Patent Publication Number: US-7723904-B2

Title: Resonator, unit having resonator, oscillator having unit and electronic apparatus having oscillator

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application is a continuation of application Ser. No. 11/607,276 filed Nov. 30, 2006, now U.S. Pat. No. 7,342,352 which is a continuation of application Ser. No. 10/875,114 filed Jun. 23, 2004 and now U.S. Pat. No. 7,193,354, which are hereby incorporated by reference, and priority thereto for common subject matter is hereby claimed. 

   FIELD OF THE INVENTION 
   The present invention relates to a resonator, a unit having the resonator, an oscillator having the unit and an electronic apparatus having the oscillator. 
   BACKGROUND OF THE INVENTION 
   There are many electronic apparatus comprising a display portion and a quartz crystal oscillator at least. For example, cellular phones, wristwatches, facsimiles, digital cameras and DVD recorders comprising a quartz crystal oscillator are well known. Recently, because of high stability for frequency, miniaturization and the light weight nature of these electronic apparatus, the need for an electronic apparatus comprising a smaller quartz crystal oscillator with a frequency of high stability has arisen. For example, the quartz crystal oscillator having a quartz crystal tuning fork resonator housed in a unit, which vibrates in a flexural mode, is widely used as a time standard in an electronic apparatus such as the cellular phones, the wristwatches, the facsimiles, the digital cameras and the DVD recorders. 
   Similar to this, the same need has also arisen for an electronic apparatus comprising a contour mode resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator or a SAW (Surface Acoustic Wave) resonator or a resonator for sensing angular velocity made of a piezoelectric material such as quartz crystal, lithium tantalite (LiTaO 3 ), lithium niobate (LiNbO 3 ) and ceramics. 
   Heretofore, however, it has been impossible to obtain an electronic apparatus comprising a smaller quartz crystal oscillator with a miniature quartz crystal tuning fork resonator of the prior art, capable of vibrating in a flexural mode, and having a frequency of high stability, a small series resistance and a high quality factor. This is the reason why, when miniaturized, the quartz crystal tuning fork resonator of the prior art, capable of vibrating in a flexural mode has a smaller electromechanical transformation efficiency. As a result, the resonator has a frequency of low stability, a large series resistance and a reduced quality factor. 
   Additionally, there has been a big problem in the quartz crystal oscillator of the prior art having the quartz crystal tuning fork resonator of the prior art, such that a frequency of a fundamental mode of vibration of the tuning fork resonator which is an output signal of the oscillator jumps to a second overtone mode of vibration thereof by shock or vibration. 
   Similarly, however, it has been impossible to obtain an electronic apparatus comprising a smaller quartz crystal oscillator with a contour mode resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator or a SAW resonator or a resonator for sensing angular velocity having a frequency of high stability, a small series resistance and a high quality factor because, when miniaturized, each resonator has a small electromechanical transformation efficiency, as a result, a frequency of low stability, a large series resistance and a low quality factor, and also is not strong against shock. 
   It is, therefore, a general object of the present invention to provide embodiments of a quartz crystal resonator, a quartz crystal unit, a quartz crystal oscillator and an electronic apparatus of the present invention, which overcome or at least mitigate one or more of the above problems. 
   SUMMARY OF THE INVENTION 
   The present invention relates to a resonator, a unit, an oscillator and an electronic apparatus comprising a display portion and a plurality of oscillators, one of which comprises a quartz crystal oscillator comprising a quartz crystal oscillating circuit having an amplification circuit and a feedback circuit, and in particular, relates to a quartz crystal resonator capable of vibrating in a flexural mode, a quartz crystal unit having the quartz crystal resonator and a quartz crystal oscillator having the quartz crystal unit and having an output signal of a frequency of high stability for a fundamental mode of vibration of the quartz crystal resonator, and also to a quartz crystal oscillator having a suppressed second overtone mode of vibration of the quartz crystal resonator, in addition, relates to a quartz crystal oscillator comprising an another contour mode resonator such as a length-extensional mode resonator, a width-extensional mode resonator and a Lame mode resonator or a thickness shear mode resonator, each made of quartz crystal or a SAW resonator or a piezoelectric resonator for sensing angular velocity. The quartz crystal oscillator is, therefore, available for the electronic apparatus requiring a miniature quartz crystal oscillator with high time accuracy and shock proof. 
   It is an object of the present invention to provide a miniature quartz crystal resonator, capable of vibrating in a flexural mode, and having a high electromechanical transformation efficiency. 
   It is an another object of the present invention to provide a miniature quartz crystal unit with a quartz crystal resonator, capable of vibrating in a fundamental mode of vibration of a flexural mode, and having a high electromechanical transformation efficiency. 
   It is a further object of the present invention to provide a quartz crystal oscillator with a miniature quartz crystal resonator, capable of vibrating in a flexural mode, and having a frequency of high stability, a small series resistance R 1  and a high quality factor Q 1 , whose nominal frequency for a fundamental mode of vibration is within a range of 10 kHz to 200 kHz. Especially, a frequency of about 32.768 kHz is very available for a time standard of a frequency signal. 
   It is a still another object of the present invention to provide an electronic apparatus comprising a display portion and a plurality of oscillators. 
   According to one aspect of the present invention, there is provided a quartz crystal resonator comprising: a plurality of vibrational arms, each of the vibrational arms having a first main surface and a second main surface and side surfaces; and a base portion to which the vibrational arms are attached, in which the resonator has a piezoelectric constant e′ 12  in the range of 0.1 C/m 2  to 0.19 C/m 2  in the absolute value. 
   According to a second aspect of the present invention, there is provided a quartz crystal unit comprising: a quartz crystal resonator having a base portion and a plurality of vibrational arms attached to the base portion; a case for housing the quartz crystal resonator; and a lid for covering an open end of the case, each of the vibrational arms having a first main surface and a second main surface opposite the first main surface and side surfaces, in which the quartz crystal resonator has a cutting angle in the range of ZY1wt(−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectric constant e′ 12  of the resonator is within a range of 0.1 C/m 2  to 0.19 C/m 2  in the absolute value. 
   According to a third aspect of the present invention, there is provided a quartz crystal oscillator comprising: a quartz crystal oscillating circuit comprising; an amplification circuit comprising a CMOS inverter and a feedback resistor, and a feedback circuit comprising a quartz crystal resonator capable of vibrating in a flexural mode, a plurality of capacitors and a drain resistor, the quartz crystal resonator being housed in a package comprising a case for housing the quartz crystal resonator and a lid for covering an open end of the case, and comprising: a plurality of vibrational arms, each of the vibrational arms having a first main surface and a second main surface opposite the first main surface and side surfaces; and a base portion to which the vibrational arms are attached, in which the quartz crystal resonator has a cutting angle in the range of ZY1wt (−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectric constant e′ 12  of the resonator is within a range of 0.1 C/m 2  to 0.19 C/m 2  in the absolute value. 
   According to a fourth aspect of the present invention, there is provided an electronic apparatus comprising a display portion and a plurality of oscillators, one of the oscillators being a quartz crystal oscillator comprising: a quartz crystal oscillating circuit comprising; an amplification circuit having a CMOS inverter and a feedback resistor, and a feedback circuit having a quartz crystal tuning fork resonator capable of vibrating in a flexural mode of an inverse phase, a plurality of capacitors and a drain resistor, the quartz crystal tuning fork resonator comprising a tuning fork base and a plurality of tuning fork arms connected to the tuning fork base, each of the tuning fork arms having a first main surface and a second main surface opposite the first main surface and side surfaces, the quartz crystal tuning fork resonator being housed in a package comprising a case for housing the resonator and a lid for covering an open end of the case, in which the quartz crystal tuning fork resonator has a fundamental mode of vibration and a second overtone mode of vibration and the amplification circuit of the quartz crystal oscillating circuit has negative resistances −RL 1  and −RL 2  for the fundamental mode of vibration and the second overtone mode of vibration of the quartz crystal tuning fork resonator, in which an absolute value of the negative resistances is defined by |−RL 1 | and |−RL 2 | and a ratio of the |−RL 1 | and R 1  is greater than that of the |−RL 2 | and R 2 , where R 1  and R 2  represent a series resistance of the fundamental mode of vibration and the second overtone mode of vibration of the quartz crystal resonator, respectively, in which an output signal of the quartz crystal oscillating circuit has an oscillation frequency of the fundamental mode of vibration of the quartz crystal tuning fork resonator and is a clock signal which is used to display time at a display portion of the electronic apparatus, and in which the quartz crystal tuning fork resonator has a cutting angle in the range of ZY1wt (−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectric constant e′ 12  of the resonator is within a range of 0.1 C/m 2  to 0.19 C/m 2  in the absolute value. 
   Preferably, the piezoelectric constant e′ 12  is within a range of 0.12 C/m to 0.19 C/m 2  in the absolute value. 
   Preferably, mounting arms protruding from the base portion comprise two. 
   Preferably, each of the vibrational arms has a groove having a first stepped portion and a second stepped portion. 
   Preferably, each of the vibrational arms has a through groove. 
   Preferably, a merit value M 2  of a second overtone mode of vibration of the quartz crystal resonator having vibrational arms and a base portion is less than 30. 
   Preferably, the quartz crystal oscillator with the quartz crystal resonator is constructed so that a ratio of an amplification rate α 1  of the fundamental mode of vibration and an amplification rate α 2  of the second overtone mode vibration of the amplification circuit is greater than that of a feedback rate β 2  of the second overtone mode vibration and a feedback rate β 1  of the fundamental mode vibration of the feedback circuit, and a product of the amplification rate α 1  and the feedback rate β 1  of the fundamental mode vibration is greater than 1. 
   The present invention will be more fully understood by referring to the following detailed specification and claims taken in connection with the appended drawings. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a general view of a quartz crystal plate from which a quartz crystal resonator of the present invention is formed; 
       FIG. 2  shows a plan view of a quartz crystal resonator of a first embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode; 
       FIG. 3  shows an A-A′ cross-sectional view of the vibrational arms of the quartz crystal resonator in  FIG. 2 ; 
       FIG. 4  shows an A-A′ cross-sectional view of another embodiment of the vibrational arms of the quartz crystal resonator in  FIG. 2 ; 
       FIG. 5  shows a plan view of a quartz crystal resonator of a second embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode; 
       FIG. 6  shows a plan view of a quartz crystal resonator of a third embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode; 
       FIG. 7  shows a B-B′ cross-sectional view of the vibrational arms of the resonator in  FIG. 6 ; 
       FIG. 8  shows a plan view of a quartz crystal resonator of a fourth embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode; 
       FIG. 9  shows a plan view of a quartz crystal resonator of a fifth embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode; 
       FIG. 10  shows a plan view of a quartz crystal resonator of a sixth embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode; 
       FIG. 11  shows a plan view of a quartz crystal resonator of a seventh embodiment of the present invention, and comprising a quartz crystal tuning fork resonator capable of vibrating in a flexural mode; 
       FIG. 12  shows a plan view of a width-extensional mode quartz crystal resonator constructing an electronic apparatus of the present invention; 
       FIG. 13(   a ) and  FIG. 13(   b ) show a plan view of a thickness shear mode quartz crystal resonator constructing an electronic apparatus of the present invention and a F-F′ sectional view of the resonator; 
       FIG. 14  shows a plan view of a Lame mode quartz crystal resonator constructing an electronic apparatus of the present invention; 
       FIG. 15(   a ) and  FIG. 15(   b ) show a plan view of a resonator for sensing angular velocity constructing an electronic apparatus of the present invention and a G-G′ sectional view of the resonator; 
       FIG. 16(   a ) and  FIG. 16(   b ) show a plan view of a quartz crystal resonator of an eighth embodiment of the present invention and comprising a quartz crystal tuning fork resonator, and a J-J′ sectional view of the resonator; 
       FIG. 17  shows a plan view of a quartz crystal unit of a first embodiment of the present invention and omitting a lid; 
       FIG. 18  shows a plan view of a quartz crystal unit of a second embodiment of the present invention and omitting a lid; 
       FIG. 19  shows a cross-sectional view of a quartz crystal unit of a third embodiment of the present invention; 
       FIG. 20  shows a cross-sectional view of a quartz crystal oscillator of a first embodiment of the present invention; 
       FIG. 21  shows a diagram of an embodiment of a quartz crystal oscillating circuit constructing a quartz crystal oscillator of the present invention; 
       FIG. 22  shows a diagram of the feedback circuit of  FIG. 21 ; and 
       FIG. 23  shows a block diagram of an embodiment of an electronic apparatus of the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Referring now to the drawings, the embodiments of the present invention will be described in more detail. 
     FIG. 1  is a general view of a quartz crystal plate  1  from which a quartz crystal resonator of the present invention is formed, and particularly, a relationship of cutting angles θ x , θ y  and θ z  of the quartz crystal plate  1  and its coordinate system is illustrated in  FIG. 1 . The coordinate system has original point o, electrical axis x, mechanical axis y and optical axis z of quartz crystal and o-xyz is constructed. 
   First, a quartz crystal plate perpendicular to z axis, so called, Z plate quartz crystal is taken. The Z plate quartz crystal has a dimension of Width W 0 , length L 0  and thickness T 0  corresponding to a respective direction of x, y and z axes. 
   Next, this Z plate quartz crystal is, first, rotated with an angle θ y  about the y axis, second, rotated with an angle θ x  about x′ axis which is a new axis of the x axis, and third, rotated with an angle θ z  about z″ axis which is a new axis of the z axis. In this case, each of the x, y and z axes changes to x″, y″ and z″ axes, respectively, because each axis is rotated twice about two axes. A quartz crystal resonator of the present invention is, therefore, formed from the quartz crystal plate with the rotation angles. 
   In other words, according to an expression of IEEE notation, a cutting angle of the quartz crystal resonator of the present invention can be expressed by ZY1wt(θ y )/(θ x )/(θ z ), and each of the angles θ y , θ x , θ z  will be described later in detail according to resonators of the present invention. 
     FIG. 2  shows a plan view of a quartz crystal resonator  10  of a first embodiment of the present invention and which is a quartz crystal tuning fork resonator. The resonator  10  comprises vibrational arms  20  and  31  and a base portion  40  attached to the vibrational arms, and the base portion  40  has mounting arms  36  and  37  protruding from the base portion, each of which is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms  20  and  31  has a first main surface and a second main surface opposite the first main surface and side surfaces, and the vibrational arms  20  and  31  have grooves  21  and  27 , respectively, each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, the resonator  10  has cutting angles θ y , θ x  and θ z  which are within a range of −20° to +20°, −25° to +25° and −18° to +18°, respectively, namely, a cutting angle of the resonator is within a range of ZY1wt(−20° to +20°)/(−25° to +25°)/(−18° to +18°). In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode of a fundamental mode of an inverse phase, and which is one of a contour mode quartz crystal resonator. 
   In more detail, the groove  21  is constructed to include a portion of a central linear line  41  of the arm  20 , and the groove  27  is similarly constructed to include a portion of a central linear line  42  of the arm  31 . Each of the grooves  21  and  27  has a width W 2 , and the width W 2  including a portion of the central linear lines  41  and  42 , is preferable because a large moment of inertia occurs at the arms  20  and  31  and the arms can vibrate in a flexural mode easily. As a result, the quartz crystal tuning fork resonator capable of vibrating in a fundamental mode can be obtained with a small series resistance R 1  and a high quality factor Q 1 . 
   In addition, when each of the vibrational arms  20  and  31  has part widths W 1  and W 3 , an arm width W of the arms  20  and  31  has a relationship of W=W 1 +W 2 +W 3 , and the part widths W 1  and W 3  are constructed so that W 1 ≧W 3  or W 1 &lt;W 3 . In addition, the width W 2  is constructed so that W 2 ≧W 1 , W 3 . In this embodiment, also, the grooves are constructed at the arms so that a ratio W 2 /W of the width W 2  and the arm width W is greater than 0.35 and less than 1, preferably, within a range of 0.35 to 0.95 and a ratio t 1 /t is less than 0.79, where t 1  and t are a thickness of the groove and the vibrational arms, as shown in  FIG. 3 , to obtain a very large moment of inertia of the vibrational arms. That is, the quartz crystal tuning fork resonator, capable of vibrating in the fundamental mode, and having a frequency of high stability can be provided with a small series resistance R 1 , a high quality factor Q 1  and a small capacitance ratio r 1  because it has a very large electromechanical transformation efficiency. 
   Likewise, each of the vibrational arms  20  and  31  has a length L and each of the grooves  21  and  27  has a length l 0  (not shown here). In this embodiment, a ratio of the length l 0  and the length L is within a range of 0.3 to 0.8 to get a quartz crystal tuning fork resonator with series resistance R 1  of a fundamental mode of vibration smaller than series resistance R 2  of a second overtone mode of vibration. In other words, the length l 0  is within a range of 30% to 80% to the length L. In general, the length l 0  is within a range of 0.45 mm to 1.25 mm. Also, when a plurality of grooves are formed in at least one of upper and lower faces of the arms and divided into the length direction of the arms, the length l 0  is a total length of the grooves. 
   In addition, electrodes  25  and  26  are disposed on side surfaces of the vibrational arm  20  and an electrode  23  is disposed on a surface of the groove  21 , which extends into the mounting arm  36  having a connecting portion  34 . Similar to this, electrodes  32  and  33  are disposed on side surfaces of the vibrational arm  31  and an electrode  29  is disposed on a surface of the groove  27 , which extends into the mounting arm  37  having a connecting portion  35 . Each of the connecting portions  34  and  35  has a length L 2  and a width W S , and each of the mounting arms has a length L 3  and a width W 6 . Also, the electrode  23  is connected to the electrodes  32  and  33 , and the electrode  29  is connected to the electrodes  25  and  26 . 
   In this embodiment, the length l o  of the groove corresponds to a length l d  of the electrode disposed inside each of the grooves, when the length l d  of the electrode is less than the length l 0  of the groove, namely, the length l 0  is of the length l d  of the electrode. In addition, the base portion  40  has a length L 1  and a width W H , the length L 1  is less than 0.5 mm, preferably, within a range of 0.015 mm to 0.49 mm, and a total length L t  (=L+L 1 ) in this embodiment is less than 2.1 mm, preferably, within a range of 1.02 mm to 1.95 mm to obtain a miniature quartz crystal resonator. Also, when a distance W 4  between the vibrational arms is taken, a total width W 5  (=2W+W 4 ) is less than 0.53 mm, preferably, within a range of 0.15 mm to 0.52 mm, and the width W 5  is equal to or less than the W H  which is less than 0.55 mm, preferably, within a range of 0.15 mm to 0.53 mm. 
   In addition, the length L 3  is greater than or equal to the length L 2  and also, the length L 1  is greater than or equal to the length L 2  or the length L 1  is less than the length L 2 . In actual, a value of L 1 -L 2  is within a range of −0.1 mm to 0.32 mm, especially, when W H  is greater than W 5 , a distance L 4  between an edge of the connecting portion and an outer edge of the vibrational arm is within a range of 0.012 mm to 0.38 mm. Also, the width W 6  is less than 0.45 mm and the length L 3  is less than 2.1 mm, preferably, within a range of 0.3 mm to 1.85 mm to reduce a leakage energy by vibration, and also, the width W S  is less than 0.41 mm and the length L 2  is within a range of 0.04 mm to 0.5 mm to get a shock proof quartz crystal resonator having a reduced leakage energy by vibration. 
   Moreover, the distance W 4  and the width W 2  are constructed so that W 4 ≧W 2 , and more, the distance W 4  is within a range of 0.045 mm to 0.65 mm and the width W 2  is within a range of 0.02 mm to 0.12 mm, because it is easy to form a resonator shape and grooves formed at the vibrational arms separately by a photo-lithographic process and an etching process, consequently, a frequency stability for a fundamental mode of vibration of the resonator gets higher than that for a second overtone mode of vibration thereof. In this embodiment, a quartz wafer having a thickness t of 0.045 mm to 0.35 mm is used. 
   For example, in order to get a smaller-sized quartz crystal tuning fork resonator, capable of vibrating in a flexural mode, it is necessary that the width W 2  of the groove is less than 0.07 mm and the arm width W is less than 0.18 mm, and preferably, the W is greater than 0.05 mm and less than 0.1 mm. Also, the thickness t 1  of the groove is within a range of 0.01 mm to 0.085 mm approximately, and the part widths W 1  and W 3  are less than 0.021 mm, respectively, preferably, less than 0.015 mm. In addition, a groove provided on at least one of an obverse face and a reverse face of the vibrational arms of this embodiment may be a through hole, namely, the thickness of the hole t 1 =0. 
   In more detail, to obtain a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode, and having a frequency of high stability which achieves high time accuracy, it is necessary to obtain the resonator whose resonance frequency is not influenced by shunt capacitance because quartz crystal is a piezoelectric material and the stability for frequency is very dependent on the shunt capacitance. In order to decrease the influence on the resonance frequency by the shunt capacitance, a merit value M i  plays an important role. Namely, the merit value M i  that expresses inductive characteristics, an electromechanical transformation efficiency and a quality factor of a quartz crystal tuning fork resonator, is defined by a ratio Q i /r i  of a quality factor Q i  and capacitance ratio r i , namely, M i  is given by M i =Q i /r i , where i shows a vibration order of the resonator, and for example, when i=1 and 2, merit values M 1  and M 2  are for a fundamental mode of vibration of the resonator and a second overtone mode of vibration thereof, respectively. 
   Also, a frequency difference Δf of resonance frequency f s  of mechanical series independent on the shunt capacitance and resonance frequency f r  dependent on the shunt capacitance is inversely proportional to the merit value M i . The larger the value M i  becomes, the smaller the difference Δf becomes. Namely, the influence on the resonance frequency f r  by the shunt capacitance decreases because it is close to the resonance frequency f s . Accordingly, the larger the M i  becomes, the higher the stability for frequency of the quartz crystal tuning fork resonator becomes because the resonance frequency f r  of the resonator is almost never dependent on the shunt capacitance. Namely, the quartz crystal tuning fork resonator can be provided with a high time accuracy. 
   In detail, the quartz crystal tuning fork resonator can be obtained with the merit value M 1  of the fundamental mode of vibration greater than the merit value M 2  of the second overtone mode of vibration by providing the above-described tuning fork shape, grooves and dimensions. That is to say, a relationship of M 1 &gt;M 2  is obtained. As an example, when a resonance frequency of a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is about 32.768 kHz for a fundamental mode of vibration and the resonator has a value of W 2 /W=0.5, t 1 /t=0.34 and l 1 /l=0.48, though there is a distribution in production, the resonator has a merit value of M 1 &gt;65 for the fundamental mode of vibration and a merit value of M 2 &lt;30 for the second overtone mode of vibration, respectively. 
   Namely, the quartz crystal tuning fork resonator can be provided with high inductive characteristics, good electromechanical transformation efficiency (small capacitance ratio r 1  and small series resistance R 1 ) and a high quality factor. As a result, a stability for frequency of the fundamental mode of vibration becomes higher than that of the second overtone mode of vibration, and simultaneously, the second overtone mode of vibration can be suppressed because capacitance ratio r 2  and series resistance R 2  of the second overtone mode of vibration become greater than capacitance ratio r 1  and series resistance R 1  of the fundamental mode of vibration, respectively. In particular, r 2  has a value greater than 1500 in this embodiment. 
   Therefore, the resonator capable of vibrating in the fundamental mode vibration can be provided with a high time accuracy because it has the frequency of high stability. Consequently, a quartz crystal oscillator comprising the quartz crystal tuning fork resonator of this embodiment outputs an oscillation frequency of the fundamental mode vibration as an output signal, and the frequency of the output signal has a very high stability, namely, excellent time accuracy. In other words, the quartz crystal oscillator of this embodiment has a remarkable effect such that a frequency change by ageing becomes extremely small. Also, an oscillation frequency of the quartz crystal resonator of this embodiment is adjusted so that a frequency deviation is within a range of −100 PPM to +100 PPM to a nominal frequency, e.g. 32.768 kHz, after mounting it on a mounting portion of a case for housing the resonator. 
   In addition, the groove thickness t 1 , shown in  FIG. 3 , of the present invention is the thinnest thickness of the grooves because quartz crystal is an anisotropic material and the groove thickness t 1  has a distribution when it is formed by a chemical etching method. In detail, a groove shape of the sectional view of vibrational arms in  FIG. 3  has a rectangular shape, but the groove shape has an about U shape actually. In the above-described embodiments, though the grooves are constructed at the arms, this invention is not limited to this, namely, a relationship of the merit values M 1  and M 2  can be applied to the conventional quartz crystal tuning fork resonator and a relationship of a quartz crystal oscillating circuit comprising an amplification circuit and a feedback circuit can be also applied to the conventional quartz crystal tuning fork resonator to suppress a second overtone mode vibration and to get a high frequency stability for a fundamental mode of vibration of the quartz crystal tuning fork resonator. 
     FIG. 3  shows an A-A′ cross-sectional view of the vibrational arms  20  and  31  of the quartz crystal resonator  10  in  FIG. 2 , and electrode construction within the grooves. The vibrational arm  20  has grooves  21  and  22  cut into it, which include a portion of central linear line of the arm  20 . The grooves  21  and  22  have a first set of electrodes  23  and  24  of the same electrical polarity, while the side surfaces of the arm  20  have a second set of electrodes  25  and  26  having an opposite electrical polarity to the first set of electrodes  23  and  24 . The vibrational arm  31  has grooves  27  and  28  constructed in a similar manner as the vibrational arm  20 . The grooves  27  and  28  have a third set of electrodes  29  and  30  of the same electrical polarity, and the side surfaces of the vibrational arm  31  have a fourth set of electrodes  32  and  33  with the opposite electrical polarity to the third electrodes  29  and  30 . The electrodes disposed on the vibrational arms  20  and  31  are connected as shown in  FIG. 3 , namely, two electrode terminals of different electrical polarity C-C′ are obtained. 
   In detail, the first set of electrodes  23  and  24  disposed on the grooves  21  and  22  of the vibrational arm  20  have the same electrical polarity as the fourth set of electrodes  32  and  33  disposed on both side surfaces of the vibrational arm  31 , while the second set of electrodes  25  and  26  disposed on both side surfaces of the vibrational arm  20  have the same electrical polarity as the third set of electrodes  29  and  30  disposed on the grooves  27  and  28  of the arm  31 . When a direct voltage is applied between the electrode terminals C-C′, an electric field Ex occurs along the arrow direction inside the vibrational arms  20  and  31 . As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. As a result, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R 1  and a high quality factor Q because even when miniaturized there is a very large electromechanical transformation efficiency for the resonator. 
     FIG. 4  shows an A-A′ cross-sectional view of another embodiment of the vibrational arms  20  and  31  of the quartz crystal resonator  10  in  FIG. 2 . The vibrational arm  20  has a through groove  21   a , which include a portion of central linear line of the arm  20 . The through groove  21   a  has a first set of electrodes  23   a  and  24   a  of the same electrical polarity, while the side surfaces of the arm  20  have a second set of electrodes  25  and  26  having an opposite electrical polarity to the first set of electrodes  23   a  and  24   a . The vibrational arm  31  has a through groove  27   a  constructed in a similar manner as the vibrational arm  20 . The through groove  27   a  has a third set of electrodes  29   a  and  30   a  of the same electrical polarity, and the side surfaces of the vibrational arm  31  have a fourth set of electrodes  32  and  33  with the opposite electrical polarity to the third electrodes  29   a  and  30   a . The electrodes disposed on the vibrational arms  20  and  31  are connected as shown in  FIG. 4 , namely, two electrode terminals of different electrical polarity E-E′ are obtained. 
   In detail, the first set of electrodes  23   a  and  24   a  disposed on the through groove  21   a  of the vibrational arm  20  have the same electrical polarity as the fourth set of electrodes  32  and  33  disposed on both side surfaces of the vibrational arm  31 , while the second set of electrodes  25  and  26  disposed on both side surfaces of the vibrational arm  20  have the same electrical polarity as the third set of electrodes  29   a  and  30   a  disposed on the through groove  27   a  of the arm  31 . When a direct voltage is applied between the electrode terminals E-E′, an electric field Ex occurs along the arrow direction inside the vibrational arms  20  and  31 . As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. As a result, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R 1  and a high quality factor Q because even when miniaturized there is a very large electromechanical transformation efficiency for the resonator. 
     FIG. 5  shows a plan view of a quartz crystal resonator  50  of a second embodiment of the present invention, and which is a quartz crystal tuning fork resonator capable of vibrating in a flexural mode. The resonator  50  comprises vibrational arms  60  and  71  and a base portion  80  attached to the vibrational arms, and the base portion  80  has a mounting arm  77  protruding from the base portion, and the mounting arm  77  is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms  60  and  71  has a first main surface and a second main surface and side surfaces, and the vibrational arms  60  and  71  have grooves  61  and  67 , respectively, each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, the resonator  50  has the same cutting angles θ y , θ x  and θ z  and the same dimensions W 1 , W 2 , W 3 , W 4 , W 5 , W, L 1 , L 2 , L 3 , L 4  and L as the resonator of  FIG. 2 . 
   In more detail, the groove  61  is constructed to include a portion of a central linear line  81  of the arm  60 , and the groove  67  is similarly constructed to include a portion of a central linear line  82  of the arm  71 . Each of the grooves  61  and  67  has a width W 2 , and the width W 2  includes a portion of the central linear lines  81  and  82  because a large moment of inertia occurs at the arms  60  and  71 . In this embodiment, the resonator is a quartz crystal tuning fork resonator capable of vibrating in a flexural mode and which can vibrate in a fundamental mode of an inverse phase easily. As a result, the quartz crystal tuning fork resonator capable of vibrating in a fundamental mode of an inverse phase can be obtained with a small series resistance R 1  and a high quality factor Q 1 . 
   In addition, electrodes  65  and  66  are disposed on side surfaces of the vibrational arm  60  and an electrode  63  is disposed on a surface of the groove  61 , and which is connected to electrodes  72  and  73  disposed on side surfaces of the vibrational arm  71 . Similar to this, an electrode  69  is disposed on a surface of the groove  67 , which extends into the mounting arm  77  having a connecting portion  75 . Also, the electrode  69  is connected to the electrodes  65  and  66 . 
     FIG. 6  shows a plan view of a quartz crystal resonator  90  of a third embodiment of the present invention, which is a quartz crystal tuning fork resonator capable of vibrating in a flexural mode. The resonator  90  comprises vibrational arms  91  and  92  and a base portion  95  attached to the vibrational arms, and the base portion  90  has a mounting arm  96  protruding from the base portion. In this embodiment, the mounting arm  96  is between the vibrational arms  91  and  92  and is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms  91  and  92  has a first main surface and a second main surface opposite the first main surface and side surfaces, and the vibrational arms  91  and  92  have grooves  93  and  94 , respectively, each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, the resonator  90  has the same cutting angles θ y , θ x  and θ z  as the quartz crystal resonator  10  of  FIG. 2 . In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode of a fundamental mode of an inverse phase. 
   In detail, similar to the resonator of  FIG. 2 , the groove  93  is also constructed to include a portion of a central linear line of the arm  91 , and also, the groove  94  is similarly constructed to include a portion of a central linear line of the arm  92 . Each of the grooves  93  and  94  has a width W 2 , and the width W 2  includes a portion of the central linear lines because a large moment of inertia occurs at the vibrational arms  91  and  92  and which can vibrate in a flexural mode easily. As a result, the quartz crystal tuning fork resonator capable of vibrating in a fundamental mode can be obtained with a small series resistance R 1  and a high quality factor Q 1 . 
   In addition, the base portion  95  has a length L 1  and a width W H . Also, each of the vibrational arms  91  and  92  has a width W, part widths W 1  and W 3  and a width W 2  of the groove, namely, there is a relationship of W=W 1 +W 2 +W 3 . In detail, the resonator  90  has the same dimensions L 1 , W H , W 1 , W 2 , W 3 , and W and the same relationship as the resonator  10  of  FIG. 2 . Moreover, the mounting arm  96  has a width W 11  and there is a distance W 10  between the vibrational arm  91  or the vibrational arm  92  and the mounting arm  96 . In order to get a quartz crystal tuning fork resonator with reduced leakage energy by vibration of the vibrational arms, the distance W 10  is within a range of 0.032 mm to 0.21 mm and the width W 11  is within a range of 0.21 mm to 0.88 mm. In addition, to get a shockproof quartz crystal tuning fork resonator, W 11  has a relationship of (1.2 to 7.6)×W. In this embodiment, a total width W t (=2W+2W 10 +W 11 ) is less than 1.3 mm, preferably, within a range of 0.52 mm to 1.2 mm, to get a miniature quartz crystal tuning fork resonator. 
     FIG. 7  shows a B-B′ cross-sectional view of the vibrational arms  91  and  92  of the resonator  90  in  FIG. 6 . The vibrational arm  91  has grooves  93  and  97  cut into it, and the grooves  93  and  97  have a first set of electrodes  100  and  101  of the same electrical polarity, while the side surfaces of the arm  91  have a second set of electrodes  99  and  102  having an opposite electrical polarity to the first set of electrodes  100  and  101 . The vibrational arm  92  has grooves  94  and  98  constructed in a similar manner as the vibrational arm  91 . The grooves  94  and  98  have a third set of electrodes  106  and  107  of the same electrical polarity, and the side surfaces of the vibrational arm  92  have a fourth set of electrodes  105  and  108  with the opposite electrical polarity to the third electrodes  106  and  107 . The electrodes disposed on the vibrational arms  91  and  92  are connected as shown in  FIG. 7 , namely, two electrode terminals of different electrical polarity D-D′ are obtained. 
   In detail, the first set of electrodes  100  and  101  disposed on the grooves  93  and  97  of the vibrational arm  91  have the same electrical polarity as the fourth set of electrodes  105  and  108  disposed on both side surfaces of the vibrational arm  92  and an electrode  104  disposed on the mounting arm  96 , while the second set of electrodes  99  and  102  disposed on both side surfaces of the vibrational arm  91  have the same electrical polarity as the third set of electrodes  106  and  107  disposed on the grooves  94  and  98  of the arm  92  and an electrode  103  disposed on the mounting arm  96 . When a direct voltage is applied between the electrode terminals D-D′, an electric field Ex occurs along the arrow direction inside the vibrational arms  91  and  92 . As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. Consequently, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R 1  and a high quality factor Q because even when miniaturized, there is very large electromechanical transformation efficiency for the resonator. 
     FIG. 8  shows a plan view of a quartz crystal resonator  110  of a fourth embodiment of the present invention, which is a quartz crystal tuning fork resonator. The resonator  110  comprises vibrational arms  111  and  112  and a base portion  113  attached to the vibrational arms, and the base portion  113  has mounting arms  114  and  115  protruding from the base portion, each of which is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, the vibrational arms  111  and  112  have grooves  116  and  117 , respectively. In more detail, each of the vibrational arms  111  and  112  has an end portion connected to the base portion and a free end portion, when a distance measured from the end portion to the free end portion is a length L, each of the vibrational arms has a width W between the end portion and half a length L/2 of each of the vibrational arms and a width W e  between half the length L/2 and the length L of the free end portion of each of the vibrational arms, and the width W is less than the width W e . In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode and vibrate in a fundamental mode of an inverse phase. 
   Similar to this, a quartz crystal tuning fork resonator  120  of a fifth embodiment of the present invention is shown in  FIG. 9  showing a plan view thereof. The resonator  120  comprises vibrational arms  121  and  122  and a base portion  123  attached to the vibrational arms, and the base portion  123  has a mounting arm  124  protruding from the base portion. In this embodiment, the mounting arm  124  is between the vibrational arms  121  and  122  and the vibrational arms  121  and  122  have grooves  125  and  126 , respectively. For this case the width W is also less than the width W e  similar to that of  FIG. 8 . 
     FIG. 10  shows a plan view of a quartz crystal resonator  130  of a sixth embodiment of the present invention, which is a quartz crystal tuning fork resonator. The resonator  130  comprises vibrational arms  131  and  132  and a base portion  133  attached to the vibrational arms, and the base portion  133  has mounting arms  134  and  135  protruding from the base portion, each of which is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, the vibrational arms  131  and  132  have grooves  136  and  137 , respectively. In more detail, each of the vibrational arms  131  and  132  has an end portion connected to the base portion and a free end portion, when a distance measured from the end portion to the free end portion is a length L, each of the vibrational arms has a width W between the end portion and half a length L/2 of each of the vibrational arms and a width W e  between half the length L/2 and the length L of the free end portion of each of the vibrational arms, and the width W is less than the width W e . In this embodiment, the quartz crystal tuning fork resonator can vibrate in a flexural mode and vibrate in a fundamental mode of an inverse phase. 
   Similar to this, a quartz crystal tuning fork resonator  140  of a seventh embodiment of the present invention is shown in  FIG. 11  showing a plan view thereof. The resonator  140  comprises vibrational arms  141  and  142  and a base portion  143  attached to the vibrational arms, and the base portion  143  has a mounting arm  144  protruding from the base portion. In this embodiment, the mounting arm  144  is between the vibrational arms  141  and  142  and the vibrational arms  141  and  142  have grooves  145  and  146 , respectively. For this case the width W is also less than the width W e  similar to that of  FIG. 10 . 
   Especially, the quartz crystal tuning fork resonator of the fourth embodiment to the seventh embodiment can be miniaturized with a small series resistance R 1  and a high quality factor Q 1  because it has a width W e  greater than a width W, and the width W e  operates as a mass. 
   Next, a value of a piezoelectric constant e′ 12  is described, which is of great importance and necessary to excite a quartz crystal tuning fork resonator capable of vibrating in a flexural mode of the present invention. The larger a value of the piezoelectric constant e′ 12  becomes, the higher electromechanical transformation efficiency becomes. The piezoelectric constant e′ 12  of the present invention can be defined by a function of the cutting angles θ y , θ x  and θ z  shown in  FIG. 1 , and piezoelectric constants e 11 =0.171 C/m 2  and e 14 =−0.0406 C/m 2  of quartz crystal. In order to obtain a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode and having a small series resistance R 1  and a high quality factor Q, the piezoelectric constant e′ 12  of the present invention is within a range of 0.1 C/m 2  to 0.19 C/m 2  in the absolute value. It is, therefore, easily understood that this value is enough large to obtain the quartz crystal tuning fork resonator with a small series resistance R 1  and a high quality factor Q. 
   Especially, in order to obtain a quartz crystal tuning fork resonator capable of vibrating in a flexural mode with a much smaller series resistance R 1 , the piezoelectric constant e′ 12  is preferably within a range of 0.12 C/m 2  to 0.19 C/m 2  in the absolute value. 
   In addition, as an example, a quartz crystal tuning fork resonator comprises a plurality of vibrational arms having a first vibrational arm and a second vibrational arm, and a groove having a first stepped portion and a second stepped portion is formed in at least one of a first main surface and a second main surface of each of the first and second vibrational arms, in which a first electrode is disposed on the first stepped portion of the groove, a second electrode is disposed on the second stepped portion of the groove, and a third electrode is disposed on each of side surfaces of each of the first and second vibrational arms, in which the piezoelectric constant e′ 12 (=e′ 12i ) is between the first electrode and the third electrode disposed opposite to the first electrode, and the piezoelectric constant e′ 12 (=e′ 12o ) is between the second electrode and the third electrode disposed opposite to the second electrode, and in which the piezoelectric constants e′ 12i  and e′ 12o  are within the range of 0.12 C/m 2  to 0.19 C/m 2  in the absolute value, respectively, and a product of the e′ 12i  and the e′ 12o  is greater than 0. 
   As an another example, a quartz crystal tuning fork resonator comprises a plurality of vibrational arms having a first vibrational arm and a second vibrational arm, and a through hole having a first side surface and a second side surface is formed in each of the first and second vibrational arms, in which a first electrode is disposed on the first side surface of the through hole, a second electrode is disposed on the second side surface of the through hole, and a third electrode is disposed on each of side surfaces of each of the first and second vibrational arms, in which the piezoelectric constant e′ 12 (=e′ 12i ) is between the first electrode and the third electrode disposed opposite to the first electrode, and the piezoelectric constant e′ 12 (=e′ 12o ) is between the second electrode and the third electrode disposed opposite to the second electrode, and in which the piezoelectric constants e 12i  and e 12o  are within the range of 0.12 C/m 2  to 0.19 C/m 2  in the absolute value, respectively, and a product of the e′ 12i  and the e′ 12o  is greater than 0. 
   Therefore, the quartz crystal tuning fork resonator described above has a small series resistance R 1  and a high quality factor Q, and also a frequency of high stability. 
     FIG. 12  shows a plan view of a width-extensional mode quartz crystal resonator  150  constructing an electronic apparatus of the present invention, and which vibrates in a width-extensional mode. The width-extensional mode resonator  150  comprises a vibrational portion  151 , connecting portions  152 ,  152   a  and supporting portions  153  and  154  connected to a mounting portion  155  constructing the supporting portions. Namely, the vibrational portion  151  is connected to the supporting portions  153 ,  154  having the mounting portion  155  through the connecting portions  152 ,  152   a . In addition, an electrode  151   a  is disposed on an obverse surface of the vibrational portion  151  and an electrode  151   b  (not visible) is disposed on a reverse surface of the vibrational portion  151 . 
   In more detail, the electrode  151   a  disposed on the obverse surface of the vibrational portion  151  is connected to an electrode  153   a  disposed on the mounting portion  155 , while the electrode  151   b  disposed on the reverse surface of the vibrational portion  151  is connected to an electrode  154   a  disposed on the mounting portion  155  through an electrode  154   b  disposed on a side surface of the mounting portion. Namely, a pair of electrodes is disposed on the obverse and reverse surfaces of the vibrational portion  151 . Also, the vibrational portion  151  has a width W 0  and a length L 0 . In general, a ratio W 0 /L 0  is less than 0.35. In addition, the mounting portion  155  is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, a cutting angle of the resonator is within a range of ZY1wt (80° to 100°)/(−10° to +10°)/(75° to +115°) and a piezoelectric constant e′ 31  of the resonator is within a range of 0.1 C/m 2  to 0.19 C/m 2  in the absolute value to obtain the resonator with a small series resistance R 1  and a high quality factor Q. 
   Similar to this, a length-extensional mode quartz crystal resonator can be obtained by replacing the width W 0  with the length L 0 . In this case the connecting portions are formed in the width direction. In this embodiment, a cutting angle of the length-extensional mode resonator is within a range of ZY1wt (80° to 100°)/(−10° to +10°)/(−35° to +35°) and a piezoelectric constant e′ 32  of the resonator is within a range of 0.12 C/m 2  to 0.19 C/m 2  in the absolute value to obtain the resonator with a small series resistance R 1  and a high quality factor Q. 
     FIG. 13(   a ) and  FIG. 13(   b ) show a plan view of a thickness shear mode quartz crystal resonator  156  constructing an electronic apparatus of the present invention and a F-F′ sectional view of the thickness shear mode resonator  156  capable of vibrating in a thickness shear mode. The resonator  156  comprises a vibrational portion  157  having a width W 0 , a length L 0  and a thickness T 0 , and electrodes  158  and  159  are disposed on an obverse surface and a reverse surface so that the electrodes have opposite electrical polarity each other. Namely, a pair of electrodes is disposed on the vibrational portion  157 . Also, the resonator  156  is housed in a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, a cutting angle of the resonator is within a range of ZY1wt (−5° to +5°)/±(37° to 58°)/(85° to 95°) and a piezoelectric constant e′ 34  of the resonator is within a range of 0.055 C/m 2  to 0.14 C/m 2  in the absolute value to obtain the resonator with a small series resistance R 1  and a high quality factor Q. In order to obtain much smaller series resistance R 1 , the piezoelectric constant e′ 34  of the resonator is preferably within a range of 0.085 C/m 2  to 0.12 C/m 2  in the absolute value. 
     FIG. 14  shows a plan view of a Lame mode quartz crystal resonator  210  constructing an electronic apparatus of the present invention, and vibrating in a Lame mode. The Lame mode resonator  210  comprises a vibrational portion  211 , connecting portions  212 ,  213  and supporting portions  214 ,  215  having mounting portions  216  and  217 . Namely, the vibrational portion  211  is connected to the supporting portions  214 ,  215  through the connecting portions  212 ,  213 . In addition, an electrode  218  is disposed on an obverse surface of the vibrational portion  211  and an electrode  219  (not visible) is disposed on a reverse surface of the vibrational portion  211 . 
   In more detail, the electrode  218  disposed on the obverse surface of the vibrational portion  211  is extended into the mounting portion  217 , while the electrode  219  disposed on the reverse face of the vibrational portion  211  is extended into the mounting portion  216 . Namely, a pair of electrodes is disposed on the obverse and reverse surfaces of the vibrational portion  151 . Also, the vibrational portion  211  has a width W 0  and a length L 0 . In general, a ratio L 0 /W 0  is approximately equal to m for a fundamental mode of vibration and an overtone mode of vibration of the resonator, where m is an order of vibration of the resonator and an integer. For example, when a Lame mode quartz crystal resonator has one of m=1, 2, 3 and n, the resonator vibrates in a fundamental mode for m=1, a second overtone mode for m=2, a third overtone mode for m=3 and an n th  overtone mode for m=n. 
   Also, the m has a close relationship to the number of electrodes disposed on the vibrational portion. For example, when an electrode is disposed opposite each other on both of an obverse surface and a reverse surface of the vibrational portion, this is called “the number of two electrodes”, in other words, “a pair of electrodes”. Namely, the vibrational portion has the number of p electrodes, where p is an even number such as 2, 4, 6, 8 and 10. As an example, when p of the vibrational portion comprises 6, the vibrational portion vibrates in a third overtone mode. In this example, three pairs of electrodes are disposed on the vibrational portion. Namely, the third overtone mode of vibration is a principal vibration. As is apparent from this relationship, there is a relationship of m=p/2. 
   Therefore, the principal vibration vibrates in the order of vibration corresponding to the number of an electrode pair or electrode pairs. For example, the principal vibration vibrates in a fundamental mode of vibration, a second overtone mode of vibration and a third overtone mode of vibration, respectively, for an electrode pair, two electrode pairs and three electrode pair disposed on the vibrational portion. In detail, when me electrode pairs are disposed on the vibrational portion, the principal vibration vibrates in an n th  overtone mode of vibration, and me corresponds to the n, where me is an integer. It is needless to say that this concept can be applied to a width-extensional mode quartz crystal resonator and a length-extensional mode quartz crystal resonator. 
   In more detail, an even number of electrodes are disposed on the obverse and reverse surfaces of the vibrational portion and the electrodes disposed opposite each other has an opposite electrical polarity. In addition, each of the mounting portions  216 ,  217  is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In this embodiment, a cutting angle of the resonator is within a range of ZY1wt (−5° to +5°)/±(35° to 60°)/(40° to 50°) and a piezoelectric constant e′ 32  of the resonator is within a range of 0.045 C/m 2  to 0.13 C/m 2  in the absolute value to obtain the resonator with a small series resistance R 1  and a high quality factor Q. 
     FIG. 15(   a ) and  FIG. 15(   b ) show a plan view of a resonator for sensing angular velocity constructing an electronic apparatus of the present invention and a G-G′ sectional view of the resonator. In this embodiment, the resonator  220  comprises a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode and comprising vibrational arms  221 ,  222  and a base portion  223  attached to the vibrational arms, the base portion  223  is mounted on a mounting portion of a package comprising a case for housing the resonator and a lid for covering an open end of the case. In addition, each of the vibrational arms  221  and  222  has a first main surface and a second main surface opposite the first main surface and side surfaces, and the vibrational arm  221  has grooves  224 ,  230 , while the vibrational arm  222  has grooves  225 ,  237 , each of which has stepped portions comprising a first stepped portion and a second stepped portion. Also, a cutting angle of the resonator is within a range of ZY1wt (−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectric constant e′ 12  of the resonator is within a range of 0.1 C/m 2  to 0.19 C/m 2  in the absolute value. The resonator of this embodiment can vibrate in a flexural mode of a fundamental mode and an inverse phase. 
   In more detail, electrodes  226  and  227  which are of the same electrical polarity are disposed on the side surfaces such that an electrode terminal H is defined, while an electrode  228  is disposed inside the groove  224  and an electrode  229  which is of the same electrical polarity to the electrode  228  is disposed inside the groove  230  such that an electrode terminal H′ of opposite electrical polarity to the electrode terminal H is defined. Namely, two electrode terminals H-H′ for an input signal are constructed. On the other hand, electrodes  231 ,  232 ,  233  which are of the same electrical polarity are disposed on the side surfaces and inside the grooves  225 ,  237  such that an electrode terminal I is defined, while electrodes  234 ,  235 ,  236  which are of the same electrical polarity are disposed on the side surfaces and inside the grooves  225 ,  237  such that an electrode terminal I′ of opposite electrical polarity to the electrode terminal I is defined. Namely, two electrode terminals I-I′ for an output signal are constructed. The resonator of this embodiment is made of quartz crystal, but may be made of a piezoelectric material such as lithium tantalite, lithium niobate and ceramics. 
     FIG. 16(   a ) and  FIG. 16(   b ) show a plan view and a J-J′ sectional view of a quartz crystal resonator  230  of an eighth embodiment of the present invention, and which is a quartz crystal tuning fork resonator. The resonator  230  comprises vibrational arms  231  and  232  and a base portion  233  attached to the vibrational arms. In addition, each of the vibrational arms  231  and  232  has a first main surface and a second main surface opposite the first main surface and side surfaces, and the first and second main surfaces have central linear portions  242 ,  243 , respectively. The vibrational arm  231  has grooves  234 ,  235  and the vibrational arm  232  has grooves  235 ,  237 . In addition, the grooves  234  and  236  are formed between the central linear portion  242  and an outer edge of the vibrational arm  231 , respectively. Similar to this, the grooves  235  and  237  are formed between the central linear portion  243  and an outer edge of the vibrational arm  232 , respectively 
   Moreover, each of the grooves  234 ,  235 ,  236  and  237  has a width W 8 , and a width W 7  including a portion of the central linear lines  242  and  243  is formed in each of the vibrational arms  231  and  232 . In addition, a distance W 9  is formed in the width direction of the vibrational arms  231  and  232  between an outer edge of the groove and an outer edge of the vibrational arms. In detail, a width W of the arms  231  and  232  has generally a relationship of W=W 7 +2W 8 +2W 9 , and the width W 8  is constructed so that W 8 ≧W 7 , W 9 . In this embodiment, also, the grooves are constructed at the arms so that a ratio W 8 /(W/2) of the width W 8  and a half width of the width W is greater than 0.35 and less than 1, preferably, within a range of 0.35 to 0.95. In addition, the width W 7  is less than 0.05 mm, preferably, less than 0.03 mm and the width W 8  is within a range of 0.015 mm to 0.05 mm to obtain a very large moment of inertia of the vibrational arms. That is, the quartz crystal tuning fork resonator, capable of vibrating in a fundamental mode, and having a frequency of high stability can be provided with a small series resistance R 1 , a high quality factor Q 1  and a small capacitance ratio r i  because it has a very large electromechanical transformation efficiency. 
   In  FIG. 16(   b ), the vibrational arm  231  has grooves  234 ,  236 ,  238  and  240  cut into it. The grooves  234 ,  236 ,  238  and  240  have a first set of electrodes  256 ,  257 ,  258  and  259  of the same electrical polarity, while the side surfaces of the arm  231  have a second set of electrodes  244  and  245  having an opposite electrical polarity to the first set of electrodes  256 ,  257 ,  258  and  259 . The vibrational arm  232  has grooves  235 ,  237 ,  239  and  241  constructed in a similar manner as the vibrational arm  231 . The grooves  235 ,  237 ,  239  and  241  have a third set of electrodes  252 ,  253 ,  254  and  255  of the same electrical polarity, and the side surfaces of the vibrational arm  232  have a fourth set of electrodes  250  and  251  with the opposite electrical polarity to the third electrodes  252 ,  253 ,  254  and  255 . The electrodes disposed on the vibrational arms  231  and  232  are connected as shown in  FIG. 16(   b ), namely, two electrode terminals of different electrical polarity K-K′ are obtained. 
   In detail, the first set of electrodes  256 ,  257 ,  258  and  259  disposed on the grooves  234 ,  236 ,  238  and  240  of the vibrational arm  231  have the same electrical polarity as the fourth set of electrodes  250  and  251  disposed on both side surfaces of the vibrational arm  232 , while the second set of electrodes  244  and  245  disposed on both side surfaces of the vibrational arm  231  have the same electrical polarity as the third set of electrodes  252 ,  253 ,  254  and  255  disposed on the grooves  235 ,  237 ,  239  and  241  of the arm  232 . When a direct current voltage is applied between the electrode terminals K-K′, an electric field Ex occurs along the arrow direction inside the vibrational arms  231  and  232 . As the electric field Ex occurs perpendicular to the electrodes disposed on the vibrational arms, as shown in the arrow signs, the electric field Ex has a very large value and a large distortion occurs at the vibrational arms. As a result, a quartz crystal tuning fork resonator capable of vibrating in a flexural mode is obtained with a small series resistance R 1  and a high quality factor Q because even when miniaturized there is a very large electromechanical transformation efficiency for the resonator. 
     FIG. 17  shows a plan view of a quartz crystal unit of a first embodiment of the present invention and omitting a lid. The quartz crystal unit  160  comprises the quartz crystal tuning fork resonator  10  shown in  FIG. 2 , a case  165  for housing the resonator and a lid for covering an open end of the case (not shown here). Also, the resonator  10  has mounting arms  36  and  37 , each of which is mounted on a mounting portion  166  and a mounting portion  167  of the case  165 . In detail, an electrode disposed on the mounting arm  36  is connected to an electrode disposed on the mounting portion  166  by adhesives  168  or a metal such as solder, and similarly, an electrode disposed on the mounting arm  37  is connected to an electrode disposed on the mounting portion  167  by adhesives  169  or a metal. 
     FIG. 18  shows a plan view of a quartz crystal unit of a second embodiment of the present invention and omitting a lid. The quartz crystal unit  170  comprises the quartz crystal tuning fork resonator  50  shown in  FIG. 5 , a case  175  for housing the resonator and a lid for covering an open end of the case (not shown here). Namely, the resonator comprises vibrational arms and a base portion. Also, the case  175  has mounting portions  176  and  177  and the resonator  50  has a mounting arm  77  protruding from the base portion, which is mounted on the mounting portion  177  of the case  175 . In detail, an electrode disposed on the mounting arm  77  is connected to an electrode disposed on the mounting portion  177  by adhesives  179  or a metal such as solder, and similarly, an electrode disposed on the base portion of the resonator  50  is connected to an electrode disposed on the mounting portion  176  by adhesives  178  or a metal such as solder. 
     FIG. 19  shows a cross-sectional view of a quartz crystal unit of a third embodiment of the present invention. The quartz crystal unit  180  comprises a contour mode quartz crystal resonator  185  or a thickness shear mode quartz crystal resonator  185 , a case  181  and a lid  182 . In more detail, the resonator  185  is mounted on a mounting portion  184  of the case  181  by conductive adhesives  76  or solder. Also, the case  181  and the lid  182  are connected through a connecting member  183 . For example, the contour mode resonator  185  in this embodiment is the same resonator as one of the quartz crystal tuning fork resonators  10 ,  50 ,  90 ,  110 ,  120 ,  130 ,  140 ,  220  and  230  described in detail in  FIG. 2-FIG .  11  and  FIG. 15-FIG .  16 . 
   In this embodiment, circuit elements are connected at outside of the quartz crystal unit to get a quartz crystal oscillator. Namely, only the quartz crystal tuning fork resonator is housed in the unit and also, it is housed in the unit in vacuum. In this embodiment, the quartz crystal unit of a surface mounting type is shown, but the quartz crystal tuning fork resonator may be housed in a tubular type, namely, a quartz crystal unit of the tubular type. In addition, the quartz crystal unit of the present invention includes any shape of a quartz crystal unit comprising a quartz crystal resonator, a case and a lid to house the quartz crystal resonator in vacuum. 
   Also, instead of the quartz crystal tuning fork resonator and the thickness shear mode quartz crystal resonator, one of a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator which are a contour mode quartz crystal resonator, respectively, or a SAW (Surface Acoustic Wave) resonator or a piezoelectric resonator for sensing angular velocity (angular velocity sensor) made of quartz crystal or ceramics may be housed in the unit. 
   In addition, a member of the case and the lid is ceramics or glass and a metal or glass, respectively, and a connecting member is a metal or glass with low melting point. Also, a relationship of the resonator, the case and the lid described in this embodiment is applied to a quartz crystal oscillator of the present invention which will be described in  FIG. 20 . 
     FIG. 20  shows a cross-sectional view of a quartz crystal oscillator of a first embodiment of the present invention. The quartz crystal oscillator  190  comprises a quartz crystal oscillating circuit, a case  191  and a lid  192 . In this embodiment, circuit elements constructing the oscillating circuit are housed in a quartz crystal unit comprising a contour mode quartz crystal resonator  195  or a thickness shear mode quartz crystal resonator  195 , the case  191  and the lid  192 . Also, the oscillating circuit of this embodiment comprises an amplifier  197  including a feedback resistor, the resonator  195 , a plurality of capacitors (not shown here) and a drain resistor (not shown here), and a CMOS inverter is used as the amplifier  197 . 
   In addition, in this embodiment, the resonator  195  is mounted on a mounting portion  194  of the case  191  by conductive adhesives  196  or solder. As described above, the amplifier  197  is housed in the quartz crystal unit and mounted on the case  191 . Also, the case  191  and the lid  192  are connected through a connecting member  193 . 
     FIG. 21  shows a diagram of an embodiment of a quartz crystal oscillating circuit constructing a quartz crystal oscillator of the present invention. The quartz crystal oscillating circuit  201  comprises an amplifier (CMOS inverter)  202 , a feedback resistor  204 , a drain resistor  207 , a plurality of capacitors  205 ,  206  and a quartz crystal resonator  203 . Namely, the oscillating circuit  201  comprises an amplification circuit  208  having the amplifier  202  and the feedback resistor  204 , and a feedback circuit  209  having the drain resistor  207 , the capacitors  205 ,  206  and the quartz crystal resonator  203 . The quartz crystal resonator  203  is one of the resonators already described above. For example, when the resonator  203  is a quartz crystal tuning fork resonator capable of vibrating in a flexural mode, an output signal of the oscillating circuit  201  is outputted through a buffer circuit (not shown in  FIG. 21 ), and is an oscillating frequency of a fundamental mode of vibration of the resonator. 
   In other words, the oscillation frequency of the fundamental mode of vibration of the quartz crystal tuning fork resonator is outputted from the oscillating circuit through the buffer circuit as an output signal. According to the present invention, a nominal frequency of the fundamental mode of vibration of the quartz crystal tuning fork resonator is within a range of 10 kHz to 200 kHz. Especially, a frequency of 32.768 kHz is very available for use in an electronic apparatus of the present invention. In general, the output signal has an oscillation frequency which is within a range of −100 PPM to +100 PPM to the nominal frequency, e.g. 32.768 kHz. 
   In more detail, the quartz crystal oscillator in this example comprises a quartz crystal oscillating circuit and a buffer circuit, namely, the quartz crystal oscillating circuit comprises an amplification circuit and a feedback circuit, and the amplification circuit comprises an amplifier (CMOS inverter) and a feedback resistor and the feedback circuit comprises a quartz crystal tuning fork resonator capable of vibrating in a flexural mode, a drain resistor and a plurality of capacitors. Also, the quartz crystal tuning fork resonator already described in  FIG. 2-FIG .  11  and  FIG. 15-FIG .  16  is used in a quartz crystal oscillator of the present invention. Instead of the quartz crystal tuning fork resonator, an another contour mode quartz crystal resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator or a resonator for sensing angular velocity may be used. 
     FIG. 22  shows a diagram of the feedback circuit of  FIG. 21 . In this embodiment, the feedback circuit has a quartz crystal tuning fork resonator capable of vibrating in a flexural mode. Now, when angular frequency ω i  of the quartz crystal tuning fork resonator  203 , a resistance value R d  of the drain resistor  207 , capacitance values C g , C d  of the capacitors  205 ,  206 , crystal impedance R ei  of the quartz crystal resonator  203 , an input voltage V 1 , and an output voltage V 2  are taken, a feedback rate β i  is defined by β i =|V 2 | i /|V 1 | i , where i shows a vibration order, for example, when i=1 and 2, β 1  and β 2  are a feedback rate for a fundamental mode of vibration and a second overtone mode of vibration of the resonator, respectively. 
   In addition, load capacitance C L  is given by C L =C g  C d /(C g +C d ), and when C g =C d =C gd  and R d &gt;&gt;R ei , the feedback rate β i  is given by β i =1/(1+kC L   2 ), where k is expressed by a function of ω i , R d  and R ei . Also, R ei  is approximately equal to series resistance R 1  of the resonator. 
   Thus, it is easily understood from a relationship of the feedback rate β i  and load capacitance C L  that the feedback rate of a resonance frequency for the fundamental mode of vibration and the overtone mode vibration becomes large, respectively, according to decrease of load capacitance C L . Therefore, when C L  has a small value, an oscillation of the overtone mode occurs very easily, instead of that of the fundamental mode. This is the reason why maximum amplitude of the overtone mode of vibration becomes smaller than that of the fundamental mode of vibration, and the oscillation of the overtone mode satisfies an amplitude condition and a phase condition simultaneously which are the continuous condition of an oscillation in an oscillating circuit. 
   As it is also one object of the present invention to provide a quartz crystal oscillator having a flexural mode, quartz crystal tuning fork resonator, capable of vibrating in a fundamental mode and having a frequency of high stability (high time accuracy) of an output signal, and having reduced electric current consumption, load capacitance C L  is less than 25 pF in this embodiment to reduce electric current consumption. To get much reduced electric current consumption, C L  is preferably less than 15 pF because the electric current consumption is proportional to C L . 
   In addition, in order to suppress a second overtone mode of vibration of the resonator and to obtain a quartz crystal oscillator having an output signal of an oscillation frequency of a fundamental mode of vibration of the resonator, the quartz crystal oscillator comprising an amplification circuit and a feedback circuit is constructed so that it satisfies a relationship of α 1 /α 2 &gt;β 2 /β 1  and α 1 β&gt;1, where α 1  and α 2  are, respectively, an amplification rate of the fundamental mode of vibration and the second overtone mode of vibration of the amplification circuit, and β 1  and β 2  are, respectively, a feedback rate of the fundamental mode of vibration and the second overtone mode of vibration of the feedback circuit. 
   In other words, the quartz crystal oscillator is constructed so that a ratio of the amplification rate α 1  of the fundamental mode of vibration and the amplification rate α 2  of the second overtone mode of vibration of the amplification circuit is greater than that of the feedback rate β 2  of the second overtone mode of vibration and the feedback rate β 1  of the fundamental mode vibration of the feedback circuit, and also a product of the amplification rate α 1  and the feedback rate β 1  of the fundamental mode of vibration is greater than 1. A description of a frequency of high stability in the quartz crystal oscillator will be performed later. 
   Also, characteristics of the amplifier of the amplification circuit constructing the quartz crystal oscillating circuit of this embodiment can be expressed by negative resistance −RL i . For example, when i=1, negative resistance −RL 1  is for a fundamental mode of vibration of the resonator and when i=2, negative resistance −RL 2  is for a second overtone mode of vibration of the resonator. In this embodiment, the quartz crystal oscillating circuit is constructed so that a ratio of an absolute value of negative resistance −RL 1  of the fundamental mode of vibration of the amplification circuit and series resistance R 1  of the fundamental mode of vibration is greater than that of an absolute value of negative resistance −RL 2  of the second overtone mode of vibration of the amplification circuit and series resistance R 2  of the second overtone mode of vibration. 
   That is, the oscillating circuit is constructed so that it satisfies a relationship of |−RL 1 |/R 1 &gt;|−RL 2 |/R 2 . By constructing the oscillating circuit like this, an oscillation of the second overtone mode can be suppressed, as a result of which a frequency of oscillation of the fundamental mode of vibration can be output as an output signal because the oscillation of the fundamental mode generates easily in the oscillating circuit. In more detail, an absolute value of the negative resistance −RL 1  is within a range of 60 kΩ to 285 kΩ, and an absolute value of the negative resistance −RL 2  is less than 105 kΩto suppress the frequency of oscillation of the second overtone mode of the quartz crystal tuning fork resonator in the oscillating circuit and to obtain the frequency of oscillation of the fundamental mode thereof. 
   In this embodiment, a quartz crystal tuning fork resonator is used, but, instead of the tuning fork resonator, an another contour mode quartz crystal resonator such as a width-extensional mode quartz crystal resonator, a length-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator may be used in a quartz crystal oscillator of the present invention. In this case, a principal vibration of the contour mode quartz crystal resonator is outputted from an oscillating circuit constructing the quartz crystal oscillator through a buffer circuit. In order to suppress a sub-vibration of the contour mode quartz crystal resonator, the quartz crystal oscillator comprising an amplification circuit and a feedback circuit is constructed so that it satisfies a relationship of α p /α s &gt;β s /β p  and α p β p &gt;1, where α p  and α s  are, respectively, an amplification rate of the principal vibration and the sub-vibration of the amplification circuit, and β p  and β s  are, respectively, a feedback rate of the principal vibration and the sub-vibration of the feedback circuit. 
   Similar to the quartz crystal tuning fork resonator, for the contour mode quartz crystal resonator, a quartz crystal oscillating circuit of the present invention is constructed so that a ratio of an absolute value of negative resistance −RL p  of the principal vibration of the amplification circuit and a series resistance R p  of the principal vibration is greater than that of an absolute value of negative resistance −RL s  of the sub-vibration of the amplification circuit and a series resistance R s  of the sub-vibration. That is, the oscillating circuit is constructed so that it satisfies a relationship of |−RL p |/R p &gt;|−RL s |/R s . By constructing the oscillating circuit like this, an oscillation of the sub-vibration can be suppressed, as a result of which a frequency of oscillation of the principal vibration can be outputted as an output signal because the oscillation of the principal vibration generates easily in the oscillating circuit. In addition, the principal vibration and the sub-vibration have the same mode of vibration and a different order of vibration each other. 
     FIG. 23  shows a block diagram of an embodiment of an electronic apparatus of the present invention, and illustrating the diagram of a facsimile apparatus. As shown in  FIG. 23 , the apparatus generally comprises a modem, a phonetic circuit, a timepiece circuit, a printing portion, a taking portion, an operation portion and a display portion. In this principle, perception and scanning of reflection light of light projected on manuscript in the taking portion are performed by CCD (Charge Coupled Device), in addition, light and shade of the reflection light are transformed into a digital signal, and the signal is modulated by the modem and is sent to a phone line (communication line). Also, in a receiving side, a received signal is demodulated by the modem and is printed on a paper in the print portion by synchronizing the received signal with a signal of a sending side. 
   In  FIG. 23 , a quartz crystal resonator is used as a CPU clock of the control portion and the printing portion, as a clock of the phonetic circuit and the modem, and as a time standard of the timepiece. Namely, the resonator constructs a quartz crystal oscillator and an output signal of the oscillator is used. Like this, a plurality of oscillators is used for the electronic apparatus. For example, it is used as a signal to display time at the display portion. In this case, a quartz crystal tuning fork resonator, capable of vibrating in a flexural mode is generally used, and e.g. as the CPU clock, a contour mode quartz crystal resonator such as a length-extensional mode quartz crystal resonator, a width-extensional mode quartz crystal resonator and a Lame mode quartz crystal resonator or a thickness shear mode quartz crystal resonator is used. To get the facsimile apparatus of this embodiment which operates normally, an accuracy signal of output of the oscillator is required for the facsimile apparatus, which is one of the electronic apparatus of the present invention. Also, a digital display and an analogue display are included in the display of the present invention. 
   In this embodiment, though the facsimile apparatus is shown as an example of an electronic apparatus of the present invention, the present invention is not limited to this, namely, the present invention includes all electronic apparatus, each of which comprises a quartz crystal oscillator and a display portion at least, for example, cellar phones, telephones, a TV set, cameras, a video set, video cameras, pagers, personal computers, printers, CD players, MD players, electronic musical instruments, car navigators, car electronics, timepieces, IC cards and so forth. In addition, the electronic apparatus may have an another oscillator comprising a piezoelectric resonator for sensing angular velocity made of quartz crystal, ceramics, lithium tantalite and lithium niobate. 
   Thus, the electronic apparatus of the present invention comprising a display portion and a quartz crystal oscillator at least may operate normally because the quartz crystal oscillator comprises the quartz crystal oscillating circuit with a frequency of high stability, namely, a frequency of high reliability. 
   Moreover, capacitance ratios r 1  and r 2  of a flexural mode, quartz crystal tuning fork resonator are given by r 1 =C 0 /C 1  and r 2 =C 0 /C 2 , respectively, where C 0  is shunt capacitance in an electrical equivalent circuit of the resonator, and C 1  and C 2  are, respectively, motional capacitance of a fundamental mode of vibration and a second overtone mode of vibration in the electrical equivalent circuit of the resonator. In addition, the resonator has a quality factor Q 1  for the fundamental mode of vibration and a quality factor Q 2  for the second overtone mode of vibration. 
   In detail, the tuning fork resonator of this embodiment is provided so that the influence on resonance frequency of the fundamental mode of vibration by the shunt capacitance becomes smaller than that of the second overtone mode of vibration by the shunt capacitance, namely, so that it satisfies a relationship of S 1 =r 1 /2Q 1   2 &lt;S 2 =r 2 /2Q 2   2 , preferably, S 1 &lt;S 2 /2, where S 1  and S 2  are called “a stable factor of frequency” of the fundamental mode of vibration and the second overtone mode of vibration. As a result, the tuning fork resonator, capable of vibrating in the fundamental mode and having a frequency of high stability can be provided because the influence on the resonance frequency of the fundamental mode of vibration by the shunt capacitance C 0  is as extremely small as it can be ignored. In this embodiment S 2  has a value greater than 0.13×10 −6  to suppress the second overtone mode of vibration of the resonator. 
   In addition, as described above, it will be easily understood that the quartz crystal resonator comprising vibrational arms and a base portion, according to the present invention, may have outstanding effects. Similar to this, the quartz crystal unit and the quartz crystal oscillator, according to the present invention, may have also outstanding effects. In addition, the electronic apparatus comprising the quartz crystal oscillator comprising the quartz crystal oscillating circuit having the quartz crystal tuning fork resonator, capable of vibrating in a flexural mode, and having novel shapes, novel electrode construction and excellent electrical characteristics, according to the present invention, may have the outstanding effects. Similar to this, it will be easily understood that the electronic apparatus comprising the quartz crystal oscillator comprising the quartz crystal oscillating circuit having the another contour mode quartz crystal resonator or the thickness shear mode quartz crystal resonator or the resonator for sensing angular velocity, according to the present invention, may have also the outstanding effect. In addition to this, while the present invention has been shown and described with reference to preferred embodiments thereof, it will be understood by those skilled in the art that the changes in shape and electrode construction can be made therein without departing from the spirit and scope of the present invention.