Exponential attenuation

It has been known for 40 years, since the initial PGSE experiment of Stejskal and Tanner [20], that NMR is an excellent tool for measuring the statistical displacements of molecules with time, usually called diffusion. The exponential attenuation a of the PGSE signal as a function of time between the PGSE pulses is given by [21]... 

Kerkhof and Moulijn [30] suggested that a supported catalyst may be modeled as a stack of sheets of support material, with cubic crystals representing the supported particles. They used this stratified layer model, illustrated in Fig. 3.9b, to calculate the intensity ratio /P//s for electron trajectories perpendicular to the support sheets, assuming exponential attenuation of the electrons in the particles and the support. 

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. 

Not surprisingly, S is in the direction of propagation. The magnitude of S, which we shall denote by the symbol I, is called the irradiance and its dimensions are energy per unit area and time. (The term intensity is often used to denote irradiance however, intensity is also used for other radiometric quantities, and we shall therefore tend to avoid this term because of possible confusion. E is now the recommended symbol for irradiance, but this hardly seems appropriate in a book where the electric field and irradiance often appear side by side.) As the wave traverses the medium, the irradiance is exponentially attenuated ... 

Underlying (3.39), and hence exponential attenuation of irradiance in particulate media, is the requirement that This condition may be... 

I f multiple scattering is negligible the irradiance of a beam of light is exponentially attenuated from /, to It in traversing a distance h through a particulate medium (see Section 3.4) ... 

Beta radiation has a finite range, whereas in theory x- and gamma rays are exponentially attenuated. It should be noted that x-radiation of each energy exhibits sharp changes in absorption coefficients for certain absorbers. 

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. 

For the Bragg case, SS is more nearly parallel to k, and the surface normal through P cuts only the a or 0 branch, or passes between them. If it passes between them, total reflection occurs and there is only an exponentially attenuated wave in the crystal. It follows that the angular width of the total reflection is directly related to the diameter of the hyperbola. 

When a beam of X-rays passes through a material the result is an exponential attenuation of the primary beam. The different possible interaction processes, as fluorescence, are characterized by a probability of occurrence per unit path length in the absorber. The flux of transmitted primary photons I after having passed through an attenuator with a thickness x (cm) and a linear attenuation coefficient called yu,(cm ) for the wavelength considered, can be calculated with respect to its initial value f and for an angle of penetration of 90 ° as follows ... 

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. 

Photons and neutrons traversing a medium show an exponential attenuation (see Chap. 4), which means that there is always a nonzero probability for a photon or a neutron to traverse any thickness of material without an interaction. As a result of this property, detectors for photons or neutrons have efficiency less than 100 percent regardless of detector size and energy of the particle. 

From the description of the Stokes problem, we conclude that the incompressible surfactant film causes an oscillating Stokes-type boundary layer to develop, which is the predominant damping mechanism. By the same argument as before, the amplitude of the wave solution satisfying Laplace s equation is modified by an exponential attenuation exp(-)3 Here, however, we take the attenuation length to be given by the viscous boundary layer thickness of Eq. (10.5.34), from which, by comparison with Eq. (10.5.32),... 

The exponential la,w for absorbing radiation energy as given in Eq. (10-1) is not exact for gamma photons, principally because the scattering of some of the photons in the beam and subsequent return is not true adsorption. The attenuation of beam intensity is always less than predicted by Eq. (10-1) and this physical phenomenon is taken into account by modifying the exponential attenuation law to give... 

The typical curve of beta-particle attenuation in aluminum absorbers, shown in Fig. 2.6, at lower energies resembles the exponential attenuation observed for gamma rays (see Section 2.4.4). The final part of the line curves downward to reach the distinct range associated with Fmax- Attenuation curves for the various beta-decay radionuclides differ because of the different beta-particle energy spectra, but both the characteristic range and the approximately exponential attenuation have been used to estimate maximum beta-particle energies (Evans 1955). 

For beta particles, reasonable estimates of the curve s/eq for counting efficiency as function of mass have been based on their approximately exponential attenuation... 

Exponential attenuation in the case of electromagnetic radiation, a = alpha P = beta X = X-ray... 

Figure 2.3 Impact of ionizing radiation on polymers. (A) Simplified scheme of radiation-induced events in polymers. (B and C) Comparison of the interaction of VVphotons and electrons with polymers. Photon beams are exponentially attenuated along the beam direction, whereas electrons are scattered and slowed down along their path through the polymer fdm. Source Adapted from Drobny [2], with permission from Elsevier Ltd.

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