( M.eq. )

1. -the eigenfrequencies of free-electron gas filling a spherical cavity of radius R (the frequencies of the filed oscillations), - the damping of oscillations. - Bessel, Hankel and Neuman cylindrical functions of the standard type defined according to the convention used e.g. in [5]. where: Continuity relations of tangential components of E and B + nontrivialityof solutions for amplitudes Almand Blm Dispersion relation for TM and TE field oscillations. 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 where: 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 wl(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: w0,l=wp(l/(2l+1))1/2,l=1,2,...10. We show, that w0,l are not the values of w0,l in the limit R -›0 as usually assumed, but w0,l wl(R= Rmin,l) = wini,l(Rmin,l). So wini,l are the frequencies of plasmon oscillation for the smallest particle radius Rmin,l 0 still possessing an eigenfrequency for given polarity l. Rmin,l can be e.g.: Rmin,l=4 = 6 nm, but it can be as large Rmin,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 wp. [1] K. Kolwas, S. Demianiuk, M. Kolwas, J. Phys. B 29 4761(1996). [2] K. Kolwas, S. Demianiuk, M. Kolwas, Appl. Phys. B 65 63 (1997). [3] K. Kolwas, Appl. Phys. B 66 467 (1998). [4] K. Kolwas, M. Kolwas, Opt. Appl. 29 515 (1999). [5] M.Born, E.Wolf. Principles of Optics. Pergamon Press, Oxford, 1975. Dispersion relation for TM mode: Self-consistent Maxwell equations describing fields due to known currents and charges: • Two independent solution of the vectorial equation: • TM mode (''transverse magnetic'': If: wp, g - plasma frequency andrelaxation rate of the free electron gas accordingly. We allow the imaginary solutions for given R: No external sources: • TE mode (''transverse electric'': We are concerned with transverse solutions only (E = 0). For harmonic fields (M.eq.) reducesto the Helmholtz equation: Let's define a function DlTM(zl) of the complex arguments zl(l,R): Solution of the scalar equation in spherical coordinates: We are interested in zeros of DlTM(zl) as a function of l and R: 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. R E S U L T S g=0 b) a) b) a) a) Resonance frequencies and b) radiative damping of plasmon oscillations as a function of the radius of sodium particle for different values of l(g=0). g = 0 g = 0 Radiative decay of plasmon oscillations in sodium particle for different values of l and for relaxation rates of the free electron gas: a) g = 0.5 eV; b) g = 1 eV • 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 Rmin,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. Comparison of plasmon frequencies and damping rates resulting from the exact and the approximated approach: The smallest particle radii Rmin,l, still possessing an eigenfrequency of given polarity l as a function of l Frequencies of plasmon oscillation wini,l as a function of the smallest particle radius Rmin,l for different relaxation rates  of free electron gas Exact: Approximated (irrespective R value ): for: Exact Riccati-Bessel functions: l and l(and their derivatives l’ and l’ in respect to the corresponding argument zB and zH) were calculated exactly using the recurrence relation: Approximated Riccati-Bessel functions “for small arguments”: with the two first terms of the series in the form: l = 1 l = 8 Using the approximated Riccati-Bessel functions in the dispersion relation, one gets: l = 1 l = 8 irrespective the value of the sphere radius R. l = 1 l = 8 l = 1 l = 8 Variation ranges of the arguments zB,l(R)=c-1 (R)R and zH,l(R)= c-1 ( ())1/2(R)R of l (zB(R)) and l (zH(R)) functions due to the dependence (R)=(R)+”(R) resulting from the dispersion relation; the example forl=1 and l=8. Legend: l=8 l=1 l=8 l=1 Re(ψl(zB)) Re(xl(zB)) Im(zB) Im(zB) Re(zB) Re(zB) Re(zH) Re(zH) Im(zH) Im(zH) or Im(ψl(zB)) Im(xl(zB)) Re(zH) Re(zH) Im(zH) Im(zH) Im(zB) Im(zB) Re(zB) Re(zB) Variation ranges of the functions l (zB(R)) and l (zH(R)) due to the dependence (R)=(R)+”(R) resulting from the dispersion relation; the example forl=1 and l=8. 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 32/46 02-668 Warsaw, Poland (M.eq.)