Thin-absorber approximation

The relative absorption depth of the Mossbauer line is determined by the product of the recoU-free fraction/s of the Mossbauer source and the fractional absorption z t) of the sample, abs = fs-e f), where c(t) is a zeroth-order Bessel function ((2.32) and Fig. 2.8). Since c(t) increases Unearly for small values of t, the thin absorber approximation, c(t) t/2, holds up to t 1. On the other hand, values as small as t = 0.2 may cause already appreciable thickness broadening of the Mossbauer lines, according to (2.31), Fexp + 0.135t). In practice, therefore the sample... 

Relative area. The relative areas of the quadrupole doublets are related to the relative abundance of each component, which can be estimated according to the thin absorber approximation (Eqn. 13, this Chapter) in cases where the approximation is valid. Rancourt et al. (1993) provide a means of estimating how closely the thin absorber approximation is satisfied for a given absorber based on the attenuation of spectral areas for individual lines (see Fig. 3 in their paper). McConnell et al. (2000) used an iron density in the high temperature experiments corresponding to an unenriched concentration of 5 mg Fe/cm, which is equivalent to a dimensionless effective thickness of 2.0. The attenuation of spectral... 

In the simplest (thin-source, thin-absorber) approximation, a simple transmission Mossbauer spectrum (O Fig. 25.6) consists of one single Lorentzian peak the half width of which corresponds to twice of the natural line width F = h ln2/Ti/2 (in energy unit). (The natural line width F is characteristic of both the recoilless emission in the source and the resonant absorption in the absorber. So doubling is actually the result of adding up. ) The peak width Wq = 2cF/Ey (in speed unit) thus obtained is an asymptotic minimum to which (normally) actual experimental peak widths are compared. For the 3/2 —> 1/2 (see Fig. 25.7) transition of Fe, e.g., Wq 0.2 mm/s is the minimum (or natural) peak width. 

As it has turned out, the peak shape - in this thin-source, thin-absorber approximation - is indeed Lorentzian, and the FWHM of the peak (Wo), apart from the conversion factor between Doppler speed and energy, is twice the natural line width F. 

In the thin-absorber approximation, an inhomogeneous linear transformation is used to make connection between the vector of the Mossbauer parameters Pm (set up from isomer shifts, quadrupole splittings, magnetic splittings, etc.) and the vector of the peak parameters Ppeak ... 

In the thin-absorber approximation (ta < 0-1), the peak area is proportional to the Mossbauer atom concentration as shown earlier ... 

Although Lorentzian line shapes should be strictly expected only for Mossbauer spectra of thin absorbers with effective thickness t small compared to unity, Margulies and Ehrman have shown [9] that the approximation holds reasonably well for moderately thick absorbers also, albeit the line widths are increased, depending on the value of t (Fig. 2.7). The line broadening is approximately... 

The intensity of a Mossbauer spectrum depends not only on the recoil-free fractions of the source and the absorber and on the number of absorbing nuclei, but also on the linewidth of the absorption lines and on whether or not saturation effects occur. The following approximate expression is valid for relatively thin absorbers [17] ... 

Equations (4)-(6) hold for a bulk (non- or weakly absorbing) rare medium 2 (case (a) in Fig. 3). In the case of a thin film with thickness <7<<7p in contact with the IRE (case (b) in Fig. 3), the thin-layer approximation gives good results (5). Here it is assumed that the electric field is determined by the IRE and the bulk medium above the thin film. The thin film is then considered as a dielectric in this field, and equations similar to Eqs. (4)-(6) can be derived. 

The molecules form a thin absorbing layer of thickness d, with a complex dielectric constant n(w) = n](exact solution and various approximations.3. For thin films, Bell has obtained a reasonably simple approximation for the absorption of an SEW due to the overlying film. If the SEW are propagated a distance D on the metal substrate, with and without the thin film, the ratio of the SEW intensities is given by... 

In the thin source and absorber approximation, the area of the subspectrum corresponding to the /th site can be written... 

A detailed optical study of this cell structure indicates that the optical path enhancement in this arrangement is only approximately fivefold. Thus this cell should be considered as a moderate version of an extremely thin absorber cell, in which the transport length has been reduced by a factor 10-15 and the optical path is enhanced by a factor of five. 

Let us compare the thin-film approximation formulas for (1) the transmissivity (1.98) (2) the reflectivities for the external reflection from this film deposited onto a metallic substrate (1.82) (3) the internal reflection at (p > (pc (1.84) and (4) the external reflection from this film deposited on a transparent substrate (dielectric or semiconducting) (1.81) (Table 1.2). In all cases s-polarized radiation is absorbed at the frequencies of the maxima of Im( 2), vroi (1.1.18°), whereas the jo-polarized external reflection spectrum of a layer on a metallic substrate is influenced only by the LO energy loss function Im(l/ 2) (1.1.19°). The /7-polarized internal and external reflection spectra of a layer on a transparent substrate has maxima at vro as well as at vlq. Such a polarization-dependent behavior of an IR spectrum of a thin film is manifestation of the optical effect (Section 3.1). 

Another important observation from Fig. 3.14 is that at small thicknesses the intensity of the vlo band depends linearly on the film thickness, in agreement with the thin-fllm approximation [Eq. (1.82)]. This linear region extends to approximately half the Berreman thickness and depends upon the angle of incidence in the same way as dg. The linear range of the band intensity in absorbance is somewhat broader than that in reflectance, as illustrated by comparing curves 1 and 4. This implies that absorbance units should be used in any quantitative analysis of thin films, rather than reflectivity units. 

In the case of a thin absorber layer such that the electric field in the device is uniform, an analytical approximation can be used to describe the voltage dependent photocurrent in thin-film solar cells at least around short circuit and for small forward bias. The equation for the photocurrent... 

Hansen s thin-film approximation is not applicable to films thicker than 100 nm. For such thick films, an exact analysis based on the transfer-matrix method is needed. If the substrate absorbs infrared radiation ( 3 0), Equations (9.13a)-(9.13c) cannot be used for simulating ER spectra of thin films on such a substrate. This means that these equations are not useful for analyzing the RA spectra of thin films on metals, which are to be discussed in Chapter 10, as metals are infrared-absorbing materials. 

Conclusions Quantitative analysis, including orientation measurements, has been shown to be straightforward when the formalism based on Harrick s weak absorber approximation is applied. For thin adsorbed layers, such as the DPPA monolayer under discussion, the results are fairly good. Application to bulk materials may introduce systematic errors as discussed above. If the weak absorber approximation is still to be applied, one should take care to work with an angle of incidence which is at least 15° larger than the critical angle, in order to avoid significant band distortions. In many cases it is possible to use quantitative data from transmission experiments to check the validity of the formalism applied to ATR data. 

The energy densities of laser beams which are conventionally used in the production of thin films is about 10 — 10 Jcm s and a typical subsU ate in the semiconductor industry is a material having a low drermal conductivity, and drerefore dre radiation which is absorbed by dre substrate is retained near to dre surface. Table 2.8 shows dre relevant physical properties of some typical substrate materials, which can be used in dre solution of Fourier s equation given above as a first approximation to dre real situation. 

---