Valence electrons collective oscillations

Valence electrons also can be excited by interacting with the electron beam to produce a collective, longitudinal charge density oscillation called a plasmon. Plas-mons can exist only in solids and liquids, and not in gases because they require electronic states with a strong overlap between atoms. Even insulators can exhibit... 

The most simple, but general, model to describe the interaction of optical radiation with solids is a classical model, due to Lorentz, in which it is assumed that the valence electrons are bound to specific atoms in the solid by harmonic forces. These harmonic forces are the Coulomb forces that tend to restore the valence electrons into specific orbits around the atomic nuclei. Therefore, the solid is considered as a collection of atomic oscillators, each one with its characteristic natural frequency. We presume that if we excite one of these atomic oscillators with its natural frequency (the resonance frequency), a resonant process will be produced. From the quantum viewpoint, these frequencies correspond to those needed to produce valence band to conduction band transitions. In the first approach we consider only a unique resonant frequency, >o in other words, the solid consists of a collection of equivalent atomic oscillators. In this approach, coq would correspond to the gap frequency. 

Let us now analyze the interaction of a light wave with our collection of oscillators at frequency two- In this case, the general motion of a valence electron bound to a nucleus is a damped oscillator, which is forced by the oscillating electric field of the light wave. This atomic oscillator is called a Lorentz oscillator. The motion of such a valence electron is then described by the following differential equation ... 

Here p is the frequency of plasmon oscillations in a system of free electrons (3.7). The oscillator strengths ft introduced previously differ from the usual fm (see Section IV) in their normalization (Efl, / = 1). A method for calculating the thus defined oscillator strengths from experimental values of e2 is presented in Ref. 89. Since the energy range essential for collective oscillations is ho> < 30 eV, the electrons of inner atomic shells can be disregarded. Thus, the value of ne is determined by the density of valence electrons only, and only the transitions of these electrons should be taken into account in the sum over i in formula (3.15). A convenient formula for calculating the frequencies molecular liquids is presented in ref. 89 ... 

Plasmon scattering. The incident electrons lose energy by exciting collective oscillations (called plasmons) of the valence electrons. The energy loss is of the order of 15 eV, and plasmon loss peaks are prominent in the low-loss region of electron energy-loss spectra. 

Optical properties of copper nanoparticles are quite remarkable because the energy of the dipolar mode of surface collective electron plasma oscillations (surface plasmon resonance or SPR) coincides with the onset of interband transition. Therefore, optical spectroscopy gives an opportunity to study the particle-size dependence of both valence and conduction electrons. The intrinsic size effect in metal nanoparticles, caused by size and interface damping of the SPR, is revealed experimentally by two prominent effects a red shift of the surface plasmon band and its broadening. 

Lauritsch et al. [65] obtained the resonance energies of the collective dipole oscillations of the valence electrons from the approximation ... 

These expressions can be used to derive the nearly total reflectance of metals below their plasma frequency. A similar characteristic frequency dependence of a(o)) and e((o) may be seen in semiconductors where oip depends the electron density in the fliled valence band. The conduction electrons can oscillate as a collective mode (plasma oscillation). A plasmon is a quantized plasma oscillation. The frequency and wavevector dependence of plasmons in one-dimensional metals have been predicted (458, 576) to be qualitatively different from those of three-dimensional metals. Recent direct measurements (552) of plasmons in the one-dimensional organic metal tetrathiofulvalinium-tetracyano-quinodimethanide (TTF)(TCNQ) are qualitatively consistent with some of the predictions assuming a tight-binding band (576). 

By comparing experimental or accurate theoretical results with others based on approximate models, it is possible to determine which among those models offers the best approximate constants of the motion and quantum numbers to describe particular states. This approach is used to evaluate and compare the extent of validity of independent-particle, Hartree-Fock and collective, molecule-like descriptions of atoms with two valence electrons. The comparisons are made on the basis of overlaps, oscillator strengths, momentum correlation and quadrupole moments. The criterion for each evaluation is the extent of agreement with results obtained from well-converged Sturmian Configuration Interaction wave functions. 

The energy range of 0-100 eV is mostly dominated by the so-called plasmon peaks and is called the low-loss region. Typical for this range is a broad peak at Ep (typically 10-30 eV) that is repea ted with decreased intensity at 2Ep, 3Ep, etc., until the peaks are no longer visible. The origin of these plasmon peaks lies in the excitation of the so-called plasmons (collective oscillations of the valence electrons) by the primary electrons. The position and width of these plasmon peaks are related to the electronic structure of the sample material and can be used to discriminate between different materials. 

Low-loss region Electrons in the low-loss region (up to 50 eV) correspond to the excitation of weakly bound outer-sheU electrons of the atoms. They are often delocalized and extend over several atomic sites. The low-loss region is dominated by collective, resonant oscillations of the valence electrons known as plasmons. Since the plasmon generation is the most frequent inelastic interaction of electron with the sample, the intensity in this region is relatively high. Intensity and number... 

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