Practical Consequences of Piezoelectric Stiffening

Abstract

Piezoelectric stiffening can be exploited to probe the sample’s electrical impedance. Conversely, the electrical impedance of the sample and the circuitry around the crystal can influence on the resonance frequency. Some precautions must be taken to avoid the corresponding artifacts.

Glossary

Variable

Definition (Comments)

A

(Effective) area of the resonator plate

C ₀

Parallel capacitance

d _q

Thickness of the resonator (d _q  = m _q /ρ _q  = Z _q /(2ρ _q f ₀))

e ₂₆

Relevant component of the e-tensor (piezoelectric stress coefficient, e ₂₆ = 9.65 × 10⁻² C/m² for AT-cut quartz)

f ₀

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

G _q

Shear modulus of AT-cut quartz (G _q  ≈ 29 × 10⁹ Pa for AT-cut quartz. G _q  is the piezoelectrically stiffened modulus)

k _t

Piezoelectric coupling coefficient(k _t  = (e ₂₆/(ε _q ε₀ G _q ))^1/2, k ² _t  = 0.8 % for AT-cut quartz)

PE

As an index: PiezoElectric stiffening

q

As an index: quartz resonator

ref

As an index: reference

R

Resistance

R _bulk

Resistance of a bulk liquid

R ₁, R ₂, R ₃

Elements of the pi-network (Fig. 14.2)

$\tilde{Y}_{ex}$

External electrical admittance ($\tilde{Y}_{ex} = 1/\tilde{Z}_{ex}$)

$\tilde{Z}_{ex}$

External electrical impedance

Z _q

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

Δ

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

$\tilde{\upvarepsilon }_{q} ,\,\upvarepsilon_{q}$

Dielectric constant of AT-cut quartz (ε _q is the clamped dielectric constant, ε _q  = 4.54 for AT-cut quartz)

ε₀

Dielectric permittivity of vacuum (ε₀ = 8.854 × 10⁻¹² C/(Vm))

ϕ

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

ρ _q

Density of crystalline quartz (ρ _q  = 2.65 g/cm³)

ω

Angular frequency