Energy Trapping and Its Consequences

Abstract

Most quartz resonators apply energy trapping. By giving the electrodes the shape of a keyhole, the acoustic thickness of the plate is made larger in the center than at the rim. Such a plate can be viewed as an acoustic cavity, where the surfaces are shaped such that they focus the acoustic energy to the center. Energy trapping has numerous consequences, among them the flexural contributions to the vibration pattern, which lead to the emission of compressional waves.

Glossary

Variable

Definition (Comments)

A

Effective area of the resonator plate

$\tilde{c}$

Speed of propagation

C ₁

Motional capacitance

$\bar{C}_{1}$

Motional capacitance of the 4-element circuit (Piezoelectric stiffening and the load have been taken into account)

$\hat{\varvec{D}}$

Electric displacement

d _f

Film thickness

d _q

Thickness of the resonator

d ₂₆

Piezoelectric strain coefficient (d ₂₆ = 3.1 × 10⁻¹² m/V for AT-cut quartz)

e ₂₆

Piezoelectric stress coefficient (e ₂₆ = 9.65 × 10⁻² C/m² for AT-cut quartz)

f

Frequency

f

As an index: film

f _n

Resonance frequency at overtone order n

f _r

Resonance frequency

f ₀

Resonance frequency at the fundamental (f ₀ = Z _q /(2m _q ) = Z _q /(2ρ _q d _q ))

G _el

Electric conductance

G _q

Shear modulus of AT-cut quartz (G _q  ≈ 29 × 10⁹ Pa)

$\hat{I}$

Electric current

k

Wavenumber

mot

As an index: motional

m _f

Mass per unit area of a film

m _q

Mass per unit area of the resonator (m _q  = ρ _q d _q  = Z _q /(2f ₀))

n

Overtone order

PP

As an index: Parallel Plate

q

As an index: quartz resonator

Q

Quality factor

r

Distance to the axis of a beam

r

Position

ref

As an index: reference state of a crystal in the absence of a load

R ₁

Motional resistance

S

As an index: Surface

$\hat{u}$

Displacement, more general, a field variable obeying the wave equation

$\hat{u}_{c}$

Amplitude in the center of a Gaussian amplitude distribution

$\hat{u}_{E}$

Slowly varying envelope

$\hat{U}$

Voltage

${\hat{\text{v}}}$

Velocity (${\hat{\text{v}}} = {\text{i}}\upomega\hat{u}$)

z

Spatial coordinate along the surface normal

Z _q

Acoustic wave impedance of AT-cut quartz (Z _q  = 8.8 × 10⁶ kg m⁻² s⁻¹)

Γ

Imaginary part of a resonance frequency

δ

Depth of penetration of a shear wave

δ _L

Loss angle (tan(δ _L ) = G′′/G′ = J′′/J′)

Δ

As a prefix: A shift induced by the presence of the sample

ε

A small quantity (In Taylor expansions)

φ

Azimuthal angle

ϕ

Factor converting between mechanical and electric quantities in the Mason circuit (ϕ = Ae ₂₆/d _q )

λ

Wavelength

σ _G

Standard deviation of a Gaussian distribution

ρ

Density

ω

Angular frequency