- Email: [email protected]
Acoustic vibrations of embedded spherical nanoparticles
ARTICLE IN PRESS
Physica E 26 (2005) 417–421 www.elsevier.com/locate/physe
Acoustic vibrations of embedded spherical nanoparticles Daniel B. Murraya,, Lucien Saviotb a
Department of Physics and Astronomy, Okanagan University College, Kelowna, British Columbia, Canada V1V 1V7 Laboratoire de Recherche sur la Re´activite´ des Solides, UMR 5613 CNRS—Universite´ de Bourgogne, Dijon, France
b
Available online 23 November 2004
Abstract A solid-matrix-embedded spherical nanoparticle has acoustic vibrational frequencies which are shifted and damped relative to modes of a free sphere. Not only the longitudinal plane wave acoustic impedances, but also the Poisson ratios of nanoparticle and matrix are important in determining the Q-factor of the ‘‘breathing’’ mode, for which frequencies and Q-factors with different material combinations are presented. High matrix sound speed (e.g. silica, titania, alumina, diamond) increases Q. r 2004 Elsevier B.V. All rights reserved. PACS: 63.22.+m; 62.25.+g; 78.30.j Keywords: Nanoparticles; Phonons; Matrix; Raman scattering
A free elastic sphere has vibrational modes whose lowest frequency is on the order of the speed of sound divided by the sphere radius. This classic problem of elastic mechanics was solved by Lamb in 1882 [1] for the case of a homogeneous sphere with isotropic elasticity. Lamb concluded his paper by assuming that the earth is made of solid steel and estimated the period of its lowest vibrational mode to be 1 h and 18 min. Ninety years later, this solution found application in explaining anomalies of the specific heat of Corresponding author. Fax: 250-491-9380.
E-mail addresses: [email protected], [email protected] (D.B. Murray).
powders of spherical lead particles 2–4 nm in size [2]. Direct observations of vibrational modes of spinel microcrystallites 22–35 nm in diameter [3] became possible by shining an intense laser beam onto the sample and carefully analyzing the scattered light to detect scattered frequencies slightly above and slightly below the laser frequency. The frequency shift of the light is equal to a vibrational frequency of the spinel microcrystallite. Since then, Raman and Brillouin scattering have remained powerful tools in the characterization of ‘‘nanoparticles’’—particles with diameters of the order of a few to a few dozen nanometers. The nanoparticles in these experiments are often not free, but rather embedded in solid materials
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.08.091
ARTICLE IN PRESS 418
D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421
such as glass. The Lamb solution does not directly apply since it assumes zero traction force boundary conditions of the sphere (i.e. a free surface). Initial attempts [4,5] to generalize the calculation to an isotropic elastic continuum sphere embedded in an infinite elastic matrix are in error [6]. But an earlier calculation [7] in the geophysics literature had correctly solved the problem and showed that an embedded sphere has complex-valued frequencies o where ReðoÞ is the center frequency of the Raman peak and ImðoÞ is the half-width at halfmaximum (HWHM). Correct application of this solution to nanoparticles was made only recently [8,9,11]. Vibrations of an embedded sphere are described in terms of a displacement field ~ uð~ r; tÞ which equals ~ where R ~ is the equilibrium position of a ~ rR material point and ~ r is its displaced position. Due to the spherical symmetry, it is convenient to choose the origin at the center of the sphere and use spherical coordinates r, y and f related to Cartesian coordinates through x ¼ r sinðyÞ cosðfÞ and so on. The time dependence is expðiotÞ where o is a complex number. In general, a vector ~ and a zero field ~ u is the sum of a zero-curl field rF ~~ divergence field r X where Fð~ rÞ is a scalar field and ~ Xð~ rÞ is a vector field. In the situation of an isotropic, homogeneous elastic medium with transverse speed of sound vT and longitudinal speed of sound vL ; normal modes are found by solving two uncoupled equations ðr2 þ k2L ÞF ¼ 0
(1)
and ðr2 þ k2T Þ~ X ¼~ 0;
(2)
where kL ¼ o=vL and kT ¼ o=vT and ~ ~ ð~ X¼r r cÞ þ ~ r w:
(3)
For convenience, the original Lamb solution will be called the free sphere model (FSM). The homogeneous isotropic sphere embedded in an infinite matrix will be called the complex frequency model (CFM) [6]. Whether for FSM or CFM, there are some ~ ~ solutions of the equation of motion with r u ¼ 0: Lamb [1] called these the ‘‘first class’’ and the modern word is ‘‘torsional’’ (TOR). Apart from
TOR modes, an important special case is radial or ‘‘breathing’’ modes, in which ~ u is purely along the r-direction and depends only on r and not on y or ~ f: In this case ~ u is of the form rFðrÞ: The remaining modes of what Lamb called the ‘‘second class’’, now called ‘‘spheroidal’’ (SPH), have a more complicated form ~ þr ~ ðr ~ ð~ ~ u ¼ rF rcÞÞ;
(4)
where F and c are scalar fields. Apart from the breathing modes, SPH modes are neither zero curl nor zero divergence. It can be said that TOR modes are purely transverse in nature. As a result, their frequencies depend only on vT : The breathing modes are purely longitudinal, but have frequencies depending on both vT and vL : The remaining SPH modes are a mixture of motions, but in most cases their frequencies depend strongly on vT and only weakly on vL ; so they are predominantly transverse in character. For a typical ordinary material with Poisson ratio 13 (in which case vL ¼ 2vT ) 80% of all modes, counting both TOR and SPH, are primarily transverse, while 20% are more longitudinal. To systematically label FSM or CFM modes, it is convenient to use index q 2 f1; 2g where TOR=1 and SPH=2. In addition, modes have an angular momentum number ‘ 2 f0; 1; 2 . . .g and a z-angular momentum m where ‘pmp‘: For brevity, we consider only the m ¼ 0 case below. But it is important to keep in mind that FSM and CFM modes have degeneracy 2‘ þ 1: There is also a solution index n 2 f0; 1; 2 . . .g: The scalar field F has the form Fðr; yÞ ¼ j ‘ ðkL rÞP‘ ðcos yÞ;
(5)
where j ‘ ðÞ are spherical Bessel functions and P‘ ðÞ are Legendre polynomials. The scalar field c has the form cðr; yÞ ¼ j ‘ ðkT rÞP‘ ðcos yÞ:
(6)
A given normal mode, whether FSM or CFM, could be denoted (q, ‘; m, n) and its frequency is or oCFM oFSM q‘n q‘n : But we will normally prefer to denote TOR modes as (TOR, ‘; m, n) and SPH modes as (SPH, ‘; m, n). The first applications of the correct CFM solution for an isotropic continuum sphere
ARTICLE IN PRESS D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421
embedded in a matrix were restricted to SPH modes [8,9]. Group theoretical arguments [10] show that in the absence of any kind of anisotropy or non-sphericity, and in the limit that the nanoparticle is much smaller than the wavelength of the laser light, only the modes (SPH, ‘ ¼ 0) and (SPH, ‘ ¼ 2) can lead to Raman scattering. Correct calculations of the CFM torsional mode frequencies were reported later [11]. Even so, the interpretation of complex-valued frequencies of normal modes is problematic. The corresponding displacement fields are not orthonormalizable and these modes cannot be quantized in the same way that FSM modes can [12]. Clarification of the significance of CFM calculations was obtained through an exhaustive calculation of all of the realvalued frequencies of an isotropic spherical nanoparticle embedded in a large but finite matrix with a free spherical outer surface of radius Rm [6]. This is called the core-shell model (CSM). Taking the limit Rm ! 1; the number of CSM modes in a given range of frequency becomes infinite, and has the Debye density of states. However, assuming uniform random phase excitation of the CSM modes, the mean squared displacement within the nanoparticle exhibits peaks whose positions agree closely with the real part of CFM modes, and whose HWHM agree closely with the imaginary parts of CFM modes. Thus, CFM calculations have a solid basis and utility, even if the corresponding CFM modes must be treated with caution. Both FSM and CFM have limitations of applicability to real systems: (1) nanoparticles are not truly spherical in shape, although the dynamics of most elaboration processes make them approximately that way and transmission electron microscopy (TEM) images confirm this; (2) for small nanoparticles, the wavelength of even the low-lying modes is not long compared to interatomic spacing, so that we are not close to the center of the Brillouin zone and the acoustic limit; (3) there will be deviations of the properties of the nanoparticle material from the same material in the bulk due to surface effects; (4) in a given sample, the radii of nanoparticles will have a distribution, rather than a single value; (5) the interface between the nanoparticle and the matrix
419
is not correctly described in terms of continuity of the displacement field and components of the stress tensor [13]; and (6) the speed of sound within the nanoparticle depends on the direction of propagation if the elasticity of the nanoparticle material is anisotropic. It is traditional and convenient to express mode frequencies in terms of either of the dimensionless variables Z ¼ oRp =vTp and x ¼ oRp =vLp ; where Rp is the radius of the nanoparticle. Table 1 refers to the lowest CFM (SPH, ‘ ¼ 0) mode for selected combinations of nanoparticle and matrix materials. The quality factor of the resonance, Q= 12 ReðxÞ=ImðxÞ; is presented in the upper half of each cell with Reðx) in the lower half. Longitudinal plane wave acoustic impedance (LPWAI) is Z0 ¼ rvL : Table 1 also indicates LPWAIs for selected nanoparticle and matrix materials. We have explored the lowest (SPH, ‘ ¼ 0) mode (ignoring highly damped ‘‘matrix modes’’ when they are present) in the parameter space of rp ; rm ; vLp ; vTp ; vLm and vTm and found some general patterns. Notably, good impedance matching between nanoparticle and matrix as exhibited in very strong lowest (SPH,0) mode damping is an unusual occurrence requiring special conditions. Closeness of the LPWAI’s alone is not sufficient. If nm np (where nm and np are the Poisson ratios of the matrix and particle, respectively) then there is never strong damping, but we can have strong damping if nm np such as for a liquid matrix. Assuming the Poisson ratios are close, it is usually necessary that rm Xrp and vLm pvLp : In addition, if nm and np are both close to 13 then strong damping is still possible if rm is as little as 0:8 rp : Conversely, light damping is always seen when any of the following conditions is satisfied: (1) np nm ; (2) rm 0:8 rp ; and (3) vLm 1:2 vLp : The ability to choose the matrix so as to have nanoparticles with lightly damped vibrational modes is important for several reasons. First, by reducing homogeneous broadening, it allows inhomogeneous broadening effects to stand out clearly. This results in a clearer picture for femtosecond pump–probe experiments where the
ARTICLE IN PRESS D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421
420
Table 1 Each cell corresponding to a combination of nanoparticle (listed along the top) and matrix (listed along the left side) material shows Q (above) and ReðxÞ (below) for the CFM (SPH, ‘ ¼ 0; n ¼ 0) mode for the indicated combination of materials (Densities are in g/cc. Longitudinal speeds of sound are given in km/s. n is the Poisson ratio.) np ! r vL Z0
Matrix
n Diamond
3.5
TiO2
4.1
Al2 O3
4.0
SiO2
2.2
GeO2
3.6
BaOP2 O5
3.8
Y2 O3
5.0
ZrO2
6.1
MgF2
3.2
Water
1.0
Ethanol
0.79
Hg
14
18 64 8.6 35 11 44 6.0 13 3.4 12 4.6 18 6.8 34 6.1 37 7.4 24
0.07
1.5 1.5 1.2 0.91 1.5 20
0.50
0.22 0.23 0.17 0.18 0.27 0.30 0.22 0.28
0.50 0.50
damping time is longer but also in low-frequency Raman and Brillouin inelastic scattering experiments where a better separation of scattering lines is possible. Solid-matrix-embedded nanoparticles always have Q factor lower than free ones. However, free nanoparticles suffer from their own practical drawbacks. For example, surface adsorption can affect their properties. Also it is often technologically easier to deal with matrixembedded nanoparticles. Where solid embedding is technologically desirable, our results provide guidance about how the matrix can be optimally chosen.
Au
Ag
Si
19 3.3 64 0.42
11 3.8 39 0.36
2.3 9.0 21 0.22
4.8 4.3 20 0.38
35 4.1 5.9 3.3 7.6 3.5 9.4 3.1 7.6 3.0 5.9 3.0 4.1 3.1 3.8 3.2 6.3 3.1
29 4.2 3.3 3.4 4.6 3.6 5.0 3.0 4.1 2.9 3.1 2.9 2.0 3.1 2.0 3.4 3.3 3.1
6.6 4.3 3.1 4.6 3.9 4.4 1.4 2.7 1.3 2.6 0.73 2.7 3.1 4.5 3.5 4.5 1.7 4.6
30 4.3 2.8 4.1 4.1 4.1 2.7 3.2 2.2 3.1 1.5 3.2 4.4 2.6 3.2 4.3 1.5 3.4
380 4.5 23 4.4 35 4.4 8.3 4.4 8.1 4.5 11 4.5 20 4.5 24 4.5 14 4.4
55 2.9 88 2.9 3.8 2.9
29 2.8 47 2.8 1.9 2.8
16 2.8 25 2.8 0.80 3.8
1.9 2.7 3.5 2.7 14 4.5
11 2.5 19 2.5 1.6 5.3
CdS
Na
0.97 3.3 3.2 0.34
References [1] H. Lamb, Proc. London Math. Soc. 13 (1881–1882) 189–212. [2] V. Novotny, P.P.M. Meincke, J.H.P. Watson, Phys. Rev. Lett. 28 (1972) 901–903. [3] E. Duval, A. Boukenter, B. Champagnon, Phys. Rev. Lett. 56 (1986) 2052–2055. [4] A. Tamura, K. Higeta, T. Ichinokawa, J. Phys. C 15 (1982) 4975–4991. [5] N.N. Ovsyuk, V.N. Novikov, Phys. Rev. B 53 (1996) 3113–3118. [6] D.B. Murray, L. Saviot, Phys. Rev. B 69 (2004) 094305. [7] V.A. Dubrovskiy, V.S. Morochnik, Izv., Acad. Sci. USSR, Phys. Solid Earth 17 (1981) 494–504.
ARTICLE IN PRESS D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421 [8] N. Del Fatti, C. Voisin, F. Chevy, F. Valle´e, C. Flytzanis, J. Chem. Phys. 110 (1999) 11484–11487. [9] P. Verma, W. Cordts, G. Irmer, J. Monecke, Phys. Rev. B 60 (1999) 5778–5785. [10] E. Duval, Phys. Rev. B 46 (1992) 5795–5797.
421
[11] L. Saviot, D.B. Murray, M. del C. Marco de Lucas, Phys. Rev. B 69 (2004) 113402. [12] T. Takagahara, J. Lumin. 70 (1996) 129–143. [13] C.W. Bettenhausen, W.C. Bowie, M.R. Geller, Phys. Rev. B 68 (2003) 035431.
Physica E 26 (2005) 417–421 www.elsevier.com/locate/physe
Acoustic vibrations of embedded spherical nanoparticles Daniel B. Murraya,, Lucien Saviotb a
Department of Physics and Astronomy, Okanagan University College, Kelowna, British Columbia, Canada V1V 1V7 Laboratoire de Recherche sur la Re´activite´ des Solides, UMR 5613 CNRS—Universite´ de Bourgogne, Dijon, France
b
Available online 23 November 2004
Abstract A solid-matrix-embedded spherical nanoparticle has acoustic vibrational frequencies which are shifted and damped relative to modes of a free sphere. Not only the longitudinal plane wave acoustic impedances, but also the Poisson ratios of nanoparticle and matrix are important in determining the Q-factor of the ‘‘breathing’’ mode, for which frequencies and Q-factors with different material combinations are presented. High matrix sound speed (e.g. silica, titania, alumina, diamond) increases Q. r 2004 Elsevier B.V. All rights reserved. PACS: 63.22.+m; 62.25.+g; 78.30.j Keywords: Nanoparticles; Phonons; Matrix; Raman scattering
A free elastic sphere has vibrational modes whose lowest frequency is on the order of the speed of sound divided by the sphere radius. This classic problem of elastic mechanics was solved by Lamb in 1882 [1] for the case of a homogeneous sphere with isotropic elasticity. Lamb concluded his paper by assuming that the earth is made of solid steel and estimated the period of its lowest vibrational mode to be 1 h and 18 min. Ninety years later, this solution found application in explaining anomalies of the specific heat of Corresponding author. Fax: 250-491-9380.
E-mail addresses: [email protected], [email protected] (D.B. Murray).
powders of spherical lead particles 2–4 nm in size [2]. Direct observations of vibrational modes of spinel microcrystallites 22–35 nm in diameter [3] became possible by shining an intense laser beam onto the sample and carefully analyzing the scattered light to detect scattered frequencies slightly above and slightly below the laser frequency. The frequency shift of the light is equal to a vibrational frequency of the spinel microcrystallite. Since then, Raman and Brillouin scattering have remained powerful tools in the characterization of ‘‘nanoparticles’’—particles with diameters of the order of a few to a few dozen nanometers. The nanoparticles in these experiments are often not free, but rather embedded in solid materials
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.08.091
ARTICLE IN PRESS 418
D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421
such as glass. The Lamb solution does not directly apply since it assumes zero traction force boundary conditions of the sphere (i.e. a free surface). Initial attempts [4,5] to generalize the calculation to an isotropic elastic continuum sphere embedded in an infinite elastic matrix are in error [6]. But an earlier calculation [7] in the geophysics literature had correctly solved the problem and showed that an embedded sphere has complex-valued frequencies o where ReðoÞ is the center frequency of the Raman peak and ImðoÞ is the half-width at halfmaximum (HWHM). Correct application of this solution to nanoparticles was made only recently [8,9,11]. Vibrations of an embedded sphere are described in terms of a displacement field ~ uð~ r; tÞ which equals ~ where R ~ is the equilibrium position of a ~ rR material point and ~ r is its displaced position. Due to the spherical symmetry, it is convenient to choose the origin at the center of the sphere and use spherical coordinates r, y and f related to Cartesian coordinates through x ¼ r sinðyÞ cosðfÞ and so on. The time dependence is expðiotÞ where o is a complex number. In general, a vector ~ and a zero field ~ u is the sum of a zero-curl field rF ~~ divergence field r X where Fð~ rÞ is a scalar field and ~ Xð~ rÞ is a vector field. In the situation of an isotropic, homogeneous elastic medium with transverse speed of sound vT and longitudinal speed of sound vL ; normal modes are found by solving two uncoupled equations ðr2 þ k2L ÞF ¼ 0
(1)
and ðr2 þ k2T Þ~ X ¼~ 0;
(2)
where kL ¼ o=vL and kT ¼ o=vT and ~ ~ ð~ X¼r r cÞ þ ~ r w:
(3)
For convenience, the original Lamb solution will be called the free sphere model (FSM). The homogeneous isotropic sphere embedded in an infinite matrix will be called the complex frequency model (CFM) [6]. Whether for FSM or CFM, there are some ~ ~ solutions of the equation of motion with r u ¼ 0: Lamb [1] called these the ‘‘first class’’ and the modern word is ‘‘torsional’’ (TOR). Apart from
TOR modes, an important special case is radial or ‘‘breathing’’ modes, in which ~ u is purely along the r-direction and depends only on r and not on y or ~ f: In this case ~ u is of the form rFðrÞ: The remaining modes of what Lamb called the ‘‘second class’’, now called ‘‘spheroidal’’ (SPH), have a more complicated form ~ þr ~ ðr ~ ð~ ~ u ¼ rF rcÞÞ;
(4)
where F and c are scalar fields. Apart from the breathing modes, SPH modes are neither zero curl nor zero divergence. It can be said that TOR modes are purely transverse in nature. As a result, their frequencies depend only on vT : The breathing modes are purely longitudinal, but have frequencies depending on both vT and vL : The remaining SPH modes are a mixture of motions, but in most cases their frequencies depend strongly on vT and only weakly on vL ; so they are predominantly transverse in character. For a typical ordinary material with Poisson ratio 13 (in which case vL ¼ 2vT ) 80% of all modes, counting both TOR and SPH, are primarily transverse, while 20% are more longitudinal. To systematically label FSM or CFM modes, it is convenient to use index q 2 f1; 2g where TOR=1 and SPH=2. In addition, modes have an angular momentum number ‘ 2 f0; 1; 2 . . .g and a z-angular momentum m where ‘pmp‘: For brevity, we consider only the m ¼ 0 case below. But it is important to keep in mind that FSM and CFM modes have degeneracy 2‘ þ 1: There is also a solution index n 2 f0; 1; 2 . . .g: The scalar field F has the form Fðr; yÞ ¼ j ‘ ðkL rÞP‘ ðcos yÞ;
(5)
where j ‘ ðÞ are spherical Bessel functions and P‘ ðÞ are Legendre polynomials. The scalar field c has the form cðr; yÞ ¼ j ‘ ðkT rÞP‘ ðcos yÞ:
(6)
A given normal mode, whether FSM or CFM, could be denoted (q, ‘; m, n) and its frequency is or oCFM oFSM q‘n q‘n : But we will normally prefer to denote TOR modes as (TOR, ‘; m, n) and SPH modes as (SPH, ‘; m, n). The first applications of the correct CFM solution for an isotropic continuum sphere
ARTICLE IN PRESS D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421
embedded in a matrix were restricted to SPH modes [8,9]. Group theoretical arguments [10] show that in the absence of any kind of anisotropy or non-sphericity, and in the limit that the nanoparticle is much smaller than the wavelength of the laser light, only the modes (SPH, ‘ ¼ 0) and (SPH, ‘ ¼ 2) can lead to Raman scattering. Correct calculations of the CFM torsional mode frequencies were reported later [11]. Even so, the interpretation of complex-valued frequencies of normal modes is problematic. The corresponding displacement fields are not orthonormalizable and these modes cannot be quantized in the same way that FSM modes can [12]. Clarification of the significance of CFM calculations was obtained through an exhaustive calculation of all of the realvalued frequencies of an isotropic spherical nanoparticle embedded in a large but finite matrix with a free spherical outer surface of radius Rm [6]. This is called the core-shell model (CSM). Taking the limit Rm ! 1; the number of CSM modes in a given range of frequency becomes infinite, and has the Debye density of states. However, assuming uniform random phase excitation of the CSM modes, the mean squared displacement within the nanoparticle exhibits peaks whose positions agree closely with the real part of CFM modes, and whose HWHM agree closely with the imaginary parts of CFM modes. Thus, CFM calculations have a solid basis and utility, even if the corresponding CFM modes must be treated with caution. Both FSM and CFM have limitations of applicability to real systems: (1) nanoparticles are not truly spherical in shape, although the dynamics of most elaboration processes make them approximately that way and transmission electron microscopy (TEM) images confirm this; (2) for small nanoparticles, the wavelength of even the low-lying modes is not long compared to interatomic spacing, so that we are not close to the center of the Brillouin zone and the acoustic limit; (3) there will be deviations of the properties of the nanoparticle material from the same material in the bulk due to surface effects; (4) in a given sample, the radii of nanoparticles will have a distribution, rather than a single value; (5) the interface between the nanoparticle and the matrix
419
is not correctly described in terms of continuity of the displacement field and components of the stress tensor [13]; and (6) the speed of sound within the nanoparticle depends on the direction of propagation if the elasticity of the nanoparticle material is anisotropic. It is traditional and convenient to express mode frequencies in terms of either of the dimensionless variables Z ¼ oRp =vTp and x ¼ oRp =vLp ; where Rp is the radius of the nanoparticle. Table 1 refers to the lowest CFM (SPH, ‘ ¼ 0) mode for selected combinations of nanoparticle and matrix materials. The quality factor of the resonance, Q= 12 ReðxÞ=ImðxÞ; is presented in the upper half of each cell with Reðx) in the lower half. Longitudinal plane wave acoustic impedance (LPWAI) is Z0 ¼ rvL : Table 1 also indicates LPWAIs for selected nanoparticle and matrix materials. We have explored the lowest (SPH, ‘ ¼ 0) mode (ignoring highly damped ‘‘matrix modes’’ when they are present) in the parameter space of rp ; rm ; vLp ; vTp ; vLm and vTm and found some general patterns. Notably, good impedance matching between nanoparticle and matrix as exhibited in very strong lowest (SPH,0) mode damping is an unusual occurrence requiring special conditions. Closeness of the LPWAI’s alone is not sufficient. If nm np (where nm and np are the Poisson ratios of the matrix and particle, respectively) then there is never strong damping, but we can have strong damping if nm np such as for a liquid matrix. Assuming the Poisson ratios are close, it is usually necessary that rm Xrp and vLm pvLp : In addition, if nm and np are both close to 13 then strong damping is still possible if rm is as little as 0:8 rp : Conversely, light damping is always seen when any of the following conditions is satisfied: (1) np nm ; (2) rm 0:8 rp ; and (3) vLm 1:2 vLp : The ability to choose the matrix so as to have nanoparticles with lightly damped vibrational modes is important for several reasons. First, by reducing homogeneous broadening, it allows inhomogeneous broadening effects to stand out clearly. This results in a clearer picture for femtosecond pump–probe experiments where the
ARTICLE IN PRESS D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421
420
Table 1 Each cell corresponding to a combination of nanoparticle (listed along the top) and matrix (listed along the left side) material shows Q (above) and ReðxÞ (below) for the CFM (SPH, ‘ ¼ 0; n ¼ 0) mode for the indicated combination of materials (Densities are in g/cc. Longitudinal speeds of sound are given in km/s. n is the Poisson ratio.) np ! r vL Z0
Matrix
n Diamond
3.5
TiO2
4.1
Al2 O3
4.0
SiO2
2.2
GeO2
3.6
BaOP2 O5
3.8
Y2 O3
5.0
ZrO2
6.1
MgF2
3.2
Water
1.0
Ethanol
0.79
Hg
14
18 64 8.6 35 11 44 6.0 13 3.4 12 4.6 18 6.8 34 6.1 37 7.4 24
0.07
1.5 1.5 1.2 0.91 1.5 20
0.50
0.22 0.23 0.17 0.18 0.27 0.30 0.22 0.28
0.50 0.50
damping time is longer but also in low-frequency Raman and Brillouin inelastic scattering experiments where a better separation of scattering lines is possible. Solid-matrix-embedded nanoparticles always have Q factor lower than free ones. However, free nanoparticles suffer from their own practical drawbacks. For example, surface adsorption can affect their properties. Also it is often technologically easier to deal with matrixembedded nanoparticles. Where solid embedding is technologically desirable, our results provide guidance about how the matrix can be optimally chosen.
Au
Ag
Si
19 3.3 64 0.42
11 3.8 39 0.36
2.3 9.0 21 0.22
4.8 4.3 20 0.38
35 4.1 5.9 3.3 7.6 3.5 9.4 3.1 7.6 3.0 5.9 3.0 4.1 3.1 3.8 3.2 6.3 3.1
29 4.2 3.3 3.4 4.6 3.6 5.0 3.0 4.1 2.9 3.1 2.9 2.0 3.1 2.0 3.4 3.3 3.1
6.6 4.3 3.1 4.6 3.9 4.4 1.4 2.7 1.3 2.6 0.73 2.7 3.1 4.5 3.5 4.5 1.7 4.6
30 4.3 2.8 4.1 4.1 4.1 2.7 3.2 2.2 3.1 1.5 3.2 4.4 2.6 3.2 4.3 1.5 3.4
380 4.5 23 4.4 35 4.4 8.3 4.4 8.1 4.5 11 4.5 20 4.5 24 4.5 14 4.4
55 2.9 88 2.9 3.8 2.9
29 2.8 47 2.8 1.9 2.8
16 2.8 25 2.8 0.80 3.8
1.9 2.7 3.5 2.7 14 4.5
11 2.5 19 2.5 1.6 5.3
CdS
Na
0.97 3.3 3.2 0.34
References [1] H. Lamb, Proc. London Math. Soc. 13 (1881–1882) 189–212. [2] V. Novotny, P.P.M. Meincke, J.H.P. Watson, Phys. Rev. Lett. 28 (1972) 901–903. [3] E. Duval, A. Boukenter, B. Champagnon, Phys. Rev. Lett. 56 (1986) 2052–2055. [4] A. Tamura, K. Higeta, T. Ichinokawa, J. Phys. C 15 (1982) 4975–4991. [5] N.N. Ovsyuk, V.N. Novikov, Phys. Rev. B 53 (1996) 3113–3118. [6] D.B. Murray, L. Saviot, Phys. Rev. B 69 (2004) 094305. [7] V.A. Dubrovskiy, V.S. Morochnik, Izv., Acad. Sci. USSR, Phys. Solid Earth 17 (1981) 494–504.
ARTICLE IN PRESS D.B. Murray, L. Saviot / Physica E 26 (2005) 417–421 [8] N. Del Fatti, C. Voisin, F. Chevy, F. Valle´e, C. Flytzanis, J. Chem. Phys. 110 (1999) 11484–11487. [9] P. Verma, W. Cordts, G. Irmer, J. Monecke, Phys. Rev. B 60 (1999) 5778–5785. [10] E. Duval, Phys. Rev. B 46 (1992) 5795–5797.
421
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