Patent Publication Number: US-9413332-B2

Title: Resonator element, resonator, oscillator, electronic device, and moving object

Description:
CROSS REFERENCE 
     The entire disclose of Japanese Patent Application No. 2013-149981 filed Jul. 18, 2013 is expressly incorporated by reference herein. 
     BACKGROUND 
     1. Technical Field 
     The present invention relates to a resonator element, a resonator, an oscillator, an electronic device, and a moving object. 
     2. Related Art 
     A so-called two-legged tuning fork type quartz crystal resonator is known as a resonator (for example, refer to JP-A-2004-266871). For example, as disclosed in JP-A-2004-266871, a resonator element included in the resonator includes a base portion and two vibrating arms extending from the base portion so as to be parallel to each other, and fluxurally vibrates the two vibrating arms in an in-plane direction so that the arms alternately repeat mutual approach and separation. 
     A pair of grooves are formed, along a longitudinal direction, in each vibrating arm of the resonator element according to JP-A-2004-266871 so that a cross-section of each vibrating arm has a substantially H shape. Further, as disclosed, in JP-A-2004-266871, it is possible to reduce so-called vibration leakage in which vibrations of the vibrating arms leak to the base portion side, by providing a notch in the base portion. Here, from dimensions of parts disclosed in JP-A-2004-266871, a Q value of the resonator element is 31,573 as shown in the following table. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Plate 
                 Groove 
                 Bank 
                 Arm 
                   
                   
               
               
                   
                 thickness 
                 Depth 
                 portion 
                 width 
               
               
                   
                 [μm] 
                 [μm] 
                 width [μm] 
                 [μm] 
                 η 
                 Q value 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 JP-A-2004- 
                 100 
                 43 
                 15 
                 100 
                 0.300 
                 31,573 
               
               
                 266871 
               
               
                   
               
            
           
         
       
     
     Meanwhile, when a width of an arm portion of the vibrating arm is set to W 1  and a width of each of principal surfaces (hereinafter, referred to as banks) which are positioned on both sides of the groove, when seen in a plan view, is set to W 3 , η can be expressed by the relation of η=2×W 3 /W 1 . So-called η is an occupancy rate of a width of the bank with respect to the width of the arm portion. 
     However, in recent years, the need for a further improvement in a Q value has increased. Consequently, in a resonator element according to JP-A-2011-199331, as means for improving a Q value, a vibrating arm includes a first arm portion (arm portion) and a second arm portion (weight portion) which is provided at a distal end of the first arm portion and has a width larger than that of the first arm portion. Thus, it is possible to increase a Q value. 
     Further, as disclosed in JP-A-2011-199331, in the resonator element, an occupancy rate of a length of a weight portion with respect to a length of the vibrating arm is set to equal to or greater than 30% and a ratio of a width of the weight portion (second arm portion) to a width of an arm (first arm portion) is set to equal to or greater than 2, and thus it is possible to prevent a vibration of a second harmonic and to reduce crystal impedance (CI value, equivalent series resistance). However, when the length of the vibrating arm is made constant and a resonance frequency of a fundamental wave is set to, for example, 32.768 (kHz), in a case where the vibrating arm is merely configured to include the first arm portion and the second arm portion (weight portion), the frequency may be reduced from 32.768 (kHz) due to a weight effect of the weight portion. Accordingly, it is necessary to increase the width of the first arm portion of the vibrating arm in order to maintain a constant resonance frequency, for the purpose of avoiding the reduction in the frequency. Meanwhile, there is a concern that stress may concentrate on a bonding portion between the first arm portion and the second arm portion (weight portion). Thus, it is necessary to optimize a relationship between the width of the first arm portion and the width of the second arm portion (weight portion) so that damage does not occur in a case where excessive external force is applied thereto, and to improve impact resistance while maintaining the frequency of 32.768 (kHz). 
     JP-A-2011-019159 is an example of the related art. 
     SUMMARY 
     An advantage of some aspects of the invention is to provide a resonator element capable of increasing a Q value, and a resonator, an oscillator, an electronic device, and a moving object which include the resonator element. 
     The invention can be implemented as the following application examples. 
     APPLICATION EXAMPLE 1 
     This application example is directed to a resonator element including a vibrating arm that has a fixed end and a free end which are lined up along a first direction. The vibrating arm includes a weight portion that is disposed on the free end side, and an arm portion that is disposed on the fixed end side. A groove is provided along the first direction in at least one principal surface of a first principal surface and a second principal surface of the vibrating arm which serve as front and back sides, respectively. In the principal surface provided with the groove, when a width of the arm portion along a second direction crossing the first direction is set to W 1 , a width, along the second direction, of a portion interposed between a side surface, which connects the first principal surface and the second principal surface, and an outer edge of the groove which is lined up with the side surface along the second direction is set to W 3 , and 2×W 3 /W 1  is set to η, a relation of 14.2%&lt;η&lt;100% is satisfied. 
     According to the resonator element described above, it is possible to increase a Q value. As a result, it is possible to decrease a crystal impedance (CI) value. 
     In particular, since a ratio of the width of the groove with respect to the width of the arm portion of the vibrating arm is optimized, it is possible to efficiently increase the strength of the vibrating arm and a Q value. 
     Thus, it is possible to realize a CI value which is smaller than 50 kΩ while achieving a reduction in size. 
     APPLICATION EXAMPLE 2 
     In the resonator element according to this application example, it is preferable that a relation of 20%≦η≦50% is satisfied. Thus, it is possible to more efficiently increase the strength of the vibrating arm and a Q value. 
     APPLICATION EXAMPLE 3 
     In the resonator element according to this application example, it is preferable that a relation of 30%≦η≦40% is satisfied. Thus, it is possible to further efficiently increase the strength of the vibrating arm and a Q value. 
     APPLICATION EXAMPLE 4 
     In the resonator element according to this application example, it is preferable that the weight portion has a width along the second direction which is larger than that of the arm portion. Thus, since the weight portion is provided at the distal end of the vibrating arm, it is possible to increase the strength of the vibrating arm by increasing the width of the arm portion of the vibrating arm, and to increase a Q value by reducing thermoelastic loss. As a result, it is possible to increase impact resistance and to decrease a crystal impedance (CI) value. 
     APPLICATION EXAMPLE 5 
     In the resonator element according to this application example, it is preferable that, when a width of the weight portion along the second direction is set to W 2 , W 2 /W 1  is equal to or greater than 1.43 and equal to or less than 1.79. 
     Thus, since it is possible to reduce the concentration of stress on a bonding portion between the arm portion and the weight portion, it is possible to more efficiently increase the strength of the vibrating arm and a Q value. 
     APPLICATION EXAMPLE 6 
     In the resonator element according to this application example, it is preferable that, when a length of the vibrating arm along the first direction is set to L and a length of the weight portion along the first direction is set to Lx, Lx/L is equal to or greater than 20% and equal to or less than 60%. 
     Thus, it is possible to more efficiently increase the strength of the vibrating arm and a Q value without increasing the length of the vibrating arm. 
     APPLICATION EXAMPLE 7 
     In the resonator element according to this application example, it is preferable that a thickness of the arm portion is equal to or greater than 115 μm and equal to or less than 150 μm. Accordingly, the arm portion of the vibrating arm becomes relatively thicker, and thus it is possible to increase the strength of the vibrating arm. In addition, since it is possible to secure a large area of the vibrating arm to which an electric field is applied by the relatively large thickness of the arm portion of the vibrating arm, it is possible to increase electric field efficiency. Thus, it is possible to decrease a CI value. 
     APPLICATION EXAMPLE 8 
     In the resonator element according to this application example, it is preferable that the groove includes a first groove provided in the first principal surface and a second groove provided in the second principal surface so that the first and second grooves overlap each other when seen in a plan view, and when a thickness of the arm portion is set to t and a distance between a bottom of the first groove and a bottom of the second groove is set to t 1 , t 1 /t is equal to or greater than 10% and equal to or less than 40%. 
     Accordingly, it is possible to secure a large area of the vibrating arm to which an electric field is applied, while increasing the strength of the vibrating arm. Thus, it is possible to increase electric field efficiency. 
     APPLICATION EXAMPLE 9 
     In the resonator element according to this application example, it is preferable that W 3  is equal to or greater than 15 μm and equal to or less than 30 μm. 
     Accordingly, it is possible to increase a Q value and to decrease a CI value, while increasing the strength of the vibrating arm. 
     APPLICATION EXAMPLE 10 
     In the resonator element according to this application example, it is preferable that the resonator element further includes an excitation electrode provided in the vibrating arm and the groove includes an electrode-formed region provided with the excitation electrode, and a non-electrode-formed region which is disposed on the weight portion side and is not provided with the excitation electrode. 
     Thus, it is possible to decrease a CI value while reducing an equivalent series capacitance C 1 . In addition, it is possible to generate a vibration in a main mode while reducing the generation of a vibration in a high-order mode. 
     APPLICATION EXAMPLE 11 
     In the resonator element according to this application example, it is preferable that, when a length of the groove along the first direction is set to Ld and a length of the electrode-formed region along the first direction is set to L 1 , L 1 /Ld is equal to or greater than 72.8% and equal to or less than 82.8%. 
     Thus, it is possible to efficiently decrease a CI value while reducing an equivalent series capacitance C 1 . 
     APPLICATION EXAMPLE 12 
     In the resonator element according to this application example, it is preferable that an end of the non-electrode-formed region on the base portion side is disposed between the weight portion and a position serving as a node of a harmonic vibration occurring in the vibrating arm. 
     Thus, it is possible to efficiently suppress the generation of a vibration in a high-order mode. 
     APPLICATION EXAMPLE 13 
     In the resonator element according to this application example, it is preferable that the resonator element further includes an excitation electrode provided in the vibrating arm, the excitation electrode includes a base layer, and a covering layer that is laminated on the base layer on an opposite side to the vibrating arm side, when a thickness of the base layer is set to Tb and a thickness of the covering layer is set to Tc, a relation of Tb/Tc≦4/7 is satisfied, the base layer is formed of chromium, and a thickness of the base layer is equal to or greater than 50 Å and equal to or less than 700 Å. Thus, even when the base layer has a phase transition point on the low-temperature side in an operation temperature range, it is possible to decrease a CI value on the low-temperature side in the operation temperature range. 
     In spite of having an excellent function as a base layer, chromium has a phase transition point at a relatively low temperature. 
     Consequently, the thickness of the base layer formed of chromium is reduced to approximately equal to or greater than 50 Å and equal to or less than 700 Å. Thus, it is possible to suppress an increase in a CI value due to a phase transition of chromium of the base layer in a low temperature region, while achieving a function as the base layer. 
     APPLICATION EXAMPLE 14 
     In the resonator element according to this application example, it is preferable that the vibrating arm is constituted by a pair of vibrating arms which are lined up along the second direction, and includes a base portion connected to the fixed end. 
     Thus, it is possible to realize a resonator element having a high Q value. 
     APPLICATION EXAMPLE 15 
     In the resonator element according to this application example, it is preferable that, when a distance between the free end of the vibrating arm and an end of the base portion on an opposite side to the vibrating arm side, when seen in a plan view, is set to La and a distance between a centroid of a system, which includes the pair of vibrating arms and the base portion, and the end on the opposite side is set to Lg, Lg/La is equal to or less than ⅔. 
     Thus, when the resonator element is fixed to a package using an adhesive or the like, the fixation is facilitated. 
     APPLICATION EXAMPLE 16 
     In the resonator element according to this application example, it is preferable that, when a width, along the second direction, between outer edges on an opposite side to facing sides of the weight portions of the pair of vibrating arms, when seen in a plan view, is set to Wa and a width of the base portion along the second direction is set to S, a relation of 85%≦Wa/S≦115% is satisfied. 
     Thus, it is possible to increase the strength of the vibrating arm and a Q value while reducing the width of the resonator element along the second direction. 
     APPLICATION EXAMPLE 17 
     This application example is directed to a resonator including the resonator element according to the application example and a package that accommodates the resonator element. 
     Thus, it is possible to provide the resonator having an excellent reliability. 
     APPLICATION EXAMPLE 18 
     This application example is directed to an oscillator including the resonator element according to the application example and a circuit. 
     Thus, it is possible to provide the oscillator having an excellent reliability. 
     APPLICATION EXAMPLE 19 
     This application example is directed to an electronic device including the resonator element according to the application example. 
     Thus, it is possible to provide the electronic device having an excellent reliability. 
     APPLICATION EXAMPLE 20 
     This application example is directed to a moving object including the resonator element according to the application example. 
     Thus, it is possible to provide the moving object having an excellent reliability. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention will be described with reference to the accompanying drawings, wherein like numbers reference like elements. 
         FIG. 1  is a plan view showing a resonator according to an embodiment of the invention. 
         FIG. 2  is a cross-sectional view taken along line A-A of  FIG. 1 . 
         FIG. 3  is a plan view of a resonator element included in the resonator shown in  FIG. 1 . 
         FIG. 4A  is a cross-sectional view taken along line B-B of  FIG. 3 , and  FIG. 4B  is a cross-sectional view taken along line C-C of  FIG. 3 . 
         FIG. 5  is a plan view showing a resonator element of the related art. 
         FIG. 6A  is a cross-sectional view (cross-sectional view taken along line D-D of  FIG. 5 ) of a vibrating arm of the resonator element of the related art, and  FIG. 6B  is a diagram showing an example of a cross-section of a vibrating arm of the resonator element according to the invention. 
         FIG. 7  is a graph showing a relationship between a ratio of a width of a bank portion with respect to a width of an arm portion and a Q value. 
         FIG. 8  is a graph showing a relationship between a weight portion occupancy rate and a high performance index. 
         FIG. 9  is a cross-sectional view showing an example of an oscillator according to the invention. 
         FIG. 10  is a perspective view showing a configuration of a mobile (or notebook) personal computer to which an electronic device including the resonator element according to the invention is applied. 
         FIG. 11  is a perspective view showing a configuration of a mobile phone (PHS is also included) to which an electronic device including the resonator element according to the invention is applied. 
         FIG. 12  is a perspective view showing a configuration of a digital still camera to which an electronic device including the resonator element according to the invention is applied. 
         FIG. 13  is a perspective view showing a configuration of a moving object (vehicle) to which an electronic device including the resonator element according to the invention is applied. 
     
    
    
     DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Hereinafter, a resonator element, a resonator, an oscillator, an electronic device, and a moving object according to the invention will be described in detail with reference to preferred embodiments shown in the diagrams. 
     1. Resonator 
     First, the resonator according to the invention (resonator including the resonator element according to the invention) will be described. 
     First Embodiment 
       FIG. 1  is a plan view showing a resonator according to an embodiment of the invention.  FIG. 2  is a cross-sectional view taken along line A-A of  FIG. 1 . In addition,  FIG. 3  is a plan view of a resonator element included in the resonator shown in  FIG. 1 .  FIG. 4A  is a cross-sectional view taken along line B-B of  FIG. 3 , and  FIG. 4B  is a cross-sectional view taken along line C-C of  FIG. 3 . 
     Hereinafter, for convenience of description, an X-axis, a Y-axis, and a Z-axis perpendicular to each other are shown in  FIGS. 1 to 4B . A distal end side of each arrow shown in the drawing is set to a “+ (positive) side” and a base end side is set to be a “− (negative) side”. In addition, a direction parallel to the X-axis (second direction) is referred to as an “X-axis direction”, a direction parallel to the Y-axis (first direction) is referred to as a “Y-axis direction”, and a direction parallel to the Z-axis is referred to as a “Z-axis direction”. In addition, a +Z-side (upper side in  FIG. 2 ) is referred to as a “top”, and a −Z—side (lower side in  FIG. 2 ) is referred to as a “bottom”. In the following description, for convenience of description, the X-axis, the Y-axis, and the Z-axis shown in the drawings conform with an X-axis (electrical axis), a Y-axis (mechanical axis), and a Z-axis (optical axis) of quartz crystal constituting a quartz crystal substrate  3  to be described later. 
     A resonator  1  shown in  FIGS. 1 and 2  includes a resonator element  2  (resonator element according to the invention) and a package  9  that accommodates the resonator element  2 . Hereinafter, the resonator element  2  and the package  9  will be sequentially described in detail. 
     Resonator Element 
     First, a configuration of the resonator element  2  will be briefly described. 
     As shown in  FIGS. 3 and 4B , the resonator element  2  includes the quartz crystal substrate  3 , and a first driving electrode  84  and a second driving electrode  85  which are formed on the quartz crystal substrate  3 . Meanwhile, in  FIG. 3 , the first driving electrode  84  and the second driving electrode  85  are partially not shown for convenience of description. 
     The quartz crystal substrate  3  is constituted by a Z-cut quartz crystal plate. Thus, the resonator element  2  can exhibit excellent vibration characteristics. The Z-cut quartz crystal plate refers to a quartz crystal substrate having a Z-axis (optical axis) of quartz crystal as its thickness direction. Meanwhile, it is preferable that the Z-axis of quartz crystal conforms with the thickness direction of the quartz crystal substrate  3 . However, the Z-axis may be inclined slightly (approximately less than fifteen degrees) with respect to the thickness direction. 
     As shown in  FIG. 3 , the quartz crystal substrate  3  includes a base portion  4  and a pair of vibrating arms  5  and  6  extending from the base portion  4 . 
     In other words, the vibrating arm includes a fixed end and a free end which are lined up along the Y-axis direction, and the fixed end is connected to the base portion  4 . 
     The base portion  4  has a plate shape that extends on the XY plane, which is a plane parallel to the X-axis and the Y-axis, and that has the Z-axis direction as its thickness direction. The base portion  4  has a shape in which a midway portion of the base portion  4  in the Y-axis direction is constricted when seen in a plan view from the Z-axis direction (hereinafter, simply referred to as “when seen in a plan view”). That is, the base portion  4  includes a first base portion  41  from which the vibrating arms  5  and  6  extend, a second base portion  42  which is provided on the opposite side to the vibrating arms  5  and  6  with respect to the first base portion  41 , and a connecting portion  43  that connects the first base portion  41  and the second base portion  42 . 
     Here, the connecting portion  43  has a width along the X-axis direction which is smaller than that of the first base portion  41 . Thus, it is possible to reduce vibration leakage while reducing the length of the base portion  4  along the Y-axis direction. 
     The vibrating arms  5  and  6  extend in the +Y-axis direction from the base portion  4  so as to be lined up in the X-axis direction and to be parallel to each other. Each of the vibrating arms  5  and  6  has an elongated shape. The base end of each of the vibrating arms is a fixed end, and the distal end thereof is a free end. In addition, the distal ends of the vibrating arms  5  and  6  are provided with weight portions  59  and  69  having a width along the X-axis direction which is larger than those of the base ends of the vibrating arms  5  and  6 . 
     As shown in  FIGS. 4A and 4B , the vibrating arm  5  includes a pair of principal surfaces  51  and  52 , which are on the XY plane, and a pair of side surfaces  53  and  54  which are on the YZ plane and connect the pair of principal surfaces  51  and  52 . In addition, the vibrating arm  5  includes a bottomed-groove (groove portion)  55  opened to the principal surface  51  and a bottomed-groove (groove portion)  56  opened to the principal surface  52 . The grooves  55  and  56  extend in the Y-axis direction. The cross-sections of portions of the vibrating arm  5  in which the grooves  55  and  56  are formed have a substantially H shape. 
     Similarly to the vibrating arm  5 , the vibrating arm  6  includes a pair of principal surfaces  61  and  62 , which are on the XY plane, and a pair of side surfaces  63  and  64  which are on the YZ plane and connect the pair of principal surfaces  61  and  62 . In addition, the vibrating arm  6  includes a bottomed-groove (groove portion)  65  opened to the principal surface  61  and a bottomed-groove (groove portion)  66  opened to the principal surface  62 . The grooves  65  and  66  extend in the Y-axis direction. The cross-sections of portions of the vibrating arm  6  in which the grooves  65  and  66  are formed have a substantially H shape. 
     As shown in  FIGS. 4A and 4B , it is preferable that the grooves  55  and  56  are formed to be symmetrical (symmetrical in the vertical direction) with respect to a segment dividing the thickness of the vibrating arm  5  into two parts in the cross-section. Similarly, it is preferable that the grooves  65  and  66  are formed to be symmetrical (symmetrical in the vertical direction) with respect to a segment dividing the thickness of the vibrating arm  6  into two parts in the cross-section. Thus, it is possible to suppress unnecessary vibrations of the vibrating arms  5  and  6  and to efficiently vibrate the vibrating arms  5  and  6  in an in-plane direction of the quartz crystal substrate  3 . 
     A pair of first driving electrodes  84  and a pair of second driving electrodes  85  are formed in the vibrating arm  5 . Specifically, one of the first driving electrodes  84  is formed on the inner surface of the groove  55 , and the other first driving electrode  84  is formed on the inner surface of the groove  56 . In addition, one of the second driving electrodes  85  is formed on the side surface  53 , and the other second driving electrode  85  is formed on the side surface  54 . 
     Similarly, a pair of first driving electrodes  84  and a pair of second driving electrodes  85  are formed in the vibrating arm  6 . Specifically, one of the first driving electrodes  84  is formed on the side surface  63 , and the other first driving electrode  84  is formed on the side surface  64 . In addition, one of the second driving electrodes  85  is formed on the inner surface of the groove  65 , and the other second driving electrode  85  is formed on the inner surface of the groove  66 . 
     When an alternating voltage is applied between the first driving electrode  84  and second driving electrode  85 , the vibrating arms  5  and  6  vibrate with a predetermined frequency in an in-plane direction (direction of the XY plane) so that the vibrating arms alternately repeat mutual approach and separation. 
     Materials of the first driving electrode  84  and the second driving electrode  85  are not particularly limited. For example, the electrodes can be formed of a metal material such as gold (Au), a gold alloy, platinum (Pt), aluminum (Al), an aluminum alloy, silver (Ag), a silver alloy, chromium (Cr), a chromium alloy, nickel (Ni), copper (Cu), molybdenum (Mo), niobium (Nb), tungsten (W), iron (Fe), titanium (Ti), cobalt (Co), zinc (Zn), or zirconium (Zr), or a conductive material such as indium tin oxide (ITO). 
     Until now, the configuration of the resonator element  2  has been described briefly. Hereinafter, characteristic parts of the resonator element  2  will be described in detail. 
     As described above, the vibrating arm  5  is provided with the grooves  55  and  56 . Similarly, the vibrating arm  6  is provided with the grooves  65  and  66 . That is, the vibrating arm  5  includes the bottomed-grooves (groove portions)  55  and  56  which are formed in the two principal surfaces  51  and  52  of the vibrating arm  5  along the X-axis direction and the Y-axis direction and which extend in the Y-axis direction. Similarly, the vibrating arm  6  includes the bottomed-grooves (groove portions)  65  and  66  which are formed in the two principal surfaces  61  and  62  of the vibrating arm  6  along the X-axis direction and the Y-axis direction and which extend in the Y-axis direction. 
     According to the vibrating arms  5  and  6  in which the grooves  55 ,  56 ,  65 , and  66  are formed, heat generated in association with the in-plane flexural vibration of the vibrating arms  5  and  6  is not likely to diffuse (heat conduction). Thus, it is possible to suppress thermoelastic loss in an adiabatic region which is a region where a flexural vibration frequency (mechanical flexural vibration frequency) f is greater than a thermal relaxation frequency f 0  (f&gt;f 0 ). That is, by the relation of f/f 0 &gt;1 being satisfied, it is possible to set the vibrating arms  5  and  6  as adiabatic regions and to reduce a decrease in a Q value. 
     In general, when a groove is formed in a principal surface of a vibrating arm, the mechanical strength of the vibrating arm is decreased, and thus it is possible to increase the strength of the vibrating arm and to reduce thermoelastic loss by increasing the width or thickness of the vibrating arm. Here, a resonance frequency f of a fundamental wave of a tuning fork type resonator element is known to satisfy the following expression.
 
f∝W/L 2   (1)
 
     Meanwhile, W denotes a width of an arm portion, and L denotes a length of the arm portion. 
     That is, when the resonance frequency of the fundamental wave is made constant, a relationship is established in which the smaller the length L of the arm portion of the resonator element, the smaller the width W of the arm portion. 
     Consequently, when the length of the vibrating arm is made constant and the resonance frequency of the fundamental wave is set to, for example, 32.768 (kHz) and when the vibrating arm is constituted by a first arm portion and a second arm portion (weight portion) having a width which is larger than that of the first arm portion in order merely to improve a Q value, the frequency is reduced from 32.768 (kHz) due to a weight effect of the weight portion. Accordingly, it is necessary to increase the width of the first arm portion of the vibrating arm in order to maintain a constant resonance frequency, for the purpose of avoiding the reduction in the frequency. On the other hand, since there is a concern that stress may concentrate on a bonding portion between the first arm portion and the second arm portion (weight portion), it is necessary to optimize a relationship between the width of the first arm portion and the width of the second arm portion (weight portion) so that damage does not occur in a case where excessive external force is applied thereto, and to improve impact resistance while maintaining the frequency of 32.768 (kHz). 
     Consequently, the vibrating arms  5  and  6  have configurations to be described below and have optimized dimensions. 
     As described above, the distal ends of the vibrating arms  5  and  6  are provided with the weight portions  59  and  69  having a width along the X-axis direction which is larger than those of the base ends of the vibrating arms  5  and  6 . That is, the vibrating arms  5  and  6  include arm portions  50  and  60  extending from the base portion  4 , and the weight portions  59  and  69  which are provided on the opposite side to the base portion  4  with respect to the arm portions  50  and  60  and have a width along the X-axis direction which is larger than those of the arm portions  50  and  60 . 
     In this manner, by the weight portions  59  and  69  being provided on the distal end sides of the vibrating arms  5  and  6 , a Q value is improved, and the widths of the arm portions  50  and  60  of the vibrating arms  5  and  6  are increased, and thus it is possible to maintain a constant resonance frequency of flexural vibration. Accordingly, it is possible to increase the strength of the vibrating arms  5  and  6  and to increase a Q value by reducing thermoelastic loss. As a result, it is possible to increase impact resistance and to decrease an equivalent series resistance CI value. 
     In particular, when each of widths of the arm portions  50  and  60  along the X-axis direction (hereinafter, referred to as “widths of the arm portions  50  and  60 ” or “arm widths”) is set to W 1 , each of widths (hereinafter, referred to as “bank widths”) of portions positioned on both sides of each of the grooves  55 ,  56 ,  65 , and  66  of the vibrating arms  5  and  6  (hereinafter, referred to as “bank portions”) is set to W 3 , and 2×W 3 /W 1  is set to η, the relation of 14.2%&lt;η&lt;100% (100% indicates a state where a groove is not formed in the arm portion) is satisfied. 
     In this manner, a ratio of each of the widths of the grooves  55 ,  56 ,  65 , and  66  along the X-axis direction (hereinafter, also simply referred to as “widths of the groove  55  and the like”) with respect to each of the widths of the arm portions  50  and  60  of the vibrating arms  5  and  6  is optimized, and thus it is possible to efficiently increase the strength of the vibrating arms  5  and  6  and a Q value. Here, the width W 3  of the bank portion of the vibrating arm  5  is a width of each of two portions (side portions)  51   a  and  51   b  along the X-axis direction with the groove  55  interposed therebetween in the principal surface  51  and a width of each of two portions (side portions)  52   a  and  53   b  along the X-axis direction with the groove interposed therebetween in the principal surface  52 . Meanwhile, the width W 3  of the bank portion of the vibrating arm  6  is the same as the width W 3  of the bank portion of the vibrating arm  5 . 
     Hereinafter, η will be described in detail. 
     The inventor has verified, by a simulation, a Q value in a case where the ratio η of the width of the bank portion with respect to each of the widths of the arm portions  50  and  60  is changed while increasing the widths of the arm portions  50  and  60 .  FIG. 5  is a plan view showing the resonator element disclosed in JP-A-2004-266871 mentioned in the related art.  FIG. 6A  is a cross-sectional view (cross-sectional view taken along line D-D of  FIG. 5 ) of the vibrating arm of the resonator element disclosed in JP-A-2004-266871 of the related art, and  FIG. 6B  is a diagram showing an example of a cross-section of a vibrating arm of the resonator element according to the invention. 
     In this simulation, the resonator element  2  having a plan-view shape (with a weight portion) which is shown in  FIG. 3  and a resonator element  2 X having a plan-view shape (without a weight portion) which is shown in  FIG. 5  are used. In the resonator element  2 X used in this simulation, cross-sections of vibrating arms  5 X and  6 X are set to have shapes and dimensions as shown in  FIG. 6A  so that a resonance frequency is set to 32.768 kHz. On the other hand, the resonator element  2  used in this simulation is set to have the same resonance frequency, total length, and total width as those of the resonator element  2 X, and the cross-sections of the vibrating arms  5  and  6  of the resonator element  2  are set to have shapes and dimensions as shown in  FIG. 6B . 
     The dimensions of the vibrating arms  5  and  6  of the resonator element  2  have been evaluated in the following ranges.
         Widths W 2  of the weight portions  59  and  69 : 215 (μm)   Widths W 1  of the arm portions  50  and  60 : 130 to 141 (μm)   Widths W 3  of the bank portions: 10 to 65 (μm)   Thicknesses t of the arm portions  50  and  60 : 120 (μm)   Depths of the grooves  55 ,  56 ,  65 , and  66 : 50 (μm)       

     The results of this simulation are shown in Table 2 and  FIG. 7 . 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Plate 
                   
                   
                   
                   
                   
               
               
                 thickness 
                 Groove 
                 Bank portion 
                 Arm width 
               
               
                 [μm] 
                 Depth [μm] 
                 width [μm] 
                 [μm] 
                 η 
                 Q value 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 120 
                 50 
                 10 
                 141 
                 0.142 
                 75.834 
               
               
                   
                   
                 15 
                 141 
                 0.213 
                 71.213 
               
               
                   
                   
                 20 
                 136 
                 0.294 
                 62.267 
               
               
                   
                   
                 25 
                 132 
                 0.379 
                 55.341 
               
               
                   
                   
                 30 
                 130 
                 0.462 
                 50.755 
               
               
                   
                   
                 35 
                 130 
                 0.538 
                 44.943 
               
               
                   
                   
                 40 
                 130 
                 0.615 
                 42.788 
               
               
                   
                   
                 45 
                 130 
                 0.692 
                 41.272 
               
               
                   
                   
                 50 
                 130 
                 0.769 
                 38.714 
               
               
                   
                   
                 57.5 
                 130 
                 0.885 
                 35.961 
               
               
                   
                   
                 65 
                 130 
                 1 
                 35.168 
               
               
                   
               
            
           
         
       
     
     Meanwhile, a Q value shown in Table 1 is calculated as a value in a case where the resonance frequency is set to 32.768 kHz by correcting a slight deviation of the resonance frequency. 
       FIG. 7  is a graph showing a relationship between a ratio of the width of the bank portion with respect to the width of the arm portion and the Q value. 
     As shown in  FIG. 7 , it is confirmed that the shape “with a weight portion” is improved in excess of the Q value (31,573) of JP-A-2004-266871 in the range of 14.2%&lt;η&lt;100%. 
     In addition, as shown in  FIG. 7 , it can be observed that the Q value becomes larger as η becomes smaller. However, it is confirmed, by a simulation, that the Q value starts falling when the width of the bank portion becomes smaller than 7 (μm). Further, when the width of the bank portion is 7 (μm), the ratio of the η value is reduced to 9.9% (=7×2/141) and 10.7% (=7×2/130). Thus, a decrease in the mechanical strength of the vibrating arm and a so-called reduction in impact resistance have been problems. 
     Further, even when the width W 1  of each of the arm portions  50  and  60  and the width W 3  of each bank portion are set to be in the following ranges, it is confirmed that the Q value is greater than the Q value (31,573) in JP-A-2004-266871.
         Widths W 1  of the arm portions  50  and  60 : 120 to 150 (μm)   Width W 3  of the bank portion: 15 to 30 (μm)       

     At this time, η is as follows.
 
η=2 ×W 3 /W 1=2×15 μm/120 μm=25.0%
 
η=2 ×W 3 /W 1=2×30 μm/120 μm=50.0%
 
η=2 ×W 3 /W 1=2×15 μm/150 μm=20.0%
 
η=2 ×W 3 /W 1=2×30 μm/150 μm=40.0%
 
     Therefore, η satisfies the following relation.
 
20.0%≦η≦50.0%
 
     Further, a ratio of the width W 2  of each of the weight portions  59  and  69  with respect to the width W 1  of each of the arm portions  50  and  60  is as follows.
 
 W 2 /W 1=215/120=1.79
 
 W 2 /W 1=215/150=1.43
 
     Therefore, W 2 /W 1  satisfies the following relation.
 
1.43 ≦W 2 /W 1≦1.79
 
     Here, W 2 /W 1  is set to have the above-described relation, and thus it is possible to reduce stress concentrating on bonding portions between the arm portions  50  and  60  and the weight portions  59  and  69 . Accordingly, even when an excessive external force is applied, it is possible to improve impact resistance without causing any damage. 
     From the above-described relationship between and the Q value, the relation of 14.2%&lt;η&lt;100% is satisfied. Thus, it is possible to efficiently increase the strength of the vibrating arms  5  and  6  and the Q value without increasing the lengths of the vibrating arms  5  and  6 . However, in order to increase the effects, it is preferable to satisfy the relation of 20%≦η≦50% and more preferable to satisfy the relation of 30%≦η≦40% in consideration of the impact resistance of the resonator elements. 
     Modification Example of Resonator Element 
     Next, a resonator element according to a modification example of the invention will be described. 
     Hereinafter, the resonator element according to the modification example will be described focusing on the differences from the resonator element included in the resonator according to the first embodiment described above, and a description of the same matters will be omitted. In the above-described embodiment, a configuration is adopted in which one groove is provided in each principal surface of each vibrating arm. However, the number of grooves is not particularly limited, and two or more grooves may be provided. For example, two grooves lined up along the X-axis direction may be provided in each principal surface. 
     The resonator element according to this modification example is the same as the resonator element according to the first embodiment described above except that the number of grooves provided in each principal surface of each vibrating arm is different. 
     The resonator element according to the modification example is provided with two bottomed-grooves opened to one principal surface of each vibrating arm and is provided with two bottomed-grooves opened to the other principal surface. A portion interposed between one side surface, which connects the one principal surface and the other principal surface, and an outer edge of the groove lined up with the one side surface along the X-axis direction is set to one bank portion, and a width of the one bank portion along the X-axis direction is set to W 3 . 
     In addition, a portion interposed between the other side surface, which connects the one principal surface and the other principal surface, and an outer edge of the groove lined up with the other side surface along the X-axis direction is set to the other bank portion, and a width of the other bank portion along the X-axis direction is set to W 3 . 
     Accordingly, since a width of the arm portion along the X-axis direction is W 1 , 2×W 3 /W 1  is set to η. 
     In addition, when the entire length of each of the vibrating arms  5  and  6  along the Y-axis direction (hereinafter, also simply referred to as “length of each of the vibrating arms  5  and  6 ”) is set to L and a length of each of the weight portions  59  and  69  along the Y-axis direction (hereinafter, also simply referred to as “length of each of the weight portions  59  and  69 ”) is set to Lx, it is preferable that Lx/L is equal to or greater than 20% and equal to or less than 60%. 
     In this manner, Lx/L is optimized, and thus it is possible to more efficiently increase the strength of the vibrating arms  5  and  6  and a Q value without increasing the lengths of the vibrating arms  5  and  6 . 
     Hereinafter, Lx/L and W 2 /W 1  will be described in detail.  FIG. 8  is a graph showing a relationship between a weight portion occupancy rate and a high performance index. 
     In  FIG. 8 , a horizontal axis represents a weight portion occupancy rate (%) which is a ratio (Lx/L) of a length Lx of each of the weight portions  59  and  69  with respect to the length L of each of the vibrating arms  5  and  6 , and a vertical axis represents a high performance index which is defined as a value obtained by multiplying a low frequency index by a high Q value index. Here, the low frequency index is an index indicating that a low frequency of the resonator element (in other words, a reduction in the size of the resonator element) can be efficiently achieved as a value becomes larger. In addition, the high Q value index is an index indicating that a decrease in the Q value of the resonator element can be efficiently reduced as a value becomes larger. 
     As shown in  FIG. 8 , when the weight portion occupancy rate is 38%, the high performance index is set to a maximum value (1.0). That is, when Lx/L is 38%, it is possible to most efficiently achieve both a low frequency (reduction in size) of the resonator element and a high Q value. 
     Accordingly, in the resonator element  2  according to the invention, Lx/L is set to equal to or greater than 20% and equal to or less than 60% based on approximately 38% so that the high performance index is set to equal to or greater than 0.8 as shown in  FIG. 8 , and thus it is possible to efficiently achieve a reduction in size and a high Q value. 
     In addition, it is preferable that Lx/L is equal to or greater than 35% and equal to or less than 41%. Thus, it is possible to set the high performance index to a value close to a maximum value as shown in  FIG. 8  and to more efficiently achieve a reduction in size and a high Q value. 
     When the width of each of the weight portions  59  and  69  along the X-axis direction (hereinafter, also simply referred to as “width of each of the weight portions  59  and  69 ”) is set to W 2  as described above in realizing the high performance index from the above-described relationship between the weight portion occupancy rate and the high performance index, it is preferable that W 2 /W 1  is equal to or greater than 1.43 and equal to or less than 1.79. 
     In this manner, W 2 /W 1  is optimized, and thus it is possible to produce effects (weight effects) obtained by providing the weight portions  59  and  69 , while suppressing an increase in the entire width of the resonator element  2  along the X-axis direction and maintaining impact resistance. 
     In addition, the width W 1  of each of the arm portions  50  and  60  is preferably equal to or greater than 120 μm and equal to or less than 150 μm, and is more preferably equal to or greater than 131 μm and equal to or less than 140 μm. Thus, it is possible to increase the strength of the vibrating arms  5  and  6  while maintaining a constant frequency of the resonator element  2 , and to reduce thermoelastic loss. 
     In addition, a thickness t of each of the arm portions  50  and  60  is preferably equal to or greater than 100 μm and equal to or less than 300 μm, is more preferably equal to or greater than 110 μm and equal to or less than 200 μm, and is further preferably equal to or greater than 115 μm and equal to or less than 150 μm. Accordingly, the arm portions  50  and  60  of the vibrating arms  5  and  6  become relatively thicker, and thus it is possible to increase the strength of the vibrating arms  5  and  6 . In addition, it is possible to secure large areas of the vibrating arms  5  and  6  to which an electric field is applied by the relatively large thicknesses of the arm portions  50  and  60  of the vibrating arms  5  and  6 . That is, since it is possible to increase the size of an area of an excitation portion interposed between the first driving electrode  84  and the second driving electrode  85 , electric field efficiency can be increased, and it is possible to decrease an equivalent series resistance CI value. 
     In contrast, when the thickness t of each of the arm portions  50  and  60  is too small, there is a tendency for the strength of the vibrating arms  5  and  6  to be considerably reduced. On the other hand, when the thickness t of each of the arm portions  50  and  60  is too large, it becomes difficult to form the external form of the resonator element  2  only by wet etching. 
     In addition, when a thickness of each of the arm portions  50  and  60  is set to t and each of a distance between the grooves  55  and  56  and a distance between the grooves  65  and  66  (that is, the thickness of the central portion of each of the vibrating arms  5  and  6  in the width direction) is set to t 1 , t 1 /t is preferably equal to or greater than 10% and equal to or less than 40%, is more preferably equal to or greater than 15% and equal to or less than 30%, and is further preferably equal to or greater than 15% and equal to or less than 20%. Thus, it is possible to secure large areas of the vibrating arms  5  and  6  to which an electric field is applied, while increasing the strength of the vibrating arms  5  and  6 . 
     In addition, the width W 3  of each of the bank portions of the vibrating arms  5  and  6  is preferably equal to or greater than 15 μm and equal to or less than 30 μm, and is more preferably equal to or greater than 18 μm and equal to or less than 25 μm. Thus, it is possible to decrease a CI value while increasing the strength of the vibrating arms  5  and  6 . 
     In addition, as shown in  FIGS. 4A and 4B , the groove  55  includes an electrode-formed region  551  which is a portion where the first driving electrode (excitation electrode)  84  is provided, and a non-electrode-formed region  552  which is positioned on the opposite side to the base portion  4  with respect to the electrode-formed region  551  and which is a portion where a driving electrode is not provided. Similarly, the groove  56  includes an electrode-formed region  561  and a non-electrode-formed region  562 , the groove  65  includes an electrode-formed region  651  and a non-electrode-formed region  652 , and the groove  66  includes an electrode-formed region  661  and a non-electrode-formed region  662 . 
     By the non-electrode-formed regions  552 ,  562 ,  652 , and  662  (hereinafter, also referred to as “non-electrode-formed region  552  and the like”) being formed, it is possible to generate a vibration (fundamental wave) in a main mode in which an equivalent series capacitance C 1  is reduced, while suppressing the generation of a vibration (harmonic) in a high-order mode. 
     Here, a relationship between a fundamental wave CI value and a CI value ratio and a relationship between a fundamental wave CI value and a length of a groove will be described. 
     The resonator element  2  is configured to oscillate a signal with a fundamental wave frequency by the vibration thereof, but has a characteristic of simultaneously oscillating the same signal also with a harmonic frequency. In addition, there is a concern that the picking-up of this harmonic signal being performed by mistake for a signal having a fundamental wave frequency in a device such as a resonator may result in an abnormality in the device. 
     Consequently, a design method performed on the basis of a CI value ratio is used as a method of forestalling such a situation. 
     The CI value ratio is a ratio obtained by dividing a high-frequency CI value (equivalent series resistance) by a fundamental wave CI value (equivalent series resistance) (high-frequency CI value/fundamental wave CI value). That is, when the high-frequency CI value becomes greater than the fundamental wave CI value and the CI value ratio is set to equal to or greater than 1.0, a signal is not likely to oscillate with a high frequency. Accordingly, there is little concern that a device or the like may erroneously pick up a harmonic signal, and thus a high-performance resonator element is obtained. 
     When a length of each of the grooves  55 ,  56 ,  65 , and  66  along the Y-axis direction is set to Ld and a length of each of the electrode-formed regions  551 ,  561 ,  651 , and  661  (hereinafter, also referred to as “electrode-formed region  551  and the like”) along the Y-axis direction (hereinafter, also referred to as “length of each of the electrode-formed regions  551  and the like”) is set to L 1 , the relations of Ld=900 (μm) and L=700 (μm) are established in this embodiment. 
     Although a ratio of the electrode-formed region L 1  with respect to the length Ld of the groove is set to 77.8%, the ratio is preferably equal to or greater than 72.8% and equal to or less than 82.8% in consideration of ±5% as an allowable variation, and is more preferably equal to or greater than 74.8% and equal to or less than 81.8%. Thus, it is possible to efficiently decrease a CI value while reducing an equivalent series capacitance C 1 . 
     In addition, it is preferable that ends of the non-electrode-formed regions  552  and  562  on the base portion  4  side are positioned between the weight portion  59  and a node of a harmonic vibration of the vibrating arm  5 . Similarly, it is preferable that ends of the non-electrode-formed regions  652  and  662  on the base portion  4  side are positioned between the weight portion  69  and a node of a harmonic vibration of the vibrating arm  6 . Thus, it is possible to efficiently suppress the generation of a vibration in a high-order mode. In addition, when a distance between the ends of the vibrating arms  5  and  6  on the opposite side to the base portion  4  and the end of the base portion  4  on the opposite side to the vibrating arms  5  and  6  (that is, the total length of the resonator element  2  along the Y-axis direction) is set to La and a distance between the centroid of a structure (that is, the quartz crystal substrate  3 ) which is constituted by the pair of vibrating arms  5  and  6  and the base portion  4  and the end of the base portion  4  on the opposite side to the vibrating arms  5  and  6  is set to Lg, it is preferable that Lg/La is equal to or less than ⅔. Thus, when the resonator element  2  is fixed to the package  9  using an adhesive or the like, the fixation is facilitated. 
     In addition, as described above, the base portion  4  includes the first base portion  41 , the second base portion  42 , and the connecting portion  43 . Thus, it is possible to reduce vibration leakage while reducing the length of the base portion  4  along the Y-axis direction. 
     Here, when a width of the first base portion  41  along the X-axis direction is set to S 1  and a minimum width of the connecting portion  43  along the X-axis direction is set to S 2 , it is preferable that S 2 /S 1  is equal to or greater than 0.3 and equal to or less than 0.6. Thus, it is possible to efficiently reduce vibration leakage while reducing the length of the base portion  4  along the Y-axis direction. 
     In this embodiment, the first base portion  41  includes a width-decreasing portion  411  which is provided at the end of the first base portion  41  on the connecting portion  43  side and which has a width along the X-axis direction decreasing continuously or intermittently toward the connecting portion  43 . Thus, it is possible to effectively suppress the deformation of the first base portion  41  which is associated with the flexural vibration of the pair of vibrating arms  5  and  6  which mutually approach and separate in the same plane. As a result, even when the length of the first base portion  41  along the Y-axis direction is reduced, it is possible to suppress the deformation of the first base portion  41  which is associated with the flexural vibration of the pair of vibrating arms  5  and  6  alternately repeating mutual approach and separation substantially in a plane and to suppress vibration leakage to the outside from the base portion  4 . In this embodiment, an outer edge  412  of the width-decreasing portion  411  and an outer edge  431  of the connecting portion  43  form a continuous curved line that does not include a corner portion when seen in a plan view. Thus, it is possible to reduce stress occurring in the base portion  4  and to reduce thermoelastic loss. 
     When an angle between a tangent of a portion of the curved line on the first base portion  41  side and a segment parallel to the X-axis direction is set to θ, it is preferable that the relation of 0°&lt;θ&lt;90° is satisfied and that the relation of 6°≦θ≦70° is satisfied. Thus, it is possible to produce preferable effects by the width-decreasing portion  411  and the connecting portion  43 . 
     In addition, as described above, in a case where a resonance frequency f of the resonator element  2  is set to become greater than a thermal relaxation frequency f 0 , when a base portion bending width which is a shortest distance between the outer edge  431  of the connecting portion  43  and an outer edge  413  of a crotch portion formed between the pair of vibrating arms  5  and  6 , when seen in a plan view, is set to Wb and when effective arm widths which are arm widths of the vibrating arms, obtained by replacing the cross-sectional shapes of the vibrating arms  5  and  6  in a surface perpendicular to the longitudinal directions of the vibrating arms  5  and  6  with rectangular shapes having the same thickness with equivalent thermoelastic loss, are set to We, it is preferable to satisfy the relation of Wb&gt;We. Thus, since thermoelastic loss occurring in the cross-section of the first base portion  41  becomes smaller than the thermoelastic loss occurring in the vibrating arms  5  and  6 , the degradation in a Q value of the resonator element  2  due to the thermoelastic loss of the first base portion  41  is suppressed. Accordingly, it is possible to improve the Q value. 
     In addition, when a width of the entirety of the pair of vibrating arms  5  and  6  along the X-axis direction is set to Wa and a width of the base portion  4  along the X-axis direction is set to S, it is preferable that Wa/S (more specifically, Wa/S 1 ) is equal to or greater than 85% and equal to or less than 115%. Thus, it is possible to reduce the width of the resonator element  2  along the X-axis direction and to increase the strength of the vibrating arms  5  and  6  and a Q value. In addition, from the viewpoint of increasing the conductivity of the driving electrodes  84  and  85  and adhesiveness with respect to the quartz crystal substrate  3 , it is preferable that each of the driving electrodes  84  and  85  includes a base layer and a covering layer which is laminated on the base layer on the opposite side to the vibrating arms  5  and  6 . In this case, when a thickness of the base layer is set to Tb and a thickness of the covering layer is set to Tc, it is preferable that the relation of Tb/Tc≦4/7 is satisfied. Thus, even when the base layer has a phase transition point on the low-temperature side in an operation temperature range, it is possible to decrease a CI value on the low-temperature side in the operation temperature range. 
     From the viewpoint of securing the conductivity and corrosion resistance of the driving electrodes  84  and  85 , it is preferable to use a metal having excellent conductivity and corrosion resistance, for example, gold as a material of the covering layer. On the other hand, from the viewpoint of increasing adhesiveness between the covering layer and the quartz crystal substrate  3 , it is preferable to use, for example, chromium as a material of the base layer. 
     When the base layer is formed of chromium, the thickness of the base layer is preferably equal to or greater than 50 Å and equal to or less than 700 Å. In spite of having an excellent function as a base layer, chromium has a phase transition point at a relatively low temperature. Consequently, the thickness of the base layer formed of chromium is reduced to approximately equal to or greater than 50 Å and equal to or less than 700 Å. Thus, it is possible to suppress an increase in a CI value due to a phase transition of chromium of the base layer in a low temperature region, while achieving a function as the base layer. 
     In addition, when a fundamental wave Q value is set to Q 1 , a harmonic Q value is set to Q 2 , the crystal impedance of a fundamental wave is set to R 1 , and the crystal impedance of a harmonic is set to R 2 , it is preferable to satisfy the relation of Q 1 &lt;Q 2  and the relation of R 1 &lt;R 2 . Thus, it is possible to efficiently reduce a vibration in a high-order mode. 
     Package 
     The package  9  shown in  FIGS. 1 and 2  includes a box-shaped base  91  having a concave portion  911 , which is opened on the top surface, and a plate-shaped lid  92  bonded to the base  91  so as to close the opening of the concave portion  911 . The package  9  has an accommodation space formed by closing the concave portion  911  with the lid  92 , and the resonator element  2  is accommodated in the accommodation space in an airtight manner. In addition, the concave portion  911  is provided with a stepped portion  912 , and the resonator element  2  is fixed to the stepped portion  912  through a conductive adhesive  11 , which is formed by mixing a conductive filler into an epoxy-based or acrylic-based resin, for example, in the second base portion  42  of the base portion  4 . 
     Meanwhile, the accommodation space may be in a decompressed (preferably, vacuum) state, or inert gas, such as nitrogen, helium, and argon, may fill the accommodation space. Thus, the vibration characteristics of the resonator element  2  are improved. 
     Materials of the base  91  are not particularly limited and various ceramics, such as aluminum oxide, can be used. In addition, although materials of the lid  92  are not particularly limited, it is preferable to use a member having a linear expansion coefficient similar to that of the material of the base  91 . For example, when the above-described ceramic is used as a material of the base  91 , it is preferable to use an alloy, such as Kovar. Meanwhile, bonding of the base  91  and the lid  92  is not particularly limited. For example, the base  91  and the lid  92  may be bonded to each other through an adhesive or may be bonded to each other by seam welding or the like. In addition, connecting terminals  951  and  961  are formed in the stepped portion  912  of the base  91 . Although not shown in the drawing, the first driving electrodes  84  of the resonator element  2  are drawn up to the second base portion  42  of the base portion  4 , and are electrically connected to the connecting terminal  951  through the conductive adhesive  11  in the corresponding portion. Similarly, although not shown in the drawing, the second driving electrodes  85  of the resonator element  2  are drawn up to the second base portion  42  of the base portion  4 , and are electrically connected to the connecting terminal  961  through the conductive adhesive  11  in the corresponding portion. 
     In addition, the connecting terminal  951  is electrically connected to an external terminal (not shown), which is formed on the bottom surface of the base  91 , through a through electrode (not shown) passing through the base  91 . The connecting terminal  961  is electrically connected to an external terminal (not shown), which is formed on the bottom surface of the base  91 , through a through electrode (not shown) passing through the base  91 . 
     Materials of the connecting terminals  951  and  961 , the through electrode, and the external terminals are not particularly limited as long as the materials have conductivity. For example, the terminals and the electrode can be formed of a metal coating that is formed by laminating a coat, such as nickel (Ni), gold (Au), silver (Ag), or copper (Cu), on a metallization layer (base layer), such as chromium (Cr) or tungsten (W). 
     According to the resonator element  2  described above, the distal ends of the vibrating arms  5  and  6  are provided with the weight portions  59  and  69 . Thus, it is possible to increase the strength of the vibrating arms  5  and  6  by increasing the widths of the arm portions  50  and  60  of the vibrating arms  5  and  6  without increasing the lengths of the vibrating arms  5  and  6 , and to increase a Q value by reducing thermoelastic loss. As a result, it is possible to increase impact resistance while achieving a reduction in size, and to decrease a crystal impedance (CI) value. 
     In particular, since η is optimized, it is possible to efficiently increase the strength of the vibrating arms  5  and  6  and a Q value. 
     Thus, in the resonator element  2 , it is possible to increase impact resistance while achieving a reduction in size, and to realize a CI value which is smaller than 50 kΩ. 
     In addition, the resonator  1  including such a resonator element  2  has an excellent reliability. 
     2. Oscillator 
     Subsequently, an oscillator to which the resonator element according to the invention (oscillator according to the invention) is applied will be described. 
       FIG. 9  is a cross-sectional view showing an example of the oscillator according to the invention. 
     An oscillator  10  shown in  FIG. 9  includes the resonator element  2  and an IC chip (chip component)  80  for driving the resonator element  2 . Hereinafter, the oscillator  10  will be described focusing on the differences from the resonator described above, and a description of the same matters will be omitted. 
     A package  9 A includes a box-shaped base  91 A having a concave portion  911 A, and a plate-shaped lid  92  that closes the opening of the concave portion  911 A. 
     The concave portion  911 A of the base  91 A is provided with a stepped portion  912 A. 
     A connecting terminal (not shown) is formed in the stepped portion  912 A. In addition, the IC chip  80  is disposed on the bottom surface of the concave portion  911 A (surface on the bottom side with respect to the stepped portion  912 A). The IC chip  80  includes a driving circuit (oscillation circuit) for controlling the driving of the resonator element  2 . When the resonator element  2  is driven by the IC chip  80 , it is possible to extract a signal having a predetermined frequency. In addition, a plurality of internal terminals (not shown) electrically connected to the IC chip  80  through a wire are formed on the bottom surface of the concave portion  911 A. The plurality of internal terminals include terminals electrically connected to an external terminal (not shown), which is formed on the bottom surface of the package  9 A, through a via hole which is not shown in the drawing and is formed in the base  91 A, and terminals electrically connected to a connecting terminal (not shown) through a via hole and a wire which are not shown in the drawing. 
     Meanwhile, in  FIG. 9 , the configuration in which the IC chip  80  is disposed within an accommodation space has been described, but the arrangement of the IC chip  80  is not particularly limited. For example, the IC chip may be disposed on the outer side of the package  9 A (on the bottom surface of the base). According to such an oscillator  10 , it is possible to exhibit an excellent reliability. 
     3. Electronic Device 
     Subsequently, an electronic device to which the resonator element according to the invention is applied (electronic device according to the invention) will be described in detail with reference to  FIGS. 10 to 12 . 
       FIG. 10  is a perspective view showing a configuration of a mobile (or notebook) personal computer to which an electronic device including the resonator element according to the invention is applied. In  FIG. 10 , a personal computer  1100  is configured to include a main body  1104  having a keyboard  1102  and a display unit  1106  having a display portion  100 , and the display unit  1106  is supported so as to be rotatable with respect to the main body  1104  through a hinge structure. The resonator  1  that functions as a filter, a resonator, a reference clock, and the like is built into the personal computer  1100 .  FIG. 11  is a perspective view showing a configuration of a mobile phone (PHS is also included) to which an electronic device including the resonator element according to the invention is applied. In  FIG. 11 , a mobile phone  1200  includes a plurality of operation buttons  1202 , an earpiece  1204 , and a mouthpiece  1206 , and a display portion  100  is disposed between the operation buttons  1202  and the earpiece  1204 . The resonator  1  that functions as a filter, a resonator, and the like is built into the mobile phone  1200 . 
       FIG. 12  is a perspective view showing the configuration of a digital still camera to which an electronic device including the resonator element according to the invention is applied. Meanwhile, connection with an external device is simply shown in  FIG. 12 . Here, a silver halide photography film is exposed to light according to an optical image of a subject in a typical camera, while the digital still camera  1300  generates an imaging signal (image signal) by performing photoelectric conversion of an optical image of a subject using an imaging element, such as a charge coupled device (CCD). 
     A display portion is provided on the back of a case (body)  1302  in the digital still camera  1300 , so that display based on the imaging signal of the CCD is performed. The display portion functions as a viewfinder that displays a subject as an electronic image. In addition, a light receiving unit  1304  including an optical lens (imaging optical system), a CCD, and the like is provided on the front side (back side in  FIG. 12 ) of the case  1302 . 
     When a photographer checks a subject image displayed on the display portion and presses a shutter button  1306 , an imaging signal of the CCD at that point in time is transferred and stored in a memory  1308 . In addition, in the digital still camera  1300 , a video signal output terminal  1312  and an input/output terminal for data communication  1314  are provided on the side surface of the case  1302 . In addition, as shown in  FIG. 12 , a television monitor  1430  is connected to the video signal output terminal  1312  and a personal computer  1440  is connected to the input/output terminal for data communication  1314  when necessary. Further, an imaging signal stored in the memory  1308  may be output to the television monitor  1430  or the personal computer  1440  by a predetermined operation. The resonator  1  that functions as a filter, a resonator, and the like is built into the digital still camera  1300 . 
     Meanwhile, the electronic device including the resonator element according to the invention can be applied not only to the personal computer (mobile personal computer) shown in  FIG. 10 , the mobile phone shown in  FIG. 11 , and the digital still camera shown in  FIG. 12  but also to an ink jet type discharge apparatus (for example, an ink jet printer), a laptop type personal computer, a television, a video camera, a video tape recorder, a car navigation apparatus, a pager, an electronic organizer (an electronic organizer with a communication function is also included), an electronic dictionary, an electronic calculator, an electronic game machine, a word processor, a workstation, a video phone, a television monitor for security, electronic binoculars, a POS terminal, medical equipment (for example, an electronic thermometer, a sphygmomanometer, a blood sugar meter, an electrocardiographic measurement device, an ultrasonic diagnostic apparatus, and an electronic endoscope), a fish detector, various measurement apparatuses, instruments (for example, instruments for vehicles, aircraft, and ships), and a flight simulator. 
     4. Moving Object 
     Subsequently, a moving object to which the resonator element according to the invention is applied will be described in detail with reference to  FIG. 13 . 
       FIG. 13  is a perspective view showing a configuration of a moving object (vehicle) to which an electronic device including the resonator element according to the invention is applied. In  FIG. 13 , a moving object  1500  includes a car body  1501  and four wheels  1502 , and is configured to rotate the wheels  1502  using a power source (engine), not shown in the drawing, which is provided in the car body  1501 . The oscillator  10  (resonator element  2 ) is built into the moving object  1500 . 
     According to such an electronic device, it is possible to exhibit an excellent reliability. 
     Meanwhile, the moving object including the resonator element according to the invention is not limited to a vehicle, and can also be applied to, for example, other vehicles such as a motorcycle or a railroad car, an aircraft, a ship, a spacecraft, and the like. 
     While the resonator element, the resonator, the oscillator, the electronic device, and the moving object according to the invention have been described with reference to the embodiments shown in the drawings, the invention is not limited thereto, and the configuration of each portion may be replaced with an arbitrary configuration having the same function. In addition, other arbitrary structures may be added to the invention. In addition, the embodiments may be appropriately combined. In addition, the resonator element is not limited to the oscillator, and can also be applied to, for example, a sensor such as a gyro sensor.