Exponential attenuation factor

The second term in (6-9) expresses that nearest and next nearest neighbors dominate scattering contributions to the EXAFS signal, while contributions from distant shells are weak. The dependence of the amplitude on 1/r2 reflects that the outgoing electron is a spherical wave, the intensity of which decreases with the distance squared. The term exp(-2r/X) represents the exponential attenuation of the electron when it travels through the solid, as in the electron spectroscopies of Chapter 3. The factor 2 is there because the electron has to make a round trip between the emitting and the scattering atom in order to cause interference. 

A low-frequency electric field emanating from body A will be screened by the salt solution with an exponential attenuation typical of double layers between parallel-planar bodies. That is, it will die as e x/> D versus the distance x from the interface at which the signal enters medium m from body A. By the time the signal travels the distance / to body B it will be screened to an extent e l/> D. The response of B back to A will also suffer a screening by a factor e l/> D. 

The efficiency of long-distance energy transfer and its gradual decrease with intercomponent distance is usually described by an exponential decay law of the type Fab = F) exp (-yJtAe), where y is denoted the attenuation factor and 1 ab is the spatial separation between the redox sites. In the case of ligand-bridged complexes is set as the distance between the metal centres. 

Table 7.2.1 Echo attenuation factors for molecular diffusion and mono-exponential relaxation. Experiments and variables are defined in Fig. 7.2.13...

Practically this means that high signal frequencies are detected at bright illumination while poor illumination results in low-pass filtering in each pixel. Log amplification inserts an amplification noise proportional to the amplification factor. In contrast, exponential attenuation results in a constant SNR over most of the range because the noise is proportional to the square root of Af Fig. 7.8.6 shows the difference in dynamic behavior of a CCD and a nonlinear CMOS sensor. 

The exponential damping factor in Equation (6) accounts for attenuation effects within the crystal, in which case the effective absorption coefficient is defined as... 

In EXAFS experiments as well as in other EXAFS-like methods, variations in the sample temperature are well described by the Debye-Waller factor and lead to the exponential attenuation of line structure when the sample temperature increases. The temperature dependence of the SEFS spectrum is also described by the Debye-Waller factor— more exactly, by two Debye-Waller factors corresponding to the interference terms of the final and intermediate states [Eq. (38)). Since these interference terms are determined by different wave numbers, p and q, the change of the sample temperature results in a change of the relative intensity of the oscillating terms, which reveals itself in the unusual dependence behavior of SEFS. 

For narrow collimation, the self-absorption correction factor for a slab of thickness d is obtained by integrating the exponential attenuation function ... 

XPS is surface-sensitive due to the mean free path A of the photoelectrons within a specimen which is in the range of several nanometers. Their inelastic losses lead to an exponential attenuation of the signal with depth. Thus only those electrons that originate at a depth not larger then 3 A may contribute to the signal. The intensity 7 of a component i with a particle concentration AT of a thin layer follows Eq. (1-45) with an exponential self attenuating factor which approaches 1 for a large thicknesses. ... 

A second factor (which could potentially affect ultraviolet initiators as well) is the attenuation of light through the sample. Depending on the thickness of the sample, the molar absorptivity of the initiator (e), and the concentration of the initiator ([A]), the differences between conversion at the surface and in the bulk of the sample can be appreciably different. These differences are the result of an exponential decay in the light intensity as a function of depth in the sample. 

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