Reflection from specularly reflecting spheres

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 4.5a. Scattering diagram for a specularly reflecting sphere that is large compared with the wavelength of incident radiation (from Siegel and Howell, 1981).

This approximation uses equation (18). The mathematical problem reduces to the matrix exponentiation. A computer program was worked out recently to calculate Px] for rectilinear trajectories and the polarization interaction in the isolated-state approximation (66). A similar approach has been used taking into account the coupling between 2PV2 and 2P3/2 states for a rectilinear trajectory (63) and for a trajectory corresponding to specular reflection from a hard sphere (69). The S matrix calculated according to equation (18) satisfies the symmetry relation... 

Detector geometry is important to optimize the collection of diffuse reflectance energy in favor of specular reflectance energy that contains no chemical absorbance information. Two approaches are commonly used. One uses an integrating sphere that provides a double-beamlike instrument, whereby the arrangement of the detectors within the reflective sphere enables a reference and a sample beam to be employed. The reference spectrum is subtracted from... 

The strict solution for the problem of the resistance to the motion of a small sphere moving through gas has been obtained by Baines et al. (1965). They considered both specular and diffuse reflection of the molecules at the surface of the sphere mass of which is large in comparison with the mean mass of gas molecules and the radius to be small compared with the mean free path of gas molecules. All these assumptions are applicable for circumstellar outflows. Fadeyev and Henning (1987) used these solutions for calculation of momentum transfer from silicate dust grains to gas molecules in cool 0-rich red giants... 

The accommodation coefficient or represents the fraction of the gas molecules that leave the surface in equilibrium with the surface. The fraction I — cr is specularly reflected such that the velocity normal to the surface is reversed. As in the case of Stokes law, the drag is proportional to the velocity of the spheres. However, for the free molecule range, the friction coefficient is proportional to dj whereas in the continuum regime dp ip), it is proportional to dp. The coefficient a must, in general, be evaluated experimentally but is usually near 0.9 for momentum transfer (values differ for heat and mass transfer). The friction coefficient calculated from (2.19) is only 1% of that from Stokes law for a 20-A particle. 

Conversely, if those two samples are measured with a sphere instrument with the specular component included, the instrument results will be very similar. By including all of the reflection from the sample, the instrument cannot detect differences in samples due to texture or gloss and will identify the color with the same pigmentation, as being the same. Even though... 

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. 

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. 

Sphere geometries vary considerably to acconunodate different instruments and specimen types. Figure 6 shows a typical example of a sphere s optical design considerations required to include or exclude a specimen s specular reflectance component. Convenient cross-tables of applications and instruments are available from some instrument companies (12, 13). Useful guides to help one choose and use the right instrument geometry to measure transmittance or reflectance of specific specimen types are available from ASTM (14). If spectral measurements must be made at angles to a specimen s surface, a standard practice is available from ASTM (15). 

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