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The calculation of the Raman depolarization ratios for the breathing vibrant modes of pyridine adsorbed on silver sol particles
L470
Surface Science 171 (1986) L470-L478 North-Holland, A m s t e r d a m
S U R F A C E SCIENCE LETTERS T H E C A L C U L A T I O N OF T H E RAMAN D E P O L A R I Z A T I O N R A T I O S F O R THE BREATHING VIBRANT M O D E S OF PYRIDINE A D S O R B E D O N SILVER S O L P A R T I C L E S Ping J I A N G , Chunping Z H A N G and Guangyin Z H A N G Department of Physics, Nankai University, Tianjin, People's Rep. of China Received 18 July 1985; accepted for publication 31 October 1985
In this paper, under the approximate condition of a static electrical field, the R a m a n depolarization ratios for the breathing vibrant modes of pyridine adsorbed on silver sol particles are calculated. Bispherical coordinates are used. The results indicate that the distortion of the incident field which is caused by the colloid aggregation contributes to an increase in the R a m a n depolarization ratio. It is also indicated by calculation that although the variation of the distance between two spheres has a strong effect on the depolarization ratio, the orientation of the adsorbed pyridine has a minimal effect when the chemical adsorption is not considered. The calculated results are in good agreement with the experimental results.
In the pyridine normal Raman spectra, the breathing vibrant modes (992 and 1028 cm -1) exhibit depolarization ratios near zero (p < 0.05); whereas in the experiment of surface enhanced R a m a n scattering (SERS), the depolarization ratios of the pyridine molecule adsorbed on metal surface are P > 0.5. The reasons for this increase are not yet clear. In ref. [1] it is considered that the magnitude of depolarization ratio is related to the orientation of pyridine adsorbed on a metal surface; however, Creighton inferred that the increase in the depolarization ratio is due to the depolarizing effect of the anisotropy of the colloid aggregates [2]. We can see there is a contradiction between these two papers. The conclusion of ref. [2] is that pyridine is adsorbed face-on to the metal surface, it is considered in ref. [1], however, that pyridine in a face-on adsorption configuration will have a depolarization ratio of zero for a totally symmetric mode. We must state that all these opinions are based on conjecture, they are not explained clearly by theory. Using classical theory, and simplifying each sol particle to be a small isolated metal sphere (radius R << wavelength )~), Creighton has calculated the depolarization ratio of the pyridine molecules adsorbed on the colloid particles [2]. The results of his calculation show O = 0.125 (corresponding to an isotropic breathing vibration of all = a22 = 0~33 ). This is much smaller than the experimental value of O > 0.5. It can be seen by comparing the transmission electron 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
P. Jiang et a L / Pyridine on silver sol particles
L471
micrographs of silver sol [3] before and after adding pyridine, that the colloids with pyridine aggregate into strings of particles rather than a single sphere. In this paper, considering two spheres each with a radius of R (R << ?~), we utilize bispherical coordinates. According to Aravind's method [4], we inspect the effect of the colloid aggregates on the depolarization ratio, and examine whether or not the orientation of molecules adsorbed on the surface of silver sol particles affect the Raman depolarization ratio. From the pyridine SER spectra we can infer that the symmetry of adsorbed molecules is not in any obvious way distorted, therefore chemisorption is not considered. The free pyridine Raman tensor is used in our calculations. The bispherical coordinate system (~, 7, ~) is given in ref. [5]. In ref. [4] the solution of the Laplace equation for bispherical systems is given. The components of the electric field of the bispherical system E,, En, E~ are given in the appendix. The pyridine molecule is considered to be located at the spherical surface point M, as shown in fig. 1. ~(us) is the complex dielectric constant of the metal at the scattered light frequency uS. c m is the dielectric constant of the surrounding medium. The molecular dipole p = a v E , where a T is the Raman tensor of the free pyridine molecule and E is the local field of the bispherical system at point M. The dipole moment of the system sphere 2 plus a molecule is p' =A(us) p [6], where
A(vs) = g ( v s )
[1
]
-1 2
is the enhancement factor of a single sphere system [2]. The intensity of the Raman scattering field is given by Es=kA(vs)axE, where k = k ' v 2, and k ' = const. In order to make the calculated results comparable with the experimental results, the coordinate components for the scattered field must be transformed,
E=
E~ = T2T1 En ,
Er
(1)
IE J
where T1 and T2 (see appendix) are the transformation matrix from bispherical coordinates (kt, r/, ¢p) to Cartesian coordinates (x, y, z) and from Cartesian coordinates to spherical coordinates (0, ~, r), respectively. It should be noted that on the surface of sphere 2/~ =/~0As shown in fig. 1, the orientation of the adsorbed molecules on the surface of sphere 2 is such that the angle between the molecular C2-axis and the
L472
P. Jiang et al. / Pyridine on silver sol particles
/~*
z
~,
--- -:~ . . . . . . . . . . . . . .
7
S\/,, . ÷.,,,.,,'e' i
1o
,y
Z
'/ 0
,y
Fig. 1. Bispherical system. The pyridine molecule is adsorbed at point M on the surface of sphere 2. MP stands for the molecular plane.
P. Jiang et al. / Pyridine on silver sol particles
L473
n o r m a l of the surface is ~, and the angle between the molecular plane and the direction of e 0 (see fig. 1) is ~. It is known that the R a m a n tensor c o m p o n e n t s of a I pyridine modes, in the coordinates by which the molecular C2-axis is defined as z-axis are a~x, c%,,,, ' , O~33 ' here. In spherical coordinates (0, % r) the azz, are n a m e d 0/11' , O~22 R a m a n tensor is n a m e d a v.
a T=T
i
ct~2
T,
a~3
T= /c°s~sin [
-sin ~
~b
sin sin c°s
cos~
sin ~ s i n q~
0
cos (2)
iP is the transposed matrix of T. In Cartesian coordinates (x, y, z) the intensity of scattered fields is given by
ei ,
G.
Esl
Er
(3)
The coordinates x, y, z are fixed on the bispherical system, with the z-axis chosen along the line passing through the centers of the two spheres, shown in fig. 2. The electric vector of the external field E 0 is directed along the polar angle 00 and the azimuthal angle ~0. The observation direction is perpendicular to the direction of p r o p a g a t i o n of the incident laser beam. The electric vector of the incident b e a m is perpendicular to the scattered plane. The depolarization ratio is given by P± (~r/2) = I E,, [2/1 E± I 2 = I,,/It_ ;
(4)
E~x, Esy, Esz, which are solved in eq. (3), are projected on the direction perpendicular to and parallel to the scattered plane; obtained, respectively:
E± and E,, can be
I,, = [ E , I 2 = sin%0E}x + cosaq~0E2y - 2 sin q~o cos eOoEsxEs.~,,
(5a)
1± = I E± I 2 = (sin 0o cos q~0Es~) 2 + 2 sin200 sin ~0 cos eOoE~Es~ + ( s i n 0o sin ~0Esy) 2 + 2 sin 00 cos 00 cos q,oE~xEs.: + 2 sin 0o cos 0o sin q~oE~yEsz + (cos 0oEsz) 2.
(5b)
The following hypotheses are m a d e in our calculations: (1) The pyridine molecules are adsorbed on the surface of the silver particles in a single layer and homogeneously.
P. Jiang et al. / Pyridine on silver sol particles
L474
Z
~Y
"oB X
O~
Fig. 2. Geometry of Raman scattering. 01 and 0 2 are the centers of sphere 1 and 2. The direction of observation (marked by OB in the figure) is perpendicular to the direction of propagation of the incident laser beam (marked by i in the figure).
(2) The angle + can be taken as an arbitrary value from 0 to 27r, i.e. the molecular plane on the surface has a r a n d o m orientation. (3) 0o and '/'o can be taken as arbitrary values from 0 to ~r and 0 to 2Tr, respectively, i.e. the orientation of the line passing through the centers of the two spheres with respect to the external field E o is random. Based on these hypotheses we can average I,~ and I ± by integrating them over 0, ~0, ~b, 0o, 4~o. f 2 ~ f~r r21r r ~r r27r
L JoL JoJo ' 'n0sin0°d 0d d0°d °) I ± sin 0 sin 0o d~b dO dq0 d0 o \"0
t/0 1/0
dq~o/R/(47r) 3 ]
JO */0
(6) The results of our calculations are shown in fig. 3.
P. Jiang et al, / Pyridine on silver sol particles
L475
~J 0.5
0.5
0.5
(c)
0.1
Fig. 3. T h e c a l c u l a t e d results of p versus D / R for t h r e e d i f f e r e n t i n c i d e n t laser f r e q u e n c i e s , a f t = a~2 = a~3. (a) ~ = 1.9 eV; (b) ~ = 2.4 eV; (c) ~0 = 2.95 eV.
In aqueous sol, the dielectric constant of water is taken to be c,~ = 1.78. The complex dielectric constant of silver is taken from ref. [7]. In order to inspect the effect of the colloid aggregates on the value of p, the variation of p with the ratio D/R (for the definition of D and R see fig. 1) is calculated. In figs. 3a and 3b we plot p as a function of D/R for different wavelengths of the incident field, a ~ = a~2 -- a~3 is assumed in the calculation. It can be seen from fig. 3, when the ratio D/R is big enough, the depolarization ratio P reaches the limit 0.14, this is approximately the same value as a single sphere, 0.125. When the ratio D/R reaches the limit 0.5, then p reaches its maximum value p >1 0.5. When the ratio D/R <0.5, then p tends to decrease. It can be seen from the transmission electron rnicrographs of silver sol in ref. [3] that the distance between spheres is not constant and has a definite
P. Jiang et al. / Pyridine on sih~er sol particles
L476
f 0.8 0.6 ii.
o.. 0.2 .............................................................
5
0
10
15 =
OAR
Fig. 4. p versus D/R for two different molecular orientations: = 90°; 2a~1= ~ 2 2 = ¢~_33 is assumed.
) ~=o°:
(------)
distribution. Therefore we estimate that, for the bispherical system, the existence of metal spheres leads to a depolarization ratio increasing to ~ 0.4. This value is much closer to the experimental results than the single spherical value. This result proves that Creighton's assumption [2] was correct. We can also calculate the change of p as a function of the ratio D / R for two extreme molecular orientations, ~ = 0, ~r/2, i.e. when the molecular C2-axis is parallel to the normal of the spherical surface and perpendicular to the normal. The calculated results are shown in fig. 4. The curves in fig. 4 seem to imply that for the bispherical system, the molecular orientation has no obvious effect on the R a m a n depolarization ratio. The following conclusions are indicated from our calculated results: (a) The colloid aggregates cause the depolarization ratio to increase. (b) The molecular orientation has no obvious effect on the depolarization ratio when the chemisorption is not considered (the R a m a n tensor used in the calculation is that of free pyridine molecules). (c) For different excitation frequencies, p remains basically constant.
Appendix In ref. [4], the potential outside the spheres 1 and 2, • is given. We can calculate the electric field outside the spheres from 1 ~0 Ei
h i 8i '
i = l ~ , 7, ¢P.
L477
P. Jiang et al. / Pyridine on silver sol particles
Therefore i
F
~"
+.=1
AO
sinh
+
cosh
[sinh_~o s i n h ( ( n + ½ l g o )
A°LF1/2
+2(. + ½)r '/2 cosh((. + ~)~0) Y°(cos */, ~) +2A'.
(2n+l)l/2[sinh#o 4¢rn(n + 1)
F 1/2
+2(n+½)F 1/2 sinh((n En I
~=~o
= E~
I ~=~o
F [
+ ½)/%)] P" c o s ( ~ - q~o)]},
0 sin 71 sinh
a 1 A° F1/-----~
/sin*/
A°~ -~-/yY°+
+ n=l ~
OY° \ 2F'~ ~ I
+ 2 A',, ~/sin*/ ~--~ P,~, +
0./ ] sinh((,, + -~),0)
2 F , / 2 ~ _ ~ )cos(q ~ )(
X c°sh((n + ½)g° _
i
Ewl~'=t'°-Ewl"=t'°
cosh((n+½)b%)
epo)
2n+1 )1/2]} 4','rn (n + 1) '
+ 4F3/2 ~ ( 2 n + 1 ) , / 2 as,Vn*/,,=a
4¢rn(n+l)
A;c°sh((n+½)#°)
× P " sin(~ - q~0), where F = cosh #o - cos */. The transformation matrices T 1 and T2 are given by sinh ~ sin */cos ¢# cosh ~ - cos 7/ sinh ~ sin */sincp cosh/.t - cos ~ 1 - cosh g cos 7/ cosh g - cos */
=
T:
=
cosScos9~ ¢p sin 0 cos cp
-sin
(cosh/~ cos * / - 1) cos q~ cosh g - cos */ (cosh g cos * / - 1) cos q0 cosh g - cos */ sinh g sin 7/ - cosh # - cos */
cosO sinrp cos cp sin O sin cp
-sin8 0 ]. cos O
-
sin
cp
COS qO ,
0
L478
P. Jiang et al. / l~vridine on sih:er sol particles
References [1] D.L. Jeanmaire and R.P. Van Duyne, J. Electroanal. Chem. 84 (1977) 1. [2] J.A. Creighton, Surface Sci. 124 (1983) 209. [3] T.E. Furtak and R.K. Chang, Eds., Surface Enhanced Raman Scattering (Plenum, New York, 1982) p. 318. [4] P.K. Aravind, Surface Sci. 110 (1981) 189. [5] R. Ruppin, Phys. Rev. B26 (1982) 3440. [61 Ref. [3], p. 99. [7] P.B. Johason and R.W. Christy, Phys. Rev. B6 (1972) 4370.
Surface Science 171 (1986) L470-L478 North-Holland, A m s t e r d a m
S U R F A C E SCIENCE LETTERS T H E C A L C U L A T I O N OF T H E RAMAN D E P O L A R I Z A T I O N R A T I O S F O R THE BREATHING VIBRANT M O D E S OF PYRIDINE A D S O R B E D O N SILVER S O L P A R T I C L E S Ping J I A N G , Chunping Z H A N G and Guangyin Z H A N G Department of Physics, Nankai University, Tianjin, People's Rep. of China Received 18 July 1985; accepted for publication 31 October 1985
In this paper, under the approximate condition of a static electrical field, the R a m a n depolarization ratios for the breathing vibrant modes of pyridine adsorbed on silver sol particles are calculated. Bispherical coordinates are used. The results indicate that the distortion of the incident field which is caused by the colloid aggregation contributes to an increase in the R a m a n depolarization ratio. It is also indicated by calculation that although the variation of the distance between two spheres has a strong effect on the depolarization ratio, the orientation of the adsorbed pyridine has a minimal effect when the chemical adsorption is not considered. The calculated results are in good agreement with the experimental results.
In the pyridine normal Raman spectra, the breathing vibrant modes (992 and 1028 cm -1) exhibit depolarization ratios near zero (p < 0.05); whereas in the experiment of surface enhanced R a m a n scattering (SERS), the depolarization ratios of the pyridine molecule adsorbed on metal surface are P > 0.5. The reasons for this increase are not yet clear. In ref. [1] it is considered that the magnitude of depolarization ratio is related to the orientation of pyridine adsorbed on a metal surface; however, Creighton inferred that the increase in the depolarization ratio is due to the depolarizing effect of the anisotropy of the colloid aggregates [2]. We can see there is a contradiction between these two papers. The conclusion of ref. [2] is that pyridine is adsorbed face-on to the metal surface, it is considered in ref. [1], however, that pyridine in a face-on adsorption configuration will have a depolarization ratio of zero for a totally symmetric mode. We must state that all these opinions are based on conjecture, they are not explained clearly by theory. Using classical theory, and simplifying each sol particle to be a small isolated metal sphere (radius R << wavelength )~), Creighton has calculated the depolarization ratio of the pyridine molecules adsorbed on the colloid particles [2]. The results of his calculation show O = 0.125 (corresponding to an isotropic breathing vibration of all = a22 = 0~33 ). This is much smaller than the experimental value of O > 0.5. It can be seen by comparing the transmission electron 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
P. Jiang et a L / Pyridine on silver sol particles
L471
micrographs of silver sol [3] before and after adding pyridine, that the colloids with pyridine aggregate into strings of particles rather than a single sphere. In this paper, considering two spheres each with a radius of R (R << ?~), we utilize bispherical coordinates. According to Aravind's method [4], we inspect the effect of the colloid aggregates on the depolarization ratio, and examine whether or not the orientation of molecules adsorbed on the surface of silver sol particles affect the Raman depolarization ratio. From the pyridine SER spectra we can infer that the symmetry of adsorbed molecules is not in any obvious way distorted, therefore chemisorption is not considered. The free pyridine Raman tensor is used in our calculations. The bispherical coordinate system (~, 7, ~) is given in ref. [5]. In ref. [4] the solution of the Laplace equation for bispherical systems is given. The components of the electric field of the bispherical system E,, En, E~ are given in the appendix. The pyridine molecule is considered to be located at the spherical surface point M, as shown in fig. 1. ~(us) is the complex dielectric constant of the metal at the scattered light frequency uS. c m is the dielectric constant of the surrounding medium. The molecular dipole p = a v E , where a T is the Raman tensor of the free pyridine molecule and E is the local field of the bispherical system at point M. The dipole moment of the system sphere 2 plus a molecule is p' =A(us) p [6], where
A(vs) = g ( v s )
[1
]
-1 2
is the enhancement factor of a single sphere system [2]. The intensity of the Raman scattering field is given by Es=kA(vs)axE, where k = k ' v 2, and k ' = const. In order to make the calculated results comparable with the experimental results, the coordinate components for the scattered field must be transformed,
E=
E~ = T2T1 En ,
Er
(1)
IE J
where T1 and T2 (see appendix) are the transformation matrix from bispherical coordinates (kt, r/, ¢p) to Cartesian coordinates (x, y, z) and from Cartesian coordinates to spherical coordinates (0, ~, r), respectively. It should be noted that on the surface of sphere 2/~ =/~0As shown in fig. 1, the orientation of the adsorbed molecules on the surface of sphere 2 is such that the angle between the molecular C2-axis and the
L472
P. Jiang et al. / Pyridine on silver sol particles
/~*
z
~,
--- -:~ . . . . . . . . . . . . . .
7
S\/,, . ÷.,,,.,,'e' i
1o
,y
Z
'/ 0
,y
Fig. 1. Bispherical system. The pyridine molecule is adsorbed at point M on the surface of sphere 2. MP stands for the molecular plane.
P. Jiang et al. / Pyridine on silver sol particles
L473
n o r m a l of the surface is ~, and the angle between the molecular plane and the direction of e 0 (see fig. 1) is ~. It is known that the R a m a n tensor c o m p o n e n t s of a I pyridine modes, in the coordinates by which the molecular C2-axis is defined as z-axis are a~x, c%,,,, ' , O~33 ' here. In spherical coordinates (0, % r) the azz, are n a m e d 0/11' , O~22 R a m a n tensor is n a m e d a v.
a T=T
i
ct~2
T,
a~3
T= /c°s~sin [
-sin ~
~b
sin sin c°s
cos~
sin ~ s i n q~
0
cos (2)
iP is the transposed matrix of T. In Cartesian coordinates (x, y, z) the intensity of scattered fields is given by
ei ,
G.
Esl
Er
(3)
The coordinates x, y, z are fixed on the bispherical system, with the z-axis chosen along the line passing through the centers of the two spheres, shown in fig. 2. The electric vector of the external field E 0 is directed along the polar angle 00 and the azimuthal angle ~0. The observation direction is perpendicular to the direction of p r o p a g a t i o n of the incident laser beam. The electric vector of the incident b e a m is perpendicular to the scattered plane. The depolarization ratio is given by P± (~r/2) = I E,, [2/1 E± I 2 = I,,/It_ ;
(4)
E~x, Esy, Esz, which are solved in eq. (3), are projected on the direction perpendicular to and parallel to the scattered plane; obtained, respectively:
E± and E,, can be
I,, = [ E , I 2 = sin%0E}x + cosaq~0E2y - 2 sin q~o cos eOoEsxEs.~,,
(5a)
1± = I E± I 2 = (sin 0o cos q~0Es~) 2 + 2 sin200 sin ~0 cos eOoE~Es~ + ( s i n 0o sin ~0Esy) 2 + 2 sin 00 cos 00 cos q,oE~xEs.: + 2 sin 0o cos 0o sin q~oE~yEsz + (cos 0oEsz) 2.
(5b)
The following hypotheses are m a d e in our calculations: (1) The pyridine molecules are adsorbed on the surface of the silver particles in a single layer and homogeneously.
P. Jiang et al. / Pyridine on silver sol particles
L474
Z
~Y
"oB X
O~
Fig. 2. Geometry of Raman scattering. 01 and 0 2 are the centers of sphere 1 and 2. The direction of observation (marked by OB in the figure) is perpendicular to the direction of propagation of the incident laser beam (marked by i in the figure).
(2) The angle + can be taken as an arbitrary value from 0 to 27r, i.e. the molecular plane on the surface has a r a n d o m orientation. (3) 0o and '/'o can be taken as arbitrary values from 0 to ~r and 0 to 2Tr, respectively, i.e. the orientation of the line passing through the centers of the two spheres with respect to the external field E o is random. Based on these hypotheses we can average I,~ and I ± by integrating them over 0, ~0, ~b, 0o, 4~o. f 2 ~ f~r r21r r ~r r27r
L JoL JoJo ' 'n0sin0°d 0d d0°d °) I ± sin 0 sin 0o d~b dO dq0 d0 o \"0
t/0 1/0
dq~o/R/(47r) 3 ]
JO */0
(6) The results of our calculations are shown in fig. 3.
P. Jiang et al, / Pyridine on silver sol particles
L475
~J 0.5
0.5
0.5
(c)
0.1
Fig. 3. T h e c a l c u l a t e d results of p versus D / R for t h r e e d i f f e r e n t i n c i d e n t laser f r e q u e n c i e s , a f t = a~2 = a~3. (a) ~ = 1.9 eV; (b) ~ = 2.4 eV; (c) ~0 = 2.95 eV.
In aqueous sol, the dielectric constant of water is taken to be c,~ = 1.78. The complex dielectric constant of silver is taken from ref. [7]. In order to inspect the effect of the colloid aggregates on the value of p, the variation of p with the ratio D/R (for the definition of D and R see fig. 1) is calculated. In figs. 3a and 3b we plot p as a function of D/R for different wavelengths of the incident field, a ~ = a~2 -- a~3 is assumed in the calculation. It can be seen from fig. 3, when the ratio D/R is big enough, the depolarization ratio P reaches the limit 0.14, this is approximately the same value as a single sphere, 0.125. When the ratio D/R reaches the limit 0.5, then p reaches its maximum value p >1 0.5. When the ratio D/R <0.5, then p tends to decrease. It can be seen from the transmission electron rnicrographs of silver sol in ref. [3] that the distance between spheres is not constant and has a definite
P. Jiang et al. / Pyridine on sih~er sol particles
L476
f 0.8 0.6 ii.
o.. 0.2 .............................................................
5
0
10
15 =
OAR
Fig. 4. p versus D/R for two different molecular orientations: = 90°; 2a~1= ~ 2 2 = ¢~_33 is assumed.
) ~=o°:
(------)
distribution. Therefore we estimate that, for the bispherical system, the existence of metal spheres leads to a depolarization ratio increasing to ~ 0.4. This value is much closer to the experimental results than the single spherical value. This result proves that Creighton's assumption [2] was correct. We can also calculate the change of p as a function of the ratio D / R for two extreme molecular orientations, ~ = 0, ~r/2, i.e. when the molecular C2-axis is parallel to the normal of the spherical surface and perpendicular to the normal. The calculated results are shown in fig. 4. The curves in fig. 4 seem to imply that for the bispherical system, the molecular orientation has no obvious effect on the R a m a n depolarization ratio. The following conclusions are indicated from our calculated results: (a) The colloid aggregates cause the depolarization ratio to increase. (b) The molecular orientation has no obvious effect on the depolarization ratio when the chemisorption is not considered (the R a m a n tensor used in the calculation is that of free pyridine molecules). (c) For different excitation frequencies, p remains basically constant.
Appendix In ref. [4], the potential outside the spheres 1 and 2, • is given. We can calculate the electric field outside the spheres from 1 ~0 Ei
h i 8i '
i = l ~ , 7, ¢P.
L477
P. Jiang et al. / Pyridine on silver sol particles
Therefore i
F
~"
+.=1
AO
sinh
+
cosh
[sinh_~o s i n h ( ( n + ½ l g o )
A°LF1/2
+2(. + ½)r '/2 cosh((. + ~)~0) Y°(cos */, ~) +2A'.
(2n+l)l/2[sinh#o 4¢rn(n + 1)
F 1/2
+2(n+½)F 1/2 sinh((n En I
~=~o
= E~
I ~=~o
F [
+ ½)/%)] P" c o s ( ~ - q~o)]},
0 sin 71 sinh
a 1 A° F1/-----~
/sin*/
A°~ -~-/yY°+
+ n=l ~
OY° \ 2F'~ ~ I
+ 2 A',, ~/sin*/ ~--~ P,~, +
0./ ] sinh((,, + -~),0)
2 F , / 2 ~ _ ~ )cos(q ~ )(
X c°sh((n + ½)g° _
i
Ewl~'=t'°-Ewl"=t'°
cosh((n+½)b%)
epo)
2n+1 )1/2]} 4','rn (n + 1) '
+ 4F3/2 ~ ( 2 n + 1 ) , / 2 as,Vn*/,,=a
4¢rn(n+l)
A;c°sh((n+½)#°)
× P " sin(~ - q~0), where F = cosh #o - cos */. The transformation matrices T 1 and T2 are given by sinh ~ sin */cos ¢# cosh ~ - cos 7/ sinh ~ sin */sincp cosh/.t - cos ~ 1 - cosh g cos 7/ cosh g - cos */
=
T:
=
cosScos9~ ¢p sin 0 cos cp
-sin
(cosh/~ cos * / - 1) cos q~ cosh g - cos */ (cosh g cos * / - 1) cos q0 cosh g - cos */ sinh g sin 7/ - cosh # - cos */
cosO sinrp cos cp sin O sin cp
-sin8 0 ]. cos O
-
sin
cp
COS qO ,
0
L478
P. Jiang et al. / l~vridine on sih:er sol particles
References [1] D.L. Jeanmaire and R.P. Van Duyne, J. Electroanal. Chem. 84 (1977) 1. [2] J.A. Creighton, Surface Sci. 124 (1983) 209. [3] T.E. Furtak and R.K. Chang, Eds., Surface Enhanced Raman Scattering (Plenum, New York, 1982) p. 318. [4] P.K. Aravind, Surface Sci. 110 (1981) 189. [5] R. Ruppin, Phys. Rev. B26 (1982) 3440. [61 Ref. [3], p. 99. [7] P.B. Johason and R.W. Christy, Phys. Rev. B6 (1972) 4370.