Energy Trapping and Its Consequences

  • Diethelm Johannsmann
Part of the Soft and Biological Matter book series (SOBIMA)


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.


Compressional Wave Gaussian Beam Amplitude Distribution Nodal Line Paraxial Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.




Definition (Comments)


Effective area of the resonator plate

\( \tilde{c} \)

Speed of propagation


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


Film thickness


Thickness of the resonator


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


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




As an index: film


Resonance frequency at overtone order n


Resonance frequency


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


Electric conductance


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

\( \hat{I} \)

Electric current




As an index: motional


Mass per unit area of a film


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


Overtone order


As an index: Parallel Plate


As an index: quartz resonator


Quality factor


Distance to the axis of a beam




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


Motional resistance


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

\( \hat{U} \)


\( {\hat{\text{v}}} \)

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


Spatial coordinate along the surface normal


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


Imaginary part of a resonance frequency


Depth of penetration of a shear wave


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


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


A small quantity (In Taylor expansions)


Azimuthal angle


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




Standard deviation of a Gaussian distribution




Angular frequency


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Physical ChemistryClausthal University of TechnologyClausthal-ZellerfeldGermany

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