Integrating sphere error

Reference and sample measurements are performed consecutively, and the resultant (sample) spectrum is obtained as the ratio of the two photon fluxes onto the detector. In a single-beam spectrometer, there are no other options in a double-beam spectrometer, the photon fluxes of the sample and reference beam path are compared. When an integrating sphere is used with two ports and a white standard in the reference position, the photon fluxes are comparable to each other, and no problems occur. Note that the ports are part of the sphere and that any material change in the reference or sample position will change the average sphere reflectance pave. The reference measurement should be conducted with exactly the same components (windows) as the sample measurement otherwise, "substitution errors" may occur. 

For any integrating sphere, the sphere error (Eg) due the variance in sphere efficiencies between the sample and the standard when both are measured at the identical port is given by... 

A fundamental difference between the conic mirror and integrating sphere devices is in how absolute reflectance values are determined. Absolute measurements in conic mirror reflectometers are direct and simple. Sources of error, as discussed in Section V, need to be accounted for. However, absolute reflectance values from measurements made with integrating sphere reflectometers are based on integrating sphere theory. Also, integrating sphere theory is based on a variety of assumptions including that of a perfect Lambertian inner wall coating. The effects of deviations from the assumptions of the theory are difficult to quantify (43). This dependence of absolute results on the sphere theory restricts the use of most integrating sphere reflectometers in the infrared primarily to relative measurements. 

Currently, the large body of theoretical examinations of measurement errors associated with integrating spheres is unmatched by a comparable body of knowledge for conic mirror devices. The intent of the next section is to heuristically review the major sources of measurement error in collecting mirror reflectometers and provide a foundation for further improvements to the performance of conic mirror instruments. 

Analytical work has been published on the optical behavior of composite materials. Edwards et al. (1962) analyzed the spectral reflectance and transmittance of imperfectly diffuse samples in an integrating sphere and provided calculations of the measurement error resulting from the nonideal material. Scattering in composites has been predicted in recent works by Varadan (1991), White et al, and Lee et al. Individually, each fiber scatters the incident radiation in a forward-facing cone centered around the incident direction. 

The accuracy of the experimental values of the photoluminescence quantum yield can be increased if the standard and the sample (either solid or film) are placed in an integrating sphere, which collects nearly the entire emitted photoluminescence intensity and is therefore less sensitive to the particular shape of the sample or the reference [182,183]. Recent comparative experiments with the circular cuvet geometry and also with an integrating sphere confirmed the quantum yield data presented in Table 30.1 within about the same experimental error [179,183]. 

It was not possible to determine quantitatively the two-photon cross-seclion of the [Tb(DPA)3] because of its low efficiency and because the use of ns-pulsed laser induced important error in the ct2pa determination. Using a fs Ti Sa laser as source and an integration sphere, Btinzli et al. [62] determined the three-photon cross-section by comparison of the 3PA and 1 PA luminescence. This absolute method does not require any standard and the [Pg.208]

Figure 7.13 Distributions p for unit-diameter hard spheres at densities of p = 0.35 (filled circles) and 0.8 (open circles). The dashed lines are the primitive quasi-chemical theory of Pratt et al. (2001), Eq. (7.27), p. 158, and the solid lines correspond to the present ME theory. Note the marked break-away of the n = 0 point from the primitive quasi-chemical curve (Pratt et al, 2001). The errors on the high n side of these distributions might reflect the fact that the present ME theory doesn t explicitly treat pair correlations. Those correlations enter only through the integrals (Pratt and Ashbaugh, 2003).

The function ffjl is derived analytically from the hard-sphere-collision integral, and readers interested in the exact forms are referred to Tables 6.1-6.3 of Chapter 6. One crucial issue is the description of the equilibrium distribution with QBMM. In fact, since the nonlinear collision source terms that drive the NDF and its moments to the Maxwellian equilibrium are approximated, the equilibrium is generally not perfectly described. The error involved is generally very small, and is reduced when the number of nodes is increased, but can be easily overcome by using some simple corrections. Details on these corrections for the isotropic Boltzmann equation test case are reported in Icardi et al. (2012). 

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