Electron Matter-Waves as Electromagnetically Induced Crystal Oscillator Effects


 The famed Davisson-Germer Experiments demonstrated the wave phenomenon of electrons similarly to X-Ray scattering from Sir Lawrence Bragg’s X-ray experimentations on crystals c. 1913. Their empirical deduction of electrons behaving as waves (i.e. oscillatory) ignores the possibility of an electron beam behaving harmonically upon elastic collision with a diffraction grating - represented by nickel crystal - in their experiment. However, it is well established in the electrical engineering science that crystals possess piezoelectric effects and are used ubiquitously in electronic circuit designs for causing stable harmonic oscillation responses to direct current voltages. In light of this, the current mathematical model proposes the Davisson-Germer results to be the effect of a nickel crystal oscillator circuit which amplifies a direct voltage source – the electron beam – causing the phenomenon of inductance from the resultant electrical feedback with the crystal atom’s electromagnetic field.


II. Crystal Lattice Structure
The Bragg's experimental history was inspired by the newfound discovery of X-rays and the conceivability of empirical phenomena on the quantum scale. Bragg et al determined the atomic spacing of their experimental crystals effecting "special reflection angles" of "homogenous x-rays": 1 "It often happens that the rays emerging from the bulb slit and falling on the crystal contain a large preponderance of rays of a given quality which can only be reflected at a certain angle…When…we see a maximum persisting in the same angular position…for several successive positions of the crystal, we know that we have a case of this special reflection. There is a relatively large quantity of very homogeneous radiation of a certain kind present in the radiation from the bulb." 2 (emphasis added) Bragg proposed an explanation for this homogeneity by conceiving of the crystal as a lattice of atoms: "There is strong evidence for supposing that the atoms of a cubic crystal-like rock-salt, containing two elements of equal valency, are arranged parallel to the planes {100} in planes containing equal numbers of sodium and chlorine atoms. The atoms in any one plane are arranged in alternate rows of each element, diagonal to the cube axes, successive planes having these rows opposite ways." 3 Yet their static geometrical optics relationship (1), diagraming what has been determined to be an elastic collision phenomenon, does not sufficiently describe the dynamics of energy transfer in the observable system. In fact, it is admitted that the "special reflections" are distinct between crystals: "The angles at which the special reflections of these rays take place are not the same for all crystals, nor for all faces of the same crystal." 4 "It was left to William Lawrence Bragg to demonstrate that the diffraction pattern was due to the reflection of the 'white' Bremsstrahlen of the primary beam on the crystal planes that selected certain wavelengths for the diffraction pattern by what became known as the 'Bragg condition '." 8 In other words, the question of whether the X-rays were an effect, i.e. "manufactured", of electromagnetic energy released from the lattice electrons during the X-ray beam collision was answered: the crystal was acting like a diffraction grating for the primary monochromatic X-ray beam, with a precise relation to its atomic lattice spacing. 9 Laue's own Nobel Address further remarks on this curiosity: 10 "I was suddenly struck by the obvious question of the behaviour of waves which are short by comparison with the lattice-constants of the space-lattice. And it was at that point that my intuition for optics suddenly gave me the answer: lattice spectra would have to ensue. The fact that the lattice-constant in crystals is of an order of 10 -8 cm was sufficiently known from the analogy with other interatomic distances in solid and liquid substances, and, in addition, this could easily be argued from the density, molecular weight and the mass of the hydrogen atom which, just at that time, had been particularly well determined. The order of X-ray wavelengths was estimated by Wien and Sommerfeld at 10 -9 cm. Thus, the ratio of wavelengths and lattice-constants was extremely favourable if X-rays were to be transmitted through a crystal. I immediately told [Paul Peter] Ewald that I anticipated the occurrence of interference phenomena with X-rays." Yet, upon such a conclusion, there is no consideration of the natural harmonics of the "lattice-space hypothesis" medium of transmission in Braggs' nor Laue's analysis. Indeed, it is self-evident there are crystal lattice atomic movements which occur as vibrations (lattice vibrations). 1112 And due to the regularity of the atoms, it is intuitive to hypothesize the uniform, i.e. isotropic, motion of atoms as a piezoelectric oscillator, as has been proposed. 13 This intuition was apparent during the time of these experiments. The animation of the atoms by de facto thermal motion is what Peter Debye aimed to characterize in the X-ray crystallography experiments in 1913. 1415 However, it must be mentioned that Debye's model assumes photon-electron energy transfer, and thus includes by necessity an elastically determined electrodynamical effect on the x-ray beam: "It was at this time that Mark and Wierl presented a preliminary description of their investigations showing that the Debye formula descriptive of the scattering of x rays by a gas…could also be applied to describe the scattering of electron rays by gases. Physically, there is one difference. The electron interferences provide information about the positions of the atomic nuclei themselves, while the x-ray interferences reveal the locations of the centers of gravity of the electron clouds about them. What is really ascertained in either case is the position of the atom centers, the desired quantity." 16 (emphasis added) In other words, when including the "thermal fluctuations" within the crystal atom in interpreting the diffraction results, the X-ray (or electron ray) atomic collision in a lattice crystal can still be conceived of as classical elastic energy transfer. Yet still, a piezoelectric oscillator, i.e. resonator, model of the crystal lattice initialized by an electromagnetic radiation disturbance (e.g. the X-ray beam) was not published by any of the researchers. Indeed, neither was there a consideration, assuming an isotropic crystal vibration, of the electric polarization effecting the motion of the electrons or x-rays.
This, despite the advancements in the science of x-ray diffraction which incorporates polarization corrections to the scattering angle, i.e. Lorentz-Polarization Correction, and the calculation of dipole effects on light scattering. 1718

III. De Broglie's Matter-Wave Hypothesis
Meanwhile, de Broglie's intuition was predicated on his curiosity of Max Plank's resolution of the UV catastrophe in black body radiators with the introduction of the "quantum" phenomenon: "Planck was led to assume that only certain preferred motions, quantized motions, are possible or at least stable, since energy can only assume values forming a discontinuous sequence. This concept seemed rather strange at first but its value had to be recognized because it was this concept which brought Planck to the correct law of black-body radiation and because it then proved its fruitfulness in many other fields." 19 He then investigated a relativistic relation between the wavelength of an electromagnetic wave and linear momentum out of necessity of taking into account the frequency nature of electromagnetic phenomena: When I started to ponder these difficulties two things struck me in the main. Firstly the light-quantum theory cannot be regarded as satisfactory since it defines the energy of a light corpuscle by the relation W = hv which contains a frequency v. Now a purely corpuscular theory does not contain any element permitting the definition of a frequency. This reason alone renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity. I thus arrived at the following overall concept which guided my studies: for both matter and radiations, light in particular, it is 17 https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Modules_and_Websites_(Inorganic_Chemistry)/Crysta llography/X-rays/Lorentz-polarization_correction 18 ECE 695S Lecture 04: Light Interaction with Small Structures (https://nanohub.org/resources/19363/supportingdocs) necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. 20 (emphasis added) It is this consideration which helped him derive a rectilinear measurement of periodical motion from theorizing the linear momentum of a light-quantum to move with a longitudinal phase velocity: "The quantity; λ is the distance between two consecutive peaks of the wave, i.e. the 'wavelength'. Hence: λ=h⁄ρ This is a fundamental relation of the theory." 21 This is the wave-momentum relation mathematically induced by de Broglie that corresponded to the scattering experienced and published by Davisson and Germer in their experiments on electron beams. In that, the Bragg relation of x-ray wavelength and crystal atomic spacing was determinant according to the similar geometric scattering angle of the x-ray energy in Bragg's crystal experiment: λ=2dsinθ (1) Thus, Davisson and Germer analytically arrived at the hypothetical electron wavelength within de Broglie's relation when computing Bragg's relation -itself a priori assuming the geometrical optics of an X-ray wave reflectance -with the experimentally measured scattering angle of the electron. This was computed with a classical kinetic energy relation when applied to electrodynamical voltages: This experiment, which presupposes the electron diffraction approximation between Bragg's scattering angle determination and de Broglie's a priori wavelength derivation, resulted in two Nobel 20 Ibid. 21 Ibid. prizes in 10 years: one for the theoretician and one for the experimentalist. From Davisson's Nobel Address: "It was brought out that light imparts energy to individual electrons in amounts proportional to its frequency and finally that the factor of proportionality between energy and frequency is just that previously deduced by Planck from the black-body spectrum. The idea of pressing the witness on the latter point had come from Einstein who outplancked Planck in not only accepting quantization, but in conceiving of light quanta as actual small packets or particles of energy transferable to single electrons in toto." 22 And this was further confirmed by Compton: The case for a corpuscular aspect of light, now exceedingly strong, became overwhelmingly so when in 1922 A. H. Compton showed that in certain circumstances light quantaphotons as they were now called -have elastic collisions with electrons in accordance with the simple laws of particle dynamics. 23 The experiments furthermore coincided with other discoveries of the wave motion behavior of matter, such as with helium and neutrons. 26 Yet despite the matriculation of the concept of "matterwaves" from these series of experiments, there is a lack of discourse on the proper mathematical characterization of the motion, i.e. mechanics, in the observation system. Namely, Bragg's relation in equation (1)

IV. Piezoelectric Effects
To more richly diagram the mechanics of the system, and to support the hypothesis that Davisson- "It is as a consequence of their asymmetrical (electrical) structure that most molecules possess a permanent dipole moment; the magnitude of this characteristic entity is a quantitative measure of the polarity of the molecule… Debye reasoned that such asymmetric molecules must possess finite and permanent electrical moments and that their total electrical polarizations result from two contributions, a displacement of electrons and atoms in the molecule and an orientation in the electrical field of the molecule as a whole.   "The concept of the orientation of dipolar molecules in an electric field, this time an alternating one, was applied by Debye (1913) in the explanation of the behavior of the two dielectric constants, real and imaginary, that are to be observed. (Permittivity and loss factor are better terms when frequency dependence is involved.) The basic principle is that when the field is applied, or released, a finite time will be required for the molecules to come to their equilibrium orientation because there is a viscous resistance to these rotatory motions. The range of frequency over which the real dielectric constant is variable extends from the static field to one that oscillates so rapidly as not to provide for any rotational motion of the polar molecule at all; the theory thus describes a typical molecular relaxation process." 38 (emphasis added) Thus, the crystal lattice atomic polarity can be theorized as behaving analogously to a Lorentz Harmonic Oscillator piezoelectric material in the presence of an electro-mechanical disturbance, as a classical mechanical model of the electron and X-ray collisions (i.e. corpuscular theory) can.
As such, there is the introduction of electromagnetic theory to classical forces in piezoelectric phenomena providing the bridge between the classical and the quantum in helping to interpret the experimental observations of Davisson and Germer and explain the successful advances of the electrical engineering sciences, e.g. the Fast-Fourier Transform. Namely, the intensity diagrams demonstrate the quantum collisions are dependent on the natural frequency, i.e. resonance, caused by a driven electromagnetic force (the X-Ray or electron). It is precisely at the juncture of electromagnetic polarity where the dynamics can be electro-mechanically equated and diagrammed using electrical circuit equivalence.
We can comparatively mathematically characterize the oscillation effects within an electromagnetic, i.e. frequency-dominant ergo time-invariant, model. In this manner, we can ascertain predictability in the observation of harmonic patterns with the theoretical electro-mechanically induced resonant frequency of the crystals, including the existence of overtone frequencies to explain the multiplicity of special reflection angles. Hypothetical resonant frequency effects from the Lorentz Harmonic Oscillator approximations will indicate the presence of dipole moments during photon (e.g. X-ray) and electron crystal elastic collisions. And it further suggests linear time-invariant radiation transfer, i.e. frequency-domain causality, universally in all physical phenomena.

V. Electromechanical Equivalent Circuit
A traditional Piezoelectric crystal oscillator circuit model is diagrammed below: Next consider its collision with another electrical body extending with center B at a selfcertain occurrence diagrammed at midpoint C. This collision can be found, for instance, in a piezoelectric oscillation event, and diagrammed as a linear time-invariant operator. Our analysis can be informed deeper with the understanding of the complex frequency number, a necessarily periodical and numerically complex self-certain agreement between the position of the point moving uniformly on the circle and its center, 42 to inspect the uniform motion of the mass of the electrical body to be mechanically rotating around a fixed center, circularly moving invariantly to time, with known magnitude as a purely numerical measure of its quantity of motion in volts. It is through this understanding where we can mathematically completely diagram the collision at point C as an electromagnetic radiation construction phenomenon using complex conjugate summation to determine the amplitude extension of the phase-shift oscillation as the length CF: In other words, it is claimed electron elastic collisions, which can be applied to the Davisson- z = a + jb = rcosw + jrsinw = re jw |z| = sqrt(a 2 + b 2 ) = r z = |z|e jw < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 O A g h N / z S o J h 3 1 x c s A / A 3 F 5 b b W U = " > A A A C L H i c b V B b S w J B G J 2 1 m 9 n N 6 r G X J Q k M Q X Y t q J c F y Z c e D f I C r s r s O K u j s 7 P b z G y i q z + o l / 5 K E D 0 k 0 W u / o / E S l H Z g 4 H D O + f j m O 0 5 A i Z C G M d F i a + s b m 1 v x 7 c T O 7 t 7 + Q f L w q C z 8 k C N c Q j 7 1 e d W B A l P C c E k S S X E 1 4 B h 6 D s U V p 1 e Y + p V H z A X x 2 b 0 c B L j u w T Y j L k F Q K q m Z L A w t m O k 6 F k e + i P r j T F d F m S I W x 4 2 o 2 x / b i d F w Z I k H L t O w k c s 4 j d y 5 x e 3 E 0 F L y T 6 K Z T B l Z Y w Z 9 l Z g L k g I L F J v J V 7 v l o 9 D D T C I K h a i Z R i D r E e S S I I r H C T s U O I C o B 9 u 4 p i i D H h b 1 a H b s W D 9 T S k t 3 f a 4 e k / p M / T 0 R Q U + I g e e o p A d l R y x 7 U / E / r x Z K 9 7 o e E R a E E j M 0 X + S G V J e + P m 1 O b x G O k a Q D R S D i R P 1 V R x 3 I I Z K q 3 2 k J 5 v L J q 6 S c y 5 o X 2 d z d Z S p / s 6 g j D k 7 A K U g D E 1 y B P L g F R V A C C D y B F / A O J t q z 9 q Z 9 a J / z a E x b z B y D P 9 C + v g F v B 6 h I < / l a t e x i t > amplitude gain caused by electromagnetic induction as the empirical phenomenon of extending imaginary quantities of magnetic energy into real electrical quantities measurable in the continuous-time domain. This measurement can be ideally diagrammed as a phasor in a polar coordination system with a complex exponential evaluated at the special reflectance angles in the crystal experiments. It is conjectured this is the principle phenomenon of all radiation transfer in electrical feedback circuitsand indeed in all electromagnetic constructive interference. This is most especially relevant in biological tissue which is principally piezoelectric in nature. operations, contrary to the experimenters judgments of the electron phenomenon being optical.
Such electromagnetic phenomena include biological physical matter which has been extensively demonstrated to be piezoelectric and electroconductive. including DNA, amino acids, etc. 44 This empirical evidence justifies inspecting cellular biological organisms as stable electromagnetic radiation transmission systems, with their continuous-time adaptations mathematically formalized as feedback control responses. It follows that there is a correspondence between feedback order and cellular biological fitness or the quantitatively measurable ability for the cellular organism to replicate its BIBO stable radiation propagation in time.