Mechanical resonance dispersion (MRD) has been applied to such diverse systems as whale blubber [1], violin wood [2], human cancellous bone, natural rubber, synthetic polymers, and a variety of single crystal and polycrystalline solids, both elements and compounds (other references below in context), with the goal of understanding the mechanical response of the solid to periodic stress in the audiofrequency range. According to Edwin R. Fitzgerald [3], the complex (hence "dispersion") shear compliance, J*=J^'-iJ^'', with J^' equal to the (energy) storage compliance and J^'' equal to the (energy) loss compliance. This is the manifestation of a periodic shear stress, α, resulting in a periodic strain, a, with both in phase and out of phase components: a = s₀

In a typical Fitzgerald experiment [3], the sample disk is clamped between piezoelectric transducers and is subjected to periodic shear stress at specific narrowly spaced frequencies. J' and J" are calculated at each frequency from knowledge of the electrical circuit [1], yielding a "spectrum" for a particular material. The details of the spectrum are a function of the chemical composition, including impurities, the crystallinity, the previous processing (e.g., sintering, fusion, quenching, or annealing), and to a lesser extent the static stress on the sample and the duration of the experiment.

The classical mechanical model predicts a single resonance frequency, but nearly all spectra, including those of elements, show multiple resonances. To explain these and other details, Fitzgerald [4] invoked a momentum transfer equation for a one dimensional lattice of N atoms (1, 2, ...N) of mass m and velocities, v₁,v₂,...,v_n-1,v_nv_n-1,...v_N,,

where K_p is the momentum transfer constant, found to be equal to $\mathrm{ih}/\left(4\pi d^2\right)$, with h equal to Planck's Constant, m the atomic mass, and d the interatomic spacing.

Applying boundary conditions to standing particle wave solutions to (1), the Planck-Einstein Equation, E = hv, plus the deBroglie Relation, pλ = h, Fitzgerald derived the particle-in-a-box normal mode frequencies

from which we obtain the operational equation for interpreting MRD spectra,

Here S equals Nd, the crystal mosaic size, and q a quantum number 1, 2, ... N-1. A standing particle wave is thus fixed between the endpoints of the mosaic separated by S and can have N-1 wavelengths from 2S to 2d. Since momentum is conserved in each of three dimensions, Eq. (1) is applicable to any row of like atoms in a crystal. An identical result, Eq. (2), was obtained [4, 5] by applying the time-dependent Schrödinger Equation.

For many materials a common dimension for crystal mosaics is on the order of 1 μm. This adds a further criterion for correct spectral assignments according to Eq. (3):S ≃ 1 μm. Of course, elements chosen, which determine m, must be known to be present in the sample or likely to be present based on sample origin and history.

Interpretation of a spectrum of resonance frequencies by this model will consist of choosing an element or an isotope for a highly resolved spectrum and a value of q for each resonant frequency such that a common value of S is calculated by Eq. (3) for all resonant frequencies in the spectrum.

This has been done only rarely previously for multiple resonances [6-9] where frequency series with ratios 1: 4: 9: 16…q² were found (twice retrospectively). Such a complete spectrum is not observed in any of the 38 spectra interpreted below, and never without additional resonances, hence the need to identify previously unrecognized sample impurities in the large majority of MRD spectra. The purpose of this article is to apply this formula (Eqs. (2) and (3)) to previously uninterpreted MRD spectra and to address some of the earlier criticism of Fitzgerald's work.

Experimental resonant frequencies V listed in the following tables were either specifically mentioned in text or tables by the authors of the respective references or were taken from their spectra figures by this author. Eq. (3) was used to calculate S from each frequency and the assigned values of atomic or isotopic mass and q. The average value of S, S_av, for each sample was used to calculate V_c from Eq. (2) for comparison with experimental values, V. Throughout the following tables, an element symbol implies no resolution of isotopes or a monoisotopic element and the use of the atomic weight [10] in determining m for Eq. (3). A specific isotope designation implies use of the specific isotopic mass [11] in Eq. (3).

One of Fitzgerald's critics, R. W. Leonard, made the argument [35] that he could produce normal (elastic) resonances in the audiofrequency range with an 8 ft lead rod, and therefore there need be no other explanation for resonances in this frequency range. He believed the Fitzgerald resonances to be artifacts of the apparatus, especially since so many occurred in the 2800-3000 Hz window.

Table 26 gives Leonard's data (V_obs) read off his semi log strip chart as reproduced in Refs. [21] and [35]. Standing waves were assigned in column 1, and the speed of sound (C_S) was calculated using the standing wave formulae, Eqs. (4) and (5). A consistent value of C_S was obtained (C_{S av} = 1041 m/s), compared to the value of 1190 m/s for annealed lead from Ref. [36]. Thus it is proven that these are natural frequencies due to standing wave modes. However, the appearance of an additional resonance at 205 Hz is attributed to a particle wave mode (q = 1), with additional such modes, q = 2,3 q = 2, 3, coincident with the fourth (850 Hz), and ninth (1900 Hz) standing wave modes, respectively. These three inelastic modes give a consistent value of S (Sav = 1.07 μm) from Eq. (3). Elastic frequencies, $v_c^E$ (Hz), are calculated from Eq. (4) using C_{S av} = 1041 m/s. Particle wave frequencies, $v_c^p$ (Hz), are calculated from Eq. (2) and S_av as in all the above Tables.

Thus it is seen that a large enough object (8 ft !) can give elastic mode resonances in the audiofrequency range, but that, in spite of some overlaps, these are distinct from the inelastic particle wave resonances in the same range and in no way disprove the existence of the latter.

In arguing that the Fitzgerald resonances were instrumental artifacts, Leonard stated [35] that "The results on all hard crystalline materials yield a resonance at a frequency very near 2890 cps ... ." (emphasis ours) We summarize our results by element for the range 2790-2990 Hz (2890± 3.5% in Table 27. In only 20 (not "all") of 38 samples analyzed here were resonances found in that range. Copper was responsible in 10 cases, but in all of these copper was a major constituent or known impurity in the sample. Iron was the next most commonly observed element (three samples). Since copper and iron are components of the apparatus, the suspicion of Leonard might be warranted. However, in 18 of 38 samples no resonance was found in the target range. Due to the variety of elements in Table 27, many of which are not found in the seven different apparati used, and the 18 occurrences of no resonances at all in the target range, it seems unlikely that an apparatus constant (causing resonance at near 2890 Hz) is involved.

Leonard also noted that fused quartz is amorphous, while the Fitzgerald Theory predicts resonances for crystalline solids with distinct rows of equally spaced atoms. More recent research [37] shows that fused quartz has medium range order, 20 Å, which might be preferentially aligned to allow for much longer linear ordering. Obviously, a complete study of all forms of SiO₂, including glasses, with measured impurities would be revealing. It is proposed here that crystallinity is not required for these resonances, only a field free region which extends along a line without interruption of fixed particles. As noted below, more research in amorphous and partially ordered materials is recommended.

Another of Fitzgerald's critics, I. L. Hopkins, measured resonances of steel balls of various diameters in the audiofrequency range, and like Leonard he concluded that quantum mechanics was not necessary to explain elastic phenomena in the audiofrequency range [38, 39]. However, calculations in Table 28 indicate that the alnico magnet driver in his apparatus may be the source of inelastic resonances of the Fitzgerald type at these frequencies.

The composition of alnico (Al-Ni-Co) alloys is typically 8–12% Al, 15–26% Ni, 5–24% Co, up to 6% Cu, up to 1% Ti, and the balance Fe (31-72%). Precipitation hardening in the production process can cause residual stress in the host lattice, either tensile or compressive. Either could affect the value of S. Resonances could be due to either host or precipitate phase, which might have different mosaic sizes and hence different values of S. In view of this, the assignments in Table 28 must be considered tentative. The case of the 0.687 inch ball is a bit unusual in that lower acoustic modes (q = 2, 3) for Co were apparently not observed, although Hopkins admits that the alnico magnet is part of the same resonant system as the steel ball, the base plate, the solenoid, and the rest of his apparatus, all of which contribute resonances in the same range, and that filtering out the relevant ones is not straight forward. We do not know which resonances he may have discarded and not reported. In any case, Fitzgerald never measured resonances in steel balls, so these experiments are hardly relevant to the particle wave theory. And, except for the Federal Communications Commission, nobody owns the audiofrequency band.

1. Interpreting MRD spectra of 38 widely different samples - single crystal, polycrystalline - metals, salts, natural and synthetic polymers - according to the particle wave theory of E. R. Fitzgerald has shown excellent agreement between measured and calculated resonance frequencies by Eq. (2).
2. Standard deviations from the average value of crystal mosaic size calculated for each frequency within a sample phase by Eq. (3) varied from 0.08 %S_av to 1.78 %S_av, with a mean of 0.77 %S_av for 41 sample phases. By comparison, for Ca and P in human cancellous bone, the corresponding value was 2.27%S_av for ⁴⁰Ca and P [6] and 0.64 %S_av for ⁴⁴Ca and P (alternate interpretation) [8].
3. The average % difference, 100 $\left(v–v_c\right)/v_c$, for 238 resonance frequencies in 41 sample phases was 1.3%. Of these, 189 (79%) were less than 2%.
4. Overall, the majority of resonances were the result of impurities in the lattice. The great majority of these was either documented in the original sample analyses or was highly likely due to chemical similarities, and/or sample origin, and/or sample processing.
5. Fig. (1) shows mass numbers M as a function of frequency over the range of this study (0-5600 Hz) for S = 1 μm, calculated from Eq. (6). N₀ is Avogadro's number. Each curve represents a different q-value, 1 to 4. Frequency was chosen as the independent variable here to parallel the interpretation process. Physical reality is the opposite: mass determines frequency.

Fig.(1). M vs. v , Eq. (6).

Fig. (2) shows the number of mass numbers within a ± 2% frequency window as a function of frequency according to Eq. (7) [obtained by differentiating Eq. (6)], with $\left(v\right)/v=\mathrm{00.2.}$.

Fig.(2). |ΔM| in a ±2% frequency window vs. v, Eq. (7).

Again, S = 1 μm. The discontinuities are the result of sequential addition of q-values from 1 to (1 and 2) to (1 and 2 and 3) to (1 and 2 and 3 and 4) with increasing frequency.

From Figs. (1 and 2) it is apparent that there are multiple choices of isotopic masses to satisfy Eq. (2) within a ±2% frequency window and that the number of choices increases with increasing frequency. This demonstrates the need for very complete and highly sensitive chemical analyses to guide interpretation. In this work, the (arbitrary) ±2% frequency window was met for 195 of 238 experimental frequencies. Better fits were possible in some cases, but only by introducing unlikely elements as impurities. Only about a fourth of all elements are "common," which constrains realistic assignments.

(6) Previous elastic mode interpretations of some of the audiofrequency resonances in lead, 2S Aluminum and in Teflon^® failed to predict all the frequencies and no more. In these cases the particle wave theory predicted all the nonelastic frequencies (including overlaps with elastic frequencies) and no more. Only in the alnico samples did elastic theory predict all the frequencies (but only 2 in each of 2 samples) and no more in the frequency range of the experiment, but even here some frequencies were apparently discarded.

The following conclusions are drawn from the present work and careful reading of previously reported work:

1. The Particle Wave Theory of plastic deformation in crystalline solids has been upheld by detailed quantitative interpretations of past MRD spectra of 38 samples obtained by six different research individuals or groups operating independently with seven different apparati.
2. In larger samples, elastic normal modes are found in the audiofrequency range, where they occasionally overlap particle wave frequencies. Alternate interpretations according to various elastic theory expressions are either partially coincidental and inadequate as shown here or flawed as suggested by Fitzgerald [21].
3. In addition to atoms and monatomic ions, interpretation of some spectra requires that molecules or molecular ions be proposed, especially for very low frequency resonances, necessitating a high mass particle (Eq. 2).
4. In complex, multiphase systems such as some alloys, multiple crystal mosaic sizes, S, corresponding to different phases, are necessary to interpret the spectra. This applies in particular to precipitation hardened alloys.
5. Fitzgerald's requirement that for particles to emerge in the field-free region they must be aligned in a row of evenly spaced like isotopes [8], seems unnecessary. The large number of isotopes and impurities in some materials and the imposition of maximum entropy (which favors mixing over segregation) make this segregation improbable or impossible. In fact, a simple particle-in-a-box model, without reference to a lattice, would appear to suffice for the interpretation of MRD spectra. This would also allow for a particle-wave interpretation of spectra of some amorphous or only slightly ordered structures with lengthy, open, field-free regions.
6. The overall simplification of spectra with time, with increased static stress, and with annealing suggests that, under the influence of periodic shear stress, impurities move out of the primary phase into precipitates, dislocations, or grain boundaries. In this view, the odd substituent or interstitial atom would be more likely to enter the field free region because it would be out of phase (and hence in a higher energy state) with lattice vibrations, due to its different mass.

Recommendations are made for future experiments in MRD:

1. Repeat many of the original experiments on the same materials, this time analyzed for impurities by the current best techniques, including electron microprobe analysis (also see (3)). Keep sample size small enough to prevent interference from natural (elastic) frequencies. Measure higher frequencies (to 10 kHz or more) [40] to confirm assignments (same element, higher q-value). Always measure a no load or minimum load spectrum for comparison with other results and to complete the analysis in (5) below.
2. Measure the resonance spectra of some materials with exchangeable cations, such as clays, zeolites, and ion exchange resins. Replace the ions and redetermine the spectra to see if frequencies shift predictably, i.e., v₂/V₁ = m₁/m₂ for masses m₁ of original ion and m₂ of replacement ion. We will then know for certain the origin of both frequencies.
3. Integrate modern microscopy techniques on the same samples to determine distribution of impurities, their composition, and crystal mosaic size, before and after prolonged periodic stressing.
4. Measure complete spectra in far less time than was required in the past (minutes instead of days or weeks) employing, for example, a more recent apparatus such as described in [40]. If equilibration is slow, this may not be possible, but it would open the possibility of studying the dynamics of equilibration.
5. With (1) - (4) in hand, do more detailed time studies on changing samples to determine the fate of impurity particles under periodic stress (e.g., do they migrate to dislocations or grain boundaries, and do they form separate phases?). The application of periodic audiofrequency stress is a potential means of strengthening materials and otherwise affecting their properties, because it affects the distribution of impurities.
6. Investigate amorphous and partially ordered materials, including quasi-crystals, to determine whether crystallinity is an absolute requirement for particle wave resonances.
7. Confirm and explain the release of P atoms in bone apatite and Si and other atoms in quartz where they are tightly bonded to four oxygen atoms.
8. Investigate the interaction of electromagnetic induction and radiation with solids exhibiting mechanical resonance dispersion. Potentially this could lead to detection and identification of materials at great distances, up to km. Further, because of its long wavelength, radiation in this audiofrequency region penetrates all common materials except metals, yielding opportunities for detection of hidden objects.

The author confirms that this article content has no conflict of interest.