Sound attenuation and anharmonic damping in solids with correlated disorder
We study via self-consistent Born approximation a model for sound waves in a disordered environment, in which the local fluctuations of the shear modulus G are spatially correlated with a certain correlation length ξ. The theory predicts an enhancement of the density of states over Debye's ω2 l...
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| Cite this: | Sound attenuation and anharmonic damping in solids with correlated disorder / W. Schirmacher, C. Tomaras, B. Schmid, G. Baldi, G. Viliani, G. Ruocco, T. Scopigno // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23606: 1-6. — Бібліогр.: 21 назв. — англ. |
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| author | Schirmacher, W. Tomaras, C. Schmid, B. Baldi, G. Viliani, G. Ruocco, G. Scopigno, T. |
| author_facet | Schirmacher, W. Tomaras, C. Schmid, B. Baldi, G. Viliani, G. Ruocco, G. Scopigno, T. |
| citation_txt | Sound attenuation and anharmonic damping in solids with correlated disorder / W. Schirmacher, C. Tomaras, B. Schmid, G. Baldi, G. Viliani, G. Ruocco, T. Scopigno // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23606: 1-6. — Бібліогр.: 21 назв. — англ. |
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| description | We study via self-consistent Born approximation a model for sound waves in a disordered environment, in which the local fluctuations of the shear modulus G are spatially correlated with a certain correlation length ξ. The theory predicts an enhancement of the density of states over Debye's ω2 law (boson peak) whose intensity increases for increasing correlation length, and whose frequency position is shifted downwards as 1/ξ. Moreover, the predicted disorder-induced sound attenuation coefficient Γ(k) obeys a universal scaling law ξ Γ(k) = f(kξ) for a given variance of G. Finally, the inclusion of the lowest-order contribution to the anharmonic sound damping into the theory allows us to reconcile apparently contradictory recent experimental data in amorphous SiO2.
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Condensed Matter Physics 2010, Vol. 13, No 2, 23606: 1–6
http://www.icmp.lviv.ua/journal
Sound attenuation and anharmonic damping in solids
with correlated disorder
W. Schirmacher1,2, C. Tomaras3, B. Schmid2, G. Baldi4, G. Viliani5 , G. Ruocco6,7,
T. Scopigno6,7
1 Physik-Department E13, Technische Universität München, James-Franck-Strasse 1,
D–85747 Garching, Germany
2 Fachbereich Physik, Universität Mainz, Germany
3 Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München,
Theresienstr. 37, D–80333 München, Germany
4 ESRF Grenoble
5 Dipt. di Fisica, Universitá di Trento, Italy
6 Dipt. di Fisica, Universitá di Roma, Italy
7 IPCF-CNR, Sezione di Roma, c/o Sapienza Universita’ di Roma, Italy
Received April 3, 2010, in final form April 14, 2010
We study via self-consistent Born approximation a model for sound waves in a disordered environment, in
which the local fluctuations of the shear modulus G are spatially correlated with a certain correlation length
ξ. The theory predicts an enhancement of the density of states over Debye’s ω2 law (boson peak) whose
intensity increases for increasing correlation length, and whose frequency position is shifted downwards as
1/ξ. Moreover, the predicted disorder-induced sound attenuation coefficient Γ(k) obeys a universal scaling
law ξΓ(k) = f(kξ) for a given variance of G. Finally, the inclusion of the lowest-order contribution to the
anharmonic sound damping into the theory allows us to reconcile apparently contradictory recent experimental
data in amorphous SiO2.
Key words: sound attenuation, vibrational properties of disordered solids, boson peak, anharmonic
interactions
PACS: 63.50.-x, 43.20.+g, 65.60.+a
The vibrational properties of disordered solids at THz frequencies are subject of an enormous
attention both from the experimental and from the theoretical side [1], due to the related anomalies
observed in the specific heat and thermal conductivity of glasses [2]. The origin of an excess of the
vibrational density of states (DOS) over the Debye prediction (“boson peak”) at THz frequencies
and its relation to the rather anomalous sound attenuation in the same frequency regime is in focus
of a lively debate [3–5]. In particular, the origin of the dispersion and attenuation of sound-like
excitation in the THz frequency range has been a matter of controversy quite recently [6, 7]. In spite
of this debate, nowadays many authors agree that the boson peak and the frequency dependence
of the sound attenuation are the related phenomena dictated by the structural disorder, rather
than by anharmonic interactions, and occur at frequencies at which the wavevector k looses its
significance for labeling the transverse vibrational states (Ioffe-Regel regime) [8].
The way the disorder controls the thermophysical properties, however, is still not fully un-
derstood, and the experimental phenomenology is not completely reproduced by the current
fluctuating-elastic-constant (FE) approaches [6, 9, 10], which are based on zero-range correlati-
ons.
The FE approach has been recently applied to low-frequency Raman scattering and for the
first time accounted for the experimentally observed frequency dependence, which is different from
incoherent neutron scattering [11]. In this study and in a recent comparison of a scalar FE model
with a simulation of a simple model having correlated disorder [12] it was realized that the finite
c© W. Schirmacher, C. Tomaras, B. Schmid, G. Baldi, G. Viliani, G. Ruocco, T. Scopigno 23606-1
http://www.icmp.lviv.ua/journal
W. Schirmacher et al.
correlation length ξ of the elasticity fluctuations plays an important role discussing the vibrati-
onal anomalies. A similar conclusion was drawn from molecular-dynamics simulations of quenched
Lennard-Jones Argon and SiO2 [13]. In the latter study it emerged that these correlations exist
over an extended number of inter-atomic distances. This verifies a conjecture of Elliott [14] quite
a time ago.
In the present paper we use the full vector FE approach [10] to study the effect of a finite
correlation length and the presence of anharmonic interactions. We find that the BP becomes
strongly enhanced with increasing correlation length, a result which emphasizes the importance
of this latter property in real systems. We further demonstrate that ξ times the disorder-induced
sound attenuation constant is a universal function of ξk (where k is the wavenumber).
We consider an elastic continuum, in which the shear modulus G fluctuates in space around
its mean value G0 (the bulk modulus Ko is supposed to be constant): G(r) = G0 + ∆G(r), we
also assume that the correlation function of ∆G(r), C(r) = 〈∆G(r + r0)∆G(r0)〉 and its Fourier
transform are of the forms
C(r) = 〈∆G2〉e−r/ξ,
C(k) =
〈
∆G2
〉
(8π/ξ)
[
k2 + ξ−2
]−2
. (1)
The corresponding mean-field equation for the low-wavenumber self energy Σ(ω) = Σ(q = 0, ω) (self-
consistent Born approximation, SCBA) takes the form [11, 12, 15, 16]
Σ(ω) =
γ
2ϕ3 〈∆G2〉
∫
|k|<kD
(
dk
2π
)3
C(k)
[
χL(k, ω) + χT (k, ω)
]
with
χL(k, ω) = k2
[
− ω2 + k2(v2
L,0 − 2Σ(ω))
]−1
, (2)
χT (k, ω) = k2
[
− ω2 + k2(v2
T,0 − Σ(ω))
]−1
. (3)
Here γ = 〈∆G2〉ϕ3/v4
0 . is the “disorder parameter”, vL,0, vT,0, are the (unrenormalized) sound
velocities, and ϕ3 =
∫
|k|<kD
(dk/2π)3C(k) is a normalization constant. kD is the usual Debye-cutoff,
typical values have been elaborated in [6].
The DOS, given by
g(ω) =
2ω
3π
∫
|k|<kD
(
dk
2π
)3 1
k2
Im {χL(k, ω) + 2χT (k, ω)} (4)
is reported in figure 1. Here we have plotted the “reduced DOS” g(ω)/gD(ω) (where gD is Debye’s
DOS) for three values of γ and five values of ξ. First we notice that, similarly to the uncorrelated
case (ξ → 0) [6, 9, 10] there exists a critical amount of disorder γc, beyond which the system becomes
unstable. The “boson peak” becomes more pronounced and is located at lower frequencies as this
value is approached. However, at variance with the experimental data, the maximum amplitude
of the boson peak predicted from the ξ = 0 case is a factor larger than the Debye expectation.
The interesting feature appearing in the correlated case (ξ > 0) is that the boson peak amplitude
increases and its position shifts towards lower frequencies with increasing ξ. In figure 1 – where we
have taken the frequencies scaled by vT /ξ in order to show that the boson peak frequencies scales
as 1/ξ – one can see that the boson peak amplitude can reach the value as high as ten with still
reasonable ξ value.
The sound attenuation constant (i.e., the width of the Brillouin line in χ′′
L(k, ω)) is given in
terms of the imaginary part of the self-energy Σ′′(ω) as [6]
Γ(k) = 2
k
vL
Σ′′(ω = k/vL), (5)
23606-2
Sound attenuation and anharmonic damping in solids with correlated disorder
Figure 1. Reduced DOS g(ω)/gD(ω) against scaled frequency ωξ/vT,0 for ξ = 1/kD (full lines)
ξ = 5/kD (dashed lines), ξ = 10/kD (dotted lines), ξ = 15/kD (dash-dotted lines), for three
disorder parameters (from left to right) γ − γc = 0.0001 (black), 0.001 (blue) and 0.01 (red).
Inset: Scaled Brillouin linewidth Γξ/v0 against scaled wavenumber qξ for the same parameters
and with the same line and color codes.
where vL = [v2
L,0 − 2Σ′(ω = 0)]1/2 is the renormalized longitudinal sound velocity. In particular,
the relation between Γ and the excess DOS derived for ξ = 0 (see equation (5) in [6]) – and this
has been found to be in agreement with the experimental data [6] – remains the same also for
ξ > 0. If one now introduces the dimensionless variables Σ̃ = Σ/v2
T,0, ω̃ = ωξ/vT,0 and k̃ = kξ and
inserts these into the SCBA equation (2) one realizes that the correlation length only appears via
the upper cutoff k̃D = kDξ. This means that for large enough ξ (in which case the cutoff does not
play any role) Σ̃ and Γ̃ does not depend on ξ. This implies, in turn, that Γ̃ = Γξ/vT,0 is - for a
fixed disorder parameter γ – a universal function of k. Σ̃ (and, in fact Σ as well) is – within our
approach – a universal function of ξω/vL or ξk.
In the inset of figure 1 we have plotted Γ̃ against k̃ for d = 3, for the three γ values of figure 1
and for ξ = 1/kD, 5/kD, 10/kD, and 15/kD. We see that the scaling is obeyed except for ξ = 1/kD
as expected. The crossover from the Rayleigh regime Γ̃ ∝ k̃4 to a behavior Γ̃ ∝ k̃s with s ≈ 2,
occurring at the low frequency edge of the boson peak [6, 10], is now understood to be a universal
feature independent of the magnitude of the correlation length (see figure 1).
In real amorphous solids, the relation Γ(k) ∝ k2 holds – at frequencies around the boson peak
frequency – for almost all glasses. This is in agreement with the present theory, which predicts
Γ(k) ∝ k2 in the boson peak for all values of γ and ξ. On the contrary, a clear-cut Rayleigh
scattering law (Γ(k) ∝ k4) is seldomly observed. Generally, at much lower frequencies, one finds a
sound attenuation proportional to k2, which increases with temperature and is therefore attributed
to anharmonic effects. Some evidence of a possible transition between k2 and k4 behavior has been
found for vitreous silica [17].
In order to account for this contribution we consider a cubic anharmonic interaction of the
form [19]
Van = λ0
∑
ij
uiivijvij + G(x)
∑
ij`
uijvi`v`j + g1
∑
i
u3
ii
(6)
with the linearized strain- uij = 1
2
(∂jui + ∂iuj) and rotation vij = 1
2
(∂jui − ∂iuj) tensor and
g1 a phenomenological mode-Grüneisen-like parameter. Using a replica field formalism at finite
temperature T , a set of mode-coupling-like self consistency equations for the full dynamical sus-
23606-3
W. Schirmacher et al.
ceptibilities has been derived [20]. This treatment involving an Akhiezer-like ’mode-coupling’ pa-
rameter g(〈∆G2〉, ξ, kBT ) = (1 + g2
1)〈∆G2〉kBT/3ξ3π4ρ3v̄4
T . In a perturbative regime, the relevant
anharmonic contribution to the attenuation of density fluctuations can be reduced to
Σ′′
an,i(0, ω̃) = g
∫ ∞
0
∫ ξkD
0
dk̃k̃2
∫ ξkD
0
dq̃q̃2 d˜̄ω
˜̄ω
χ̃′′
T (q̃, ˜̄ω+)
[
χ̃′′
T (k̃, ˜̄ω + ω̃) − χ̃′′
T (k̃, ˜̄ω − ω̃)
]
, (7)
where the full susceptibilities are replaced with their harmonic disorder-modified counterparts (2).
The specific form of the correlation function enters the anharmonic sound attenuation function
only indirectly through the harmonic susceptibilities. The latter ones, are only weakly effected by
a change from exponential to Gaussian correlations. At room temperature, the integral (7) scales
like Tω for ω � ωB. This is already suggested by the specific form of the integral kernel, and has
been confirmed numerically [20].
Figure 2. Brillouin linewidth, divided by frequency Σ̃′′
·α including disorder and anharmonicity
for ξ = 2/kD, γ/γc = 0.99 , α = (vL/vT )2 = 2.52 for different temperatures T/T0=0 (full,black),
0.001 (long dashes, red), 0.005 (short dashes, green), 0.01 (dash-dots, blue). T0 is defined by
Σ̃′′
an(ω̃) = (T/T0)ω̃ in the linear region.
Figure 2 shows the full Brillouin linewidth due to disorder and anharmonicity with the known
parameters of SiO2 [6] γ = 0.99, ∆G2
c = 0.401, ρ2v4
T , ξ = 2/kD, (vL/vT )2 = 2.52,kD = 1.6×1010/m,
T = 300 K, g1 ≈ 0. The anharmonic corrections already lead to deviations from the disordered
contribution slightly below the shoulder of the BP, the highest frequency, where the Akhiezer-like
behavior is present in SiO2, is therefore predicted to be one order of magnitude below the BP, i.e.,
in the 100 GHz regime.
This compares qualitatively with some recent inelastic UV light scattering data of vitreous
SiO2 by Masciovecchio et al. [17]. They observed the crossover from a ω2 to a very steep increase
– compatible with a ω4 law – around 100 GHz.
More recently Devos et al. [18] reported on the results of the measurements of sound absorption
on thin films of SiO2 produced by chemical vapor deposition (CVD). These Γ(k) data are found
to be different from the data of [17]. The authors of [18], implicitly assuming that the structure
of the materials investigated in the two sets of experiments are the same, claim that the data sets
reported in [17] must be in error.
As it is well known, the structure of disordered solids crucially depends on the preparation
method [21]. A glass, which is quenched from the melt, is known to have a much more relaxed
structure than an amorphous material evaporated onto a cold substrate from the gaseous phase.
Therefore, we believe that both the anharmonic coupling strength as well as the correlation length
of differently prepared materials can differ considerably. This may be the reason that the Akhiezer-
Rayleigh-crossover might occur at different frequencies in differently prepared materials.
23606-4
Sound attenuation and anharmonic damping in solids with correlated disorder
In conclusion, we have combined a model in which the shear modulus exhibits spatially corre-
lated fluctuations with an anharmonic interaction. By this we are able to discuss these phenomena
in terms of the structure of the materials and to solve an apparent contradiction recently reported
in the literature. More importantly, we have at hand a theory capable of quantitatively reproducing
most of the phenomena related to the high frequency vibrations in glasses, such as the existence
of a boson peak and the puzzling k dependent behavior of the sound attenuation.
References
1. Binder K., Kob W., Glassy Materials and Disordered Solids. World Scientific, Singapore, 2005.
2. Ruocco G., Nature Mater., 2008, 7, 842.
3. Horbach J., Kob W., Binder K., Eur. Phys. J. B, 2001, 19, 531.
4. Nakayama T., Rep. Prog. Phys., 2002, 65, 1195.
5. Gurevich V.L., Parshin D.A., Schober H.R., Phys. Rev. B, 2003, 67, 094203.
6. Schirmacher W., Ruocco G., Scorpigno T., Phys. Rev. Lett., 2007, 98, No. 2, 025501.
7. Rufflé B., Parshin D.A., Courtens E., Vacher R., Phys. Rev. Lett., 2008, 100, No. 1, 015501.
8. Shintani H., Tanaka H., Nature Mater., 2008, 7, 7, 670.
9. Maurer E., Schirmacher W., J. Low-Temp. Phys., 2004, 137, 453.
10. Schirmacher W., Europhys. Lett., 2006, 73, 892.
11. Schmid B., Schirmacher W., Phys. Rev. Lett., 2008, 100, 137402.
12. Schirmacher W., Schmid B., Tomaras C., Viliani G., Baldi G., Ruocco G., Scorpigno T., Phys. Stat.
Sol. (c), 2008, 5, 862.
13. Leonforte F., Tanguy A., Wittmer J.P., Barrat J.-L., Phys. Rev. Lett., 2006, 97 055501.
14. Elliott S.R., Europhys. Lett., 1991, 19, 201.
15. John S., Stephen M.J., Phys. Rev. B, 1983, 28, 6358.
16. Ignatchenko V.A., Felk V.A., Phys. Rev. B, 2006, 74, 174415.
17. Masciovecchio C., Baldi G., Caponi S., Comez L., Di Fonzo S., Fioretto D., Fontana A., Gessini A.,
Santucci S.C., Sette F., Viliani G., Vilmercati P., Ruocco G., Phys. Rev. Lett., 2006, 97, 035501.
18. Devos A., Foret M., Ayrinhac S., Emery P., Rufflé B., Phys. Rev. B, 2008, 77, 100201.
19. Landau L.D., Lifshitz E.M., Theory of Elasticity. Butterworth, Oxford, 1986.
20. Tomaras C., Schmid B., Schirmacher W., Phys. Rev. B, 2010, 81, 104206.
21. Elliott S.R., Physics of Amorphous Materials, 2nd edition. Longman Scientific, London, 1990.
23606-5
W. Schirmacher et al.
Послаблення звуку та ангармонiчне загасання у твердих
тiлах зi скорельованим безладом
В. Шiрмахер1,2, К. Томарас3, Б. Шмiд2, Дж. Бальдi4, Дж. Вiлiанi5 , Дж. Руокко6,7,
T. Скопiньйо6,7
1 Вiддiлення фiзики E13, Технiчний унiверситет Мюнхена, Ґархiнґ, Нiмеччина
2 Факультет фiзики, Унiверситет Майнца, Майнц, Нiмеччина
3 Центр теоретичної фiзики Арнольда Зоммерфельда, Мюнхенський унiверситет iм.
Людвiґа-Максимiлiана, Мюнхен, Нiмеччина
4 ЄЦСР, Ґренобль, Францiя
5 Факультет фiзики, Унiверситет Тренто, Тренто, Iталiя
6 Факультет фiзики, Римський унiверситет, Рим, Iталiя
7 Центр SOFT-INFM-CNR, Italy
За допомогою самоузгодженого наближення Борна ми дослiджуємо модель для звукових хвиль у
невпорядкованому оточеннi, де локальнi флуктуацiї модуля зсуву G є просторово скорельованi з
певною кореляцiйною довжиною ξ. Теорiя передбачає зростання густини станiв понад законом Де-
бая ω2 (бозонний пiк), чия iнтенсивнiсть зростає iз збiльшенням кореляцiйної довжини та частота
зсувається в область низьких частот як 1/ξ. Крiм того, передбачено, що коефiцiєнт викликаного
безладом послаблення звуку Γ(k) задовiльняє унiверсальному закону скейлiнґу ξΓ(k) = f(kξ) для
заданої варiацiї G. Врештi, врахування внеску найнижчого порядку до ангармонiчного загасання зву-
ку в теорiї дозволяє нам узгодити наявнi суперечливi недавнi експериментальнi данi для аморфного
SiO2.
Ключовi слова: послаблення звуку, вiбрацiйнi властивостi невпрорядкованих твердих тiл,
бозонний пiк, ангармонiчнi взаємодiї
23606-6
|
| id | nasplib_isofts_kiev_ua-123456789-32096 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T19:01:46Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Schirmacher, W. Tomaras, C. Schmid, B. Baldi, G. Viliani, G. Ruocco, G. Scopigno, T. 2012-04-08T16:12:55Z 2012-04-08T16:12:55Z 2010 Sound attenuation and anharmonic damping in solids with correlated disorder / W. Schirmacher, C. Tomaras, B. Schmid, G. Baldi, G. Viliani, G. Ruocco, T. Scopigno // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23606: 1-6. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 63.50.-x, 43.20.+g, 65.60.+a https://nasplib.isofts.kiev.ua/handle/123456789/32096 We study via self-consistent Born approximation a model for sound waves in a disordered environment, in which the local fluctuations of the shear modulus G are spatially correlated with a certain correlation length ξ. The theory predicts an enhancement of the density of states over Debye's ω2 law (boson peak) whose intensity increases for increasing correlation length, and whose frequency position is shifted downwards as 1/ξ. Moreover, the predicted disorder-induced sound attenuation coefficient Γ(k) obeys a universal scaling law ξ Γ(k) = f(kξ) for a given variance of G. Finally, the inclusion of the lowest-order contribution to the anharmonic sound damping into the theory allows us to reconcile apparently contradictory recent experimental data in amorphous SiO2. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Sound attenuation and anharmonic damping in solids with correlated disorder Послаблення звуку та ангармонічне загасання у твердих тілах зі скорельованим безладом Article published earlier |
| spellingShingle | Sound attenuation and anharmonic damping in solids with correlated disorder Schirmacher, W. Tomaras, C. Schmid, B. Baldi, G. Viliani, G. Ruocco, G. Scopigno, T. |
| title | Sound attenuation and anharmonic damping in solids with correlated disorder |
| title_alt | Послаблення звуку та ангармонічне загасання у твердих тілах зі скорельованим безладом |
| title_full | Sound attenuation and anharmonic damping in solids with correlated disorder |
| title_fullStr | Sound attenuation and anharmonic damping in solids with correlated disorder |
| title_full_unstemmed | Sound attenuation and anharmonic damping in solids with correlated disorder |
| title_short | Sound attenuation and anharmonic damping in solids with correlated disorder |
| title_sort | sound attenuation and anharmonic damping in solids with correlated disorder |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32096 |
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