The Absorption Coefficient

In the previous section, we have mentioned that a light beam becomes attenuated after passing through a material. Experiments show that the beam intensity attenuation dl after traversing a differential thickness dx can be written as 

From a microscopic point of view of the absorption process, we can assume a simple two energy level quantum system for which N and N are the ground and excited state population densities (the atoms per unit volume in each state). The 

In fact, the transition cross section can be written in terms of a line-shape function g(v) (with units of Hz ) in the following way  

The hne-shape function gives the profile of the optical absorption (and emission) band and contains important information about the photon-system interaction. Let us briefly discuss the different mechanisms that contribute to this function, or the different line-broadening mechanisms. 

The ultimate (minimum) linewidth of an optical band is due to the natural or lifetime broadening. This broadening arises from the Heisenberg s uncertainty principle, AvAt U2jt, Av being the full frequency width at half maximum of the transition and the time available to measure the frequency of the transition (basically, the life- 

Free atoms in the ground state can absorb radiation energy of exactly defined frequency hv). The absorption coefficient kfj at a discrete frequency is defined as  

According to equation 32, the amount of absorption is directly proportional to the oscillator strength ff and to the number of absorbing atoms (Nj). 

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The sensitivity of the luminescence IP s in the systems employed here decreases with increasing x-ray energy more strongly than in the case of x-ray film. Therefore, this phenomenon must be compensated by using thicker lead front and back screens. The specific contrast c,p [1,3] is an appropriate parameter for a comparison between IP s and film, since it may be measured independently of the spatial resolution. Since the absorption coefficient p remains roughly constant for constant tube voltage and the same material, it suffices to measure and compare the scatter ratio k. Fig. 2 shows k as a function of the front and back screen thickness for the IP s for 400 keV and different wall thicknesses. The corresponding measured scatter ratios for x-ray films with 0,1 mm front and back screens of lead are likewise shown. The equivalent value for the front and back screen thicknesses is found from the intersection of the curves for the IP s and the film value. 

Fig. 2 shows the response of a C2 film system on a step wedge (wall thickness range 2. .. 18 mm) exposed with a X-ray tube at 160 kV. For the exposure withy-rays (Irl92 or Co60) corresponding linear relationships are obtained. From this linear relationship it is followed, that the influence of the scattered radiation and the energy dependence of the absorption coefficient can be considered by an effective absorption coefficientPcff in equation (1). 

The quantity e is called the absorption coefficient or extinction coefficient, more completely the molar decadic absorption coefficient it is a characteristic of the substance and the wavelength and to a lesser extent the solvent and temperature. It is coimnon to take path length in centimetres and concentration in moles per... 

In an EXAFS experiment the measurable quantity is the absorption coefficient a of Equation (8.16). If Qq is the absorption coefficient in the absence of EXAFS, deduced from the steeply falling background shown in Figure 8.32, then x k), the fractional change of a due to EXAFS, is given by... 

Internal redection starts by consideration of an interface between two media, a denser transparent medium with refractive index n, and a rarer medium with a complex refractive index (= where is the absorption coefficient of the medium) as shown in Figure 23. If of the rarer... 

Transmission. The spectral transmission of glass is determiaed by reflectioa at the glass surfaces and the optical absorption within the glass. Overall transmission of a flat sample at a particular wavelength is equal to (1 — R), where P is the absorption coefficient, t the thickness of glass, and... 

The sound absorption of materials is frequency dependent most materials absorb more or less sound at some frequencies than at others. Sound absorption is usually measured in laboratories in 18 one-third octave frequency bands with center frequencies ranging from 100 to 5000 H2, but it is common practice to pubflsh only the data for the six octave band center frequencies from 125 to 4000 H2. SuppHers of acoustical products frequently report the noise reduction coefficient (NRC) for their materials. The NRC is the arithmetic mean of the absorption coefficients in the 250, 500, 1000, and 2000 H2 bands, rounded to the nearest multiple of 0.05. 

Eor reverberation room tests of some irregularly shaped items, such as items of furniture, the number of sabins of absorption per item is commonly reported, rather than the absorption coefficient. It is important that the number and arrangement of the items also be reported because both of these factors can affect the results of the test. 

Thickness. The traditional definition of thermal conductivity as an intrinsic property of a material where conduction is the only mode of heat transmission is not appHcable to low density materials. Although radiation between parallel surfaces is independent of distance, the measurement of X where radiation is significant requires the introduction of an additional variable, thickness. The thickness effect is observed in materials of low density at ambient temperatures and in materials of higher density at elevated temperatures. It depends on the radiation permeance of the materials, which in turn is influenced by the absorption coefficient and the density. For a cellular plastic material having a density on the order of 10 kg/m, the difference between a 25 and 100 mm thick specimen ranges from 12—15%. This reduces to less than 4% for a density of 48 kg/m. References 23—27 discuss the issue of thickness in more detail. 

Reverse saturable absorption is an increase in the absorption coefficient of a material that is proportional to pump intensity. This phenomenon typically involves the population of a strongly absorbing excited state and is the basis of optical limiters or sensor protection elements. A variety of electronic and molecular reorientation processes can give rise to reverse saturable absorption many materials exhibit this phenomenon, including fuUerenes, phthalocyanine compounds (qv), and organometaUic complexes. 

Sihcon cells are hundreds of micrometers ( -lm) thick in order to faciUtate handling with minimal breakage, although most solar radiation is absorbed in the first 20—30 p.m. Light penetration decreases exponentially, proportional to, where d is the absorption coefficient of a material and T is its thickness. The values of (X for a given material vary with the wavelength of incident radiation in siUcon, (X is 10 —10 /cm over most of the range of usable solar radiation. 

Using the complex refractive index N = n + iK where i =- f—1 and Kis the absorption coefficient, the reflectivity R of metals and alloys is given by ... 

Nx = / nowhere Ix is the radiation flux, n is the number of gas molecules in the path of the beam per cm of projected area, and a is the cross-section of absorption. Alternatively, the absorption coefficient, is defined drrough the Beer-Lambert equation... 

The im inary component of the refractive index is associated with absorption. In general, the absorption for thin films is not significant and, consequently, P can be ignored. However, for materials containing the elements Li, B, Cd, Sm, or Gd, where the absorption coefficient is large, P must be taken into account and the refiacdve index is im inary. 

The refractive index of a film or a substrate material can be measured with a sensitivity better than 5 x 10, the best available for non-invasive optical measurement methods, especially for thin films. The extinction coefficient can be measured with almost the same sensitivity, which corresponds to a lower limit of 10-100 cm for the absorption coefficient of the material. 

In this expression, z is the distance from the surface into the sample, a(z) is the absorption coefficient, and S, the depth of penetration, is given by Eq. 2. A depth profile can be obtained for a given functional group by determining a(z), which is the inverse Laplace transform of A(S), for an absorption band characteristic of that functional group. 

Expressions of this type can be written for both gas and liquid films in which the absorption coefficients are the gas- and liquid-film coefficients, respectively. The driving force across the gas film is given by the difference between the actual partial pressure of the soluble gas and that at the interface, v/hile the driving force across the liquid film is given by the difference between the concentration of the soluble gas at the interface and that in the main bulk of liquid. 

In Eq. (4-29) jc is the distance traveled by the wave, and a is the absorption coefficient. Sound absorption can occur as a result of viscous losses and heat losses (these together constitute classical modes of absorption) and by coupling to a chemical reaction, as described in the preceding paragraph. The theory of classical sound absorption shows that a is directly proportional to where / is the sound wave frequency (in Hz), so results are usually reported as a//, for this is, classically, frequency independent. 

EXAFS is observed as a modulating change in the absorption coefficient caused by the ejected electron wave back-scattering from the surrounding atoms, resulting in interference between ejected and back-scattered waves. It is defined as ... 

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