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Light scattering by low-energy excitations in glasses
ELSEVIER
Physica B 219&220 (1996) 251 254
Light scattering by low-energy excitations in glasses A.P. Sokolov Institute of Automation and ElectromeoT, Russian Academy of Sciences, Universitets~, pr.1, 630090, Novosibirsk, Russian Federation
Abstract
It is well-known that the low-frequency Raman spectra in disordered materials have two contributions: vibrational, the so-called boson peak, and relaxational quasielastic scattering. Comparisons of the Raman spectra with neutron scattering and heat capacity data are presented. It shows that there is some intrinsic relation between the quasielastic scattering and the boson peak vibrations. The results are described in the framework of the model of damped oscillators.
It is well-known that low-energy (v < 100 c m - 1) excitations spectra of glasses have two anomalous (in comparison with the Debye model) contributions: anharmonic at very low energies, which is usually ascribed to motion in double wells, and harmonic ascribed to quasilocal vibrations. In Raman or inelastic neutron scattering spectra the former appears as a quasielastic scattering (QES) and the latter as the so-called boson peak. The nature of both kinds of excitations and the mechanism of the light scattering are still not clear. It was suggested in Ref. [1] that the light scattering intensity in disordered systems is proportional to the density of the vibrational states ,q(v): I(v) = .q(v) C(v)(n + 1)/v.
(1)
Here (n + 1) is a temperature dependent Bose factor and C(v) is a light-to-excitations coupling coefficient. It depends on the way how the excitations modulate the dielectric constant of the medium. In particular, it was shown that for acoustic plane waves C ( v ) ~ v 2 [2]. In contrast it was suggested in the framework of a softpotential model that C(v) is constant for the low-energy excitations localised in soft potentials and is negligibly small for the sound waves [3]. Fig. l(a) shows the Raman and neutron scattering data for SiO2 at few temperatures. In both spectra there are
low-frequency boson peaks at Vmax ~3(~50 cm 1, which varies almost harmonically with T, and the QES contribution at a lower frequency, which strongly increases with T. One can see that the Raman and neutron spectra are similar in the frequency range where the QES contribution dominates (high T, low frequencies), but differ strongly at frequencies where the vibrational contribution dominates (v > Vrnax), Fig. l(b) shows the effective coupling coefficient Cert(V) obtained from the comparison of the light and neutron scattering spectra (Eq. (1)). For vibrational excitations (low T or high v) Ceff(V ) ~ V, but Ceff(V ) ~ const, in the region where the QES contribution dominates. In order to point out the frequency dependence of the vibrational coupling coefficient Cv(v), Fig. 2 shows a comparison of the neutron data with the Raman spectrum for lowest T, where the QES contribution is strongly suppressed. One can see that the assumption of Cv(v) ~v (Fig. 2) gives a reasonably good approximation and reproduces the total ,q(v) from the Raman data. The same conclusion can be done from comparison of the Raman data with heat capacity measurements at low T (Fig. 3) [4]. Similar results about a nearly linear Cv(v) for the vibrational excitations have been obtained also for Se, As2S3, B203, DGEBA, PB, PS, P M M A and Sm-P O glasses [4-7]. Thus the latter seems to be a general behaviour for the vibrational Cv(v).
0921-4526/96/$15.00 ~c 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 1 9 5 ) 0 0 7 1 0 - 5
252
A.P. Sokolov / Physica B 219&220 (1996) 251-254
0.1
. . . . . .
a.,
1
,
•
quasielastic s c a t t e r ~ 10"2 2 7 0 K 1 ~
]
t-'~lk
~t102 ~
~"
~[, o0K°
1 0 -3
the boson peak ~ t
. . . . . . .
b r
.
~;.
.
21-
~. ~:P~l~r"
Experiment O Heralux A Spectrosil Calculated (Herasil) I h (10K) 1L0
.
T [K]
"7.
i max
&
3"
1'5
20
Fig. 3. Comparison of the heat capacity data with estimations from the Raman spectra, assuming Cv(v) oc v [4].
g L) 10 4
160K! U°%v
o
0.1
1
50K
v [THz]
Fig. 1. (a) Low-frequency Raman (lines) and neutron (symbols) spectra of SiO2 at different T. The QES and the boson peak contributions are shown. (b) Effective coupling coefficient Cell(V) = Iv/g0, + 1) at different T. Line shows a linear dependence (data from Ref. [7]).
.
1 0 -2-
.
.
.
.
SiO2 o
.
.
.
i
~
"~" ~ +
an (10K) neutrons (50K)
~' ~
:-"
1 0 .3 0.1
. . . . . . . .
v [THz]
Fig. 2. Comparison of the depolarised Raman spectra (line), divided by additional v, with 9(v) (o).
At the frequency range, where the QES contribution dominates, CefdV) is nearly frequency independent (Fig. l(b)). This means that the coupling coefficient for QES Cqos(v) ~ const. As it was shown in Ref. [7], the weak frequency and temperature dependence of Coff(v) at v < Vm,x(Fig. l(b)) can be explained as a superposition of
two contributions to our spectra: QES with C q e s = const and vibrations with Cv(v) ~ v. It was also found in Ref. [7] that Cqe.~~ Cv(Vmax).This result has been obtained for all three glasses PB, PS and SiO2 analysed in Ref. [7], although the Ymaxdiffers ~3 times in these substances. This result shows that there is some relation between these two contributions to the light scattering spectra. The results presented above show that the light-tovibrations coupling coefficient Cv(v) has a strong frequency dependence. However, it is not v2, like it is expected for acoustic plane waves with weak damping, but it varies rather linearly with v. This means that vibrational excitations in the frequency range around the boson peak are not plane waves. It is important that in all glasses analysed so far a reasonable agreement has been found between Raman spectra and total g(v) (Figs• 2, 3) [4-7]. Thus one can tentatively conclude that at least for the light scattering process all vibrations around the boson peak have the same properties and there is no reason for their separation into sound waves and quasilocal vibrations. The nature of this linear dependence for Cv(v) is still not clear, although there are some suggestions [5, 8]. An important result of the light scattering data analysis is the fact that Cqc s ~ Cv(vmax) (Fig. l(b)) [7]. This means that the quasielastic contribution appears in the Raman spectra of glasses with the same strength as the vibrations around the boson peak maximum. This is an additional support of the idea suggested in Refs. [9, 10], that the anharmonic (the quasielastic) and harmonic (the boson peak) peculiarities of the low-energy spectrum in glasses are interrelated. First support of this idea came from the analysis of the depolarisation ratio p, which is defined as a ratio of depolarised to polarised light scattering intensities. It was found that p varies from ~0.25 for ZBLAN
A.P. Sokolov/Physica B 219&220 (1996) 251 254
glass, ~0.3 for SiO2 up to maximum value ~0.75 in most of the organic systems. But in all cases it was found to be the same for the boson peak and for the quasielastic scattering. The model suggested in Refs. [9, 10] assumes that the vibrations in glasses are moving like in viscous media due to intensive relaxation processes. In this case one can write for the elastic response function of the vibration with frequency f2: D(Q,o)) = [,,)2 _ ~22[1 _ M(e),z)] ) 1,
(2)
where M(u), r) is a memory function, which was chosen in Refs. [9, 10] as M(o), z} = i &oL/(1 - i(or).
(3)
Here b describes the strength of the relaxation channel which damps the vibration, and r is its characteristic relaxation time. The microscopic nature of the memory function is not specified in the model. It can be generalised for the case of a distribution of relaxation times r. The most important consequence of the assumption (Eq. (2))is that the scattering spectra I(u)) oc Im{D(~2, m)} will have two contributions: an inelastic one at (o ~ and a quasielastic one with intensity ~ 6 and width ~ r -1 [10]. Assuming that all vibrations around the boson peak are damped by a similar mechanism (Eq. (2)), one can write for the Raman scattering intensity: I((,)) ~_c.f d/2 C,j(O).q(Q) Im {D((2, (,))}.
(4)
Here {j are polarisation indexes, and for the neutron scattering: S(q, u)) -~ j"dr2 .q(g2)Im {D(gL e))}.
(5)
Here the integration is over all vibrations contributing to the boson peak. At small damping IM(& c,))l ~ £22, Eqs. (2)-(5) give for the quasielastic intensities at lower frequencies:
/qes(¢')) ~ (57;/(1 + ~2)2"C2) ~d(2 Cij(~))q(~.Q)/~C22
(6)
and
Sqcs( q, ( o ) ~ 6L/(I +
(;)2.[.2)I d'Qg((2)/Q2.
(7)
The coupling coefficient for the quasielastic (anharmonic) part of the spectra can be written now as a ratio of Eqs. (6) and (7):
Cqcs((O ) =
Iqes((o),/Sqcs(q, (o)
253
vibrational excitations around the boson peak maximum (Fig. l(b)) and also the depolarisation ratio for QES. According to this model the QES intensity is proportional not only to the relaxation strength 3, but also to the integrated boson peak intensity (Eqs. (6) and (7)). This can explain a contradiction which has been found for Sm P O glasses in Ref. [1 l]: the authors observed that the QES intensity increases with increasing Sm content, although the relaxation strength (for acoustic waves) decreases. The Raman spectra presented in Ref. [11] show a strong increase of the boson peak intensity with increasing Sm content. This increase is stronger than the decrease of the relaxation strength and as a result the QES intensity increases. The results presented above clearly show that there is some intrinsic relation between harmonic (vibrational) and anharmonic (quasielastic, TLS) contributions to the dynamic spectra of glasses. The model [9, 10] ascribes both contributions to the same type of motion which has a harmonic and an anharmonic part (Eq. (2)). The latter appears due to an anharmonicity of the glassy structure (strong relaxation processes, permanent structure variations, etc.). This anharmonicity increases with temperature and depends strongly on the chemical composition. In particular, it was found in Ref. [12] that the ratio of the anharmonic to the harmonic contributions to the spectra correlates with the degree of fragility of the system: it is very small in 'strong' glassformers (like SiO2), which show a nearly Arrhenius-like temperature variation of the viscosity ~1 around Tg, and very high for "fragile" systems (like Ca K NO3) where q shows strongly non-Arrhenius behaviour. Thus the analysis of the light scattering spectra and their comparison with the neutron and heat capacity data shows that there is some relation between the quasielastic scattering and the boson peak. This relation can be explained in the framework of the model of a damped oscillator [9, 10], which assumes that the low-frequency vibrations are moving in the glassy structure like in a viscous medium. The comparison also shows that the vibrational contribution to the Raman spectra can be well described assuming that the total density of the vibrational states contributes to the light scattering spectra with C~(v)~ v. This result means that at least from the light scattering experiments there is no reason to separate the vibrations into sound waves and quasilocal excitations. However, the nature of this linear frequency dependence of C,.(v) is still not clear.
_ fd~ GA~),q(~)/~ ~
dr2 g(f2)/Q2 = const ~
Ci,i(~'~max).
(8)
Thus the model [-9, 10] predicts the relation observed in Ref. [7] between Cqes and the coupling coefficient for the
References [11 R. Shuker and R.W. Gammon, Phys. Rev. Lett. 25 (1970) 222.
254
A.P. Sokolov/Physica B 219&220 (1996) 251-254
[2] J. Jackle, in: Amorphous Solids: Low-Temperature Properties, ed. W.A. Phillips (Springer, Berlin, 1981) pp. 135-160. [3] V.L. Gurevich, D.A. Parshin, J. Pelous and H.R. Schober, Phys. Rev. B 48 (1993) 16318. [-4] A.P. Sokolov, A. Kisliuk, D. Quitmann and E. Duval, Phys. Rev. 13 48 (1993) 7692. I-5] E. Duval et al., J. Chem. Phys. 99 (1993) 2040, 2046.
[6] A. Brodin et al., Phys. Rev. Lett. 73 (1994) 2067. I-7] A.P. Sokolov, U. Buchenau et al., Phys. Rev. 13 52 (1995) R9815. [8] V.N. Novikov, Soy. JETP Letters 51 (1990) 77. [-9] P. Fulde and H. Wagner, Phys. Rev. Lett. 27 (1971) 1280. [10] G. Winterling, Phys. Rev. B 12 (1975) 2432. [11] C. Carini et al., Phys. Rev. 13 47 (1993) 3005. [12] A.P. Sokolov et al., Phys. Rev. Lett. 71 (1993) 2062.
Physica B 219&220 (1996) 251 254
Light scattering by low-energy excitations in glasses A.P. Sokolov Institute of Automation and ElectromeoT, Russian Academy of Sciences, Universitets~, pr.1, 630090, Novosibirsk, Russian Federation
Abstract
It is well-known that the low-frequency Raman spectra in disordered materials have two contributions: vibrational, the so-called boson peak, and relaxational quasielastic scattering. Comparisons of the Raman spectra with neutron scattering and heat capacity data are presented. It shows that there is some intrinsic relation between the quasielastic scattering and the boson peak vibrations. The results are described in the framework of the model of damped oscillators.
It is well-known that low-energy (v < 100 c m - 1) excitations spectra of glasses have two anomalous (in comparison with the Debye model) contributions: anharmonic at very low energies, which is usually ascribed to motion in double wells, and harmonic ascribed to quasilocal vibrations. In Raman or inelastic neutron scattering spectra the former appears as a quasielastic scattering (QES) and the latter as the so-called boson peak. The nature of both kinds of excitations and the mechanism of the light scattering are still not clear. It was suggested in Ref. [1] that the light scattering intensity in disordered systems is proportional to the density of the vibrational states ,q(v): I(v) = .q(v) C(v)(n + 1)/v.
(1)
Here (n + 1) is a temperature dependent Bose factor and C(v) is a light-to-excitations coupling coefficient. It depends on the way how the excitations modulate the dielectric constant of the medium. In particular, it was shown that for acoustic plane waves C ( v ) ~ v 2 [2]. In contrast it was suggested in the framework of a softpotential model that C(v) is constant for the low-energy excitations localised in soft potentials and is negligibly small for the sound waves [3]. Fig. l(a) shows the Raman and neutron scattering data for SiO2 at few temperatures. In both spectra there are
low-frequency boson peaks at Vmax ~3(~50 cm 1, which varies almost harmonically with T, and the QES contribution at a lower frequency, which strongly increases with T. One can see that the Raman and neutron spectra are similar in the frequency range where the QES contribution dominates (high T, low frequencies), but differ strongly at frequencies where the vibrational contribution dominates (v > Vrnax), Fig. l(b) shows the effective coupling coefficient Cert(V) obtained from the comparison of the light and neutron scattering spectra (Eq. (1)). For vibrational excitations (low T or high v) Ceff(V ) ~ V, but Ceff(V ) ~ const, in the region where the QES contribution dominates. In order to point out the frequency dependence of the vibrational coupling coefficient Cv(v), Fig. 2 shows a comparison of the neutron data with the Raman spectrum for lowest T, where the QES contribution is strongly suppressed. One can see that the assumption of Cv(v) ~v (Fig. 2) gives a reasonably good approximation and reproduces the total ,q(v) from the Raman data. The same conclusion can be done from comparison of the Raman data with heat capacity measurements at low T (Fig. 3) [4]. Similar results about a nearly linear Cv(v) for the vibrational excitations have been obtained also for Se, As2S3, B203, DGEBA, PB, PS, P M M A and Sm-P O glasses [4-7]. Thus the latter seems to be a general behaviour for the vibrational Cv(v).
0921-4526/96/$15.00 ~c 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 1 9 5 ) 0 0 7 1 0 - 5
252
A.P. Sokolov / Physica B 219&220 (1996) 251-254
0.1
. . . . . .
a.,
1
,
•
quasielastic s c a t t e r ~ 10"2 2 7 0 K 1 ~
]
t-'~lk
~t102 ~
~"
~[, o0K°
1 0 -3
the boson peak ~ t
. . . . . . .
b r
.
~;.
.
21-
~. ~:P~l~r"
Experiment O Heralux A Spectrosil Calculated (Herasil) I h (10K) 1L0
.
T [K]
"7.
i max
&
3"
1'5
20
Fig. 3. Comparison of the heat capacity data with estimations from the Raman spectra, assuming Cv(v) oc v [4].
g L) 10 4
160K! U°%v
o
0.1
1
50K
v [THz]
Fig. 1. (a) Low-frequency Raman (lines) and neutron (symbols) spectra of SiO2 at different T. The QES and the boson peak contributions are shown. (b) Effective coupling coefficient Cell(V) = Iv/g0, + 1) at different T. Line shows a linear dependence (data from Ref. [7]).
.
1 0 -2-
.
.
.
.
SiO2 o
.
.
.
i
~
"~" ~ +
an (10K) neutrons (50K)
~' ~
:-"
1 0 .3 0.1
. . . . . . . .
v [THz]
Fig. 2. Comparison of the depolarised Raman spectra (line), divided by additional v, with 9(v) (o).
At the frequency range, where the QES contribution dominates, CefdV) is nearly frequency independent (Fig. l(b)). This means that the coupling coefficient for QES Cqos(v) ~ const. As it was shown in Ref. [7], the weak frequency and temperature dependence of Coff(v) at v < Vm,x(Fig. l(b)) can be explained as a superposition of
two contributions to our spectra: QES with C q e s = const and vibrations with Cv(v) ~ v. It was also found in Ref. [7] that Cqe.~~ Cv(Vmax).This result has been obtained for all three glasses PB, PS and SiO2 analysed in Ref. [7], although the Ymaxdiffers ~3 times in these substances. This result shows that there is some relation between these two contributions to the light scattering spectra. The results presented above show that the light-tovibrations coupling coefficient Cv(v) has a strong frequency dependence. However, it is not v2, like it is expected for acoustic plane waves with weak damping, but it varies rather linearly with v. This means that vibrational excitations in the frequency range around the boson peak are not plane waves. It is important that in all glasses analysed so far a reasonable agreement has been found between Raman spectra and total g(v) (Figs• 2, 3) [4-7]. Thus one can tentatively conclude that at least for the light scattering process all vibrations around the boson peak have the same properties and there is no reason for their separation into sound waves and quasilocal vibrations. The nature of this linear dependence for Cv(v) is still not clear, although there are some suggestions [5, 8]. An important result of the light scattering data analysis is the fact that Cqc s ~ Cv(vmax) (Fig. l(b)) [7]. This means that the quasielastic contribution appears in the Raman spectra of glasses with the same strength as the vibrations around the boson peak maximum. This is an additional support of the idea suggested in Refs. [9, 10], that the anharmonic (the quasielastic) and harmonic (the boson peak) peculiarities of the low-energy spectrum in glasses are interrelated. First support of this idea came from the analysis of the depolarisation ratio p, which is defined as a ratio of depolarised to polarised light scattering intensities. It was found that p varies from ~0.25 for ZBLAN
A.P. Sokolov/Physica B 219&220 (1996) 251 254
glass, ~0.3 for SiO2 up to maximum value ~0.75 in most of the organic systems. But in all cases it was found to be the same for the boson peak and for the quasielastic scattering. The model suggested in Refs. [9, 10] assumes that the vibrations in glasses are moving like in viscous media due to intensive relaxation processes. In this case one can write for the elastic response function of the vibration with frequency f2: D(Q,o)) = [,,)2 _ ~22[1 _ M(e),z)] ) 1,
(2)
where M(u), r) is a memory function, which was chosen in Refs. [9, 10] as M(o), z} = i &oL/(1 - i(or).
(3)
Here b describes the strength of the relaxation channel which damps the vibration, and r is its characteristic relaxation time. The microscopic nature of the memory function is not specified in the model. It can be generalised for the case of a distribution of relaxation times r. The most important consequence of the assumption (Eq. (2))is that the scattering spectra I(u)) oc Im{D(~2, m)} will have two contributions: an inelastic one at (o ~ and a quasielastic one with intensity ~ 6 and width ~ r -1 [10]. Assuming that all vibrations around the boson peak are damped by a similar mechanism (Eq. (2)), one can write for the Raman scattering intensity: I((,)) ~_c.f d/2 C,j(O).q(Q) Im {D((2, (,))}.
(4)
Here {j are polarisation indexes, and for the neutron scattering: S(q, u)) -~ j"dr2 .q(g2)Im {D(gL e))}.
(5)
Here the integration is over all vibrations contributing to the boson peak. At small damping IM(& c,))l ~ £22, Eqs. (2)-(5) give for the quasielastic intensities at lower frequencies:
/qes(¢')) ~ (57;/(1 + ~2)2"C2) ~d(2 Cij(~))q(~.Q)/~C22
(6)
and
Sqcs( q, ( o ) ~ 6L/(I +
(;)2.[.2)I d'Qg((2)/Q2.
(7)
The coupling coefficient for the quasielastic (anharmonic) part of the spectra can be written now as a ratio of Eqs. (6) and (7):
Cqcs((O ) =
Iqes((o),/Sqcs(q, (o)
253
vibrational excitations around the boson peak maximum (Fig. l(b)) and also the depolarisation ratio for QES. According to this model the QES intensity is proportional not only to the relaxation strength 3, but also to the integrated boson peak intensity (Eqs. (6) and (7)). This can explain a contradiction which has been found for Sm P O glasses in Ref. [1 l]: the authors observed that the QES intensity increases with increasing Sm content, although the relaxation strength (for acoustic waves) decreases. The Raman spectra presented in Ref. [11] show a strong increase of the boson peak intensity with increasing Sm content. This increase is stronger than the decrease of the relaxation strength and as a result the QES intensity increases. The results presented above clearly show that there is some intrinsic relation between harmonic (vibrational) and anharmonic (quasielastic, TLS) contributions to the dynamic spectra of glasses. The model [9, 10] ascribes both contributions to the same type of motion which has a harmonic and an anharmonic part (Eq. (2)). The latter appears due to an anharmonicity of the glassy structure (strong relaxation processes, permanent structure variations, etc.). This anharmonicity increases with temperature and depends strongly on the chemical composition. In particular, it was found in Ref. [12] that the ratio of the anharmonic to the harmonic contributions to the spectra correlates with the degree of fragility of the system: it is very small in 'strong' glassformers (like SiO2), which show a nearly Arrhenius-like temperature variation of the viscosity ~1 around Tg, and very high for "fragile" systems (like Ca K NO3) where q shows strongly non-Arrhenius behaviour. Thus the analysis of the light scattering spectra and their comparison with the neutron and heat capacity data shows that there is some relation between the quasielastic scattering and the boson peak. This relation can be explained in the framework of the model of a damped oscillator [9, 10], which assumes that the low-frequency vibrations are moving in the glassy structure like in a viscous medium. The comparison also shows that the vibrational contribution to the Raman spectra can be well described assuming that the total density of the vibrational states contributes to the light scattering spectra with C~(v)~ v. This result means that at least from the light scattering experiments there is no reason to separate the vibrations into sound waves and quasilocal excitations. However, the nature of this linear frequency dependence of C,.(v) is still not clear.
_ fd~ GA~),q(~)/~ ~
dr2 g(f2)/Q2 = const ~
Ci,i(~'~max).
(8)
Thus the model [-9, 10] predicts the relation observed in Ref. [7] between Cqes and the coupling coefficient for the
References [11 R. Shuker and R.W. Gammon, Phys. Rev. Lett. 25 (1970) 222.
254
A.P. Sokolov/Physica B 219&220 (1996) 251-254
[2] J. Jackle, in: Amorphous Solids: Low-Temperature Properties, ed. W.A. Phillips (Springer, Berlin, 1981) pp. 135-160. [3] V.L. Gurevich, D.A. Parshin, J. Pelous and H.R. Schober, Phys. Rev. B 48 (1993) 16318. [-4] A.P. Sokolov, A. Kisliuk, D. Quitmann and E. Duval, Phys. Rev. 13 48 (1993) 7692. I-5] E. Duval et al., J. Chem. Phys. 99 (1993) 2040, 2046.
[6] A. Brodin et al., Phys. Rev. Lett. 73 (1994) 2067. I-7] A.P. Sokolov, U. Buchenau et al., Phys. Rev. 13 52 (1995) R9815. [8] V.N. Novikov, Soy. JETP Letters 51 (1990) 77. [-9] P. Fulde and H. Wagner, Phys. Rev. Lett. 27 (1971) 1280. [10] G. Winterling, Phys. Rev. B 12 (1975) 2432. [11] C. Carini et al., Phys. Rev. 13 47 (1993) 3005. [12] A.P. Sokolov et al., Phys. Rev. Lett. 71 (1993) 2062.