(M.eq.) Size dependence of the number, frequencies and radiative decays of plasmon modes in a spherical free-electron cluster K.Kolwas, A.Derkachova and S.Demianiuk Institute of Physics, Polish Acadamy of Sciences, Al. Lotników 32/ Warsaw, Poland A B S T R A C T Nanoscale metal particles are well known for their ability to sustain collective electron plasma oscillations - plasmons. When we talk of plasmons, we have in mind the eigenmodes of the self-consistent Maxwell equations with appropriate boundary conditions. In [1-4] we solved exactly the eigenvalue problem for the sodium spherical particle. It resulted in dipole and higher polarity plasmon frequencies dependence  l (R), l=1,2,...10 (as well as the plasmon radiative decays) as a function of the particle radius R for an arbitrarily large particle. We now re-examine the usual expectations for multipolar plasmon frequencies in the "low radius limit" of the classical picture:  0,l =  p (l/(2l+1)) 1/2, l=1,2, We show, that  0,l are not the values of  0,l in the limit R -› 0 as usually assumed, but  0,l   l (R= R min,l ) =  ini,l (R min,l ). So  ini,l are the frequencies of plasmon oscillation for the smallest particle radius R min,l  0 still possessing an eigenfrequency for given polarity l. R min,l can be e.g.: R min,l=4 = 6 nm, but it can be as large R min,l=10 = 87.2 nm. The confinement of free-electrons within the sphere restricts the number of modes l to the well defined number depending on sphere radius R and on free-electron concentration influencing the value of  p. [1] K. Kolwas, S. Demianiuk, M. Kolwas, J. Phys. B (1996). [2] K. Kolwas, S. Demianiuk, M. Kolwas, Appl. Phys. B (1997). [3] K. Kolwas, Appl. Phys. B (1998). [4] K. Kolwas, M. Kolwas, Opt. Appl (1999). [5] M.Born, E.Wolf. Principles of Optics. Pergamon Press, Oxford, Self-consistent Maxwell equations describing fields due to known currents and charges: No external sources: We are concerned with transverse solutions only (   E = 0 ). For harmonic fields (M.eq.) reduces to the Helmholtz equation: Solution of the scalar equation in spherical coordinates: Continuity relations of tangential components of E and B + nontriviality of solutions for amplitudes A lm and B lm Dispersion relation for TM and TE field oscillations. Two independent solution of the vectorial equation: TM mode (''transverse magnetic'': TE mode (''transverse electric'': F O R M U L A T I O N OF T H E E I G E N V A L U E P R O B L E M: P L A S M O N F R E Q U E N C I E S A N D R A D I A T I V E D A M P I N G R A T E S We allow the imaginary solutions for given R: - the eigenfrequencies of free-electron gas filling a spherical cavity of radius R (the frequencies of the filed oscillations), - the damping of oscillations. Let's define a function D l TM (z l ) of the complex arguments z l (  l,R): We are interested in zeros of D l TM (z l ) as a function of  l and R: Dispersion relation for TM mode: If:  l in given l is treated as a parameter to find, R is outside parameter with the successive values changed with the step R  2nm up to the final radius R=300nm.  p,  - plasma frequency and relaxation rate of the free electron gas accordingly. R E S U L T S a) b) Radiative decay of plasmon oscillations in sodium particle for different values of l and for relaxation rates of the free electron gas: a)  = 0.5 eV; b)  = 1 eV The smallest particle radii R min,l, still possessing an eigenfrequency of given polarity l as a function of l Frequencies of plasmon oscillation  ini,l as a function of the smallest particle radius R min,l for different relaxation rates  of free electron gas Comparison of plasmon frequencies and damping rates resulting from the exact and the approximated approach: Approximated (irrespective R value ): Exact: for: Conclusions: If the sphere is too small, there is no related values of  l (R) real nor complex. For given multipolarity l the eigenfrequency  l (R) can be attributed to the sphere of the radius R starting from R min,l  0. Plasmon frequency  l (R) in given l is weakly modified by the relaxation rate , while radiative damping rate  ” (R) is strongly affected by  in the rage of smaller sphere sizes. a) Resonance frequencies and b) radiative damping of plasmon oscillations as a function of the radius of sodium particle for different values of l  =0).  b) a)  Legend : - Bessel, Hankel and Neuman cylindrical functions of the standard type defined according to the convention used e.g. in [5]. or where: Approximated Riccati-Bessel functions “for small arguments”: where: Using the approximated Riccati-Bessel functions in the dispersion relation, one gets: irrespective the value of the sphere radius R. Re(ψ l (z B )) Im(ψ l (z B )) Re(z B ) Im(z B ) Re(z B ) Im(z B ) Re(z B ) Im(z B ) Re(z B ) Im(z B ) l=1 l=8 Re(  l (z B )) Im(  l (z B )) Im(z H ) Re(z H ) Im(z H ) Re(z H ) Im(z H ) Re(z H ) Im(z H ) Re(z H ) l=1 l=8 Variation ranges of the functions  l (z B (R)) and  l (z H (R)) due to the dependence  (R)=  (R)+  ”(R) resulting from the dispersion relation; the example for l=1 and l=8.  l and  l (and their derivatives  l ’ and  l ’ in respect to the corresponding argument z B and z H ) were calculated exactly using the recurrence relation: with the two first terms of the series in the form: Exact Riccati-Bessel functions: Variation ranges of the arguments z B,l (R)=c -1  (R)R and z H,l (R)= c -1 (  (  )) 1/2  (R)R of  l (z B (R)) and  l (z H (R)) functions due to the dependence  (R)=  (R)+  ”(R) resulting from the dispersion relation; the example for l=1 and l=8. l = 1 l = 8 l = 1