AD-A±65 754 CLASSICAL DECAY RATES FOR MOLECULES IN THE PRESENCE OF i/iA SPHERICAL SURFAC..(U) STATE UNIV OF NEW YORK ATBUFFALO DEPT OF CHEMISTRY Y S KIM ET AL. OCT 97

UNCLARSSIFIED UBUFFRLO/DC/07/TR- 55 N68814-BE-K-8643 F/O 7/4 NL

mhhmohh"IEEEomos

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DXFILE Copy aSurface Science, in press

Classical Decay Rates for Molecules in the Presence of a Spherical Surface:

A Complete Treatment

In

iD Young Sik Kim*0 Department of Chemistry

University of RochesterRochester, New York 14629

0 P. T. Leung and Thomas F. GeorgeDepartments of Chemistry and Physics & Astronomy239 Fronczak HallState University of New York at BuffaloBuffalo, New York 14260

ABSTRACT

k comprehensive treatment of the classical decay rates for a molecule in

the vicinity of a spherical surface is presented through the application of the

work of van del Pol and Brenner and that of Fock. This theory takes full

advantage of the Hertz vector formalism, which is mathematically simpler than the

widely-adapted Lorenz-Mie approach which uses the complicated vector harmonic

expansions. Results are obtained for both radiative and nonradiative transfer

when the molecule is located outside or inside the surface. Numerical results

are given for the cases of a surface, cavity apd-Werosol . and comparison with

other works are made. ..

dcument has been approved"Slic releose and sole; its "

c : tion is unlimited. O

*Present address: Departments of Chemistry and Physics & Astronomy239 Tronczak HallState University of New York at BuffaloBuffalo, New York 14260

87 10 28 154

I. INTRODUCTION

Ever since the role of surface roughness was recognized as prominent in

leading to such dramatic surface phenomena as surface-enhanced Raman scattering

1and other photochemical processes, the problem of the decay rates for molecules

in the vicinity of a spherical particle has been investigated intensively.2 -5

While the classical electromagnetic approach has been followed to establish the

results for the reduced (normalized) decay rates in both the electrostatic limit2

3-5and the more exact electrodynamical treatment, we feel that a complete

analysis of such a problem has not yet been available in the literature. Given

two possible orthogonal orientations of the molecule (radial/tangential), the two

possible locations of the molecule (outside/inside the sphere) and the two

different kinds of energy transfers during the decay process

(radiative/nonradiative), there are altogether eight problems to be solved. In a

paper following the analogous treatment of Chance, Prock and Silbey for a flat

surface,6 Ruppin solved the complete problem for a molecule located outside the

sphere.3 Very recently, Chew4 published results using the energy flux method

3'6

for both the cases where the molecule is located outside and inside the sphere,

and the equivalence with the results obtained from Green's dyadic method6 is

proven in the limit where all the dielectric constants are real. Hence, all the

nonradiative transfers are ignored in Chew's treatment.4 As a matter of fact,

Eqs. (6) and (7) in Ref. 4 are identical to the results for radiative transfer

for a molecule located outside the sphere as given in Zqs. (27) and (28) in

Ruppin's work. Hence, combining the work of Ruppin 3 and Chew,4 we conclude that

six of the eight problems are solved, leaving the two problems of nonradiative

rates for a molecule inside the sphere to be solved. Moreover, all these past

7treatments have been based on the Lorenz-Mie theory established by the expansion

of the field quantities using vector spherical harmonics.

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11. TITLE Classical Decay Rates for Mo ecules in the Presence of aSpherical Surface: A Complete Treatment

12.PESOAL UTORS)Young Sik Kim, P. T. Leung and Thomas F. Geo~rge13a. TYPE OF REPORT 1311 TIME COMEE Id. DATE OF REPORT 1Yr. M.. Dityj 17 . PAGE COUNT

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* Prepared for publication in Surface Science

1?, COSATI CODES iBt SUBJECT TERMS (Coafa,,. on ,euerae Ifneeeem.,y wWl ideantify by bombh numeri

FIEILD GROUP SUBs. GPR. SPHERICAL SURFACE RADIATIVE AND NONRADIATIVENEARBY MOLECULE HERTZ VECTOR FORMALISMCLASSICAL DECAY RATE CVT N EOO

It. ABSTRACT iCon tiaue on o.vuora of aneeney end identify by block aumbarl

A comprehensive treatment of the classical decay rates for a molecule in thevicinity of a spherical surface is presented through the application of the work ofvan del Pol and Brenuier and that of Fock. This theory takes full advantage of the

*Hertz vector formalism, which is mathematically simpler than the widely-adaptedLorenz-Mie approach which uses the complicated vector harmonic expansions. Resultsare obtained for both radiative and nonradlative transfer when the molecule islocated outside or inside the surface. Numerical results are given for the cases ofa surface, cavity and "aerosol", and coinparisonswith other works are made.

24L OISTRIGUTION/AVAILASILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION

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Contract N00014-86-K-0O43

TECHNICAL REPORT No. 55

Classical Decay Rates for Molecules in the Presence of a Spherical

Surface: A Complete Treatment

by

Young Sik Kim, P. T. Leung and Thomas F. George

Prepared for Publication

in

Surface Science

Departments of Chemistry and PhysicsState University of New York at BuffaloBuffalo, New York 14260

October 1987

Reproduction in whole or in part is permitted for any purpose of theUnited States Government.

This document has been approved for public release and sale;its distribution is unlimited.

-At r ;t

'Io

- .- ' t. "

cepy "...L.

3

In this present work, we present a complete solution of the eight problems

via a different approach, namely, the Hertz vector formalism.5 '6 This formalism

has a mathematically simpler feature, since the expansions of the Hertz functions

are more of a scalar type, and one can see easily from the Green's function

property that there exists a simple transformation which relates the fields for

the outside-molecule case to those where the molecule is located inside the

sphere. Hence it is not necessary to solve again the boundary value problem for

the inside-molecule case as is done in Chew's work.4 We elaborate the theory in

Sec. II, where we shall see that for the case where the molecule is inside a

sphere with a complex dielectric constant, a straightforward application of the

methods of Ruppin3 and Chew is not appropriate, such that one mu:.t resort to

more microscopic models for the treatment of the bulk decay rates (i.e., in the

absence of the surface). Numerical examples are given in Sec. III, where we

shall illustrate that the neglect of the nonradiative rates as in Chew's

formalism can lead to serious errors in some cases. Discussion and conclusion of

the present work are given in Sec. IV.

II. Theory

It is well known that the electromagnetic scattering problem in the

presence of a dielectric sphere can in general be formulated in two different

ways. While the scattering of a p wave from a sphere has been well treated

in both the Lorenz-Hie approach7 and the Herta vector (Debye potential)

approach, it seems that the problem of diole-sphere interactions (hence the

problem of molecular lifetimes) has always been treated by the former approach

using the vector harmonics expansion.3A Here we want to solve the lifetime

problem in the Hertz vector (1) formalism using expansions in ordinary spherical

functions. For those mathematical results which have been obtained already by

ft q .

- '.A.

4

the former approach, we shall only indicate briefly how the same results may be

obtained in our present approach, with possible simplifications being emphasized.

For a dipole located outside the sphere, the problem has been solved for

both the vertical (radial)9 and the horizontal (tangential)10 dipoles. Assuming

nonmagnetic media and the region outside (inside) the sphere (of radius a) to be

characterized by the local dielectric function cI (C2), the Hertz vectors (along

the radial direction) can then be expressed in the following series expansions in

terms of the various "Nie coefficients":

For a vertical dipole located at (dO.0) with Yi = kid. gi k ia and k -

ik, I (2n+l) h(1) (kr) + B h(l)(k r) # (cos) , a < r < dr n' (ln 1 n 1n-0

k, n (2n+l)n() + Bnh(l)ll() nj (k' ) Jn (kncs ) d <

il =) n n n 1 1

at

n-0

ik 3 h (1) (Y1 )I (2n+1l[j nhi ( I O~1 (y) n in(k!OP (case) V r < ak2 n-0

(2)

where jand h (1) are the ordinary spherical Bessel and Henkel functions. The

nonvanishing field components are then given by

1 (sine (3)tr ard sinS 89 as (a

i 2

ag (ri) * (b)o ;-d I%I

HO L ik 2 a11 (30)

for a horizontal dipole located at 0d0.0), vs need tvo Hertz vectors vith one

along i and one, &long I. given by (i .1.2) 10

uIa -Cos# * 1 4

a a

aa

I 2n+l )[(kr) I(k~)P(oO , a < r< dd L n~n+1) Cn 1 a k*r)+PBnosO

* n-I

P (6)

rd nn+I 1)I~j + %C(yl)ICn(kir)Pn(cos9) , d <r

2rX72 ncoe2 D;' C'(y 1 3 2.( 0 cs *0 c (7)ad ~ )aa)Jk)10n-i

na

k 2 n~l i (y) + A(k) (y) (1)( 1 r)%cs , a < r

I nn+i a (1 IhnI

n-Ii

6

Q W k 2n~l C1 h(1) (yI j(k 2r)Pn(cOsO) 0 < r < a . (9)02 a -;-+7 n h; 1nnn-I

The nonvanishing field components are then given by

E = L2DE (l0a)r r

E 1 a2 tro En8 (10b)6 r bras rl)csinO a$

1 a2(rn1) I Wan (0c)

S-r-inO ara$ c as

H - - ! L2U (lOd) (r r

H r (10d)H ck all + 1 2rl ) (0)"

9 w sine a, r arae

H 1 *2(rli) (10f)W be r sine 3ra

with

L2 U I (SI + I D

The "Mie coefficients" in the above equations are defined as

A a Jn.(P)*,'P2) - Jn(p2),m(PI)

nP2)C;( P) . I ) n2

.2.

7

B CIJn(P1)#n(P2 ) - ¢2Jn(P2)*n(PI)

En C 1( l - el 1 ( (24(hn nP,( (1 1 n 2

" n(p 2 )C (p 1 - h~l)(p1 ),'(p 2 )-

n m 2Jn(p2)C(P) -

From this formulation and by making use of the well-known properties of Green's

functions, we can easily obtain via a simple transformation T the solution for

the reflected and transmitted fields for the case where the dipole is located

inside the sphere without having to solve again the boundary value problem.4 If

we define T by

n(In

T kie1,, k2PC 2 (12)

then it is not difficult to check that the "inside solutions" are easily obtained

from the oustide solutions through the operation with T.

Now let us apply the above results for our decay rate problem. According

to the classical approach, there are two different mthods for calculating the

decay rate of the molecule as Induced by the presence of the surface. In the

Green's dyadic approach, the dipole is modeled as a damped hermic oscillator

being driven by the surface field. The induced decay rate is then given by

Y a Y 0+ ai G (13)

8

where y0 is the decay rate in the absence of the boundary surface (i.e., with the

whole space filled with the dielectric medium in which the dipole is located), a

W e2/m. w being the emission frequency (which is assumed to change negligibly),

and In G is the imaginary part of the dyadic function defined as the surface

field (ER) reflected at the dipole site per unit dipole moment (P). The other

approach would be to calculate the rate of the energy carried away from the

molecule, which can further be divided into that radiated to infinity [radiative

(R) transfer] and that lost in the form of Joule heat into any dissipative mdium

present [nonradiative (NR) transfer]. Hence, combining these two approaches, the

total rate given by Eq. (13) can be written as

R + R (14)

with3

Y J dA r2 4. (15)

(r-.-)

and-VN 1 It2 (16)

VI

Nero I Re(xA*) is the Poynting vector, a =2- In c is the conductivity ofST 4-W

the dissipative medium, and V - w2 p2/2a is the average energy of the dipole. The

integral in Eq. (15) is over a surface at infinity, and that in 1q. (16) t over

the volume of the dissipative medium. Very often, the Joule beat integral is

3quite complicated, and one can obtain it indirectly through Eqs. (13)-(1). We

9

shall apply the fields obtained from Eqs. (1)-(10) to calculate the y's given in

Eqs. (13)-(15) for the following three cases:

Case (i): The surface problem

Here P is located in vacuum (eI W 1) outside a sphere of dielectric

constant c2 ' + ic", and Yo is then given by 1 4p /W 2ak l/3w. Using Eqs.constant I

(1), (3a) and (13), we obtain for a vertical dipole4

* ()- 1 + - Re (2n+l)n(n+l)B n[ 2" 1 (17)

n-i

Furthermore, using Eqs. (1)-(3), the Poynting vector can readily be calculated,

and from Eq. (15) we obtain3'4

YAI R W Ji (y ) + Bnh( (yl) 2

"- . 1 n(n+l)(2n+l) Il n (18)

YO 2 n-i y

Similarly, using Eqs. (4)-(10). (13) and (15). we obtain the corresponding

results for a horizontal dipole

a y M I '(Yl) 2 ) Re WI+AO,( (19)R W )'y. +

!I.- (2n+) I ) + A h1)(Y)1 + IAn(yl] }h( 1

(20)

10

The nonradiative rates can then be obtained from the difference between the total

and radiative rates.

Case (ii): The cavity problem

Here P is located inside a spherical vacuum ( 2 - 1) with a dielectric (c1

C' + ic") occupying the whole space outside the vacuum. The total decay rates

can readily be obtained from the outside solutions (Eqs. (17) and (19)) through

the transformation T (Eq. (12)) as discussed above, whereby we obtain"

3j 2n-ii

3 - J(2 2

1 + Rej (n + 1) E Y2 n 2 (22)nYi

with En and Pn given by

En -T{B) , (23)

n nF n = T{A n} (24) ]

We want to remark that in this configuration the decay rates can only be p

radiative or purely nonradiative, depending on whether the outside medium is

transparent (W" a 0) or dissipative (e" 0 0) with respect to the molecular

emission frequency. Furthermore, it is interesting to note that in the

perfectly-conducting limit (121>Ic1I) where we have4

'A

A ; - (P )/hL)(p)n 1 n i

B a-*()/(25)

both Eqs. (21) and (22) vanish. Thus, in the present classical treatment, the

molecular lifetime becomes infinite in a perfectly-conducting cavity. This is

understandable since in this approach only two mechanisms can "cause" the

molecule to decay, i.e., either for energy being brought to infinity (radiative

transfer) or dissipated into a host medium (nonradiative transfer). Since no

field can penetrate into the perfect conducting environment and hence both these

transfers cannot occur, it is not surprising to see that both decay rates vanish

in Eqs. (21) and (22). In the more exact quantum treatment, however, the

vanishing of the decay rates in a perfectly-conducting cavity can only

12restrictively occur.

Case (iii): The "aerosol" problem13

Here 1 is located inside a dielectric sphere (e2 - E' + i") with the

outside being vacuum (e1 = 1). We shall further divide this into the following

two cases for discussion:

(a) Transparent dielectric

This case has already been treated by Chew. 4 In the present approach, the

total decay rate can again be obtained directly from Eqs. (17) and (19) through

T. With yo being given by Ik 2 2 / - 2*k 3 3w 4 we obtain the same

expressions as given in Eqs. (21)-(24). The radiative transfers in this case can

similarly be obtained as in Eqs. (18) and (20) via the transformation i -

Tltout) to give4

siilrl

12

R aY.L _3 C 3/2 n(n+l)(2n+l) IJ(>2) 2 (6

2O 2 1 IP2YT (26

n=1!I C 3/ (2n+l)(1 1(y2 +2 Jnp2c) •27

n-1 ;7,1+ t2n

In this case, since £2 is real (transparent, E" = 0), there is no

dissipative (nonradiative) effects, and it has been shown explicitly by Chew that

the radiative rates in Eqs. (26) and (27) are identical to the total rates as

given in Eqs. (21) and (22).

(b) Dissipative dielectric

This is the case which has not been discussed before in the context of

"surface problems". Nevertheless, the same problem for a molecule in the bulk of

an adsorptive medium has received considerable attention in the literature.4

The difficulty lies in the evaluation of yo in Eq. (13). Obviously, the decay in

a dissipative medium of infinite extent can only be nonradiative. However, if

one applies Eq. (16) directly to calculate yo in this case, one will get

divergences in the integral due to the predominance of the near field in this

case, unless one adopts a cutoff volume (forbidden volume 14) in the lower bound

of the integral.is Moreover, this approach has been criticized by Agranovich and

Dubovskii (AD),16 who proposed a more exact microscopic treatment which shows

that the result from such macroscopic theory can be correct only under certain

restrictive conditions. In the following, we shall make use of the results of AD

to investigate the surface effects in 1q. (13).

-w|

13

According to AD, under the condition that the "Kossbauer-type" effect (in

the optical region)17 can be neglected, y0 can be expressed in a form which is.15

very similar to that obtained from Fleinberg's theory,

Y0 3Z ,,NoX3 (28)Ysp 641 4 0

where y p is the spontaneous decay of the free molecule, N is the molecular

number density of the dielectric medium, A is the emission wavelength, and Z is a

numerical factor depending on the lattice structure of the medium. A rough

estimate gives Z - 10. 16 Eq. (28) holds as long as A is much greater than the

lattice constant of the medium and the spatial dispersion of the dielectric

constant c" can be neglected.16 Using Eqs. (13), (21), (22) and (28), we finally

obtain the total decay rates for 1 in a dissipative dielectric sphere as

'r3 n(Y2 ) 2 "- 1 +,I f(w) Re(,/- (2n+l)n(n+l) E n[ -] ) (29)

YO 2[Y2n-i

nir 2 ) 0]

where f(w) a Yp /Yo is the reciprocal ratio as given in 1q. (28). The radiative

rates can analogously be obtained from Eqs. (26) and (27) by multiplying each of

thes by f(w). The nonradiative rates can then be obtained again from the

difference between the total and the radiative rates. Furthermore, for P located

at the center of the sphere, we have results analogous to those for the case of a

transparent spihere obtained by Chew4 where for a dissipative sphere are given by

[ , r ,W ... " ' V ,wt""

u-u V' Y u -,~~iVi. K\ n r WYVuV1 -V w w . -&W VV U W U WIN h( U U - - -

14

-. 1.- 1 + f(M) Re(Vr- ) , (31)N0 10 2

R R £|

Y Yo • (32)

III. Numerical Results

We have performed numerical studies for each of the above three classes of

problems. For simplicity, we treat one of the two media as a vacuum (c = 1).

The other medium, if it is transparent to the molecular emission, is taken to be

glycerol with c - 2.16, 4 ,13 and if it is dissipative, it is taken to be silver at

A - 413 nm with c - -4.42 + 0.73 i. 18 Figure 1 shows the results for a dipole

above a Ag spherical surface of radius a - 100 A. We see that while the neglect

of the nonradiative transfer (as in Chew's formulation) leads to somewhat lower

values for y for the radial dipole case, it leads to tremendous differences for

the tangential dipole case as r * a. This phenomenon also occurs in the flat-

surface case, with its origin due to the parallel (antiparallel) orientation of

the "image dipole" with respect to the normal (tangential) orientation of the

source dipole.6 Of course, the extemely large values of the rates at r = a are

not physical either, and it is known that more reasonable values for y at r 0 a

can be obtained by considering the spatial dispersion of the dielectric function

of the substrate at close distances.19 Nevertheless, we should add that in all

the numerical calculations of Ref. 4, the substrate sphere is taken either as

transparent (c" = 0) or as perfectly conducting (hence no penetration of fields

and ylR = 0), so that the neglect of the nonradiative rates does not lead to any

error in these cases.

IS

Figure 2 shows the results for a dipole in a cavity of radius a - 4000 A.The outside medium is taken to have a real I a 2.16. We observe that both

radial and tangential dipoles decay at the same rate (w I + ReK1 (see Eq. (23))

at the center, with the tangential dipole showing appreciable oscillating

behavior as the dipole is moving towards the cavity wall. We also note that both

enhanced and diminished values (with respect to the free molecule case) for y can

occur in this case. Figure 3 shows the results for the same cavity problem with

the outside medium being absorptive (Ag at X - 413 nm). We notice that there is

no oscillating behavior, and the decay for this case is very small (unless the

dipole is far away from the center) and purely dissipative in nature (we recall

that in the exteme limit where the outside medium becomes perfectly conducting,

the decay is zero as discussed in Sec. II).

Figures 4 and 5 show results for a dipole inside an "aerosol" of

transparent and dissipative media, respectively. For the transparent case

(radius a - 4000 A), we again observe similar oscillatory behavior as for the

'flat-surface case and a possible "diminution effect" for the decay rate. For the

dissipative case (radius a - 1000 A), it is interesting to note that (for both

orientations) most of the decays are nonradiative in nature, and the surface

effect is hardly noticeable except when the molecule is very close to the

surface. This means that most of the molecular deexcitation energy is dissipated

within a very small region of the medium around the molecule, which can be

understood from Eq. (28) showing that -0 is very large for molecular emissiond

wavelengths. To have appreciable surface efffects, one osut look for much higher

emission frequencies, which can occur, for example, in nuclear spectroscopy in

the y-ray range. Thus for our interest here which focuses on molecular

lifetimes, the surface effect is hardly manifested.

V&

16

IV. Conclusion

We have in this paper presented a complete treatment of the classical decay

rates of a molecule in the vicinity of a spherical surface. We have adopted a

different formulation of the dipole-sphere interaction problem via the Hertz

vector formalism, which is distinct from the more common approach via the Lorenz-

Mis expansions of the field quantities in terms of complicated vector spherical

harmonics. We have explored all the eight problems as stated in the Introduction

within this classical approach. Since this approach is limited to only two kinds

of mechanisms (radiative and dissipative transfers) through which the molecular

decay rates can be examined, some interesting features (e.g., possible enhanced

decays for a perfectly-conducting cavity)12 cannot be obtained in the present

formalism. Recently, a quantum electrodynamical formalism (QED) of the spherical200

cavity problem was published,2 0 although the electrostatic image concept was

adopted and hence the dynamical nature of the dipole field was omitted. It would

therefore be interesting to incorporate the dynamical dipole fields, as treated

in our present paper, into the QED formalism to see if it would give rise to new

interesting phenomena for the decay rate.

Acknowledgments

This research was supported by the Office of Naval Research, the Air Force

Office of Scientific Research (APSC). United States Air Force, under Contract

F49620-86-C-0009, and the National Science Foundation under Grant CHI-8620274.

The United States Government is authorised to reproduce and distribute reprints

notwithstanding any copyright notation hereon.

I0e

17

References

1. See, e.g., H. Metiu, Prog. Surf. Sci. 17, 153 (1984); M. Moskovits, Rev.

Mod. Phys. 57, 783 (1985).

2. 3. 1. Gersten and A. Nitzan, Surf. Sci. 158, 165 (1985) and referencestherein.

3. R. Ruppin, J. Chan. Phys. 76, 1681 (1982).

4. H. Chew, J. Chem. Phys. 87, 1355 (1987).

5. P. T. Leung and T. F. George, J. Chem. Thys. (in press). This articlecompares the static and the dynamic theories and gives an assessment of the

ccuracy of the former theory.

6. R. R. Chance, A. Frock and R. Silbey, Adv. Chem. Phys. 37, 1 (1978).

7. See, e.g., M. Kerker, The Scattering of Light and Other ElectromagneticRadiation (Academic, New York 1969).

8. See, e.g., A. B. Pluchino, Applied Optics 20, 2986 (1981).

9. B. van del Pol and H. Bremmer, Phil. Nag. XXV 141 (1937).

10. V. A. Pock, Soy. Phys. JET? 19, 916 (1949) (in Russian); see also V. A.lock, Electromagnetic Diffraction and Propagation Problems (Porgamon, NovYork, 1965).

11. The solutions obtained here are essentially the dynamical version for thereaction fields in the Onsager model (cf. L. Onsager, J. Am. Chem. Soc. St.1486 (1936)).

12. Sae e-g., M. Lewenstein, T. W. Mossberg and 1. J. Glauber, Phys. Rev. Lett.59, 775 (1987), and references therein.

13 See, e.g., L. M. lolan, S. Arnold and S. D. Druger, Chem. Phys. Lett. 118,322 (1985); J. Chemn. Thys. (in press).

14. See the classic monograph by V. M. Agraziovich and M. D. Galanin, Blectronic* Excitation Energy Transfer In Condensed Matter (North Holland, Amterdam,

1982), and references therein.

* 15. 1. L. Feinberg, JUT? Lett. 7,, 20 (1966).

16. V. M. Agranovich and 0. A. Dubovskii, Sow. Phys.-Solid State 1.1. 1631(1971).

17. A. Naradudin, Solid State Physics 18, 274 (1966).

18. See, e.g., 1. D. Palik, Handboolk of Optical Constants of Solids (Academic,New York, 1985).

19. G. W. Ford and V. N. Weber, Surf. Sci. 109, 451 (1981).

Or"Wo IM on I rVLW~18

20. A. 0. Sarut and J. P. Dowling, Phys. Rev. A .36. 6149 (1987).

19

Figure Captions

1. Transition rates for the two polarizations of an oscillator molecule outside

a dielectric sphere of radius a - 100 A for A w 4133 A and c2 -4.42 +

0.731. The molecule is located at a distance d from the center, and T and R

stand for the total and radiative transition rates, respectively. The solid

curve is for the radial dipole, while the dashed curve is for the tangential

dipole.

2. Transition rates for the two polarizations for an oscillator molecule in a

vacuum cavity (c2 - 1) of radius a - 4000 A for A - 4133 A and = 2.16.

The molecule is located at d from the center. The solid curve is for the

radial dipole while the dashed curve is for the tangential dipole.

3. Same as Fig. 2, except that s 1 -4.42 + 0.731.

4. Transition rates for the two polarizations for an oscillator molecule inside

a dielectric sphere of radius a - 4000 A for X = 4133 A and c2 - 2.16. The

molecule is located at d from center. The solid curve is for the radial

dipole while the dashed curve is for the tangential dipole.

5. Sam as Fig. 4 except that - -4.42 + 0.73i and a = 1000 A

2.0

10 0- FA

//

2o2

N 1.5

dom I0t

008'

0. 0* 1 10

d/a

8.0

4,0

0.00.0 0.5 1.00

d/a

1.2

0.7

0.0 0.5 1.00

d/a

1 +10 - 61.0

1 -!

0.0

0.0 0.5 1.00

d//

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