Kramers dielectric function

Once the imaginary part of the dielectric function is known, the real part can be obtained from the Kramers-Kronig relation ... 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

First, we note that the consequence of no absorption (e" = 0) at all frequencies is that the integral in (9.44) vanishes and e = 1. Optically, such a material does not exist there is no way that it can be distinguished from a vacuum by optical means. The Kramers-Kronig relations also tell us that it is a contradiction to assert that either the real or imaginary parts of the dielectric function can be independent of frequency the frequency dependence of the one implies the frequency dependence of the other. These consequences of the Kramers-Kronig relations are almost trivial, but it is disturbing how often they are blithely ignored. 

Unfortunately, radiative transfer effects and particle size distribution are not the only parameters affecting the observed millimeter slope of disk SEDs. While commonly assumed, it is not certain that the dielectric function of the particle material produces a v2 slope even for small particles in the Rayleigh limit (Draine Lee 1984). This concern can be partly alleviated by comparing the slope of the millimeter opacity in disks to that in the ISM. A value of ft close to 2 in the ISM is consistent with many observations of dense clouds (Bianchi et al. 2003 e.g. Kramer et al. 2003), and recently, it was directly determined to be (1.7 — 2.0)1q 33 by Shirley et al. (2008). 

Details of the derivation of the harmonic o.scillator dielectric function and of the Kramers-Kronig transformation are described in standard textbooks, such as (Kuzmany, 1990b Kittel, 1976). Eq. 4.8-1 is also well known as the Kramers-Heisenberg dielectric function. The integrated absorption coefficient in Eq. 4.8-5 is very often used in conventional vibronic IR spectroscopy to characterize the concentration of the absorbing species. 

This is called the Kramers-Kronig (KK) relationship, from which the dielectric function e = ej + e2 can be derived [3.25]. Since e is also a linear response function, ej and 2 are again related by the KK relationship, thus the information contained in the dielectric function can be examined by concentrating on one of the two components of the dielectric function. We choose to work with 2(m) because it is what optical (X-ray) absorption spectroscopy measures and can be directly related to the atomic polarisability Im[a(o )] that appeared in (3.5). 

The reflectivity was measured by using as a reference a gold film evaporated onto the surface of the investigated samples. This method enabled us to measure the reflectivity coefficient with an accuracy of about 0.2 %. The imaginary part curves of the dielectric function Im[s(ai,T)J were calculated, from reflectivity spectrum, by means of the Kramers-Kronig (KK) procedure with uncertainty less than 1.5%. 

The Kramers - Kronig analysis was applied to determine the position of observed lines. In Fig. 14 and 15 are shown the curves of imaginaiy p>art of dielectric function hn e((o x,y) for compositions VI and VII (see Table 1) resp>ectively, obtained by Kramers-Kroning transformation from the reflectivity curves ptresented in Fig. 4 and 5 at 30 K. The imaginary part of dielectric function for ZnxCdyHg(i-x-y)Te solid solution can be ptresented as the superposition of Lorentzians as it is follow from Eqn. (1). The fittings by the Lorentzian sums are presented in Fig.l4 and 15 too. The p>arameters of Lorentzians are presented in Tab. 5 and 6 respectively. 

By using suitable extrapolations [3] of the experimental data, beyond the upper and lower frequency limits of the measurements, and carrying out a Kramers-Kronig analysis, it is possible to obtain the dispersion of the real parts of the dielectric function and of the conductivity. The latter function is particularly suited for a comparison with the results of theoretical models, and for the determination of physical parameters by model fittings. 

The real part of the dielectric function Slmdm( ) corresponding to the localization-modified Drude model can be calculated using the Kramers-Kronig relations, giving... 

FIGURE 15.48 Real and imaginary parts of the dielectric functions (solid lines) and loss function Imf-l/s) (dotted line) from Kramers-Kronig analyses of the reflectance spectra for light polarized parallel to the chain direction for highly oriented AsFs doped trans-polyacetylene. (From Leising, G., Phys. Rev. B, 38, 10313, 1988. Reprinted from the American Physical Society. With permission.)... 

FIGURE 15.49 The frequency-dependent dielectric function for highly oriented polyacetylene doped with perchlorate from Kramers-Rronig analyses of the reflectance spectra, (a) Light polarized perpendicular to polyacetylene chain direction, (b) Light polarized parallel to polyacetylene chain directions. (From Miyamae, T., Shimizu, M., and Tanaka, J., Bull. Chem. Soc. Jptu, 67, 2407, 1994. Reprinted from the Chemical Society of lapan. With permission.)... 

Mathematically, integral Kramers-Kronig relations have very general character. They represent the Hilbert transform of any complex function s(co) = s (co) + s"(co) satisfying the condition s (co) = s(—co)(here the star means complex conjugate). In our particular example, this condition is applied to function n(co) related to dielectric permittivity s(co). The latter is Fourier transform of the time dependent dielectric function s(f), which takes into account a time lag (and never advance) in the response of a substance to the external, e.g. optical, electric field. Therefore the Kramers-Kronig relations follow directly from the causality principle. 

Figure 3 Real and imaginary parts of the dielectric function for poly(di-n-hexylsilane) calculated from the reflectivity data shown in figure 2. The negative values for 62 just below the first UV transition are artifacts of the extrapolations used in the Kramers-Kronig analysis.

Therefore, having determined both R(to) and o(w), Eqs. (4.3) and (4.4) serve to obtain the refractive and the absorption indices n and k, as well as the complex dielectric function. Alternatively, if R(b>) can be measured over a sufficiently large spectral range, a Kramers-Kronig analysis also allows computations of a series of optical constants. In these circumstances, reflectivity measurements alone suffice in obtaining the dielectric function. 

Fig. 152. (a) Real part and (b) imaginary part of the dielectric function, e(ft>) = Si + is2, of PANI-CSA obtained from a Kramers-Kronig transformation of the reflectivity data. Note that the real part of the dielectric function crosses 0 twice, at 0.8 eV and again at 0.2 eV ei (o)) is positive below 0.2 eV. Reproduced by permission of the American Physical Society from K. Lee, A. J. Heeger, and Y. Cao, Phys. Rev. B 48, 14884 (1993). Copyright 1993, American Physical Society. 

Fig. 19.11. Real part of the dielectric function of EuO, EuS, EuSe and EuTe at 300 K as determined by the Kramers-Kronig relation (after Guntherodt, 1974).

The FR is according to eq. (19.29) a dispersive effect and the CD is an absorptive effect, being proportional to the difference of the absorption indices for right hand and left hand circularly polarized light. Just as in the case of refractive index and absorption index or real and imaginary part of the dielectric function, a Kramers-Kronig relation connects also the FR and CD... 

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