Reflection from diffuse spheres

UV-VIS-NIR diffuse reflectance (DR) spectra were measured using a Perkin-Elmer UV-VIS-NIR spectrometer Lambda 19 equipped with a diffuse reflectance attachment with an integrating sphere coated by BaS04. Spectra of sample in 5 mm thick silica cell were recorded in a differential mode with the parent zeolite treated at the same conditions as a reference. For details see Ref. [5], The absorption intensity was calculated from the Schuster-Kubelka-Munk equation F(R ,) = (l-R< )2/2Roo, where R is the diffuse reflectance from a semi-infinite layer and F(R00) is proportional to the absorption coefficient. 

To describe quantitatively the diffusion-controlled tunnelling process, let us start from equation (4.1.23). Restricting ourselves to the tunnelling mechanism of defect recombination only (without annihilation), the boundary condition should be imposed on Y(r,t) in equation (4.1.23) at r = 0 meaning no particle flux through the coordinate origin. Another kind of boundary conditions widely used in radiation physics is the so-called radiation boundary condition (which however is not well justified theoretically) [33, 38]. The idea is to solve equation (4.1.23) in the interval r > R with the partial reflection of the particle flux from the sphere of radius R ... 

For some typical modes of scattering from large spherical particles (f >5), simple formulations of phase functions can be obtained. These modes include scattering from a specularly reflecting sphere, scattering from a diffuse reflection sphere, and scattering by diffraction from a sphere. 

Figure 4.5b. Scattering phase function for a diffuse reflecting sphere which is large compared with the wavelength of incident radiation and with constant reflectivity (from Siegel and Howell, 1981).

For a diffuse sphere, each surface element that intercepts incident radiation will reflect the energy into the entire 2ir solid angle above that element. Thus, the radiation scattered into a specified direction will arise from the entire region of the sphere that receives radiation and is also visible from this specified direction. Consequently, the phase function for a diffuse sphere can be obtained as [Siegel and Howell, 1981]... 

Solids can be measured in transmission or reflection (reflectance) modes. Both specular reflection and diffuse reflection are used. Diffuse reflection accessories include the Praying Mantis from Harrick Scientific Products, Inc., and a variety of integrating spheres available from most major instrument companies. Specular reflection is used for highly reflective materials diffuse reflectance for powders and rough surfaced solids. Materials characterization relies heavily on techniques like these. 

Integrating sphere n. A sphere coated inside with a highly reflective, diffuse material and used for the measurement of luminous flux. If the internal surface is perfectly diffuse, the intensity of any part of the sphere is the same. Many instruments used for reflectance measurements utilize such a device for measuring the diffuse or total reflectance from a sample material relative to a reference material. 

Another problem arises in integrating sphere measurements when the sample cannot be placed flush against the port of the sphere, ff the sample is specular, the reference should be recessed by an identical distance as the sample to be measured. For materials that are primarily diffuse in character, we have determined the measured reflectance for a lambertian material decreases by approximately 3% (absolute) per millimeter of distance of the sample from the sphere up to approximately 4 mm. Beyond that, it is difficult to predict. Measurement of the reference at a similar distance helps but is still very inexact. 

A more sophisticated way to improve the uniformity of a source is to pass the output through an integrating sphere (shown in Figure 9.5) - a hollow sphere whose inside surface is specially coated for high, diffuse reflectance. The integrating sphere has an input port and an output port, and a baffle that keeps radiation from the source from reaching the output port until it has been reflected at least once. The port assemblies are designed to make it easy to install sources and detectors. 

The overall tumbling of a protein molecule in solution is the dominant source of NH-bond reorientations with respect to the laboratory frame, and hence is the major contribution to 15N relaxation. Adequate treatment of this motion and its separation from the local motion is therefore critical for accurate analysis of protein dynamics in solution [46]. This task is not trivial because (i) the overall and internal dynamics could be coupled (e. g. in the presence of significant segmental motion), and (ii) the anisotropy of the overall rotational diffusion, reflecting the shape of the molecule, which in general case deviates from a perfect sphere, significantly complicates the analysis. Here we assume that the overall and local motions are independent of each other, and thus we will focus on the effect of the rotational overall anisotropy. 

Figure 8.11. Diffuse reflectance absorption spectra of a strongly fluorescent sample (1,6-diphenylhexatriene adsorbed on porous alumina) (a) conventional measurement w ith monochromatic irradiation and detection via an integrating sphere (b) measurement in a fluorimeter with two monochromators. Reaction spectra during Irons - cis photoisomerization are also given (adapted from Ref. 26).

For a complete definition of Eq. (53) we need to determine the constants Cnk from the conditions (17)-(19) and then calculate the Fourier integral Eq. (1) for the echo signal. To avoid the tedious algebra we compare the three published solutions numerically, but first reproduce these solutions here using our notation. Two of these solutions resulted from a calculation that included the effect of surface relaxation. To make a correct comparison we eliminate from the equations the terms due to relaxation. Then we have the following formulae for the echo intensity for diffusion in a sphere with radius a and reflecting walls ... 

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