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Raman scattering by small metallic particles in amorphous media
Volume 125, number 2,3
PHYSICS LETTERSA
2 November 1987
RAMAN SCATFERING BY SMALL METALLIC PARTICLES IN A M O R P H O U S MEDIA O.L. MALTA Departamento de Quimica Fundamental, Universidade Federal de Pernambuco. Cidade Universitaria, 50730 Reqfe, PE. Brazil
Received 16 June 1987;revised manuscript received 2 September 1987:accepted for publication 4 September 1987 Communicated by A.P. Fordy
We raise and examine the question of the possibility of observing inelastic scattering of light by small metallic particles embedded in amorphousmaterials; an observable effect is predicted.
The experimental observation of far-infrared absorption by small metallic particles embedded in amorphous materials has motivated several discussions on the mechanisms responsible for it [1-4]. Among these mechanisms, the thermal motion of these particles in a given medium has been considered [5,6]. It was shown that the variation, due to this thermal motion, of the dipole moment, produced by the charge separation between the particle surface and the nearest layer of surrounding ions in the medium, may give far-infrared absorption coefficients of the same order of magnitude as observed experimentally [5,6]. This result prompted us to study a different problem, namely, the possibility of observing inelastic scattering of light by these small particles due to their thermal motion. The aim of this letter is to examine this question. In connection with this problem, we point out that in the study of medium-range structure in amorphous solids some low frequency Raman vibrational modes have been attributed to the displacement of clusters of atoms [ 7 ]. Most of the theoretical apparatus needed here has already been developed in ref. [6]. The initial idea is to consider a small spherical particle of radius a and mass M in a spherical well ½Kr 2. The force constant K is given by the nature and the density of chemical bonds on the particle surface and the fundamental mode has a frequency given by O~o= (K/M)~/2. The use of the spherical oscillator to model this system is justified on the following grounds. Firstly, as shown by electron micrographs 80
these metallic particles are in general, to a good approximation, spherical. Secondly, they have shown to be chemically quite stable and preserve an identity which differs considerably from that of the host medium [ 8 ]. The well-known eigenstates of a spherical oscillator are specified by the quantum numbers n , / = 0 , 1, 2 .... and m = - - I . . . . . 0 . . . . . l and the eigenvalues E,i= ( q + 3 )~oo are given by the principal quantum number q = 2 n + / [6]. Thus, the total Raman cross section from an initial state q to a state ~/' is given by ~,l,1-
47cogto ' 3 C4 exp(-qDo)o/O)
× [ 1 - e x p ( -hoJo/O)] 3
×~' ~
I{n'l'm'lc~tnlm)l
~-.
(1)
t H ' t#l
In this equation ~o and o)' are the angular frequencies of the incident and scattered light, respectively, and 0 = k~T. The temperature dependent term in eq. (1) corresponds to the Boltzmann factor divided by the partition function of the spherical oscillator. The symbol ~ ' stands for summation over the values of n,/and n',l' for which 2 n + l = q a n d 2n' + l = q ' . c~ is the polarizability tensor of the system constituted by the metallic particle and the nearest layer of surrounding ions. Since we are dealing with an isotropic system, the polarizability tensor is diagonal and we may expand it as
Volume 125, number 2,3
PHYSICSLETTERSA
a-~ aeq + (V a)eq - r ,
(2)
where %q is the polarizability at the equilibrium position and r is the displacement of the metallic particle. It is convenient here to put the second term in the right-hand side of eq. (2) in its spherical tensor form. This is done by taking
( V o t ) e q ' r = r ~ ( - 1 ) q V , C~ ' ) ,
q=0,+l,
(3)
q
(4)
where fl is the polarizability of the chemical bonds between the particle surface and the nearest surrounding ions. The matrix element and the summtions to be evaluated in eq. (1) reduce to the same ones as given in ref. [6]. As expected, the selection rule is At/= ___1. The total Stokes Raman cross section ( r / ' = r / + l , co' = co -coo) is obtained by summing over all q while the anti-Stokes one (r/' = q - 1, (2)' =co+COo) is obtained by summing over q~ 0. For the former one the result is O'=48~ 2 0)0),3 ah[(Ofl/Or)eqps]2 C4 PM (2)0 × [ 1 - e x p ( --hcoo/O)] 3 × ~ G(r/) exp(-~lhcoo/0),
G ( ~ / ) = ½ [ q + 3 - ½ ( - 1 ) ~+t] x {~?+~q+3 + ½ ( q + 3 ) [ q - ½- ½ ( - 1)~+~]).
(6) The dependence of a with the particle radius, as predicted by eq. (5), is rather complicated due to the fact that (2)o depends on a:
COo= ( 3p, k/pMa) ,/2 ,
where Vq represents the spherical components of (V c~)~q and C(q~>= (4~)1/2 Ylq(~) is a Racah tensor operator of rank 1 [9]. The main point now concerns the evaluation of Vq. Firstly, due to the spherical symmetry of the system, the gradient of a should be the same in all di= rections and therefore Vq= V(q= 0, _ 1 ). Secondly, we assume that the contribution to o~ due to the isolated metallic particle is totally contained in a~q, i.e., it is unaffected by its vibrational motion. Thus, V a is given by the variation with r of the interfacial con= tribution to the polarizability. With these conditions and introducing a surface density of bonds [ 5,6 ], Ps, we may write V= ( Ofl/Or)e q Ps × 4na2 ,
2 November 1987
(5)
~=0
where PM is the metal specific mass (M=pM4~a 3) and G(q) is given by [6]
(7)
where k is a local force constant [5,6]. However, at very low temperatures eq. (5) reduces to a = 72n 2 (2)(2)' 3 ah [(Off~Or),qp,] 2
C4
(8)
PMCOo
and in this case, provided co >> coo, it may be noted that a varies as a 3 / 2 We wish to make a numerical estimation of tr. Typical values to be used are co=3.66×1015 s -1 (2=5145 A), coo=1.88×1013 s -1 (100 c m - ' ) , p s = 3 . 5 X 1 0 Is cm -2, (Ofl/Or)eq~lO -16 cm 2, a = 5 X 10 -7 cm and p u = 10.5 g/cm 3 (in the case of silver). Then, at room temperature we get from eq. (5) 0",,~ 10 -25 c m 2. This value is about four orders of magnitude higher than the Raman cross sections in benzene [ 10]. However, it should be kept in mind that the intensity of the Raman line is proportional to the concentration of scatterers and in the case of a uniform dispersion of metallic particles the concentration is usually smaller than in the case of molecular species by several orders of magnitude; typical values are 1015-10 t8 cm -3 and 102°-1023 c m - 3 respectively. In conclusion, we predict observable inelastic scattering of light by small metallic particles embedded in amorphous media. The predicted effect could be a tool to get information on particle shape and size. Another interesting point is that provided the frequency of the incident light is in resonance with the plasmon frequency of the particles, a considerable local field enhancement may occur [ 11 ]. As a consequence, at least for a sufficiently high particle concentration, the intensity of the Raman line could be considerably affected. The author is grateful to Professor R. Ferreira and 81
Volume 125, number 2,3
PHYSICS LETTERS A
Professor M.A.F. Gomes for helpful comments and t h e C N P q ( B r a z i l i a n A g e n c y ) for f i n a n c i a l s u p p o r t .
References [ 1 ] D.B. Tanner, A.J. Sievers and R.A. Buhrman, Phys. Rev. B 11 (1975) 1330. [2] C.G. Granqvist, R.A. Buhrman, J. Wyns and A.J. Sievers, Phys, Rev. Lett. 37 (1976) 625. [3] R.P. Devaty and A.J. Sievers, Phys. Rev. Lett. 52 (1984~ 1344. [4] S.-I. Lee, T. Won Noh, K. Cummings and J.R. Gaines, Phys. Rev. Lett. 55 (1985) 1626.
82
2 November 1987
[5] E. Sim~inek, Solid State Commun. 37 (1981) 97. [6] O.L. Malta and G.T. de Sagey, Chem. Phys. Lett. 134 ( 1987 ) 485. [7] J.C. Phillips. J. Non-Cryst. Solids 43 (1981) 37. [8] U. Kl-eibig and C. von Fragtein, Z. Phys. 224 (1969) 307, and references therein. [9] A.R. Edmonds, Angular momentum in quantum mechanics ( Princeton Univ. Press, Princeton, 1957 ). [ 10] T.C. Damen, R.C.C. Leite and S.P.S. Porto, Phys. Rev. Lett. 14 (1965) 9. [ 11 ] R.K. Chan and T.E. Furtak, Surface enhanced Raman scaltering ( Plenum, New York, 1982).
PHYSICS LETTERSA
2 November 1987
RAMAN SCATFERING BY SMALL METALLIC PARTICLES IN A M O R P H O U S MEDIA O.L. MALTA Departamento de Quimica Fundamental, Universidade Federal de Pernambuco. Cidade Universitaria, 50730 Reqfe, PE. Brazil
Received 16 June 1987;revised manuscript received 2 September 1987:accepted for publication 4 September 1987 Communicated by A.P. Fordy
We raise and examine the question of the possibility of observing inelastic scattering of light by small metallic particles embedded in amorphousmaterials; an observable effect is predicted.
The experimental observation of far-infrared absorption by small metallic particles embedded in amorphous materials has motivated several discussions on the mechanisms responsible for it [1-4]. Among these mechanisms, the thermal motion of these particles in a given medium has been considered [5,6]. It was shown that the variation, due to this thermal motion, of the dipole moment, produced by the charge separation between the particle surface and the nearest layer of surrounding ions in the medium, may give far-infrared absorption coefficients of the same order of magnitude as observed experimentally [5,6]. This result prompted us to study a different problem, namely, the possibility of observing inelastic scattering of light by these small particles due to their thermal motion. The aim of this letter is to examine this question. In connection with this problem, we point out that in the study of medium-range structure in amorphous solids some low frequency Raman vibrational modes have been attributed to the displacement of clusters of atoms [ 7 ]. Most of the theoretical apparatus needed here has already been developed in ref. [6]. The initial idea is to consider a small spherical particle of radius a and mass M in a spherical well ½Kr 2. The force constant K is given by the nature and the density of chemical bonds on the particle surface and the fundamental mode has a frequency given by O~o= (K/M)~/2. The use of the spherical oscillator to model this system is justified on the following grounds. Firstly, as shown by electron micrographs 80
these metallic particles are in general, to a good approximation, spherical. Secondly, they have shown to be chemically quite stable and preserve an identity which differs considerably from that of the host medium [ 8 ]. The well-known eigenstates of a spherical oscillator are specified by the quantum numbers n , / = 0 , 1, 2 .... and m = - - I . . . . . 0 . . . . . l and the eigenvalues E,i= ( q + 3 )~oo are given by the principal quantum number q = 2 n + / [6]. Thus, the total Raman cross section from an initial state q to a state ~/' is given by ~,l,1-
47cogto ' 3 C4 exp(-qDo)o/O)
× [ 1 - e x p ( -hoJo/O)] 3
×~' ~
I{n'l'm'lc~tnlm)l
~-.
(1)
t H ' t#l
In this equation ~o and o)' are the angular frequencies of the incident and scattered light, respectively, and 0 = k~T. The temperature dependent term in eq. (1) corresponds to the Boltzmann factor divided by the partition function of the spherical oscillator. The symbol ~ ' stands for summation over the values of n,/and n',l' for which 2 n + l = q a n d 2n' + l = q ' . c~ is the polarizability tensor of the system constituted by the metallic particle and the nearest layer of surrounding ions. Since we are dealing with an isotropic system, the polarizability tensor is diagonal and we may expand it as
Volume 125, number 2,3
PHYSICSLETTERSA
a-~ aeq + (V a)eq - r ,
(2)
where %q is the polarizability at the equilibrium position and r is the displacement of the metallic particle. It is convenient here to put the second term in the right-hand side of eq. (2) in its spherical tensor form. This is done by taking
( V o t ) e q ' r = r ~ ( - 1 ) q V , C~ ' ) ,
q=0,+l,
(3)
q
(4)
where fl is the polarizability of the chemical bonds between the particle surface and the nearest surrounding ions. The matrix element and the summtions to be evaluated in eq. (1) reduce to the same ones as given in ref. [6]. As expected, the selection rule is At/= ___1. The total Stokes Raman cross section ( r / ' = r / + l , co' = co -coo) is obtained by summing over all q while the anti-Stokes one (r/' = q - 1, (2)' =co+COo) is obtained by summing over q~ 0. For the former one the result is O'=48~ 2 0)0),3 ah[(Ofl/Or)eqps]2 C4 PM (2)0 × [ 1 - e x p ( --hcoo/O)] 3 × ~ G(r/) exp(-~lhcoo/0),
G ( ~ / ) = ½ [ q + 3 - ½ ( - 1 ) ~+t] x {~?+~q+3 + ½ ( q + 3 ) [ q - ½- ½ ( - 1)~+~]).
(6) The dependence of a with the particle radius, as predicted by eq. (5), is rather complicated due to the fact that (2)o depends on a:
COo= ( 3p, k/pMa) ,/2 ,
where Vq represents the spherical components of (V c~)~q and C(q~>= (4~)1/2 Ylq(~) is a Racah tensor operator of rank 1 [9]. The main point now concerns the evaluation of Vq. Firstly, due to the spherical symmetry of the system, the gradient of a should be the same in all di= rections and therefore Vq= V(q= 0, _ 1 ). Secondly, we assume that the contribution to o~ due to the isolated metallic particle is totally contained in a~q, i.e., it is unaffected by its vibrational motion. Thus, V a is given by the variation with r of the interfacial con= tribution to the polarizability. With these conditions and introducing a surface density of bonds [ 5,6 ], Ps, we may write V= ( Ofl/Or)e q Ps × 4na2 ,
2 November 1987
(5)
~=0
where PM is the metal specific mass (M=pM4~a 3) and G(q) is given by [6]
(7)
where k is a local force constant [5,6]. However, at very low temperatures eq. (5) reduces to a = 72n 2 (2)(2)' 3 ah [(Off~Or),qp,] 2
C4
(8)
PMCOo
and in this case, provided co >> coo, it may be noted that a varies as a 3 / 2 We wish to make a numerical estimation of tr. Typical values to be used are co=3.66×1015 s -1 (2=5145 A), coo=1.88×1013 s -1 (100 c m - ' ) , p s = 3 . 5 X 1 0 Is cm -2, (Ofl/Or)eq~lO -16 cm 2, a = 5 X 10 -7 cm and p u = 10.5 g/cm 3 (in the case of silver). Then, at room temperature we get from eq. (5) 0",,~ 10 -25 c m 2. This value is about four orders of magnitude higher than the Raman cross sections in benzene [ 10]. However, it should be kept in mind that the intensity of the Raman line is proportional to the concentration of scatterers and in the case of a uniform dispersion of metallic particles the concentration is usually smaller than in the case of molecular species by several orders of magnitude; typical values are 1015-10 t8 cm -3 and 102°-1023 c m - 3 respectively. In conclusion, we predict observable inelastic scattering of light by small metallic particles embedded in amorphous media. The predicted effect could be a tool to get information on particle shape and size. Another interesting point is that provided the frequency of the incident light is in resonance with the plasmon frequency of the particles, a considerable local field enhancement may occur [ 11 ]. As a consequence, at least for a sufficiently high particle concentration, the intensity of the Raman line could be considerably affected. The author is grateful to Professor R. Ferreira and 81
Volume 125, number 2,3
PHYSICS LETTERS A
Professor M.A.F. Gomes for helpful comments and t h e C N P q ( B r a z i l i a n A g e n c y ) for f i n a n c i a l s u p p o r t .
References [ 1 ] D.B. Tanner, A.J. Sievers and R.A. Buhrman, Phys. Rev. B 11 (1975) 1330. [2] C.G. Granqvist, R.A. Buhrman, J. Wyns and A.J. Sievers, Phys, Rev. Lett. 37 (1976) 625. [3] R.P. Devaty and A.J. Sievers, Phys. Rev. Lett. 52 (1984~ 1344. [4] S.-I. Lee, T. Won Noh, K. Cummings and J.R. Gaines, Phys. Rev. Lett. 55 (1985) 1626.
82
2 November 1987
[5] E. Sim~inek, Solid State Commun. 37 (1981) 97. [6] O.L. Malta and G.T. de Sagey, Chem. Phys. Lett. 134 ( 1987 ) 485. [7] J.C. Phillips. J. Non-Cryst. Solids 43 (1981) 37. [8] U. Kl-eibig and C. von Fragtein, Z. Phys. 224 (1969) 307, and references therein. [9] A.R. Edmonds, Angular momentum in quantum mechanics ( Princeton Univ. Press, Princeton, 1957 ). [ 10] T.C. Damen, R.C.C. Leite and S.P.S. Porto, Phys. Rev. Lett. 14 (1965) 9. [ 11 ] R.K. Chan and T.E. Furtak, Surface enhanced Raman scaltering ( Plenum, New York, 1982).