A Frequency Sum Rule and Spectral Moments in Lattice Dynamics

Frei, V.; Deus, P.

doi:10.1002/pssb.2221250113

Cited by 9 publications

(3 citation statements)

References 21 publications

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“…We mention, by passing, the “sum rule” for phonon frequencies introduced in ref (see also ref ). In our theory it obtains the form stemming directly from eq .…”

Section: Results and Discussionmentioning

confidence: 99%

Raman Spectra of Nonpolar Crystalline Nanoparticles: Elasticity Theory-like Approach for Optical Phonons

2018

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A simple way to investigate theoretically the Raman spectra (RS) of nonpolar nanoparticles is proposed. For this aim we substitute the original lattice optical phonon eigenproblem by the continuous Klein-Fock-Gordon-like equation with Dirichlet boundary conditions. This approach provides the basis for the continuous description of optical phonons in the same manner how the elasticity theory describes the longwavelength acoustic phonons. Together with continuous reformulation of the bond polarization model it allows to calculate the RS of nanoparticles without referring to their atomistic structure. It ensures the powerful tool for interpreting the experimental data, studying the effects of particle shape and their size distribution. We successfully fit recent experimental data on very small diamond and silicon particles, for which the commonly used phonon confinement model fails. The predictions of our theory are compared with recent results obtained within the dynamical matrix method -bond polarization model (DMM-BPM) approach and an excellent agreement between them is found. The advantages of the present theory are its simplicity and the rapidity of calculations. We analyze how the RS are affected by the nanoparticle faceting and propose a simple power law for Raman peak position dependence on the facets number. The method of powder RS calculations is formulated and the limitations on the accuracy of our analysis are discussed. arXiv:1806.08100v1 [cond-mat.mes-hall]

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“…We mention, by passing, the “sum rule” for phonon frequencies introduced in ref (see also ref ). In our theory it obtains the form stemming directly from eq .…”

Section: Results and Discussionmentioning

confidence: 99%

Raman Spectra of Nonpolar Crystalline Nanoparticles: Elasticity Theory-like Approach for Optical Phonons

2018

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“…This model would obey the simple form of the Blackman Sum Rule 16 . Deviations from Blackman's sum rule have been discussed by several authors [18][19][20] . The left-hand side of equation ( 20) has been calculated along the principal directions of several important non-primitive crystals by Rosenstock 19,20 .…”

Section: The Blackman Sum Rulementioning

confidence: 99%

Interatomic Forces, Phonons, the Foreman–lomer Theorem and the Blackman Sum Rule

Stewart

2001

Int. J. Mod. Phys. B

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Foreman and Lomer proposed in 1957 a method of estimating the harmonic forces between parallel planes of atoms of primitive cubic crystals by Fourier transforming the squared frequencies of phonons propagating along principal directions. A generalized form of this theorem is derived in this paper and it is shown that it is more appropriate to apply the method to certain combinations of the phonon dispersion relations rather than to individual dispersion relations themselves. Further, it is also shown how the method may be extended to the non-primitive hexagonal close packed and diamond lattices. Explicit, exact and general relations in terms of atomic force constants are found for deviations from the Blackman sum rule which itself is shown to be derived from the generalized Foreman-Lomer theorem.

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“…For physical purposes, namely in treating the phonons in cubic crystals in terms of gradually enlarged unit cells (Frei & Deus, 1984;Frei & Pol~fk, 1984;Frei, Mandula & Slanina, 1985), cubic sublattices E of cubic lattices L have been examined, especially those with the maximal common point group P* =/5 n P = m3m (i.e. the klassengleiche sublattices).…”

Section: Introductionmentioning

confidence: 99%

On inclined cubic sublattices of cubic lattices

Frei¹

1990

Acta Cryst Sect A

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regions than others. The centroid of the structurefactor probability distribution is obtained, in this case, by taking the Fourier transform of the expected electron-density function. In other terms, each atom of a molecular replacement model would be smeared over its distribution of possible positions. It is assumed that there is a sufficient number of independent contributions to the difference in the structure factors, so that the central limit theorem applies and the probability distribution is a Gaussian about the centroid estimate.For a model of a crystal structure, it is preferable to consider the average effect of a specific set of errors on a set of structure factors, in other words to consider the reciprocal-space vector as the random variable. The probability distributions underlying the differences between the model and the true structure enter through the frequencies of the errors over all the atoms. Essentially the same probability distributions of structure factors arise as in the previous case, because of the symmetry between real and reciprocal space in the Fourier transform.Considered in terms of normalized structure factors, all sources of error have the same effect, which can be summarized in a single parameter, o¥. This parameter plays the same role in the probability distributions as O" A in the distributions of Srinivasan & Ramachandran (1965). Therefore, the methods suggested previously to estimate phase probabilities and to calculate electron-density maps (Read, 1986) are still valid. However, the interpretation of the parameter ere is different. In particular, the variation of o¥ with resolution cannot be attributed entirely to coordinate error. Methods such as the Luzzati (1952) plot and the o'a plot (Read, 1986) (1987). Protein Eng. 1,377-384. WILSON, A. J. C. (1949). Acta Cryst. 2, 318-321. WILSON, A. J. C. (1976). Acta Cryst. A32, 781-783. WOOLFSON, M. M. (1956). Acta Cryst. 9, 804-810. Acta Cryst. (1990

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A Frequency Sum Rule and Spectral Moments in Lattice Dynamics

Cited by 9 publications

References 21 publications

Raman Spectra of Nonpolar Crystalline Nanoparticles: Elasticity Theory-like Approach for Optical Phonons

Raman Spectra of Nonpolar Crystalline Nanoparticles: Elasticity Theory-like Approach for Optical Phonons

Interatomic Forces, Phonons, the Foreman–lomer Theorem and the Blackman Sum Rule

On inclined cubic sublattices of cubic lattices

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