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REVIEW 2 major objections 5 minor 43 references

Atomic Vibrations in Glasses

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that the boson peak in glasses consists of quasi-localized acoustic modes created when disorder destroys plane waves at nanometre wavelengths, possibly hybridized with low-lying optic modes.

desk verdict An honest review chapter, not a research contribution; its value is in the synthesis and pedagogy, and its central boson-peak/Ioffe-Regel identification is explicitly labelled as a belief, not a demonstrated result. read the letter →

arxiv 1908.08321 v1 pith:GLXCWIBI submitted 2019-08-22 cond-mat.dis-nn

B. Hehlen , B. Rufflé This is my paper · ORCID
classification cond-mat.dis-nn
keywords bosonpeakglassesquasi-localizedmodesIoffe-Regelcrossoverthermalconductivityplateauvibrationaldensityofstatesrigid-unitacousticplanewaves
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review chapter argues that the excess low-frequency vibrations of glasses, known as the boson peak, are not a separate family of modes but the remnants of acoustic plane waves destroyed by disorder at nanometre wavelengths. In a simple disordered chain, the authors show that the density of states barely changes while the eigenmodes progressively lose their plane-wave character and become quasi-localized. They identify these quasi-localized excitations, possibly hybridized with low-lying optic (rigid-unit) modes, as the boson-peak modes. If this is right, the boson peak is a genuine glass-specific feature, and the same modes explain both the end of propagating sound and the plateau in thermal conductivity near 10 K.

What carries the argument

The argument is carried by the disordered one-dimensional diatomic chain, in which random spring constants leave the vibrational density of states nearly unchanged while the eigenmodes' wave-vector spectral density broadens: low-frequency modes remain plane-wave-like, but higher-frequency modes become quasi-localized vibrations spread over a few atomic distances. This picture is extended to three-dimensional glasses through the Ioffe-Regel crossover, the frequency at which the acoustic mean free path shrinks to the wavelength, marking the end of propagating sound. The paper uses this machinery to connect the quasi-localized modes to the boson peak and to the thermal-conductivity plateau.

What would settle it

Measure the transverse acoustic mean free path in a glass such as vitreous silica up to and beyond the Ioffe-Regel crossover and compare that frequency with the maximum of the reduced density of states g(ω)/ω²; a systematic mismatch between the two frequencies would falsify the central identification.

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Extended reading notes

Core claim

The paper's central claim is that the boson peak in glasses originates from a renormalization and redistribution of the acoustic branches caused by the destruction of plane waves of nanometre wavelength, with optional hybridization with low-lying optic vibrations such as the librations of rigid SiO4 tetrahedra. In the authors' picture, the boson peak is not the sum of all scattering channels but a specific set of quasi-localized modes that account for solid-state properties unique to glasses, including the plateau in thermal conductivity and the Ioffe-Regel crossover where sound waves stop propagating. The claim is supported by comparing the boson-peak frequency with the Ioffe-Regel crossover frequency for transverse acoustic excitations, a correspondence the authors state is believed but not yet demonstrated experimentally.

Load-bearing premise

The argument assumes, without direct experimental proof, that the boson-peak frequency coincides with the Ioffe-Regel crossover for transverse acoustic waves in every glass.

Editorial extensions

If this is right

  • The boson peak is a universal glass-specific vibrational feature, not a spectroscopic artifact.
  • The thermal-conductivity plateau follows from the same disorder-induced collapse of the acoustic mean free path that ends plane-wave sound.
  • The boson-peak frequency should match the transverse Ioffe-Regel crossover frequency in every glass, a prediction the authors note is still untested.
  • Because quasi-localized modes have ill-defined wavevectors, vibrational selection rules are relaxed, so Raman, infrared, hyper-Raman and neutron spectra each weight the boson peak differently.
  • Amorphous silicon, which lacks low-lying optic modes, provides a case where the purely acoustic mechanism can be isolated.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the transverse Ioffe-Regel identification holds, a direct measurement of the transverse mean free path across the crossover in one glass would give a parameter-free check of the boson-peak frequency.
  • The same quasi-localization mechanism predicts a characteristic nanometre length scale (roughly 1-3 nm) that should be visible in simulations of eigenmode spatial profiles and in hypersound attenuation data.
  • A systematic comparison of boson-peak frequencies with transverse Ioffe-Regel frequencies across network, metallic and molecular glasses would reveal whether the correspondence is universal or limited to network glasses.
  • The authors' distinction between boson-peak modes and other scattering processes implies that reported boson-peak intensities in Raman and neutron data may need to be decomposed before being compared with microscopic theories.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This manuscript is a review chapter on atomic vibrations in glasses, centered on the nature of the boson peak. It introduces a disordered diatomic linear chain to show how disorder converts plane-wave acoustic modes into quasi-localized modes, then reviews the connection between vibrational anomalies (heat capacity, thermal conductivity plateau) and the Ioffe-Regel crossover, and surveys spectroscopic evidence in v-SiO2, v-B2O3, and amorphous silicon. The central claim, stated in Sec. 7, is that boson-peak modes mostly originate from the destruction of acoustic plane waves at nanometer wavelengths, with possible hybridization with low-lying optic/rigid-unit modes, and that this mechanism is causally linked to the thermal conductivity plateau.

Significance. If established, the thesis would provide a unified explanation of several glass-specific anomalies. The chapter is a useful scholarly review: it is honest about open questions, documents the Ioffe-Regel crossover evidence for longitudinal acoustic modes, and explicitly acknowledges that the transverse correspondence is not experimentally demonstrated. However, the central thesis is presented more confidently in the abstract and Sec. 7 than the evidence warrants, and that gap is load-bearing for the chapter's main conclusion.

major comments (2)
  1. [Sec. 5.2 / Sec. 7] The statement in Sec. 7 that boson-peak modes 'mostly originate from a renormalization and a redistribution of the modes of the acoustic branches due to the destruction of plane waves of nanometer wavelengths' is presented as the conclusion of the chapter, but the evidence reviewed in Sec. 5.2 concerns longitudinal acoustic excitations only (lithium diborate, densified silica, glycerol), and the text explicitly says that no experimental data have demonstrated the transverse Ioffe-Regel/boson-peak correspondence. In Sec. 6.2, the transverse crossover in amorphous silicon is reported to 'raise some doubts about its physical meaning.' The chapter therefore does not provide direct support for the central causal link. I recommend that the abstract and Sec. 7 be revised to present this as a working hypothesis, and that the longitudinal/transverse discrepancy be discussed explicitly.
  2. [Sec. 2.1 / Sec. 2.2] The 1D chain calculation with δK/K0 = 0.25 and M/m = 2 illustrates the qualitative change in mode character with frequency, but Sec. 2.2 concedes that the extension to 3D network glasses 'has been much debated.' Since the chain result is invoked in Sec. 7 as the foundation for the acoustic-plane-wave-destruction mechanism, the manuscript should either cite quantitative 3D evidence that the same mechanism produces the excess modes at the boson-peak frequency, or further qualify the status of the argument. As written, the leap from the 1D toy model to 3D network glasses is an unsupported extrapolation.
minor comments (5)
  1. [Section numbering] Section 2.2 is labeled '1.2 Real amorphous solids'; the section number should be corrected to 2.2.
  2. [Eq. (4)] Equation (4) appears to be missing its right-hand side; the displayed equation is blank, so the thermal conductivity expression cannot be read.
  3. [Fig. 6 caption] The caption of Fig. 6 is identical to that of Fig. 5 and refers to v-SiO2, while the text in Sec. 5.1 explicitly describes Fig. 6 as the spectrum of v-B2O3; the caption should be corrected.
  4. [Acknowledgments] The Acknowledgments contain a typo: 'gratefully tanked' should be 'gratefully thanked.'
  5. [References] Reference [14] misspells the first author's name as 'Greabner'; the correct spelling is 'Graebner.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chapter's interpretive synthesis is supported by independent experimental data; its explicitly unverified Ioffe-Regel premise is an empirical hypothesis, not a constructional identity.

full rationale

The chapter presents a review and interpretive synthesis, not a derivation in which an output is forced by its own inputs. The 1D chain model in Section 2.1 uses openly illustrative parameters (δK/K0 = 0.25, M/m = 2) to show how disorder produces quasi-localized modes; these parameters are not fitted to boson-peak data, and the model does not generate a quantitative prediction that is then compared with the boson peak. The load-bearing physical premise, that the boson-peak frequency equals the transverse Ioffe-Regel crossover frequency in all glasses, is explicitly stated in Section 5.2 as an unverified belief: 'It is believed that the boson peak frequency actually corresponds to the Ioffe-Regel crossover frequency for the transverse acoustic excitations in all glasses, but no experimental data have demonstrated the validity of this statement yet.' An unverified hypothesis is a correctness risk, not circularity; the paper does not redefine the boson peak in terms of the Ioffe-Regel frequency, nor does any equation make the two quantities equal by construction. The self-citations in the reference list (Rufflé et al. 2003, 2006; Hehlen et al. 2000; Hehlen and Neuville 2015) are to published, externally falsifiable experimental measurements of acoustic damping, hyper-Raman spectra, and Raman responses; they are used as empirical evidence, not as an authority that forbids alternatives. The paper also candidly notes that generalization of the 1D chain picture to real 3D glasses 'has been much debated' (Section 2.2), and that the transverse Ioffe-Regel measurement in amorphous silicon 'raises some doubts about its physical meaning' (Section 6.2). These admissions confirm that the central claim is an interpretation resting on identifiable, independently checkable evidence rather than a step that reduces to its own definition or to a fitted parameter. No circular step can be quoted from the manuscript.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The toy model parameters are illustrative free choices, not fits to the boson peak. The argument rests on several domain assumptions, the strongest being the unverified identification of the boson peak with the transverse Ioffe-Regel crossover. The chapter introduces no new physical entities.

free parameters (2)
  • delta-K/K0 = 0.25 = 0.25
    Standard deviation of the spring-constant distribution in the illustrative 1D disordered chain (Section 2.1). It is chosen by hand, not fitted to glass data, and tunes how strongly the chain is disordered.
  • M/m = 2 = 2
    Mass ratio in the illustrative diatomic chain (Section 2.1). It determines the acoustic/optic gap in the toy model and is not fitted to a real glass.
assumptions (4)
  • domain assumption Harmonic approximation for low-frequency glass vibrations
    The review describes vibrations as eigenmodes of a dynamical matrix and derives heat capacity from harmonic oscillator states (Eq. 1, Section 1); anharmonicity is treated only as a damping mechanism (Section 5.2).
  • domain assumption 1D disordered-chain results carry over to 3D network glasses
    Section 2.1 uses a 1D chain to show quasi-localised modes; Section 2.2 admits 'To what extent such a simple picture can be generalized has been much debated', yet the chapter uses this picture to explain the boson peak in real glasses.
  • domain assumption Boson peak frequency equals transverse Ioffe-Regel crossover frequency
    Section 5.2 explicitly calls this a belief with no supporting experimental data. It is load-bearing for linking quasi-localisation to the boson peak.
  • domain assumption Rayleigh scattering l^-1 proportional to omega^4 dominates near the Ioffe-Regel crossover
    Section 3.2 and 5.2 use this trend to connect the drop in mean free path to the thermal conductivity plateau; it is carried from prior literature (refs 14, 29, 30), not derived in the chapter.

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Cite this review

Pith. "Pith review of Atomic Vibrations in Glasses." pith.science (2026). https://pith.science/paper/GLXCWIBI

@misc{pith2026190808321,
  author       = {Pith},
  title        = {Pith review of: Atomic Vibrations in Glasses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLXCWIBI}},
  note         = {Machine review of arXiv:1908.08321}
}
abstract

In glasses, atomic disorder combined with atomic connectivity makes understanding of the nature of the vibrations much more complex than in crystals or molecules. With a simple model, however, it is possible to show how disorder generates quasi-local modes on optic branches as well as on acoustic branches at low-frequency. The latter modes, possibly hybridizing with low-lying optic modes in real glasses, lead to the excess, low-frequency excitations known as {\it boson-peak modes}, which are lacking in crystals. The spatially quasi-localized vibrations also explain anomalies in thermal conductivity and the end of the acoustic branches, two other specific features of glasses. Together with the quasi-localization of the modes at the nanometric scale, structural disorder lifts the crystalline or molecular spectroscopic selection rules and makes interpretation of experiments difficult. Nevertheless, vibrations in simple glasses such as vitreous silica or vitreous boron oxide are nowadays rather well described. But a comprehensive understanding of the boson peak modes remains a highly debated issue as illustrated by three archetypal glass systems, vitreous SiO$_2$ and B$_2$O$_3$ and amorphous silicon.

Figures

Figures reproduced from arXiv: 1908.08321 by the authors.

Figure 4
Figure 4. Raman spectra of v-SiO2, α-quartz and polycrystalline α-cristobalite. The lack of long-range order prevents vibrations in glasses from being described in a unique way. A first possibility is to consider the atomic-displacements (eigenmodes) of an elementary structural unit such as the SiO4 tetrahedron of Td -symmetry or the Si-O-Si bridge of C2v -symmetry in v-SiO2, and the BO3 triangle of D3h-symmetry or the B-O-B … view at source ↗
Figure 8
Figure 8. Vibrational excitations in v-SiO2. (a) Dispersion curve of acoustic modes and low￾lying optic branches along the Γ → M (plain lines) and Γ → Z (dot dashed lines) directions of the Brillouin zone; squares Raman data. (b) Raman spectra of α-cristobalite and neutron spectra of the soft mode of β-quartz at 1250 K whose position at Tg is indicated by the dashed line at around 36 cm-1 .(c) Spectroscopic signatures of the … view at source ↗

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Taraskin and S.R

    N. Taraskin and S.R. Elliott, Propagation of plane-wave vibrational excitations in disordered systems, Phys. Rev. B, 61 (2000) 12017-30

    work page 2000

  2. [2]

    Dean, Vibrations of glass-like disordered chains, Proc

    P. Dean, Vibrations of glass-like disordered chains, Proc. Phys. Soc., 84 (1964) 727-44

    work page 1964

  3. [3]

    Rothenfusser, W

    M. Rothenfusser, W. Dietsche and H. Kinder, Linear dispersion of transverse high- frequency phonons in vitreous silica, Phys. Rev. B, 27 (1983) 5196-98

    work page 1983

  4. [4]

    Bilir and W.A

    N. Bilir and W.A. Phillips, Phonons in SiO2: low-temperature heat capacity of cristobalite, Phil. Mag., 32 (1975) 113-22

    work page 1975

  5. [5]

    Dove, M.J

    M.T. Dove, M.J. Harris, A.C. Hannon, J.M. Parker, I.P. Swainson and M. Gambhir, Floppy modes in crystalline and amorphous silicates, Phys. Rev. Lett., 78 (1997) 1070-73

    work page 1997

  6. [6]

    Sigaev, E.N

    V.N. Sigaev, E.N. Smelyanskaya, V.G. Plotnichenko, V.V. Koltashev, A.A. Volkov and P. Pernice, Low-frequency band at 50 cm-1 in the Raman spectrum of cristobalite : identification of similar motifs in glasses and crystals of similar composition, J. Non-Cryst. Solids, 248 (1999) 141-45

    work page 1999

  7. [7]

    Ramos and U

    M.A. Ramos and U. Buchenau, Low-temperature thermal conductivity of glasses within the soft-potential model, Phys. Rev. B, 55 (1997) 5749-54

    work page 1997

  8. [8]

    Phillips, Tunneling states in amorphous solids, J

    W .A. Phillips, Tunneling states in amorphous solids, J. Low Temp. Phys., 7 (1972) 351-60

    work page 1972

Show all 43 references
  1. [9]

    Anderson, B.I

    P.W. Anderson, B.I. Halperin and C.M. Varma, Properties of glasses and spin glasses, Phil. Mag, 25 (1972) 1-12

    1972

  2. [10]

    Safarik, R.B

    D.J. Safarik, R.B. Scwartz and M.F. Hundley, Similarities in the Cp/T3 peaks in amorphous and crystalline metals, Phys. Rev. Lett., 96 (2006) 195902

    2006

  3. [11]

    Chumakov et al., Role of disorder in the thermodynamics and atomic dynamics of glasses, Phys

    A.I. Chumakov et al., Role of disorder in the thermodynamics and atomic dynamics of glasses, Phys. Rev. Lett., 112 (2014) 025502

    2014

  4. [12]

    Gurarie and J.T

    V. Gurarie and J.T. Chalker, Some generic aspects of bosonic excitations in disordered systems, Phys. Rev. Lett., 89 (2002) 136801. 28

    2002

  5. [13]

    Debye, Zustandsgleichung und Quantenhypothese mit einem Anhang über Wärmeleitung, in Vorträge über die kinetische Theorie der Materie und der Elektrizität, vol

    P. Debye, Zustandsgleichung und Quantenhypothese mit einem Anhang über Wärmeleitung, in Vorträge über die kinetische Theorie der Materie und der Elektrizität, vol. VI (Leipzig: G. Teubner, 1914)

    1914

  6. [14]

    Greabner, B

    J.E. Greabner, B. Golding and L.C. Allen, Phonons localization in glasses, Phys. Rev. B, 34 (1986) 5696 -5701

    1986

  7. [15]

    boson peak

    W. Schirmacher, Thermal conductivity of glassy materials and the “boson peak”, Europhys. Lett., 73 (2006) 892-98

    2006

  8. [16]

    Beltukov, C

    Y.M. Beltukov, C. Fusco, D.A. Parshin and A. Tanguy, Boson peak and Ioffe-Regel criterion in amorphous silicon-like materials: effect of bond directionality, Phys. Rev. E, 93, (2016) 023006

    2016

  9. [17]

    Hayes and R

    W. Hayes and R. Loudon, Scattering of Light by Crystals (New York, Wiley, 1978)

    1978

  10. [18]

    Martin and W

    A.J. Martin and W. Brenig, Model for Brillouin scattering in amorphous solids, Phys. Stat. Sol., 64 (1974) 163-72

    1974

  11. [19]

    Kirk, Quantitative analysis of the effect of disorder -induced mode coupling on infrared absorption in silica, P hys.Rev

    C.T. Kirk, Quantitative analysis of the effect of disorder -induced mode coupling on infrared absorption in silica, P hys.Rev. B, 38 (1988) 1255-73

    1988

  12. [20]

    Taraskin and S

    S.N. Taraskin and S. R. Elliott, Nature of vibrational excitations in vitreous silica, Phys. Rev. B, 56 (1997) 8605

    1997

  13. [21]

    Hehlen and G

    B. Hehlen and G. Simon, The vibrations of vitreous silica observed in hyper-Raman scattering, J. Raman Spectrosc., 43 (2012) 1941-50

    2012

  14. [22]

    Galeener, A.J

    F.L. Galeener, A.J. Leadbetter and M.W. Stringfellow, Comparison of the neutron, Raman, and infrared spectra of vitreous SiO2, GeO2, and BeF2, Phys. Rev., B27, (1983) 1052- 78

    1983

  15. [23]

    Wilson, P

    M. Wilson, P. A. Madden, M. Hemmati and C. A. Angell, Polarization effects, network dynamics, and the infrared spectrum of amorphous SiO2, Phys. Rev. Lett., 77 (1996) 4023-26

    1996

  16. [24]

    Umari, X

    P. Umari, X. Gonze and A. Pasquarello, Concentration of small ring structures in vitreous silica from a first-principles analysis of the Raman spectrum, Phys. Rev. Lett., 90 (2003) 027401. 29

    2003

  17. [25]

    Hehlen, Inter-tetrahedra bond angle of permanently densified silicas extracted from their Raman spectra, J

    B. Hehlen, Inter-tetrahedra bond angle of permanently densified silicas extracted from their Raman spectra, J. Phys.: Condens. Matter, 22 (2010) 025401

    2010

  18. [26]

    Maris, Interactions of sound waves with thermal phonons in dielectric crystals, p

    H.J. Maris, Interactions of sound waves with thermal phonons in dielectric crystals, p. 279- 345 in W.P. Mason and R.N. Thurston (eds), Physical Acoustics: Principles and Methods (New York: Academic Press, 1971)

    1971

  19. [27]

    Anderson and H.E

    O.L. Anderson and H.E. Bommel, Ultrasonic absorption in fused silica at low-temperature and high frequencies, J. Amer. Ceram. Soc., 38 ( 1955) 125-131

    1955

  20. [28]

    Hunklinger and W

    S. Hunklinger and W. Arnold, pp. 155-215 in W.P. Mason and R.N. Thurston (eds) Physical Acoustics Vol. XII, (New York: Academic Press, 1976)

    1976

  21. [29]

    Rufflé, M

    B. Rufflé, M. Foret, E. Courtens, R. Vacher and G. Monaco, Observation of the onset of strong scattering on high frequency acoustic phonons in densified silica glass, Phys. Rev. Lett., 90 (2003) 095502

    2003

  22. [30]

    Rufflé, G

    B. Rufflé, G. Guimbreti`ere, E. Courtens, R. Vacher and G. Monaco, Glass specific behavior in the damping of acousticlike vibrations, Phys. Rev. Lett., 96 (2006) 045502

    2006

  23. [31]

    Malinovsky and A.P

    V.K. Malinovsky and A.P. Sokolv, The nature of the boson peak in Raman scattering in glasses, Solid State Commun., 57 (1986) 757-61

    1986

  24. [32]

    Novikov and A.P

    V.N. Novikov and A.P. Sokolov, Poisson’s ratio and the fragility of glass forming liquids, Nature, 431 (2004) 161-63, and Ref. therein

    2004

  25. [33]

    Swainson, M.T

    I.P. Swainson, M.T. Dove and D.C. Palmer, Infrared and Raman spectroscopy studies of the α-β phase transition in cristobalite, Phys. Chem. Minerals, 30 (2003) 353-65, and Ref. therein

    2003

  26. [34]

    Buchenau, M

    U. Buchenau, M. Prager, N.Nu¨cker, A.J. Dianoux, N. Hamad and W.A. Philips, Low- frequency modes in vitreous silica, Phys. Rev. B, 34 (1986) 5665

    1986

  27. [35]

    Hehlen, E

    B. Hehlen, E. Courtens, R. Vacher, A. Yamanaka, M. Jataoka and K. Inoue, Hyper- Raman scattering observation of the boson peak in vitreous silica, Phys. Rev. Lett., 84 (2000) 5355-58. 30

    2000

  28. [36]

    Weigel et al., Polarized Raman spectroscopy of v-SiO2 under rare-gas compression, Phys

    C. Weigel et al., Polarized Raman spectroscopy of v-SiO2 under rare-gas compression, Phys. Rev. B, 93 (2016) 224303

    2016

  29. [37]

    Hehlen and D

    B. Hehlen and D. Neuville, Raman response of network modifier cations in alumino silicate glasses, J. Phys. Chem. B, 119 (2015) 4093

    2015

  30. [38]

    Temple and C.E

    P.A. Temple and C.E. Hathaway, Multiphonon Raman spectrum of silicon, Phys. Rev. B, 7 (1973) 3685-97

    1973

  31. [39]

    Zwick and R

    A. Zwick and R. Carles, Multiple-order Raman scattering in crystalline and amorphous silicon, Phys. Rev. B, 48 (1993) 6024-32

    1993

  32. [40]

    Christie, S.N

    J.K. Christie, S.N. Taraskin and S.R. Elliott, Vibrational behavior of a realistic amorphous- silicon model, J. Non-Cryst. Solids, 253 (2007) 2272-79

    2007

  33. [41]

    Kamitakahara, C.M

    W.A. Kamitakahara, C.M. Soukoulis, H.R. Shanks, U. Buchenau and G.S. Grest, Vibrational spectrum of amorphous silicon: experiment and computer simulation, Phys. Rev. B, 36 (1987) 6539-42

    1987

  34. [42]

    B.L. Zink, R. Pietri and F. Hellman, Thermal conductivity and specific heat of thin-film amorphous silicon, Phys. Rev. Lett., 96 (2006) 055902

    2006

  35. [43]

    Mizuno, H

    H. Mizuno, H. Shiba and A. Ikeda, Continuum limit of the vibrational properties of amorphous solids, Proc. Natl. Acad. Sci., 114 (2017) E9767. Appendix : supplementary information for figure captions Fig. 2. Data from F. Birch and H. Clark, Amer. Jour. Sci., 238 (1940) 529-58....

    2017

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