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
IFROM _ TO ____j October 1987 I 251S. SUPPLEMENTARY NO0TATION
* 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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SECUR4ITY CLASSIFICATION OF TIlS PAGE
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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//
II
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