Resonator element, resonator, electronic device, electronic apparatus, and mobile object

A resonator element comprising:

• a substrate including a first principal surface and a second principal surface respectively forming an obverse surface and a reverse surface of the substrate, and vibrating in a thickness-shear vibration mode;
• a first excitation electrode disposed on the first principal surface;
• and
• a second excitation electrode disposed on the second principal surface, and being larger than the first excitation electrode in a plan view,
• wherein the first excitation electrode fits into an outer edge of the second excitation electrode in the plan view, and
• the following relational expressions are fulfilled:
• M=K×(hx/2×ts)×√Δ
• Δ=(fs−fe)/fs
• fs=R/[ts+te2×(ρe/ρx)]
• fe=R/[ts+te×(ρe/ρx)]
• 15.5≦M≦36.7
• where:
• M denotes an energy trap confficient;
• K denotes an anisotropy factor of the substrate;
• hx denotes a length of the first excitation electrode along a thickness-shear vibration direction;
• ts denotes a thickness of the substrate;
• Δ denotes an amount of frequency drop;
• fs denotes a cut-off frequency of the substrate;
• fe denotes a frequency when the excitation electrodes are disposed on the substrate;
• R denotes a frequency constant of the substrate;
• te denotes a sum of a thickness of the first excitation electrode and a thickness of the second excitation electrode;
• te2 denotes the thickness of the second excitation electrode;
• ρe denotes a density of the excitation electrodes;
• and
• ρx denotes a density of the substrate.