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Optical Properties of Metallic Nanoparticles

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 232))

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Abstract

The unification of the theories describing electric and magnetic aspects of our world was one of the great scientific achievements in the nineteenth century [1–3] and brought us a very successful part of theoretical physics: classical field theory. A detailed overview about the historical evolution from René Descartes up to Maxwell and Lorentz can be found in Whittaker [4]. After the revolution of our understanding of the basic forces and constituents of matter in the last 100 years, classical electrodynamics found its place in a sector of the unified description of particles and interactions known as the standard model [5].

The work of James Clerk Maxwell changed the world forever

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Notes

  1. 1.

    A good introduction into the topic of QED can be found in [6] for example.

  2. 2.

    Decoherence is the keyword when it comes to the transition from the quantum to the classical world , an excellent review about that concept can be found in [9], for example (also see [10]). A very interesting debate about the meaning of quantum mechanics has been published by the same author in [11], by the way, where seventeen physicists and philosophers, all deeply concerned with understanding quantum mechanics, share their opinion on what comes next and how to make sense of the theory’s strangeness.

  3. 3.

    Because of energy and momentum conservation at least the time averaged electromagnetic field can still be treated in a classical way, the same goes for clearly quantum mechanical processes like spontaneous emission [5]. A detailed discussion of this topic is also given in the introductory chapters of [12].

  4. 4.

    Without the mathematical disguise the spheres took the less charming form of canon balls on a war vessel in the original seventeenth-century formulation. C. F. Gauss introduced the first attempts of a formal proof of this conjecture in 1831, but it was not until 1988 that the mathematician T. C. Hales was able to finally proof the conjecture by computational methods [17] (it was published after a 7 year long review process).

  5. 5.

    This relation connects the microscopic response with a macroscopic field, a more detailed discussion follows in Sect. 3.2.3.

  6. 6.

    Nonlinear effects may arise at interaction with fiber glass or certain magnetic materials and many other systems.

  7. 7.

    1 eV \(= 1.602176565 \times 10^{-19}\,\mbox{ kg m}^{2}/\mbox{ s}^{2}\), see [27] .

  8. 8.

    Also see comments to Feynman’s unpublished attempt in [31], where he mixes classical and quantum mechanical concepts by starting with Newton’s law of motion and implying the commutation relation between position and velocity. The discussion of a heuristic derivation based on symmetry, charge conservation, superposition, the existence of electromagnetic waves and the Lorentz force can be found in [18], for example.

  9. 9.

    Stokes’ theorem

    $$\displaystyle{\int _{\varOmega }\left (\boldsymbol{\nabla }\times \boldsymbol{ F}\right ) \cdot \hat{\boldsymbol{ n}}\,\mathrm{d}S =\oint _{\partial \varOmega }\boldsymbol{F}\mathrm{d}\boldsymbol{r},}$$

    relates the curl of a vector field \(\boldsymbol{F}\) integrated over a surface to the line integral of the vector field at the boundary. Written in a more general formulation it also contains the divergence theorem or Green’s theorem as special cases.

  10. 10.

    Since a redefinition of the symbol \(\sigma\) and \(\boldsymbol{j}\) will become necessary later on, the additional bar over the symbols has been introduced to avoid confusions.

  11. 11.

    Born 10th May 1788 in Broglie, Eure (Haute-Normandie); † 14th July 1827 in Ville-d’Avray, Hauts-de-Seine.

  12. 12.

    Another method sometimes discussed in literature models the atomic or molecular structure explicitly. There one relates the dipole moment per unit volume to that of an atomic or molecular constituent and writes this as a Taylor series similar to our approach, see [35, 36] and references therein.

  13. 13.

    The electrical analogs of ferromagnets, which possess a spontaneous magnetization per unit volume, are the so-called ferroelectrics . In these materials the dipole moment P (0) in the absence of an electric field is nonzero and leads to the presence of a static, macroscopic electric field, E (0)(r). Such time independent effects may be analyzed by the methods of electrostatics and can be accounted for by including an effective charge density \(\varrho\) \(_{p} = -\boldsymbol{\nabla }\cdot \mbox{ $\mathbf{\mathit{P}}$}^{(0)}\), for example.

  14. 14.

    This is the case for metals like Au or Ag, for the semiconductors Si and Ge as well as for liquids, gases, and for a number of other common crystals. The interested reader may find a very useful compilation of the nonzero elements of \(\boldsymbol{\chi }^{(i)}\) (2) and \(\boldsymbol{\chi }^{(i)}\) (3) for crystals of various symmetry in [36].

  15. 15.

    It was Helmholtz’ student Heinrich Hertz who subsequently provided proof for their existence in the laboratory through his famous experiments with oscillating charges and currents.

  16. 16.

    The most common form as a set of four equations expressed in the language of vector calculus was independently proposed by Heaviside and Hertz as a concise version of Maxwell’s original set of equations [18].

  17. 17.

    Equations (3.25) and (3.37) are examples of the convolution theorem of Fourier integrals [5]:

    $$\displaystyle{c(\omega ) = a(\omega )b(\omega )\quad \longleftrightarrow \quad C(t) = \frac{1} {\sqrt{2\pi }}\int \limits _{-\infty }^{\infty }\mathrm{d}t'\,A(t')B(t - t').}$$

    A convolution in the time domain is translated to a product in the frequency domain and vice versa. The nonlocal connection between D and E therefore is only visible for the time dependent representation.

  18. 18.

    Born 21st August 1789 in Paris; † 23rd May 1857 in Sceaux.

  19. 19.

    Born 14th July 1793 in Sneinton, Nottingham; † 31st May 1841 in Nottingham. See e.g. [42] or [43] for more details on his life.

  20. 20.

    Remember that within Maxwell’s theory we deal with abrupt interfaces and we additionally assume homogeneous media where \(\varepsilon\) only depends on the frequency ω.

  21. 21.

    The redefinition of \(\bar{\sigma }\) does not change the unit of the potential, since we still follow Eq. (3.53). Therefore the units of the electromagnetic fields or other quantities also remain unchanged.

  22. 22.

    By the way, as for all forced oscillations you will require a phase shift of 90° between excitation and the swinging to build up a resonance (also see Fig. 2.9).

  23. 23.

    Essentially the interaction with the electron beam is a much more complicated process (see Sect. 5.2.2), but the analogy is handy after all.

  24. 24.

    Born 30th November 1756 in Wittenberg, Electorate of Saxony; † 3rd April 1827 in Breslau, Prussia.

  25. 25.

    It is also possible to establish a numerical approach based on the electromagnetic fields instead of the potentials. But in contrast to our simple collocation scheme, see Sect. 4.4, this usually requires more complex numerical implementations. In the potential-based BEM approach we have to invert matrices of the order N × N, whereas in the field-based BEM approach the matrices are of the order 3N × 3N.

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    Andreas Trügler

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Trügler, A. (2016). Theory. In: Optical Properties of Metallic Nanoparticles. Springer Series in Materials Science, vol 232. Springer, Cham. https://doi.org/10.1007/978-3-319-25074-8_3

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