# Piezoelectricity and the Lorentz Harmonic Oscillator?

achillesheels
TL;DR Summary
Anyone familiar with the Material Polarization mathematics of piezoelectricity? Why can't it be explained more elegantly with a simple harmonic motion mathematical model?
Hi!

As I outlined in my https://www.physicsforums.com/threads/hello-reality-anyone-familiar-with-the-davisson-germer-experiment.985063/post-6305937, I'm curious to ask if there is anyone with knowledge on the theory of the piezoelectric effect on this forum? I think it's fascinating how a crystal oscillator can be diagrammed with an RLC network circuit. I ran into this in trying to understand the Davisson-Germer experiment without geometrical optics but as an electromagnetic disturbance. It piqued my interest into understanding the piezoelectric material science of crystals.

I naturally approached this with an electromagnetic perspective - because of the electrical circuit analogy (duh ). So I could not help but intuitively perceive the intrinsic piezoelectric motion to be like a simple harmonic oscillator, or a resonance effect of energy transfer. And it turns out there is a frequency-dependent mathematical model which fits consistently with the electrical circuit diagramming called the Lorentz Harmonic Oscillator model. Does anyone much about the linear model of the piezo-electric effect (https://en.wikipedia.org/wiki/Piezoelectricity#Mathematical_description)? I think the Lorentz Harmonic Oscillator model is much more elegant and can explain deeper the harmonic oscillation effects of crystal oscillator circuits. I also like the frequency-dependent mathematics which the piezoelectric model does not use (it incorporates tensors to describe the mechanical stress and it does not quantify any lattice vibrations.)

Anyways, I'd love to learn more about how the piezoelectric science can be used to explain the electromagnetic effects ability to use crystals in electronic circuits. I've also attached a couple of "fundamentals of piezoelectricity" summaries I found online which presents the current mathematics in the science. Thank you for your time.

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• fundamentals of piezoelectricity.pdf
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• (more fundamentals of piezo) 2013_Bookmatter_RoboticTactileSensing.pdf
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Gold Member
I would recommend "Linear Piezoelectric Plate Vibrations" by H. F. Tiersten pub. Springer, as a detailed and highly readable source.

The sources you've given are quite nice in and of themselves. I'm not certain what your question is exactly. The RLC network approximation for garden variety quartz resonators works only in the neighborhood of the main resonate frequency, Typical crystals I've played[1] with exhibit "spurious" resonances which correspond to more complex plate modes than the fundamental bulk shear mode.

[1] typically 2-10MHz AT cut crystals.

achillesheels
achillesheels
I would recommend "Linear Piezoelectric Plate Vibrations" by H. F. Tiersten pub. Springer, as a detailed and highly readable source.

The sources you've given are quite nice in and of themselves. I'm not certain what your question is exactly. The RLC network approximation for garden variety quartz resonators works only in the neighborhood of the main resonate frequency, Typical crystals I've played[1] with exhibit "spurious" resonances which correspond to more complex plate modes than the fundamental bulk shear mode.

[1] typically 2-10MHz AT cut crystals.

Thank you for the resource! I guess my question is if the physical nature of the quartz resonance has ever been treated as a linear harmonic oscillator according to the dipole moments in the crystal in the presence of an electric field disturbance? The ad hoc equivalence circuit approximation does not seem to have a physically mechanical justification for why it is a mathematically effective model?

Gold Member
I guess my question is if the physical nature of the quartz resonance has ever been treated as a linear harmonic oscillator according to the dipole moments in the crystal in the presence of an electric field disturbance?

I don't think that's the mechanism. The harmonic oscillation is entirely mechanical and driven by the geometry of the crystal. In an AT cut crystal these are most often 1/2 wave volume shear waves through the thickness of the crystal plate. The oscillation frequency is determined by the plate thickness and the velocity of acoustic waves in the crystal.[1] The only thing happening at the atomic level is the stress introduced by the applied field.

[1] I've dissected crystals[2] and measured the plate thickness. It's right on the expected value.

[2] This is why I can't have nice things.

Klystron and hutchphd
achillesheels
Oh that’s interesting, the oscillation is an acoustic effect and not electrical? I would expect the (inverse) piezoelectricity to require polarization of the material on the atomic level to, say, cause a phase shift in a DC input to make an AC waveform.

Gold Member
Oh that’s interesting, the oscillation is an acoustic effect and not electrical?
It's electro mechanical, right. One drives oscillation in the crystal plate electrically and there is a back reaction on the drive circuit.

achillesheels
achillesheels
That's great insight on oscillation mechanics, thanks. I found another resource this time titled "the piezoelectric quartz resonator" p216 (attached):

This description of the reverse current effects of the piezoelectric stress-and-strain (to produce AC) is what I was curious to learn about...I don't think this can be caused by acoustics because it necessarily involves an electromagnetic effect...I'm wondering if the current flow through the crystal causes the crystal vibrations, and are ultimately the effect of simple harmonic motion from the elastic collisions of the current's electrons with the crystal atoms' dipole radiation field? How else can self-induction be explained in a piezoelectric material?

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• AM30_214.pdf
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Gold Member
How else can self-induction be explained in a piezoelectric material?
I think you are over thinking it and incorrect.

I've done tests where a short pulses drive a 4 MHz crystal. These pulses are issued at 10s of kHz rep rates. Each pulse is much shorter (5 to 10 ns) than the 250 ns crystal period. The Q-value is such that the ring down of the crystal is quite long, much more that 100 crystal cycles between pulses. This signal voltage between drive pulses is generated by the free vibration of the crystal mass and the piezoelectric nature of the crystal material.

The coupling between the crystal vibration and the drive voltage is weak. In these tests, the frequency of the pulses must accurately match a sub harmonic of the fundamental mode in order that sufficient vibration is induced.

achillesheels
I think you are over thinking it and incorrect.

I see. And the author is in agreement with you - that it is a mechanically resonant - dominant behavior. From p220 of the previous attachment:

So the elastic collisions from the current cause mechanically stored potential energy (hence the resonance) which can operate as a frequency-selective filter. But can this mechanical resonance cause a 360 degree phase shift, i.e. simple harmonic oscillation, to a DC driven-input? Isn't this one type of useful application?

Gold Member
But can this mechanical resonance cause a 360 degree phase shift, i.e. simple harmonic oscillation, to a DC driven-input?
I don't follow. At DC a typical AT cut resonator looks like a 2pF capacitor or so. All this follows from the RLC equivalent circuit. You seem to have some fixation with elastic collisions? My suggestion is read the references. Mechanically there are the Hook's law potential energy (associated with the C) and the vibrational kinetic energy (associated with the L) all weakly coupled to the applied field (associated with the applied sinusoidal voltage).

achillesheels
You seem to have some fixation with elastic collisions?

Lol I do 😅

Elastic collisions were very critical in the development of crystallography diffraction experiments last century (Bragg, Laue, Debye, Rutherford, Davisson, etc.) and I'm trying to develop a deeper mathematical understanding of their scattering patterns which do not rely on geometrical optics (treating everything as classically mechanical waves) and instead on electromagentics.

I appreciate your help, piezoelectricity is fascinating! How did you get into playing around with them?

Gold Member
How did you get into playing around with them?
I was looking at quartz as a potential radio frequency gravitational wave detectors. There is a group in Australia that tried cryogenically cooled quartz resonators and obtained obscenely high Q-values. My approach to the problem has been quite different. At any rate I've spent (some might say wasted) much time looking at the physics of garden variety crystals.

achillesheels
achillesheels
FYI for anyone interested in this subject matter, I found a great YouTube video which explains visually the dipole moment that is created in a piezoelectric material:

I wish he shared his references, but this is what I was curious to learn more about. It seems polar bonds make the dielectric effect possible under mechanical stress.