prevent chipping in case the crystals should be excited at very great amplitudes of oscillation. As in prior art thickness-mode oscillators and resonators, the shape or configuration of the finished crystal (i. e. whether circular, polygonal, rectangular or other shapes) has little er no effect upon oscillation performance.

Although certainspecic ways and means for complishing the objects of the invention have en set forth it is to be understood that they have en given for the purpose of explaining the inventive concept and should not be construed as limitations to the scope of the invention. Neither is it to be understood that any statements hereinmade in regard to the value or relationships between frequency constants, angles of orientation and temperature coeicients of frequency are other than close approximations. The invention, therefore, is not to be limited except insofar as is necessitated by the prior art and by the spirit of the appended claims.

What is claimed is:

1. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having an electrode face parallel to a reference plane which is rotated about a Y|0 axis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of one of said apex surfaces, said iZ-l-H axis being normal to a W-axis which lies at least 1 removed from a Y-axis in a plane normal to the Z-axis, the exact angle formed by the intersection of said reference plane with said Z-axis being determined, in substantial agreement with the chart of Fig. '7, by the temperature coefficient of frequency desired and by the particular apex surface of the mother crystal toward which said electrode face is inclined.

2. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated not less than substantially 14 and no more than substantially 54 about a Y-l-H axis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a major apex surface, said Y-i-o axis being normal to a `lfi-axis which lies not less than substantially 15 and no more than substantially 29 removed from an X- axis in a plane normal to the Z-axis.

3. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated from substantially 14 to substantially 54 about a Y-laxis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a major apex surface, said Y-laxis being normal to a W-axis which lies between an X-axis and a Y-axis in a plane normal to the Z-axis, the angle formed by the intersection of said W -axis and said X-axis being between the limits of substantially 15 and not more than 29, whereby said element exhibits a temperature coefficient of frequency within substantially i5 cycles per million per degree centigrade when vibrated at a fundamental thickness-mode frequency.

fi. The invention as set forth in claim 3 further characterized in that the thickness of said elen ment expressed in thousandths of an inch is equal to .Li f

where f is a fundamental thickness-mode frequency expressed in megacycles and K is a number of the order of substantially to 103.

5. A quartz piezoelectric element cut from a mother crystal having 'major and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated from substantially 20 to substantially 49 about a Y-iaxis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a major apex surface, said Y|0 axis being normal to a W -axis which lies between an Xaxis and a Y-axis in a plane normal to the Z-axis, the angle formed by the intersection of said 'VV-axis and said X-axis being between the limits of substantially 15 and not more than substantially 29, whereby said element exhibits a substantially zero temperature coeflicient offrequency when vibrated at a fundamental thick-t ness-mode frequency.

6. The invention as set forth in claim 5 further characterized in that the thickness of said element expressed in thousandths of an inch is equal tol f where f is a fundamental thickness-mode frequency expressedin megacycles and K is a number of the order of substantially 88 to 100.6.

7. A quartz piezoelectric element cut from al mother crystal having maj or and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated from substantially 14 to substantially 44 about a Y-l-H axis in a direction away from parallelism with the LZ-axis and toward parallelism. with the plane of a major apex surface, said Y-laxis being normal to a Wfaxis which lies between an Xfaxis and a Y-axis in a plane normal to the Z-axis, the angle formed by the intersection of said W-axis and said X-axis being between the limits of substantially 15 and not more than substantially 29, whereby said element exhibits a temperature coeiiicient of frequency of substantially |5 cycles per million per degree centigrade when Vibrated at a fundamental thickness-mode frequency.

8. The invention as set forth in claim 7 further characterized in that the thickness of said element expressed in thousandths of an inch is equal to where f is a fundamental thickness-inode frequency expressed in megacycles and K is a number of the order of substantially S5 to 99.

9. A quartz piezoelectric element cut from a mother crystalhaving major and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated from substantially 25 to substantially 54 about a Y-i-)v axis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a major apex surface, said I4-0 axis being normal'to a W -axis which lies between an X-axis and a Y-axis in a plane normal to the Z-axis, the angle formed by the intersection of said W-axs and said X-axis being between the limits of substantially 15 and not more than substantially 29, whereby said element exhibits a temperature coefficient of frequency of substantially 5 cycles further characterized in that the thickness of said element expressed in thousandths of an inch is equal to i f where f is a fundamental thickness-mode frequency expressed in megacycles and K is a number of the order of substantially 90 to 103.`

11. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having its velectrode, faces parallel to a plane which is rotated not less than substantially 27 and no more than substantially 42 about a Y-l-H axis in a` direction away from parallelism with the Z-axis and toward parallelism with the plane of a minor apex surface, said Y-laxis being normal to a W-axis which lies not less than substantially 13 andA no more than substantially 29. removed from an X-axis ina plane normal to the Z-axis.

12. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having its electrode faces parallel to a plane .which is rotated from substantially 27.to substantially 42 about a Y-l-f'axis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a minor apex surface, said Y-faxis being normal to a W-axis which lies between an X-axis and a Y-axis in a plane normal to the Z-axis, the angle formed' by the intersection of said W- axis and said X-axis being `between the limits of substantially 13 and not more than substantially 29, whereby said element exhibits a, temperature coefficient of frequency within substantially 215 cycles `per million per degree centigrade when vibrated at a fundamental thickness-mode frequency.

13. The invention as set forth in claim 12 further characterized in that the thickness of said element expressed in thousandths of an inch is equal to "HIN where f is a fundamental thickness-mode frequency expressed in megacycles and K is a number of the order of substantially 70.2 to 66.9.

14. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated from substantially 32 to substantially 37 about a Y-l-0 axis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a minor apex surface, said Y-iaxis being normal to a W-axis which lies between an X- axis and a Y-axis in a plane normal to the Z- axis, the angle formed by the intersection of said W-axis and said X-axis being between the limits of substantially 13 and not more than substantially 29, whereby said element exhibits a substantially zero temperature coefficient of frequency when vibrated at a fundamental thickness-mode frequency.

l5. The invention as set forth in claim 14 further characterized in that the thickness of said element expressed in thousandths of an inch is equal to 1S f where j is a fundamentalthickness-mode 4fre-` quency expressed in megacycles and K is a number of the order of substantially 68.2 to 66.6.

16. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated from substantially 27 to substantially 32 about a Y-l-H axis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a minor apex surface, said Y-laxis being normal to a W-axis which lies between an X- axis and a Y-axis in a plane normal to the Z- axis, the angle formed by the intersection of said W-axis and said X-axis being between the limits of substantially 13 and not more than substantially 29, whereby said element exhibits a temperature coefficient of frequency of substantially +15 cycles per million per degree centigrade when vibrated at a fundamental thickness mode frequency.

17. The invention as set forth in claim 16 further characterized in that the thickness of said element expressed in thousandths of an inch is equal to where f is a fundamental thickness-mode frequency expressed in megacycles and K is a number of the order of substantially 68.6 to 66.9.

18. A quartz piezoelectric element cut from a mother crystal having major and minor apex surfaces, said element having its electrode faces parallel to a plane which is rotated from substantially 37 to substantially 42 about a Y|6 axis in a direction away from parallelism with the Z-axis and toward parallelism with the plane of a minor apex surface, said Y+0 axis being normal to a W-axis which lies between an X- axis and a Y-axis in a plane normal to the Z- axis, the angle formed by the intersection of said W-axis and said X-axis being between the limits of substantially 13 and not more than substantially 29, whereby said element exhibits a temperature coefficient of frequency of substantially l5 cycles per million per degree centigrade when vibrated at a fundamental thickness-mode frequency.

19. The invention as set forth in claim 18 further characterized in that the thickness of said element expressed in thousandths of an inch is equal to Where f is a fundamental thickness-mode frequency expressed in megacycles and K is a number of the order of substantially 70.2 to 68.6.

20. A quartz piezoelectric element of low temperature coefficient of frequency adapted to vibrate in a shear mode at a frequency determined substantially by its thickness or smallest dimension which is perpendicular to its vsubstantially parallel major surfaces, said major surfaces having an edge axis inclined substantially 90 degrees with respect to the Z axis and substantially l0 degrees with respect to an X axis, and having another edge axis disposed substantially perpendicular to said first-mentioned edge axis and inclined with respect to the Z axis substantially +36 degrees or toward parallelism with the plane of a minor apex face of the crystal from which said element is cut.

21. A quartz piezoelectric element of low temperature coeiiicient cf frequency adapted to vibrate in a shear inode at a frequency determined substantially by its thickness or smallest dimension which is perpendicular to its substantially parallel major surfaces, said major surfaces having an edge axis inclined substantially 90 with respect to the Z axis and substantially 5 with respect to an X axis, and having another edge axis disposed substantially perpendicular to said first mentioned edge axis and inclined with respect to the Z axis substantially +35.5 or t0- ward parallelism with the plane of a minor apex face of the crystal from which said element is cut.

CHARLES F. BALDWIN. SAMUEL A. BOKOVOY.

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