Rayleigh waves are finding numerous modern applications including: universal quantum transducers (Schuetz et al., 2015), carrier mobility studies in diamond-based transistors (Singh, et al., 2020; Bonomo et al., 2020), and acoustic preparation of spin states (Whiteley et al., 2019), as examples. Moreover, there is substantial interest in using nanomechanical systems (Cleland, 2003) to drive EPR interactions in a context where phonon effects are important.

The classical theory of Rayleigh waves has been presented correctly for many years (Auld, 1973; Graff, 1991; Landau, 1986; Love, 1944), but there have been many unnecessary approximations and much confusion in the literature when normalizing Rayleigh waves.  In part, this confusion stems from the different but equivalent Rayleigh expressions in the literature (Auld, 1973; Graff, 1991; Hassan and Nagy, 1998; Steg and Klemens, 1974).  In a very insightful contribution, Hassan and Nagy (1998) have demonstrated the equivalence of the family of Rayleigh modes and how to convert between these modes. In the present contribution, Rayleigh wave phonons for the Auld-type, Graff-type, and Steg-Klemens-type are properly normalized (Stroscio and Dutta, 2001) by including the necessary integral over the mode vertical profile such that the mechanical energy in the modes is equated with the energy of a phonon modes. A portion of the results to be presented may be found on Stroscio and Dutta (2025).

The phonon mode normalizations to be presented have many contemporary applications.

Research supported, in part, under AFOSR award FA9550-19-1-0282.

References

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Bonomo, G., et al., Diamond and Related Materials, 101, 107650 (2020).

Cleland, A. N., Foundations of Nanomechanics. (Springer, Berlin, Heidelberg, 2003).

Graff, K. F., Wave Motion in Elastic Solids (Dover, New York, 1991); see page 324.

Hassan, Waled, and Peter B. Nagy, J. Acous, Soc. of Am.,104(5), 3107 (November 1998).

Landau, L. D., and E. M. Lifshitz, Theory of Elasticity (Pergamon Press, Oxford, 1986).

Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, (Dover, New York, 1944).

Schuetz, M. J. A., et al, Phys. Rev. X, 031031:1–30, (2015).

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Stroscio, Michael A., and Mitra Dutta. Phonons in Nanostructures, (Cambridge University Press, Cambridge, 2001).

Stroscio, M.A, and Dutta, M., J. Acous. Soc. Am., 158(2), 1072 (August 2025).

Stroscio, M. A., et al., J. Phys.: Cond. Matter, 8, 2143 (1996).

Whiteley, Samuel J., et al., Nature Physics, 15, 490–495 (2019).