question: What is it exactly that determines the frequency of a piezoelectric material?
comment: piezoelectric crystals physically resonate much like a tuning fork, a large tuning fork can fit a longer wavelength, and hence will have a lower frequency than a small one.
passages: BULLET::::- Barium titanate (BaTiO)—Barium titanate was the first piezoelectric ceramic discovered.
  
  BULLET::::- Lead zirconate titanate (Pb[] with 0 ≤ "x" ≤ 1)—more commonly known as PZT, lead zirconate titanate is the most common piezoelectric ceramic in use today.
  
  BULLET::::- Potassium niobate (KNbO)
  
  BULLET::::- Sodium tungstate (NaWO)
  
  BULLET::::- BaNaNbO
  
  BULLET::::- PbKNbO
   In 2000, it was shown that an actuator can be made of paper. The wood fibers, called cellulose fibers, are aligned so that there is a dipolar orientation. This allows the material to become piezoelectric. To enhance the effect, the material can be poled for further alignment. D33 responses for cellular polypropylene are around 200 pC/N. Some applications of cellular polypropylene are musical key pads, microphones, and ultrasound-based echolocation systems.
  
  Section::::Materials.:Aromatic short peptide self-assemblies.
   The semiconductors where the strongest piezoelectricity is observed are those commonly found in the wurtzite structure, i.e. GaN, InN, AlN and ZnO. ZnO is the most used material in the recent field of piezotronics.
  
  Since 2006, there have also been a number of reports of strong non linear piezoelectric effects in polar semiconductors.
  
  Such effects are generally recognized to be at least important if not of the same order of magnitude as the first order approximation.
  
  Section::::Materials.:Polymers.
   Usually the resistance change in metals is mostly due to the change of geometry resulting from applied mechanical stress. However, even though the piezoresistive effect is small in those cases it is often not negligible. In cases where it is, it can be calculated using the simple resistance equation derived from Ohm's law;
  
  where
   Section::::Mechanism.
   BULLET::::- The best-known application is the electric cigarette lighter: pressing the button causes a spring-loaded hammer to hit a piezoelectric crystal, producing a sufficiently high-voltage electric current that flows across a small spark gap, thus heating and igniting the gas. The portable sparkers used to ignite gas stoves work the same way, and many types of gas burners now have built-in piezo-based ignition systems.
   BULLET::::- Piezoelectric actuator. A single crystal or a number of crystals are expanded by applying a voltage for moving and controlling a mechanism or system.
  
  Section::::Application.:Frequency standard.
  
  The piezoelectrical properties of quartz are useful as a standard of frequency.
   Piezoelectric materials also show the opposite effect, called the converse piezoelectric effect, where the application of an electrical field creates mechanical deformation in the crystal.
  
  Section::::Mechanism.:Mathematical description.
  
  Linear piezoelectricity is the combined effect of
  
  BULLET::::- The linear electrical behavior of the material:
  
  BULLET::::- Hooke's Law for linear elastic materials:
  
  These may be combined into so-called "coupled equations", of which the strain-charge form is:
  
  In matrix form,
   BULLET::::- Bismuth ferrite (BiFeO) is also a promising candidate for the replacement of lead-based ceramics.
  
  BULLET::::- Sodium niobate NaNbO
  
  BULLET::::- Barium titanate (BaTiO)—Barium titanate was the first piezoelectric ceramic discovered.
  
  BULLET::::- Bismuth titanate BiTiO
  
  BULLET::::- Sodium bismuth titanate NaBi(TiO)
  
  So far, neither the environmental effect nor the stability of supplying these substances has been measured.
  
  Section::::Materials.:III–V and II–VI semiconductors.
   Section::::Application.
  
  Currently, industrial and manufacturing is the largest application market for piezoelectric devices, followed by the automotive industry. Strong demand also comes from medical instruments as well as information and telecommunications. The global demand for piezoelectric devices was valued at approximately US$14.8 billion in 2010. The largest material group for piezoelectric devices is piezoceramics, and piezopolymer is experiencing the fastest growth due to its low weight and small size.
  
  Piezoelectric crystals are now used in numerous ways:
  
  Section::::Application.:High voltage and power sources.
  
  Direct piezoelectricity of some substances, like quartz, can generate potential differences of thousands of volts.
   More recently, there is growing concern regarding the toxicity in lead-containing devices driven by the result of restriction of hazardous substances directive regulations. To address this concern, there has been a resurgence in the compositional development of lead-free piezoelectric materials.
   Piezoresistive effect
  
  The piezoresistive effect is a change in the electrical resistivity of a semiconductor or metal when mechanical strain is applied. In contrast to the piezoelectric effect, the piezoresistive effect causes a change only in electrical resistance, not in electric potential.
  
  Section::::History.
  
  The change of electrical resistance in metal devices due to an applied mechanical load was first discovered in 1856 by Lord Kelvin. With single crystal silicon becoming the material of choice for the design of analog and digital circuits, the large piezoresistive effect in silicon and germanium was first discovered in 1954 (Smith 1954).
  
  Section::::Mechanism.
   Piezotronics
  
  Piezotronics effect is using the piezoelectric potential (piezopotential) created in materials with piezoelectricity as a “gate” voltage to tune/control the charge carrier transport properties for fabricating new devices. Neil A Downie showed how simple it was to build simple demonstrations on a macro-scale using a sandwich of piezoelectric material and carbon piezoresistive material to make an FET-like amplifying device and put it in a book of science projects for students in 2006. The fundamental principle of piezotronics was introduced by Prof. Zhong Lin Wang at Georgia Institute of Technology in 2007.
   BULLET::::- IEC 444 (1973) Basic method for the measurement of resonance freq & equiv series resistance of quartz crystal units by zero-phase technique in a pi-network
  
  BULLET::::- IEC 302 (1969) Standard Definitions & Methods of Measurement for Piezoelectric Vibrators Operating over the Freq Range up to 30 MHz
  
  Section::::External links.
  
  BULLET::::- Piezoelectric cellular polymer films: Fabrication, properties and applications
  
  BULLET::::- Piezo motor based microdrive for neural signal recording
  
  BULLET::::- Research on new Piezoelectric materials
  
  BULLET::::- Piezo Equations
  
  BULLET::::- Piezo in Medical Design
  
  BULLET::::- Video demonstration of Piezoelectricity
  
  BULLET::::- DoITPoMS Teaching and Learning Package – Piezoelectric Materials
   Some metals display piezoresistivity that is much larger than the resistance change due to geometry. In platinum alloys, for instance, piezoresistivity is more than a factor of two larger, combining with the geometry effects to give a strain gauge sensitivity of up to more than three times as large than due to geometry effects alone. Pure nickel's piezoresistivity is -13 times larger, completely dwarfing and even reversing the sign of the geometry-induced resistance change.
  
  Section::::Mechanism.:Piezoresistive effect in bulk semiconductors.
   BULLET::::- DNA
  
  BULLET::::- Viral proteins, including those from bacteriophage. One study has found that thin films of M13 bacteriophage can be used to construct a piezoelectric generator sufficient to operate a liquid crystal display.
  
  Section::::Materials.:Synthetic crystals.
  
  BULLET::::- Langasite (LaGaSiO), a quartz-analogous crystal
  
  BULLET::::- Gallium orthophosphate (GaPO), a quartz-analogous crystal
  
  BULLET::::- Lithium niobate (LiNbO)
  
  BULLET::::- Lithium tantalate (LiTaO)
  
  Section::::Materials.:Synthetic ceramics.
  
  Ceramics with randomly oriented grains must be ferroelectric to exhibit piezoelectricity. The macroscopic piezoelectricity is possible in textured polycrystalline non-ferroelectric piezoelectric materials, such as AlN and ZnO. 
  
  The family of ceramics with perovskite, tungsten-bronze and related structures exhibits piezoelectricity:
   For the next few decades, piezoelectricity remained something of a laboratory curiosity, though it was a vital tool in the discovery of polonium and radium by Pierre and Marie Curie in 1898. More work was done to explore and define the crystal structures that exhibited piezoelectricity. This culminated in 1910 with the publication of Woldemar Voigt's "Lehrbuch der Kristallphysik" ("Textbook on Crystal Physics"), which described the 20 natural crystal classes capable of piezoelectricity, and rigorously defined the piezoelectric constants using tensor analysis.
  
  Section::::History.:World War I and post-war.
   Of the 32 crystal classes, 21 are non-centrosymmetric (not having a centre of symmetry), and of these, 20 exhibit direct piezoelectricity (the 21st is the cubic class 432). Ten of these represent the polar crystal classes, which show a spontaneous polarization without mechanical stress due to a non-vanishing electric dipole moment associated with their unit cell, and which exhibit pyroelectricity. If the dipole moment can be reversed by applying an external electric field, the material is said to be ferroelectric.
  
  BULLET::::- Polar crystal classes: 1, 2, m, mm2, 4, , 3, 3m, 6, .
   As very high electric fields correspond to only tiny changes in the width of the crystal, this width can be changed with better-than-µm precision, making piezo crystals the most important tool for positioning objects with extreme accuracy—thus their use in actuators.
  
  Multilayer ceramics, using layers thinner than , allow reaching high electric fields with voltage lower than . These ceramics are used within two kinds of actuators: direct piezo actuators and Amplified piezoelectric actuators. While direct actuator's stroke is generally lower than , amplified piezo actuators can reach millimeter strokes.
   In 2015, Cambridge University researchers working in conjunction with researchers from the National Physical Laboratory and Cambridge-based dielectric antenna company Antenova Ltd, using thin films of piezoelectric materials found that at a certain frequency, these materials become not only efficient resonators, but efficient radiators as well, meaning that they can potentially be used as antennas. The researchers found that by subjecting the piezoelectric thin films to an asymmetric excitation, the symmetry of the system is similarly broken, resulting in a corresponding symmetry breaking of the electric field, and the generation of electromagnetic radiation.
   There are a number of different dielectric mechanisms, connected to the way a studied medium reacts to the applied field (see the figure illustration). Each dielectric mechanism is centered around its characteristic frequency, which is the reciprocal of the characteristic time of the process. In general, dielectric mechanisms can be divided into relaxation and resonance processes. The most common, starting from high frequencies, are:
  
  Section::::Dielectric mechanisms.:Electronic polarization.
  
  This resonant process occurs in a neutral atom when the electric field displaces the electron density relative to the nucleus it surrounds.
   The use of piezoelectricity in sonar, and the success of that project, created intense development interest in piezoelectric devices. Over the next few decades, new piezoelectric materials and new applications for those materials were explored and developed.
   A series of electronic devices have been demonstrated based on this effect, including piezopotential gated field-effect transistor, piezopotential gated diode, strain sensors, force/flow sensors, hybrid field-effect transistor, piezotronic logic gates, electromechanical memories, etc. Piezotronic devices are regarded as a new semiconductor-device category. Piezotronics is likely to have important applications in sensor, human-silicon technology interfacing, MEMS, nanorobotics and active flexible electronics.
  
  Section::::Mechanism.
   The strain-charge for a material of the 4mm (C) crystal class (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO) as well as the 6mm crystal class may also be written as (ANSI IEEE 176):
  
  where the first equation represents the relationship for the converse piezoelectric effect and the latter for the direct piezoelectric effect.
   The first practical application for piezoelectric devices was sonar, first developed during World War I. In France in 1917, Paul Langevin and his coworkers developed an ultrasonic submarine detector. The detector consisted of a transducer, made of thin quartz crystals carefully glued between two steel plates, and a hydrophone to detect the returned echo. By emitting a high-frequency pulse from the transducer, and measuring the amount of time it takes to hear an echo from the sound waves bouncing off an object, one can calculate the distance to that object.
  