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Energy Trapping and Its Consequences

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

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.

Keywords

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.

Notes

Glossary

Variable

Definition (Comments)

A

Effective area of the resonator plate

\( \tilde{c} \)

Speed of propagation

C1

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

df

Film thickness

dq

Thickness of the resonator

d26

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

e26

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

f

Frequency

f

As an index: film

fn

Resonance frequency at overtone order n

fr

Resonance frequency

f0

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

Gel

Electric conductance

Gq

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

\( \hat{I} \)

Electric current

k

Wavenumber

mot

As an index: motional

mf

Mass per unit area of a film

mq

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

R1

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

Zq

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

δL

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

σG

Standard deviation of a Gaussian distribution

ρ

Density

ω

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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