Patent Publication Number: US-2023152269-A1

Title: Sensor element and gas sensor

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a sensor element and a gas sensor. 
     2. Description of the Related Art 
     A sensor element that detects the concentration of a specific gas such as NOx in a measurement-object gas such as an exhaust gas of an automobile is a known art (for example, Patent Literature 1). The sensor element in Patent Literature 1 includes: an elongate element body; an outer electrode disposed on the upper surface of the element body; an outer lead portion; a connector electrode; and a porous layer that covers the outer electrode and the outer lead portion. The outer electrode, the outer lead portion, and the connector electrode are connected in this order and are electrically continuous with each other, and the connector electrode is electrically connected to the outside. The sensor element in Patent Literature 1 further includes a dense layer disposed so as to divide the porous layer in the longitudinal direction of the element body. The dense layer covers the outer lead portion. The dense layer does not easily allow moisture to pass therethrough. Therefore, even when moisture in the measurement-object gas moves through the porous layer by capillary action, the presence of the dense layer prevents the moisture from reaching the connector electrode. A method for producing the above sensor element that is described in Patent Literature 1 includes: forming electrodes, green porous layers, and green dense layers on a plurality of green ceramic sheets corresponding to the element body by screen printing; stacking the plurality of green ceramic sheets; and firing the stacked green ceramic sheets. 
     CITATION LIST 
     Patent Literature 
     
         
         PTL1: International Publication No. WO2019/155865 
       
    
     SUMMARY OF THE INVENTION 
     As for the sensor element including the dense layer as described in Patent Literature 1, cracking may occur in the sensor element. It is therefore desirable to reduce the occurrence of cracking in the sensor element. 
     The present invention has been made to solve the foregoing problem, and it is a main object to further reduce the occurrence of cracking in a sensor element. 
     To achieve the above main object, the present invention employs the following means. 
     The sensor element of the present invention is a sensor element for detecting a specific gas concentration in a measurement-object gas, the sensor element including: an elongate element body that includes a solid electrolyte layer and has a shape including at least one side surface extending in a longitudinal direction and forward and rear ends that are ends opposite to each other in the longitudinal direction; at least one connector electrode that is disposed on a rear end side of any of the at least one side surface and provided for electrical continuity with the outside of the sensor element; a porous layer that has a porosity of 10% or more and covers at least a forward end side of the side surface on which the connector electrode is disposed; a dense layer that is disposed on the side surface so as to divide the porous layer in the longitudinal direction or to be located rearward of the porous layer, is located forward of the connector electrode, and has a porosity of less than 10%; and an intermediate layer disposed at least between the dense layer and the element body, wherein, when thermal expansion coefficients of the solid electrolyte layer, the dense layer, and the intermediate layer in a temperature range of from 20° C. to 1360° C. are denoted by thermal expansion coefficients Ea, Eb, and Ec, respectively, the ratio Ea/Eb is more than 1.0 and 5.0 or less, and Ea&gt;Ec&gt;Eb is satisfied. 
     The sensor element includes the solid electrolyte layer, the dense layer, and the intermediate layer. The ratio Ea/Eb of the thermal expansion coefficient Ea of the solid electrolyte layer to the thermal expansion coefficient Eb of the dense layer is more than 1.0 and 5.0 or less, and the thermal expansion coefficient of the solid electrolyte layer is relatively close to the thermal expansion coefficient of the dense layer. Moreover, the intermediate layer is present at least between the dense layer and the solid electrolyte layer, and the thermal expansion coefficient Ec of the intermediate layer satisfies Ea&gt;Ec&gt;Eb. Specifically, the intermediate layer whose thermal expansion coefficient Ec is between the thermal expansion coefficient of the solid electrolyte layer and the thermal expansion coefficient of the dense layer is present between them. Since the solid electrolyte layer, the dense layer, and the intermediate layer satisfy the above positional relation and the relations between the thermal expansion coefficients Ea to Ec, the intermediate layer reduces stress caused by the difference between the thermal expansion coefficient Ea of the solid electrolyte layer and the thermal expansion coefficient Eb of the dense layer when the sensor element is heated during use. When the stress is generated in the sensor element, cracking is likely to occur. However, in the above sensor element, since the stress is reduced, the occurrence of cracking is reduced. 
     In the sensor element of the present invention, when the mean value of the thermal expansion coefficient Ea and the thermal expansion coefficient Eb is denoted by Ed (=(Ea+Eb)/2), formula (1) below may be satisfied. In this case, the thermal expansion coefficient Ec is relatively close to the median Ed of the thermal expansion coefficients Ea and Eb. Specifically, the thermal expansion coefficient Ec is not excessively close to the thermal expansion coefficient Ea and not excessively close to the thermal expansion coefficient Eb. Therefore, the stress generated when the sensor element is heated is further reduced, and the occurrence of cracking is further reduced. 
         Ed− 0.8×( Ed−Eb )&lt; Ec&lt;Ed+ 0.8×( Ea−Ed )  (1)
 
     In the sensor element of the present invention, the ratio Ea/Eb may be 3.0 or less. In this case, the thermal expansion coefficient Ea of the solid electrolyte layer and the thermal expansion coefficient Eb of the dense layer are closer to each other, so that the occurrence of cracking in the sensor element is further reduced. 
     In the sensor element of the present invention, the intermediate layer may have a thickness T of 1 μm or more. In this case, the effect of the presence of the intermediate layer in reducing the occurrence of cracking in the sensor element is obtained more reliably. The thickness T of the intermediate layer may be 10 μm or less. 
     In the sensor element of the present invention, the solid electrolyte layer may contain zirconia as a main component, and the dense layer may contain alumina as a main component. The intermediate layer may contain zirconia and alumina. The main component as used herein means a component with the highest content and is specifically a component with the highest volume ratio. 
     In the sensor element of the present invention, the sensor element may include: a detection portion including a plurality of electrodes disposed on a forward end side of the element body and used to detect the specific gas concentration in the measurement-object gas; and an outer lead portion that is disposed on the side surface on which the connector electrode is disposed and provides electrical continuity between any of the plurality of electrodes and the connector electrode. The porous layer may cover at least part of the outer lead portion. In this case, the porous layer may fully cover a portion of the outer lead portion that is not covered with the dense layer. The sensor element of the present invention may include an outer electrode that is one of the plurality of electrodes included in the detection portion and that is electrically continuous with the connector electrode through the outer lead portion and disposed on the side surface on which the connector electrode is disposed. In this case, the porous layer may cover the outer electrode. 
     The gas sensor of the present invention includes the sensor element in any of the above modes. Therefore, the gas sensor has the same effect as the effect of the above-described sensor element of the present invention, e.g., the effect of reducing the occurrence of cracking in the sensor element. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a cross-sectional view showing the manner of attaching a gas sensor  10  to a pipe  58 . 
         FIG.  2    is a perspective view of a sensor element  20 . 
         FIG.  3    is a cross-sectional view taken along A-A in  FIG.  2   . 
         FIG.  4    is a top view of the sensor element  20 . 
         FIG.  5    is a bottom view of the sensor element  20 . 
         FIG.  6    is an enlarged view around an intermediate layer  98  in  FIG.  3   . 
         FIG.  7    is a partial cross-sectional view around the intermediate layer  98  in a cross-section taken along B-B in  FIG.  4   . 
         FIG.  8    is a partial cross-sectional view showing an intermediate layer  98  in a modification. 
         FIG.  9    is a partial cross-sectional view showing an intermediate layer  98  in a modification. 
         FIG.  10    is a partial cross-sectional view showing an intermediate layer  98  in a modification. 
         FIG.  11    is a perspective view of a sensor element  20  including an intermediate layer  99 . 
         FIG.  12    is a bottom view showing a second dense layer  95  and gap regions  96  in a modification. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Next, embodiments of the present invention will be described using the drawings.  FIG.  1    is a vertical cross-sectional view showing the manner of attaching, to a pipe  58 , a gas sensor  10  including a sensor element  20  in an embodiment of the present invention.  FIG.  2    is a perspective view of the sensor element  20  when it is viewed from the upper right front.  FIG.  3    is a cross-sectional view taken along A-A in  FIG.  2   .  FIG.  4    is a top view of the sensor element  20 .  FIG.  5    is a bottom view of the sensor element  20 .  FIG.  6    is an enlarged view around an intermediate layer  98  in  FIG.  3   .  FIG.  7    is a partial cross-sectional view around the intermediate layer  98  in a cross-section taken along B-B in  FIG.  4   . In the present embodiment, as shown in  FIGS.  2  and  3   , the longitudinal direction of an element body  60  of the sensor element  20  is defined as a forward-rearward direction (lengthwise direction) of the element body  60 , and the stacking direction (thickness direction) of the element body  60  is defined as an upward-downward direction. A direction perpendicular to the forward-rearward direction and the upward-downward direction is defined as a left-right direction (width direction). 
     As shown in  FIG.  1   , the gas sensor  10  includes an assembly  15 , a bolt  47 , an external cylinder  48 , a connector  50 , lead wires  55 , and a rubber stopper  57 . The assembly  15  includes the sensor element  20 , a protective cover  30 , and an element-sealing member  40 . The gas sensor  10  is attached to the pipe  58  such as an exhaust gas pipe of a vehicle and used to measure the concentration of a specific gas (a specific gas concentration) such as NOx or O 2  contained in the exhaust gas used as a measurement-object gas. In the present embodiment, the gas sensor  10  measures the concentration of NOx as the specific gas concentration. The sensor element  20  has opposite ends (forward and rear ends) in the longitudinal direction, and the forward end side is the side exposed to the measurement-object gas. 
     As shown in  FIG.  1   , the protective cover  30  includes a bottomed cylindrical inner protective cover  31  that covers the forward end side of the sensor element  20  and a bottomed cylindrical outer protective cover  32  that covers the inner protective cover  31 . A plurality of holes for allowing circulation of the measurement-object gas are formed in each of the inner and outer protective covers  31  and  32 . An element chamber  33  is formed as a space surrounded by the inner protective cover  31 , and a fifth surface  60   e  (forward end surface) of the sensor element  20  is disposed inside the element chamber  33 . 
     The element-sealing member  40  is a member for sealing and fixing the sensor element  20 . The element-sealing member  40  includes: a cylindrical member  41  including a main metal fitting  42  and an inner cylinder  43 ; insulators  44   a  to  44   c ; powder compacts  45   a  and  45   b ; and a metal ring  46 . The sensor element  20  is located on the center axis of the element-sealing member  40  and pierces through the element-sealing member  40  in the forward-rearward direction. 
     The main metal fitting  42  is a cylindrical metallic member. The main metal fitting  42  has a thick-walled portion  42   a  located on the forward side and having an inner diameter smaller than that of the rear side. The protective cover  30  is attached to a portion of the main metal fitting  42  that is on the same side as the forward end of the sensor element  20  (i.e., the forward side). The rear end of the main metal fitting  42  is welded to a flange portion  43   a  of the inner cylinder  43 . A part of the inner circumferential surface of the thick-walled portion  42   a  is formed as a bottom surface  42   b  that is a step surface. The bottom surface  42   b  bears the insulator  44   a  such that the insulator  44   a  does not protrude forward. 
     The inner cylinder  43  is a cylindrical metallic member and has the flange portion  43   a  at its forward end. The inner cylinder  43  and the main metal fitting  42  are welded and fixed to each other so as to be coaxial with each other. The inner cylinder  43  has a reduced diameter portion  43   c  for pressing the powder compact  45   b  in a direction toward the center axis of the inner cylinder  43  and a reduced diameter portion  43   d  for pressing the insulators  44   a  to  44   c  and the powder compacts  45   a  and  45   b  in a downward direction in  FIG.  1    through the metal ring  46 . 
     The insulators  44   a  to  44   c  and the powder compacts  45   a  and  45   b  are disposed between the inner circumferential surface of the cylindrical member  41  and the sensor element  20 . The insulators  44   a  to  44   c  serve as supporters for the powder compacts  45   a  and  45   b . Examples of the material of the insulators  44   a  to  44   c  include ceramics such as alumina, steatite, zirconia, spinel, cordierite, and mullite and glass. The powder compacts  45   a  and  45   b  are formed, for example, by molding a powder and each serve as a sealing medium. Examples of the material of the powder compacts  45   a  and  45   b  include talc and ceramic powders such as alumina powder and boron nitride powder, and the powder compacts  45   a  and  45   b  may each contain at least one of these materials. The powder compact  45   a  is filled between the insulators  44   a  and  44   b , sandwiched therebetween from opposite sides (forward and rear sides) in the axial direction, and pressed by the insulators  44   a  and  44   b . The powder compact  45   b  is filled between the insulators  44   b  and  44   c , sandwiched therebetween from opposite sides (forward and rear sides) in the axial direction, and pressed by the insulators  44   b  and  44   c . The insulators  44   a  to  44   c  and the powder compacts  45   a  and  45   b  are sandwiched between the bottom surface  42   b  of the thick-walled portion  42   a  of the main metal fitting  42  and both the reduced diameter portion  43   d  and the metal ring  46  and pressed from the forward and rear sides. The pressing force applied by the reduced diameter portions  43   c  and  43   d  causes the powder compacts  45   a  and  45   b  to be compressed between the cylindrical member  41  and the sensor element  20 , and the powder compacts  45   a  and  45   b  close the communication between the element chamber  33  in the protective cover  30  and a space  49  in the external cylinder  48  and fix the sensor element  20 . 
     The bolt  47  is fixed to the outer side of the main metal fitting  42  so as to be coaxial with the main metal fitting  42 . The bolt  47  has a male thread portion formed on the outer circumferential surface of the bolt  47 . The male thread portion is inserted into a fixing member  59  that is welded to the pipe  58  and has a female thread portion formed on the inner circumferential surface of the fixing member  59 . In this manner, the gas sensor  10  can be fixed to the pipe  58  with the forward end side of the sensor element  20  of the gas sensor  10  and the protective cover  30  protruding into the pipe  58 . 
     The external cylinder  48  is a cylindrical metallic member and covers the inner cylinder  43 , the rear end side of the sensor element  20 , and the connector  50 . An upper portion of the main metal fitting  42  is inserted into the external cylinder  48 . The lower end of the external cylinder  48  is welded to the main metal fitting  42 . The plurality of lead wires  55  connected to the connector  50  are drawn from the upper end of the external cylinder  48  to the outside. The connector  50  is in contact with and electrically connected to upper connector electrodes  71  and lower connector electrodes  72  that are disposed on rear end portions of respective surfaces of the sensor element  20 . The lead wires  55  are electrically continuous with electrodes  64  to  68  and a heater  69  of the sensor element  20  through the connector  50 . The gap between the external cylinder  48  and the lead wires  55  is sealed by the rubber stopper  57 . The space  49  inside the external cylinder  48  is filled with a reference gas. A sixth surface  60   f  (rear end surface) of the sensor element  20  is disposed inside the space  49 . 
     As shown in  FIGS.  2  to  7   , the sensor element  20  includes the element body  60 , a detection portion  63 , the heater  69 , the upper connector electrodes  71 , the lower connector electrodes  72 , a protective layer  80 , a first dense layer  92 , a second dense layer  95 , and the intermediate layer  98 . The element body  60  includes a layered body prepared by stacking a plurality of oxygen-ion-conductive solid electrolyte layers. As shown in  FIG.  3   , in the present embodiment, the element body  60  includes six solid electrolyte layers  78   a  to  78   f . The solid electrolyte layers  78   a  to  78   f  are made of a ceramic containing zirconia (ZrO 2 ) as a main component. The element body  60  has an elongate rectangular parallelepiped shape whose longitudinal direction extends in the forward-rearward direction and has first to sixth surfaces  60   a  to  60   f  that are the upper, lower, left, right, forward, and rear outer surfaces of the element body  60 . The first to fourth surfaces  60   a  to  60   d  are surfaces extending in the longitudinal direction of the element main body  60  and correspond to the side surfaces of the element main body  60 . The fifth surface  60   e  is the forward end surface of the element body  60 , and the sixth surface  60   f  is the rear end surface of the element body  60 . As for the dimensions of the element body  60 , for example, the length may be from 25 mm to 100 mm inclusive. The width may be from 2 mm to 10 mm inclusive, and the thickness may be from 0.5 mm to 5 mm inclusive. The element body  60  has formed therein: a measurement-object gas inlet  61  having an opening on the fifth surface  60   e  to introduce the measurement-object gas into the element body  60 ; and a reference gas inlet  62  having an opening on the sixth surface  60   f  to introduce the reference gas (air in the present embodiment) used as a reference for detection of the specific gas concentration into the element body  60 . 
     The detection portion  63  is used to detect the specific gas concentration in the measurement-object gas. The detection portion  63  includes a plurality of electrodes disposed on a forward end side of the element body  60 . In the present embodiment, the detection portion  63  includes an outer electrode  64  disposed on the first surface  60   a  and further includes an inner main pump electrode  65 , an inner auxiliary pump electrode  66 , a measurement electrode  67 , and a reference electrode  68  that are disposed inside the element body  60 . The inner main pump electrode  65  and the inner auxiliary pump electrode  66  are disposed on the inner circumferential surface of an internal space of the element body  60  and each have a tunnel-like structure. 
     The principle of the detection of the specific gas concentration in the measurement-object gas by the detection portion  63  is well known, and its detailed description will be omitted. The detection portion  63  detects the specific gas concentration, for example, in the following manner. The detection portion  63  pumps oxygen in the measurement-object gas around the inner main pump electrode  65  to the outside (the element chamber  33 ) or pumps oxygen from the outside according to a voltage applied between the outer electrode  64  and the inner main pump electrode  65 . Moreover, the detection portion  63  pumps oxygen in the measurement-object gas around the inner auxiliary pump electrode  66  to the outside (the element chamber  33 ) or pumps oxygen from the outside according to a voltage applied between the outer electrode  64  and the inner auxiliary pump electrode  66 . This allows the measurement-object gas whose oxygen concentration has been adjusted to a prescribed value to reach the measurement electrode  67 . The measurement electrode  67  functions as a NOx reduction catalyst and reduces the specific gas (NOx) in the measurement-object gas that has reached the measurement electrode  67 . Then the detection portion  63  generates an electric signal corresponding to an electromotive force generated between the measurement electrode  67  and the reference electrode  68  according to the oxygen concentration in the reduced gas or corresponding to a current flowing between the measurement electrode  67  and the outer electrode  64  according to the electromotive force. The electric signal generated by the detection portion  63  is a signal indicating a value corresponding to the specific gas concentration in the measurement-object gas (a value from which the specific gas concentration can be derived) and corresponds to the detection value detected by the detection portion  63 . 
     The heater  69  is an electric resistor disposed inside the element body  60 . When electric power is supplied to the heater  69  from the outside, the heater  69  generates heat and heats the element body  60 . The heater  69  can heat the solid electrolyte layers  78   a  to  78   f  included in the element body  60 , can keep them hot, and can adjust their temperature to the temperature at which the solid electrolyte layers  78   a  to  78   f  are activated (e.g., 800° C.) 
     The upper connector electrodes  71  and the lower connector electrodes  72  are disposed on rear end-side portions of side surfaces of the element body  60  and are electrodes that allow electrical continuity between the element body  60  and the outside. The upper and lower connector electrodes  71  and  72  are not covered with the protective layer  80  and are exposed. In the present embodiment, the upper connector electrodes  71  include four upper connector electrodes  71   a  to  71   d  arranged in the left-right direction and disposed on the rear end side of the first surface  60   a . The lower connector electrodes  72  include four lower connector electrodes  72   a  to  72   d  arranged in the left-right direction and disposed on the rear end side of the second surface  60   b  (lower surface) opposite to the first surface  60   a  (upper surface). Each of the connector electrodes  71   a  to  71   d  and  72   a  to  72   d  is electrically continuous with a corresponding one of the heater  69  and the plurality of electrodes  64  to  68  of the detection portion  63 . In the present embodiment, the upper connector electrode  71   a  is electrically continuous with the measurement electrode  67 , and the upper connector electrode  71   b  is electrically continuous with the outer electrode  64 . The upper connector electrode  71   c  is electrically continuous with the inner auxiliary pump electrode  66 , and the upper connector electrode  71   d  is electrically continuous with the inner main pump electrode  65 . The lower connector electrodes  72   a  to  72   c  are electrically continuous with the heater  69 , and the lower connector electrode  72   d  is electrically continuous with the reference electrode  68 . The upper connector electrode  71   b  is electrically continuous with the outer electrode  64  through an outer lead wire  75  disposed on the first surface  60   a  (see  FIGS.  3  and  4   ). Each of the other connector electrodes is electrically continuous with a corresponding electrode or the heater  69  through a lead wire disposed inside the element body  60 , a through hole, etc. 
     The outer lead wire  75  is a conductor containing a noble metal such as platinum (Pt) or a high-melting point metal such as tungsten (W) or molybdenum (Mo). Preferably, the outer lead wire  75  is a cermet conductor containing a noble metal or a high-melting point metal and the oxygen-ion-conductive solid electrolyte contained in the element body  60  (zirconia in the present embodiment). In the present embodiment, the outer lead wire  75  is a cermet conductor containing platinum and zirconia. The porosity of the outer lead wire  75  may be, for example, from 5% to 40% inclusive. The line width of the outer lead wire  75  (its thickness, i.e., the width in the left-right direction) is, for example, from 0.1 mm to 1.0 mm inclusive. An unillustrated insulating layer for insulation between the outer lead wire  75  and the solid electrolyte layer  78   a  of the element body  60  may be disposed between the outer lead wire  75  and the first surface  60   a  of the element body  60 . 
     The protective layer  80  includes inner porous layers  81  and an outer porous layer  85 . The inner porous layers  81  are porous bodies that cover at least the forward end side of the side surfaces of the element body  60  on which the upper and lower connector electrodes  71  and  72  are disposed, i.e., of the first and second surfaces  60   a  and  60   b . In the present embodiment, the inner porous layers  81  cover the first and second surfaces  60   a  and  60   b . The outer porous layer  85  is a porous body that covers the forward end side of the element body  60 . The outer porous layer  85  is disposed on the outer side of the inner porous layers  81 . 
     The inner porous layers  81  include a first inner porous layer  83  that covers the first surface  60   a  and a second inner porous layer  84  that covers the second surface  60   b . The first inner porous layer  83  covers the entire region, from the forward end to the rear end, of the first surface  60   a  on which the upper connector electrodes  71   a  to  71   d  are disposed, except for the regions in which the first dense layer  92  and the upper connector electrodes  71  are present (see  FIGS.  2  to  4   ). The width of the first inner porous layer  83  in the left-right direction is the same as the width of the first surface  60   a  in the left-right direction, and the first inner porous layer  83  covers the first surface  60   a  so as to extend from the left edge of the first surface  60   a  to its right edge. Since the first dense layer  92  is present, the first inner porous layer  83  is divided in the longitudinal direction into a forward end-side portion  83   a  located forward of the first dense layer  92  and a rear end-side portion  83   b  located rearward of the first dense layer  92 . The first inner porous layer  83  covers at least partially the outer electrode  64  and the outer lead wire  75 . In the present embodiment, as shown in  FIGS.  3  and  4   , the first inner porous layer  83  covers the entire outer electrode  64  and covers the entire portion of the outer lead wire  75  in which the first dense layer  92  is not present. The first inner porous layer  83  protects the outer electrode  64  and the outer lead wire  75  from components of the measurement-object gas such as sulfuric acid and plays a role in preventing corrosion of the outer electrode  64  and the outer lead wire  75 . 
     The second inner porous layer  84  covers the entire region, from the forward end to the rear end, of the second surface  60   b  on which the lower connector electrodes  72   a  to  72   d  are disposed, except for the regions in which the second dense layer  95  and the lower connector electrodes  72  are present (see  FIGS.  2 ,  3 , and  5   ). The width of the second inner porous layer  84  in the left-right direction is the same as the width of the second surface  60   b  in the left-right direction, and the second inner porous layer  84  covers the second surface  60   b  so as to extend from the left edge of the second surface  60   b  to its right edge. The presence of the second dense layer  95  divides the second inner porous layer  84  into a forward end-side portion  84   a  located forward of the second dense layer  95  in the longitudinal direction and a rear end-side portion  84   b  located rearward of the second dense layer  95 . 
     The outer porous layer  85  covers the first to fifth surfaces  60   a  to  60   e . The outer porous layer  85  covers the inner porous layers  81  to thereby cover the first surface  60   a  and the second surface  60   b . The length of the outer porous layer  85  in the forward-rearward direction is shorter than that of the inner porous layers  81 . Unlike the inner porous layers  81 , the outer porous layer  85  covers only the forward end of the element body  60  and a region around the forward end. In this case, the outer porous layer  85  covers a portion of the element body  60  that surrounds the electrodes  64  to  68  of the detection portion  63 , i.e., a portion of the element body  60  that is disposed inside the element chamber  33  and exposed to the measurement-object gas. In this manner, the outer porous layer  85  plays a role in preventing the occurrence of cracking in the element body  60  that are caused by adhesion of, for example, moisture etc. in the measurement-object gas. 
     The protective layer  80  is formed of, for example, a ceramic porous material such as an alumina porous material, a zirconia porous material, a spinel porous material, a cordierite porous material, a titania porous material, or a magnesia porous material. In the present embodiment, the protective layer  80  is formed of an alumina porous material. The thickness of the first inner porous layer  83  and the thickness of the second inner porous layer  84  may be, for example, 5 μm or more and may be 14 μm or more. The thickness of the first inner porous layer  83  and the thickness of the second inner porous layer  84  may be 40 μm or less and may be 23 μm or less. The thickness of the outer porous layer  85  is, for example, from 40 μm to 800 μm inclusive. The porosity of the protective layer  80  is 10% or more. The protective layer  80  covers the outer electrode  64  and the measurement-object gas inlet  61 . However, when the porosity is 10% or more, the measurement-object gas can pass through the protective layer  80 . The porosity of the inner porous layers  81  may be from 10% to 50% inclusive. The porosity of the outer porous layer  85  may be from 10% to 85% inclusive. The porosity of the outer porous layer  85  may be higher than the porosity of the inner porous layers  81 . 
     The porosity of the inner porous layers  81  is a value derived as follows using an image (SEM image) obtained by observation using a scanning electron microscope (SEM). First, the sensor element  20  is cut in the thickness direction of the inner porous layers  81 , and a cross section of one of the inner porous layers  81  is used as an observation surface. The cross-section is embedded in a resin and polished to obtain an observation sample. Next, the magnification of the SEM is set to 1000× to 10000×, and an image of the observation surface of the observation sample is captured to thereby obtain an SEM image of the inner porous layer  81 . Next, the image obtained is subjected to image analysis, and a threshold value is determined by a discriminant analysis method (Otsu&#39;s binarization) using a brightness distribution obtained from the brightness data of pixels in the image. Using the determined threshold value, the pixels in the image are binarized and classified into object portions and pore portions, and the area of the object portions and the area of the pore portions are computed. Then the ratio of the area of the pore portions to the total area (the total area of the object portions and the pore portions) is computed as a porosity (unit: %). The porosity of the outer porous layer  85  and the porosities of the first dense layer  92 , the second dense layer  95 , and the intermediate layer  98  described later are computed in the same manner as described above. 
     The first dense layer  92  and the second dense layer  95  serve as water intrusion preventing portions that prevent capillary action of water in the longitudinal direction of the element body  60 . The first dense layer  92  is disposed on the first surface  60   a  on which the upper connector electrodes  71  and the first inner porous layer  83  are disposed. The first dense layer  92  is disposed on the first surface  60   a  so as to divide the first inner porous layer  83  into forward and rear portions in the longitudinal direction as described above. The first dense layer  92  is disposed closer to the forward end of the element body  60  than the upper connector electrodes  71 , i.e., disposed forward of the upper connector electrodes  71 . The first dense layer  92  is disposed rearward of the outer electrode  64 . The first dense layer  92  is disposed rearward of all the plurality of electrodes  64  to  68 , including the outer electrode  64 , included in the detection portion  63  (see  FIG.  3   ). The first dense layer  92  is disposed at a position that overlaps the insulator  44   b  in the forward-rearward direction (see  FIG.  1   ). In other words, a region extending from the forward end of the first dense layer  92  to its rear end is located within a region extending from the forward end of the insulator  44   b  to its rear end. The first dense layer  92  plays a role in preventing moisture moved rearward through the forward end-side portion  83   a  by capillary action from passing through the first dense layer  92  to thereby prevent the moisture from reaching the upper connector electrodes  71 . The first dense layer  92  is a dense layer with a porosity of less than 10%. The width of the first dense layer  92  in the left-right direction is the same as the width of the first surface  60   a  in the left-right direction, and the first dense layer  92  covers the first surface  60   a  so as to extend from the left edge of the first surface  60   a  to its right edge. The first dense layer  92  is adjacent to the rear end of the forward end-side portion  83   a . The first dense layer  92  is adjacent to the forward end of the rear end-side portion  83   b . As shown in  FIG.  4   , the first dense layer  92  covers part of the outer lead wire  75 . 
     The second dense layer  95  is disposed on the second surface  60   b  on which the lower connector electrodes  72  and the second inner porous layer  84  are disposed. The second dense layer  95  is disposed on the second surface  60   b  so as to divide the second inner porous layer  84  into forward and rear portions in the longitudinal direction as described above. The second dense layer  95  is disposed closer to the forward end of the element body  60  than the lower connector electrodes  72 , i.e., disposed forward of the lower connector electrodes  72 . The second dense layer  95  is disposed rearward of the outer electrode  64 . The second dense layer  95  is disposed rearward of all the plurality of electrodes  64  to  68 , including the outer electrode  64 , included in the detection portion  63  (see  FIG.  3   ). The second dense layer  95  is disposed at a position that overlaps the insulator  44   b  in the forward-rearward direction (see  FIG.  1   ). In other words, a region extending from the forward end of the second dense layer  95  to its rear end is located within a region extending from the forward end of the insulator  44   b  to its rear end. The second dense layer  95  plays a role in preventing moisture moved rearward through the forward end-side portion  84   a  by capillary action from passing through the second dense layer  95  to thereby prevent the moisture from reaching the lower connector electrodes  72 . The second dense layer  95  is a dense layer with a porosity of less than 10%. The width of the second dense layer  95  in the left-right direction is the same as the width of the second surface  60   b  in the left-right direction, and the second dense layer  95  covers the second surface  60   b  so as to extend from the left edge of the second surface  60   b  to its right edge. The second dense layer  95  is adjacent to the rear end of the forward end-side portion  84   a . The second dense layer  95  is adjacent to the forward end of the rear end-side portion  84   b.    
     The length Le of each of the first dense layer  92  and the second dense layer  95  in the longitudinal direction (see  FIGS.  4  and  5   ) is preferably 0.5 mm or more. When the length Le is 0.5 mm or more, the passage of moisture through the first dense layer  92  and the second dense layer  95  can be prevented sufficiently. The length Le may be 5 mm or more. The length Le may be 25 mm or less and may be 20 mm or less. In the present embodiment, the length Le of the first dense layer  92  and the length Le of the second dense layer  95  are the same but may be different values. 
     The first dense layer  92  and the second dense layer  95  differ from the protective layer  80  in that their porosity is less than 10%. However, a ceramic composed of any of the materials exemplified for the protective layer  80  described above can be used. Specifically, the first dense layer  92  may be a ceramic porous body containing, as a main component, at least one type of ceramic particles selected from alumina particles, zirconia particles, spinel particles, cordierite particles, titania particles, and magnesia particles. In the present embodiment, the first dense layer  92  and the second dense layer  95  are both formed of a ceramic containing alumina as a main component. The thickness of the first dense layer  92  and the thickness of the second dense layer  95  may each be, for example, from 1 μm to 40 μm inclusive. The thickness of the first dense layer  92  and the thickness of the second dense layer  95  may each be 20 μm or less, may be 9 μm or less, and may be 3 μm or less. The porosity of the first dense layer  92  and the porosity of the second dense layer  95  are each preferably 8% or less and more preferably 5% or less. The smaller the porosity, the further the first dense layer  92  and the second dense layer  95  can reduce the capillary action of water in the longitudinal direction of the element body  60 . 
     As shown in  FIGS.  1  to  3 ,  6 , and  7   , the intermediate layer  98  is disposed between the first dense layer  92  and the element body  60 . Although the details will be described later, the intermediate layer  98  plays a role in reducing the occurrence of cracking in the sensor element  20 . As shown in  FIG.  7   , the intermediate layer  98  is located between the first dense layer  92  and the outer lead wire  75  and covers the outer lead wire  75 . Therefore, the first dense layer  92  covers the outer lead wire  75  with the intermediate layer  98  interposed therebetween. As shown in  FIG.  6   , in the present embodiment, the length of the intermediate layer  98  in the forward-rearward direction is the same as the length Le of the first dense layer  92 . Specifically, the intermediate layer  98  is disposed only on the lower side of the first dense layer  92  and not disposed between the first inner porous layer  83  and the element body  60 . As shown in  FIG.  7   , the width of the intermediate layer  98  in the left-right direction is the same as the width of the first surface  60   a  in the left-right direction. Moreover, the width of the intermediate layer  98  in the left-right direction is the same as the width of the first dense layer  92  in the left-right direction. The thickness T of the intermediate layer  98  is, for example, from 1 μm to 40 μm inclusive. Like the inner porous layer  81 , the intermediate layer  98  may have a porosity of 10% or more, i.e., may be a porous body. The porosity of the intermediate layer  98  may be 50% or less. Like the first dense layer  92 , the intermediate layer  98  may have a porosity of less than 10%, i.e., may be dense. The porosity of the intermediate layer  98  may be 8% or less and may be 5% or less. In the present embodiment, the intermediate layer  98  is dense. When the thickness T of the intermediate layer  98  is 1 μm or more, the effect of the presence of the intermediate layer  98  in reducing the occurrence of cracking in the sensor element  20  can be obtained more reliably. The thickness T of the intermediate layer  98  may be 10 μm or less. 
     The thickness T of the intermediate layer  98  may be the thickness T1 of the thinnest portion of the intermediate layer  98  (for example, a portion located directly above the outer lead wire  75  as shown in  FIG.  7   ) or the overall average thickness T2 of the intermediate layer  98 . Even when the thickness T used is the thickness T1 or the thickness T2, the effect of reducing the occurrence of cracking can be obtained more reliably when the thickness T is 1 μm or more. 
     The intermediate layer  98  may be formed, for example, of a ceramic containing, as a main component, at least one type of ceramic particles selected from alumina particles, zirconia particles, spinel particles, cordierite particles, titania particles, and magnesia particles. The intermediate layer  98  may be formed of a noble metal such as platinum. The intermediate layer  98  may be formed of cermet containing the above-described ceramic particles and noble metal particles. Preferably, the intermediate layer  98  contains, as main components, the main component of the solid electrolyte layers  78   a  to  78   f  and the main component of the first dense layer  92 . In the present embodiment, the intermediate layer  98  is a ceramic containing, as main components, zirconia used as the main component of the solid electrolyte layers  78   a  to  78   f  and alumina used as the main component of the first dense layer  92 . 
     A method for producing the gas sensor  10  having the above-described structure will be described below. First a method for producing the sensor element  20  will be described. The method for producing the sensor element  20  includes a production step of producing a green sensor element that is the sensor element  20  before firing and a firing step of firing the green sensor element. In the present embodiment, the outer porous layer  85  is formed by plasma spraying after the firing step. Therefore, the green sensor element produced in the production step does not include a green outer porous layer  85 , and the sensor element  20  after the firing step does not include the outer porous layer  85 . 
     [Production Step] 
     In the production step, the green sensor element that is the sensor element  20  before firing is produced. In the production step, first, six ceramic green sheets (green solid electrolyte layers) corresponding to the solid electrolyte layers  78   a  to  78   f  included in the element body  60  are prepared. The ceramic green sheets are produced, for example, by mixing a solvent, a binder, etc. with a raw material powder containing the material of the solid electrolyte layers  78   a  to  78   f  (a zirconia powder in the present embodiment) to obtain a paste containing the material of the raw material powder as a main component and then forming the paste into a sheet shape. If necessary, through holes, grooves, etc. are punched in the ceramic green sheets to form portions that later become inner spaces of the element body  60  through firing. Next, patterns for green electrodes, green lead wires, green connector electrodes, a green heater, etc. are formed by screen printing on the ceramic green sheets to be used as the solid electrolyte layers  78   a  to  78   f . The green electrodes later become the above-described electrodes  64  to  68  of the detection portion  63  through firing. The green lead wires later become, through firing, the lead wires that connect the electrodes to the upper connector electrodes  71  and the lower connector electrodes  72 . The green lead wires include a lead wire that later becomes the outer lead wire  75  through firing. The green connector electrodes later become the upper connector electrodes  71  and the lower connector electrodes  72  through firing. The green heater later becomes the heater  69  through firing. Moreover, patterns for a green intermediate layer that later becomes the intermediate layer  98  through firing, a green first dense layer that later becomes the first dense layer  92  through firing, and a green first inner porous layer that later becomes the first inner porous layer  83  through firing are formed by screen printing on a surface of the ceramic green sheet that later becomes the solid electrolyte layer  78   a  through firing (a surface that later becomes the first surface  60   a  of the element body  60 ). Similarly, patterns for a green second dense layer that later becomes the second dense layer  95  through firing and a green second inner porous layer that later becomes the second inner porous layer  84  through firing are formed by screen printing on a surface of the ceramic green sheet that later becomes the solid electrolyte layer  78   f  through firing (a surface that later becomes the second surface  60   b  of the element body  60 ). Next, the six ceramic green sheets with the patterns formed thereon are stacked to form a layered body. The layered body is cut into small layer bodies having the same size as the size of the sensor element  20 . These small layered bodies are green sensor elements. The patterns for the green first inner porous layer, the green second inner porous layer, the green intermediate layer, the green first dense layer, and the green second dense layer may be printed after the production of the layered body described above. 
     The paste used to form the green first inner porous layer is, for example, a paste that is prepared by mixing a raw material powder composed of the material of the above-described first inner porous layer  83  (an aluminum powder in the present embodiment), a binder, a solvent, a pore-forming material, etc. and that contains, as a main component, the material of the raw material powder. The paste forming the green second inner porous layer is prepared in the same manner as described above. The paste used to form the green first dense layer is, for example, a paste that is prepared by mixing a raw material powder composed of the material of the above-described first dense layer  92  (an aluminum powder in the present embodiment), a binder, a solvent, etc. and that contains, as a main component, the material of the raw material powder. To control the porosity of the first dense layer  92 , a pore-forming material may be added to the paste. The paste for forming the green second dense layer is prepared in the same manner as above. The paste used to form the green intermediate layer is, for example, a paste that is prepared by mixing a raw material powder composed of the materials of the above-described intermediate layer  98  (an aluminum powder and a zirconia powder in the present embodiment), a binder, a solvent, etc. and that contains, as main components, the materials of the raw material powder. To control the porosity of the intermediate layer  98 , a pore-forming material may be added to the paste. 
     The green first inner porous layer and the green second inner porous layer may be formed using the same paste or using pastes prepared using different raw material powders. The green first dense layer and the green second dense layer may also be formed using the same paste or using pastes prepared using different raw material powders. 
     [Firing Step] 
     Next, the firing step of firing the green sensor element obtained in the production step is performed. In the firing step, the green sensor element is fired at a prescribed firing temperature (e.g., 1360° C.±50° C.), and then the temperature is lowered to room temperature (e.g., 20° C.) after firing. In this manner, the six ceramic green sheets become the solid electrolyte layers  78   a  to  78   f , and the green electrodes become the electrodes  64  to  68 . The green lead wires become the plurality of wires including the outer lead wire  75 , and the green connector electrodes become the upper connector electrodes  71  and the lower connector electrodes  72 . Moreover, the green heater becomes the heater  69 . The green intermediate layer becomes the intermediate layer  98 , and the green first dense layer becomes the first dense layer  92 . The green first inner porous layer becomes the first inner porous layer  83 , and the green second dense layer becomes the second dense layer  95 . The green second inner porous layer becomes the second inner porous layer  84 . The sensor element  20  is obtained through the firing step. 
     In the present embodiment, after the firing step has been performed to produce the sensor element  20 , the outer porous layer  85  is formed by plasma spraying. The plasma spraying can be performed, for example, in the same manner as in plasma spraying described in Japanese Unexamined Patent Application Publication No. 2016-109685. Then the gas sensor  10  equipped with the sensor element  20  is produced. First, the sensor element  20  is caused to pierce axially through the cylindrical member  41 , and the insulator  44   a , the powder compact  45   a , the insulator  44   b , the powder compact  45   b , the insulator  44   c , and the metal ring  46  are placed in this order between the inner circumferential surface of the cylindrical member  41  and the sensor element  20 . Next, the metal ring  46  is pressed to compress the powder compacts  45   a  and  45   b . With this state maintained, the reduced diameter portions  43   c  and  43   d  are formed to thereby produce the element-sealing member  40 , and the gap between the inner circumferential surface of the cylindrical member  41  and the sensor element  20  is thereby sealed. Then the protective cover  30  is welded to the element-sealing member  40 , and the bolt  47  is attached to obtain the assembly  15 . Then the lead wires  55  piercing through the rubber stopper  57  and the connector  50  connected to the lead wires  55  are prepared, and the connector  50  is connected to the rear end side of the sensor element  20 . Then the external cylinder  48  is welded and fixed to the main metal fitting  42  to thereby obtain the gas sensor  10 . 
     In the sensor element  20  in the present embodiment, when the thermal expansion coefficients of the solid electrolyte layers  78   a  to  78   f , the first dense layer  92 , and the intermediate layer  98  in the temperature range of from 20° C. to 1360° C. are denoted by thermal expansion coefficients Ea, Eb, and Ec, respectively, the ratio Ea/Eb is more than 1.0 and 5.0 or less, and the relation Ea&gt;Ec&gt;Eb is satisfied. The thermal expansion coefficients Ea to Ec are not volume expansion coefficients but are linear expansion coefficients. Let the median of the thermal expansion coefficient Ea and the thermal expansion coefficient Eb be Ed (=(Ea+Eb)/2). Then it is preferable that the thermal expansion coefficient Ec satisfies formula (1) below. The thermal expansion coefficients Ea, Eb, and Ec are referred to also as thermal expansion coefficients A′, B′, and C′. The median Ed is referred to also as the median D′. 
         Ed− 0.8×( Ed−Eb )&lt; Ec&lt;Ed+ 0.8×( Ea−Ed )  (1)
 
     The thermal expansion coefficient Ea of the solid electrolyte layers  78   a  to  78   f  is measured by thermomechanical analysis (TMA) as follows. First, the sensor element  20  is cut such that a portion including the solid electrolyte layers  78   a  to  78   f  of the element body  60  is cut out to thereby obtain a measurement piece. Next, the measurement piece is placed in a container, and the expansion coefficient of the measurement piece when the temperature is changed from 20° C. to 1360° C. is measured under an applied load of 1 g. Specifically, a dimension a1′ of the measurement piece at 20° C. is measured. Next, a dimension a2′ of the measurement piece heated to 1360° C. while a load of 1 g is applied is measured. Then the thermal expansion coefficient Ea is computed using the formula: the thermal expansion coefficient Ea[%]=(a2′−a1′)/a1′×100. The dimensions a1′ and a2′ of the measurement piece are measured as dimensions in the forward-rearward direction, i.e., the longitudinal direction, of the sensor element  20 . Similarly, the thermal expansion coefficient Eb is computed using a measurement piece that is cut from the sensor element  20  so as to include part of the first dense layer  92 , and the thermal expansion coefficient Ec is computed using a measurement piece that is cut from the sensor element  20  so as to include part of the intermediate layer  98 . When thermal expansion coefficients of the solid electrolyte layers  78   a  to  78   f  are not the same, e.g., when the materials of the solid electrolyte layers  78   a  to  78   f  are not the same, the thermal expansion coefficient of a layer closest to the first dense layer  92  and the intermediate layer  98  (the solid electrolyte layer  78   a  in the present embodiment) is used as the thermal expansion coefficient Ea. 
     The thermal expansion coefficients Ea to Ec of the sensor element  20  can be controlled as follows. For example, the thermal expansion coefficient Ea of the solid electrolyte layers  78   a  to  78   f  can be controlled by changing the material of the raw material powder contained in the paste for forming the ceramic green sheets. The thermal expansion coefficient Eb of the first dense layer  92  can be controlled by chaining the material of the raw material powder contained in the paste for forming the green first dense layer. The thermal expansion coefficient Ec of the intermediate layer  98  can be controlled by changing the material of the raw material powder contained in the paste for forming the green intermediate layer. Therefore, the ratio Ea/Eb can be set to be more than 1.0 and 5.0 or less by selecting an appropriate combination of the material of the raw material powder of the solid electrolyte layers  78   a  to  78   f  and the material of the raw material powder of the first dense layer  92 . For example, the thermal expansion coefficient of zirconia at 40° C. to 400° C. is 10.5×10 −6 /° C., and the thermal expansion coefficient of alumina at 40° C. to 400° C. is 7.2×10 −6 /° C. Therefore, the thermal expansion coefficient of zirconia is larger than the thermal expansion coefficient of alumina. When the raw material powder of the solid electrolyte layers  78   a  to  78   f  is zirconia and the raw material powder of the first dense layer  92  is alumina, the thermal expansion coefficient Ea is larger than the thermal expansion coefficient Eb, and the ratio Ea/Eb can be set to be more than 1.0 and 5.0 or less. When cordierite (thermal expansion coefficient: less than 0.1×10 −6 /° C.) or silicon nitride (thermal expansion coefficient: 2.8×10 −6 /° C.), which are materials having a smaller thermal expansion coefficient than alumina, is used as the raw material powder of the first dense layer  92 , the value of the ratio Ea/Eb can be larger than that when alumina is used (for example, a value of about 5.0) while the value of the ratio Ea/Eb is set to be more than 1.0 and 5.0 or less. When a material whose thermal expansion coefficient is between the thermal expansion coefficient of the material of the raw material powder of the solid electrolyte layers  78   a  to  78   f  and the thermal expansion coefficient of the material of the raw material powder of the first dense layer  92  is appropriately selected as the material of the raw material powder of the intermediate layer  98 , Ea&gt;Ec&gt;Eb can be satisfied, and formula (1) can be satisfied. Alternatively, the raw material powder of the intermediate layer  98  may contain both the material of the raw material powder of the solid electrolyte layers  78   a  to  78   f  and the material of the raw material powder of the first dense layer  92 . In this case also, Ea&gt;Ec&gt;Eb can be satisfied, and formula (1) can be satisfied. By appropriately controlling the volume ratio of the material of the raw material powder of the solid electrolyte layers  78   a  to  78   f  and the volume ratio of the material of the raw material powder of the first dense layer  92  in the raw material powder of the intermediate layer  98 , the thermal expansion coefficient Ec can be controlled while Ea&gt;Ec&gt;Eb is satisfied, and this allows formula (1) to be satisfied. 
     Next, an example of the use of the thus-produced gas sensor  10  will be described below. When the measurement-object gas flows through the pipe  58  with the gas sensor  10  attached to the pipe  58  as shown in  FIG.  1   , the measurement-object gas flows through the protective cover  30  and into the element chamber  33 , and the forward end side of the sensor element  20  is exposed to the measurement-object gas. Then, with the sensor element  20  heated by the heater  69 , the measurement-object gas passes through the protective layer  80 , reaches the outer electrode  64 , and also reaches the sensor element  20  through the measurement-object gas inlet  61 , and the detection portion  63  generates an electrical signal corresponding to the NOx concentration in the measurement-object gas as described above. By outputting this electrical signal through the upper and lower connector electrodes  71  and  72 , the NOx concentration is detected based on the electrical signal. 
     In this case, the measurement-object gas may contain moisture, and the moisture may move through the protective layer  80  by capillary action. When the moisture reaches the exposed upper and lower connector electrodes  71  and  72 , rust or corrosion may occur in the upper and lower connector electrodes  71  and  72  due to components such as water and sulfuric acid dissolved in water, or a short circuit may occur between adjacent ones of the upper and lower connector electrodes  71  and  72 . However, in the present embodiment, even when moisture in the measurement-object gas moves through the protective layer  80  (in particular, the first inner porous layer  83  and the second inner porous layer  84 ) toward the rear end of the element body  60  by capillary action, the moisture reaches the first dense layer  92  or the second dense layer  95  before it reaches the upper and lower connector electrodes  71  and  72 . Since the porosity of the first dense layer  92  is less than 10%, the capillary action of water in the longitudinal direction of the element body  60  is unlikely to occur. In this case, the first dense layer  92  can prevent moisture from passing through the first dense layer  92  from the forward end-side portion  83   a  side and reaching the upper connector electrodes  71  (the upper connector electrodes  71   a  to  71   d ). Therefore, in the sensor element  20 , the occurrence of the above-described problem caused by water adhering to the upper connector electrodes  71  can be reduced. Similarly, the second dense layer  95  can prevent moisture from passing through the second dense layer  95  from the forward end-side portion  84   a  side and reaching the lower connector electrodes  72  (the lower connector electrodes  72   a  to  72   d ). Therefore, in the sensor element  20 , the occurrence of the above-described problem caused by water adhering to the lower connector electrodes  72  is reduced. Preferably, the length Le of the first dense layer  92  in the longitudinal direction is 0.5 mm or more because the passage of moisture through the first dense layer  92  can be reduced sufficiently. Similarly, the length Le of the second dense layer  95  is 0.5 mm or more. 
     Moreover, the sensor element  20  has a ratio Ea/Eb of more than 1.0 and 5.0 or less and satisfies Ea&gt;Ec&gt;Eb as described above. Therefore, the intermediate layer  98  reduces the stress caused by the difference between the thermal expansion coefficient Ea of the solid electrolyte layers  78   a  to  78   f  and the thermal expansion coefficient Eb of the first dense layer  92  when the sensor element  20  is heated by the heater  69  during use. When stress is generated in the sensor element  20 , cracking tends to occur. However, in the sensor element  20  in the present embodiment, the stress generated during heating is reduced, so that the occurrence of cracking in the sensor element  20  is reduced. Moreover, since formula (1) above is satisfied, the stress during heating of the sensor element  20  is further reduced, and the occurrence of cracking during heating of the sensor element  20  is further reduced. The ratio Ea/Eb is preferably 3.0 or less. When the ratio Ea/Eb is 3.0 or less, the thermal expansion coefficient Ea of the solid electrolyte layers  78   a  to  78   f  is closer to the thermal expansion coefficient Eb of the first dense layer  92 , so that the occurrence of cracking in the sensor element  20  is further reduced. 
     When cracking occurs in the sensor element  20 , particularly in the first dense layer  92  or the second dense layer  95 , the function of the first dense layer  92  or the second dense layer  95  as the water intrusion preventing portion described above may deteriorate. However, since the occurrence of cracking in the sensor element  20  in the present embodiment is reduced, the function of the first dense layer  92  and the second dense layer  95  as the water intrusion preventing portions is unlikely to deteriorate, and thus the occurrence of the above-described problem caused by water adhering to the upper connector electrodes  71  and the lower connector electrodes  72  is reduced. 
     The correspondences between the components in the present embodiment and the components in the present invention will be clarified. The solid electrolyte layers  78   a  to  78   f  in the present embodiment correspond to the solid electrolyte layer in the present invention, and the element body  60  corresponds to the element body. The upper connector electrodes  71   a  to  71   d  correspond to the connector electrode, and the first surface  60   a  corresponds to the side surface on which the connector electrode is disposed. The first inner porous layer  83  corresponds to the porous layer, and the first dense layer  92  corresponds to the dense layer. The intermediate layer  98  corresponds to the intermediate layer. The detection portion  63  corresponds to the detection portion, and the outer lead wire  75  corresponds to the outer lead portion. The outer electrode  64  corresponds to the outer electrode. 
     In the gas sensor  10  in the present embodiment described above in detail, as for the thermal expansion coefficients Ea, Eb, and Ec of the solid electrolyte layers  78   a  to  78   f , the first dense layer  92 , and the intermediate layer  98  of the sensor element  20  in the temperature range of from 20° C. to 1360° C., the ratio Ea/Eb is more than 1.0 and 5.0 or less, and Ea&gt;Ec&gt;Eb is satisfied. In this case, the intermediate layer  98  reduces the stress caused by the difference between the thermal expansion coefficient Ea of the solid electrolyte layers  78   a  to  78   f  and the thermal expansion coefficient Eb of the first dense layer  92  when the sensor element  20  is heated during use. Therefore, the occurrence of cracking in the sensor element  20  during heating is reduced. Moreover, since the thermal expansion coefficients Ea to Ec in the sensor element  20  satisfy formula (1) above, the stress generated when the sensor element  20  is heated is reduced, and the occurrence of cracking is further reduced. Moreover, since the ratio Ea/Eb in the sensor element  20  is 3.0 or less, the occurrence of cracking is further reduced. In the sensor element  20 , since the thickness T of the intermediate layer  98  is 1 μm or more, the above-described effect of the presence of the intermediate layer  98  in reducing the occurrence of cracking in the sensor element  20  is obtained more reliably. 
     The present invention is not limited to the embodiment described above. It will be appreciated that the present invention can be implemented in various forms so long as they fall within the technical scope of the invention. 
     For example, in the above embodiment, the length of the intermediate layer  98  in the forward-rearward direction is the same as the length Le of the first dense layer  92  as shown in  FIG.  6   , but this is not a limitation. It is only necessary that the intermediate layer  98  be disposed at least between the first dense layer  92  and the element body  60 . For example, as shown in  FIG.  8   , the length of the intermediate layer  98  may be larger than the length Le, and the intermediate layer  98  may be present also between the first inner porous layer  83  and the element body  60 . In  FIG.  8   , the intermediate layer  98  extends frontward and rearward from the first dense layer  92 . Specifically, the intermediate layer  98  is present between the forward end-side portion  83   a  and the element body  60  and also between the rear end-side portion  83   b  and the element body  60 . The intermediate layer  98  may be present over a region extending from the forward end of the first surface  60   a  to its rear end. However, when the intermediate layer  98  is dense, it is preferable that the intermediate layer  98  is disposed so as to avoid overlapping the outer electrode  64  such that the intermediate layer  98  does not cover the outer electrode  64 . The length of the intermediate layer  98  may be smaller than the length Le, and a region in which the intermediate layer  98  is not present may be present between the first dense layer  92  and the element body  60 . In other words, it is only necessary that, in a cross-section taken in the forward-rearward direction, the intermediate layer  98  be present between at least part of the first dense layer  92  and the element body  60  (the solid electrolyte layer  78   a ). However, it is preferable that the intermediate layer  98  is present between the first dense layer  92  and the solid electrolyte layer  78   a  at least in a region extending from the forward end of the first dense layer  92  to its rear end, as shown in  FIGS.  6  and  8   . In this manner, the occurrence of cracking in the sensor element  20  can be further reduced. The length of the intermediate layer  98  in the forward-rearward direction may be, for example, from 0.5 mm to 55 mm inclusive. 
     In the embodiment described above, as shown in  FIG.  7   , the width of the intermediate layer  98  in the left-right direction is the same as the width of the first surface  60   a  in the left-right direction and the width of the first dense layer  92  in the left-right direction, but this is not a limitation. It is only necessary that, in a cross section taken in the left-right direction, the intermediate layer  98  be present between at least part of the first dense layer  92  and the element body  60  (the solid electrolyte layer  78   a ), and the width of the intermediate layer  98  in the left-right direction may be smaller than that in  FIG.  7   . However, it is preferable that the intermediate layer  98  is present at least between the element body  60  and a region of the first dense layer  92  that is located at the center in the left-right direction and has a width of 50% (this region is hereinafter referred to as a central region). As shown in  FIG.  9   , the central region includes two left and right central regions among four regions obtained by dividing the left-right width We of the first dense layer  92  into quarters. It is preferable that the intermediate layer  98  is present at least between the central region of the first dense layer  92  and the element body  60 , as shown in  FIG.  9   . In the cross section in the left-right direction in the example in  FIG.  9   , the intermediate layer  98  is present only between the central region of the first dense layer  92  and the element body  60 , and the width Wc of the intermediate layer  98  in the left-right direction is the same as the width of the central region of the first dense layer  92  (We/2). Preferably, the width Wc of the intermediate layer  98  is equal to or larger than the width of the central region (We/2). As shown in  FIG.  10   , the intermediate layer  98  may be divided into left and right portions and may include an intermediate layer  98   a  located on the left side and an intermediate layer  98   b  located on the right side. In this case also, it is preferable that the width Wc of the intermediate layer  98  (the sum of the width of the intermediate layer  98   a  and the width of the intermediate layer  98   b ) is equal to or more than We/2. In  FIG.  10   , the sum of the width of the intermediate layer  98   a  and the width of the intermediate layer  98   b  is equal to We/2. From the viewpoint of reducing the occurrence of cracking in the sensor element  20  described above, the intermediate layer  98  in  FIG.  9    that is present between the central region of the first dense layer  92  and the element body  60  is preferred to the intermediate layer  98  in  FIG.  10   . 
     In the embodiment described above, the intermediate layer  98  is present between the element body  60  and the first dense layer  92  disposed on the first surface  60   a  side (the upper side) of the element body  60 , but this is not a limitation. As shown in  FIG.  11   , an intermediate layer  99  may be present between the element body  60  and the second dense layer  95  disposed on the second surface  60   b  side (the lower side) of the element body  60 . In this case, it is preferable that, when the thermal expansion coefficients of the solid electrolyte layers  78   a  to  78   f , the second dense layer  95 , and the intermediate layer  99  in the temperature range of from 20° C. to 1360° C. are denoted by thermal expansion coefficients Ea, Eb, and Ec, respectively, the ratio Ea/Eb is more than 1.0 and 5.0 or less and Ea&gt;Ec&gt;Eb is satisfied, and it is more preferable that formula (1) above is satisfied. The above-described various modes of the intermediate layer  98  can be applied to the intermediate layer  99 . Preferably, the thermal expansion coefficient of the first dense layer  92  and the thermal expansion coefficient of the second dense layer  95  are the same, and the thermal expansion coefficient of the intermediate layer  98  and the thermal expansion coefficient of the intermediate layer  99  are the same. 
     In the embodiment described above, the second surface  60   b  of the element body  60  may have an exposed portion on which the second inner porous layer  84  and the second dense layer  95  are not present.  FIG.  12    shows an example in which gap regions  96  are disposed rearward and forward of the second dense layer  95  so as to be adjacent thereto. The gap regions  96  in  FIG.  12    include a forward gap region  96   a  disposed between the forward end-side portion  84   a  and the second dense layer  95  and a rear gap region  96   b  disposed between the rear end-side portion  84   b  and the second dense layer  95 . In portions in which the gap regions  96  are present, the second surface  60   b  is exposed. The gap regions  96  are spaces in which the second inner porous layer  84  is not present, so that the capillary action of water in the longitudinal direction of the element body  60  is unlikely to occur. Therefore, the gap regions  96  also serve as water intrusion preventing portions that prevent moisture moving in the longitudinal direction of the element body  60  from reaching the lower connector electrodes  72 , as does the second dense layer  95 . As for the gap regions  96 , only one of the forward gap region  96   a  and the rear gap region  96   b  may be provided. The length Lg of the gap regions  96  in the longitudinal direction is preferably 1 mm or less. When the gap regions  96  include the forward gap region  96   a  and the rear gap region  96   b  as in  FIG.  12   , the length Lg is the sum of the length Lg 1  of the forward gap region  96   a  in the longitudinal direction and the length Lg 2  of the rear gap region  96   b  in the longitudinal direction. A gap region may be present on the first surface  60   a  side of the element body  60 . However, in the embodiment described above, the outer lead wire  75  is disposed on the first surface  60   a . It is therefore preferable, from the viewpoint of protecting the outer lead wire  75 , that no gap region is present on the first surface  60   a  side. 
     In the embodiment described above, the outer porous layer  85  is formed by plasma spraying after the firing step, but this is not a limitation. For example, a green outer porous layer that later becomes the outer porous layer  85  through the firing step may be formed by, for example, dipping in the production step. In this case, the green outer porous layer becomes the outer porous layer  85  through the firing step. In the embodiment described above, the protective layer  80  may not include the outer porous layer  85 . 
     In the embodiment described above, the first dense layer  92  divides the first inner porous layer  83  in the longitudinal direction into the forward end-side portion  83   a  and the rear end-side portion  83   b , but this is not a limitation. The first dense layer  92  may be located rearward of the protective layer  80 . For example, in the embodiment described above, the first inner porous layer  83  may not include the rear end-side portion  83   b . Similarly, the second dense layer  95  may not divide the second inner porous layer  84  and may be located rearward of the protective layer  80 . However, when the first inner porous layer  83  does not include the rear end-side portion  83   b , part of the outer lead wire  75  is exposed. It is therefore preferable that the first inner porous layer  83  includes the rear end-side portion  83   b.    
     In the embodiment described above, the first dense layer  92  and the second dense layer  95  are disposed at respective positions overlapping the insulator  44   b  in the forward-rearward direction, but this is not a limitation. For example, the first dense layer  92  and the second dense layer  95  may be disposed at positions overlapping the insulator  44   a  or the insulator  44   c  in the forward-rearward direction or may be disposed rearward of the metal ring  46 . In the embodiment described above, the first dense layer  92  and the second dense layer  95  are disposed at positions that are not exposed to the element chamber  33 . However, at least one of the first dense layer  92  and the second dense layer  95  may be disposed at a position exposed to the element chamber  33 , i.e., a position exposed to the measurement-object gas. For example, at least one of the first dense layer  92  and the second dense layer  95  may be disposed at a position located rearward of the outer porous layer  85  and exposed to the element chamber  33 . 
     In the embodiment described above, the sensor element  20  may not include the second inner porous layer  84 , and the second surface  60   b  may not be covered with the second inner porous layer  84 . In this case, the sensor element  20  may not include the second dense layer  95 . It is only necessary that a dense layer be disposed on at least one side surface on which connector electrodes and a porous layer are disposed (the first and second surfaces  60   a  and  60   b  in the embodiment described above) among the side surface of the element body (the first to fourth surfaces  60   a  to  60   d  in the embodiment described above). In this case, moisture is prevented from reaching the connector electrodes at least on the side surfaces on which the dense layer is disposed. Moreover, it is only necessary that the intermediate layer be disposed between the element body and the dense layer. 
     In the embodiment described above, the first inner porous layer  83  covers a region of the first surface  60   a  that extends from the forward end of the first surface  60   a  to its rear end except for the regions in which the first dense layer  92  and the upper connector electrodes  71  are present, but this is not a limitation. For example, the first inner porous layer  83  may cover a region extending from forward end of the first surface  60   a  to the forward ends of the upper connector electrodes  71   a  to  71   d  except for the region in which the first dense layer  92  is present. Alternatively, the first inner porous layer  83  may cover at least a region extending from the forward end of the first surface  60   a  to a position rearward of the first dense layer  92  except for the region in which the first dense layer  92  is present. The same applies to the second inner porous layer  84 . 
     In the embodiment described above, the element body  60  has a rectangular parallelepiped shape, but this is not a limitation. For example, the element body  60  may be cylindrical or columnar. In this case, the element body  60  has one side surface. 
     In the embodiment described above, the gas sensor  10  detects the NOx concentration as the specific gas concentration, but this is not a limitation. The concentration of a different oxide may be used as the specific gas concentration. In the case where the specific gas is an oxide, when the specific gas itself is reduced near the measurement electrode  67 , oxygen is generated, as in the embodiment described above. Therefore, the specific gas concentration can be detected based on the value that is detected by the detection portion  63  and corresponds to the oxygen. The specific gas may be a non-oxide such as ammonia. When the specific gas is a non-oxide, the specific gas is converted to an oxide, for example, near the inner main pump electrode  65  (for example, ammonia is oxidized and converted to NO). When the oxide produced by conversion is reduced near the measurement electrode  67 , oxygen is generated, and the specific gas concentration can be detected based on the value that is detected by the detection portion  63  and corresponds to the oxygen. As described above, even when the specific gas is an oxide or a non-oxide, the gas sensor  10  can detect the specific gas concentration based on oxygen derived from the specific gas and generated near the measurement electrode  67 . 
     Examples will next be described. In each Example, sensor elements were actually produced. However, the present invention is not limited to the following Examples. 
     Example 1 
     A sensor element that was the same as the sensor element  20  shown in  FIGS.  2  to  4 ,  6 , and  7    was produced, except that the gap regions  96  (the forward gap region  96   a  and the rear gap region  96   b ) were present on the second surface  60   b  side of the element body  60  as shown in  FIG.  12    and that the protective layer  80  did not include the outer porous layer  85 . The sensor element produced was used as Example 1. The sensor element  20  in Example 1 was produced as follows. First, the production step was performed as follows. Zirconia particles containing 4 mol % of yttria serving as a stabilizer, an organic binder, and an organic solvent were mixed, and the mixture was used to prepare six ceramic green sheets by tape molding. The green sheets were subjected to punching processing as needed, and patterns for the green electrodes, the green lead wires, the green connector electrodes, the green heater, the green intermediate layer, the green first dense layer, the green first inner porous layer, the green second dense layer, and the green second inner porous layer were formed by screen printing. The green first inner porous layer and the green second inner porous layer were formed using a paste prepared by mixing a raw material powder (alumina powder), a binder solution (polyvinyl acetal and butyl carbitol), a solvent (acetone), and a pore-forming material. A paste for the green first dense layer and the green second dense layer was prepared such that the porosities of the first and second dense layers  92  and  95  were 0%. Specifically, the same paste as the paste for the green first inner porous layer except that no pore-forming material was added and the amount of the solvent added was changed to adjust the viscosity was used as the paste for the green first dense layer and the green second dense layer. The green intermediate layer was formed using a paste prepared by mixing raw material powders (the alumina powder and the zirconia powder), a binder solution (polyvinyl acetal and butyl carbitol), and a solvent (acetone). No pore-forming material was added to the paste for the green intermediate layer, so that the porosity of the intermediate layer  98  was 0%. When screen printing was performed, the green first dense layer was formed by printing after the formation of the green intermediate layer such that the green intermediate layer was present between the green first dense layer and the ceramic green sheet that later became the solid electrolyte layer  78   a . Then the six green sheets were stacked to obtain a layered body, and the layered body was cut to obtain small layered bodies, i.e., green sensor elements. Next, the firing step was performed in which each green sensor element was fired at 1360° C.±50° C. and the fired sensor element was cooled to room temperature (20° C.). A sensor element  20  including the element body  60 , the first dense layer  92 , the intermediate layer  98 , etc. was thereby produced and used as a sensor element  20  in Example 1. The thickness T of the intermediate layer  98  was 5 The value of the thickness T used was the thickness T1 measured at the thinnest portion of the intermediate layer  98  (the portion located directly above the outer lead wire  75 ). 
     Examples 2 to 10 and Comparative Examples 1 to 3 
     Sensor elements  20  with the relations between the thermal expansion coefficients Ea, Eb, and Ec different from those in Example 1 were produced and used as Examples 2 to 10 and Comparative Examples 1 to 3. In Examples 2 to 10 and Comparative Examples 1 to 3, the solid electrolyte layers  78   a  to  78   f  were the same as those in Example 1. 
     Specifically, the value of the thermal expansion coefficient Ea was the same for Examples 1 to 10 and Comparative Examples 1 to 3. In Examples 2 to 10 and Comparative Examples 1 to 3, the thermal expansion coefficient Eb of the first dense layer  92  was changed by selecting the raw material powder of the green dense layer from alumina, cordierite, and silicon nitride. In Examples 2 to 10 and Comparative Examples 1 to 3, the thermal expansion coefficient Ec of the intermediate layer  98  was changed by adjusting the volume ratio of the material of the raw material powder of the solid electrolyte layers  78   a  to  78   f  contained in the green intermediate layer and the volume ratio of the material of the raw material powder of the first dense layer  92  contained in the green intermediate layer. In Examples 2 to 8 and Comparative Examples 1 to 3, the thickness T of the intermediate layer  98  was set to 5 μm, which was the same as the thickness in Example 1. In Example 9, the thickness T of the intermediate layer  98  was set to 10 μm. In Example 10, the thickness T of the intermediate layer  98  was set to 1 μm. The thickness T of the intermediate layer  98  was controlled by changing the amount of the solvent contained in the paste for the green intermediate layer to thereby adjust the viscosity or adjusting the number of screen printing operations for printing the green intermediate layer. 
     [Measurement of Thermal Expansion Coefficients Ea to Ec] 
     The thermal expansion coefficients Ea to Ec in each of Examples 1 to 10 and Comparative Examples 1 to 3 were measured using the method described above. A thermomechanical analysis apparatus (type: TMA4000SA) manufactured by NETZSCH was used for the measurement. 
     [Evaluation of Cracking Resistance] 
     For each of Examples 1 to 10 and Comparative Examples 1 to 3, ten sensor elements  20  were produced, and each sensor element  20  was subjected to a test in which the sensor element  20  was exposed to high-temperature high-pressure vapor for a prescribed time to thereby evaluate cracking resistance. This test was performed using a method according to JIS A 1509-8:2014. First, the sensor element  20  was placed in an autoclave. Then the pressure inside the autoclave was gradually increased so as to reach 1 MPa or higher over about 1 hour, and the increased pressure was maintained for 1 hour or longer. Then the pressure was reduced to normal pressure as fast as possible, and the sensor element  20  was left to cool. The cooled sensor element  20  was visually checked to determine whether or not cracking occurred in the first dense layer  92 . When the number of cracked sensor elements  20  out of the ten sensor element  20  was zero, the cracking resistance was rated “excellent (A).” When the number of cracked sensor elements  20  was one, the cracking resistance was rated “good (B).” When the number of cracked sensor elements  20  was two or more, the cracking resistance was rated “fail (F).” The high-temperature high-pressure state in the autoclave during the test is severer than a normal use environment of the sensor element  20  attached to a vehicle. 
     Table 1 summarizes the thermal expansion coefficients Ea to Ec in each of Examples 1 to 10 and Comparative Examples 1 to 3 and the results of the evaluation of the cracking resistance of the sensor elements  20 . In Table 1, the values of the thermal expansion coefficients Ea to Ec are the ratios with respect to the thermal expansion coefficient Ea with the value of the thermal expansion coefficient Ea used as a reference (value: 1). Table 1 also shows the ratio Ea/Eb, the magnitude relation between Ea to Ec, the median Ed of the thermal expansion coefficient Ea and the thermal expansion coefficient Eb, the value of the left-hand side of formula (1), i.e., (Ed−0.8×(Ed−Eb)), the value of the right-hand side of formula (1), i.e., (Ed+0.8×(Ea−Ed)), whether the thermal expansion coefficients Ea to Ec satisfy formula (1), and the thickness T of the intermediate layer  98 . 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Ratio Ea 
                 Ratio Ec 
                 Ratio Eb 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                 of thermal 
                 of thermal 
                 of thermal 
               
               
                   
                 expansion 
                 expansion 
                 expansion 
                   
                   
                   
                   
                   
                   
                 Thickness 
               
               
                   
                 coefficient 
                 coefficient 
                 coefficient 
                   
                   
                   
                   
                   
                 Formula 
                 of inter- 
               
               
                   
                 of solid 
                 of inter- 
                 of 1st 
                   
                   
                 Median Ed 
                   
                   
                 (1)*2 is 
                 mediate 
               
               
                   
                 electrolyte 
                 mediate 
                 dense 
                 Ratio 
                 Magnitude 
                 (=(Ea + 
                 Ed − 0.8 
                 Ed + 0.8 
                 satisfied 
                 layer T 
                 Evalu- 
               
               
                   
                 layer※1 
                 layer※1 
                 layer※1 
                 Ea/Eb 
                 relationship 
                 Eb)/2) 
                 (Ed − Eb) 
                 (Ea − Ed) 
                 or not 
                 [μm] 
                 ation 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Example 1 
                 1 
                 0.85 
                 0.7 
                 1.43 
                 Ea &gt; Ec &gt; Eb 
                 0.85 
                 0.73 
                 0.97 
                 OK 
                 5 
                 A 
               
               
                 Example 2 
                 1 
                 0.71 
                 0.7 
                 1.43 
                 Ea &gt; Ec &gt; Eb 
                 0.85 
                 0.73 
                 0.97 
                 NG 
                 5 
                 B 
               
               
                 Example 3 
                 1 
                 0.99 
                 0.7 
                 1.43 
                 Ea &gt; Ec &gt; Eb 
                 0.85 
                 0.73 
                 0.97 
                 NG 
                 5 
                 B 
               
               
                 Example 4 
                 1 
                 0.75 
                 0.7 
                 1.43 
                 Ea &gt; Ec &gt; Eb 
                 0.85 
                 0.73 
                 0.97 
                 OK 
                 5 
                 A 
               
               
                 Example 5 
                 1 
                 0.9 
                 0.7 
                 1.43 
                 Ea &gt; Ec &gt; Eb 
                 0.85 
                 0.73 
                 0.97 
                 OK 
                 5 
                 A 
               
               
                 Example 6 
                 1 
                 0.6 
                 0.35 
                 2.86 
                 Ea &gt; Ec &gt; Eb 
                 0.68 
                 0.42 
                 0.94 
                 OK 
                 5 
                 A 
               
               
                 Example 7 
                 1 
                 0.8 
                 0.3 
                 3.33 
                 Ea &gt; Ec &gt; Eb 
                 0.65 
                 0.37 
                 0.93 
                 OK 
                 5 
                 B 
               
               
                 Example 8 
                 1 
                 0.95 
                 0.90 
                 1.11 
                 Ea &gt; Ec &gt; Eb 
                 0.95 
                 0.91 
                 0.99 
                 OK 
                 5 
                 A 
               
               
                 Example 9 
                 1 
                 0.85 
                 0.7 
                 1.43 
                 Ea &gt; Ec &gt; Eb 
                 0.85 
                 0.73 
                 0.97 
                 OK 
                 10 
                 A 
               
               
                 Example 10 
                 1 
                 0.85 
                 0.7 
                 1.43 
                 Ea &gt; Ec &gt; Eb 
                 0.85 
                 0.73 
                 0.97 
                 OK 
                 1 
                 A 
               
               
                 Comparative 
                 1 
                 0.7 
                 0.7 
                 1.43 
                 Ea &gt; Ec = Eb 
                 0.85 
                 0.73 
                 0.97 
                 NG 
                 5 
                 F 
               
               
                 Examples 1 
               
               
                 Comparative 
                 1 
                 1.3 
                 1.5 
                 0.67 
                 Eb &gt; Ec &gt; Ea 
                 1.25 
                 1.45 
                 1.05 
                 NG 
                 5 
                 F 
               
               
                 Examples 2 
               
               
                 Comparative 
                 1 
                 0.6 
                 0.19 
                 5.10 
                 Ea &gt; Ec &gt; Eb 
                 0.60 
                 0.27 
                 0.92 
                 NG 
                 5 
                 F 
               
               
                 Examples 3 
               
               
                   
               
               
                 ※1The coefficients of thermal expansion Ea to Ec are shown as ratios to Ea with Ea as the reference (value 1). 
               
               
                 ※2 Formula (1): Ed − 0.8 × (Ed − Eb) &lt; Ec &lt; Ed + 0.8 × (Ea − Ed) 
               
            
           
         
       
     
     As can be seen from Table 1, in Examples 1 to 10 in which the ratio Ea/Eb was more than 1.0 and 5.0 or less and Ea&gt;Ec&gt;Eb was satisfied, the evaluation of the cracking resistance of the sensor elements  20  was “excellent (A)” or “good (B),” and the occurrence of cracking was reduced. However, in Comparative Examples 1 and 2 in which Ea&gt;Ec&gt;Eb was not satisfied and in Comparative Example 3 in which the ratio Ea/Eb was more than 5.0, the evaluation of the cracking resistance of the sensor elements  20  was “fail (F).” This confirms that, when the ratio Ea/Eb is more than 1.0 and 5.0 or less and Ea&gt;Ec&gt;Eb is satisfied, the occurrence of cracking in the sensor element  20  can be reduced. In Examples 2 and 3 in which formula (1) was not satisfied, the evaluation of the cracking resistance was “good (B).” However, in Examples 1, 4 to 6, and 8 to 10 in which formula (1) was satisfied, the evaluation of the cracking resistance was “excellent (A).” This confirms that, when formula (1) is satisfied, the occurrence of cracking in the sensor element  20  can be further reduced. In Example 7 in which the ratio Ea/Eb was more than 3.0, the evaluation of the cracking resistance was “good (B).” However, in Examples 1, 4 to 6, and 8 to 10 in which the ratio Ea/Eb was 3.0 or less, the evaluation of the cracking resistance was “excellent (A).” This confirms that, when the ratio Ea/Eb is 3.0 or less, the occurrence of cracking in the sensor element  20  can be further reduced. The results in Example 10 confirm that the effect of reducing the occurrence of cracking is obtained when the thickness T of the intermediate layer  98  is in the range of 1 μm or more. The results in Example 9 confirm that the effect of reducing the occurrence of cracking is obtained when the thickness T of the intermediate layer  98  is in the range of 10 μm or less. 
     The present application claims priority based on U.S. Patent Application No. 63/211,665 filed on Jun. 17, 2021, and the entire contents of which are incorporated herein by reference.