Mean free path damping

The damping factors take into account 1) the mean free path k(k) of the photoelectron the exponential factor selects the contributions due to those photoelectron waves which make the round trip from the central atom to the scatterer and back without energy losses 2) the mean square value of the relative displacements of the central atom and of the scatterer. This is called Debye-Waller like term since it is not referred to the laboratory frame, but it is a relative value, and it is temperature dependent, of course It is important to remember the peculiar way of probing the matter that EXAFS does the source of the probe is the excited atom which sends off a photoelectron spherical wave, the detector of the distribution of the scattering centres in the environment is again the same central atom that receives the back-diffused photoelectron amplitude. This is a unique feature since all other crystallographic probes are totally (source and detector) or partially (source or detector) external probes , i.e. the measured quantities are referred to the laboratory reference system. 

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

The mean square variation in Rj is represented by the Debye-Waller factor, aj] X is the elastic mean-free path of the photoelectron and it is the damping term [exp(-2Rj/X)] that invariably limits back-scattering contributions to <4 A from a metal atom in a biological system. The jth. neighbor makes an angle d, with the polarization vector of the incident X ray and the term 3 cos dj averages to 1 for solutions and polycrystalline samples. 

F is the bulk collision constant, A is a positive dimensionless factor, Vf is the Fermi velocity and R the particle radius. From a classical point of view, this modification is supported by the fact that, when the radius is smaller than the bulk mean free path of the electrons, there is an additional scattering factor at the particle surface. This phenomenon, known as the mean free path effect, is abundantly discussed in [19]. In a quantum approach, the boundary conditions imposed to the electron wave functions lead to the appearance of individual electron-hole excitations (Fandau damping) [21] resulting in the broadening of the SPR band proportional to the inverse of the particle radius as in Eq. (8) [22]. A chemical interface damping mechanism has also been considered, leading to the l/R dependence of F [23]. 

Bui there are further differences between the methods, which enn he seen by comparison of the integrands in equations (10.24) and (10.25). The atom pair correlation function in equation (10.24) is multiplied by expt 2i/a. ). which takes into account the effect of the liniie lifetime of the photo-electron and the hole generated by the absorption of the X-rays. Owing to the mean free path term expt--2r//.,). the pair correlation funciions are asymmetrical and damped with increasing distance. This effect can clearly be seen in ihe Fourier transform of the EXAFS function. 

The EXAFS amplitude falls off as 1 /R. This reflects the decrease in photoelectron amplitude per unit area as one moves further from the photoelectron source (i.e., from the absorbing atom). The main consequence of this damping is that the EXAFS information is limited to atoms in the near vicinity of the absorber. There are three additional damping terms in Equation (2). The 5 q term is introduced to allow for inelastic loss processes and is typically not refined in EXAFS analyses. The first exponential term is a damping factor that arises from the mean free path of the photoelectron (A(k)). This serves to limit further the distance range that can be sampled by EXAFS. The second exponential term is the so-called Debye-Waller factor. This damping reflects the fact that if there is more than one absorber-scatterer distance, each distance will contribute EXAFS oscillations of a... 

In this chapter, absorption and scattering efficiencies spectra will be presented for silver nanoparticles (NPs) with different shapes and dimensions. All the spectra are calculated in the discrete dipole approximation framework (see Chapter 2), with the Palik complex dielectric function e(lattice dispersion relation (LDR) prescription for the polarizability (see Sec. 2.4.3.2). For dimensions of the NPs smaller than the mean free path of the conduction electrons, the surface damping correction A sd is added to the Palik dielectric function (see Sec. 2.3), as several works have shown that the dielectric constant is strongly dependent on the size and the shape of the nanoparticle [21-23]. 

Moreover, in considering the effects of the size in the optical response of a metallic nanoparticle, we must put in evidence that in the case of particles with dimensions comparable or smaller than the mean free path of its oscillating electrons (i.e. for gold and silver particles of radius o < 10 nm) the dielectric function of the particles becomes strongly size-dependent and the additional surface damping must be considered for a correct treatment of the resonances intensity. 

One can imagine from these experimental results that PDA chains inside poly(DCHD)shell may contact with some points on the surface of Ag core [47,49]. These contact points at hetero nano-interface might be anchors to disturb and depress the plasma oscillation, and then diminish the mean-free-path of conduction free electrons in VB of Ag core. In other words, the LSP of Ag core would damp simply without changing the resonance frequency, owing to the reduction of electronic conductive domain induced by locally and strongly physicochemical interaction at the core/shell interface in the hybridized NCs [103,105]. 

This design is sometimes a twin-piston design, in which a second piston is 180° out of phase with the first. This means that when one piston is in its forward stroke, the other is in its backward stroke. The result is a flow that is free of pulsations. With the single-piston design, a pulse-damping device positioned in the flow path following the pump is used. 

---