- #1
achillesheels
- 8
- 1
- How did you find PF?
- Searching for the Lorentz Harmonic Oscillator
Greetings,
I'm happy to find such an enthusiastic community with an encyclopedic knowledge and mathematical rigor. I'm a Biomedical Engineering Researcher that's had to breach into the world of condensed matter physics to better understand the physical principles of the piezoelectric crystal oscillator. My goal is to understand these physical principles using linear classical mechanical modeling in order to find a mathematical correspondence/consistency with the electrical engineering model of the crystal's electromechanical effects (I like to be thorough).
I was unfamiliar with the physics behind the mathematics of the engineering sciences (linear time-invariant systems theory) and went down a very interesting path to understand the mathematical science's ability to model physical reality: applying it to the Davisson-Germer experiment (see Nobel Prize address and experiment summary from Physics Today). Upon inspection, I learned that the original experimenters did not much consider the thermal or radiation effects of the crystal in the interaction with the electron beam - they used a geometrical optics frame of reference of the electron "wave" phenomenon, i.e. "Bragg's Law". My further literature review (I was enlightened to learn crystallography was very cutting edge coinciding the quantum discoveries of the early 20th century!
) did not have a satisfactory mathematical model of the atomic electromagnetic effects in the crystallography science. There is some mathematical effort at harmonic analysis, i.e. reciprocal space, to realize the self-evident "lattice" vibrations, and thermodynamic considerations, i.e. The Debye Model, but not a mathematical model which can justify the electromechanical effects of matter when disturbed by an electromagnetic force, i.e. physical causality independent of time (or "frequency-dependent" causes).
I am on this site to help better understand the feasibility of the Lorentz Harmonic Oscillator model and its application in explaining the periodical intensity pattern from the Davisson-Germer experiment. This compels a more sophisticated observation of the phenomenon which involves a classical mechanical diagram of the polarization of the nickel crystal upon electron elastic collision (or a "driven damped harmonic oscillator" of the dipole moment). This permits the interpretation of the Davisson-Germer electron diffracted intensity pattern as simple harmonic motion caused by a linear restoring force of the electromechanical material resonance (by the nickel ions) when effected by a DC input (the electron beam) and motivates the general understanding of the crystal harmonic oscillator phenomenon as a linear time-invariant event.
This interpretation can motivate a more general representation of physical causes as theoretically linear time-invariant electromechanical frequency responses due to the theoretically scalar (linear) nature of the mathematical graphical diagramming (block diagram representation). And this dielectric material interpretation permits modeling the DNA molecule as an analog filter (indeed DNA conductance and polarity has been demonstrated in the scientific literature). This would be valuable to my research's objective of formalizing a mathematical model of biological cellular replication, i.e. evolutionary biology, as bounded discrete-time control systems.
I've attached some of the references I have compiled in better understanding the classical mechanical model and would appreciate any feedback on its application in the presented hypothesis.
Sincerest Regards,
Joseph A. Hazani
I'm happy to find such an enthusiastic community with an encyclopedic knowledge and mathematical rigor. I'm a Biomedical Engineering Researcher that's had to breach into the world of condensed matter physics to better understand the physical principles of the piezoelectric crystal oscillator. My goal is to understand these physical principles using linear classical mechanical modeling in order to find a mathematical correspondence/consistency with the electrical engineering model of the crystal's electromechanical effects (I like to be thorough).
I was unfamiliar with the physics behind the mathematics of the engineering sciences (linear time-invariant systems theory) and went down a very interesting path to understand the mathematical science's ability to model physical reality: applying it to the Davisson-Germer experiment (see Nobel Prize address and experiment summary from Physics Today). Upon inspection, I learned that the original experimenters did not much consider the thermal or radiation effects of the crystal in the interaction with the electron beam - they used a geometrical optics frame of reference of the electron "wave" phenomenon, i.e. "Bragg's Law". My further literature review (I was enlightened to learn crystallography was very cutting edge coinciding the quantum discoveries of the early 20th century!
) did not have a satisfactory mathematical model of the atomic electromagnetic effects in the crystallography science. There is some mathematical effort at harmonic analysis, i.e. reciprocal space, to realize the self-evident "lattice" vibrations, and thermodynamic considerations, i.e. The Debye Model, but not a mathematical model which can justify the electromechanical effects of matter when disturbed by an electromagnetic force, i.e. physical causality independent of time (or "frequency-dependent" causes).I am on this site to help better understand the feasibility of the Lorentz Harmonic Oscillator model and its application in explaining the periodical intensity pattern from the Davisson-Germer experiment. This compels a more sophisticated observation of the phenomenon which involves a classical mechanical diagram of the polarization of the nickel crystal upon electron elastic collision (or a "driven damped harmonic oscillator" of the dipole moment). This permits the interpretation of the Davisson-Germer electron diffracted intensity pattern as simple harmonic motion caused by a linear restoring force of the electromechanical material resonance (by the nickel ions) when effected by a DC input (the electron beam) and motivates the general understanding of the crystal harmonic oscillator phenomenon as a linear time-invariant event.
This interpretation can motivate a more general representation of physical causes as theoretically linear time-invariant electromechanical frequency responses due to the theoretically scalar (linear) nature of the mathematical graphical diagramming (block diagram representation). And this dielectric material interpretation permits modeling the DNA molecule as an analog filter (indeed DNA conductance and polarity has been demonstrated in the scientific literature). This would be valuable to my research's objective of formalizing a mathematical model of biological cellular replication, i.e. evolutionary biology, as bounded discrete-time control systems.
I've attached some of the references I have compiled in better understanding the classical mechanical model and would appreciate any feedback on its application in the presented hypothesis.
Sincerest Regards,
Joseph A. Hazani