• Radiative reaction correction

• Dynamic depolarization

• Surface scattering corrections to the dielectric function of nanoparticles

• Decay rates of atoms and molecules in a proximity of small (nano)particles

• Multilayered nanoshells

Theoretically, the radiation damping term is necessary for the extinction cross section to be the sum of the scattering and absorption cross sections, which is sometime misleadingly referred to as unitarity in the dipole approximation (see also Appendix C of [5]).

References:

1. A. Wokaun, J. P. Gordon, and P. F. Liao, Radiation Damping in Surface-Enhanced Raman Scattering, Phys. Rev. Lett. 48, 957-960 (1982) (published 5 April 1982).
2. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Oxford, Pergamon, 1959) p 186.
3. K. Ohtaka and M. Inoue, Electromagnetic aspects of the enhanced Raman scattering by a molecule adsorbed on a polarisable sphere, J. Phys. C: Solid State Phys. 15 6463-6480 (1982) (received 21 May 1982).
4. B. T. Draine, The discrete-dipole approximation and its application to interstellar graphite grains, Astrophys. J. 333, 848-872 (1988).
5. A. Moroz, Depolarization field of spheroidal particles,
   J. Opt. Soc. Am. B 26(3), 517-527 (2009) [pdf].
   Belongs to the most-downloaded publications of J. Opt. Soc. Am. B in the field of plasmonics.
   (Shows that the hypothesis of a uniform polarization within a spheroidal particle is incompatible with the exact limit of polarizability in the small particle limit up to the order x**2 when the size parameter x--> 0. Accompanying F77 code sphrd.f to determine extinction efficiency in various long-wavelength approximations discussed in the article.
   See here for on-line supplementary material.
   See also my worksheet involving some intermediary calculations.)

From the monograph by Bohren&Huffman [see Eq. (12.13) in Sec. 12.1.1], the surface plasmon position of a spherical particle shows the following size dependence:

where epsilon_r is the dielectric constant of a sphere at the resonance position and epsilon_h is the dielectric constant of the surrounding medium.

The above size dependences cannot be accounted for by a polarizability comprising only the k**3-dependent radiative reaction term. One needs in addition a k**2-dependent dynamic depolarization term [1,3].

All the terms in the electron dipole polarizability are included if the electron dipole polarizability is determined directly from the dipole T-matrix element on using the relation

where the first (second) expression holds in the SI (gauss) units. (A free T-matrix code for metallic particles is available in Ref. [4] herein below.)

As shown by Dungey and Bohren [5], the use of such a polarizability brings a significant improvement within the coupled/discrete dipole approximation (CDA/DDA) over the schemes incorporating only the radiative reaction term.

It is worth of reminding here that the static sphere polarizability

which is the Rayleigh (x ---> 0) limit of the expression (R2), is not unitary (see also Appendix C of [6]) and is insufficient in describing essential features of small metal naparticles, such as the size-dependence of a red-shift of the surface plasmon resonance (SPR) (see Eq. (R1) and Sec. 12.1.1. of [2])

The Meier and Wokaun recipe [1] is based upon two hypothesis. The first hypothesis is that the polarization is homogeneous over the volume of the particle. The second hypothesis, which can be viewed as a continuation of the first one, is that the same dipole moment is assigned to each volume element of the particle. Given the above two hypothesis, Ref. [6] provides an exact solution for the case of a spheroid and, in doing so, it also amends the original Meier and Wokaun result for a sphere [1].

In Ref. [6], the dynamic k**2-dependent depolarization component of E_d has been shown to depend on dynamic geometrical factors, which can be expressed in terms of the standard geometrical factors of electrostatics. The Meier and Wokaun recipe itself was shown to be equivalent to a long-wavelength limit of the Green's function technique. The resulting Meier and Wokaun long-wavelength approximation (MWLWA) exhibits a red-shift compared against exact T-matrix results. The red-shift is a hallmark of the Meier and Wokaun long-wavelength approximation: it will persist unless the assumption of a uniform field E_i inside a particle is relaxed [6]. For a sphere it is possible to get rid of the red-shift by assuming rather simple form of a weak nonuniformity of the field E_i inside a particle that can be fully accounted for by a renormalization of the dynamic geometrical factors [6].

Interestingly enough, for electric field oriented along the spheroid rotational axis, the work by Kuwata et al [7] has anticipated dynamic depolarization factors by purely empirical formula obtained by best fit to numerical results, in which case dynamic depolarization factors were fitted by a polynom of the 3rd order in the standard geometrical factor of electrostatics L_z. In contrast, analytic results of Ref. [6] suggest that a linear L_z dependence, but with eccentricity-dependent coefficients, may be enough.

The long-wavelength approximations of Refs. [6,7] accurately match the SPR position, height, and linewidth of the exact results for noble particles with an equivalent-volume-sphere radius of up to 50 nm in the visible. Note in passing that the linewidth Gamma directly determines the plasmon dephasing time T_2=2 hbar/Gamma, where hbar is the Planck constant, the quality factor Q of the resonance at the SPR frequency omega_r via the formula Q = omega_r/Gamma, and the local field enhancement factor |f| (in a harmonic model |f| = Q).

Recently, the results of Ref. [6] have been extended to provide an approximation for the polarizability for large metallic spheroidal nanoshells [8].

References:

1. M. Meier and A. Wokaun, Enhanced fields on large metal particles: dynamic depolarization, Opt. Lett. 8, 581-583 (1983).
2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1998).
3. M. Kerker, D.-S. Wang, and H. Chew, Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles: errata, Appl. Opt. 19, 4159-4174 (1980).
4. A. Moroz, Improvement of Mishchenko's T-matrix code for absorbing particles,
   Appl. Opt. 44(17), 3604-3609 (2005). [pdf]
   (Accompanying F77 code is available here together with routines zger.f and zsur.f; see also routines zge.f and zsu.f for multiplication from the other side.)
5. C. E. Dungey and C. F. Bohren, Light scattering by nonspherical particles: a refinement to the coupled-dipole method, JOSA A 8, 81-87 (1991).
6. A. Moroz, Depolarization field of spheroidal particles, J. Opt. Soc. Am. B 26(3), 517-527 (2009).
   (Accompanying F77 code sphrd.f to determine extinction efficiency in various long-wavelength approximations discussed in the article. See here for on-line supplementary material. )
7. H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation, Appl. Phys. Lett. 83(22), 4625-4627 (2003).
8. H. Y. Chung, P. T. Leung, and D. P. Tsai, Dynamic modifications of polarizability for large metallic spheroidal nanoshells, J. Chem. Phys. 131, 124122 (2009).

The free-path effect can be incorporated in the dielectric constant of free-electron metals according to the recipe by Kreibig and Fragstein [1]. Although the effect of the mean free path limitation on the real part of the dielectric function is slight, the effect on the imaginary part is often substantial. For instance, for small silver particles near a SPR, the imaginary part is enhanced by more than 10% for particles of radius 100 nm, whereas the imaginary part is twice the bulk value for the radius of about 11.5 nm. The effect of the decreased mean free path is to increase the width and lower the peak height of the surface plasmon absorption (see Sec. 12.1.6 of Bohren&Huffman [4]).

For a homogeneous sphere of a radius R, essentially two different classical surface scattering models were proposed: (i) diffusive and (ii) isotropic scattering models.

• Euler diffusive surface scattering model [5] is the oldest one. The model is concerned with the trajectories immediately after individual scattering events. The electron reflexion on the boundary surface is taken to be fully isotropic, meaning there is equal probability for electron to be scattered at any angle with respect to the surface inward normal.

• Isotropic scattering is, in contrast to the Euler model [5], concerned with the trajectories immediately before scattering events. No explicit assumption is made about individual scattering events at the enclosure boundary except that the scattering events conspire to maintain a homogeneous and isotropic particle distribution in the electron enclosure [6].

• Billiard scattering model [7,8] presumes a specular reflection at the enclosure boundary, i.e., the angle of incidence equals the angle of reflection. This allows one to unambiguously follow any given electron trajectory. In contrast to previous models, further considerations in the billiard case follow in the phase space on using the concept of invariant probability measures. The billiard measure comprises an extra cos(theta)-factor with regard to the Euler measure, where theta is the polar angle measured with regard to the inward normal at the particle surface.

In Ref. [10], the mean free path for a spherical shell geometry was calculated (i) on assuming Euler diffusive surface scattering and (ii) under the assumption of a homogeneous and isotropic electron distribution within a shell. Further, the general billiard formula of Refs. [7,8] was confirmed in the nanoshell case by an explicit averaging procedure. The extra cos(theta)-factor in the billiard measure has been interpreted as that the (electron) enclosure surface is a Lambertian surface. The latter is distinguished by a fact that the radiant intensity observed from it is directly proportional to the cosine of the angle theta between the observer's line of sight and the surface normal, and hence when an area element on the surface is viewed from any angle, it has the same radiance. (An example a perfect Lambertian radiator is a black body, and, to a large extent, the Sun in the visible spectrum.) Consequently, the billiard measure can be interpreted as that the electron scattering events on the domain surface conspire to produce a diffusive Lambertian surface. The condition of specular reflections can then be abandoned.

Let D be the shell thickness and R the outer shell radius. Preliminary experimental results, which were performed exclusively within the range D/R < 0.25 and which shown a linear dependence of L on D, are compatible with the billiard, or Lambertian, scattering model and rule out both the Euler diffusive scattering and the isotropic scattering. The billiard model characterized by the Lambertian scattering on the electron enclosure surface is distinguished by the following properties in the nanoshell case [10]:

• It is the only model which yields a linear behavior of the mean free path L on the shell thickness D in the thin shell limit (up to D/R approx 0.3). Compare that with L_d ~ (D/2) ln (R/D) in the diffusive case, and with L_i ~ 14(2RD)**(1/2)/[3 ln (R/D)] in the isotropic case.
• For any shape of a bounded and connected electron enclosure and in any space dimension, the surface scattering mean free path L is straightforwardly determined by substituting the surface S and volume V of the enclosure into the general billiard formula [7,8,10], which in 3D reduces to
  L = 4*V/S,                   (C1)

  and in 2D reduces to

  L = pi* V/S.                   (C2)

• The billiard measure is the only measure which yields a finite result when the integral for the slab geometry is performed. It goes without saying that the result coincides with the thin-shell limit in the spherical shell geometry [10].
• On using the Birkhoff-Khinchin theorem, the billiard mean free path can be interpreted as the time average of the free paths along typical trajectories [7,8].
• Gives the best agreement with the experimental data and agrees with the experimental findings of Ref. [11] without any fitting parameter.
• Appears to agree with quantum-mechanical calculations.

Quite amazingly, although Lermé et al [13] did not emphasize it, their experimental data regarding silver nanospheres fit nicely to the billiard model [10]. The latter yields for the mean-free path the value of 4R/3 (e.g. eq. 141 of Ref. [10]) which, when compared with eq 1 of [13], would have yielded g=3/4=0.75. The latter value is almost in the middle of an overall experimental value of determined by Lermé et al [13]. Compared to the billiard model, the diffusive model of Euler and Kreibig would yield for the mean-free path the value of R, or g=1 [10].

As suggested to me by Professor C. F. Bohren, the history of the above Eqs. (C1) and (C2) goes much further and derives from early results by Cauchy, who studied the average projected area of convex bodies [14,15]. The relevant Cauchy theorem states that in three spatial dimensions [14,15]

                  (C3)

where S is the corresponding surface area of a convex body. The average is taken over all possible orientations in space. Other way round, the average projection of a convex body onto a plane is equal to one quarter of its surface area. In reactor and in other branches of physics the mean-free path of a single particle bouncing off the walls of a convex body has been then heuristically determined as [16-19]

                  (C4)

One can verify that on substituting for from (C3) into (C4) one recovers (C1). This relation has also been used in the derivation of the Sabine equation in acoustics, using a geometrical approximation of sound propagation.

In two dimensions, the average projected area of a convex body reduces to an average projected length [20]

                  (C5)

The average chord length is thus, with V the area of the figure and S the circumference of the figure, given by

                  (C6)

which is identical to (C2).

However, a core-shell problem becomes the problem of calculating the mean-free path in a circular shell region that is (i) neither convex nor (ii) simply connected. This means that the early results by Cauchy [14,15], Czuber [16], Dirac [17], Weinberg and Wigner [18], and others [9,18-20], which yield the mean free path in terms of the geometrical parameters of a convex domain, can no be longer applied. Nevertheless, the validity of the geometric formula (C6) can still be established in the case of the billiard scattering [21]. In analogy with the 3D case [10], a different result is then obtained in the diffusive case [21].

References:

1. U. Kreibig, Electronic properties of small silver particles: the optical constants and their temperature dependence, J. Phys. F: Met. Phys. 4, 999-1014 (1974).
2. U. Kreibig and C. V. Fragstein, The limitation of electron mean free path in small silver particles, Z. Physik A 224(4), 307-323 (1969).
3. F. W. Reynolds and G. R. Stilwell, Mean Free Paths of Electrons in Evaporated Metal Films, Phys. Rev. 88, 418-419 (1952).
4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1998).
5. J. Euler, Infrared properties of metals and the mean free paths of conduction electrons, Z. Phys. 137, 318-332 (1954).
6. U. Kreibig, Small silver particles in photosensitive glass: Their nucleation and growth, Appl. Phys. 10(3), 255-264 (1976).
7. N. Chernov, Entropy, Lyapunov Exponents and Mean Free Path for Billiards, J. Stat. Phys. 88(1-2), 1-29 (1997). [ps]
8. N. Chernov and R. Markarian, on-line book Introduction to the Ergodic Theory of Chaotic Billiards.
9. E. A. Coronado and G. C. Schatz, Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach, J. Chem. Phys. 119(7), 3926-3934 (2003).
10. A. Moroz, Electron mean-free path in a spherical shell geometry, J. Phys. Chem. C 112(29), 10641-10652 (2008).
    (The article has been supplemented with a detailed Electronic Supporting Information. See also an accompanying F77 code fsc.f to calculate the mean-free path for various model cases discussed in the article.)
11. G. Raschke, S. Brogl, A. S. Susha, A. L. Rogach, T. A. Klar, J. Feldmann, B. Fieres, N. Petkov, T. Bein, A. Nichtl, and K. Kürzinger, Gold Nanoshells Improve Single Nanoparticle Molecular Sensors, Nano Lett. 4(10), 1853-1857 (2004).
12. H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, Width of cluster plasmon resonances: Bulk dielectric functions and chemical interface damping, Phys. Rev. B 48, 18178-18188 (1993) [pdf].
13. J. Lermé, H. Baida, C. Bonnet, M. Broyer, E. Cottancin, A. Crut, P. Maioli, N. Del Fatti, F. Valle, and M. Pellarin, Size Dependence of the Surface Plasmon Resonance Damping in Metal Nanospheres, J. Phys. Chem. Lett. 1(19), 2922-2928 (2010).
14. A. Cauchy, Memoire sur la rectification des courbes et la quadrature des surfaces courbes (Paris, 1832). Reproduced in Oevres Complčtes, 1st series, Vol. 2, p. 167 (Gauthiers Villars, Paris 1908).
15. A. Cauchy, C. R. Acad. Sci., Paris, 13, 1060 (1841). A short account of his results [14] in the form of theorems was later published in the Comptes rendus, but no formal proof was given.
16. E. Czuber, Zur Theorie der geometrischen Wahrscheinlichkeiten, Sitzungsberichte der mathematisch-naturwissenschaftliche Klasse der Kaiserlichen Akademie der Wissenschaften Wien 90(2), 719-742 (1884).
17. P. A. M. Dirac, Approximate rate of neutron multiplication for a solid of arbitrary shape and uniform density, British Report MS-D-5, Part I, 1943.
18. V. Vouk, Projected area of convex bodies, Nature 162, 330-331 (1948).
19. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, (The University of Chicago Press, Chicago, 1958).
20. W. J. M. de Kruijf and J. L. Kloosterman, On the average chord length in reactor physics, Ann. Nucl. Energy 30(5), 549-553 (2003).
21. A. Moroz, Electron mean-free path in metal coated nanowires, J. Opt. Soc. Am. B 28(5), 1130-1138 (2011).
    (The article has been supplemented with a detailed Supporting Information. See also an accompanying F77 code fsc2d.f to calculate the mean-free path for various model cases discussed in the article.)

I'm proud to provide in my articles [8,9] an extension for the classical electromagnetics problem of a dipole radiating inside and outside a stratified sphere consisting of an arbitrary number of concentric spherical shells. There is no limitation on the dipole position, the number of the concentric shells, the shell medium, nor on the sphere radius. Electromagnetic fields were determined anywhere in the space, the time-averaged angular distribution of the radiated power, the total radiated power, Ohmic losses due to an absorbing shell, and Green's function were calculated. The results were applied to inelastic light scattering (fluorescence and Raman), the radiative and nonradiative normalized decay rates, and frequency shift. Using correspondence principle, the radiative decay rate was calculated from the Poynting vector, whereas the nonradiative decay rate was calculated from the Ohmic losses inside a sphere absorptive shell. Fast and numerically stable transfer-matrix solution has been incorporated into the respective F77 codes CHEW and CHEWFS, which are freely available here and here, respectively.

In Ref. [9] frequency shifts, radiative decay rates, the Ohmic loss contribution to the nonradiative decay rates, fluorescence yield, and photobleaching of a two-level atom radiating anywhere inside or outside a complex spherical nanoshell with a dielectric core and three alternating gold and silica concentric spherical shells were determined. An application of the above codes for a homogeneous metal nanoparticle can be found in Ref. [10].

Coincidentally, one of the core-shell particle geometries studied in my article [8] correspond to that of a recently reported spaser-based nanolaser [11].

References:

1. B. van der Pol and H. Bremmer, Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radio telegraphy and the theory of the rainbow I, Phil. Mag. 24, 141-176 (1937).
   (The very first calculations of radiative decay rates for vertical (radial) dipole).
2. B. van der Pol and H. Bremmer, Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with application to radio telegraphy and theory of the rainbow. II, Phil. Mag. 24, 825-864 (1937).
   (The very first calculations of radiative decay rates for vertical (radial) dipole).
3. V. A. Fock, The fields of a vertical and horizontal dipole above the earth surface, Soviet. Phys. JETP 19, 916-929 (1946).
   (The very first calculations of radiative decay rates for horizontal (tangential) dipole).
4. R. Ruppin, Decay of an excited molecule near a small metal sphere, J. Chem. Phys. 76, 1681-1684 (1982).
5. H. Chew, Transition rates of atoms near spherical surfaces, J. Chem. Phys. 87, 1355-1360 (1987).
We study the transition rates of atoms inside and outside dielectric spheres
6. H. Chew, Radiation and lifetimes of atoms inside dielectric particles, Phys. Rev. A 38, 3410-3416 (1988).
7. Y. S. Kim, P. T. Leung, and T. F. George, Classical decay rates for molecules in the presence of a spherical surface: A complete treatment, Surf. Sci. 195, 1-14 (1988).
8. A. Moroz, A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere, Ann. Phys. (NY) 315(2), 352-418 (2005). (published online on 7 October 2004) [story behind this article]
   (accompanying F77 code CHEW and limited MS Windows executable)
9. A. Moroz, Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell, Chem. Phys. 317(1), 1-15 (2005). (published online on 9 August 2005) [quant-ph/0412094] (see its on-line Appendix A for a MS Windows executable; source F77 code CHEWFS)
10. A. Moroz, Non-radiative decay of a dipole emitter close to a metallic nanoparticle: Importance of higher-order multipole contributions,
    Opt. Commun. 283(10), 2277-2287 (2010). [arXiv:0909.4878]
    (accompanying F77 code CRMNT and limited MS Windows executable)
11. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, Demonstration of a spaser-based nanolaser, Nature 460, 1110-1112 (2009).

up to almost all significant digits in double precision!!!

This pathological behaviour can easily be demonstrated on the example of j_0 and y_0. Recall that j_0=(e^{iz}-e^{-iz})/(2iz) and y_0=-(e^{iz}+e^{-iz})/(2z). Thus, for sufficiently large Im z >0, each of j_0 and y_0 would be dominated by the e^{-iz}/(2z) term, resulting in j_0 ~ i e^{-iz}/(2z) and y_0 ~ - e^{-iz}/(2z) that implies the above relation (R3). Consequently, if the spherical Hankel functions h_l of the first kind are formed as the sum [2]

their precision may be drastically compromised. Indeed, the above relation would then for Im z >> 1 determine a much smaller number (recall h_0=-ie^{iz}/z << 1) from the sum of two much bigger numbers (recall j_0 ~ i e^{-iz}/(2z) >> 1 and y_0 ~ - e^{-iz}/(2z) >> 1). Therefore, it is always recommended to determine h_l by a direct independent recurrence, such as that proposed by Mackowski et al [3] [see recurrences (63),(64) therein].

Otherwise (see Sec. 4.6 and model calculations for a nanoshell C in Ref. [4]) the radiative decay for the interior of a nanoshell may differ by up to four-orders of magnitude from the correct one. If one can perform calculations in an extended precision, this pathological behaviour can be largely overcome, yet the independent recurrence by Mackowski et al [3] is still highly recommended.

References:

1. A. Alu and N. Engheta, Multifrequency optical iInvisibility cloak with layered plasmonic shells, Phys. Rev. Lett. 100, 113901 (2008).
2. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1973)
3. D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, Internal absorption cross sections in a stratified sphere, Appl. Opt. 29(10), 1551-1559 (1990).
4. A. Moroz, Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell, Chem. Phys. 317(1), 1-15 (2005). (published online on 9 August 2005) [quant-ph/0412094] (see its on-line Appendix A for a MS Windows executable; source F77 code CHEWFS)

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© Alexander Moroz, January 2, 2009 (last updated January 19, 2012)